A P P L I E D AE R O D Y N AM I C S
By L B AI RSTO W,
Roya Co ege o f Sci en ce i n
Sch o ar ; Fe l l o w and M e
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A er onau t i ca Soci e ty, et c
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A E RO PL AN E
By A
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Mecha n i cs ; Wh i t w o rt h
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S T R U C TU RES
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M Sc . Ar o M
SUTTO N PI PPARD,
Fell ow of t h e Roya l A eronau t i ca l Soc i e t ,
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L AU RE N C E Pa n c u u n. a t e R A l
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A ero na u t i ca Soci et y
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TH E A E RO E N G I N E
Majo r A T EV A N S a nd Cap t ai n W
By
A D AM S,
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G RYLLS
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D E S I G N O F SC R EW
P RO PE L L E RS
LO NG MA N S, G R EEN AN D
N EW YO RK . LO N DO N ,
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BO M B A Y, C AL L UTT A , A N D
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M A D RA S
A PPL I E D A E R O D Y N A M I C S
BY
L EO N ARD B AI RSTO W ,
A I ON
TO
MI N I T R M M R
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I N N T IO N OM MI TT A C I EN TS I N E T A T IO N C M TT R
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B ic
PREFACE
wo rk ai ma a t th e extr ac ti o n of pr inc i p l es of fl ight f ro m and th e
ill us tr atio n of th e u se of de ta il ed i n fo rmatio n o n aero na uti cs n ow
avai l abl e fr o m many sou rce s; n ota bly th e p ubli ca tio n s of th e Advi so ry
Co mmittee fo r Ae r o na uti cs
Th e ai n outlines of th e th eo ry of fl ight
are si m
p l e but th e stag e of app li ca tio n n ow r eac h ed n ecess ita tes ca r e ful
exam i na tion of sec o ndary featu re s T h is boo k i s ca st with thi s dis ti nc tion
in V i e w and starts with a des c r i p tio n of th e var iou s c l a sse s of ai rcr a ft
both h eavi er and lig hte r tha n a i r and th en proceed s to deve lo p th e
la ws of s teady fl ight on e l eme n tar y pri nci p l e s
p l e te
L a ter c h ap te rs co m
th e de tai l as kn own a t th e pre sen t ti me an d c over pred i c t io n s and
a na lyses of perfo rmance aerop l ane acrob ati c s and th e general p r obl em s
of co n t rol and s ta bility Th e s ubj ec t of ae rod yn ami cs i s al m os t wholly
based o n exper i me nt and me thod s are de scri bed of obta i n in g ba s i c
i n fo rma t io n f ro m tes ts on ai rcra ft i n fl ight o r f ro
tests i n a wi nd
c h anne l on m od e l s of ai rcra ft and ai rcra ft par ts
Th e a utho r i s anxiou s to ackn owl edg e h i s par ti c ul ar i nde btedn e ss to
th e Ad vi so ry Co mm itt ee fo r Ae r ona uti cs fo r perm i ss io n to make u se of
repo rts is su ed under i ts a uthor ity Exten s i ve re ference i s made to those
rep o rts whic h p r io r to th e war were i ss u ed ann ua lly ; it i s under stood
th a t all repo r ts approved for is s u e be fo re th e begi n n i n g of 19 19 are n ow
read y fo r p ubli ca t io n T o thi s mate ri a l th e a utho r h a s had acces s but
it Wi ll be u nders tood by a ll in ti ma te ly
ua i u t ed with th e repo r t s th a t
th e co n ten ts can n ot be fully r ep resen
by ext rac t s Th e prese n t
volume i s n ot an a ttemp t a t c oll ec ti o n of th e re sults of r e searc h but a
co n tr ibutio n t o th e i r app li ca tio n to ind u st ry
Fo r th e l a st y ear of th e wa r th e autho r was res po n sibl e to th e
De par t men t of Ai rcra ft P rod u c ti o n fo r th e c o nd uc t of aer ody nam i c
re sea rc h o n ae rop l anes i n fl ight and h i s th anks are d u e fo r perm
i ss io n
to make u se of i nfo rma tio n acqui r ed Fo r pe r m
i ss io n to r epro d u ce
p hotog rap h s acknowled en t i s made to th e Admi ra l ty Ai rs hi p Depar t
men t Mess rs H an d l ey a ge an d Cc th e B r iti s h and Colo ni al Ae r o p l an e
o Cc Mess rs D N ap i er and Cc a n d EL M
Cc t h e Pli w n ix Dyn am
Sta ti one r y Ofii ce
Tn a
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mm
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L
Wren
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October 6t h , 19 19
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B AI B STOW
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vi ii
CONT ENT S
AE R I AL
BI ANO EU V R E S AND TH E EQUATI O N 0 1 MO TI O N
"
CONT ENT S
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—L eve l
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G en e ra
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CONT ENT S
L I ST OF PL ATES
F o u r t ee n t e n s
F ig h t i n g
k
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of
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B i p la n e S
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E ngi ne
N ea r l y
et e
Ex p e ri
co
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r-coo e
Wai t e r coo l ed
-
i n fl i gh t
c ou t
Aerop
R o t a ry Engi n e
er
S t a t i o na ry E n gi ne
d Rig id Ai rs hi p
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Aerop
lan e arran ged t o s how Au t
o ro t a t i o n
dd i es b eh i n d Cy l i nder
l ow of Wat er pas t an In cli n ed Pl t e L ow nd H i gh Speed
Lo w a n d H i gh Spe ed s
F l o w of Ai r p as t an I n cli n ed P la t e
Sl i gh t l y St ab le Mod el
V ery St a bl e Mod e l—
St a b l e Mod e l wi t h t w o Re l F i n — Mode l w h i h d e l p
O ci l l a t i o n —Mod e l w h i ch i ll us t rat e L a te r al I n t a bili t i
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Uns ta b
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CHAP TER I
GENERAL DESCRI PTI ON OF STANDARD FORMS OF AI RCRA FT
I n r aon v crron
'
th e Openi n g re ferences to a ircra ft as rep r es en ted by p hotogr ap h s of
modern types bo t h h eavi er th an a i r and light er th an a ir a tt en tion will
be mo re esp ec i a lly d irec t ed to thos e p oi n t s whi c h spec ifica lly rel a t e to th e
i cs
subj ec t ma tt er of this boo k i e to app li ed aerOdyn a
St ri c tly i n
t erp ret ed th e wo rd aer o dy nami cs i s used o nly fo r th e stu d y of th e fo rces
ot i on t h rough th e a i r but fo r man y reas o ns it i s
on bo d i es d u e to th e i r
not c o n ven i en t to adh ere too c losely to t his defi n itio n In th e cas e of
hea vi er th an ai r cra ft one of th e aero d ynam i c fo rces i s r e qui red to c ou n t er
bal ance th e w eight oi th e ai rcra ft an d i s th ere fo re di rec tly r el a t ed to a
i c fo rce I n light er th an ai r cra ft s iz e depend s dir ec tly o n
non d y n a m
the w eigh t to be carri ed but th e w eight it self i s b al anced by th e buoy ancy
of a mas s of en t rapped hy drogen whi c h aga i n h as n o dy nami c o rigin As
the si ze o f ai r cra ft increas es th e resis ta nce to m otion a t any prede t ermin ed
speed in creas es an d th e ae r o dyn ami c fo rces fo r light er th an ai r c r a ft
depend u p o n and are c onditioned by n o n dyna i c fo r ces
Th e i n t er rela tio n indi ca t ed a bo ve b etw ee n aero dyna i c and st a ti c
forces h as ex t ens ions whi ch aflect th e ex t e rna l fo rm t aken by ai rcra ft
One of th e m os t i mp o rt an t it ems i n a i rcra ft design i s th e ec o n o mi ca l
d ist ributio n of ma t eri al so as to pro d u ce a suffi c i en t margin of st ren gth
fo r th e l eas t w ei ght of ma t eri al Accep tin g th e st a t emen t th a t add ition a l
a c o nse qu ence of increased w eight it will be apprec i a t ed th a t
th e probl em of ex t erna l fo rm cann ot b e de t ermi ned sol ely f ro m aero dynami c
co ns i dera tio ns As an exam
p l e of a si mp l e ty pe of c ompro mi se ay b e
ins t anced th e probl em of wi ng fo rm Th e grea t est lift fo r a gi ven resi stance
i s obt a i ned by th e u se of si n gl e lo n g an d narr ow p l anes th e advan t a ge b e in g
l ess and l ess marked as th e ra tio of l en gth to b read th inc reas es but r ema i nin g
Most aerOp lan es h ave this as pec t
a pprec i a bl e wh en th e ra tio i s t en
ra tio m o re n ea rly e qua l to six th en t en and i nst ead of t h e sin gl e p lane
a doubl e arrangemen t i s pre ferred th e eflect of th e d oubli n g b ein g an
app rec i a bl e los s of aero dynami c efli ci en cy Th e rea so ns whi c h h ave l ed
t o this resu lt are partly acc oun t ed fo r by a speci a l c o nven i ence i n fi ghtin g
which acc o mpan i es th e use of sho rt plan es but a fac tor of grea t er i m
po rtance i s th a t ari sin g fro m th e st ren gt h desi dera t a Th e w eight of
wings of la r g e aspec t ra tio i s grea ter fo r a gi ven liftin g capac ity th an th a t
of sho rt win gs and th e ex t erna l su pp o rt necessary i n all ty pes of a erOplan e
is m
o re diffi c ult to ac hi eve with aero dynami c econ omy fo r a sin gl e th a n
for a d oubl e p l an e
i ca lly a li mit i s fixed to th e w eight
Aero dyn a m
In
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APP L IED AER ODYN AMIC S
2
carri ed by a win g a t a chosen speed an d fo r sa f e a lightin g th e t endency
This gi ves a
h as been to fix thi s speed a t a littl e o ver fo rty m il es an hou r
lower li mit t o th e win g ar ea of an aerOplane whi c h h as to carry a sp ec ifi ed
w e ight Th e gener al experi ence of desig i ers h as b een th a t this li mit i s
a serious res t ri c tio n in th e design of a m on op l an e but o ffers very littl e
bi p lane I n a few cases three p lanes h ave b een su perp osed
but th e ty pe h as n ot rece i ved any genera l deg ree of accep t ance Fo r
sma ll aerO p lan es th e fu rth er loss of ae ro d ynami c e ffi c i enc y i n a t ri p l an e
h as b een accep t ed fo r th e sake of th e g rea t er rap i di ty of m
an aau vr e whi c h
can b e made to acc o mpan y red uced span and c ho rd whils t i n very l arg e
a erOp lan es th e c hi e f advan t ag e of th e t ri p lane i s a red u c tion of th e o ver a ll
d i mens ions Up to th e presen t ti me it appears tha t an advan ta ge r emain s
with th e bi p l ane ty pe of co nst ru c tion although very goo d m
onOp lan es and
t ri p l anes h ave b een built
Th e illus t ra tio n shows tha t ai rcraft h ave en t ered th e st a ge of en gineer
in g as di stinc t fro m a ero dyn ami ca l sci ence i n th a t th e fina l pro d u c t
i s de t erm i n ed by a n um b er of co nsi dera tions whi c h are m utu ally res t ri c ti v e
and i n whi c h th e prac tica l kn owl ed ge of us age i s a very i mp ortan t f ac to r
i n th e a tta inmen t of th e b es t r esult
aero dynami cs i t
Although a i r i s th e fl ui d in d i ca t ed by th e ter m
h as b een found th a t man y of th e p h en o mena of fl ui d m otion are in dependen t
of th e parti c ul ar fl ui d m o v ed Advan t age h as be en t aken of thi s f ac t in
arran gin g experi men ta l wo rk and in a la t er ch ap t er a st ri k in g Op ti ca l
illus t ra tion of th e t ruth of t h e above ob se rva tion i s gi ven T h e di stin c tion
b e tw een aero dyn ami cs and th e d ynami cs of flui d m otion tends to di sa ppear
preh en si ve t r ea t men t of th e subj ec t
i n a ny c o m
I n th e co nsi dera tio n of aeria l man oeu v r es an d st a b ility th e aero
dyn ami cs of th e m otio n m us t b e rela t ed to th e dynami c s of th e m o v in g
masses It i s u su a l to as su me th a t a i rcra ft are rigi d bo di es fo r th e p u rp oses
i n di ca ted and i n gen era l th e assu m i on i s j u stifi a bl e I n a fe w cas es as
in cert ain fin s of a i rshi p s whi c h deflec t under lo ad grea t er refin emen t may
be necessary as th e sc i ence of aerona uti cs dev eIOp s
It will readily b e un ders too d th a t aer o dyn am
i cs in i t s st ri c t i n t er
a t i on h as littl e di rec t c o nnec tio n with th e i n t erna l c o ns t ru c tio n of
re
t
p
a i r cr a ft th e i mp o rtan t it ems bein g th e ex t erna l fo rm an d th e c h an ges of
it whi c h gi ve th e p ilot c o n t rol o ver t h e m otion As th e s ubj ec t i s i n it self
ex t ens i ve an d as th e in t erna l st ru c ture i s b ein g dea lt with by oth er w rit ers
th e pre sen t boo k a i ms o n ly a t su pp lyi n g th e i nfor ma tio n by mean s of
whi c h th e fo rces on a i rcra ft i n m otion may b e ca l c ul a t ed
Th e sc i ence of ae ro dynami cs i s still ve ry youn g and it i s thi rt een y e ars
o n ly si nce th e fi rs t lo n g h ep o n an aero p l ane was made in p ubli c by Sa n to s
Du mon t Th e c i rc ui t of th e Ei ffe l T owe r i n a di rigibl e b alloon p r eceded
thi s fea t by o nly a sho rt perio d of ti me Aeronau ti cs a tt r ac t ed th e
a tt en tio n of n umero u s thinkers d u rin g past cen tu ri es and man y his to ri ca l
a ccou n ts are ex t an t d ea lin g wi th th e r esult s of t h e i r la bou rs Fo r man y
reasons ea rly a tt emp t s a t fli ght all fell sho r t of prac ti ca l su ccess although
th ey adv anced th e th eo ry of th e subj ec t i n variou s de grees Th e p r esen t
ep oc h of avi a tion may b e sa i d to h ave b e gun with th e p ubli ca tion of th e
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FOR MS OF AIRCR AFT
ST AND ARD
3
experi men ts m
ade by Langley i n Ameri ca in th e peri od 18 90 to 1900
Th e appara tus used was a whi r li ng arm fi tt ed with various con t ri van ces
for th e meas uremen t of th e fo rces o n fla t p l a t es m o ved th rough th e a i r a t
the end of th e arm
One lin e of exper imen t may perh ap s be describ ed b ri efly A n u m b er
of p la tes of e qua l ar ea w ere made and arran ged to h ave th e same tota l
we igh t a ft er whi c h th ey w ere cons t ra in ed to rem
a in ho ri zon t a l and to
fall dow n verti ca l gu i des a t th e en d of th e whir li n g arm Th e ti me of f a ll
of th e p l a te s thro ugh a gi ven d is ta n ce was mea su red and found to depend
not o nly on th e Speed of th e p l a t e t h rough th e a ir but als o on i t s sh ape
At th e s ame speed it was foun d th a t th e p l a t es with th e gr ea t e st d i mens ion
all er a spec t ra tio Fo r
ac ross th e wi nd f e ll m o r e slowly th an tho se of sm
sm
a l l ve lo c iti e s of f a ll th e tim
e of fa ll increased markedly with th e Speed
of th e p l a t e t hr ough th e ai r B y a c hange of experi m
en t i n whi c h th e
p la t es w ere h e l d o n th e whi rli n g arm a t an i nc li na tio n to th e ho ri zon t a l
a nd by r u nn i n g th e arm a t i ncreas in g speeds t h e va lu e of th e l a tt er wh en
th e p l a t e j us t lift ed it self was found R epe ti ti o n of thi s exper i men t
show ed th a t a par ti c ul ar i nc li na tio n gave l ess res is t an ce th an an y oth er
fo r th e co nditio n th a t th e p la t e shoul d j us t b e a i r borne
F r o m Lan gl ey s experi men ts it was ded uced th a t a p la t e w eighin g t wo
p ounds p e r s qu are foot c ould b e su pp o rt ed a t 8 5m p h if th e inc lin a tion
Th e resi st an ce was th en on e si x th of th e w eight
was ma d e e ight de grees
and ma ki n g a llow an ce fo r oth er parts of an aero p l ane it was c onc lu ded
tha t a t o t a l w eight of 750 lbs coul d b e carri ed fo r th e expen ditu re of 25
ho rs ep ow er Early experimen t ers set th ems e l ves th e t ask of buil d i n g a
co mp l e t e s t ru c ture withi n th es e li mit a tions and su c ceeded i n pro d u ci n g
ai rcraf t whi c h lift ed th ems el ves
L ang l ey p ut hi s experi men t al result s to th e t es t of a flight fr o m th e
O wi ng to acc i den t th e aer o
t Op of a hous e bo a t o n th e Poto mac ri ver
p lan e di v ed in to th e ri ver an d b ro u ght th e experi m
en t t o a very early end
I n En gland Maxi m a ttemp t ed th e des ign of a l arg e aerOplan e and
engi ne and achi eved a n ota bl e result wh en h e built an en gi ne exclusi ve
of boilers and w a t er whi c h w ei gh ed 180 lbs an d develo ped 8 60 ho rs e
p ow er To avoi d th e di ffi c ulti es of dea lin g with s t a bility i n flight th e
ae rO pla n e was made cap ti ve by fi x i n g wh ee ls b e tw een u pper and low er
ra ils Th e experi men ts carri ed out w ere very fe w i n n um b er but a lift
of
lbs was obtained be fo re o ne of th e wh ee l s carri ed a w ay a ft er
co n t ac t with th e u pper r a il
Fo r so me t en y ears aft er th ese experi men t s avi a tion too k a ne w
direc tio n and a tt emp ts to gai n kn owl ed g e of co n t rol by th e us e of aer o
p lan e gli ders w ere made by Pilch er L ili en tha l an d Ch an ut e F ro m a hill
bui lt fo r th e p u rp ose L ili en th a l made n umerous gli des b efo re b e in g cau ght
in a powerfu l gus t whi c h h e was una bl e to negoti a t e a n d whi c h co st hi m
I n th e course of hi s experi men t s h e di sco vered th e grea t su perio rity
hi s lif e
of a c ur ved wi n g o ver th e p l anes o n whi c h L angl ey co nd uc t ed his t est s
By a suit a bl e c hoi ce of c u rved win g it i s p ossi bl e to red uce th e resi st ance
to l ess than h alf th e va lu e es t i ma t ed fo r fla t p la t es of th e same c arryin g
n
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APP L IED AERODYN AMI CS
4
th a t wh i c h c oul d b e pro d u ced by m o vi n g th e bo dy of t h e aeron aut
i n a d i rec tio n to c oun t erac t th e e flec ts of th e wind fo rces
In th e same perio d very ra p i d p rogr ess wa s m
ade i n th e dev eIOp m
vnt
of th e light pe t rol m
oto r for a uto m
obil e r o ad t ransp o r t a n d b etween 19 00
and 1908 it became c l ear th a t th e p ro sp ec ts of mec h ani ca l flight h a d
mat eri ally i mp roved Th e fi rst achi evemen ts of p ower d riven a eroplanes
to ca ll fo r genera l a t t en tion th roughout th e worl d w e r e tho se of two
a n an d B l eriot wh o mad e n u me r ous sho r t fl igh t s
F renc hm
en H enri B arm
whi c h w er e li m
it ed by l ac k of ade q u a t e c on t rol Th ese two p ion eers t oo k
O pp o sit e v i e w s as to th e p ossibiliti es of th e bi p l a n e and m o n o p l a n e but
i n th e end th e fi rst pro d u ced an aero p l an e whi c h b eca m
e very p o p ul ar
as a t ra i n i n g aero p l an e fo r ne w p ilot s whils t th e s ec o nd h ad th e ho n o u r
of th e fi rst cro ssin g of th e Engli sh Ch annel fro m F rance t o Do ver
Th e l ack of c o n t rol re ferred t o exi st ed c hi efly in th e l a t era l b a l an ce of
th e aero p l anes it b e in g di fli cu lt to keep th e wi n gs ho riz on t a l by means
of th e ru dder a lon e Th e r ev olution ar y st ep cam
e fro m th e Broth ers Wright
A
i n Ameri ca as th e res ult of a pa ti en t s tu dy of th e probl ems of gli di n g
l a t era l c on t r ol was developed whi c h depen ded o n th e twis tin g o r warp i n g
of th e aero p l an e win gs so th a t th e lift on t h e dep r ess ed wi n g could b e
i ncreased i n o rder to ra i se it with a co rrespo n d i n g decr ease of li ft
As th e c h an g es of lift d u e to w arp i ng w e re acc ompani ed
on th e oth er wi ng
by ch an ges of dra g whi ch t ended to t u rn th e a ero p l ane t h e B roth ers
Wright co nnec t ed th e warp and ru dder c on t rols so as to keep th e aer op l an e
o n a st raight c ou rse d u ring th e w arp in g Th e princ i p l e of i n c r ea sin g th e
lift on th e lower win g by a speci a l c on t r ol 18 n ow u ni versa lly app li ed but
th e ru dder 1s n ot c onnec t ed to th e win g flap c on t rol whi c h h as t aken th e
p l ace of win g w arp in g F ro m th e ti e of th e Wright s fi rs t p ubli c fligh t s
cr ease t h e d ura tion
i n Eur o pe i n 1908 th e avi a to rs of th e wo r l d b egan to m
of th e ir flight s fro m m
in ut es to hours P rogress b ecame very rap i d and
th e speed of flight h as ris en fro m th e 8 5 m p h of th e Henri F arman to
near ly 140 m p h i n a m o dern fightin g sc out Th e range h as b een
i nc r eased t o o ver 2000 mil es i n th e bo m bin g c l ass of aerop l ane and th e
Atl an ti c O cean h as r ecen tly b een cro ssed fro m N ewfound land to I re l and by
th e V i ckers V i my bo m ber
As soo n as th e probl ems of sus t a in i n g th e w eight of an ae plane a n d
of c o n t rolli ng th e motion thr ough th e ai r h ad b een sol ved many i nvestiga
tio ns w ere a tt e m
p t ed of st a bili ty so as to elu c i da t e th e re qui remen ts i n
P arti a l a tt emp t s
an ae ro p l ane whi c h woul d render it a bl e to c o n t rol it se lf
were made in Fr a nce fo r th e aero p l an e by F er b er See an d oth ers but t h e
most satisfacto ry t rea tmen t i s du e t o Bryan Star ting in 1908 i n c olla bora
s B ry a n app li ed th e st andard ma th ema ti ca l e qu a tio n s
tion with Willi a m
of motion of a ri gi d bo dy to th e di stur bed m
otions of an aerop lane and th e
c ul m
i nation of thi s wo rk appeared i n 1911 Th e m
ath ema ti ca l th eo ry
remain s fun da m
en t all y i n th e fo rm pro p o sed by B ry an but c h an ges h av e
b een ade in th e metho d of app li ca tion as th e result of th e develo pmen t
of experimen t a l res earc h u nder th e Advi so ry Co mmitt ee for Ae rona u t i c s
Th e ma th ema ti ca l th eo ry i s founded o n a set of n u m bers obt a in ed f ro m
exp e ri ment an d it i s c hi efly i n th e de t ermin a tion of th es e n u m
b ers th a t
w as
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FOR MS OF AIRCRAF T
ST AND ARD
5
develo pmen t h as taken p lace i n recen t y ears Some ex t ens ions of th e
ma thema ti ca l th eo ry h ave b een made to c o ver fligh t i n a na tu ra l wind
9
and in sp i ra l pa ths
Experi men ta l wo rk o n st a bility o n th e m o de l scal e a t th e N a tiona l
Physi ca l L a bo ra to ry was cc o rdi na ted with flyin g experi men t s a t th e
Roy al Ai rcr a ft F ac to ry an d th e res ults of th e ma th em
a ti ca l th eo ry of
s t a bility w ere app li ed by B us k i n th e p r o d u c tio n of th e R E 20 aero p l ane
whi c h with c on t rol on th e ru dder o nly was fl own fo r dis t ances of 60 o r 70
mil es on several occasions By this ti me 1914 th e ma i n foundations of
avi a tio n as w e n ow kn ow i t h ad b een l ai d Th e l a t er his to ry is l argely
th a t of de t a il ed develo pmen t under st r ess of th e Grea t War
Th e hi sto ry of a i rs hi p s h as followed a di fferen t cou rse Th e probl em
of s u pp o rt n ever aro se i n th e same w ay as fo r aero p l anes and seap l anes
a s b alloons h ad b ee n kn own fo r m
an y y ea rs be fore th e adven t of th e ai r
s hi p
Th e firs t c h an ge fr o m th e free b a lloo n was littl e mo re th an th e
a tt ac hmen t of an engin e in o r der to gi ve it i ndependen t motion through
th e ai r and th e p ower ava ila bl e was very small Th e sp h eri ca l b a lloo n
h as a high resi s tan ce i t s c ou rse i s n ot easily di rec t ed and th e di rigibl e
b alloo n became elon ga t ed a t i t s ear li est s tag es Th e lon g cigar sh aped
fo rms ad0pt ed b rought th ei r own speci al di ffi c ulti es as th ey too are difli cu lt
to s t eer an d are inc lined to buckl e an d co llap se unl ess su fli ci en t preca utions
a re ta k en
en t h as been a tt ained in all cases by
St eerin g an d mana gem
th e fi ttin g of fin s both ho riz o n t a l an d verti ca l to th e rear of th e air shi p
en velope an d th e probl em of a ffi xin g fins of suffi c i en t ar ea t o th e flexi bl e
e n velope of an ai rs hi p h as i mp osed en gin eeri n g li mit a tions whi c h p reven t
a si mp l e app li ca tio n of aero dynam
i c kn owl ed ge
Th e proble mof ma in t ena nce of fo rm of an air shi p envelope h as l ed to
s evera l solutio ns of very differen t na tures
I n th e n on rigi d a i rshi p th e
en ve lo pe i s kep t i nfla t ed by th e pro vision of suffi c i en t in terna l pressur e
e ither by a uto ma ti c val ves whi c h li mit th e maxim u m pressur e or by th e
p ilot who lim
its th e M
u m Th e i n t erio r of th e enve lo pe i s d i vi ded
by gastight fa b ri c in to two o r thr ee c o m
p art men ts th e l arges t of whi c h
i s fi ll ed with hy drogen and th e sm
all er on es are fu lly o r parti a lly in fla t ed
with a ir e ith er fro m th e sli p s t ream of an ai rscre w o r by a Spec i a l
f an As th e a irsh i p ascends in to a i r a t lower pressure th e valves to th e
a i r c h am b ers o pen and a llow a i r to e sc a pe as th e hy dr og e n expan ds a n d
so lo ng as thi s i s p ossi bl e l oss of lift i s av oi ded
Th e g rea t est h eight to
whi c h a n on rigi d a irshi p can go without loss of hy dr oge n i s th a t fo r whi c h
th e ai r c h am bers o r balloon et s are emp ty and h ence th e si ze of th e
b a lloonet s i s pre p o rtion ed by t h e ce ili n g of th e ai rshi p
If th e car of an a i rshi p i s sus pen ded near i t s cen t re th e e nvelope a t
res t has gas forces ac tin g o n it whi c h t end to ra is e th e t ail and h ead Th e
undersi de of th e envelope i s th en i n tension on ac coun t of th e gas lift
whils t th e u pper si de i s in co m
pres sion As fa b ri c cann ot withstand
c ompression s uffi c i en t in terna l p ressu re i s app li ed to c oun t erac t th e e ffec t
of th e li ft i n p ro d u c in g co mpression
Th e car of th e n on rigi d ai rshi p i s a ttac h e d by ca bl es to th e undersi de
of th e env e lo pe an d as th ese ar e i nc li n ed an i n w ar d p ull i s exert ed whi c h
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APP L IED AER ODYN AMI CS
6
t ends to n eut ra li se th e t e nsio n i n th e fa b ri c Fo r so m
e par tic ul ar in t erna l
pressu re th e fa b ri c will t end to p u cker and Spec i a l exp eri m
en t s are made
to de t ermine thi s p r essure and to d i st ribut e th e p ull i n th e ca bl es so as
to make th e p re ss ur e as sma ll as p o ssibl e be for e p u ckeri ng occu rs Th e
experi m
en t i s made o n a mo de l ai rshi p whi c h i s i nvert ed and fi ll ed with
Th e lo ads i n th e ca bl es th e i r p ositions and th e p r es sur e are a ll
wa t e r
and th e neces sary measuremen t s are eas ily made Th e
u n der c o n t r ol
th eo ry of th e experi ment i s dea lt with in a l a t er c h ap t er
I n flight th e ex t erio r of th e envelo pe i s s ubj ec t ed to aer o dy nami c
re ssu re s whi c h are in t ens e near th e n o se
but
whi
c
h
f
a
ll
off ve ry
p
F ro ma t en den cy of th e n ose to blow
rap i dly a t p oi n t s b eh in d th e n os e
i n u nder p ositi ve p r essur e a c h an ge o cc u rs to a t endency to su ck out a t
a d i st an c e of l ess th an h alf th e d i ame ter of th e a i rs hi p b e hi nd th e n o se
and thi s su c tion i n varyi n g degre es persi st s o ver th e grea ter part of th e
en v elo pe
At h igh Sp eeds th e t endency of th e n o se to blow i n i s ver y
grea t a s co mpared with th e in t erna l press ur e nec essary t o re tai n th e fo rm
of th e rest of th e enve lo pe a n d a red uc tio n i n th e weight of fa b ri c u sed i s
obt a i ned if th e n o se i s re i n fo rced loca lly ins t ea d of ma in t ain in g i t s sh ape
by i n t ern a l press ure a lone I n on e of th e p hotograp h s of thi s c h ap t er th e
r e i n fo rcemen t of th e n o se i s very c l ear ly shown
Th e p r obl em of th e m ai n t e n ance of fo rm of a n on rigi d a irshi p i s
apprec i a bly si m
p lifi ed if th e w eight to b e ca rri ed i s n ot a ll c oncen t ra t ed
i n on e car
I n th e semi rigi d a i rshi p th e enve lo pe i s s till of f a b ri c mai n t a ined to
fo rm by i n t erna l pressu r e but b e tween th e en ve lo pe and car i s in t er posed
a lo n g gi rder whi c h di st ribut es th e c o ncen t rat ed l oad of th e car o ver th e
whol e su rface of th e en velo pe Th i s ty pe of a i rshi p h as b een u sed i n
F rance but h as recei ved m ost develo pmen t i n It aly it i s n ot us ed i n thi s
c oun t ry
Rigi d a i rshi p s depen d u po n a me t a l framewo rk fo r th e ma in t enance
of th ei r fo rm and i n German y w ere develop ed to a very high degree of
e ffi c i en cy by Cou n t Z eppel i n Th e l a rgest a i rshi p s are of rigi d con st ru c tion
and h ave a gro ss lift of nearly seven ty tons Th e f ramewo rk i s u su a lly
ay
of a light a lu in ium a lloy o cca sionally of w oo d an d i n th e futu re s tee l m
p ossibly b e u sed Th e st ru c tu re i s a light l a tti cewo rk sy s t emof gi r ders
r u nn i n g a lon g and around th e enve lo pe an d b raced by wi res in to a st i ff
fra m
e I n m o dern ty pes a keel gi r der i s p ro vi ded i n si de th e en velo pe a t
th e botto m whi c h serves to di st ribut e th e lo ad f ro m th e ca rs and al s o
fu rni sh es a co mm uni ca tio n w ay Th e n um b er of ca rs m
ay b e fou r o r m o re
and th e b endin g un der th e lift of th e hy drog en 1s kep t s mall by a ca re ful
c hoi ce of th ei r p osition s Som
e of th e t r an sverse girders a re b raced i n si de
th e enve lo pe by a n u m
b er of r a d i a l wi res th e cen t re s of whi c h ar e j oi ned
by a wi re r u n ni n g t h e whol e l en gth of th e ai rshi p a lon g i t s a xi s I n th e
com
par tm
en t s so pro d u c ed th e gas c on t a iners are fl o a t ed an d th e lift i s
e by th e pre ssu re o n a ne ttin g of sma ll c ord
t rans ferr ed t o th e rigi d fram
Th e l atti ce wo rk i s c o vere d by fa b ri c i n o rder to p ro d uce a s ooth
u n b ro ke n s u r face an d so keep d ow n th e res i st a n ce
Speed s of 75 m p h
h ave b een reach ed in th e l a t est B riti sh ty pes of rigi d a irshi p and th e re turn
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APP L I ED AER ODYN AMI CS
8
th e di rec tion of Pr and tl of wh i ch n o results h ave b een obt a ined i n thi s
coun t ry
S o me of th e German w rit ers on st a bi lity were followin g c losely
a lon g parall e l lin es to thos e of B ry an in Brit ain and h ad prior to 1914
arri ved a t t h e i dea of max im uml at er a l st a bility
Th e oth er Eur opean l a bor a to ry of n ot e was a t K ou t chi no n ea r Mos cow
with D Ri abou ehi nsky as d i recto r Thi s l a bo ra to ry appears to have b een
a pri va t e est a blis hmen t an d p l ay ed a y ery useful par t i n th e develop m
en t
of some of th e fun dam
en ta l th eo ri es of fl ui d m
otion Th e p rac ti ca l dem
an d
o n th e ti m
e of th e experi men t ers appears to h ave b een l ess s evere than i n
th e m ore West ern c oun t ri es
A N a tion a l Advi so ry Co mm itt ee fo r Ae ron a uti cs was fo rmed a t
Wash in gton on Apri l 2 1915 by th e Presi den t of th e Unit ed Sta t es
R ep o rts of wor k h ave appea red fr omti me to t i me whi c h l ar ge ly follow
th e lin es of th e ol der Briti sh Co mm itt ee and add to th e growin g stock of
val ua bl e aeronauti ca l da t a
Be fo re dea ling with Spec ific cases of a ircra ft it may b e u seful to c o mpa re
a n d c on t r a s t ma n s e ffo r t s with th e m os t near ly c orr e sp on di n g p r o d u c t s
of nature Betw een th e bird s and th e man carryin g a er op l ane th er e are
p oin t s of si m
i l arity and d ifferen ce whi c h st ri ke an ob server i mmed i a t e ly
B oth h ave win gs those i n th e bird b ei n g m o va bl e so as to allow of fla pp in g
whils t those i n th e aero p lane ar e fi xed to th e bo dy B oth th e bird a nd
th e a er op l an e h ave bo di es whi c h car ry th e moti ve p ower i n one cas e
m usc ul a r and i n th e oth er mec hani ca l B oth h ave th e i n t elligence fac to r
i n th e bo dy th e aerOplan e as a p ilot Th e aero p lane bo dy i s fi tt ed with
an a i rsc r e w a n o rg an wholly u nrep r ese n t ed in bi r d and an i m
a l lif e th e
p ro p u ls io n of th e bi r d th rough th e air as w ell as i t s su pp or t b ein g ac hi ev e d
by th e flapp in g of i t s wings In bo th ca s es th e bo di es t ermina t e i n thin
su rf aces o r t a i ls whi c h are used fo r c o n t ro l but whils t th e aer OpIa n e h as
a verti ca l fin th e bi r d h as n o su c h o r gan Th e w i n gs of a bird ar e so m
obil e
a t will th a t m
an ce u vr es of grea t co m
p l exity can b e m
ade by a lt er in g th ei r
p os itio n and sh ape m
an ce u v r es whi c h are n ot p o ss ibl e with th e rigi d win gs
of an aero p lane I n a d dition to th e di flerence b e tw een a irs cr ew and flap
p ing wings aero p lanes and bir ds di ffer grea tly i n th e arran gem
en t s for
a lighti ng th e sk i ds and wh ee ls of th e aerop l ane b e in g tot a lly d issi mil ar
to th e l e gs of th e b i rd
Th e stu dy of bi r d flight as a b as is fo r av i a tion h as c l early h ad a marked
i n fl uence on th e pa rti c u l ar fo rm whi ch m
o de rn a er0plan es h ave t aken
a n d n o me tho d of aer o dy na m
i c su pp o rt i s k no wn w h i ch h as th e sam e
valu e as t ha t obt a in ed fro m win gs si mi lar to those of bi rds Th e fac t th a t
flapp in g motion h as n ot b een adop t ed a t l eas t fo r ex t ensi ve t ri al appea rs
to b e d u e e n tir ely to mec h ani ca l di ffi c ulti es I n thi s respec t na tur a l
developmen t i nd i ca t es so m
e li mita tion to th e s ize of b i rd whi c h can fly
Th e sma ll er bi rd s fly with ease an d with a very ra p i d flapp i ng of th e wi n gs
l arger bi rds Spen d lon g perio ds on t h e wi ng bu t genera l i n fo rma tion
i ndi ca t es th a t th ey a re so ari ng bi r d s t ak i n g a dva n t age of u p c u rren t s
u an d
b ehi n d cli fls or a l arge s t eam
e r W ith th e still l arge r bi r d s t h e em
o st ri ch flight i s n ot p ossibl e Th e hi st o ry of bi rd life i s i n s t rxct acco r d an c e
with th e m
ec h a n i c a l p r i n ci p l e th a t s t ru c tu re s o f a si mil a r n a tu r e ge t
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ST AND ARD
~
F OR MS OF AIRCR AFT
9
re la tively w eaker as th ey ge t l arger Man although h e h as st ee l and a
l a rge se l ec tion of oth er ma t eri als a t h is d isp osal h as n ot found an ythi ng
so m uc h b e tt er th an th e m us cl e of t h e bir d as to make th e probl em of
s u pp o rti n g l ar g e w ei ght s by flapp in
r o mi si n g th an th e
fl
ight
an
y
m
o
re
p
g
results fo r t h e l ar gest bi rds I n loo ki n g fo r an a lt erna ti ve to flapp i ng
th e scre w pro pell er as develo ped fo r st eamshi p s h as b een m
o difi ed fo r aeri al
u se and a t pr esen t i s th e u ni versa l i ns t ru men t of p ro p uls ion
Th e ad ep tio n of rigi d wi ngs i n large fly i ng m ac hines in o rder to obta in
s u ffi c i en t st ren gth a ls o b rought ne w me tho d s o f c on t rol
Mec h an i cal
an uau vr e Show
p ri nc i p l es re l a t i n g to th e efi ect of siz e on th e capac ity fo r m
th a t reco very fro m a d is tur b ance i s slow er fo r th e l arger co nst ru ction
Th e gus ts enc oun t ered are m u c h th e s ame fo r b i rds and aero p lane an d
th e slown ess of reco very of th e aem
make
it
i
mpr
ob
a
bl
e
th
a
t
th
e
n
l
a
e
s
p
b e a uti ful ev olutions of a bi rd i n c oun t er i n g th e e ffec ts of a gus t will ever
b e i mita t ed by a man carryin g aero p l ane In o ne respec t th e aero p l ane
h as a d is tinc t advan t a ge : i t s Speed th r ough th e a i r is grea t er th an th a t
o f th e birds an d Speed i s it self o ne of th e m os t e ffec ti ve means of c o m b a t i n g
th e e ffec t of gus ts
F u rth er re f er en ce to bir d fl ight i s fo reign t o th e p ur p os e of thi s boo k
whi c h re l a t es t o i nfo rma tion obt a ined without sp ec i a l a tt en tion to th e
s tu d y of bi rd fl ight
Th e a i rs hi p envelo pe and th e s ub marine h ave mo re resem bl ance to
th e fish es than to an y oth er li v ing crea tur es Generally Spea kin g th e fo rm
of th e l ar ger fi sh es pro v i des a very goo d b asis fo r th e fo rm of a irshi p s
It is c urious th a t t h e fins of th e fish are usu a lly verti ca l as d i stinc t f ro m
th e ho ri z o n ta l t ai l f ea th ers of th e b i rd and th e fins o ver a n d under th e
cen t ra l bo dy h ave n o c o u n t erpar t i n th e a i rshi p B oth th e a r t ific i al and
li v i n g cra ft obta in su pp o rt by di sp la cemen t of th e mediu m i n whi c h th ey
a r e s ub merged and ris in g and f a llin g can b e pr o d u ced by m o der a t e c h a n g es
of v ol u me Th e resem bl ance b e tw een th e fish es and a i rshi p s i s far l ess
c los e th an th a t be tween th e bi rds and aero p lan es
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G ENERAL D ESCRI P TI ON
Pa a r rcvnxa Axa cna r r
or
n um ber of p ho t ograp h s of m o dern air cra ft and aero en gi nes a re
repro d u ced as ty p ica l of th e subj ec t of aero na utic s Th ey will b e used to
defi ne those part s whi c h are i mp o rt an t i n eac h type Th e de t a il s of th e
m otio n of a ircra ft a re th e subj ec t of l a t er c h ap t ers in whi c h th e c ond ition s
of s t eady m otion and s t a bility ar e deve lo ped and d iscu ssed
Th e Ant oni n a —Th e fro n ti sp i ece show s a l arg e aer0p lan e i n fli ght
B uilt by Messrs Handl ey P a ge
Cc th e aerOplan e i s th e h eav i est y e t
fl own and w eighs a bout
I t s en gi nes deve lo p
lbs wh en fully lo aded
15
00 ho rsep ow er an d p r o pe l th e aero p l ane a t a Speed of a bout 100 m il es
an hour It i s of n o rm
a l bi p lane c o nst ru c tion for i t s win gs th e Spec i a l
c ha rac t eri sti cs b ei n g i n th e box t ail an d i n th e a rran gemen t of i t s four
engi n es Each engine h as i t s own a i rscrew th e p ow er un it s b e in g d i vi ded
i n to two by th e bo dy of th e a er0p lane each h alf co ns isti n g of a pa i r of
en gi nes arranged b ack to b ack One a irscre w of each pa ir i s wor k in g i n
A
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APP L IED AERODYN AMI CS
10
th e dra ught of th e fo rw ard scre w and thi s t andem arrangemen t i s as y e t
ewha t n o vel
so m
—Fig 1shows a Si ngl e sea ter fi ghtin g sc o ut th e SE 5
B i p lane ( Fig
m u c h u sed i n th e la ter st ages of th e w ar I t s four wi n gs are of e qua l l ength
a n d fo rm
th e two p la n es whi c h gi ve th e nam
e to th e ty pe Th e lower win gs
a re a tt ac h ed to th e un dersi de of th e bo d y b e hi nd th e a i rscre w an d en gine
c owl whi lst th e u pper win gs are j oined to a shor t ce n t re sec tion su pp o rt ed
fro mth e bo dy on a fr a m
ewo rk of st ruts and wir es Away fro m th e bo dy th e
u pper an d lower p la nes are su pp o rt ed by wi ng st r ut s and wi re b rac i ng
a n d th e whol e fo r m
I n fli ght th e lo ad fro mth e w in gs i s
s a sti ff gi r der
t r ansmitt ed to th e bo dy th r ough th e wi ng st ru t s a n d th e wi res f ro m th ei r
u pper ends to th e u ndersi de of th e bo dy Th es e wir es are fre qu en tly
referred to as lift wi res Th e d ownw ar d load o n th e wi n gs w hi c h a ec om
s
n
a
n
i
s
th
e
r
u
n
n
i
n
g
of
th
e
aer
o
p
l
ane
o
ver
r
ough
g
r
ou
nd
i
t
aken
by
a
ti
e
p
li ft wires whi ch ru n fro mth e low er en ds of th e win g st rut s to th e cen t re
sec tion of th e u ppe r p l a n e
I n th e di rec tio n of m
otion of th e aer o p la ne i n flight ar e a n um b er of
b rac in g wi res fro m th e bottomof th e var ious st ru t s to th e to p of th e
n eighbou r i n g m
em
ber Th es e wi res s t i ffen th e win gs i n a w ay whi c h
main ta ins th e c orr ec t angl e to th e bo dy of th e aerop lane an d are kn ow n
Th e b r ac i n g sy st emi s r ed un dan t
on e o r m o re
as i nc i de nce wi res
mem b ers may b reak without ca usin g th e c oll ap s e of th e st ru c ture
Th e wi n gs of eac h p l ane wi ll b e seen fro m th e p hotograp h to b e b en t
u p w ard s i n wh at i s kn own as a dih edra l angl e th e obj ect of whi ch i s to
as sist in obt aini n g l a t er a l st a bility Fo r th e l at e ra l co nt rol win g flap s
ar e pr o vi ded th e ex t e n t of whi c h can b e seen o n th e wi n gs on th e l e ft of
th e fi gu re On th e lower flap th e l ever fo r a tt ac h men t of th e o per a tin g
ca bl e i s vi sibl e th e l att er bei ng l ed i n to th e wi n g a t th e fro n t spar an d
h ence by p ull ey s to th e p ilot s c ockp it Th e po sition s of th e fron t an d rear
spars ar e i n di ca t ed by th e e n ds of th e win g s t rut s i n th e fo r e an d a ft
di rec tion an d r u n a lon g th e win gs para ll e l to th e l ead in g ed ge s
Th e bo dy res t s o n th e Spars of th e botto m p l an e an d car r i es th e en gin e
and ai rscre w i n th e fo rward end Th e en gine i s w a t er cool ed and t h e
n ecess ary r adi ato rs are m
oun t ed mth e n os e i m
medi ately behi n d th e ai r
b to
sc r e w
B lin ds sho wn clo sed ar e r e qui red i n a er 0plan es whi c h c li m
g rea t h eights si n ce th e t e m
peratur e i s th en w ell be low th e freezin g p oin t
of wa t er a n d unrest ri c t ed flow of a i r through th e radi a to r d ur in g a gli de
woul d l ead to th e freez i ng of th e wa t er an d to lo ss of c on t rol of th e en gine
Th e bli nd s can b e adj ust ed to gi ve i nt e rm
ed i a t e de gr ees of c oolin g to
c o rres p ond with en gin e p owe rs i nt er medi a t e be tween gli din g a n d th e
maxi m
u m p ossibl e
Alo n gs i de t h e b ody and st ret c hi n g b ack b ehi n d th e p ilot s sea t i s o ne
of th e ex ha u st p i pes whic h carry th e hot gas e s w ell to t h e r ear of th e aero
p l a ne Th e p ilot s sea t i s ju s t b ehi n d th e t ra ilin g ed g e of th e u pper win g
Abo ve th e exh a ust p i pe a n d nea r th e fro n t of th e bo dy i s a c o ver o ve r th e
c yli n ders on o ne si de of th e en gi ne th e c o ver b ei ng us ed to r ed uce th e a i r
r es i st a n ce
Th e airsc re w i s in th e ex t rem
e fo rwa r d p ositio n on th e aero p l an e and
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Dig tiz e d by
G o ogle
FOR MS OF AIRCR AFT
ST AND ARD
11
fo u r bl ades Th e di a m
e t er i s fi xed i n this case by th e h igh Speed of
the a i rsc r e w Sh a ft an d n ot as i n many cases by th e ground c l earance
re quir ed fo r sa fe ty wh en run n i ng o ver th e groun d
Below th e bo dy and under th e wings of th e low er p l ane i s th e l and in g
ch assis Th e frame c onsis t s of a pai r of vee Sh aped st rut s b as ed on th e
bo dy an d j oined a t th e botto m ends by a cross tub e Th e st ru ct u r e i s
s u pp o rt ed by a d i ago na l cro ss b rac ing of wi re s
Th e wh ee ls an d axl e are
h el d to th e undercarri age by bin din gs of rubber c o rd so as to pro v i de
fle xibili ty . Th e shocks of l andi ng ar e t aken par tly by thi s rubber cord
a n d pa r tly by th e p n e u ma ti c ty re s o n th e wh ee ls
With th e a er0p lane
bo dy nea r ly ho riz on tal th e wh ee l axl e is a h ea d of th e cen t re of g ravity
of the aer o p lane so tha t th e e ffec t of th e fi rs t co n tac t with th e gro u nd i s
to throw u p th e n ose increas in g th e angl e of inc i den ce an d drag If th e
Sp eed of a li ghti n g i s t oo g rea t th e lift may i ncrease su ffi c i en tly to rai s e
the aerop lane off th e ground Th e art of makin g a co rrec t l andin g is on e of
th e m os t di ffi c ult p arts to be l earn t by a p ilot
Th e ta il of th e aero p l ane is n ot c l ear ly Shown i n th is fi gu re and
h as
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W ith an en gi ne develo p ing 210ho rs ep ower and a load b ri n gin g th e gross
n
i
s
we ight of th e aem
l
a
e
to
lb
th
e
l
a
n
e
illu
s
t
ra
t
ed
capa bl e
s
r
20
0
a
0
e
O
p
p
of a s peed of o ver 180 m p h and can c li m
b to a h eight of
f ee t
Th e li m
it to th e height to whi c h a ir cra ft can c li m b i s us ually call ed t h e
ceilin g
— Th e m ost st ri k in g differ en ce fro m Fig 1 i s th e
Monop lane (Fig
c hang e f ro m two p lanes to a si ngl e on e a n d i n o r der to su pp o r t th e wi n gs
a ga ins t l an di n g Sh o cks a p yra m
i d of st ru t s o r ca b an e h as b een bui lt
o ver th e bo dy Fro m th e apex of th e p y rami d b raci n g wir es ar e carri ed
to p oin t s o n th e u pper si de s of th e fron t an d rea r sp ars Th e lower b racin g
wires go fro m th e Spars to th e undersi de of th e bo dy and eac h i s d u p li ca t ed
o me t er us ed as part of
On th e right wi ng near th e ti p i s a tub e anem
th e e qui p m
en t for measu rin g th e speed of th e aero p l ane In bi p lan es th e
a n em
om
et er i s us ua lly fi xed to o ne of th e win g st rut s as th e e ffec t of th e
p resence of th e win g o n th e r ea d i ng i s l ess mar ked th an i n th e case n ow
i llus tr a t ed
In thi s ty pe of aerOp lane made by th e Briti sh dz Coloni al Aero p la n e
i n e r ot a t es and th e a i rscre w h as a s o m
oy
th
e
en
g
e
w
h a t un u su a l f ea tur e
C
i n th e
sp inner
whi c h i s a tt ac h ed to it Th e a irsc rew h as two bl ades
o nly an d thi s ty pe of cons tr u c tion h as b ee n m
or e c o mm on th an th e fou r
bladed type for r e aso ns of eco n o my of ti m
b er Th e d i fferen ces of
e ffi c i ency are n ot marked an d e ith er ty pe can b e made to gi ve goo d
s ervi ce th e c hoi ce b ein g de t ermin ed i n so m
e cases by th e Speed of rot a tion
of th e ai rscre w sh a ft of an ava il a bl e en gi n e
Th e undercarri ag e i s very si m
il ar to th a t shown i n Fig 1 On on e
of th e fron t s t ruts i s a s ma ll windm
ill whi c h dri ves a p M p for th e pe t rol
feed W indm
i lls are n ow f re qu e n tly u sed for a u xilia ry se r v i c es su c h a s
th e e l ec t ri ca l h ea ti n g of c lothi ng and t h e g enera tion of c u r r en t for t h e
wi rel ess t ran smi ssion of messages
Th e t ail i s c l early v isibl e an d un dern ea th t h e ex t reme e n d o f t h e bo dy
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APPL IED AERODYN AM I CS
12
th e t ail ski d T his ski d is hin ged to th e bo dy and i s s ec u red by rubber
c o r d a t i t s i nner en d so as to dec reas e th e shock of c o n t ac t with th e g round
Th e ho riz o n t a l p l ane a t th e t ai l is seen to b e di vi ded th e fro n t part o r
t a il p l ane be in g fi xed whils t th e rear part o r e l eva to r i s m o va bl e a t th e
p ilot s wi sh Th e c o n t rol ca bl es go insi de th e fuse l ag e a t th e root of th e
t a il p l ane Un dernea th th e t a il p l ane 18 seen to be b raced to th e bo dy
a bo ve th e b rac in g wir es are a tt ac h ed to th e fin whi ch li ke th e t a il p lane
is fi xe d to th e b o d y
T h e ru dder i s hi dden b ehin d th e fin but th e ru dder
l ever fo r a tt ac h men t of th e con t rol ca bl e can b e seen a bout ha lfway u p
th e fin
Th e p ilot sit s under th e ca b ane an d hi s d own w ard v i ew i s h el ped
by hol es th rough th e wi n gs I mmedia tely i n fro n t of hi m i s a wi nd screen
a nd a ls o i n thi s ins t ance a mac hi ne gu n whi c h fi re s t h rough th e a i rscre w
— T h e difi er ence of Sh ape fro th e land ty pes i s
Fly i ng-boat ( Fi g
marked in severa l di rec tio ns as wi ll b e seen f ro m th e illus t ra tion rel a ti ng to
th e Phoen ix Co rk fly ing boa t P 5 Th e parti c ular fea ture whi c h gi ves
i t s name t o th e ty pe i s th e bo a t s t ru c tu re under th e low er win g an d thi s
rep l ace s th e wh ee l undercarri a ge of th e aero p lan e in o r der to ren der p ossibl e
a lightin g o n w a t er T h e flyin g boa t i s shown m ou n t ed o n a t roll ey d u rin g
t rans it fro m th e sh eds t o th e wa t e r On th e unders i de of th e bo a t j us t
b ehind th e na tio n a lity c irc l es i s a s t ep whi c h p l ay s an i mp o r t an t part i n
th e pre li in ary ru n o n th e wa t er A sec ond st ep occ urs un der th e win gs
a t th e p lace of l a st c on t ac t with th e sea d u rin g a flight but i s hi dden by
th e deep Shad ow of th e low er win g
Undernea th th e low er win g a t th e out er st rut s i s a win g fl oa t whi c h
k eep s th e wi n g out Of th e w a t er i n any Sli ght roll Th e win g st ru c tu re i s
m u c h l ar ger th an thos e of Figs 1 and 2 and th ere are si x pa i rs Of in t er
p lane st rut s T h e u pper p l ane i s apprec i a bly lon ger th an th e lower th e
ex t ensio ns b ein g b raced fro m th e f ee t of th e out er st rut s Th e l evers o n
th e win g flap s o r a il erons ar e n ow very c l ear ly shown
owi n g to th e
pro xi m ity of w aves to th e lower wi n g a ilero ns are n ot fi tt ed to th em
Th e t a il i s ra ised high a bo ve th e boa t and i s i n th e Sli p s t reams fr o m
th e t wo a irs c r e ws As th e cen tre line of th e ai rscrews i s far a bo ve th e
ce n t re O f gravi ty swit chi n g on th e en gine woul d t end to make th e flyi n g
bo a t d i ve w ere it n ot so arran ged th a t th e sli p st ream e ffec t on th e t ail
i s arran ged to gi ve an Opp o sit e t endenc y
Th e fi n and ru dder a r e c l early
sho wn as a r e a ls o th e l evers on th e ru dder and e l eva to rs
Bes i des h av in g
a d i h edra l a n gl e on th e wi n s s
a
l
h
ave
b
een
tt
ed
a
bo
ve
th
e
to
p
l
fi
n
s
fi
g
wi n gs as part of th e l a t eral b a l ance Of th e fly i ng bo a t
T h e en gines are bui lt on st rut s b e tw ee n th e win gs an d eac h en gine
dri ve s a t rac to r a i rscre w Th e e ngines are ru n i n th e sa m
e d i rec tion
a lthough a t a n early s t a g e of deve lo pmen t of fl i n
s
e
fl
bo
a
t
th
e
e
t
of
s
c
y g
i
S
n
ac
tio
n
of
th
e
r
ot
a
to
r
y
a
i
r
cre
w
w
ere
e
i
m
i
na
t
ed
by
arra
gi
n
g
O
O
c
l
e
s
s
p
g
fo r rot a tion in O pp osit e di r ec tions Thi s was fo u n d to b e unn ecessary
Th e t a i l of th e fly i ng boa t h as b een espec i a lly arran ged to c o m
e i nto th e
sli p s t rea mo f t h e a i rscre ws
but i n a e ro p l anes thi s o cc u rs withou t
Spec i a l pr o v is io n o r de si re
Not o n ly d o es th e ai rscre w i nc rea se th e ai r
Speed o ver th e t a il but it a lt ers th e an gl e of i n c i d enc e and blow s t h e t a il
is
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m
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m
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Fl o
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4
.
—Cock p i t
l
o f a n ae r o p ane.
D g t z ed b y
G
FOR MS OF
ST AND ARD
AIRCR AF T
18
t
s
s
s
s
i
o
r
d
ow
n
depend
i
n
g
o
n
i
e
tti
n
g
T
h
ere
a
l
o
a
tw
i
s
t
i
n
th
e
li
p
t
r
am
S
s
e
p
w hi c h is fre qu en tly unsym e t ri ca lly p l aced with reSp ect t o t h e fin and ru dder
a n d t e n d s to pro d u ce tu rn in g
T h e e ffec ts of s wit c hin g th e en gin e o n and
O ff may be very c o mp l ex
I n o r der to ease th e p ilo t s e ffo r t s man y aero p l an es are fi tt ed with an
a dj us ta bl e t ai l p l ane and if th ey are st a bl e th e adju st m
en t can b e made
s o as to gi ve an y c hos en flyi n g sp eed without th e a pp li ca tio n of fo rce to
th e c o n t rol sti ck
’
—
Th e p hotogr ap h of th e P an th er was t aken
Pilot s Cock p i t ( Fig
f r o m a bo ve th e aer o p l an e loo ki n g d own and fo rward At th e botto m of th e
fi gure i s th e ed ge of th e sea t whi c h res t s o n th e to p of th e pe t rol t ank Alon g
t h e cen t re of th e figu re is th e c on t rol c olumn hin g ed a t th e botto m to a rock
B y suit a bl e
i n g sh a ft so tha t th e p ilot i s a bl e to m o ve it i n any di r ec tion
ca bl e c o nnectio ns it i s arran ged th a t fo re and a ft m o vemen t dep r esses o r
ra ises th e el eva to rs whilst m o vemen t t o right o r l e ft ra i ses o r lowers t h e
ri ght ail er ons So me of th e co nn ec tio ns can b e seen ; b ehin d th e co n t r ol
co lu m
n i s a l ever a tt ac h ed to th e rocki n g sh a ft and h av in g a t i t s ends th e
ca bl es fo r th e a il erons Th e ca bl es can b e seen passin g i n inc lined di r ec tions
i n f ron t of th e pe t rol t an k O n th e near Si de of th e c on trol c olu mn but
par tly hi dden by th e sea t i s th e link whi c h o pera tes th e el eva to rs
In th e cas e of each co n t r ol th e m otion of th e c olu n re qui red i s th a t
whi c h woul d b e made w ere it fi xed to th e aerop lane and th e p ilot h el d
i ndependen tly and h e a tt emp t ed to p ull th e aero p l ane in to an y des ir ed
s
o
itio
n
I
n
oth
er
wo
rd
if
th
e
p
ilot
p
ull
s
th
e
s
ti
ck
tow
ard
s
hi
m
th
e
n
o
s
e
s
p
of th e aerop l ane c o mes u p whi ls t m o v in g th e c olumn to th e right b rin gs
th e l eft win g u p
On th e to p of th e c on t r ol c ol u mn i s a sma ll swit c h whi c h i s us ed by th e
p ilot to c ut out th e en gi ne t em
p o rarily an O pera tion whi c h i s fre qu en tly
re qui red with a rot ary en gine j us t be fo re l andin g
Acr oss th e p hot ogr ap h an d a littl e b e low th e en gin e c on t rol swit c h es
i s th e ru dder b ar th e hi n g e of whi c h is verti ca l and b ehi n d t h e con t rol
col u mn
Th e two ca bl es to th e ru dder ar e seen to c ome st ra ight b ack
In th e ru dder c o n t rol th e p i lot p ush es th e ru dder
u nder th e p ilot s sea t
ba r to th e right i n o rder to tu rn to th e right
In th e t a p l eft
Severa l inst ru men ts ar e shown i n th e p hotograp h
c o rner i s th e an eroi d b aro me te r whi c h gi ves th e p ilot an appro x i ma t e
i dea O f hi s h ei gh t In th e cen t re i s th e compass an i ns t ru m
en t Spec i a lly
des igned fo r a ircra ft wh ere th e co ndition s of u se are n ot very f av ou r a bl e
to goo d results I mmed ia t ely b e low th e c ompas s and partly hi dden by
i t i s th e a i rs peed i n di ca to r whi c h i s us u ally c onnec t ed to a tub e a n em o met er
s u c h as was shown i n Fi g 2 on th e ed g e O f th e wi n g
Still low er o n t h e
inst ru en t boar d and b e hin d th e c o n t rol co lu mn i s th e cro ss l eve l whic h
indi ca t es to a p ilot wh e th er h e i s si de Sli pp in g o r n ot T o th e right of
th e cross l eve l are th e st ar tin g swit c h es fo r th e en gin e two magne to s b ei ng
us ed as a p reca utionary measu re Below and to t h e r ight of t h e ru dder
bar is t h e en gine revolution i nd i ca tor
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APP L IED AER ODYN AMIC S
14
ENGI NES
O
—
In thi s ty pe of en gi ne th e R R: 2
Ai
th e ai rscre w i s bolt ed to th e cran k c as e and cyli nders and th e whol e th e n
rota t es a bout a fi xed c r an ksh a ft Th e cy linders ni ne in n um b er develo p a
n et b rake ho rs ep ow e r of a bout 280a t a Speed of 1100 to 18 00 rev olutio ns p er
mi n ut e Th e cylinders are p r o vi ded with gills whi ch grea tly assis t th e c ool
in g of th e cylinder d u e to th e i r m otion thr ough th e ai r W ithout an y fo rwa r d
m otio n of th e aerOplane c oolin g i s pro vi ded by th e rot a tio n O f th e cylinders
and an apprec i a bl e par t of th e ho rs ep ow er developed i s a b so r be d i n tu rnin g
th e en gin e aga in s t i t s ai r resi st an ce Ai r and pe t rol are adm it t ed th rough
p i p es shown a t th e si de of eac h cylinder and both th e in l e t an d e xh a us t
va l ve s are m ec h ani ca lly o pera t ed by th e ro d s fro m th e h ead of th e cylinder
to th e crank c ase Th e cam mec h an is m fo r o pera tin g th e ro ds i s in si de
th e crank case Th e hub for th e a tt ac hmen t of th e a i rscre w i s shown in
th e ce n t re
ilar appearance has st a tio nary
A ty pe of en gi ne of generally s im
cylinders and i s kn own as radi a l
It i s prob a bl e th a t th e c oolin g lo sses
in a radi a l en gi ne are l ess th an tho se i n a ro ta ry en gine of th e same n et
p ow er but n o direc t co m
pari son appears to h ave b een made Th e
e ffec ti ven ess of an en gine cann ot b e di ssoci a ted fro m th e means t aken to
c ool i t s cylin ders Th e re sis t ance of cylinders i n a ra d ia l en gin e and
ra di a to rs in a w a t er c ool ed engine shoul d b e estima t ed and a llow ed fo r
b e fo re c om
pa ri so n can b e made with a rot ary en gine th e losses of whi ch
h ave al ready b een ded uc t ed i n th e en gin e t es t b ed fi gu res Fo r en gin es
with st a tion ary cyli nders t es t b ed fi gur es us ua lly gi ve b rake ho rs ep ow er
without a llow an ce for aer o dyn ami c c oolin g losses
—
5
b
n
s
a
s
A
i
r
o
l
E
n
i
n
Fig
Th
e
e
gi
ne
how
n
tw
e
l
ve
h
c
o
e
d
e
Vac typ e
g
(
)
cylinders develo p s a bout 24 0 ho rsep ow er an d i s kn own as th e
Th e c yli nders are arran ged a bo ve th e cran k ca se i n
R A F 4d
two rows of si x with an a n gl e b e tw een th em h en ce th e name gi ven
to th e ty pe In o r der to cool th e cylin ders a c owl h as b een pro vi ded
so th a t th e fo rwa r d m otio n of th e aerOp lan e fo rc es a i r b e tw een th e
cylinders an d o ver th e cylin der h eads At th e ex t reme l e ft of th e p hoto
grap h i s th e ai rscre w hub and in th i s parti c u l ar en gin e th e a i rs cr e w is
geared so as to tur n a t h a lf th e Sp eed of th e cran ksh a ft th e la tt er mak in g
To th e right of an d below th e a i rscrew hub i s on e of
1800 to 2000 r p m
th e ma gnetos wi th i t s d is t ributin g wi re s fo r th e c o rrec t ti min g Of th e
exp lo sio ns i n th e sever a l cyli n ders At th e botto m of th e p hotograp h are
th e i n l e t p i pes ca r bure tto rs pe t rol p i pes and th rottl e va l ves
—Wa t er c ool ed en gin es h ave b een used
Water-cooled Eng i ne ( Fig
m or e th an any oth er type i n both aero p l an es and a irshi p s Th e two
p hotograp h s of t h e N ap i er 4 50 h p engi ne show wh a t an in t ri ca t e
mec h an i sm th e aer o en gine may b e Th e cyli nders ar e arranged i n th ree
rows of four eac h one be in g s urro un ded by a w a t er j acke t Th e feed
p i pes of th e w at er ci r c ula tin g syst em can b e seen in Fig 6b goin g fr om
th e w a ter p u mp a t th e botto m of th e p i c ture t o th e low er en d s of th e
cylinder j acke t s whi ls t a bo ve th em are th e p i pes whi c h co nnec t th e
a)
led Rotary Eng i ne ( Fig 5
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F la
F la
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5( b )
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5 (a )
—Air
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Ro t ary
l
e n gi ne
coo ed s t at i onary
.
e ngi ne .
Dignt iz ed b y G o c
APP L IED AERODYN AMIC S
16
th e shi p and rin gs runn in g round it Two ty pes of rin g are vis ibl e on e
of whi c h i s wholly c o mp osed of si m
p l e gi rders whi ls t th e sec on d h as k in g
pos ts as s t i fleners on th e i n si de Fr o m th e co rners of thi s seco nd frame
radi a l wi res p ass to th e cen t r e of th e envelo pe an d fo rm o ne of th e di v ision s
of th e a i rshi p Th e ce n t res of th e various rad ia l di v is ions are co nnec t ed
by an axi a l w i re whi c h t akes th e en d press ure of th e gas b ags i n th e cas e
of defla tion of on e of th em o r of i nc lina tion of th e a irs hi p T h e co r d ne ttin g
a ga in st whi c h th e gas b a gs rest can b e s een very c l ea rly T h e ai rs hi p i s
i ralty by Messrs Bear dm o re
on e bu i lt fo r th e Ad
—
Th e n on rigi d type of c on s t ru c tion
Th e Non ri gi d Airshi p (Fig
i s essen ti a lly d ifferen t f ro m th a t d escribe d a bo ve th e sh ape of th e enve lo pe
b e in g a in t ained wholly by th e in terna l gas press ure Th e N S type of
a i rshi p ill ust ra t ed i n Fig 9 h as a gr oss w eight of 11 tons and
t r ave ls a t a li ttl e m o re than 55 m p h T h e l en gth i s 262 fee t and th e
maxi m u m wi d th of th e envelo pe 57 fee t Fi g 9b gi ves th e b est i dea of
th e cro ss sec tio n of thi s type of a i rshi p and shows thr ee lob es mee tin g in
we ll defin ed c o rners Th e ty pe was o rigi na t ed in Spai n by T o rres Qu evedo
It c o n ta in s an i n terna l
an d develo ped i n P aris by th e Ast ra Co mpan y
ro pes and fa b ri c b e tw een th e c o rners
Th e
sa tis fac t o ry d i st ribution of lo ads o n th e f a b ri c d u e to th e w eight of th e
car and en gi n es i s p o ssibl e with thi s c onst ru c tio n without necessit a tin g
Fi g 90 t aken
suspens io n f ar be low th e low er s u r f ace of th e enve lo pe
fr o m b elow th e airshi p shows th e wires f ro m th e car to th e j un c tio n of
th e lob es a t th e botto m of th e envelo pe and th es e t ake th e whol e load
To b r ace th e car agai ns t ro llin g wires ar e
u nder l eve l keel con di t i on s
carri ed out o n e ith er si de and fi xed to th e lob es a t so me d i st ance f ro m th e
p lane of symme t ry of th e a irshi p Th e princ i p l e of reli e f of st ress by
d ist ribution of lo ad h as b ee n utilis ed i n thi s s hi p th e car and en gine
nace ll es b eing su pp or ted as separa t e u n its Co mm uni ca tio n i s per i tt ed
acro ss a gan gway whi c h add s n othi n g of valu e to th e dis t ribution of
lo ad
T h e en gines are two in n um b er situ a t ed be hin d th e obs erva tion car
and eac h is pro vi d ed with i ts own a irscr e w Benea th th e en gin es and a ls o
b e low th e car are bump in g b ags fo r u se o n alightin g
As th e sh ape of th e a i rshi p i s dependen t o n th e in t erna l gas pressu re
spec i a l arran geme n t s are made to c o n t rol thi s qu an tity an d th e f a b ri c p i p es
sho wn i n Fig ; 90 show how a i r i s a dmit t ed fo r thi s p u rp o se to enc lo sed
p ortions of th e envelo pe T h e enve lo pe i s d i vi ded in si de by gastight
fa b ri c so th a t i n th e lower lob es both of th e fo re and rear parts of th e
a i rshi p small ch am
b ers o r b alloonet s are fo rmed in to whi ch a ir can b e
p um
ped or f ro mwhi c h it can b e r e l eas ed Th e p ositio n of th es e balloon et s
ca n b e see n i n Fig 90 a t th e end s of th e pa i r of lo n g ho ri z o n t a l f ee d
p i p es ; th ey are c ross c o nnec t ed by f a b ri c tub es whi ch are a lso c l early
Th e hi gh pressu re ai r i s obta in ed f ro m sc oop s lowered in to th e
v i sibl e
sli p s t reams f r o m th e a i rscre ws th e sc oop s b e i n g v i s ibl e i n all th e fi gures
but a r e fol ded a ga in s t th e envelo pe i n Fig 9a V a l ves are pro vi ded i n
th e feed p i pes fo r use by th e p ilot who infla t es o r deflat es t h e b alloo net s
as r e qui r ed to a llow for c h an g es i n v olu m
e of th e hy drogen d u e to vari a tion s
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STAND ARD
of h eight
Auto ma ti c
.
FOR M S OF AIR CR AFT
17
va l ves are arranged to re lease a ir if th e pressure
Th e we ight of f a b ri c necessary to withst and th e pressur e of t h e gas
i s grea tly r ed u ced by rei n fo rc in g th e n o se of th e a i rshi p as s how n i n
Fig 9b Th e m axi m um ex t ern a l ai r force d u e to m
otion o cc u rs a t t h e
n ose of th e a i rsh i p a n d a t high speeds b ec o mes gr ea t er th an th e in t er na l
press ur e usu a lly pro v i ded Th e region of hi gh pressure i s ext remely l ocal
an d by th e add itio n of s ti ffeni n g rib s th e exce ss of pres sur e o ver th e
in t erna l pressure i s t rans mitt ed back to a p a rt of th e enve lope wh ere it i s
eas ily s upp o rt ed by a sma ll in t erna l press ure Occasionally th e n o se of
a n a i rs hi p 1s blown i n a t h igh Spe ed but with th e arran gemen t s ad o p t ed
th e consequ ence s are uni mp or t an t and th e c o rrec t sh ape 18 reco vered by
an increas e of b a lloo n et p r essu re
Th e in fla tio n of one b a lloo ne t and th e defla tio n of th e oth er i s a c on t r ol
by means of wh i c h th e n ose of th e a i rshi p can b e ra is ed o r low ered and so
e ffec t a c ha nge of t rimbut th e usu a l c on t rol i s by e l eva to rs and ru dders
I n th e N 8 ty pe of ai rsh i p th e r u dder i s co nfin ed to th e low er s ur f ace a n d
t h e u pper fi n i s of red uced size
This th e l ar ges t of th e n o n rigi d a irshi p s
i s th e pro d u c t of th e Admir a lty Ai rs i p D ep ar t men t fr o m th ei r st a ti on
h
a t K in gs n o rth and h as seen m uc h servi ce as a sea sc out
—Th e ear ly kit e b alloo n was p r ob a bly a Germ
an
K i te B alloons ( Fig
ty pe with a s t r in g of parac hut es a ttac h ed to th e t a il in o rder to keep
th e balloo n poin tin g in t o th e wi nd Th e lift on a k it e b a lloon i s p art ly
d u e to buoy ancy an d partly d u e to dynam
i c lift th e la tt er be i n g l argely
ay
pred o minan t in winds of 40 o r 50 m p h Th e b all oo n i s cap ti ve and m
e ith er be sen t a loft in a na tura l wind o r b e towed fro m a shi p Two ty pes
of modern k it e b alloon are shown i n Fi g 10 (a) and ( b) showi n g th e l a t est
and m os t s u ccessful develo pmen t T o th e ta il of th e b a lloon are fi xed
t hr ee fins wh i c h are kep t infla t ed i n a wind by th e pr ess ure of a ir in a
sw ap a tt ac h ed to th e lower fi n
With thi s arran gemen t th e b alloo n
sw in gs s lowly b ack w ar ds a n d fo rw ards a bout a verti ca l ax is an d t rave ls
si de way s as an acc o mpan yi n g m o vemen t
Th e kit e wi re i s shown i n Fi g l ob as c o in g to a m oto r bo a t Th e
s ec o nd ro pe whi c h d i p s in to th e sea i s an a uto m
a ti c dev i ce fo r m
a in t a ini n g
th e h eight of th e b a ll oo n The g en er a l st eadin es s of th e b a lloon depen ds
o n th e p oin t of a tt ach men t of th e k it e wi re and th e i mp o r t a n t di fference
illus t ra t ed by th e ty p es Fig 10 ( a) and ( c) is th a t th e l att er b eco me s
lo ngitu d i nally uns t a bl e a t h igh wi nd speed s and t en ds to b reak a way
th e form
er d oes n ot b ecome un sta bl e T h e genera l di sp osition
of th e rigging is shown most c l early m Fig 10a wh ere a riggi n g b and
i s shown fo r th e a tt ac hmen t of th e car and kit e lin e
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CHAPT ER 11
THE P RI NCI PLES OF FL I GH T
A
p
n
o
s
u
a
n
x
i
T
n
s
()
I n develo p i n g th e ma tt er under th e a bo ve h eadin g an en dea v ou r will b e
m ade to av oi d th e finer de tai ls both of ca l c ula tio n and of experi men t I n
th e l a ter st ages of any en gi neering deve lo pmen t th e am ou n t of ti me dev ot ed
to th e de t ails i n order t o pro du ce th e b es t results is ap t to d ull th e sens e
t al and co mm on to a ll
of those imp ortan t f ac t o rs wh i c h are fu nda
d isc u ss ions of the su bj ec t
It us u ally fa lls t o a f ew p ioneers to est a blis h
the ma in princi p l es and av i a tio n follows th e r ul e T h e re la tions b e tw een
lift resis t ance and ho rsep ower b ecame th e subj ec t of genera l di s c ussio n
am ongs t en thu si asts i n the pe rio d 1896 1900ma i nly o wing to th e res ear c h es
of Lan gl ey Maxi m made an ae ro p lan e embo dyin g hi s vi e ws and w e can
n ow see tha t o n the subj ec t s of w eight and horsep ow er these
es t a blish ed th e fun da en t al t ru t hs Me tho ds of
da ta and of maki n g ca l c ula tions have i
o ved and h ave b een exten ded
to co ver poin ts n ot arisin g in th e ear ly days of flight and one ext ensio n
i s the c onsi dera tio n of fl ight a t a lti tu des of man y thou sands of f ee t
Th e mai n fr ame wo rk of the presen t c hap t er i s th e rela tin g of expo
men t al dat a to th e co n ditions of fli ght and th e experi men t a l da t a will be
t aken for gran t ed L a ter c hap t ers i n th e boo k t ake u p th e exam ina tio n
of th e experi men t a l dat a and the fin er de ta ils of the ana lysis and pred i c tio n
of aer op lane per fo rman ce
Wings —Th e m ost pro inen t i mpo r tan t part s of an aer o p lane are the
win gs and the ir fu nc tion i s th e su ppo r tin g of the aer o p l ane a gains t grav i t a
t i ona l a tt rac tio n
T he for ce o n th e wi n gs arises fro m m otio n through th e
a i r and is acco p ani ed by a d own ward motio n of th e air o ver whi c h th e
wi n gs have passed Th e princ i p l e of d yna i c s u pp o rt in a fl ui d has b een
called th e sacrifi ci a l princ i p le ( by L o rd Rayl eigh I b eli eve) and st at ed
b road ly exp r ess es the fac t tha t if you d o n ot wish to fa ll you rse lf you m ust
make so methi n g els e f all in thi s case ai r
If AB Fig 11; b e t aken to represen t a wing ovi u g in the di rectio n of th e
arr ow it will mee t a ir a t r est a t C and will l eave it a t EE end u ed with a
d own ward m otion Now fro m N ewt o n s la ws of m otio n it is kn own tha t
the ra t e a t whi c h d own ward m o men tu m is gi ven to th e fl ui d i s e qu al to
the su pp or tin g fo rce o n the win gs and if w e kne w the exac t motion of
the a i r round th e win g th e u p ward for ce coul d b e cal c ula t ed T h e p r obl em
i s however too d iffic ult for th e p r esen t s t a t e of ma th ema ti ca l kn owl e dge
and ou r info rma tio n i s a lmo st en tire ly b ased o n th e result s of tests o n
m o de ls of wi n gs in an ar tifici al a i r cur ren t
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18
THE
PRINC I P L ES OF F L I GH T
19
T h e d irec t measuremen t of the s us t a i n in g fo rce i n thi s w ay d o es n ot
inv ol ve any ne cess ity for kn owledge of the de t ai ls of th e flow It i s us ua l
to d i v i d e the res u lt an t fo rce R i n to two c o mponen t s L th e lift and D
the dra g but th e essen ti a l e as u rement s i n the ai r c urr en t are the ma gni
tu de o f R and i t s dir ec ti o n 7 th e la tt er b ein g reckoned fr o m th e n o rma l
t o t h e d irec tio n of motio n
The resolutio n i n to lift and dra g i s n ot the
only use fu l fo rm and it will be fou nd la t er tha t i n so me ca l c ula tio ns it i s
co nven i en t to u s e a li ne fi xed re la ti ve to the wi n g as a b asis fo r resol u tio n
ra t h er t h an th e directio n of motion
No a t ter by wha t means th e res ults are obt ained it i s found tha t th e
su ppo r tin g fo rce o r lift of an aer o p l ane wi n g ca n b e repres en t ed by c u rves
su c h as t hos e of Fi g 12
T h e lift i n g fo rce depen ds o n the an gl e a ( Fig 11)
t h e a erofoil ma kes
the re l a ti ve wi nd and it is in t eres tin g to
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tha t t he
b e p o siti ve whe n a is nega ti ve i e when the
relati ve win d i s apparen tly blowin g o n the u pper sur face T he c ho rd i s
t he straight l i ne tou c hin g the wi n g o n th e u nder s u rf ace i s i nc lin ed d own
wards a t 8 o r m o re be fo re a wi n g of us u a l fo rm ceases to lift
T he li ft on t he win g depends n ot on ly o n th e an gl e of i nc i dence and
of course t h e area but also on th e ve lo c ity rel a ti ve to th e ai r and fo r
full scale ae ro p l anes th e li ft i s pro p o rtio na l to th e s qu are of th e Speed a t
the sa e angl e of i nc i dence Of cou rse i n any gi ven flyin g mac hine t he
weight of th e achi ne i s fi xed an d there fo re th e lift i s fi xed and it follows
fro th e a bo ve sta t e en t tha t only o ne speed of flight can co rres po nd
t h a gi ven angl e of i nc i dence and th a t th e speed and an gl e of inc i dence
ust c han ge toge ther i n su c h a wa y t h a t th e lift i s c ons t an t Thi s r e l a tio n
can easily b e s een by re ference to Fig 12
T h e c urve AB CD E i s obt ai ned
by exper i en t as follows A win g ( i n prac ti ce a m o de l of it i s used and
find
.
,
.
.
.
,
,
°
.
,
m
m
m
m
m
m
m
,
-
.
,
,
,
.
.
.
.
APP L IED AER ODYN AMI CS
20
m ulti p lyin g f a c to rs app li ed ) i s mo v ed thr ough t h e a i r a t a s p ee d of 4 0 mp h
I n o ne expe ri men t t he an gl e of in c i den ce i s made zero and th e meas u red
This gives t he p oin t P of Fi g 12 When th e an gl e of
lift i s 840 lbs
i n ci den ce i s 5 th e lift i s 900 lbs an d so on In th e co u rse of s uc h an
en t th er e i s r eac h ed an an gl e of in ci den ce a t whi c h th e lift i s a
e x perim
maxim uman d this i s s hown a t D in Fig 12 fo r an an gl e of inci d ence of
For a ngl es of inc i dence grea t e r t han t h is it i s n ot p o ss ibl e t o
17 o r
ca rry so m uc h lo ad a t 40 m p h Without a ny f u r ther ex peri men t s it i s
n ow po ssibl e to dr a w th e r em
ainde r:of th e c ur v es of Fi g 12 At B th e lif t
”
fo r 40 m p h has b ee n fo u nd to b e 610 lbs At B 1 it will b e 610X GS) lbs
.
.
.
,
.
.
.
°
.
.
,
,
.
,
°
.
.
.
.
.
.
.
.
.
.
,
243 0 0
‘ Qfl fi fl
B Sa
-
s
A
‘ A
o
m
cu
na r lo n
Fro l 2
.
.
o r
C HO R D
—Wing lift
( D EG R E ES )
and s peed .
lbs an d so on th e li ft fo r a gi ven an gl e b e in g pro p o r ti o na l
t
to the s qua re of the s peed
Now s uppose tha t the win gs fo r whi ch Fig 12 was prepared a rc to b e
used o n an ae ro p lane w eighin g 2000 l
At 8 5 p h t h e win gs ca nn ot b e
ma de to ca rry ore t han 1580 lbs and co ns e qu e ntly the aero p l an e wi ll
n eed to ge t u p a speed of mo re tha n 8 5 p h be fo re it can l eave the ground
At 40
as we see a t D th e w eight can j us t b e li ft ed and this co n
s t i t u t es th e s lowes t p os s ibl e fl yin g s p eed of t h a t a e r o p l a n e
The a n gl e of
inci den ce i s then 17 t o 18 d egree s If th e s p ee d i s in cr e ased t o 50 p h
t h e r e qui r ed lift i s obta in ed a t an an gl e of inc i den ce ra ther l es s th an
a n d so o n un til if th e en in e i s po w er ful en ough to d r i ve t h e a erO la n e a t
g
p
100 m p h t he an gl e of i nc i d enc e h as a s all n egati v e va lu e
a
610X (fig
,
.
m
.
m
.
.
,
.
m
.
.
.
.
.
.
,
,
m
.
.
,
.
.
.
.
m
.
.
.
APP L IED AERODYN AM ICS
22
It i s n ow p o ssibl e to ma k e T a bl e ] showi n g th e resist ance of th e aero
p lane at various spee ds and to est i Me th e ne t ho rsepow er re qui red t o
p ro pe l a n aero p l ane weighin g 2000 lbs Th e lo sses in the o rgans of p r o
p uls io n will n ot b e c o nsi dered a t this p oin t b u t will b e dea lt with al mo s t
di a t ely wh en de t ermin i ng th e ho rsepow er av ail a bl e
A rough i dea of th e b rake ho rs ep ower of th e en gin e re qui red fo r
m
,
.
,
.
ANG L E O F
Fro 13
.
I N C I D EN C E
—Wing d
D EG R EES
rag and sp ee
.
d
.
ho riz o n t al fli gh t can b e obt a ined by assu m in g a pro pell er
of
60 per cen t i n a ll cas es
It will th en be seen tha t the aer o p l ane woul d
j u s t b e a bl e to fly with an en gine of 4 5 ho rsepower a t a s peed o f
appro xi ma t e ly 50 m p h At 70 m p h th e b rake ho rsepow er of th e
en gine woul d need to b e near ly 80 whils t to fly a t 100 m p h woul d
need n o l ess than 225ho rsep ow er B y variou s m o difi ca tio ns of win g ar ea
th e ho rs ep ow er for a gi ven spee d can b e vari ed co nsi d er a bly but th e
examp l e gi v en illus t r a t es fa i rly accu ra t ely th e li mits of s peed of an
.
.
.
.
.
.
,
.
.
.
.
.
.
.
THE
PRINCIPL ES OF F L I GHT
28
aero p l ane of th e weight assu med ; c g an engi ne develo pi n g 100 ho rse
p ow er ma y b e expec t ed to gi v e a fli gh t s peed r an g e of fro m 40 m p h
to 80 m p h to an aer o p lane weighi ng 2000 lbs
.
.
-
.
.
.
.
.
.
.
Tota l
Resi sta nce of reat of
aerop lane ( lbs ),
resi st ance
( lbs ) .
mm
Th e Prop ull lve [ co
Up to th e presen t th e ca l c ul a tio ns h ave
re ferred to t he b e hav iou r of th e aer o p l ane, without de tail ed re ference to
th e mea ns by wh i c h motion through th e air i s pro d u ced
pro posed to show how th e n ecess ary horsepow er i s esti mat ed i n o rder tha t
the aero p lane ma y fly T hi s es t i ni at e in volv es th e c onsi der a tio n of th e
-
.
’
.
ai rscre w
An a irs cre w ac t s o n the ai r i n a mann er so mewha t si milar to tha t of
a wi n g an d th rows a i r b ackwar ds in a con tin uous st ream in o rder to
pro d u ce a for w ard thrus t The thrust i s obt ai ned fo r th e l east ex
r
i
t
s
n
a
r
of
p
ow
er
o
ly
wh
en
th
e
rev
olutio
n
of
th
e
e
gi
ne
e
a
ve
r
y
n
i
n
u
e
e
n
d
p
speci a l rel a tio n to th e fo rw ard s peed
In cr ease of th e speed of rev olutio n without a lt era tion of th e fo rw ar d
s peed of th e aero p l an e l eads to i ncreased th rus t but th e l a w of i ncreas e i s
co mp l ex In cr ea sin g th e speed of th e aero p l ane usually h as th e effect of
d ecreasi n g th e thru st a ga i n i n a manner whi c h it i s n ot easy to exp ress
simp ly
Cal cu l a tio ns can b e made to show wha t th e ai rscre w will d o
un der an y c i rc u ms t ances but th e di s c us sio n wi ll b e l e ft to a spec i a l c h ap t er
i
h
r
l
a
w
can
how
ever
b
e
ded
u
ced
f
r
o
m
th
e
b
e
a
iou
r
of
v
One si m
e
a
w
scr e ws and is of m u c h th e same na ture as tha t a l read y poi n ted out fo r th e
s u ppo rt i n g s u rf a ces
It was s ta ted tha t if th e an gl e of inc i dence i s kep t
co nstan t th e lift and dr a g of a win g increase i n pro p o rtio n to th e squ are
of th e s peed Now i n th e a i rscre w it will he found tha t th e an gl e of
i nc i dence of eac h bl ade sec tio n i s kep t c o nst an t if th e rev olutions are
incr eased i n th e same pre po rtio n as the fo rwar d speed and tha t u nder
s uc h c o nd iti o ns th e t h rus t and t o r qu e both vary as th e s qu ar e of th e
speed
If fr o m a fo rw ard speed of 40 m p h and a rot a tio na l speed of
600 r p m th e fo rw ar d s peed b e incr ease d to 80 m p h an d th e
r ot a tiona l s peed to 1200
th e t h r us t wi ll b e in creased four ti mes
Gi ven a t a bl e of figures su c h as T a bl e 2 whi c h shows the thrus t of
an ai rscre w a t sev eral Speeds of rot atio n when tr avellin g a t 40 m p h
throu gh th e a i r results can b e ded u ced fo r the th ru st a t other values of
th e fo rwar d speed i n the m
anner d escrib ed b elow
.
,
.
.
,
.
,
.
.
,
,
,
,
.
,
,
,
.
,
.
.
.
.
.
.
.
.
.
.
.
,
,
.
,
.
m
ean t t h e power necessary t o dri ve
hors epower i s h ere
e ry
t
i
i
i
o
n
s
re
v
f
l
s
d
h
d
f
e
c
e
fi
c
i
e
n
t
ea
n
s
f
ro
o
n
e
x
i
s
e
co
n
t
er
l
u
t
e
o
a
T
r
p
p
p
an aer o p lane whe n gl i din g;
B
ne t
g
m
.
.
.
APP L IED AERODYN AMIC S
24
Th e figu r es i n T a bl e 2 woul d be obtai ned eith er by cal c ul a tio n o r by
an experi men t T es t s o n ai rscr e ws are fre qu en tly mad e a t t h e end of a
lo n g armwhi c h can be rot a t ed so gi v i n g the ai rscre w i t s fo rw ar d m otio n
Ac tua l a i rsc re ws may b e t es t ed on a l ar ge w h ir lin g ar mo r a m o de l a i r
s cre w ma y h e us ed i n a wi n d c ha n ne l and m ulti p lyi n g f ac to rs emp loy ed
to a ll ow fo r th e c han ge of sca l e
.
.
,
,
.
For ward sp eed 40
Revs per
.
m
i nu te
mp h
.
.
.
Th rust
.
It wi ll b e
noti ced fro m T a bl e 2 tha t the ai rscre w gi ves n o th rus t u n ti l
rot a tin g fast er th an 500 r p m At lower sp eeds than this th e ai rscr ew
woul d o pp os e a resistance to th e fo rward motion an d woul d t en d to b e
tu rn in g as a wind ill Wh en th e subj ec t i s en tered in to i n m ore d et a il
it will be foun d tha t the n u m b er of rev olutio ns necessary be fo re a thrus t
i ned by t he
p it c h of th e a irscre w Th e ter
i s pro d u ced i s de t erm
p it c h i s obta i ned fro m an ana logy be tween an airscre w and a screw
th e advan ce of the la t t er alo n g i t s a xis fo r o ne c o mp le t e rev olutio n bein g
kn own as th e p i t c h
W hils t th ere are ob vious mech an i c al differences
b e tw een a soli d scre w tu rn i n g i n i ts n ut and an ai rscre w m ov i n g i n a
m obil e flui d th e expre ss io n h as man y advan ta ges i n th e la tt er cas e and
wi ll b e ref er red to fr e qu en tly Fo r th e presen t it i s n ot necessary to kn ow
how p it c h i s defin ed
Th e n u m b ers gi ven i n T a bl e 2 c o rres p o nd with th e c u rve marke d
AB C i n Fig 14
T o ded u ce tho se fo r an y oth er sp eed say 60
th e
firs t colu mn i s m ulti p li ed by 38 and th e seco nd by
gi v in g th e
followi n g t a bl e :
.
.
.
m
,
.
.
m
,
”
.
,
.
.
.
.
,
,
Forward
Revs p er
.
i
m
t
nu e .
sp eed
00
mh
.
p
.
.
Th rust ( M )
.
m
It will b e n oti ce d tha t th e a i rscr ew mus t n ow b e r ot a tin g mu c h o re
rap i dly th an b efore i n o rder to pro d u ce a thru s t T h e remain ing c u rves
of Fig 14 were pro d u ced i n a si m ilar way and relat e t o speeds of t h e
.
.
,
THE PRINCIP LES OF FL I GH T
25
Th e th ru st neces sary to su ppo rt t h e aero p lane in th e a ir a t
0 60 70 a n d 100 m p h h as b een obt a i ned i n T a bl e 1 an d
spee ds of 40 5
us in g Fig 14 it i s now po ssibl e to obta i n the pro pe ller rev olutions whi c h
Th e poin t s are mar ked
ar e n e cess ary to pro d u ce thi s re qui red thr u st
C C, 03 C3 and 04 T o pro d u ce a th ru s t of 610 lb s a t 40 p h th e
pro pe ll e r mus t b e t u rnin g a t a bout 18 80
as s how n a t th e p oi n t 0
0 m p h th e en gi ne ma y b e shut d own very appr o
s p ee d r is es t o 5
th e rev olutio n s b ei n g o nly 980 Fo r high er velo c iti es of fli ght the
in g 2000 lbs
.
,
,
,
.
.
.
,
.
m
.
,
,
.
,
.
.
.
.
.
.
.
.
.
A l R S C R EW
F10 lea
.
R EV O L UT I O N S
—Thrust and
s
peed
0
.
rev olutio ns i ncreas e st eadily un til a t 100m p h th e ra t e of rota
tion 18 o v e r 1600 r p m
T h e en gin e ma y however n ot b e p owerful en o u gh
t o dri ve t h e p r o pe ll er a t th e s e ra t es and it i s n ow nec e ss ary to esti ma t e
in a mann er s i mil ar to th a t fo r thr us t how m u c h ho rsep ow er 18 re qui r ed
The i n iti a l dat a gi ven i n T a bl e 4 ar e a ga i n as su med to hav e b een
.
,
.
.
.
,
.
.
,
,
,
.
,
TAB LE 4
.
—Aras onxw
H oas nr ow xn
Forward sp eed 40
mp h
.
.
.
mS m
ar
ea
).
APP L I ED AERODYN AMIC S
26
m
obt ained experi men t ally and the fi gures fr o this t a ble ar e p lotted i n
Fig 15i n th e c u rve AB C To obt ai n th e c u rve fo r 60 m p h t h efi rs t
c olu n of T a ble 4 i s m ulti p lied by $23 an d th e sec ond by ( fi
ob t ai
ng
g
the n um b ers gi ven in Ta bl e 5
,
.
m
'
.
.
.
.
m
.
r
A
I RS
C R EW
p
.
m
R EV O L U T I O N S
.
Th e c u rves so obt ained fo r various flight spee ds indi ca te
p ower b e fo r e the a irscre w has st ep p ed T h e Speeds ar e lower th an thos e
fo r whic h the thrus t h as b ec o me zero and indi ca t e th e poin ts a t whi c h
the ai rsc r e w b ec o mes a wind ill In an aer o p l ane how ever th e resi st anc e
to tu rni n g of th e en gi ne woul d grea tly r edu ce the speed a t whi c h th e win d
ill b eco mes eflect i ve be low tha t i nd i ca t ed fo r n o ho rsep ow er and s to pp ag e
of the pe t rol su pply to t h e en gi ne wou l d often r esult i n th e st o ppage of t h e
ai rscre w
.
m
,
.
m
,
,
’
-
,
.
PRI NCIP LES
THE
FL I GHT
OF
27
F ro m Figs 14 and 15 it 18 n ow easy to fin d th e b rake hors epower of
th e en gi n e w hi ch woul d b e necessary to dri ve th e aero pl ane through the
a t spee ds fro m 40 to 100 p h Fro m Fig 14 it i s found tha t th e
aer o p l ane wh en t rave lli n g a t 5
0 m p h t hr ou gh th e a i r nee ds an a i rs cr e w
s peed of 98 0 r p m
To dri ve th e ai rscr e w a t thi s speed 18 seen fr o m Fi g
poin t C, to need 89 ho rs epower Fo r other Speeds the ho rsepower i s
indi ca t ed by th e poin ts C 02 C3 and C4 an d th e coll ec t ed res ul ts ar e
gi ven i n T a bl e 6
.
m
.
.
.
.
.
.
.
.
.
.
,
.
,
,
,
.
m
-
l
m
m)
8M
of a
e
Ho
ewar d
neon can t or
p anc
.
On Fig 15a line OP h as b een dra wn whi c h repres en t s t he wo rk whi c h
parti cu lar en gi ne cou l d d o a t th e v arious s peeds of rot a tion this again
Th e en gine i s su ppos ed to b e giv ing 120 h p
is an experi men t a l cu rve
a t 1200 r p m It wi ll b e seen fro m Fi g 15 tha t th e engi ne i s n ot p ow er fu l
en ough to dri ve th e aer o p l ane a t e ith er th e low es t o r th e hi gh est Speed s fo r
the cal cula tio ns h ave b een made Fo r man y p urp os es th e in fo rma tion
gi ven i n Fig 15is mor e c o nveni en tly expressed in th e for m shown in Fi g 16
wh ere th e a bsc issa 18 th e flight Speed of the aerOplan e Th e c ur ve AB CD E
of th e l a tt er figu re 18 p lott ed fr o m th e poi n ts C 01 Oz 03 and C4 of Fi g 15
while the line F GH c o rresp onds with the p oin ts B E, B J B , and B 4
T he fi rs t c urv e shows the ho rsepower re quired fo r flight and th e sec ond
Fr o m th e di a gr am 1n thi s fo rm it i s easily seen
t h e ho rs ep ower availa bl e
tha t th e po in t F represen ts th e slowest speed a t whi c h th e aero p lane can
fly i n thi s case 403 m p h and tha t H shows the po ssib i lity of reachi n g a
s peed of near ly 98 m p h
Fig 16 s hows m ore than this fo r it gi ves th e res er ve horsepower a t any
This r eserve ho rsepow er is r oughly pro p o rtio na l to th e
s peed of flight
Sp eed a t w h i c h th e a e ro p lane can c li m b and th e c u rve s hows tha t th e b es t
cli m bi n g Speed i s mu c h nea rer to th e low er li mit of speed than to the
.
,
.
.
.
.
,
.
.
.
,
.
.
.
,
.
,
.
,
,
,
,
.
,
,
.
.
,
.
.
.
.
,
.
,
.
.
,
m
—
—
i
k
s
o
n
1
2
1
6
n
a
l
n
a
r
F
c
C al cula tio ns r ela tin g to t h e flight sp eed
Ge e
c
g
of an aero p lane are i llustr at ed fa irly exac tly by th e c u rves i n Fig 12 16
As th e subj ec t i s en t er ed i n to i n de t a il many s ec on dary co ns i dera tio ns will
b e s een to c o me in Th e diffic ulti es will b e foun d to co nsis t very l argely
i n t h e de t er inatio n of th e st an dard c urves mar ked AB CD E i n th e fi gur es
and the analysi s of res ults to obt a i n these dat a c onstitut es one of th e m o re
la bo rious parts of the pro cess T he c o mp li ca tio n i s very l ar gely one of
de tai l and shoul d n ot b e a llowed to ob sc ure th e c o mmo n b asi s of flight
16
co ndi tio ns fo r a ll aero p lanes as typ i fie d by th e c ur ves of Figs 12—
.
-
.
m
.
.
,
.
,
.
.
APP L IED AERODYN AM I CS
28
m
bi
—In the more general th eo ry of th e aero p l ane it i s
tio ns may b e m o d i fied t o
o f i n t ere s t t o Show how th e prev iou s ca lc ul a
The ra t e a t whi c h
in c lu de fli ght s oth er t ha n tho se in a ho r i zo n t a l p lane
a erOpla n e ca n c li m b h as a l ready b een re ferred to i nc i d en t ally i n con
n aoti e n with Fig 16
It i s cl ear fro m th e outse t tha t th e a ir fo rces ac ti ng o n th e aero p l an e
depend o n i t s s peed and an gl e of inci den ce and ar e n ot depend en t
Cli
ng
Flig h t
.
.
.
,
ac tin g
fo rces will vary with th e a ttitu de of the aer o p l ane If th e aer o p lan e
hin g th e a i rscre w t h ru st will ne ed to b e grea t er th an fo r hori z on t a l
flight whils t if descendin g th e th rust is r ed u ced and may become zero
o r ne ga ti ve Ther e i s a mi ni mu m an gl e of descen t for any a er o p l an e wh en
.
,
.
co
co
S P C ED O F
(w
F
m16 —H
.
.
ac
70
m)
RO G
pee d for leve l fl igh t
a s o u t n ou n T H
orsep ower a nd s
U
a
H
.
th e a i rscrew is gi v i n g n o t hr ust and t h is angl e i s oft en r e ferred to a s th e
an gl e of gli de for t h e aer o p lane
Mor e co rr ec tly it s houl d b e re ferr ed
to as th e l east an gl e of gli de
Th e me tho d of ca l c ula tio n of gli di n g and c li m bin g flight i s illus t rat ed
in Fi g 17 whi c h i s a d i a gram of th e for ces ac tin g o n an aero p l ane i n free
flight but with i t s flight path i nc li ned to th e hori zo n t al
I n ho ri zon t a l flyin g it wi ll b e assu med th a t th e di rec tio n of th e thrus t i s
ho ri zo nt a l i n whi c h cas e it d irec tly b alances th e resist ance of the r e ai n der
of th e aerOplan e to m otion through the ai r In th e a bo ve d i a gram thi s
s t a t emen t means th a t T = D
Si mila r ly th e w eigh t of th e a erOplane i s
exa c tly cou n t er b a lanced by th e lift on th e wi n gs i e L W Th e an gl e of
i nc i dence of th e win gs may b e vari ed by adjus t men t of t h e el eva to r in
whi c h case the thr us t woul d n o t st ri c tly li e a lon g t h e wi n d If necessary
a slight c o mp li ca tion of fo rmul a cou l d b e i n tro du ced to mee t thi s case b u t
th e e flect of thi s vari a tio n i s sma ll and i n acco rdance with the i dea o n
,
”
.
”
.
.
,
.
m
,
'
.
.
,
.
.
.
,
.
,
'
,
,
APP L IED AEROD YN AMICS
30
can n ow b e us ed to show how d i a grams 12 and 18 m
ay b e
a lt ered to allow fo r i nc li ned flight In th e fi rs t p l ace the o rdina t es of
Fig 18 whi c h aft er additio n of th e drag of th e bo dy s how the valu e
of D fo r man y angl es of i nci dence n eed to b e decreased by m u lti p lyi n g
by cos 0 to gi ve D cos 0 The effec t of t h is m ulti p li catio n is very s ma ll
as a r ul e
At 10 th e fac to r is 09 85 and a t
0940 Fo r a very
s t eep Sp i ra l gli de a t say
the di fferen ce b e tween cos 0 and unity bec o m es
i mpo rt an t cos 0 b e in g th en 0 707
To the va lu e of D cos 0i s to b e added a t er m W sin 9 i n o rder to obt a in
the t h rus t of the a irscre w wh en c li m b i ng a t an angl e 0 We may th en
ake a t a bl e as be low u sin g figu r es fr o m T a bl e 1 to obt ai n th e sec ond
m
colu mn
Eq u a tio n (2)
.
.
,
,
,
,
.
°
.
,
.
.
,
.
,
m
.
B
TA LE 7
-
.
T1
var
'
ww w
mm
a
a
.
m
lm
bi
Dr g i n b lmt
nig h t x
n al
cr
a
cos
Alr
c i
9
th ru s t wh e n
”
n g at 6 ( l bs )
v
.
an gl e of c li mb was chosen ar bit rarily a t
and to co m
p l et e th e
investiga tio n of th e p ossibiliti es of c lim b Ta bl e 7 woul d b e r epea t ed fo r
other an gl es Usin g Figs 14 and 15 fo r th e a i rsc r e w as fo r ho ri zo n t a l
fli ght we may n ow ca l c ula t e the horsep ower re quired fo r fli ght wh en
c li m bin g and so obta i n th e fi gu res of T a bl e 8
The
.
.
,
.
,
TAB LE 8
.
—H
oas xrowna
wa n
mm
t ca
a
.
is
th e lowes t and high es t speeds of th e t a bl e th e hors epow er r e quired
far grea t er than tha t av aila bl e and th e figur es ar e n ot withi n th e ran ge
of
Fi
At
’
'
,
16
.
may n ow p ro ceed to p lot th e ho rsep ower of T a bl e 8 a gains t s pee d
to obt ai n a di a gr a mc o rresp o ndin g with Fi g 16 Th e ne w c urve mar ked
A1B 101D 1 i n Fig 18 c o mpar ed with AB CD E as repro d uced fr o m Fi g 16
0 h p a t all Speeds d u e to the c li m b a t
s hows an i nc rea se of near ly 5
Th e highes t Speed of flight i s shown by th e in t ers ec tio n of Al B l c l with
F GH a t H 5 FGH is the ho rsep ower avai la bl e and i s th e same as th e
T h e highest Sp eed is 7
and
s i m il a r ly ma r ked c u r ve of Fig 16
.
.
.
.
.
.
.
,
.
.
PRI N CIP LES OF F L I GHT
THE
81
n ce th e angl e 0 i s co ns t an t a lon g Al B l c l the ra te of cli m b will be
grea tes t a t this p oin t fo r the co ndi t ions assumed Ra t e of c lim b V ” is
com
m only esti ma ted i n fee t per min ut e and w e then have
si
.
,
,
,
Max V, fo r d
5
8 8 X Va ” x sin 0
88 X
x 008 75
°
.
,
T he c alc ula tions sho wn i n T a bl es 7 and 8 have been repea te d for other
an gl es of c li m b and on e an gl e of des cen t to obt ai n co rr esp on d i ng c u rves
O D KI O
F
ir
b
i g 18
lo w
.
.
m18 —H
.
.
OI
P L I O PI Y
4,
TABLE 9
li
c
L ES
P ER H O U R )
d for cli m
bing fl igh t
H0, etc
.
Ang le
MI
ors ep ower and sp ee
The in tersec tio ns H
ct
(
.
—Ru
z or
'
then pro vi de da ta fo r T a ble
mm mS
0
.
u
ea s
»
.
m
m
t
t
i
m
b
Un i
.
Fli gh t
9
n
ra s a! cli
or l ven
I
m
b
59 15
58 2
5113
n ot
p oss i bl e
.
T a bl e 9 s hows th a t the r a t e of c li m b vari es rap i d ly with the fli ght s peed
t h e n ei gh bour hOOd 100 m p h to 80
but th a t fr o m 65 m p h to
56
p h th e va lu e of ra t e of c li m b var i es o nly fr o m 725 to 78 0 T h is
i llu s t ra tes th e w ell kn own fac t tha t th e b es t ra t e of c lim b of an aero p lane
i s n ot m u c h a ffec t ed by s ma ll i n acc urac i es i n the fli ght s pe ed
Th e t a bl e shows an oth er i n t e resti n g de t ail ; t h e maxi m u m angle of
m
.
.
.
.
.
,
.
.
.
-
.
.
.
APPLIED AERODYN AM IC S
82
cli m b i s
but th e grea tes t ra te of c li m b o ccu rs a t a sm
a ll er an gl e
Fo r reasons conn ec t ed w ith th e c on t rol of th e aero p l ane an an gle of 8 o r
therea bo uts woul d prob a bly b e c hosen by a p ilot ins t ead of th e
s hown t o b e th e be s t
Di
d i vi n g i s mean t descen t with the e n gi ne o n as
By
di sti n guis hed f ro m a gli de mw hi c h th e engin e is c ut ofl If th e engi ne b e
kep t fully on it i s found th a t th e speed of rot a tio n of the a i rscre w r i ses
hi gher and hi gh er as th e angl e of des cen t i ncreas es There i s how eve r
an u pper li m
it to th e Speed a t whi c h an aer o p l an e en gine ma y b e ru n w ith
sa fe ty and m ou r illus t r a tio n a n appr o pri a t e li m it woul d b e 1600 r p m
T he s peed of ro t a tion c o rr esp o ndin g with H4 was 1550 r p m and it wi ll
b e seen tha t the ne w res t ri c tio n wi ll co me i n to o pera t i o n fo r s t ee p er
d escen t Fig 14 if ext ended woul d n ow en a ble u s to de t erm
ine the th r us t
of th e a i rsc re w a t any 13 d
without re ference to th e ho rsep ower but it
will b e ev i den t tha t the li its of us e fulness of eac h of th e prev ious fi gu res
ha v e b een reac hed a nd an ex tensio n of experi men t a l da t a i s necessary to
c o v er the higher speeds
Th e f ac t th a t u nde r cer t a in c ir c u ms t an ces fo rces vary as th e s qu are
of fo rwa r d Speed of t h e a ero p lane suggests a m o re co mpre hensi ve fo rmof
presen t a tio n than tha t of Figs 12 18 14 and 15 and th e n ew c u rv es of
Figs 19 and 20 show an ext ens io n of t h e ol d i n fo rma tio n to co ver t h e
ne w poin t s o cc urr in g i n th e cons i dera tion of di v in g T h e va lu es of th e
ext ended p o r tion a re so sma ll tha t o n any apprec i a bl e sca l e it i s o nly
p oss ibl e to s how the range c o rresp o nd in g with sma l l an gl es of i nc i dence
and fo r sma ll va lu es of thr us t and ho rsep ower
.
°
m
.
-
,
'
.
,
,
.
.
.
.
.
.
,
.
.
.
.
,
,
'
m
,
,
.
.
,
,
,
.
.
.
TAB L E l o
.
Th e
c u rve c o n nec tin g
—A1asoa :w
thr us t
V
3
mp h
.
.
Tna usr wu s s
and speed
is
s
m
nrv
o.
ho wn i n Fig
.
21
.
.
I nst ead of e qu a tio n (2) will b e u sed the e qua tio n
V
a
ni s
h
V
.
m
z
.
T he u se of D 1 i ns tead of D cos 0 i s c o nveni en t n ow si n ce th e
l eve l flight a t high Spe eds i s n ot de t er ined i n any other cal cu l a tio n I n
—
1
1
s
s
s
10 i s c hos en and various
co mp ilin g T a bl e
o me angl e of pa th u c h a
s peed s of flight ar e a ssu med
Fro m these Speed s the t h ir d c olu n is
m
.
°
,
.
m
THE
PRINCIPL ES
FL I GHT
OF
uls t ed and gi ves on e of the qu an titi es of Fig
—
Fig
l
1
6
e
1
0
o
c
u
a
t
an
an
gl
e
of
b
rs
1
c
)
(
.
19
.
.
19
.
—Li ft
T h e valu e
.
-
0 197
and fr o m the same fi gur e
°
Fro
88
DENCE
drag of aeroplane a t very h igh s peed
ANGLE O F I NC I
a nd
s.
0 00 0 2
F
mah —Th
.
r ust and
c o rrespo ndi n g va lu e of
h orsepower of a i rs crew a t
is
r
ead
'
ofi
as
very
h i gh s peeds
0 05
06
-
.
.
Column
5
of
APP L IED AERODYN AM IC S
84
T a bl e
1
11 follows
an d
fr o m th e kn own w eight of
and the l as t c olu mn
2,
$
of
5
is
t he
a er Oplan e
the
su
colu m
ns in acc o rdance with e qu a tio n
T a bl e
11 are
° of p at h
“ 13 1
of
0
.
p lott ed i n Fig
21 and
.
mof
m
and c olu n s
th e prece din g
T
The va lu es of
fr o m
marked with the appr o pri a te va lu e
by
.
The i n t ersec tio n a t A of th e c u rve
0
addi ng 001
0
4 and 5.
.
—10 and th e cu rve
°
If2
flyi n g i n
“
“
fro m T a bl e 10 shows the s peed a t whi c h th e aer o p lane mus t b e
o rder tha t the a irscr e w shall b e gi v in g the thru st re qui r ed by e qua tio n
The results shown i n Fig 21 can b e coll ec t ed in a fo rm whi c h s hows
how th e resist ance of an aero p lane i s di v i ded b e tw een th e aer o p lane and
.
a irscr e w
an d
At A
.
the Spe ed
hence T=29Olbs
.
=2901bs +848 lbs
.
.
i s 110 m
.
p h and the valu e of 2
.
.
V2
D 1= 2901b s
Equ a tio n (8) th en shows tha t
688 lb s R epe titio n of th e pr o cess l eads
.
TABLE 12
—s
.
y
m
u rn
Du
o
wa n
”
m
i a tio n of t h
Exa
suffi c i en t
is 0 0240,
—Wsin 0
to T a bl e 12
.
.
m
t nrv
c.
m
3,3,n
wa s a
t a bl e shows th a t a mo der at e angl e of descen t i s
to p r o d u ce a c onsi der a bl e i nc rease of spe ed The ma xi mu m
n
e
.
THE
PR IN CIP LES OF FLI GHT
85
fl ight speed i s reac hed b e fore t he pa th b ec o mes verti ca l but t he va lu e i s
littl e grea t er than tha t fo r verti ca l d escen t The t er mi na l spee d of ou r
ty p i ca l aero p lane is 155m p h With the li m
it ation p laced on the a irscre w
t h a t i t s rev olutio ns s hould n ot exceed 1600 p m it will b e noticed fro m
c ol u mn (4) of Ta bl e 12 tha t the thr us t ceas es a t a bout 125 m p h a n d
tha t a t higher s peeds t he a i rscre w o ffers a resi s t ance whi ch 1s an a ppreci a bl e
fr ac tio n of th e tot al At the t er mina l veloci ty the tot a l resist ance i s
di vi ded b e tween the ai rscr e w wi n gs an d bo dy i n the pro p ortio ns 19 1 per
cen t 4 8 ? per cen t and 872 per cen t respec ti vely
If the c urve of ho rsep ower of Fi g 20 be examined a t t h e t ermina l
,
.
.
.
.
.
.
.
.
.
,
.
,
.
.
,
.
.
.
ve loc ity it will b e fo u nd th a t the valu e of
3
6) gi ves
7
to
3
a valu e
of 10 4 an d th e horse p ow
er 13 then nega ti ve Thi s means tha t the a i r
s cre w i s t endi ng to ru n as a wi ndmill and th e h o rsep ow er t en din g to dri ve
.
,
,
'
m21
F
.
.
Angl e of descen t
-
8 0°
d i n di vi ng
and sp ee
.
0
A speed m uch le ss than 155 m p h woul d prov i de
i t is a bout 15
su fi ci ent p ow er to res t ar t a sto ppe d en gi ne , si nce 80 h p woul d p r ob a bly
T his means of res tar tin g
s u fi ce to carry o v er th e firs t c o mpressio n st ro ke
a n engin e i n th e a i r i s fre qu en tly used i n experi men t a l work
Gli di ng — In o rdi nary flyin g l angu a g e
gli d in g i s d istin gui sh ed fro m
d i vi ng b y th e fac t tha t i n th e fo rmer the en gine i s swit c hed off If th e
rev olutions of th e a irscrew b e ob served th e angl e of gli de can b e cal c ula t ed
Th ere i s, how ever, o ne spec i a l ca se whi c h h as c onsi der a bl e
as b e fo re
.
.
.
.
.
.
.
.
.
.
.
i n t er es t and thi s occ urs wh en th e engine rev olution s ar e j u st su c h as to
gi ve n o th rus t fro m th e air scre w Fig 20 shows fo r ou r illus t ra tion
tha t the rev olutio ns per min ut e of the a i rscre w m us t then b e 12 5 ti mes
th e Speed of the aer Op lan e in mil es per hou r If th e rev olutions b e li mi t ed
to 1600p
Fig 20
as b e fo re t h e hi gh es t s peed permis sibl e i s 128 m p h
s hows tha t th e engi ne woul d then need to dev elo p a bout 8 5ho rsep ower
and woul d be t hrottled d own but not s wit c hed off
Th e spec i a l i n t erest of gli des with th e a i rsc r ew giv in g n o thru s t will
,
.
m
.
.
,
,
.
,
.
.
.
.
,
.
APP L I ED AERODYNAMI CS
86
be
n fr ome qu at i on ( 2) by p ut t i n g T ,
s ee
gi ves
D
W
-
:
t an
0 when
th e res t of the e qu a tio n
9
where D is the dra g in ho rizo n t al fli ght a t th e same angl e of i nc i dence
'
d uri n g t h e gli de and co nse qu en tly
is
,
A NG LE O F
.
.
the we ll kn own ra tio of lift to
-
I NC I D EN C E o r W I N G S ( DEG REES )
.
m22 —A
F
as
l
ffi ci ency and gl i ding ang le
er op ane e
.
4 5)
The nega tive sign i mp li es d own w ar d flight and w e see t ha t th e gli din g
lift
meas ur e of th e
of an aero pl ane
of 1ts
,
,
drag
aer o dynami c effi ci ency as d istinc t fr o m tha t of th e ai rscre w In prac ti ce
it is n ot possibl e to ens ure th e c o ndition of n o th rus t with suffi ci en t acc u rac y
.
APP L IED AERODYN AM I CS
88
t ra ins but o ccasio na lly an aero p l ane flyi n g a gai nst th e wi nd i s blown
b ackw a rds r ela ti ve to an ob ser v er o n the ground Flying with th e wind
the p ilot may t rave l a t speeds very mu c h great er than tho se in di ca t ed i n
ou r earli er ca l c u l a tio n s
Th e m otion of the aero p l ane may b e very
i rr e gul ar j us t as woul d be th e motion of th e t ra i n if the ra ils mo ved si de
ways and u p and dovmas w e ll as b ack wards and fo rw ar ds with t h e
di flerence tha t the c o nnec tio n b e tween th e a ir and aero p lane i s not so rigi d
as th a t b e tween a t r ai n and i t s r a il s
Th e motion of an aero p l ane i n a
gusty wind i s so mewha t c o m
p li ca t ed but metho ds of akin g th e necessary
ca lc ul a tio ns have already b een deve lo pe d and will b e re ferred to a t a m o re
advanced s t a ge
If th e rails i n th e t ra i n analogy h ad b een m ovi ng s tea di ly u p wards
with th e t rain st a tiona ry on th e r ails th e t rain might h ave b een des crib ed
as so ar i ng
Th e t ra i n woul d b e lift ed by th e source of energy liftin g th e
rai ls Si milarly if u p c urren t s o cc ur in th e a i r an aerop lane may c on t inu e
to fly wlnlst ge ttin g high er an d h igh er a bo ve th e gr oun d without u sin g
any p ower fr o mth e aerOplan e engi ne T his case i s ea sily subj ec t ed to
n um
erica l ca l c ul ation L or d R ayl eigh and P rof L angl ey h ave shown th a t
soar i n g may b e p o ssibl e without u
i
t
c
u
rre
n
t
s
if
th
e
wi
nd
i
s
gu
s
ty
o
r
if
p
h as di fl er en t Speed s a t di fl er en t h eight s
u en tly
Su c h c on di tio ns occ ur fr eq
in na ture an d birds may some ti m
es soar under su ch c on d itio ns Con ti n u ed
flight without flapp in g of th e win gs usually occ u rs o n accoun t of risin g
c u rren ts T h ese m
ay b e d u e to hot gr o u n d o r r oun d th e co a s ts m o re
fre qu en tly to th e deflec tion of sea b reez es by th e cliffs n ear th e sho re Gulls
may fr e qu en tly b e seen t ravellin g along a bo ve th e ed ges of clifl s th e pa th
followi ng som
ewha t c losely th e outlin e of th e c o ast Oth er ty pes of soarin g
are scarcely kn own m Engl an d
T o cal c ul at e th e u p ward vel oc ity of th e a i r necessa ry for soarin g i n t h e
case of th e aerop l ane a lr eady consi dered it i s o nly n ecessary to re fer b ack
to th e gli di ng a ngl es and sp ee ds of flight V a lu es obtai n ed fr o mFigs
12 and 22 are c oll ec t ed i n T a bl e 18 fo r a weight of 2000 lb s
,
.
.
,
,
'
m
.
,
,
.
,
.
-
.
,
'
,
.
.
.
-
,
'
.
,
.
.
,
.
'
,
.
.
,
.
.
.
TAB L E 13
m
lm
A11 to of lncldence .
ro
Flg 12
(
.
m
Sp eed of
p h ) , t ro
.
.
.
t
.
—Somm
e
.
I 011 § 18
mm
l fro
ang e,
22
Vert i ca l
t
l
l
ve oc i t y of
Sg?:
3t
3 “
cu t
11
l as t c olum
n of T a bl e 18 are re a d ily obt a in ed fr o m
those i n col u mns 2 an d 8 At 60 m p h a n d a gli di ng angl e of l i n 9 t h e
falli n g Speed rs Q mp h i e 6 7
h as i n colu mn 4 Th e l east veloc ity
of ris i ng wi n d i s r equi red at a sp eed j u s t b elow t h at of l eas t resi st anc e
an d m thi s cas e am oun t s to a bout 54 m p h o r n ear ly 8 f e et per sec o n d
Wi nd s h avi ng l arge u p ward com
p o n ent v elociti es a re k now n t o exi s t
Th e fi gur es i n t h e
.
.
.
.
’
.
.
.
,
.
.
.
.
,
.
.
.
.
.
THE
PRIN CIP L ES
OF
FL I GHT
89
wi nd s hav in g a ho riz on ta l c o mp o nen t of 20 m p h an up war d ve loc ity
of 6 or 7 m p h h as b een reco rded o n severa l occa sion s
In s t ro n ger
winds th e u p c u rren t s may b e grea t er but mall cas es th ey appear to b e
loca l One w ell a uth en ti ca ted t es t on th e c l im bin g Speed of an aerOplan e
sho w s th a t a ris in g c u rren t of a bout 7 mi l es an hour exi st ed o ver a
dis tance of m or e than a mil e T h e c li m bing speed of th e aer op l ane h ad
been ca l c ul a t ed by metho d s si milar to those des cri be d in th e earli er p ages
t he
of t h is boo k an d foun d to b e so m
ewh a t l ess th an 400 f ee t p er m
in
g ene ra l co rrec tness of t h is figu re was gu aran t eed by th e aver age perfo rm
ance o f th e aero p lan e On on e occasion how ever th e r ecor d in g b aro
g rap h i ndi ca t ed an i ncrea se of 1000 f ee t i n a m in ut e an d it woul d appear
th a t 600 fee t per i n ut e of this w as d u e to th e fac t th a t th e aer o p l ane
was ca rri ed bo di ly u p w ard s by th e a i r i n add itio n to i t s na tur al c li m bi n g
ra t e At 60 il es an hour th e column t r aversed per m
i n ut e i s a i le a s
a l rea d y i n d i ca t ed
T h e p ossibility of soari ng o n u p c u rren ts fo r lon g d i st ances d o es n ot
s ee
t o b e very grea t
It will b e n oti ced fro m th e metho d of cal c ul a tio n
gi ve n fo r T a bl e 18 th a t th e speed of th e u p c u rren t re qui red for su pp o rti n g
a fl yi n g machine a t a gi ven gli di ng an gl e i s prop o rtiona l to th e flyin g
s peed
Hen ce bir ds h av ing m u c h lower Speeds can so ar in l ess
s t ro n g
an aer o p lane Th e lo ca l c h arac t er of th e
u p c urren t s than
u p c u rren t s i s evi denced by th e t en dency fo r bir ds wh en so ari n g to keep
o ver th e same par t of the ea r th
In
.
.
.
.
.
.
.
-
,
-
.
.
.
~
.
,
,
m
,
m
.
m
,
.
m
-
.
,
-
,
.
-
.
-
.
mb So far th e calc ul a tio ns h ave b een
for
w eight of aero p la n e an d fo r an a t mo sp h er e
as dens e a s th a t
i n th e low er reac h es of th e ai r
It will often
happ en tha t add itio nal w e ight i s to b e ca rr i ed i n th e fo r m of ex t ra
pass eng ers o r goo ds Also d uri ng war fare m o rder to escape fromhostil e
air cra ft gu ns i t may b e necessary to c li m b m
any thou sands of fee t a bo ve
t he earth s sur face
Th e p robl em n ow to b e a tt acked i s th e m
e tho d of
es tim
a ti ng th e e ffec ts o n th e perfo rm ance of an aerop l ane of ex t r a w eight
and of red uced density Th e great est h eight y e t reac h ed by an aerOp lan e
s
is a bout 5mi l es and a t su c h h e ight th e b aro me t er st and s a t l e ss th an 10 m
of merc u ry it is c lear fr o m th e out se t th a t th e condi tion s of fli ght are
th en very differ en t fr o m those near th e ground In or der to c lim b to su ch
hei ghts the w eight of th e aerop l ane 18 kep t to a mi n im um and th e r eserve
ho rsep ower m
ade as grea t as p oss ibl e T h e pro bl em i s easily d i v i sibl e
in to two dis tinc t part s one of whi ch rel a t es to th e p ow er r equi red to
su pp o rt t h e aero p l ane m th e a i r of low er de nsity and th e oth er of whi c h
dea ls with th e red u c tion of ho rs ep ower of th e en gi ne fro m th e sam
e ca u se
Th e l a tt er of th e two ca us es i s of th e gr ea t er i mp o r ta n ce in li mitin g t h e
height of c li m b
It h as already b een p oin t ed out i n c o nn ec tion with Fig 12 th a t th e
liftin g fo rce o n any aerOplan e var i es as th e squ are of th e Speed so lon g
as th e an gl e of i nc i dence I S kep t co n st an t
Now su pp ose th a t th e w eight
whi ch
mad e
an
Aerop lane
a
fi xed
can
cli
-
.
'
.
.
,
,
’
.
.
”
.
,
.
.
,
.
.
.
.
Rep ort
of
d vi sory
t he A
mmi t t
Co
ee
for Aerona u t ics, 19 11- 12. p 3 15
.
.
APPL IED AERODYN AMI CS
40
of th e ae rop la n e i s increas ed in th e ratio M2 1 1 by the additi o n of l oad
insi de th e bo dy 1e wh ere it d o es n ot add to th e resis tan ce di r ec tly I n
or der tha t th e aerop l ane may lift wi thout a ltering i t s angl e of inci dence
it i s necessary to increase th e speed m th e pro portion of M 1 This m
erease will app ly with e qu a l exac tn ess to th e rev olutio ns of th e air scr e w
a n d th e sim
p l e rul e 18 reac h ed tha t if an aerop lane has i t s w eight i ncreased
3
in th e ra tio M :1 and i t s speeds i n th e ra tio M 1 flyi ng Will be p ossibl e
a t th e same angl e of i nci dence fo r both lo adings
Fr o m th e prev ious ana lysis it wi ll be realiz ed th a t th e increa se of
sp eed necessa ry to
ve th e g rea t er lif t i nv ol ves an increase in th e r esi st ance
2
n
prop o rtio a l to M an d to b al ance thi s an in crease of propell er th rus t als o
9
m
i
to
Th e me tho d of findi ng hors ep ower shows tha t th e
o
r
t
n
a
l
o
M
p p
8
n
i creas ed ho rsep ower is in th e ra tio of M 1 to th e ol d ho rsep ow er Lea v
ing th e vari a tion of density alon e for th e m omen t new cal c ul a tions for
oth er load s coul d be made as b efore Si nce Fi g 15 exi s ts fo r th e ol d
loadin g a simp l er me tho d may be follow ed
Th e c ur ves OP and
of Fi g 15are r ep r od u ced in Fig 28 below
with an increas e of scal e for th e airscrew rev oluti ons Th e two fu rth er
c u rves of Fig 28 marked 8000 lbs and 4000 lbs are pro du ced as shown
i n T a bl e 14 i n ac co rdance with th e l aws j us t en un cia t ed
,
.
.
.
,
.
.
,
,
.
m
.
.
,
.
.
.
,
.
.
.
.
.
.
.
B
TA LE l t
Weigh t
m
m
l n cn
Weigh t
2000 lb s
.
t
o.
8000 1b s
b y 11225
Lo
zn
mm
.
Weigh t
.
t o b y r at
.
o
.
4000 lbs
t o b y 11 14
.
.
.
u
.
.
by
Fi g 28 sh ows th a t th e aer o p l ane woul d s till fly with a tot al load of
4 000 lbs
At top speed th e air scre w sp ee d h as f all en fr o
1525 to
14 70 r p
o wi ng to th e ex t r a lo adi n g It 18 ea sy to cal c ula te th e maxi u
lo ad whi ch might b e carri ed , si nce Fig 23 shows th a t th e ai rscrew woul d
i n th e li iti ng case b e mak i g a bout 1400 r p m a n d deli verin g 18 5ho rs e
m
.
m
m
.
.
.
.
.
n
mm
.
p ower I f th en w e fin d
28 an d m
u lti p ly 2000 lb s by thi s
n um
be r 4 560lbs Will b e obt ain ed as th e li m
iti ng lead whi c h this aerop l ane
can ca rry It will b e seen th a t each 1000 lbs of lo a d carri ed n ow r e qui r e s
a bout 80 ho rsep ow er
.
,
,
,
.
.
,
.
.
.
.
.
.
THE
41
PRINCIP LES OF FL I GHT
Co rresp o ndi n g cal c ula tions b ased on Fig 28 i n an exactly anal ogous
way to thos e of Ta bl e 14 on Fig 16 have been made T h e det ai ls are n ot
.
.
.
A I R S C R EW
,
Fro 23
.
.
—Effec t
of a
ddi t i ona l we igh t
R EV O L UTI O N S
h orsepower an d airscrew re vol u t ions
on
.
gi ven but th e res ults are shown in Fi g
24 ,
and
w
e ar
M
,
.
s c r ap o r
F ro 24
.
.
-
Eff ect
ddi t i ona l weigh t
of a
.
.
it can b e se en how th e sp eed
D n.
on
d of fl i gh t
t he s pee
.
T h e c u rv es FGH and B CDE ar e rep ro du ce d fr omFig 16 whil st those
marked 8000 lbs an d 4000 lbs ar e th e r esults of th e n ew calc ul a tions
a ll d i fferen ce of
Th e fir st ve ry n oti cea bl e fea ture of Fig 24 i s t h e sm
.
.
,
.
.
.
APPL IED AERODY NAM I CS
42
p ed d u e to d oubli ng th e loa d th e f all b ein g fro m 98 m p h to 8 6
m p h Th e eflect on the slowes t Speed of flight i s very m uc h gr ea t er fo r
th e l eas t p ossibl e Speed of st eady ho riz on t al flight i s 64 mp h with a load
of 4000 lbs ins t ead of 40 m p h with a loa d of 2000 lbs Th e di fli cu lt i es
of l an ding ar e m
uc h i ncr eas ed by t h is increase of min i m u mflyi ng Speed
Fig 24 can b e used to i llus t ra t e a p oin t in th e eco n om
i cs of flight Th e
subj ec t wi ll n ot b e p u rsu ed d eep ly h e r e si nce m
or e c om
preh ensi ve metho ds
will b e develo ped l a t er
If it b e dec i ded th a t 9 Speed of 90 mp h i s
des ira ble for a gi ven servi ce it i s seen tha t 2000 lbs can b e carri ed for a n
expend iture of 129 ho rsep ower 8000 lbs for 18 8 horsep ow er an d 4000 lb s
fo r 152 ho rsep ower If th es e n um b ers are expre ssed a s ho rsep ower p er
thousand p oun ds ca rr i ed they b eco me 65 46 and 8 8 showi n g a p rogr essi ve
c h an g e i n fav o u r of th e h eavy loadi ng Th e di fference i s very gr ea t and
ob vious ly of co mm
erc i a l i n ter es t Va ri a tion of load in g i s n ot th e
only fac to r l eadi n g to ec on omy but th e im
p r ession gi ven a bo ve froma
p a rti c ular in st ance may b e accep t ed as ty p i ca l of th e aerop l an e as w e n ow
t 0p
S e
,
.
.
.
'
.
.
.
,
.
.
,
.
.
.
.
.
.
.
.
.
,
.
.
.
.
.
.
,
.
,
.
,
.
,
,
,
.
,
.
,
It sho u l d b e rem
em
b ered tha t th e presen t cal cu la tions refer to
i ncreas ed lo ad i n an exi st i ng aerOplan e Any new des ign fo r an o rigina l
w ei ght of 4000 lbs woul d differ fromth e prototy pe prob a bly both in size
and i n th e p ow er of i t s engi ne
—
i
A
F
l
f
ht
a
t
lt
i
t
d
f
t
A
t
a
h
e
i
ght
of
f
a
n
d
e
s
o
ee
t
u
e
e
g
{
f ee t th e density of th e ai r is r ela ti vely only 07 4 of tha t nea r th e
ground an d w e n ow in quire as to th e e ffec t of th e ch ange Th e exp eri
men tal l aw is a sim e one an d st at es tha t a t th e sa me a ttitu de and speed
of flight th e air force i s pro p ortional to th e air de ns ity
Th e new per fo rmance a t 10 000 i t may b e cal c ul a ted fr om th a t near
gr ound l evel by a process som
ewh a t analogous t o th e one follow ed for
v a ri a tion of w eight At th e sam
e angl e of i nc i den ce it 13 p os sibl e to p ro d u ce
th e same lift m air of di ffer en t densiti es by ch angin g th e sp eed and th e l a w
i s th a t «V2 1s c o nst an t d uri ng th e c h ang e
Th e power re quired 18 n ot th e sa m
e since th e speed has incre ased as
.
.
.
m
,
.
,
.
.
,
-
.
,
.
l
N/ and h ence
:
h as
t h e h orsep o
a lso increased
as
We then
ge t th e follo win g si m
p l e ru l e for the aer o p l an e and a irscre w tha t flight
a t red u ced densi ty 18 p ossibl e a t th e same angl e of i nci dence if th e sp eed
of flight and the speed of ro t a ti on of th e a i rscrew are increas ed 1n prop ortion
0
'
0
,
/Z
to Q
the ho rs epow er required for fl ight
18
a lso increased in
t he
pro
T a bl e 15shows how th e cal c u la tio ns are m ade
ns 8 a nd 4 of T a bl e 15th e c urve Al B l Cl of Fig 25i s dra wn
F ro m c olum
to r ep resent th e horsep ower n ecessary for flight a t
f ee t Th e
o rigi na l c u rve for un it de nsity i s shown as AB C
ass pe r u n i t v ol u m
e o f t h e fl i d or
i a t h e re l t i ve d ens i t y wh i l e p i s u ed for t h e m
a bs o l t
d ns i t y
.
.
.
.
a
v
u e
e
.
,
s
u
THE
PRIN CIP L ES
TAB LE i ii
-
Far
m
s
u
OF
Gn
F L I GH T
m Hm
r
on r s
.
i s 0 58 5
is
Flo 25
.
.
—Eflect
.
A I RS C REW REVO L U TI O NS
of va ri a t i on of
h eigh t
48
011
.
h or se pow er a nd air scre w re vol u t i ons
.
—
i
o
t
h
T h e horsep ower of an
wi
H
t
E
n
n
e
f
i
P
w
er
h
o
g
g e
eng ine i n an average a t m os p h ere falls off m o re rap i d ly th an th e density
en t a lly For a h eight
an d c u rves of v ari a t i on h ave b een de ri ved experi
of
ft th e ho rsep ower a t any gi ven Sp eed of rot a tion i s foun d to b e
Th e c ur ve O l Pl of Fi g 25 i s
06 9 of tha t wh ere th e den sity i s un ity
Vari ati on
m
‘
.
.
obt a i ned from
01 by m ul t i plyin g th e o rd i na t es by 06 9
.
.
’
.
Th e pai r of c urves
APPLI ED AERODYN AMICS
44
n ow refers to flight a t
ft and th e rev olutions of th e
engin e a t to p speed
a t B wi ll b e seen to be a littl e l ess th an those a t
th e ground Th e res erve horsep ower fo r c li m bing will be seen to b e m u c h
r ed u ce d an d i s littl e m o re t h an h alf th a t a t th e low l ev e l
Th e r e m
u s t co m
e so me p oi n t i n th e as cen t of an aerOpla ne a t whi ch
a new c u rve for OP will j us t t ou ch th e new c u rve for AB C and th e density
fo r whi c h thi s occ urs will de t ermi ne th e gr ea t est h ei ght to whi ch th e aero
p l an e can c lim
b Thi s p oin t i s t ec hni ca lly kn own as th e ceili ng
A
r epe titio n of th e ca l c ul a tio n for a h ei ght of
ft shows t his h eight
Th e dr op in ai rscrew rev olutions a t t 0p
as b ei ng very near to th e c eili n g
speed (B 11) is n ow w ell m
arked
Th e co rr es pon di ng c u rves fo r flight Speed an d ho rsep ow er have been
cal c ulat ed a n d ar e sh own in Fi g 26 Th e c urves fo r ho rsep ow er re quir ed
Ol Pl AI CI B I
.
,
,
,
.
.
,
,
”
.
.
.
.
.
”
.
.
S P EE D o r
F ro 26
.
.
—Effect
of
“
1 116 111
h eigh t
on
d
t h e Spee
of
flig h t
.
and Spee d are obtain ed fro m thos e a t ground level ( Fig 16)
bo th a bscissa e an d o r dina t es
Th e horsepowers a t max im
um
-
.
mi imum
an d
p eds ar e gi ven by th e p oin ts A1 RI A1, an d B u of Fig 25
and fix two p oin ts on eac h c urve of ho rsep ow er availa bl e and h en ce
fixgt h e ma xi m u m and mi n i m u m Speeds
Th e Speeds a t grou n d level
ft an d
ft ar e foun d to b e 98
89 m p h and 79
sh o wi n g a ma r ked f all with in creas ed h ei ght
T h e i ncr ease of th e low es t Speed of l evel s teady fl ight i s of littl e i m
p ort a nce since l an din g d o es n ot n ow nee d to b e consi dered
An oth er it emi n th e ec o n o m
i c s of fl ight i s illust rat ed by Fig 26 Th e
lo ad car ri ed i s 2000 lbs a t all h eigh t s but a t a Speed of 90 mp h t h e
ho rsep ow er s r eq u i r ed are 129 n ear th e groun d 9 9 a t
ft and 82 a t
ft
64 4 9 an d 4 1 ho rs ep ower p er 1000 lbs of lo ad car ri ed
Th e
in t ense col d a t g rea t h eight s su c h as
ft m
u st b e o ffset aga i ns t t h e
n
S e
,
,
.
,
o
.
.
.
.
,
.
.
.
.
.
.
,
.
,
.
,
.
.
.
,
.
.
.
.
APP L IED AERODYN AM IC S
46
on
O ne of th e c onditions fo r st eady flight re qui res th a t th e result an t fo rce
th e whol e aero p l ane sh a ll pass thr ough th e cen t re of gravity of th e
S P E ED M P H
.
30
4 0
4 0
50
so
60
.
80
70
70
so
S P EE D M P H
.
a er Op lan e,
whi c h
th e
.
.
it i s i mp o ssibl e to fin d any p oi nt near th e
fo r
c onditio n i s sa ti sfied a t all spee ds It wi ll be su pp os ed th a t
a nd
.
THE
PRINCIP L ES OF FL I GHT
47
th e cen tr e of gravity i s su ccessi vely a t th e p oin ts A B and C of Fig 27
and it wi ll be sh own how to pro d u ce th e desi red s h eet by m eans of a ta il
p lan e with adj us ta bl e a ngl e of i nci dence T a bl e 16 sh ows th e va lu es of
resul tan t fo rce and th e l everag es a bout th e p oin t A in term s of th e cho rd
of th e ae rofoil c an d fin ally th e co u p l e i n t erms of th e previous qu a n titi es
,
.
,
'
.
,
TAB LE 16
.
—W
13 0
Mow
r a.
mA
Di st ance trc
.
T h e m o men t s fo r th e p oin ts B an d C are obta in ed by a repetition of
th e p r o cess follow ed fo r A Th e resulting fi gu res h ave been used to dra w
th e c u rves of Fi g 28 whi c h are marked A B C
T h ese cou p l es are to be b a lanced by th e t ail p l an e an d th e fi rst p oi n t to
b e c o n s i der ed i s th e s h ee t of th e d own c u rren t of ai r f ro m th e wi ngs on th e
Th e angl e thr ough whi c h th e air i s
air f o rces ac ti ng on th e t a il p l an e
deflec t ed i s call ed th e angl e of d o wn w ash a n d is denot ed by e
.
.
.
,
,
,
.
,
.
”
,
of th e ai r i s di r ec tly rela t ed to th e lift o n th e
In c /1276 170 17 o f C/ wr d
FIG
r
e
i
p
mt
m—
.
Do wn was h
f romw i ngs
found to b e very nearly prop o rtion al to t h e lift fo r
angles of i n c i dence an d a ty p i ca l di agram showi ng d own wa sh
i n Fi g 29
en ally
vari o u s
.
it
is
,
.
APP L IED AER ODYN AMI CS
48
Th e u pper st raight line AB of Fig 29 shows th e angl e of th e cho rd
of th e wi ngs rel a ti ve to th e ai r i n fro n t of th e wi ngs whils t CD shows th e
angl e a t th e ta il Th e chord of th e t a il p l ane wi ll n ot u su a lly be parall el
to th e c hor d of th e wi ngs and i t s se tti ng i s d enot ed by a‘ Fi g 80 wi ll
make th e various quan ti ti es c l ear
.
,
.
.
,
.
.
l
0
m
d e u n s cr r
A N GLE
o
.
For an angl e of i nci den ce a a t th e win gs w e have a t th e t ail an angl e of
wi nd rela ti ve to AP of a e and th e tail p l ane b ei ng set a t an
to
AP fo r th e an gl e of i nc i dence of th e t ail p lane i s gi v
.
,
,
a
t}
I NC L IN A T I O N
0 1 TA N
—Li ft
Fro 3 1
.
.
th e medi an li ne of th e sec tion
t
tg
ll 3
.
a nd
d
pu
m
g ft
.
o
Fig
.
ns
ai
r o
l pl ane
8 1 shows
w
m
o
.
c u rves of
Vzm
ph
.
for a ty p i cal t ail p l anesuita bl e for an aerop l ane w eighi ng 2000lbs
p 11.
of gg s qf ee t area and a chor d of 4 fee t As n othi ng i s lost in th e princ i p l e
of b a l an ce by th e om
i ssion of t erms depen d in g o n th e chan ge of cen tre of
.
.
THE
PRINCIP LES OF FL I GHT
49
pressur e of th e ta il p lan e su c h t erms wi ll b e ign ored and the fo rce on th e
t a 1l p lan e W11
1a l ways b e a ssumed to pass t h rough th e p o in t P
If th e di sta nce fro mA to P b e deno t ed by l th e c q
u ati 011 for w om
;
en 1
of th e t ail a bout A is
,
,
.
momen t
or
l, { h
'
cos
(a
cos
(
)
D
e
'
si n
(
a
e
)}
m or e co nven i en tly
m om
en t
Irvi
-
;
)+
a
s1n
e
nx
Th e cal c ul a tio n pr oceed s as in T a bl e
B
TA LE l 7
a
.
—
( a e)
17
.
—Ti mMO ME N TS
.
.
—
e
—l
1
°'
6
Wh en th e aer oplan e i s i n e qui li b riumth e c oup l e gi ven i n th e l as t c olumn
m us t b e e qu al to but of opp osit e si gn to th a t o n th e wi ngs Coup les d u e
to th e t ail ar e th er e fore p lott ed i n Fig 28 with th e ir sign r eversed Th e
i n t ers ec tions of th e variou s c u rves th en show th e speed s of st eady flight
fo r various t a il se ttin gs
Th e di fferen c es b e tw een Figs 280 28b 286 c orr esp on d with th e
di fl erences i n th e p ositio n of th e ce n tr e of g ravity 1 c with A B and C
Th ey are c o nsi dera bl e and i mp o r t an t
Fi g 28a show s th a t e quilib rium i s n ot p ossibl e withi n t h e flight r a nge
4 0 m p h to 100 m
n
l
s
n g i s l ess t h a n
u
til
th
e
t
a
i
e
tti
th
e
p
d
h
s
e
e
p
5 th e Speed for e q u i lib riu m
i s 65
b ei ng th en 100 m p h Fo r 0,
4 9 mi l e s per ho u r
an d for a ,
Fig 28b shows tha t th e aerop l an e i s a l m ost i n e quilibri u ma t all Speed s
os t n ea r ly c o rrec t a t
fo r th e same sett in g a , = o th e st a t emen t b ei n g m
To c h a nge fr o m50 m p h t o 4 1 mp h
Speed s of 70 m p h to 100 m p h
the t ail p l an e se tti ng nee ds to b e alt ered fr o m+ 1 to
Fig 28 0 i s to a l ar ge ex ten t a reversa l of Fig 28a Th e a ngl e of t ail s ettin g
m ust exceed +2 to b ri ng th e e quilib rium p osition wi thin th e fli ght rang e
s
i
102
S
t
th
e
peed
fo r
4 0 m p h to 100 m
A
h
p
it is 8 1
it i s 60 mp h To red u ce t h e Speed
a n d fo r
fu rth er woul d n eed s till gr ea t e r angl es and th e t a il p l ane p asses i t s criti cal
ight no t b e p ossi bl e i n th is case to fly s tea d ily a t 4 5 mp h
a ng l e
It m
em
i ght be t ru e for a p ositio n o f t h e cen t r e of grav ity of t h e a ero
Th e sa m
p l ane fu rth er fo rw ard th an A
,
,
.
.
.
.
.
'
,
,
,
°
'
,
.
.
.
,
.
.
.
.
.
.
.
.
-
.
.
°
.
.
.
,
.
.
,
.
.
.
.
.
.
.
.
.
.
.
.
°
-
.
.
.
°
.
.
.
.
.
.
.
.
,
.
.
.
50
APP L IED AERODYNAM I CS
If w e regard th e var i a tio n of t ail setti ng as a c on trol w e s ee th a t both
A and C are p ositio ns of th e cen t re of gr av ity whi c h l ead to i nse nsiti ven ess
whi lst p ositio n B l eads to g rea t sen siti vi ty An exam
l
e
1
3
th
en
reac
h
ed
of
P
a gener al co nclusio n th a t gr eat est sensiti veness i s obtai ned fo r a par ti c ul ar
p osi tion of th e cent re of gr av ity an d tha t for or di nary wi n gs thi s p oin t i s
a bout 0 4 of th e c hord from th e l eadi n g edg e We sh all see th a t thi s
c o nclusi on 19 not grea tly m o difi ed if th e t ail p l an e b e r ed u ced 1n area
Co nsi der n ow th e a ero p l a n e with i ts cen tr e of gr avity a t A flyi n g a t
an an gl e of i n c i den ce of 8 0 and a speed of 70m p h
but with a ta il setting
of
Th e wi ngs are th en gi vi ng a c oup l e —0 080172 whi ch t end s to
,
,
.
,
.
.
,
,
,
°
.
.
.
,
,
S PEED
F
mli
.
i
MR H
—Long it u d i nal ba lan ce
wi t h
s
mll t
a
a il
p
l ane
.
decreas e th e a ngl e of i nci dence an d to p ut th e aer op l an e ma c ondition
s u it a bl e fo r hi gh er speed wh e rea s t h e e quilib rium p osi tio n for thi s t a il
er sp eed
Th e t a il i s however exerti ng a c ou p l e of
se ttin g i s at a
2
s
s
and
thi
t
end
in th e Opp osit e di rec tion an d over c omes th e
l
4
o
o
v
+
c ou p l e d u e to th e win gs It i s a l m ost cert ai n t ha t t h e a er Oplan e would b e
s t a bl e a n d settl e d own to i t s Speed of 4 9 m p h if l e ft to itself with th e
t ail plane fi xed a t
Fig 280shows th e r everse cas e ; th e win g mom
en t b ei n g grea t er th an th e
t ail m om
ent th e aerOplane wou l d b e un st a bl e It i s n ot prep osed to
disc uss sta bili ty i n det ail h ere but it shoul d b e n ot ed th a t th e si mp l e
c ri t eri a n ow em
p l oy ed ar e o nly app r oxi ma t e altho u gh r oughly corr ec t
It ca n n ow b e seen th a t gr eat es t sensiti v ity t o c ont rol occ ur s wh en t h e
,
.
,
,
'
,
.
.
.
.
.
.
,
,
,
.
THE
PRIN CIPLES
F L I GHT
OF
51
ta b i lity i s n eutr al ; p utti n g th e cen t re of gra vity forwar d red uces t h e
sens i t i vity and i n t r o d uces s t a bility whils t p utti n g th e cen t re of gr a vit
y
b ack red u ces th e sens iti vi ty a n d m
akes t h e a ero p l an e unst a bl e
Tai l Plans of Di fi er ent Si ze —For p ositi o ns A an d B of th e cen t re of
grav ity of th e a erOplane calc ul a tions have b een made fo r a t ai l a rea of
8 5 s qf ee t i ns t ead of 55 Th e effec t i s a red u c tio n of t h e m
om
en t d u e to
th e t ai l i n th e p r ep o rtion of 8 5to 55for th e sa m
e t a il s etti ng a n d a er o p l a ne
Th e result s ar e shown i n Fi g 82 For n eith er p ositio ns A n o r B
s p eed
i s th e c harac t er of th e di agram grea tly alt ered th e c hi e f ch an g es b ei n g th e
all er righti ng c ou p l e for a gi ven d isp l ac ement as show n by th e sm
all er
sm
an gl es of cr ossin g as com
pared with Fig 28 A t ail se tt i ng angl e of
10 wi t h p o sition A n ow o nly r ed u c es th e speed to 58 m p h a n d it i s
prob a bl e th a t th e t ail p l ane woul d rea c h i t s criti cal angl e a t low er spee d s
of fl ight
Fo r p ositio n B th e di agram shows a s all er res t o ring c ou pl e a t low
ewh a t grea t er d is tur bi ng cou pl e a t high speeds
speed s a n d a som
all ta il plan es ten d tow ard s i ns ta bility but th e e ffec t of si z e i s n ot
Sm
so mar ked as th e e flect of th e ce n t re of gravity c h an g es represen t ed by
A B an d C Th e c o n tr ol may n ot b e su fli ci ent to s t all th e a er Oplan e wh en
Thi s t en d s to safe ty i n fli ght
i ts cen t re of gr avity i s a t A
Elevat orL —Ma ny aer op l an es are fi tt ed with t a il p l a n es whi c h can b e
Th e m otions pro vi ded fo r thi s p urp ose ar e slow an d t h e
set i n th e ai r
co n t r ol i s n o rmally t aken by th e eleva tors Th e e ffec t of th e m
otion of
th e el eva tors i s e qui val en t to a sm
all er m otion of th e whol e t ail p l a n e
and Fig 38 shows a ty p i ca l di agr amfor vari a tion of li ft wi th v a ri a tion of
an gl e of el eva to rs th e lift b ei ng th e on ly qu an tity c o nsi d ered of suffi c i en t
im
p or t ance fo r r eprod u c tion
s
,
.
.
.
.
.
.
,
,
-
-
.
.
°
.
.
.
,
m
.
.
,
'
.
,
.
.
.
,
.
,
.
,
.
Th e
o r di na te of Fig
88
.
13
th e valu e of
11;
2
7
fo r a t a il p l ane and el eva to rs
of 55 sqfee t a rea of whi ch tot a l th e el ev a to rs form4 0 p er cen t Th e
a b sc iss ae are th e a ngl es of i nci d ence of th e t ail p l an e a n d ea ch c u rve c orr e
Th e a ngl e of t h e el eva t ors
Sp on ds wi th a gi ven se tti ng of t h e e l eva to rs
i s mea su r ed fr o mth e cent r e li n e of th e t a il p l a n e an d i s p o siti v e 111a
a kin g a n a ngl e of i n c i de n ce g re a t er th a n t h e
th e el eva to r 1s d own 18 m
—
n
n
15 a n d + 15 th e c u rv es
t a il p l a n e Fo r el eva tor a gl es b e twee
are r oughly e qu ally spaced o n angl e but a fter th a t t h e in creas e of lif t
with fu rth er n cr eas e of el eva tor a n gl e 8 m u c h red uced
ay b e u sed for n ega t i ve setti ngs by ch a ngi ng th e si gns
Th e di agr a mm
of both angl es and of th e lift T hi s follows b eca us e th e tai l p l an e h as a
sy mme t ri ca l s ec tion
F ro m th e di agram a t A it will b e seen th a t an el eva to r se ttin g of 5
.
.
,
,
.
,
,
.
.
°
°
.
,
.
.
.
°
,
prod uces an
;
v
of
0 015, and
en t of
thi s woul d a ls o b e p r o d uced by a m ovem
Fo r thi s par
th e whol e t ail p l ane and el eva tors thr ough 2 6 (B Fig
o ved thr o u gh
t i cular pre p o rtio n of el eva to r to tot a l t ail sur face th e a ngl e m
en t
by th e el eva to r 13 th en a bout twi ce as gr ea t fo r a gi v en lif t a s th e m
ov em
of th e whol e t a il surface Vari a tions of t ail p l an e s et t i ngs of 10 w ere seen
t o b e re quired ( Fig 28 ) if th e ce nt r e of gr a vi ty of th e a erop l ane was far
°
,
-
.
.
.
°
52
APPL IED AER ODYN AM IC S
fo rw ar d an d thi s woul d mean exc essive el eva to r an gl es an a ngle of o ver
20 b ein g i ndi ca t ed a t C for
Th ese el eva tors are lar ge a n d it wi ll
b e seen th a t a n a er Op la n e m
ay b e so s t a bl e th a t th e co n t r ol s ar e n ot su ffi
c i en t t o en sure fli ght o v er th e full r ang e oth erwise p oss ibl e Fo r th e cen t r e
o f gravity a t p o siti o n B Fig 28
th e el eva tor c o nt rol i s amp l e for a ll
p u rp o ses
,
,
°
,
.
.
,
,
.
O D4
-
.
-
C D6
.
O O8
.
F 101 33
.
.
—Li ft
of ta i
l p l ane a
l
n d e ev a t or
for d i fi e rent
se t ti ngs .
necessa ry to move th e El evato rs —Th e m usc ul ar efi ort req
uired o f
en t a bout th e hi ng e of th e for ces o n t h e
i n ed by t h e m
om
t h e p ilot i s de t erm
el e va to r and i t i s to r ed u ce thi s effo r t th a t a dj u st a bl e t ail p l an es are us ed
If it b e desir ed to fl y fo r lon g pe rio d s a t a sp eed of 70 m p h th e ta il p l a n e
all
om
en t on t h e hi n ge i s v e ry sm
Fonlarge aerOp lan es
i s so set th a t th e m
b alanci n g of c o nt r ol s i s res ort ed to but th e re i s a li m it to t h e a pp ro ac h
to c om
p l et e b al a nce whi ch will u lti m a tely l ea d to r el ay c o n t rol by so m
e
medi at e sc op e of thi s sec tion will be li m
mech ani cal de vi ce Th e im
it e d
to un b al anced el eva tors i n whi ch t h e size i s fi x ed a t 40 p er ce n t of th e tot a l
t ail p l a n e a n d el evat o r ar ea
It h as b een seen th a t th e li ft on th e t ai l i s t h e i m
p o rt an t fact or in longi
ay us e fully p lot hi ng e m o men t s o n th e b as i s
t u di nal b al a n ce an d s o w e m
of lift p roduced I n th e cal c ula tion a tot al ar ea of 55 sqf ee t will b e
as s umed so as t o c om
par e dir ectly wi t h th e p r evi ous cal c ul a ti on s o n t a il
s e tti n g
'
Efi ort
.
,
.
.
.
.
.
,
,
.
.
.
,
.
.
.
Th e c urv es of Fi g
t he
.
84
my b e u
a
t ail i nci d enc e are u sed w i t h
t he
d fo r n egati ve valu es of
se
v r d si g n
re e s e
.
1
if
A
L
,
2
-
6
.
an d
APP L I ED AER ODYN AMI CS
54
th e va lu es of
T a bl e
V2
M;
13
can b e de t ermined by
u se
of Fig
.
(
84
see
— and th e force
easily cal c ulat ed fr omiig
i
,
c ol u mn
on
’
th e p ilot s
h an d i s th en cal cul a t ed by a ssum
in g th a t hi s h and i s 2 fee t f ro mth e p i v ot
of h is c o n tr ol sti ck
en t a t th e el eva to r hi n g e me an s a
A p o siti ve m o m
p ull o n th e sti ck
B efo r e co m
entin g on th e c on t r ol for ces th e r e sult s of si m
il ar cal c ula
tions fo r p osi ti on s B an d C of t h e cent r e of gr avity of th e aer op l ane ar e
gi ven i n T a bl e 19 i n c o mpari son with those fo r A
m
.
.
.
B
TA LE 19
—Fons “
Poemon e
.
ox
or
Sr w x r oa
Co xr ao r.
02 111 111: or G n
m
r
Drrr xa s x r
'
.
Speed
m
Co nsi der p osition 0 fi rst ; a t 100 i l es per hou r th e p ilot i s p ullin g
h ard on h i s c on t r ol sti ck It h as al r ea dy been seen th a t t h e aero p l ane i s
u nst a bl e with th e cen t r e of gr av ity a t C , a n d o ne r esult of thi s i s a t en denc y
to d i ve w ithout c o nscious ac t of th e p ilot T h e result of a di ve i s an
i n cr ease of Speed , an d T a bl e 19 shows th a t an increase of p u ll ay b e ex
t
e
d
t
r ea t th a t
ec
A
a
o
dera
t
e
an
gl
e
of
d
i
ve
th
e
p
u
l
b
ec
o
e
s
o
g
l
a
p
y
th e p ilot i s n ot st r on g en ou gh to con t rol hi s aero p l an e , whi ch ay th en
ge t i n to a ver ti ca l di ve or p ossibly on i t s b ack A sk ilful p ilot can r ec o ver
th e c on d itio n rep r esen t ed
hi s c o rrec t flyin g a ttitu de, but th e a er Oplan e
by C i s danger ous
Position A shows th e r everse p ic tu re ; th e aer op l an e i s st a bl e and d o es
n ot t en d to di ve without c onsc ious e ffor t by th e p ilot I t need s to b e
.
m
m
m
.
m
m
.
'
m
.
.
.
p ush ed i n to a di ve and if th e force get s very gr ea t owin g to incr eas e of
ati cally st ep s th e pro cess
Speed it a u t o m
Th e a erOplane whi ch i s light es t on i t s co n t rol s i s still th a t with th e
cen t re of gravity a t B but it i s fu rth er c l ear fro m T a bl e 19 th a t an i m
p r o vem
en t wou l d b e obt a ined by a choi ce of cen t r e of gravity som
ew h er e
b etween A and B
,
.
,
.
( ) Fo s ca s
on
1
1
mF
a
noar
or
A
a
m Bo
c
ar
A di a gram illu st ra tin g th e fo rm of a very l a rge flyin g bo a t h u ll i s shown
lbs
ac hi ne b ein g
Th e
i n Fig 8 5, th e w eight of th e flyin g
design of a flyin g bo at hull h as to pro v i de for t axyi n g on th e w a ter prio r
to flight an d fo r ali ghti ng Wh en o nce i n th e ai r th e p r obl em of th e m otion
of a flying boa t d i ffe rs littl e fro th a t of an aerOp lane, th e c hi e f d i ffe rence
°
m
.
.
m
.
THE
PRI N CIP L ES
FL I GHT
OF
55
be i n g
th a t th e ai rscre ws are ra i sed high a bo ve the cen t re of grav i t y in
o r d e r t o p r o v i de goo d c l ea rance of th e a i rs crew s fr o mwa v es an d a ny
gr ee n wa t er whi c h m
ight b e thrown u p Th e p resen t sect ion of thi s
c h ap t er i s d ir ec t ed chi efly to an illustra tion of t h e fo rc es a nd c ou p l es o n
a fl yin g bo a t in th e perio d of motion through th e wa t er
Expe ri men t s on fl yi n g boa t h ulls h ave u s ua lly b e en m
ad e on m o de l s
a t t h e Willi am Frou d e N a tion a l T an k a t T eddi n gto n b u t in on e i n s t ance
a fl yin g boa t w a s t owed by a to r pedo boa t d est roy er a n d m
e a su r e m
en t s
o f re s i st an ce a n d i nc li na tion made fo r c o mpari so n with th e m
o d el s T h e
co m
p ari so n was n ot c o mp l et e but t h e gen era l a gre e m
en t b etwe en m
o de l
a n d full sca l e wa s sa ti s fac to ry
en a a s th e dep r ession o f th e
Suc h p h en o m
b o w d u e t o s witc hin g o n th e en gi ne a n d p o rp oi si ng a r e rep ro d u ced i n
t h e m o de l wi th s uffi c i en t acc ur acy fo r th e p hen o mena to b e kep t under
c o n t rol in th e design st a ges of a flyin g bo a t
In m
ak in g t est s of fl o a t s i n w a t e r F ro nde s l a w of c o rr esp ondin g Speed s
i s u s ed s in ce th e grea t er part of th e fo rce ac tin g on th e fl oa t ari se s f ro m
.
.
,
-
,
.
,
.
.
’
,
,
r u s u sr
A IRS C REW AX IS
é
6 6 -1
4
0117 111
0 1 2
4
m3
F
.
5
.
64
-
f
ee t
F lyi ng
5
boat
1
1
1111
.
th e w aves pro du ced an d if th e l a w be followed it i s k nown on th eo re ti ca l
gr oun ds th a t th e wav es i n th e mo de l will b e sim
il a r to thos e o n th e ful l
Th e la w s ta t es th a t a sca l e mo del shoul d b e towed a t a Speed e qua l
s ca l e
to th e Speed of th e fu ll sca l e fl o a t m ulti p li ed by th e s qu are root of th e sc al e
o de l of a flyin g boa t hull whi ch t axi es a t 4 0 m p h
A on e si x t een th sca l e m
rc es on th e full sc a l e
will gi ve th e s ame s h ape of w aves a t 10 m
Th
e
fo
h
p
are th en ded uced fro m those on th e mo de l by m ulti p lyin g by th e s qu ar e of
th e sca l e an d th e s qu are of th e c orr esp on din g Speed s
by th e c ub e of
th e sca l e Si m
il arly m o m
en ts vary as t h e fou r th p ower of th e l i near
d i ens io ns for t est s a t c o rr esp o ndi ng s peed s
As th e fl o a t i s r u nni ng on th e su rface of th e w a t er th e forces o n it
depend o n th e w eight su pp o rt ed by th e wa t er as w ell as on th e speed and
i nc li na tion of th e fl o a t an d thi s c o m
p l ex ity r en ders a c omp l e t e set of
experi m
en t s very excep tiona l Th e full sc h eme of fl o a t experi m
en t s
whi c h would e li mi na t e th e necessity fo r any re fer ence to th e aerody
i cs of th e su perst ru c tu re woul d gi ve th e lift d rag a n d p it c hi n g m omen t
na m
of a fl oa t for a ran g e of speeds and fo r a ra ng e of weight su pp or ted Fro m
,
.
.
-
.
.
.
.
,
m
.
,
.
,
,
.
,
.
.
.
APP L I ED AER ODYN AMI CS
56
uc h ob se r va tions a nd t h e kn own aer o dy n a mi c fo rc es and m o m en t s on th e
p l e t e c on d itions
s u pe rst r u c tur e for var iou s p o sition s of th e e l eva to r th e c o m
of e quilib riu m c oul d b e wor k ed out in a ny pa r tic ul ar case
At low a i r Speed s
A l ess co mp l et e seri es of experi men t s u su a lly su ffi c es
th e lift f ro mth e wi n gs i s n ot very grea t and a t th e Speed of gr ea t es t fl oa t
resist ance n ot so m
u c h as one quart er of th e tota l d isp l ac emen t a t rest
At hi gh er speed s but still b e for e th e e l eva to rs a re ve ry e ffec ti ve th e a ttitu de
of th e win gs i s fi xed by th e c ou p l es o n th e fl oa t an d d o es n ot v a ry gr ea tly
p ro mi se th e re fo re i s t o t ake th e an gl e of i nc i d ence
A sa ti sfac tory co m
of th e win gs wh en th e co n st an t va lu e h as been reach ed and to cal c ul at e
fr o mit an d th e kn own p ro pert i es of t h e wi ngs th e Speed a t whi ch th e whol e
loa d will b e ai r bo rne At lower speed s th e a i r bo rn e load i s t aken as p ro
p o rtiona l to t h e s qu ar e of th e a i r Speed Aft er a lit t l e experi ence thi s p art
s
,
.
.
,
.
,
,
.
,
,
.
,
-
-
.
.
6 000
5000
F
m3 6
.
.
s p ar e
ov en WATER
—
W t er re
a
w
(
e a
f a flying
s is t ance o
)
.
boa t h u ll
.
of th e c a l c ul atio n p resen ts n o ser iou s d i ffi c ulty an d th e c u rve of lift
on fl o a t
shown i n F i g 3 6 i s th e re sult for th e fl o a t u n der c o n si dera tio n
At res t o n th e w a t er th e di Spla cem
en t was
lb s a t 20
lbs ; a t 40
a ll a t 60 m
e very sm
lbs a n d h ad b ec o m
p h
Fo r th e lo ad s Sho wn by th e lift c u rve th e fl o a t too k u p a defin it e an gl e
of inc lina tion to th e wa t er whi c h i s Shown i n th e same fi gu re Th e re
Si st en ce i s a l so shown in on e of th e c ur v es of Fig 86
Th e an gl e of inc i den ce
depends gen era lly o n th e aero dynam i c c ou p l e of th e su perst ru c ture an d
th e par t of thi s d u e to a i rsc re w th ru st was represen t ed i n th e t e st s B y
m o vemen t of th e el ev ator t hi s c ou p l e i s v ari a bl e to a v ery sli ght ex t en t a t
low speeds but to a n a pp r eci a bl e ext en t a t high s p eed s
Th e fi rst n oti cea bl e f ea tur e of th e wa t er r esista n ce of th e fl o a t i s th e
w h er e it i s
rap i d growth a t low speed s fro mzer o to 5400 lbs a t 27
17 per cen t of th e tota l w e ight of th e flyi n g bo a t
At high er speed s th e
,
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THE
PRI NCIP L ES
OF
F L I GHT
57
i t an ce f alls app r ec ia bly an d will of c ours e b eco me ze ro wh en t h e lift
on th e fl o a t i s ze r o
If th e ae ro dyn am
i c e ffi c i en cy of t h e flyi n g bo a t is
8 a t th e m
om
e n t of getting o ff th e a i r re sis t a nce i s 4 000 lbs an d wi th
ay be t aken as pr o
n e lig i b l e erro r th e ai r r esi s t a n c e a t oth e r Sp ee d s m
g
p o rtion a l to th e s qu are of th e a ir sp eed Since th e a t t i t u de i s seen to be
n ear ly c o nst a n t a t th e high e r an d m
o r e im
p o rt an t speeds B y ad ditio n
of th e dra gs fo r wa t er and a i r a cu rve of tot a l r es is t an ce i s obtain ed whi c h
rea c h es a va lu e of a li t tl e o ver 6000 lbs a t a speed of 80
rises
slowly t o 6600 lb s a t 5
0
and t h en fa lls rap i dly to l ess than 5000 lbs
e com
p l et ely ai r bo rn e t h e r esi st anc e aga i n
Aft er th e flyi ng bo a t h as bec o m
i ncreas es with in crea s e of sp eed
a tio n re qui red to esti m
a t e th e drag of a seap l ane
Th e ad ditio n a l i nfo rm
be fo r e i t l eav es th e w a t e r i s thus obt ai ned an d th e m
ethod of ca l c ul a tio n
pr oceed s as for t h e aerOplan e Th e drag of t h e win gs i s es ti m
a t ed and to
r es s
.
.
,
,
,
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.
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.
-
.
,
.
c oc
c
,
o
oc o
+ I O0 0 0 0
,
2 000
m mm
m37
F
.
.
o n or F LO AT
cu
-
-
s onoo
-
| oo,ooo
( d eg r e es )
Pit c hing m
om
en t on a fly ing boat hu ll
.
added th e dr a g of th e fl oa t i nc lu di n g i t s a i r resi s t anc e To th e s um
i s f u rth er added th e r es is ta nce of th e r e m
ai n in g part s of th e a i rcr a ft
T h e ca l c ul a tion of th e Sp eed an d ho rs ep ow er of t h e a i rsc r e w follows t h e
sa m
e f u n d am
ent a l lin es a s for th e aerop l an e and di ffer s f ro mit o nly i n
th e ex t ens ion of th e a i rscr e w c urve s t o lower fo rward Sp eed s Th e sa m
e
ex t ens ion woul d b e needed for a c ons i dera tio n of t h e t axyin g of an a cr e
p lan e o ver an aero dro me Th e ex t ens ion of ai rsc r e w c h ar ac t eris ti cs i s
e as i ly obt a in ed experi m
en ta lly o r may b e c al c u l at ed a s sho wn i n a l at er
c h ap te r
Th e e vi de n ce on lo n gitu di na l b a l an c e i s n ot wholly sa t i sfac to ry but an
e xa m
p l e of a t est i s gi ve n i n Fig 8 7 whi ch shows a seri es of ob ser v ati o n s a t
a co ns ta n t Sp eed th e re si st a n ce a n d t h e p i t c hi n g m
om
en t be i n g measur e d
fo r variou s a n gl es of in c i d en ce
od el
In t h e exp eri m
en t t h e h ei ght o f t h e m
fr o m still wa t e r wa s li m
i t ed by a s t e p a n d it i s i m
p rob a bl e t h a t u nd er
t h ese c i rc u m
s t a n ces th e lo ad on th e fl o a t woul d cor rect ly su pp l emen t th e
lo ad on th e win gs T rea tin g th e d i a gram how ever as though equi lib riu m
it
is
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,
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,
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APPL IED AERODYN AM ICS
58
of ve r ti ca l lo ad h ad b een a tt ai ned it will b e n oti ced th a t th e p it chi n g
a ll er a ngl e s th e m o men t w a s
m o men t was zero a t
and th a t a t sm
p ositi ve an d thus t ended to b rin g th e fl o a t if di stur b ed b a ck to
en t c h an ged very rap i d ly but fo r
For gr ea t er an gl es of i nc i dence th e m o m
a ll er angl es th e c h an ge was very m uc h m o re grad u al and it is in t eres t
sm
i n g to c o mpare th e ma gnitu de with th a t a pp li ca bl e by s uita bl e e l eva to rs
o n th e su pers t ru c tu re Fo r th e pres en t rough illust ra tion th e aero d ynami c
p it c hin g m o men t d u e to a full u se of th e e l eva tors may be t a ken as
f
ee
t
if
b
a
l
anced
so th a t th e p ilot can u s e th e ful l
a
n
d
b
s
l
20V2m
ph
A c ou p l e of
a n gl e a c ou p l e of
lbs f ee t a t 55 m p h i s obt a i ned
thi s ma gn itu de i s s u ffi c i en t to c h an ge th e a ngl e of th e fl o a t fro m9 de grees
to 4 de grees and th e p ilot h as apprec i a bl e c on t rol o ver th e longitu d i na l
a ttitu de so me t i me b e fo re l ea vin g th e w a t er
,
,
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-
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,
-
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’
,
.
iii
L
r an
w
a
( )
-
mC
r n a n -A
RAF T
- th an - a i r cra ft obt a i n s u pp o rt fo r th e i r w e igh t by th e utilisa
ght
er
ll
A li
tion of th e differences of th e pro perti es of two gases, usu a lly a i r an d
hy drogen I n th e early day s of b a lloo ni n g th e difference i n th e densiti es
of hot and col d a i r was used to obt ain th e lift of th e fire b a lloo n , wh il s t
l a t er th e enc losed gas was obt a ined fr o m c o a l V ery recen tly , h e lium h as
b ee n c ons i dered as a p ossibility , but n one of th e co m b i na tions pro d u ce so
m u c h lift fo r a gi ven v olu me as hy dr ogen an d a i r, sin ce th e fo r er i s th e
light est gas kn own Th e ex t erna l gas is n ot a t th e c hoi ce of th e aero na ut
times as h eavy as p ur e
At th e same pressur e and t emper a tur e a i r i s
hy dr ogen , and th e lift on a w e ightl ess vesse l fi ll ed with hy drogen an d
.
.
m
.
.
i mmersed i n a i r would b e
12
1
9
7
-
of th e w eight
of th e ai r
dis p l aced
.
Heli u m is twi ce as dens e a s hy drogen whil st co al gas i s seven ti mes as
dense an d is n ever u sed fo r di rigibl e a ircra ft
So me of th e probl ems r el a ti n g to th e a i rshi p b ear a g rea t resem bl a n ce
to probl ems i n me te o rolo gy As i n th e case of th e aer op l ane th e st ra tu m
of a ir pa ssed th rough by th e a i rshi p i s very thi ck th e li mit b ein g a bout
fee t wh er e th e density h as fa ll en to n ea rly h a lf tha t a t th e su rfa ce o f
th e earth AS th e lift of an ai rs hi p depen d s on th e w eight of d i sp laced
a i r it will b e s ee n th a t th e lift m us t dec r eas e with h eight u n l ess th e v ol u me
of di sp l aced a i r can b e i n c r eased It i s th e li m
it to whi c h adj u st m
en t
of v ol u me can t ake p l ace whi c h fi xes th e grea t es t h eight to whi c h an a i rshi p
can go Th e gas c on t ai n ers i n si de a rigi d a i rshi p a r e o nly parti ally in fla t ed
a t th e grou n d an d un der red u ced pressu re th ey expan d so as to ma in t a in
a t l eas t app r o xi ma t ely a lif t whi c h i s i n depe n den t of h eight Th e process
of adj us t men t whi c h i s alm o st a uto ma ti c i n a ri gi d a i rshi p is ac hi eved by
a uto ma ti c and man u a l co n t rol i n th e non rigi d ty pe a i r f ro m th e balloonet s
b ei n g rel eased as th e hy dr ogen expan d s In both ty pes th ere fo r e , th e
apparen t defini t en es s of Sh ape d o es n ot a pp ly to in t erna l fo rm
Th e fi rs t probl e mi n ae rost a ti c s w h i c h will b e co nsi der ed i s th e e ffec t
o n th e v olu me of a mas s of gas enclosed i n a flexibl e b ag of m o vem
en t fr o m
one part of th e a t m osp h ere to an oth er Th e w e ll kn own th eo rems rel a tin g
,
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-
THE
PRI N CIP LES
OF
59
FL I GHT
th e p r o per ti es of gase s will b e a ssu med an d o n ly th e app li ca tion s d o
v elO ed
e
l
s
Th
e
a
s
u
pp
o
ed
to
b
e
i
mpr
i
s
o
ed
a
par
ti
a
l
y
i
nfla
t
d
i
s
s
n
i
n
p
g
flexi bl e b ag of s ma ll size th e l a t er c on dition b ein g in t ro d u ced so as to
e li min a t e sec o nda ry e ffec t s of c h an ges of density f ro m th e firs t examp l e
a l to th e s u r f ace w hil s t oth e r
T h e gas i nsi de th e b a g exert s a pres su r e n orm
press ur es are app li ed ex t ernally by th e su rroundi n g a i r At B Fig 88
t h e in t erna l pres su re will b e grea t er th an th a t a t A by th e am oun t n ec essary
If w b e th e w e ight of ga s per un it
to s u pp o rt th e c olu mn of gas a bo ve it
v ol u m
e th e di fference of i n t ern a l pres su r e a t B and A i s wk Sim
il a r ly if
w b e th e w eight of a i r per un it v olu me th e di ffe rence of ex t erna l pre ssu r es
i s w h and th e ver ti ca l c o mp o n en t of th e in t erna l a n d ex t erna l press ur e s a t
Now for th e sa m
e g ases ( w w) i s c ons t a n t an d th e
w) h
A an d B i s ( w
e l e men t of lift i s pro p o rtiona l to h and to th e hori z on t a l cro ss sec tion of
t h e c o lu mn whi c h s t ands o n B
Add i n g u p a ll th e e l emen t s Show s th a t
th e to ta l lift i s e qu a l to th e pro
d u c t of th e v olum
e of th e b a g an d
t h e di fferen ce of th e w eight s of un it
v olum
es of ai r an d th e enc lo sed
At o rd i nary ground pressur e
gas
and t em p era ture 2116 lbs p er sq
foot and 15 C th e valu e of w fo r
per c ubi c foot
ai r i s 00768 l b
whi ls t w for hy d roge n woul d b e
—
5
8
w fo r a i r and p u re
000 ; 10
hy dr og en woul d th ere fo re b e
l b p er c ubi c foot
I n prac tice
p u re hy dr ogen i s n ot obt ai na bl e
a n d u nder any c i rc u ms t a n ces b e
c o me s co n t amina t ed with a i r aft er
a littl e use In st ead of th e fi gur e
0071 va lu es ran gin g fro m 0064 to
F w 38
0068 are us ed depend in g o n th e
p u rity of th e e nc los ed gas
I t a suit a bl e w e ight b e hu n g to th e botto m of th e flexibl e gas b a g th e
whol e may b e made to rema in suspended a t an y parti c ul ar p l ace i n th e
a t m osp h er e Wh a t will th en h appen if th e whol e b e ra i sed so me thou sand s
of fee t an d rel eased
Will th e appar a tus ri se o r f a ll 2
Th e e ffec t of an i ncrea se of h eight i s co mp l ex I n th e firs t p la ce t h e
density of th e ai r f a lls but with a Sim ult aneous fa ll of p r es sure an d th e
hy dr ogen expan d s so lon g a s full infl a tion h as n ot occ u rred Fo r cert am
c o nditions n ot gr ea tly d i ffe r en t f r o m thos e of an o rd inary a t m o sp h er e th e
inc r eased v olu me exac tly c oun t er b al ances th e eflect of red u ced den si ty
i n v ol ve s
Th e probl em
an d e quili b riu m i s un di s tu r bed by c h an ge of h eight
th e u se of cert ai n equ a tio ns fo r th e p ro perti es of gases If p b e th e p r essu re
s
a
r
s
th
e
w
e
ight
of
u
n
i
t
v
ol
me
and
th
e
a
b
olut
e
t
empe
a
tu
re
of
a
th en
t
u
w
g
to
,
.
,
.
,
.
,
.
,
.
.
,
’
,
’
,
'
'
,
.
-
.
.
.
,
.
'
°
.
,
’
.
.
,
.
.
,
~
.
-
.
‘
,
.
,
.
'
,
.
.
,
.
,
,
p
For
a ir R
,
957,
an d for
t
hy drogen
, R
( 8)
18 75 p b e in g in lbs p er s q
foo t
,
.
.
,
APP L IE D AERODYN AMIC S
60
w i n lbs p er
c ubi c f oot
an d
t i n Cen t i gr ade
degrees on th e a b solut e sca l e o f
tem
pera ture
Wh en a gas i s expan ded both i t s t emper a tur e and pressure are ch an ged
a n d u nl ess h ea t ed o r c ool ed by ex t erna l a g e n c y d u ri n g th e p r ocess th e
add itio nal gas r el a tion is
.
,
.
,
)
1
w
P0
,
for both a i r and
wh ere y i s a p hy si ca l c ons t an t fo r th e gas a n d e qu a l to
hy d rogen 110 we and to are th e va lu es of p w an d t whi c h exi st ed a t th e
b eginnin g of th e expansion
I nsi de th e flexibl e b a g gas w eighi n g W lbs h as b een enc los ed a t a
e d i sp l a ced a t any oth er p r essu re
pressu re p o an d a den sity wo Th e v olum
.
,
,
,
,
.
.
.
is
E
w
,
an d as
was
s
een ear li er th e lift
,
on
th e b a g wh en i mmers ed
in
ai r
is
th e v olum
e di s p laced m ultip li ed by th e di fierence of th e w eight s of un it
v olu mes of a i r and hy drogen Th e e qu a tio n is th ere fo re
’
.
W
w
If th e b ag b e so sm
a ll th a t p h as sen sibly th e same valu e in si de and out
e qu a tio n ( 8) shows th a t th e w eight s of un it v olu mes of th e two gases vary
i n versely a s the i r a bsolut e t e m
pera tu res and e qu a tion ( 10) Shows th a t
th e lift i s i n depen den t of p osition i n th e a t m osp h ere if th e t emper a tu r es
of th e two are th e same If th e b ag b e h e l d in any one p lace e qu a lity of
t emper a tu res will ulti m
a t e ly b e r eac h ed but fo r rap i d c h an ges i n po sition
e qu ation ( 9) shows th e c h an ge s of t empera tu re to b e det ermi ned by th e
c h an ges of pre ss ure It i s n ow p r op osed to in ves tigat e th e l a w of va ri a tion
of p r essure with h eight w h i c h will gi ve e quilib riu m a t all h eight s fo r rap i d
c h anges of p osition
,
,
.
,
,
.
.
Cou vs cn
'
vs
EQUI L I B RI UM
If fo r the ex tern al a t mosp h er e e qu a tion ( 9) i s sa t i sfied th e gas in si de t h e
b ag expa n ds so as to keep th e lift c on st an t R ep l ace th e hy dr ogen by a i r
and in n ew surr oun d ings a t th e red u ced p r essu re rec o nsi der th e probl em of
e q u ili b ri u m It will b e fo u nd th a t t h e pres su res in si de an d out si de th e b a g
are e qu a l a t a ll p oi nt s a n d th e fa b ri c may th en b e rem o ved wi thou t
a ff ec t i n g th e c o n d ition of th e a i r Th e c on d itions ar e how ever tho se fo r
e qui lib riu m an d th e a i r woul d n ot t end to re tu rn t o i t s ol d p o sition It i s
ob v iou s th a t n o t en den cy to c onv ec tio n c urr en t s exi st s a lthough th e ai r
i s c ol de r a t gr ea t er h e ight s
Th e qu an tity whi c h de t ermi nes th e
p o ssibi lity o r oth erwi se of c o n vec tion i s c l ea rly n ot one of th e thr ee u sed
i n e qu a tio n s ( 8 ) a n d
p ot e n ti a l t empera ture i s
A qu an tity ca ll ed
em
p loy ed i n thi s c o nn ec tion an d i s th e t e m
pera tu r e t aken by a p o rtion of
as
i
s co m
whi
c
h
p r e ssed adi a b a ti c ally fro mi t s ac tu a l sta t e to one i n whi c h
g
,
,
.
.
,
.
,
,
,
.
,
.
,
APP L IED AER ODYNAMICS
62
It will be seen fro m T a bl e 20 th a t th e fa ll of t emp era tu re for c onvec ti ve
e quili b riu m i s very nearly t h ree degrees Cen tigr ade for eac h 1000 f ee t of
h eight I n th e s t an dar d a t mo sp h er e th e fa ll i s l es s th an two de gr ees
fo r eac h 1000 ft of h eight i e th e p ot en ti a l t empera ture ri ses a s th e h eight
i ncreases and i n di ca t es a co nsi der a bl e degr ee of s t a bility
Li ft on a Gas Con tai ner of Consi der ab le D i m ensi on —111 th e fi rs t
examp l e th e c o n t a iner was kep t s ma ll so th a t th e gas density was sensibly
.
.
,
.
.
.
,
th e same a t a ll parts
.
a l ar g e c on t ain er th e qu an tity i
f;
In
whi c h o cc u rs
i n eq
u ation ( 10) i s n ot c ons tan t Since fo r th e hy drogen i n th e c on t ai ner
an d fo r th e a i r i m
me di a t ely outsi de th e density var i es with th e h eight of
th e p oin t a t w h i c h it i s meas ured To develo p th e subj ec t fu rt h er c o n
vec t i v e e quilib riu m i n si de and out si de t h e gas b ag will b e a ssu m
ed and
e qu a tion ( 18 ) us ed to defin e th e r el a tion b e tw een pressure and h eight
Th e e qua tion i n ne w fo rm i s
,
.
,
-
,
.
<
:
5
1
fo r v alu es of 11 l ess th an 5000 fee t th e sec o n d t e r m i n t he b racket i s
Th e expression may th en b e expanded
s ma ll 1n c o mparis o n wi th un ity
by th e bin o i a l th e o rem and a li mited n u m
b er of t er ms re t a ined Th e
expans ion leads to
an
d
m
.
.
”0
P0
P0
1
.+
1
h?
27 ?
( 14
e c ho sen p oin t i n th e ga s
wh ere we and p o are th e va lu es of p an d w a t so m
e an d h i s measu red a bo ve an d b elow thi s p oin t
s ay i t s cen t r e of v ol u m
Fo r a difference b etw een g round l evel an d h 5000 fee t th e t erms of ( 14 )
are 1, 018 5and 00 12 and th e t erms ar e seen to c onver ge rap i d ly O n
th e di ffer en ce of p r ess ure b etw een th e two p l aces th e acc u racy of ( 14) as
gi ven i s a bout 1 per cen t Fo r an y a i rshi p y e t c o ns i der e d th e acc ur a cy
of ( 14) woul d b e m u ch grea t er th an th a t shown in th e i l lus tr a tion an d m
ay
th ere fore b e u sed a s a r el a tion b e tween p re ssu re an d h eight i n esti ma t in g
th e lift of an a i rs hi p
If p , b e th e p r essu re a t B Fig 8 8 d u e to i n t ern a l p r essu r e an d x2
a l to th e en velo pe a t B an d th e verti ca l th e
a n gl e b e tw een th e n o rm
en t of area a t B If a c olu
c o nt ributio n to th e lift i s —p 2 cos x2 X e l em
b e drawn a bo ve B th e ho riz on t a l c r o ss sec tion i s e qu a l to cos x3 x e lem
en t
of area a t B an d th e va lu e of th e la tt er qu an tity i s e qu a l to an
i n c rem
en t of v olume 8
di vi ded by h o r wh a t i s th e same t hi n g
Th e tot a l lift i s th en gi ven by th e e qu a tio n
by h
kg
,
.
,
-
.
,
.
,
.
.
,
,
,
,
.
-
,
,
.
,
,
,
,
.
gross h i t
hi
h2
8 ( v ol
.
v
ol
)
(
h,
wh ere th e pressur e s for th e a i r are in di ca t ed by da sh es
F r o m e qu a tion ( 14 ) th e n ecessary va lu es fo r u s e i n e qu a tion ( 15) can
b e ded uced Since
.
,
P1
P2
1 wO
P0
}
011 1he
PRIN CIP LES OF F L I GHT
THE
68
hy dr og en i nsi de with a si mil ar expression fo r a i r outsi de Equ ation ( 15)
bec o m
es
4
0
1 3 ° 03
8
h
gr oss lift ( wo
2)
( v ol )
t
for
.
}
'
,
w0)
v ol
.
.
70
1
Y
‘
ns i derin g
T h e t erm ( wo
o
v
ol
th
a
t
whi
c
h
woul
d
b
e
obt
a
i
ned
by
c
m
i
s
o)
th e hy dr ogen an d ai r of u ni fo rm density we and wo respec ti ve ly Th e
s ec o n d t erm depends o n th e m
ea n h e ight of th e p oin t s A and B a bo ve th e
cen t re of v olum
e an d i n a sym
me t ri ca l a i rs hi p on an even kee l th e q
u an tity
’
.
'
.
,
h 1 + h2
°
is
2
zer o
for
all pai rs of p oi n t s and th e sec ond in t egra l van ish es
.
If th e axi s of th e a ir shi p i s in c lin ed th e i n t e gra l of ( 17) m ust b e exami ned
fur th er For a fully i nfl a ted fo rm whi c h h as a verti ca l p l ane of sy mme t ry
.
th e avera g e va lu e of
hi
ha
2
for any sec tio n
is
e qu a l to a:sin
0,
a
:
b ein g th e
di st an ce fr o m th e cen t re of v olume a lon g th e axis and th e sec tion b ei n g
n o rma l to th e ax is Th e el emen t of v olu me i s t h an e qu a l to th e area of
cross s ec tion m ulti p li ed by dx and
,
.
-
,
hr
MS
[ s
-
i
G
m
1
8
( )
d
m
T his in tegra l i s easily evalu a t ed grap hi cally fo r any fo r of en velope but
for th e p u rp o s es of i ll u str a tio n a cyli nder of l en gth 21and di a e t er d will
be us ed
Th e fi rs t p oin t i s easi ly ded uced a nd shows tha t th e gro ss lift
of an in c lined cylin der i s th e same as th a t o n an even kee l Gen er a li si n g
fr o m thi s it m
ay b e s ai d th a t fo r an a i rshi p th e gr o ss lift i s n o t apprec i a bly
ay b e ca lc u l a t ed fr o m
a ffec t ed by th e inc li na tion of th e ax is and th e lift m
th e di sp la cemen t an d th e differ en ce of densiti es a t th e h e ight of th e cen t r e
of v olu e
Pi tch i ng Moment du e to I n cli n ati on of th e Ar i a — Mo m
en t s wi ll b e
t aken a bout th e cen t r e of v olu e of th e a i rshi p To d o thi s it i s on ly
—
ss
nece ary to m
ulti p ly th e lift of an el em
en t by :r b efo r e th e i n t eg ra tio n
i n ( 17) i s pe rfo rmed T h e fi rst t erm will b e zero whil st th e sec on d h as
a v a lu e e qu a l for th e cyli n der to
.
m
,
,
.
,
,
m
.
m
.
.
,
,
,
w0
(
Pitc hin g m
o m en t = _
1
Y
H
Po
'
2 1 ( wo )
3 7
z
— w g
o
A s in 0 l3
.
P0
To apprec i a t e th e sign ifi ca n c e of ( 19) c ons i der a n u m eri c a l cas e A
he ight of
f ee t in a c on v ecti ve a t m
osp h er e h as b een c hosen as c o rre
s p on di n
s
r
i
n
r
r
s
r
i
e
with
fu
ly
expanded
hy
dr
og
en
c
o
n
t
a
e
Th
e
p
e
u
e
h
e
s
s
l
g
1150 lb s per s qu ar e foo t and wo i s 004 33
Th e valu e of m
o i s of n o
im
p o r t ance An a i rshi p 70 feet i n d i ame t er an d of l en gth 650 fee t shows
for an i nc lina tio n of 15 a c ou p l e of m
lb s ft a n d to
or e th an
.
.
'
.
,
.
.
°
-
.
.
,
APP L IED AER ODYN AMI CS
64
c ou nt e rac t thi s a force of 90 lbs on th e ho riz on ta l fi n and e l evato rs wo u l d
ay
b e needed Th e coup l e m
howeve r o cc ur wh en th e a irs hi p h as n o
motio n r el a ti ve to t h e ai r in whi c h cas e it i s b a l anced by a m o men t du e
to t h e weight of th e a i r shi p whi c h in th e illus t r a tion woul d b e
A m o vemen t of 3 i ns woul d suffi ce whils t th e m o vemen t ca u sed by
lbs
a p it c h of 15 woul d b e a bout 8 f ee t Th e e ffec t i s th en e qui va l en t to a
red u c tio n of me t acen tr i c h eight of 3 p er cen t
Equ a tio n ( 19) shows th a t th e p itc hin g m o men t i ncreases r ap i dly with
th e l e ngth of th e sh i p but i n th es e cas es th e ty pe of c o ns t ru c tion adop t ed
all a m
red uce s th e m
om
en t to a sm
oun t Th e l en gth of th e ai rs hi p i s di v i ded
i n to c o m
pa r t m
en t s separa t ed by bul kh eads whi c h can su pp o rt a co nsi der
a bl e pressur e In e ac h c o m
pa rt men t i s a separ a t e hy dr ogen c on t a iner
e nt i s th ere fo re suc h th a t th e gas ca nn ot fl ow f ree ly
a n d th e a rra nge m
fro men d to en d of th e a i rshi p
Thi s grea tly r ed uces th e c h an ges of
de nsity d ue to i nc lina tion of th e axi s and so red u ces th e p it chi n g m o men t
Th e a rra n geme n t a ls o e ffec ti ve ly i n t erven es to p r even t su r gin g of th e
hy dr ogen whi c h m
ight inc r ease th e p it c hin g m omen t s as a resu lt of th e
e ffec t s of i ne rti a of th e hy drogen
It may th e re fo re be c onclu ded tha t th e result of disp l ac ing a ir by
hy d rogen i s a fo rce ac tin g u p war d s a t th e cen t r e of th e v olume of th e
di sp l aced a i r a n d with s uit a bl e p reca utio ns in l arg e a i rshi p s n o oth er
c on se quence s are of pri mary i m rt ance
.
.
,
,
,
,
.
.
,
°
.
.
,
.
,
.
.
.
,
,
.
m
,
F oa cxs
ON
AN
m
mm
m
A
.
s x rp
r o rr s
M O TI ON
TH RO UGH TH E
m
A
Th e aero dy na ics of th e a irs hi p i s f u nda m
en t a lly m u c h si m
p l er th an
th a t of th e aero p l an e Thi s follows when on ce it i s app reci a t ed th a t th e
a ttitu de re l a ti v e t o th e win d d o e s n o t d ep en d o n th e s peed of th e a i rs hi p
Th e mo st i m
p or ta n t forces a r e t he drag whi c h v a ri es a s th e s qu a r e o f
th e Sp eed a n d t h e a i rscre w th rust whi ch a lso v ari es a s th e s qu ar e of th e
Speed s i n ce it c ou n t er b a l a n c es th e d r a g
A sec ondary c onse quence of th e
vari atio n of th r u st a s th e s qu a r e of t h e speed i s th a t a t a ll sp eeds th e
ay b e wo r ki n g i n t h e c o n d ition of m
a i rs c r ew m
axim
u meffi c i en cy a st a t e
whi ch was n ot p o ssibl e i n th e aero p l an e for an a i rscrew of fi xed shape
ay b e obt a ined f r o m a n a i rshi p e n velo p e
It i s t ru e th a t dy n am
i c lift m
bu t thi s h as not th e sa m e s ig ni ficance a s i n th e c as e of th e ae ro p l ane si nce
h eight ca n b e gai ned apar t fro mth e p ow er of th e engine Th e n u m b er
of exp eri m
e n t s fro m
whi c h observa tions of dra g for a i rshi p s can b e ded u ce d
with acc u racy i s ve ry sm all a n d th e fi gur es n e w quot ed ar e b ased o n full
s ca l e o b s erva tio n s an d Sp eed a tt ai ned
together with a c ert ai n am oun t
of analy si s b a sed on m o del s of a i rshi p s both fully rigged an d p arti a lly
rigg e d
Th e two i llust ra tion s c hosen c o rresp ond with th e n on rigi d an d rigi d
a i rshi p s shown i n Figs 7—
9 Ch ap t e r I
Th e N S ty pe of n on rig i d
a i rshi p h as a l ength of 262 fee t and a m
u m wi d th of 57 fee t
a xi m
lbs an d t h e result of t h e a n a l y si s of fl ight t es t s
Th e gross lift i s
sho ws t h a t t h e d ra g i n p oun d s i s a pp r o x i m
Th e d r u g i s
a t ely
mad e u p i n this i nstance in the propo rti ons of 40 per cen t for t h e envelope
.
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-
-
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,
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,
THE
3 5per
PRINCIP LES
F L I GH T
OF
65
cen t fo r th e car and riggi ng ca bl es and 25per cen t fo r the verti c a l
and hor iz on t a l fin s ru dder and e l eva to rs Th e ho rsep ower necessary to
pro pe l th e air shi p depends o n th e e ffi c i ency of th e a irscre w
th e rel a tion
b e in g
3 75 7) B H P
(20)
,
.
.
.
,
,
.
.
.
It h as a l ready b ee n men tio n ed tha t th e a i rscre w if c o rrec tly designed
woul d a lw ay s b e wor kin g a t i t s maxi m um efii ci en cy a t a ll Speed s an d a
reaso na bl e valu e fo r th e e ffic i e ncy i s 0 75 At m ax i m
u m p ower th e two
engines of th e N 8 ty pe of a i rshi p deve lo p 520 B H P an d f ro m e q u a tion
Sp eed of th e a i rshi p i s
r eadi ly fo u nd th a t th e maxi m
m
it
th
en
u
i
s
2
0
( )
p h Th e dr ag a t thi s speed i s 2500 lbs
57 5m
F or a l ar g e ri gi d a i rshi p 698 f ee t i n l ength and with an e n velo pe 79 f ee t
a
n
d
et er th e d r ag i n lbs was 1
th e gross lif t
lbs
i n di am
1
T h e drag of th e envelope was a bout 60 per cen t of th e tot al with ca rs and
rig gin g acc ountin g fo r 8 0 per cen t and fins a nd co n t rol su rfaces for 10 per
ce n t I t will be n oti ced th a t th e envelope of th e rigi d a i rshi p h as a grea t e r
n a t e r es is t ance th an th a t of th e n on ri gi d
thi
l
a rge ly
i
o
a
n
d
s
i
s
r
t
o
r
O
p p
acc o u n t ed fo r by th e sm
a ll er r e la ti ve size of th e cars an d r iggin g in th e
.
.
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,
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,
L,
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,
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-
,
T h e rel a tio n b e tween ho rsep ower
is
an d
3
1 25V
‘
m
an d S
peed
.
a si m
il ar fo rm to
B HP
1
3 75
)
,11
h as
2
1
( )
.
.
W ith e ngin es develOp i ng 1800B H P an d an a i rscre w e ffi ci en cy of
e qu a
tio n (21) shows a ma xi m um speed of 74 m p h Th e dra g i s th en 6800 lbs
A co n ven i en t fo rm ula whi c h i s fre qu en tly used to exp r e ss th e r esi st ance
of a i rshi p s i s
2
R es i st ance in lbs
C p V
22
( )
.
.
.
.
.
.
.
.
wh er e C i s a c onst an t definin g th e qu a lity of th e a i rshi p fo r drag Th e
adva n t a g e of th e for m
ul a i s th a t C d oe s n ot depen d on t h e s i ze of t h e
ai rshi p o r i t s ve loc ity o r o n th e density of th e ai r but i s d ir ec tly a ffec t ed
by c h an ges of ex t erna l form I n th e fo rm ul a p i s th e w eight i n p oun d s
of a c u bi c foot of air di vi ded by g in fee t per sec per s ec V i s t h e
ve loc ity of th e a i rshi p in fee t per sec and v ol i s th e v olu m
e in c ubi c
f ee t of th e a ir di sp l ac ed by th e enve lope Fo r th e n on ri gi d ai rshi p a bo v e
th e va lu e of C i s 003 an d fo r th e rigi d a i rs hi p C 0016
o t io n
Longi tu di nal B alan ce of an Ai rsh ip — Fo r an a i rshi p n ot i n m
b a la n ce i s obt ai ned by s uit a bl e adj u stm
en t of th e p osition s of th e wei gh t s
carri ed A cer t a in a m
oun t of a lt e ra tion of t r i m can b e obt ain ed by
t ra ns ferrin g ai r fr o m o ne of th e b alloonet s of a n on rigi d a i rshi p to a no t h er
Fi g 9 Ch ap t er I show s th e p i pe s to th e two b alloon et s whi c h are a bout
O ne p oun d of air m o v ed fro m
th e fr o n t to th e rear p ro d u c es
120 f ee t apart
a cou p l e of 1201bs ft If th e ce nt re of buoyancy of th e hy d r ogen b e t a k en
as 10 f ee t a bo ve th e cen t re of gra vi t y a n d t h e w e i ght of t h e a i rshi p i s
lb s t h e c ou p l e n ecessary to d i s p l a ce t h e a i rshi p t h r ou gh o n e d egre e
en t of 3 5lbs of a i r f ro m
o v em
i s 4 200 lbs f ee t and woul d requi r e a m
on e
balloo n e t to th e oth er B y thi s me ans suffi c i en t adju st men t i s ava il a bl e
.
,
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.
.
,
”
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-
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,
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-
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,
,
.
-
.
-
.
.
,
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.
P
APP L I ED AERODYN AMIC S
66
fo r th e t ri mof th e a i rs hi p wh en n ot in m otion I n th e rigi d a i rshi p a
ilar co n t ro l can b e obta ined by th e m o vemen t Of wa t er b all as t f ro m
si m
p l ace to p l ac e
otion th e aero dyn a m
i c fo rces i n t ro d uce a ne w c o n di tion of
Wh en i n m
b al ance whi c h i s ma in t ai ned by m o vemen t of th e e l eva to rs Th e c ou p l es
d u e to m
o ve m
en t s of th e e l eva to rs are very m u c h grea t er th an tho s e
ari sin g fro m adj ust ment of th e a ir b e tween th e b alloonet s a rough fi gur e
2
fo r th e e l eva to rs of th e N S ty pe of a i rshi p b ein g ti l/ m l lbs feet per
At a speed of 40 m
degr ee of m o vemen t of th e e l eva to r
p h t h e c ou p l e du e
to on e de gree c h an ge of el eva to r p osition i s 8000 lbs feet an d so woul d
For a su ffi c i en tly l a rg e
tilt th e a i rshi p th rough an an gl e of a bout
m o vemen t of th e e l eva to rs c on si dera bl e i nc lina tio n of th e axi s of an ai r
shi p co ul d b e ma i n t a ined a t high spe ed s an d th e a i rshi p th e n h as an
apprec i a bl e dy n a i c lift Fo r th e N S ty pe of a i rshi p a bout 200 lbs of
dy na i c lift o r a bout 1 per cen t of th e gross lift i s obt a ined a t 4 0 m p h
fo r an i nc li na tio n of t h e ax i s of on e de gr e e
Th e various it em
s b ri efly tou c h ed o n i n c o nn ec tio n with lon gitu di n a l
b a l ance are mo re na turally deve loped in c o ns i der in g th e st a bility of
a ir sh i p s si nce it i s th e vari a tion fro mn o rma l c on dition s whi c h c o nstitut es
th e b asi s of st a bility and apar t fro m a t en dency to p itc h and y a w th e c on t rol
of an a i rshi p presen t s n o fundamen ta l di ffi c ulti es
.
-
.
.
m
,
.
.
w
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.
-
m
m
-
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,
,
.
EQU I L I B RI U M
K I TE B ALLO O NS
or
Th e c onditions fo r th e e quilib riu m of a kit e b a lloon are m ore c o mp l ex
th an those fo r th e a irshi p Th e k it e b a lloon h as i t s own buoy an cy whi c h
p o rtan t a t low wi n d speeds but u ni m
p o rt an t in hi gh wi nd s Th e
i s a ll i m
ae ro dy n am
i c fo rces of lift a n d drag an d of p it ch i n g m o m
en t are a ll o f
im
p or tanc e and i n ad dition th er e i s th e c o nst rain t of a kit e wi re It i s
n ow pre p osed to c ons i der i n de ta il th e e quilib riu mof th e two ty pes of k it e
b alloo n shown i n Fig 10 Ch ap t er I an d to exp l a in why o ne of th em i s
sa ti sfac to ry i n h igh wi nd s a n d th e ot he r un sa ti s fac to ry
of a kit e b alloo n i s shown m Fig 3 9 on whi c h ar e m
A d i a g ra m
ar ked
th e qu a n titi es use d i n ca l c ula tion Ax es of r efer en c e a r e t ak en to b e
ho ri z on t a l an d ve r ti cal with th e o rigi n a t th e cen t re of gr av ity I f
towed th e k it e b a lloo n woul d b e m o vi n g a lon g th e p o siti ve di rec tio n
of th e axi s of X whilst i n th e st a tion ary b a lloo n th e win d i s
blowin g a lo ng th e nega ti ve d i rec tio n of th e a xi s Th e axi s of Z i s
v er ti ca lly d o wn ward an d t h e p it c hing mo men t M i s p ositi ve wh en it t en ds
to r a i se th e n ose of th e ba lloon Th e kitin g e fi ect r esult s fr o m an i h
of th e axi s of th e k it e b a lloon to th e r el ati ve wi nd Th e
cli n a t i on a
buoy ancy d u e to hy dr ogen h as a resultan t F whi c h ac t s u p wards a t th e
cen t r e of v olu m
e of th e e nc lo sed ga s a p oi n t kn own a s th e c en t r e of
buoy an cy ( GB of Fig
Th e kit e wir e c o m
es to a p ull ey a t D whi c h
r un s freely i n a b ri d le a tt ach ed to th e b alloon a t th e p oin ts E a n d II
ov es i n a n elli p se of whi c h E an d H a r e t h e f oc i a n d fo r a
Th e p oi n t D m
c o ns i der a bl e ra nge of i n c li n a tion th e p o in t o f v i rtua l attac h m
en t i s a t A
t h e c en t r e of c u r va tu r e of t h e p ath of D
.
,
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,
PRI NCIP L ES OF F L I GHT
THE
67
By arra n gi n g th e riggin g d ifferen tly th e p oin t of a tt ac h men t c oul d b e
t rans f erre d t o B
T o e fi ec t thi s th e p ull ey a t D i s rem o ved th e poi n ts
E and H m
o v e d ne arer th e ax is of th e b a lloon th e wi res fro mthe m
mee ti n g
t he k it e w i r e a t B
Th e de ta ils of th e ca l c ula tion s follow t h e same r outin e
for a ll p oin t s of a ttac hm
en t and th e e ffec ts illu st ra t ed will b e those of
chan gi n g f r o mty pe Fig 10a to ty pe Fig 100 with a fi xed a tt ac h men t
an d tho s e d u e to c h a n gi ng th e p oi n t of a tt ac h me n t of ty pe Fig 110 f ro m
A to B o f F ig 8 9
Th e cc o rdinat es of th e p oin t of a tt ac h men t (o r v i r tu a l
p oin t of a t t ac h men t) of th e k it e wi r e are den ot ed by a an d c respec ti ve ly
for di s t an c es a lon g th e axi s of X an d Z
Th e l en gth of th e k it e b a lloo ns
co nsi dere d i n th es e pag es was a bout 80 fee t and th e m
axi m u m d i ame t er
27 f ee t
'
.
,
,
.
,
.
.
.
.
-
.
.
,
.
of
k i t b ll oon
39 —Eq i l i b ri u m
.
a
u
e
a
.
—
s
b
1
0
a
l
O
F
r
n
t
Fig
and
a
par
i
c
ul
ar
o
Fi
s
t
wi th h r ee
)
(
exa m
p l e of thi s ty pe th e weight of th e b a lloon st r u c tu re was 1500 lbs
Fo r
an d a t a h e ight of 2000 f ee t th e buoy a n c y fo rce F was 2085 lbs
va r i ou s angl es of in c lin a tio n of th e b a lloo n th e va lu es of th e l en gths a g
e t ry of th e b alloon Th e result s
an d f w e r e ca l c ul a t ed f ro m th e k n own g eo m
of t h e ca l c ul a tio ns are gi ve n i n T a bl e 21 b e low
a de and t e s t ed i n a win d c h ann el
A m o de l O f th e k it e b a lloo n w a s m
so th a t fo r various a ngl e s of i nc li n a tio n a th e va lu e s of th e lift dr a g and
ae r o dyn ami c p it c hi n g m o men t a bout th e cen t r e of grav ity w ere m
ea su red
Th e O b serva tio n s w ere c o n vert e d to th e full si ze by m ulti p lyin g by th e
en t s
s qu a r e of th e s ca l e fo r th e fo r ce s a n d by t h e c ub e of t h e sc ale for m o m
Ex t en sio n s of ob ser va tio ns to spe e ds high er th an tho se o f t h e wi n d c h ann el
en t i n p re p o r tio n to t h e s qu ar e
ad e by i ncrea si n g th e fo rc es a n d m
om
w ere m
o f th e w i n d Sp eed
.
.
.
,
.
.
,
,
,
,
.
.
.
APP L IED AERODYN AM I CS
68
F r om Fig 3 9 it will b e seen th a t th e c o mp onen ts Of th e t ens io n of th e
k it e wi re are very simp ly rel a t ed to th e lift and dra g of th e kit e b all oo n
Th e rel a tio ns are
T , drag
.
.
T2 = li ft + F
-
W
Th e tota l p it chi n g m omen t i s obt a ined by t ak in g mo men t s of th e for c es
a bout CG and add ing to th em th e c ou p l e fro m aer o dyn am
i c ca u se s
oth er th an lift and dr a g Th e r esult a n t m
o men t m u s t b e zero fo r an y
p ositio n of e qui lib rium an d h en c e
— T2a + Ff = 0
.
,
TAB LE 21
.
I ncli nat i on of th e axi s
of t h e ba lloo n to
h or izont a l
m
(Jo-or di na te s o f t he p osi ti o n of t h e p oi nt
‘u w h
fl t 01th e k it e wi re
mm
of
11
bet ween
.
di sta nce
ce nt re of
of
.
1
:
F W i s c onst an t a n d e qu a l to 585lbs T2 differs fro m th e lift
by a c o ns t an t am oun t an d in t a bul a tin g th e results of experi en t TI and
Th e va lu e of th e
T2 h ave b een u sed di r ec tly i ns t ead of dr a g a n d lift
a er o dy nami c mo men t a bout th e cen tr e of gr av ity t e M of e qua tion
n of T a bl e 22 fo r variou s wind Speed s whil s t
i s gi ven i n th e sec o n d c olum
th e valu e of th e whol e of th e l e ft h an d si de of (24) fo r variou s a ngl es of
i nc i dence and for a ran g e of Sp eed s i s shown in th e six th c olumn of th e
t a bl e F ro m an ex am
i n a tion of th e fi gur es i n c olumns ( 3) an d (4 ) it will
be seen th a t fo r th e same angl e of i nc i dence th e aero dyn ami c p it c hin g
mo m
en t and th e d ra g vary as th e s q
uar e of th e wi nd speed A si m il ar
resul t will b e foun d fo r T2 585
Equili b riumo cc u rs wh en th e fi gu res i n th e l as t c ol u mn of T a bl e 22
c h an g e sign and a n in sp ec tio n shows a progressi ve c h an ge of a ngl e of
for n o wi nd to a littl e m
i nc i dence f ro ma bout
o re th an 15 a t a wind
A p o siti ve m o men t t end s to p ut th e n ose of t h e b a lloo n
Speed of 80 m p h
u p an d so increase th e a ngl e of i nc i dence th e e ffec t b ei ng a t enden cy
towards th e p ositio n of e qui lib riu m
T h e fig u res for n o wi n d gi ve a measure of th e i mp o rt an ce of th e c ou p l es
d u e to r eserve buoy an cy an d by c ompari so n with those d u e to a com
bin a tio n
i c c ou p l es an d fo rc es a t 8 0 mp h it will b e
O f buoy a n cy a n d a ero d y n a m
real i sed th a t th e e q ui li briu m of a k it e b alloo n i n a hi gh wind depen ds
o s t w holly on th e aer o dy n a m
i c fo rces an d c ou p l e Thi s i s an illust r a
alm
ti on of a la w whi c h appe a rs on m
a ny occ asion s th a t efi ect s of buoy anc y
Since
.
m
,
,
.
,
.
.
,
-
.
.
.
,
°
.
.
.
,
.
,
.
.
'
,
.
.
APP L IED AERODYN AMICS
70
p h th e t en sio n i n th e k it e ca bl e h as b een i ncreas ed to m o r e
th an 14 ti mes i t s valu e fo r n o wi nd Had th e ri ggin g be en so arran g e d
th a t th e a ngl e of in c i dence fo r e quilib riu m w as
T a bl e 22 shows th a t
th e fo rce woul d h ave b een 80 per cen t grea t er th an a t
an d c o nvers e
a red u c tio n of t ensio n woul d h ave b ee n pro d u ced by riggin g th e k it e
b a lloo n so as to b e i n e qui lib riu m as a small er an gl e of inc i dence Th e
e ffec t of ch ang e of p osition of th e p oi n t of a tt ac h men t of th e k it e wi re wi l l
n ow b e disc u ssed
Th e aero dynami c p it c hi n g m o m
en t o n th e k it e b a lloo n i s seen f ro m
c olu m
n 8 of T a bl e 22 to t en d to ra i se t h e n o se of th e b a lloon a t a ll an gl e s o f
i nc i dence Th e c ou p l e d u e to buoy ancy depends o n th e p oi n t of a tt ach
men t of th e kit e wi re and th e n ose will t end to c o m
e d own as thi s p oi n t
At high speed s it h as b een seen th a t t h e
i s m o ved nearer th e n o se
buoy ancy c oup l es are u ni mp o r tan t i n th e i r e ffec t s on e quili b ri u m an d th a t
th e o nly va ri a tio n s of i m
p or t ance ar e those whi c h a ffec t th e c ou p l es d u e t o
th e t ension i n th e kit e wir e
Tza i s gr ea t er th an M a s may b e s een f ro m T a bl e 22 i t
Si nce Tl c
follows th a t to obt ai n e quilib riu m a t a l ow er an gl e of i nc i dence th e fo rm
er
qu an tity m us t b e i nc r eased Tl c — Tga i s th e m o men t of th e k it e wi r e
a bout th e cen t re of grav ity and can b e i nc r ea sed by m o vin g th e p oin t o f
a tt ac h men t fo rw ar d Ch an gin g th e verti ca l p ositio n i s m u c h l ess e ffec ti v e
si nce th e k it e wi re i s m o re nearly vert i ca l th a n ho ri z on t a l
Be fo re th e cal c ul a tion of e quilibr iu m can b e sa i d to b e c om
p l et e a n
exami na tio n of th e result an t fi gu r e t aken by th e riggin g will need to b e
made to e n su re th a t a ll c or d s are i n t en sio n I n re ference to Fig 8 9 it w ill
b e ob served th a t ED and H D wi ll b e i n t ension if th e l i ne of th e k it e wi re
p r o d uced f a ll s b etween th em A r unn i n g blo ck ensu re s thi s c on ditio n
but a j oi n t a t D might pro d u ce d iffe r en t r es ult s Th e vi r tu al p oi n t o i
a tt ac h men t woul d m
o ve to E or H if H D o r ED b ecame sl ack
Posi ti on of a Ki te Balloon relati ve t o th e Lower End of t h e K i te Wi re
W h en e qui lib riu m h as b een a tt a i ned th e p osition i n space of th e k it e b a ll oo n
i s de t erm ined by t h e l ength of k it e wi r e and i t s w e ight and by th e fo rc e s
o n th e b a lloo n Th e e qui lib r iu m of th e b a lloo n h as b een dea lt with an d
i t s co nnec tio n to th e k it e wir e i s fully de t ermi ned by th e t en sion s TI and T2
Th e wi n d fo rces o n th e wi re b e in g negligibl e t h e c u rve t aken by th e wi r e i s
a ca t enary and th e ho ri z o n t a l c o m
p onen t of t h e t en sion i n th e wi re i s
c o nst an t a t all p oin t s D e fine th e co o rd i na t es of th e u pper end of th e wi r e
rel a ti ve to th e lower end by Ean d g an d th e w e ight of th e wi re rcp e p er
un it l en gth by w Th e e qu a tio n of th e ca t enary i s th en
.
At 80 m
.
.
.
.
.
~
.
.
.
,
.
,
.
,
,
.
,
,
.
.
,
.
.
.
,
'
.
.
.
,
.
.
,
-
.
,
.
w
w
+ A)
t
cos
w
(25)
A
wh ere A i s a c onst an t so c ho sen th a t C 0 wh en E 0
th e d i st ances
are measu red f ro m th e lower end of th e k it e wi re : th e e qu a tion s fo r a
ca t enary ca n b e foun d in t ex t boo k s on el emen t ary cal c ulu s T h e l ength
of th e k it e wi re to an y p oi n t i s gi ven by
T,
w
w
ai nh
s
A
: A)
“
a
T1
t ;
:
,
-
.
g
}
THE
PRI NCIP L ES OF FL I GHT
71
and th e verti ca l c om
p onen t of th e t e nsi on i n th e wi r e i s
'
‘
As
mp l
t ak e t h e
an e xa
e
T,
88 0 1bs
mq
.
,
A)
1
q l i b i mp
e ui
r u
2300 1bs
T,
,
1
T, si n h ,— ( E
T2
.
os i t i o n at
S
,
2000 ft
.
40
,
.
mp h
.
.
.
0 15lb per ft
to
.
ru n .
.
Fro e u a t io n ( 27) and a t a bl e of h ype r bol ic s i nce t he v a l u e of 8
A i s d ed u ced
as 99 20 f eet
Us i ng bot h e u at i ons (26) a n d (27) t he va l u e of A i s foun d as 9 160 f eet ,
a nd h ence
760 f ee t
Us ing t h e v al u es of E A and A i n e u a t i on ( 25) s h ows t h at Z
1850 f eet
Th e k i t e b a lloon i s t h en 1850 f ee t u p an d 760 f ee t b ack fro t h e f oo t of t h e c a bl e
q
.
.
Had t h e ca ble
a nd
t he
q
m
.
.
u i t e s t r ai g h t i t s i nc l i na t i on t o t h e v ert ica l wou ld h av e bee n t a n
been q
h eigh t
”
t he
of
balloo n
For t h is
6
g
V Ti i Tfi
assu
b ase 715fee t
mp t i
on
t he
h eigh t
wou l
d
g
}
.
z
g
wou ld b e 2000
l
'
a nd
di s t ance b ac k
i ts
be 1870 f ee t
and
t h e d is
.
.
Fr o m th e a bo ve examp l e it may b e c onc lu ded th a t th e wi re ca bl e i s
n early s t raight and th a t a very si mp l e ca l c ul a tion su ffi ces fo r a m o dera t e
wind Si nce T a bl e 22 shows th a t th e ra tio of T1 to T2 d oes n ot ch ang e
m u c h a t high speed s it follows th a t t h e kit e b alloo n will b e blown b ack t o
a de finit e p osition a s th e result of light wi nd s but will th en m ain t ai n i t s
p ositio n as th e wi nd veloc ity i nc r ea ses
K ite B alloon wi th Lar ge Verti cal Fi n and Small Hori zon tal Fi ns ( Fig
— As th e ca l c ul a tions follow th e lin es al r ea dy i n d i ca t ed t h e resul t s will
b e gi ven with very littl e exp l an a ti on Th e obj ec t of th e cal c ul a tion s i s t o
d ra w a c o m
parison b e tw een t h e two fo rm s of k it e b a lloon an d t o sh ow t h e
d ifference d u e to fo rm of fi ns and p oin t of a t t ac h men t of t h e k it e wi re
In t h e ne w illust ra tio n th e b a lloo n will b e t ake n to h ave th e w eight
1500 lb s an d buoy an cy 208 5 lbs u sed fo r th e c a l c ul a tion s on t h e k i t e
e n t will b e t a k en
b a lloo n with th ree fi ns I n one ca se th e p oi n t of a tt ac h m
en t a t D whil st i n a
as A and will c o rr esp o n d with th e ru nn i ng a tt ac h m
Th e p oin t s A a n d B
sec o nd case an ac tu a l a tt a c h men t a t B will b e u sed
en
are m
arked o n Fig 39 an d co rr esp ond ing with th em i s th e t a bl e of di m
s io n s b e lo w
.
,
,
.
.
.
.
,
.
,
.
,
,
.
,
.
.
,
.
A
.
R u n ni ng at t ach
k i t e wi re
.
( d eg rees )
.
mt
en
of
B
.
Fi xed
at t ac h
k i t e w i re
.
mt
en
of
A
and
B
APPL IED AERODYN AMI CS
72
O nly th e valu es of p itc hin g m o m
en t an d t en sions i n th e wi re for a
se of
h
n
n
a
s
will
b
e
gi
ve
s
th
e
y
fo
r
th
e
pre
e
t
p
u
rp
o
e
s pee d of 4 0 m
s
u
c
p
illu st ra tin g th e li m
it a tion of th e ty pe
.
.
.
m
,
.
TAB LE 25
.
Tot al p i t ch i ng
(l
‘
-
mm t
o
en
.
m
B Fi xed
at t a ch
e nt
ft -l
.
.
n s of T a bl e 25will show th a t with
exami na tio n of th e l ast two c olum
en t of k it e wi re t h e angl e of e quilib riumi s
an d fo r
t h e r un n in g a tt ach m
th e fi xed a tt ac h m
en t a
B oth ang l e s are m u c h great er th an tho se
s ho wn i n T a bl e 22 fo r th e same wi n d speed an d a t high er Speed s th e r e sult s
would b e still l ess fav our a bl e to th e ty pe Th e p oin t of a tt a ch m
en t will
b e seen f ro m T a bl e 24 to h ave b een m o ved fo rward m or e th an 6 f eet
b etween p ositions A and B an d i s a l r eady i nc o nveni en tly p l aced without
h av i ng in t ro d uced suffi c i en t c orr ec tion It may th ere for e b e co nc lu ded
th a t th e ho riz o n t a l fins shown i n Fig 100 are wholly i nade qu at e fo r t h e
c o nt rol of a ki t e b alloo n i n a high wi nd
An
.
,
.
,
.
.
.
C HAP TER III
GENERAL D ESCRI P TI ON OF M ETH ODS OF M EAS UREMEN T I N A ERO
D YNAM I CS , A N D TH E P RI NCI PL ES UNDERL YI N G TH E USE OF
I NSTR UM EN TS AND SPECI AL APP ARATUS
m
m
An aon v n a cs as w e n ow kn ow it is a l m os t wh olly an exp eri en ta l
It i s proba bly n o exag gera tion to say th a t n ot a si n gl e ca se of
sci ence
flu i d m otio n rou nd an a i rcr a ft o r part is withi n th e r eac h of co mp u t a tion
Th e e ffect of fo rces act i n g o n rigi d bo di e s fo rms th e subj ect of d ynami cs ,
a tic a l sc i ence with whi c h aerona uti cs
an d i s a highly develo ped ma th e
i s in ti m a t e ly co n cerned
Su c h math ema ti ca l assis t an ce can how ever ,
on ly l ead to th e b est results if th e fo rc es ac ti n g are acc urat ely kn own , and
it i s t h e de termi na tion of th es e fo rces whi c h pr o vi des th e b asi c da t a o n
.
.
m
.
,
whi c h aer ona uti ca l kn owl ed g e rests Two ma i n me tho ds of a tt ack are
in c o mm
o n u se o ne of whi c h d eals with mea su rem
ent s o n a i rcra ft in flight
an d th e oth er with m o dels of a ircra ft i n an art ifi c ia l wi n d under la bo ra t o ry
Th e two lin es of i nvesti ga tion are re quired since th e p ossi
co n ditio ns
b ili t i es of experimen t i n th e a i r are li it ed to flyi n g cra ft and ar e un suit ed
to th e an aly si s of th e tot a l resist ance i n to th e part s d u e to wi n gs bo dy
un der carri a g e et c O n th e m o del si de th e c on t rol o ver t h e c ond itions of
experi en t i s very grea t and th e acc u racy a tt a ina bl e of a hi gh o rder
Th ere i s how ever an unce rt a in ty arisin g fro m th e small scal e whi c h
makes th e o r der of acc u racy of app li catio n to th e full sca l e l ess th an th a t
of th e meas ur emen t o n th e m o del Th e th e ory of th e u se of m ode ls is
of suffi ci en t i mport ance to w arran t a separa t e ch ap t er and th e genera l
cn t s
r es ult th ere reac h ed i s th a t with r easo n a bl e ca r e in mak in g th e exp eri
observa tio ns o n th e m o del sca l e may b e appli ed to a i rcra ft by i ncreasin g
t h e fo rces meas ured i n prep o rtio n to th e s quare of th e speed and th e squ ar e
of th e sca l e
T h e full develop m
en t of th e mean s of m
easuremen t woul d need many
c hap t ers of a boo k and will n ot b e a tt emp t ed Thi s c h ap ter a ims o nly a t
exp lai n in g t h e gen eral us e of i nstr u men t s and appara tu s an d th e preca utions
whi c h m ust b e observed i n app lyin g qui t e o rd inary in st rumen t s to experi
m en ta l work in a ircra ft As an exa m
p l e of th e need fo r care it wi ll be shown
tha t the co mm o n l evel used o n th e ground ceases to b eh ave as a l evel in
th e ai r although it h as a sufli ci ent va lu e as an ind i ca to r of si desli pp in g
fo r it to b e fitt ed to a ll aero p l anes
In very f e w of th e cases dea lt wi th are th e in st ru men t s sh own in
mec h ani ca l det a il but an a tt emp t h as b een ma de to gi ve su ci en t descri p
tio n to ena bl e th e th eo ry to be unders too d and t h e reco rds of th e ins t rumen t s
app r ecia t ed Th e par ti c ular me tho ds and appar a tu s describ ed are m ostly
B ritis h as pr od u ced fo r th e servi ce of th e Ai r Mi ni st ry but with mi n o r
.
,
,
m
.
,
,
,
m
,
.
.
,
,
,
‘
.
m
,
.
.
.
,
m
.
,
.
,
73
,
APP L IED AERODYN AM IC S
74
varia tions may b e t aken as represen t a tive of th e me tho ds and appara tus
of th e worl d s aero dy nami c l a bor ato ri es
A kn owl ed g e of th e Spee d a t whi c h
Th e Measu re ent of Ai r Vel oci ty
an ai rcra ft m o ves th rough th e a ir i s perh ap s of grea ter i mp o rta nce i n
understan din g wh a t is occ u rrin g th an an y oth er sin gl e qu antity I t s
measu remen t h as there fore rece i ved m u c h a tt en tio n and reac h ed a high
degree of acc u racy Fo r c o mp l et e a i rcra ft th e i n st ru men t s us ed can b e
ca lib rat ed by fl ight o ver m ea sured di st ances co rrec tion s fo r wi nd bein g
fou nd fr o m fl ights to and fr o in rap i d su ccession o ver t h e sam
e grou nd
Th e readi n g of th e i n st rumen t s i s foun d to depen d o n th e p o sition of c ert ai n
par ts rel a ti ve to th e a ir cra ft and in o rder to avoi d th e c om
p li ca tio n thus
int ro d u ced experi men t s will fi rs t b e describ ed under la bo ra to ry c onditions
All i n st ru men t s whi c h are used on ai rcra ft fo r mea su ri n g wi nd veloc ity
et ers
om
depend o n th e measu remen t of a d yna m
i c pres sur e
i a an e m
d i fference pro d uced in tub es h el d i n th e wind Th e small wi ndmill ty p e
of an em
o me t er u se d for m
any oth er p u rp oses h as pro perti es whi c h render
it u n suit a bl e fo r aero dyna m
i c experi m
en t s eith er in fli ght o r in th e l a bo ra
tory O ne fo rm of tub e anem omet er i s shown in Fig 40so far as i t s ess en ti a l
wo rk in g parts are i nv ol ved I t
c on sist s of an inn er tu be Open
a t o ne en d and fac in g th e a i r
c ur r en t ; the oth er end is con
h ee led to on e si de of a pressu r e
ga uge An out er tu b e i s fi xe d
c on cen t ri ca lly o ver th e i nn er o r
Pitot tub e and th e ann u lus is
em
om
ete
o pen to th e a i r a t a n u m b er of
40
T b
F m
sm
all hol es th e ann ulu s i s c o n
u set ed to th e oth er end of th e pressure g a ug e an d th e r ead i n g of t h e
gau ge i s th en a measu r e of th e Speed
Fo r th e tub e sho wn th e rel atio n b e tween pressu re and speed may b e
’
m
.
.
-
.
.
,
.
,
.
,
.
,
.
.
.
.
.
,
.
,
,
.
.
-
u
r.
e an
‘
,
.
vi ta
= 66 2Vfi
°
-
8
.
o
( 1)
wh ere v i s th e veloc ity of ai r in fee t per sec an d h is th e h ea d of w a t er i n
inc h es whic h i s r e qu ired to b al ance th e dynami c pressu re Th e rel a tion
pera t u re
shown i n ( 1) app li es a t a pre ssu r e of 760 mm of w a t er and a t e m
of
C th is h av i n g b een c ho sen as a st andard c on d ition fo r experimen t s
Fo r other pressu res and t empera tures
i n aero d ynami c l a bo ra to ri es
e qu a tio n ( 1) i s rep laced by
.
,
.
.
.
,
.
”th e
.
wh ere a is th e den sity of th e ai r rela ti ve to th e st andard c onditio n
all co rrec tion whi c h wi ll b e re ferr ed to sho rtly
Excep t fo r a very sm
th e fo rm ul a gi ven by (2) app li es t o va lu es of 0 u p to 8 00 ft s
Th e tub e an em
om
et er i llust ra t ed i n Fig 40 h as b een made th e
u a tion s
subj ec t of th e m o s t ac c ur a t e de t erm i n a tio n of th e co n st an t of eq
.
,
,
-
.
.
.
MET H OD S OF MEASUREMENT
( 1)
75
but th e exac t sh ape d o es n ot appear to b e of very grea t
an d
As a resu lt of
t h e i nn er tube i s
many experi men t s it may b e s t a t ed th a t th e pressure in
independen t of th e shape O f th e Openin g if th e t u b e h as
a l en gth of 20 o r 80 d i am
e te rs Th e ac tu al si z e may b e vari ed fr o m th e
a ll es t whi c h can be m
ade any o ne Or two hundred ths of an i nc h i n
sm
di ame ter u p to several in ch es
T h e ex t erna l tub e needs grea t er a tt en tion th e ta pered n ose sho wn i n
Fi g 40 ma y b e o mitt ed o r variou s sh apes Of sma ll c urv a ture su bstitut ed
T h e r in gs of small hol es shoul d c o me w e ll on th e paralle l part of th e tub e
e fi ve o r si x d i am
e t ers b ehi nd th e Pitot tub e Openi n g T h e d i amet er
a n d so m
o f th e hol es th emsel ves shoul d n ot excee d th r ee hundr ed th s of an i nc h i n
a tu be of 0 8 inc h di ame t er and th e n u m
b er of th em i s n ot very i m
p o rt an t Wh en dea li n g with m
easur em
en t s of fl u c tu a tin g veloc iti es it is
o ccas io na lly desi ra bl e to pre p or tion th e n u m b er of hol es to th e size of th e
ay b e t rans
O pen in g of th e Pitot tub e i n o rder th a t c h an g es Of pressur e m
mi t t ed to Opp osit e si des of th e gauge wi th e qual rap idi ty This can b e
ac hi eved by c o veri n g th e whol e of th e tub es by a flexibl e b a g to whi c h
rap i d c h an g es Of sh ape ar e gi ven by th e ti p s Of th e fin gers B y adj u st men t
of th e n u m b er of hol es th e e ffec t of th ese ch an ges o n th e pressu re gauge
can b e red uced to a very sm
all a m
oun t
T h e outsi de tub e shoul d h ave a sm ooth s urface with c l early c ut edges
all hol es but with or di nary skill ed wo rk sho p l a bour th e tubes ca n
t o t h e sm
b e r epea t ed so acc u ra t ely th a t calib ra tion i s u nn ec essary Th e in st ru m
en t
i s th er efor e very w ell adap t ed for a pri mary st andard
I ni ti al Deter mi nati on of th e Constan t of th e Pi tot Stati c Pressu r e Head
— Th e m os t c o mp l et e a bsolut e de t er mina tio n y e t m
ade i s th a t of Bra m
w e ll B ali and F age and i s describ ed in det a il in R ep o rts and Mem oranda
mittee for Aeronautics Th e an emomet er
NO 72 1912 of th e Advi sory Com
w as m oun t ed o n a whir lin g arm Of 30 fee t rad iu s rot a tin g in si de a buil d in g
easur ed fr o m th e rad i u s of
Th e spee d Of th e tub e o ver th e ground was m
th e t ube fro mth e axi s O f rot a tio n and th e sp eed O f th e rot ation of th e
arm Th e l a tt er c ould be ma in t ai n ed const an t fo r lon g perio ds so th a t
ti mi n g by sto p wa t c h gave very high percen t age ac c u racy Th e a i r in
th e buil di n g was how ever apprec ia bly d istur b ed by th e r ot a tion of th e
wh i rlin g arman d wh en st eady c onditions h ad b een re ac h ed th e vel oc ity
of th e anem o me t er thr ough th e a i r was on ly a bout 98 per cen t of th a t
o ve r th e ground A Spec i al windm
ill an em
ome t er was made fo r th e
eval u a tio n O f th e m o vemen t of a ir i n th e r oo m It c onsist ed of fo u r l arge
van es set a t 3 0 de grees to th e d i rec tio n of m otion and th e rot a tion of
th ese van es a bout a fi xed axis w as O bt ai ned by c oun t i n g th e signa ls i n
a t el ep hone recei ver d u e to c on tac t with merc ury c u p s a t eac h rot a tion
So me su c h devi ce was es sen ti al to succ ess as th e forces on th e vanes w er e
so s ma ll tha t o rd i nary me tho d s of mec h an i ca l geari n g in t ro d u ced en ough
fri c tio n to st ep th e vanes A vel oc ity Of one fee t per sec on d c ould b e
mea s ured with acc u rac y T o ca lib ra t e thi s v an e a n emo me t er it was
m oun te d o n th e whir lin g arm an d m o ved round th e buil d in g a t very low
an y erro r d u e to m otio n of a i r in th e roo m i s presen t i n su c h
Sp eed s
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-
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,
-
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'
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,
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,
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APP L IE D AERODY N AM I CS
76
cali b ra tion but as it i s a 7 per cen t co rrec tion o n a 7 per cen t di fi erence
be tween ai r speed and ground speed th e resi d u al erro r if negl ec t ed woul d
x
n ot exceed
r
cen
t
A
s
how
ever
th
e
per
cen
t
i
s
kn
ow
n
to
e
is t
e
7
p
th e actu al acc u rac y is v ery grea t if th e speed through th e air i s taken as
Th e o rder of acc u rac y
93 per cen t of tha t o ver th e fl oo r of th e buil d ing
arri ved a t was 2 o r 8 part s in 1000 o n all parts of th e measu remen t
To de t erm
i ne th e ai r m oti on m th e buil di n g d u e to th e rota t io n Of th e
whir li ng armth e tube anemo me ter was remo ved the vane anem o me t er
p laced su ccessi vely a t seven p oi nts o n i ts pa th an d th e Speed meas ured
Fo r th e mai n experim
en t th e tub e an em omet er was rep l aced a t t h e
end of th e arm and th e tub es to th e p r essure ga uge l ed a lon g th e ar t o
i t s cen t re and th ence thr ough a ro ta tin g seal m whi c h l eaka ge was preven t ed
by merc ury As a ch eck on th e c o nnec tin g p i pes th e experimen t was
r epea t ed with th e tub e c o nnec tio ns fro m th e
m
a
ug
e
to
th
e
ane
o
me
t
er
g
reversed a t th e whir li ng arm end Th e press u re di fference was mea su red
on a Ch at t ock tilt i n g ga uge des crib ed l a t er
Th e res ults of th e t ests ar e shown i n T a bl e 1 b elow
'
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-
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,
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,
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,
m
,
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B
TA LE 1
Sp eed ove r th e flo or
t h e b ui l di ng
( feet p er sec )
.
u
of
S peed of t he a i r o v e r th e
floor of t h e b u i ld i ng
S peed o f t u be ane
m
o
.
Co nnect i ng t u
bes re versed
.
10005
l
or neg ect i ng
t he
dou bt fu l rea di ng
09 99 7
Th e pressur e r ead in gs on th e ga uge w ere co nverte d i n to
_
h,
an d
29h
i
s
2
th e va lu e of
.
”
h ead of ai r
,
a di r ec t cal cul a tion fro m th e observa tio ns of
pressu re and ve loc ity I ts valu e 18 see n to b e un ity within th e accu rac y
‘
of th e experi men t s th e average valu e b e i ng l ess th an
per cen t di flerent
f ro m u nity
.
'
,
.
,0
.
APP L IED AERODYN AMIC S
78
o mpari so n of e qu a tions (8) a nd ( 11) b r in gs out th e i n t er est in g r es ult
th a t th e di ffer ence of pressur e be tw ee n th e two e n ds of th e tub e of th e
whi rli n g arm i s of th e same form as to depend ence on v e loc ity a t flyin g
spee ds as th e pressu re d ifferen ce i n a tub e anem o me t er of th e typ e
Th e veloc ity i n ( 11) is rel a ti ve to th e a i r whilst i n ( 8)
sho wn i n Fig 4 0
th e velo cit y i s re la t ed to th e fl oo r of th e buil d in g H a d th e a i r in th e
buil din g been still so th at th e two veloc iti es h ad b een e qu al th e d ifferences
of pres sur es in th e ane m
o met er and b e tween the en ds of a tub e of t h e
whi rlin g ar woul d h ave b een e qu a l to a high degr ee of app r ox ima tio n
One end of th e p r essu re ga ug e b ein g c o nnec t ed to th e ai r in a sh el t e r ed
par t of the buil di ng e qu a tio n (8) can b e us ed to e st i m
at e th e press ure i n
e i th er of t h e tub es of the a n em
om
et er Th e im
p o rt an t ob ser va tio n was
then made th a t th e ai r i nsi de th e ann ul ar space of th e tub e anemom
e t er
a t th e en d of th e armwas a t th e sam
e pressur e as th e air in a sheltered
p osition i n th e buil d in g T hi s i s a j us tifica tion fo r th e na m
e st ati c
p ressu re tub e sin ce th e p ressu re is tha t of th e st a tion ary a ir t h rough whi c h
th e tub e i s m ov in g Th e whol e p ressur e di fference d u e to veloc ity thr ough
th e air i s th en d u e t o dyn am
i c pressu re in th e Pitot tub e whi c h b rin gs th e
en tering ai r to res t A ma them
a ti ca l an aly sis of t h e p ressu re i n a st rea m
b rought to res t i s gi ven i n th e ch ap t er on dyn ami ca l si ila rity wh er e it
i s shown th a t th e in c r emen t of pressure as ca lc ul a t ed i s
A
c
.
.
,
.
,
m
.
,
.
.
"
,
.
,
m
.
,
l
3p
ed i u m an d th e seco n d
wh ere a i s th e veloc ity of sound i n th e undist u r b ed m
t erm of ( 13 ) i s th e s a ll c o rrec tion to equa tio n (2) whi c h was th ere re ferred
to At 800 ft s th e sec o nd t erm i s 16 per cen t of th e first and ( 13 ) is
th ere fore app li ca bl e with grea t acc ur acy
Th e princ i p les of dynam i ca l si mil arity (see Ch ap t er V III ) indica t e fo r
th e p r ess ur e a theo re ti ca l re la tionshi p of t h e form
m
.
,
-
-
.
.
,
,
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.
1
51
v
whi c h c o n ta ins t h e k inem
a ti c v isc o sity v n ot hith erto dea l t with an d l
whi c h defies th e size of th e tub es an d is c onst an t for any on e a n em
o me t er
v
Th e f u nctio n may in genera l h ave any fo rm but i t s depend ence on i n
,
,
,
,
.
5
thi s i nst ance h as b een shown i n e qu a tio n
Th e experim
en t s o n th e
whirli n g arm h ave shown th a t th e depen dence of th e func tion on v i sc o sity
o v er th e r an ge of Speeds p ossibl e was negligibly small Th e li m it of range
o ver whi c h (13 ) h as b een experi ment ally j u sti fi ed i n a ir i s lim
it ed to 50 ft s
p o rt ance in th e th eo ry
It i s n ot how ever t h e Speed whi c h i s o f gre at est i m
,
.
.
of th e i nst ru m
en t but th e q u an tity
,
m
3
1
If
thi s
can
-
.
b e ex te n de d by any
v ali d ity of ( 18) can b e c h eck ed to a hi gh er st age an d th e ex
om
t ens ion can b e ac hi ev e d by m
o v in g th e tub e a nem
et er th rough st i ll
es l ess th an th a t of a i r
w a t e r wh ic h h as a k in e m
a ti c v isc osity 12 o r 18 ti m
a tio n a s a v elo c i ty
A veloc i t y of 20 ft s thr ough w a t er gi ves as m u c h i n fo r m
e an s
t he
,
.
.
-
.
MET H OD S OF MEASUR EME N T
79
through ai r and th e experimen t was made a t th e W i lli a m
Frou de
Nationa l T ank a t T ed di n gton
Th e anem
om
et er was n ot of exac tly th e
sa e pa ttern as th a t shown i n Fi g 40 but d i ffered f ro m it i n m i n o r p a r t i cu
lars and h as a slightly differen t co n stan t
The res ul ts of th e expe r imen ts ar e sho wn in T a bl e 2
of25
0ft
-
.
s
.
,
m
.
.
,
.
.
TAB LE 2
.
sw a
m
.
p ar ses )
qi
E
u v
.
w
gggg fi j
a
3”
V 3
a alr
r
or
Wa te r
2 h
5
5
The va lu es of
s
hown
in
m
th e last c ol u n vary a h t t le a bo ve
be low 09 9 and th e ta bl e may b e t aken as j ustifi ca tio n fo r th e u se of
eq uation ( 13) u p to 300 ft s Th e di fference be tween 09 9 and 100 may
fairly be a tt ribut ed to c h an ges of fo rm of th e tub e anemo met er fro mth a t
shown i n Fi
s
I
n
th
e
a
e
of
w
a
t
er
th
e
ve
lo
c
ity
of
ou
nd
n
ly
c
s
i
s
4
e
ar
0
g
5
000 ft per sec and th e seco n d t erm of ( 13) i s c o mp l e t e ly negligibl e
Fro T a bl e 2 it may thus b e ded uced th a t th e c onst an t of e qu a tio n ( 1) i s
independen t of v u p to th e high est speed s a tt a ined by ai rcra ft
Effect of Incli nati on of a Tu be Ane o meter o n i ts Readi ng s — I t wo u l d
have b een an tic i pa ted fro m th e acc ur acy of calib ratio n a tta in ed th a t th e
pressure d ifference b e tw een t h e i nner and outer tub es i s n ot ex t remely
sensiti ve to th e se ttin g of th e tub es a lo n g th e wi n d
At in c li nat io n s of
an d 15 th e e rro rs of th e tube an emo me t er illust ra t ed i n Fig 40 are
1 per cen t 2 5 per cen t an d
per cen t of th e v eloc ity an d ten d to
over es t i ma tio n if n ot a llow ed fo r
and
,
-
.
.
m
.
.
.
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,
.
m
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,
.
°
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.
,
.
.
-
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,
APP LI ED AERODYNAMI CS
80
on an Aerop lan e — An em
o met ers of t h e
general ty pe describ ed in th e p r ec ed in g pages ar e u se d on aerop l an es
and a irshi p s In th e aerOplane th e tub es are fi xed on an i nt erplan e
st ru t a bout t wo thi rd s of th e w ay u p and with th e O penin g of th e Pito t
o ne foot i n fron t of th e st ru t Th e p ositio n so c hos en i s c o nveni en t since
it av oi ds damage d u r in g mo vemen ts of an a erOplan e in i t s sh ed but i s
n ot su ffic i en tly far a h ead of th e aerOplan e as to b e free fr o m th e d istur b anc e
of th e wi n gs Although th e anemo m
e ter co rrec tly i ndi ca t es th e v eloc ity
of ai r in i ts n eighbou rh oo d it d o es n ot re gis ter th e motion of th e aero p l ane
rela tive t o un dis tur bed air Th e e ffec t of d is tu r b ance i s estimat ed fo r
eac h aerOplane by flights o ver a mark ed ground c ourse an d Fig 41 ill u s
t ra t es a ty p i ca l result T h e ai r i m
m ed i a te ly i n fron t of th e aerOplane i s
p us hed fo rwar d with a Speed varyin g fro m2 p er cent of th e ae ro plane
0 m p h to 7 per cen t a t 40 m p h
speed a t 10
How i s this c o rrec tion to b e app li ed 2 Do es it depend o n t ru e Sp eed
o r o n th e indi ca to r rea di n g ? In o r der to an swer th es e qu estio ns it i s
Use
of
Tube An em o meter
.
.
-
,
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,
,
.
.
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,
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.
.
.
.
.
.
.
.
‘
Fro 4 1
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.
necessary to an ti ci pat e th e result of th e ana lysis i n l a t er c hap ters Th e
p ressure gaug e insi de t h e a erOplan e co ckp it indi cat es a qu an t i ty whi c h
may b e very di fferen t fro m th e t ru e Speed th e quan tity ac tually meas ured
Allowin g for th e in terfer en ce of
b e in g of th e ty pe shown in e qu a tion
th e aero p l a ne it i s foun d th a t th e readi n g depends o n th e den sity of th e ai r
th e Speed of th e ai rcra ft and i t s inc li n a tio n Th e i nclin a tio n of th e aero
.
,
,
,
.
plan e
i s fi xed wh en
£
i
s
3
k no wn
,
w
b ei ng th e lo ad ing of th e win gs i n lbs
.
qu ar e foot 0 the rela tive density of th e a i r a n d v the t ru e veloc ity
Th e qu an tity a t e occ urs oft en and i s ca ll ed in d i c a t ed a i r Speed o r so me
ti m
a ir Speed
Fo r aero p l anes design ed fo r a lo n g j ou rney d u ri n g
es
i on of pe t rol an d oi l i s an a pp r ec ia bl e p r ep or tio n of
whi c h th e consu
r s
e
p
,
,
.
m
”
m
.
th e tot al w eight th e co r rec tion
houl d b e app li ed to
s
v
.
Fo r an
aero p l ane whi ch fl ies with i t s tot a l w eight sensibly c onst ant it will b e
seen tha t w i s co ns t an t a n d th a t th e in c li na tio n of th e a er o p l an e i s de
2
in ed by 00 a n d it i s to this qu an tity th ere fo re t h a t th e ca lib r a tio n
t erm
c orrec tions for p o sitio n shoul d app ly
,
,
.
MET H OD S OF MEASU R EMEN T
to ear th f r o m this altitu de of
f eet occ u p i ed three qua rt ers of an
hour Th e l a g of th e b ar om et er i s show n a t th e end of th e descen t and
c o rresp o nds with an e rro r in h eight of 200 o r 800 fee t o r a bout 1 per cen t
of the maxi m u m h eight to whi c h th e aerOplane h ad cli bedx
Revoluti on I ndi cator s and Coun ters —Mo t o r car prac ti ce h as l ed to
th e in t r o d u c tion of rev olution i nd i ca to rs an d these h ave b e en ad o p t ed
in th e aero p lane Many i nst rum
en ts depen d fo r th ei r o p eratio n o n t h e
t endency of a bo dy to fly out under th e infl u ence of a cen t rifugal accel era
tion th e r ot atin g bo dy b ei ng a rin g hi n ged to a sha ft so as to h av e r el a ti ve
motion r ound a di amet er of the ri ng Th e rin g i s cons t ra ined to the shaft
by a sprin g th e a m
oun t of d istortion of whi c h i s a m
easu re of sp eed of
rot a tion of th e sh a ft Various me tho d s of ca lib ra tio n of su c h in di ca to rs
a re i n u se and th e read in gs ar e u su a lly ve ry sa ti sfa c t o ry
Fo r th e m o st
accu ra t e exp erim t al wo rk th e i nd i ca tor i s use d t o keep th e Speed of rot a
tio n co nst an t whilst th e ac tu a l Speed i s obt ained fro m a rev olution coun t er
and a s te p wa t c h
Th e a ir speed ind i ca to r th e aneroi d b aro me t er an d th e rev olutio n
i ndi cato r are th e mo st i mp o rta n t in st r umen ts carri ed in an aero p l ane
both fro m th e p oi nt of vi ew of genera l uti lity and of acc ura te r eco rd of
pe rfo r a n ce Many oth er i ns t ru men ts ar e u sed fo r Spec i a l p urp oses an d
th ose of i mp o rt ance in aero dyn ami c s will b e desc rib ed
Acceler o etcr
Th e m ost sa tis fac to ry accel ero met er fo r u se on aero
p lanes is very si mp l e i n i t s m a in i dea and i s d u e to D r Searl e F R S
worki n g a t th e Roy a l Ai rcra ft Est a bli sh men t d uri n g th e war Th e essen ti a l
part of th e i nst rum
en t is illu st rat ed i n Fig 44 an d c on si st s of a qu ar t z fi b re
b en t to a semi c irc l e and rigi d ly a tt ach ed to a b as e block a t A an d B If
th e bl ock b e gi ven an acce l e ra tion n o rma l to t h e p l an e of t h e q u ar t z fi b re
th e force o n th e l a tt er c au ses a deflec tion of th e p oi nt 0 r el ati ve to A and
-
.
,
,
m
.
-
.
,
.
.
,
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,
.
m
,
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,
-
.
-
,
,
m
,
.
m
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.
-
,
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,
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,
.
5
i
n
3
{5
Fro 4 4
.
.
.
—Accel erom
e ter
.
B an d th e deflec tion i s a meas ure of th e m
agn itu de of th e accel era tion
B y th e p rov isio n of suit abl e illu m
in ation and l en ses an ima ge of th e p oin t
C i s th rown o n to a p hotograph i c filman d th e in st ru men t b ec om
es re
en t i s si m
cord ing Th e calib ra tio n of th e in st ru m
p l e : th e c omp l et ed
inst r u me n t i s h el d with th e p l ane of t h e fi b re vertic al an d t h e v ert ex th en
li es a t C as shown in Fig 44 (b) With th e p l ane ho ri z on tal th e film
reco r d
show s 01 fo r one p ositio n an d 02 fo r th e i nver t ed p ositio n th e di fferen ce s
CC, and CC, b ein g d u e to th e w eight of th e fi b re an d th ere fo re e qu al to
t h e d eflec tion s d u e to an accel era tio n of g ti e 3 22 feet p er sec per sec
.
,
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,
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APP L IED AER ODYNAMIC S
m
Th e st i fin ess of th e fi b re i s so grea t i n c o mpariso n with i t s a ss
tha t th e period of vib ra tion is ex tremely sho rt and th e air damp ing i s
ake th e m otio n dead heat As co mpar ed with th e m oti on s
suffi c i en t to
of an aero p l ane whi ch are to be regist ered th e m otio n of th e fib re i s
Fi g 4 5
so rap i d th a t th e i ns t ru men t a l err o rs d u e to l ag ma y b e ign o red
shows so me of th e resu lt s rec o rded th e accele r o me t er h av i n g been st rap p e d
to th e knee of th e p i lot o r passen ger d u rin g aeria l anaau vres i n a n
aero p l ane
In th e rec ords repro d u ced th e uni t h as been t aken as g t e 322 fee t
per sec per sec and in t h e m ock flight b etween two aero p lan es it may b e
2
n oti ced th a t fou r u n its o r nearly 13 0 ft s was reac h ed T h e i nt erp r e
t ati en of t h e reco rds follows read ily wh en once th e g eneral pri nc i p l e i s
appreci a t ed th a t acce l erations are those d u e to t h e a ir fo rces o n th e aer o
p lan e To see this la w co nsi der th e fi b re as illust ra te d in Fi g 44 (a) wh e n
h e l d i n an aero p l ane i n st ead y fli ght th e p lane of th e fi b re be i n g ho riz o n t a l
A lin e n o rma l to th is p lane is k n own as th e accelero me t er axi s and i n t h e
examp l e i s verti cal Si nce th e aero p l ane has n o acce l era tio n a t all t h e
fi b re will bend under th e a c tion of i t s w eigh t o n ly and regis ter g ; i n
th e a bsence of l i ft th e aero p l ane woul d fa ll with acce l era tion g and th e
rec o rd may th en b e regarded as a meas ure of th e u p w ard accel era tio n
whi ch wou l d b e pro d u ced by th e lift if w eight d i d n ot exi st If th e m o t io n
of th e aero p l ane b e ch an ged to th a t of verti ca l descen t a t i t s t ermi na l
velo c ity th e accel era tion i s aga i n zero and th e wei ght of th e fi b re d o es n ot
pro d uce any deflec tion Aga i n it i s seen th a t th e accel era tio n recorded
i s th a t d u e to th e ai r fo rce a lo n g th e a ccel ero me t er axi s and thi s th eo rem
can b e gen era lis ed fo r an y motio n wh a t ever T h e reco rd th en gi ves th e
ra tio of th e ai r force a lo ng th e acce l ero met er axi s to th e m
ass of th e
m
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m
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-
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,
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,
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aerOp lan e
.
Co nsi der
th e p i lot as an acce l ero met er by re aso n of a spring a tt ac h men t
to th e sea t Hi s acce l era tions are those of th e aero p lane and hi s app aren t
w e ight as esti a t ed fro m th e co mpression of th e Sprin g of th e sea t will b e
Wh en t h e accel ero me t er
shown by th e rec o rd of an ac ce l er o me t er
in di ca t es 9 h i s apparen t w eight i s e qu al to h i s rea l w e ight
At
fou r ti mes 9 h is appar en t w eight i s four tim
es hi s real w eight whi ls t
a t zero rea ding of th e accel erome t er th e apparen t w eight is n othi ng
Nega ti ve accel era tions i nd i ca t e th a t th e p ilot is th en h e l d i n h i s sea t by
h is b elt
ind shows th a t
Exami n in g th e rec o rd s with th e a bo ve remark s in
o sc ill a tions of th e e l eva to r may b e made whi c h red uce th e p ilot s apparen t
w eight to zero and an e rro r of j u d gmen t in a d i ve mi ght thr ow a p ilot
fro m hi s sea t unl ess sec urely st rapped in In a loo p th e t endency d urin g
th e grea ter part of th e an oeu vr e i s towards fi rmer sea ti n g Genera lly
th e firs t e ffec t occ u rs i n ge tt in g i n to a di ve and th e seco nd wh en ge ttin g
out It will be n oticed th a t in t h ree mi n ut es of m ock fi ght i ng t h e grea t
prep o nderance of accel era tion ten ded to fir sea ti n g and on o nly one
occasio n d i d th e appar e n t w e ight fa ll to ze r o
Levels —Th e ac tio n of a l eve l as u sed on th e groun d depends on t h e
property of fl ui ds to ge t as low as p ossibl e under th e ac tio n of gravity
.
m
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m
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m
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m
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MET H OD S OF MEASUREMEN T
Ve r t i c a l d i v i s io ns
ls s e cond s Ho ri zon t a l l i nes or
e ve
.
MO
.
FIG H T
—Accclero mt
Fro 45
.
C K
e e r record s .
mml
u
e s or
"
g
MET H OD S OF MEASUR EMEN T
wh ere 96 i s th e inc li na tio n of th e p lan e of sy mme tr y of th e aerop l ane to
th e verti ca l
F ro m ( 19) follows a w e ll kn o wn p r operty of th e cross l evel of an ae ro
p lane fo r if th e aero p l ane i s b anked so as n ot to b e si deslipp i n g th e cross
wi nd for ce is zero an d
f
.
-
-
,
,
a
9
th e an gl e of b an k of th e aerOplan e is e qual to 0 th e inc li na tion of a
pen d ulu m to th e verti cal To an obs erver in t h e aer op l ane th e fin al
pos itio n of th e pend u lumd urin g a c o rrec tly b an ked tu rn i s th e same as
if it h ad o rigi na lly b een fi xed t o i t s axis ins t ead of b ein g fr ee to rota t e
Th e devi a tio n of a cross level fro i t s zero p o sitio n i s th en an in di ca tio n
of si des li pp in g and n ot of inclin ation of t h e wi n gs of th e aero p l an e to th e
i
s.
,
.
m
-
.
T h ere is n o ins t ru m
en t in re gular u se wh ic h ena bl es a p ilot to m ain t ai n
an eve n kee l
In c l ea r w ea th er th e ho riz on i s u sed but spec i a l t ra in in g
B y a c o m bina tio n
i s necessary in o rder t o fly th rough thi ck b anks of fog
of ins t ru en ts t h is can b e ac hi eved as follows an aerOplane can only fl y
s t ra ight with i t s win gs lev el i f th e cross l eve l read s zero an d vi ce sen d
Th e c o pass is n ot a very sa tis fac to ry i nst rumen t wh en u sed a lone as it
is n ot sensiti ve to cert a i n c h an ges of dir ec tio n and may m
om
en ta rily gi ve
an e r ron eous indi ca tio n
It is there fo r e su pp l e m
en t ed by a tu rn indi ca to r
whi c h ay e ith er b e a gy rosco p i c t 0p o r any in st r u men t whi ch measu res
th e di fferen ce of veloc ity of th e wi n gs th rough th e ai r T his i ns t rumen t
i na t e ser ious t u rni n g erro rs a n d so pro d u ce a
mak es it p ossibl e to e lim
con di t i on i n whi c h th e co mpa ss is reli a bl e
St raight flyin g and a cross
l evel rea din g zero th en ens ur es an even kee l
Aerody na i c Turn I ndi cator —Ah ins t ru en t design ed an d made
by Sir Ho rac e D arwi n depends o n th e meas uremen t of th e di flerence of
velocity b e tw ee n th e ti p s of th e wi ngs of an aerOplane as th e res ult of
turni n g Th e th eo ry i s eas ily developed by an ex te nsion of equ a tio n
wh ere it was shown th a t th e d i fference of pressure d u e to cen t r ifugal fo rce
on th e co l u mn of ai r i n a ho riz on t a l r ot a tin g tub e was
.
,
m
m
.
'
-
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,
,
m
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,
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.
m
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m
'
.
2
( 1)
h
wh ere p w a s t h e a i r dens ity a th e veloc ity of th e out er end of the p i pe of
wh i c h th e i nner end was a t th e cen t re of rot a tion Th e differ ence of
p res sur e b e tween p oin t s a t di fferen t radii is th en seen to b e
327
ave 2
,
.
22
( )
wh ere v, i s th e ve loc ity of th e inn er end of th e tub e If an aero p l ane
h as a tub e of l en gth l st re t ch ed fr o m win g ti p to w in g ti p th e di fference
of th e veloc iti es of th e inn er and out er wi n gs is wl cos 95d u e to an an gula r
veloc ity on an d e qua tion (22) b ec om
es
.
,
,
8p
=
=pcw l cos ¢
2
( 3)
wh ere v i s th e veloc ity of th e aero p l ane and tfi i s t h e a ngl e of b an k Fo r
slow tu rn i n g cos s is nea rly un ity and th e pre ssure d iffer ence b e tw een
.
,
APP L IED AERODYNAM I CS
th e win g ti p s i s p rep ort i onal to t h e rat e of turni n g of th e aerop l ane To
thi s di fference of pressure woul d b e added th e c o m
po nen t of th e we ight
of th e ai r i n th e tube d u e to b an kin g w er e thi s l att er n ot eli i nat ed by th e
arrangement of th e app ara tus Th e tub e i s o pen a t i t s end s to th e a t m
o
Sp h ere thr ough st a ti c p r essu re tub es o n swi velli n g h eads and th e pressur e
d u e to b anki ng is th en c oun t erac t ed by th e di fference of p r essur es out si de
th e ends of th e tub e Tu rn i n g of th e aero p lane woul d pro d u ce a fl ow of
a ir fro m th e i nner to th e out er win g and th e preven tion of thi s flow by a
deli ca t e pressure ga uge gi ves th e mo vemen t whi c h indi ca t es tu rni n g
—
i
r
v
t
C
o
n
t
ro
l
l
e
A
i
r
s
i
c
a
t
o
r
Th e grea t c h an ges of apparen t
Ga y
d
p eed I nd
w eight whi c h may occ ur in an aero p la n e m
ake it necessary to exa m
in e
very carefully th e ac tion of ins tr um
en ts whi ch depend fo r th e i r n o rmal
pro perti es o n th e a tt rac tio n of grav ity In th e case of th e a ccel ero
me t er an d cross l evel th e res u lt h as been to fin d very d irec t and sim
ple
u s es i n an aero p l ane a lthough th es e w ere n ot ob v ious ly c onnec t ed with
.
m
.
,
.
,
.
-
.
.
-
,
FM
Fro 4 7
.
.
—Th e acti on of a g avi t y
r
“
ld
cont rol e
( 6)
ai r - sp ee
d i n dica tor
.
t erres t ri a l uses A Sp eci al use can b e found fo r a gravity co n t roll ed a ir speed
indi ca to r but th e o r di nary in st rumen t i s Sprin g c on t roll ed to av oi d th e
T h e co mp l e t e in st rumen t n ow under
spec i a l f ea tur e n e w re f err ed to
d iscu ssio n c on sists of an anemo m
et er of th e Pi tot and st a ti c p r essure tub es
ty pe with c onn ec tin g p i pes t o a U tub e in th e p ilot s c o ckp it Th e U tub e
i s shown di a gramma ti cally i n Fig 4 7 th e li m b s of th e gauge b ein g mar ked
fo r st a ti c pressu re an d Pito t c o nnec tions Wh en th e aero p l ane i s i n
m otio n th e di fferen ce of p ressu re arisi ng ae ro dynam ically i s b al an ced by
a h ea d of fl ui d th e magnitu de of this h ead It b ein g de t ermined fo r a gi ven
aero dyna i c p r essure by th e apparen t w eight of th e fl ui d T h e two tub es
of th e gauge may b e made c oncen t ri c so as to av oi d erro rs d u e to t ilt o r
s i de w ay s acce l er a tio n and th e ca l c ul a tio n s n ow pr e p osed will t ake advan t a g e
of th e add itional simp li c ity of princi p l e resultin g fr o m th e u se of concen t ri c
tub es
T h e rel a tio n b e tween th e aero dynami c pressure and th e h ead It can
b e wr itt en as
k )?
h PA? 008 91 +1
)
wh ere k i s th e c o nst an t of th e Pitot and st a ti c pressure co m b i na tio ns as
-
.
,
.
’
-
-
.
.
,
.
m
,
.
-
,
.
m
MET H OD S OF MEASUR EMEN T
f c t ed by inc lina tion of th e aero p lan e p is th e a ir dens ity and v th e veloc ity
of th e aero p l ane O n th e oth er si de of t h e e qua tion h is th e h ea d of
fl ui d p th e weight of un it v olu me of t h e fl ui d as o rdi narily obt a ined 0,
th e i ncli na tio n of th e gau ge to th e verti cal and f th e u p war d accel era tio n
of th e ga uge glass alon g i t s own axi s In st eady flight f i s zero and cos 0,
so near ly e qu a l to uni ty tha t i t s varia tio ns
ay be ign o red
Eq
th en shows tha t h is pro p o rtion al to th e squ are of th e indi cat ed air speed
whi c h woul d b e regist ered by a sprin g c on t roll ed i ndi ca to r
T h e s pec i a l p ro per ty of th e gr avity c on t rolled a ir s pee d indi ca to r i s seen
by c o nsi deri ng un st eady m otion Fig 4 7 (b) shows th e necessary dia gram
from whi c h to est i ma t e th e va lu e of f T h e liqui d gau ge i s fi xed to th e
aer Oplah e with i t s ax i s a lo ng th e line AG and i t s i n c li na tio n to t h e verti ca l
wi ll depend on th e an gl e of c li m b 8 th e an gl e of inc i den ce a and th e
an gl e of se ttin g of th e i nst rumen t rela tive to th e c ho r d of th e win gs a o
Th e re la tio n m
ay b e
af e
,
,
.
,
,
“,
,
,
m
.
.
.
.
.
.
,
,
,
.
01 z : 0
ao
a.
m
m
a
n
d
t
a
e
r
T h e fo rces on th e aer op lane are i t s w eight
h
e
o
d
y
na
i
c
g
res ult an t R ac tin g a t an an gl e y + 90 to th e d i rec tio n of m otion It
then follows th a t
—
=
0
a
a
S
R
°
OO
08
+
1
l
(
o
v
M
f
,
,
°
.
m
9
an d
fo r h
-
003
—
a + a o)
0OS (Y
91 +f
c o m bini n g e qu a tions (24 ) and (27) giv es th e fun damen t al e qu a t ion
.
As
th e result of experimen ts o n
L
E
R
it is kn own tha t th e lift
aerOplan es
cos
y
-
z
m
k
‘
l
v
9
2
( )
S
wh ere k, is kn own as a lift c oe ffi c i en t an d depen ds only
inci dence of a gi ven win g and n ot o n i ts area S o r Speed v
can th en b e expressed as
.
m
k
cos
an gl e of
Equa tion (28)
on
t he
y
PwS
T h e fi rst fac t o r of this expres sion i s c onst an t whils t the seco nd is a
fu nc tio n o nly of th e an gl e of in c i dence if th e en gi ne and a i rsc rew are st ep p ed
If th e engine b e r unni n g th e st at emen t i s app ro xi ma t ely t ru e a small
erro r in lift b ei n g th en d u e to vari a tion of a i rscrew th rus t u nl ess t h e a ir
s cr e w sp eed b e kep t i n a de fin it e rel a tio n to th e fo rw ar d speed
Th e res u lt of th e an alysis i s to show th a t in un st eady fl ight as well as
i n s te ady fli ght th e readi n g of th e gravity c on t roll ed ai r speed ind ica to r
depends o n th e an gl e of in c i den ce of th e ae rop l ane and n ot o n th e Speed
Fo r all wi n gs th e qu an tity kn h as a grea t est valu e ; cos y an d
—
a + a ) are nea rly un ity fo r a co n si de r a bl e ran ge of an gl es and th e
cos ( y
o
ra tio re qu ired by (30) i s exactly unity wh en a o a Th e valu e of h th en
,
.
,
.
-
.
,
.
MET H OD S OF MEASUR EMEN T
ed a t th e Roy a l Air cra ft Est a blis hmen t A consi dera bl e n u m ber of
tubes is us ed eac h of whi c h co mm uni c at e s wi th a co mm o n reserv oi r a t one
end an d i s c o nn ec t ed a t t h e oth er to th e p o i n t a t whi c h press ure is to b e
m
eas u red In th e la t es t ins t rumen t th e tubes are arran ged round a h a lf
cylin der and are t hi rt y i n n um b er and th e whol e is enc losed i n a
tight bo x Behin d th e tu b es b ro m i de paper i s wound by hand and rest s
aga in s t t h e pressure gauge tub es ; exp osure i s made by swi tc hin g o n a
sma ll la m
p o n th e axi s of th e cylinder
A di agra
p repared fro mon e of the records t aken i n flight i s shown i n
Fig 4 8 wh i c h shows n ine teen tub es in us e Th e out si de tub es ar e c o nnec t ed
to t h e st a ti c press ure tub e of th e ai r s peed in di ca to r an d th e li ne j oi n in g
t h e t e ps of th e columns of fl u i d fur n ish es a da tum fr o m whi c h oth er pres
sures are m
Th e cen t ra l tub e m
arked P was c o m
monly con nec ted
eas u red
to t h e Pitot tube of th e ai r speed indi ca to r whi ls t th e tub es n u m bered 1 16
were co nnec t ed to hol es in on e of th e wi n g ribs of an ae ro p lane
T h e me tho d of experi men t i s sim
p l e : th e b ro mi de paper h avi n g b ee n
brought i nto p ositio n b ehind th e tub es th e aero pl a n e i s b rought to a st eady
st a te and ma in t a ined th ere for an apprec i a bl e time d u ri n g whi c h ti me th e
la mp i n th e c a mera i s swi t ch ed o n an d th e exp osur e made Th e p rep o rtions
of th e a ppa ra tus ar e su ffi c i en t to prod u ce damp in g and th e rec o rds are
cl ea r an d easily rea d to th e near es t one hun dr ed th of an inc h
Co ns i dera bl e u s e has b ee n made of th e i nstr u men t in de t er min in g th e
press ur es o n aereplan e wi n gs o n t a il p lanes and i n th e sli p st reams of
a i rscre ws
Ci ne a Camero —A me tho d of rec o rdi n g m o vemen t s of a i rcra ft h as
bee n deve lo ped a t th e Roy al Aircra ft Esta blishmen t by G T R Hill by
t h e a d a p t a tio n of a c in e m
a ca m
era Th e camera i s carri ed i n th e rear sea t
of a n aero p l ane and t h e filmi s dr i ven fro m a sm
a ll a u xilia ry wi ndmill
Thi s a ero p lane is fl own l eve l and st raight and th e camera is d i r ec t ed by
t h e o pe ra to r tow ard s th e aere plane whi c h i s ca r ryin g out ae r i a l man oeu vres
T h e p o ssibl e m otions of th e cam
era are rest ri c t ed to a rota tio n a bout a
ve r t i ca l a nd a ho ri z on ta l ax is and th e p osi t io n r el a t i ve to th e aero p la ne i s
rec o rded o n th e film Fr o mth e su ccessio n of p i c tu res so obt ained it is
p ossibl e to ded uce th e an gu l ar p os itio n in space of th e p u rsu i n g aereplan e
An aly t i ca lly th e p rocess i s la bo riou s but by th e u se of a glob e di v i ded i n to
a n gl es th e sp h eri cal g eo m
e t ry h as b een grea tly sim
p lifi ed and th e ca m
era
is a va lua bl e ins t ru m
en t fo r aero na uti ca l res earc h
—
f
A
o
i
i
s
A p i nhol e
er nlau e Os c ll a t on
Ca era or th e Recor di ng of
ca mera fi xed t o an aero p l an e and p oin t ed to th e sun pro vi d es a t race
of p i t c hin g o r rolli n g acco r din g to wh eth er th e aero p l ane is flyi ng t o o r
fro mth e su n o r with th e su n to one si de A m or e perfec t o p tical camera
for th e same p ur p o se h a s b een made an d used a t Martl esh am Hea th th e
p in hol e b ein g rep l aced by a cylindri ca l l ens an d a narr ow slit n o rmal to
th e line i ma ge of th e su n pro d u ced by th e l ens T h e reco rd i s t aken o n
a ro t a tin g filman d a goo d samp l e p hotograp h i s repro du ced in Fig 4 9
Th e osc i lla tion was tha t of p i t c hin g th e camera b ei n g i n th e rear sea t of
At a time ca ll ed
an aer o p l ane an d th e p ilot fl y in g a w ay fr o m th e s u n
in ut e o n th e fi gu r e th e p ilot p us h ed fo rw ard th e con t r ol co lu mn u n til
1m
us
.
,
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,
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m
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,
-
,
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-
-
,
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,
,
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,
-
.
,
m
.
n
.
.
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,
.
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,
,
.
,
.
.
,
,
m
.
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,
.
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,
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APP L IED AER ODYN AMI CS
th e aerOplan e was di ving a t an angl e of nearly 20 de grees to th e horizon ta l
a n d th en l e ft th e c o n t rol c olumn free
Th e aero p lane b ei n g st a bl e b egan
to d i ve l ess st eep ly an d p resen tly o vershot th e ho rizont al an d p ut i t s n ose
u p to a bout 11 de grees Th e osc illa tio n persist ed fo r thr ee c omp l et e
perio ds b e fo re be in g apprec i a bly d isto rt ed
by th e gustin ess of t h e a ir Th e pe rio d
was a bout 25 seco nds and suc h a r eco r d
is a gu ar an t ee of lon gitu d i n a l s ta bility
Fig 50 is a succ ession of recor d s of
th e p it c hin g of an aero p lane th e firs t of
whic h shows th e an gular mo v emen t s of
th e aero p lane wh en th e p ilot was keep in g
th e flight a s st eady a s h e w as a bl e Th e
ex t r eme devi a tion s fr o m th e mean are
a bout a de gree Th e seco n d r eco rd fol
lowed with th e aer op lan e l eft to co n t rol
itself an d th e fl uc tua tions ar e n ot of
greatly di fferen t a
i t u de to t h a t for
p ilot s co n t rol Th e periodi c ity i s how ever
more c l early marked in th e sec ond reco rd
Fro 4 9
St bi li t y record
and th e perio d is th a t n a tura l to th e aer o
p la n e Th e third recor d shows th e na tura l
perio d ; as th e res ult of p u t t i ng th e n ose of th e aero p lane u p th e reco rd
shows a w e ll da
ped osc i lla tion whi c h is repeat ed by th e reverse proce ss
of p uttin g th e n ose u p
Photograp hs of la t eral o sc i lla tio ns h ave b een t aken but fo r various
reaso ns th e rec or d s are di fli cult to in t erpre t and m uc h mor e i s necessary
,
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,
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,
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,
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m
m
,
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,
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-
a
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.
-
m
,
.
,
,
m
M
Fro 5
0
.
.
u r ss
m
M
.
—
Oont rol r ecord
ur es.
.
before th e full a dva nta ges of th e i nst ru m
en t ar e developed as a mean s of
esti ma tin g l a t era l st a bility
Pu rposes
Sp eci al Modi fi cati ons of an AerOp lan e for Exp eri en
Fi g 51 shows one of th e m ost st ri k in g m o di fica tions eve r carri ed
out on an aero p l an e an d is du e t o th e Roy a l Ai rcra ft Est a bli sh men t
Th e body of a B E2 ty pe aerop l ane was ou t j ust b ehin d th e rear c ockp it
.
mm
.
.
,
.
,
METH OD S OF M EASUR EMENT
MET H OD S OF MEASUR EMENT
95
t ands in th e mi d dle of a l arge roo m b ein g ra i sed fro m th e floo r on a light
meta l fr a m
ewo r k Th e a irfl ow i s p ro d uced by a four bl ad ed ai rsc rew
dri ven by el ec t ro m oto r and th e ai rscre w i s situ at ed in a c on e in th e cen t re
of th e c h annel th e co n e gi v in g a grad u al t rans itio n fro m th e squ are fo rw ar d
section to th e c i r c ul ar s ec tio n a t th e a i rscre w
Th e m
oto r is fi xed to th e
far w a ll of th e buil d i ng a n d c o nnect s by a line of sh a fti n g to th e ai rsc r e w
The a i rs cre w i s design ed so th a t a i r i s d ra wn i n to t h e t rum
pe t m outh
e l e ft of Fig 52 pass es thr ough a cell of t h i n p l a tes
shown a t th e ex t r e m
to b rea k u p s m
a ll v o rt i ces and th ence to th e wo rk in g sec tion n ea r th e
O pen d oo r
Jus t b e fo re th e en d of th e squ ar e t runk i s a seco nd honey c om
b
i n a t e any sm
a ll t en denc y fo r th e twis t of th e a i r n ea r th e a i rscre w
to e lim
t o spread to th e wo rk in g sec tio n
Aft er pas si n g th rough th e a i rsc re w th e
ai r i s de li vered in to a d is t r ibuto r whi c h i s a bo x with si des so per fo ra t ed
th a t th e a ir is pass ed in to the roo m a t a un ifo rm low velo city T hi s part
s
,
-
.
-
,
,
.
.
.
,
,
.
.
,
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m
Ott o
an
-
Per centage change of Veloc i ty
F ro 5
3
.
of
ai r
he
flw
o
.
wind c ha nn el
-
Th c s te ad in ess
h as
an
of
.
Wm
o Ca n cu n
t he
ai r flow
.
w i t h ou t
i n wi n
di s t r i bu t or
.
d ch annels
im
p ort an t bearin g o n th e
st
.
ead iness of
the
.
Th e speed of th e m oto r is c ont roll ed fro m a p ositio n under th e wo rkin g
se c tio n wh e r e th e appa ra tus fo r m
easu rin g fo rce s an d th e wind vel oc ity
i s a ls o ins t a ll ed
O ver th e grea t er part of th e c r oss sect i on of th e ch ann e l th e a i rfl ow
i s s tr a ight and i t s veloc ity unifo rm within th e lim
its of 1 1 per cent Th e
rap i d ity of u se depends to a l arge ex t en t on th e m
a gnitu de of th e flu ct u a
tio ns of Sp eed with tim
e an d Fi gs 53 (a ) an d 53 (b) show th e am oun t of th ese
i n a p art i cu lar case wh en th e c h anne l wa s t est ed wi thout a d is t ributo r
an d with a goo d d i s t ributo r
Without the d ist ributor th e veloc ity ch an ged
by t 5 p er cent of i t s m
ean valu e a t very fre qu en t i n t erval s and as this
woul d m
ean ch an ges of fo rce of 1 10% on any m o del h el d in th e st ream it
wo ul d follow th a t th e b al ance r ea di n g woul d b e suffi ci en tly uns t ea d y to
be un sa tis fac to ry
W ith th e d ist ributo r th e fl u c tu a tion s of veloc ity rar ely
,
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-
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,
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,
,
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APP L IED AERODYNAMI C S
96
exceeded 1 05 per cen t o r one t en th of th e am oun t i n th e previou s
illust ra tion
A gr ea t am oun t of experi men t a l wo rk ha s b een carri ed out on th e design
of wind c h annels and th e rep o rt s of th e Advi so ry Co mmittee fo r Aer o
na uti cs c on tai n th e resu lts of th ese investiga tion s Although th e res ult s
of wi nd ch ann el experi men t s fo rm b asi c ma t eri a l fo r a boo k o n aer o
dy nami cs th e det a ils of th e appara tus itself are of seco n dary i mp o rt ance
and th e in t erest ed reader i s re ferred fo r fu rth er de ta ils to th e r ep o rt s
men tioned a bo ve 1
—
B
n
o
c
s
Th e re quir emen ts fo r a l a bo ra to ry b al an ce ar e
Aer dy na i c ala e
so vari ed an d n umerou s th a t n o sin gl e p i ece of appa ratu s i s su fli ci en t to
mee t th em and Spec i a l co n t ri van c es are con tin u a lly re quired to c epe wit h
ne w probl ems Some of th e arran gemen ts of grea t est u se will b e illu st ra t e d
di agramm
ati ca lly and a gai n fo r de tails readers will b e re ferred to th e
mittee fo r Aerona uti cs Ei ffel and oth ers
rep o rt s of th e Adv iso ry Com
Th e first ob serva tio ns of forces an d m o men t s whi c h are re quir ed ar e
thos e fo r st eady m otion through th e a ir and i n many of th e probl ems
plifica tio n of th e syst em of forces to b e meas u red
sy mme t ry in t r o d u ces si m
For an a irshi p th e i mp or t an t fo rce i s th e dra g whils t fo r th e aero p lan e
lift dra g and p it c hi n g mom
en t are meas ured Fo r th e la ter probl ems o f
co n t rol an d s ta bility l a t era l force y a wi ng and r ollin g cou p l es are requi re d
wh en th e a ircra ft i s n ot sy m m
e t ri ca lly situ a t ed in respec t to i ts d i rec tio n
of m otio n thr ough th e ai r
At a still la t er sta ge th e for ce s an d cou p l es d u e to an gu l ar ve loc iti es
b ecome i mp o rt an t and fo r light er th an a ir a i rcra ft it i s necessary to mea su r e
th e c han ges of fo rce d u e to acce l era tion and th e consequ en t uns tea d y
na ture of th e a i rfl ow T h e probl ems thu s presen ted can o nly be dea lt wit h
sa tis fac to rily a ft er m u c h experi en ce i n th e u se of l a bo ra to ry appara t u s
but th e ma i n lines of a ttac k will n ow b e outlin ed
en t of Th ree Forces and On e Cou p le
Standard B ala e for th e Measur em
—
P
o
f
r
Sy met y
for a B ody h avi ng a lan e
Th e d ia gram in Fi g 54 wi ll
illust ra t e th e arrangem
en t AB AE and AF are thr ee arms m utu a lly a t
right angl es fo rm in g a rigi d c o nst ru c tio n fr ee to rot a te in an y di rec tio n
a bout a p oin t su pp o rt a t A Th e arm AB proj ec ts u p wards thr ough th e
fl oo r of th e wind c hannel an d a t i t s u pper end carri es th e m o de l th e a i r
fo rces o n whi c h are to b e measur ed Down w ard s th e arm AB i s ex t ende d
to O and this lim b carri es a wei gh t Q whi c h i s adj us t a bl e so as to b alance
th e w eight of any m o del and gi ve th e re quired de gree of sensiti vity to
th e whol e by vari a tio n of th e d ist an ce of th e cen t re of grav ity below th e
p oin t of su pp o rt a t A T h e arm AB i s d i vi ded so th a t th e u pper par t
carryin g th e m o del can be rota t ed i n th e wind an d i t s an gl e of a tt ac k
vari ed this rot a tion t akes p l ac e out si de th e ch anne l
Th e arms AE an d AF are p ro v i ded with sca l e pans a t th e end and by
the vari a tio n of th e w eights in th e sca l e pan s th e arm AB can be kep t
verti cal fo r any ai r fo rces ac tin g Th e sy st em i s th ere fo re a n ull
metho d si n ce the measu remen ts are made wi thout any d istu r b ance of th e
p ositio n of th e m o de l
Momen t a bout th e verti cal axis AB i s m easu red by a be ll cran k l ever
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-
-
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m
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APP L IE D AERODYN AM ICS
98
valu es of th e w ei ghts i n t h e scal e pans at E an d F th en co ns titute
Th e arms AE and AF are i niti ally set t o be
z ero read i n gs of drag and lift
alon g and a t ri ght angl es to th e wind d irec t io n withi n one twen ti e th degree,
whils t th e ax is AB is verti cal to one part in 5000 Th e wind is n ow pro
du ced and a t a defin it e veloc ity th e w e igh t s i n th e scal e pans a t E a n d F
whi ch are needed fo r b al ance are rec o rded ; th e di fference fro mth e zero
valu es gi ves th e lift and dra g a t th e gi ven an gle of inc i dence Th e m o de l
i s th en r ota ted and th e w e ights a t E and F a ga in c h an ged and so o n fo r
a suffi c i en t ran ge of an gl e of inc i dence say —6 to
—
t
f
e
For thi s meas urem en t th e lock to th e armII I i s
r
P
r
ssura
Cen e o
rem o ved and th e v ert i cal axis co nst rai ned by b rin gin g th e cu p J in to
c on t ac t with C Th e w eights o n the scale pans a t E an d F are th en in
o per a ti ve and th e w eights in th e scal e pan a t I b ec o me ac ti ve Fo r th e
an gl es of i nc i den ce used fo r lift an d drag a ne w seri es of ob serva tio ns i s
made of w eight s i n th e sca l e pan a t I F rom th e three read in gs a t eac h
angl e of in c i den ce th e p os itio n of th e res ult an t fo rc e rela ti ve to th e axis
Th e mo del b e in g fi xed to th e arm AB th e axis of rota
AB is cal c ula t ed
tion rela tive to th e mo de l is found by observi ng two p oin t s whi c h d o n ot
m o ve as th e m o de l is rota t ed This is ac hi eved t o th e n earest hun dr e d th
of an inch and finally th e in tersec tion of th e res ultan t fo rce and th e c ho rd
of th e aerofoil Le th e cen t re of pressure is found by cal c ul a tion fr o m th e
Th e
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°
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,
Th e pro po rtion s ad o p ted fo r t h e su pport i ng sp ind l e are de t erm
in e d
partly by a desi re to keep i t s air resi st ance very low and partly by an e ffo rt
to appro ac h rigi dity Th e fo rm ad o p t ed a t th e N a tional Physi cal La bo ra
to ry is su fli cient ly flexibl e for co rrec tio n to b e necessa ry as a r esult of th e
deflec tio n of th e aer ofoil under air lo ad Al sot th e whol e deflec tio n
occ u rs as a res u lt of th e b endin g of th e sp i nd l e and as this is round th e
p lane of deflect io n con t a ins th e resultan t fo rce A littl e co ns i dera tio n will
th en show th at th e m o men t read in g (sca l e pan a t 1) is u na flect ed by
deflec tio n and th a t th e lift and dra g are equ ally aflect ed Th e co rrec tio ns
t o lift and drag are sma ll an d very easi ly app li ed wh ereas c o rrec tio ns fo r
th e aero dyna ic eflect s o f a sp i ndle al t hough sm
a ll are very difi cu lt to
app ly As a genera l rul e it may b e st a ted th a t co rrec tion s for me tho ds
of hol d in g ar e so di fli cu lt to app ly sa tisfac to rily wh en th ey arise fro m
aero d ynam
i c i n terference th a t th e l ay out of an experi m
en t is freq
u en tly
de t erm
i n ed by th e me tho d of su pp o rt whi c h pro d u ce s l east d istur b ance
of th e ai r c u rr ent Th e experi ence o n this p oi n t i s c o nsi dera bl e an d i s
growin g an d o nly i n preli m inary investiga tio n s i s it con si der ed suffi c ien t
to make t h e rough ob v ious c o rrec tions fo r th e resist ance of th e hol d in g
sp i nd l e
Exa p l e of Use o n a Ki te B alloon — Fo r th e sym e t ri cal p ositio n of
a k ite ba lloon th e proced u re fo r th e de termina tion of lift drag and m o men t
is exac tly as fo r t h e aer ofoil th e m ode l k it e b a lloon be in g p l ace d o n i t s
si de i n o rder t o get a p l ane of sy mme try parall e l to t h e p l ane EAF
Any
observer of t h e k ite b alloo n in th e Open will h ave n oti ced th at th e craft
Not o nly
swi n gs si deways i n a wind slowly and with a regu l ar period
h as it an angl e of i nc i dence o r p itc h bu t an an gl e of y a w and th e c o nd itio n
.
m
.
,
,
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'
’
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,
m
,
'
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,
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-
,
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,
m
m
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,
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,
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,
,
METH ODS
OF
M EASUR EMEN T
99
can b e represen t ed i n th e wind c hann el by m oun t i n g th e kite b alloon
m o del in i t s ordi nary p ositio n and th en rot at i n g th e armAB T here i s
n ot n ow a p lane of sym
me t ry parallel to B AF and t h e proced ure is som
c
wh a t m o di fied Th e mo del is t reat ed as fo r th e aerofoil so far a s th e tak i ng
of readi n gs on th e scal e pans E F and I i s co ncerned aft er whic h th e arm
IE i s l ocked and th e two st ee lyards b rought in to Opera tion fo r th e measu r e
men t of u p ward force
T h e r ead in gs are n ow repea t ed with th e m o del u p si de d own in o rder to
a llow fo r th e l ac k of sy mme t ry an d t h e ne w w eight s i n th e sca l e pan s
W ith th e ai d of Fig 55th e reason fo r thi s can
E and F ar e ob served
be made c l ear A wi ll be t aken as a p oin t i n th e mo del and also on th e
axis of AB and fr om A ar e drawn lines par allel to AE and AF Th e
en ts on th e m o del ca n b e expressed
co mp l e t e sy s te m of fo rces and m o m
by a dra g along E A a cross wi n d for ce
a lon g F A a lift along A B a rollin g
c ou p l e L a bout A E t endi n g to tur n
A F t ow ar ds A R a p itchi n g cou p l e M
t endi n g to turn A E tow ar ds A B an d
a y a wi n g c ou p l e N t endin g to tu rn A F
tow ar ds A E Now consi der th e mea
su rem
Th e
en t s made o n th e b a l ance
force A B was measured di rec tly on th e
two s teelya r ds whi lst th e cou p l e N was
de termi ned by th e wei ghin g a t I
D en ot in g th e w eighi n gs a t E and F
by B , and B e with d istin guishin g dash es
it will be seen th a t
'
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-
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,
,
'
'
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’
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’
,
'
'
'
'
,
'
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’
'
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,
an d
'
B2
= L + 1 cross wind fo rce
'
-
8
1
( )
8
2
( )
wh ere l i s th e l ength AA
N eith er
Fm
5
rea ding l ea ds to a di rec t mea su re of drag
o r c r o ss wind force Inver t th e m o de l
a bout th e dra g axis A E so tha t A F b ec o m
es A F an d A B b ec o mes
As th e rot a tio n h as t aken p l ace a bout th e win d d irec tion th e
AB
fo rce s an d cou p l es rel a tive to th e m o del h ave n ot b een ch an ged in an y
way and it will follow tha t th e dra g and rollin g m omen t are un ch an ged
Th e lift cross wind fo rce p it c hin g m o men t an d y awi n g m
om
en t h ave th e
em
a gnitu de as b e fo re but th eir di rec tion i s reversed r ela ti ve to th e
sa m
ba lance Ins t ead of e qua tions (3 1) and (32) th e re ar e th en two new
e qu a tion s
M + 1 drag
(33 )
l cross win d fo rce
L
( 34 )
'
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5
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-
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'
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'
"
'
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'
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'
-
It will th en b e seen fr o m a c om
bin a tio n of t he two se ts of r ead in gs t h at
3
'
RI
1
2!
APP L IED AERODYN AMICS
Ba
'
Bz
cross wind force
-
21
I
RB
3
”
2
2
2
T he result of th e experimen t i s a comp l e te det ermina tion of the fo rces an d
cou p l es o n a m o del of unsym et ri ca l attitu de and th e generali sa tio n to
any m o de l follows a t on ce
p l et e de t er i na tio n i s co rrect th e me tho d
Although th e prin c i p l e of c o m
e tho d of findi n g L an d
as described i s n ot sa ti sf ac to ry as an experi me n t a l
M although it i s c omp l e t ely sa tisfac to ry fo r dra g and cr oss wi n d fo rce
Th e reason for thi s i s that th e m o en t l x dr ag i s grea t c ompared with
M and a small percen t age erro r in it makes a l arge per cen t a ge er ro r in
M If how ever 2 be ade zer o eq u a tions (81) and (8 2) show th a t both
L and M can be mea sur ed d irec tly and various a rran gemen ts h ave been
made to e ffec t this No uni versally satisf ac to ry me tho d has b een evol ved
and th e m ore c om
p l ex p r obl e s are dea lt with by Spec i alis ed me tho d s
suita bl e fo r eac h case
Th e b al ance ill u st ra t ed di agra mati cally i n Fig 54 i s oft en used in
c o m bina tion with oth er devi ces suc h as a roof b alan ce and various speci al
arran g emen t s will n ow b e describ ed
Drag of an Ai rshi p EnveIOp a — Fo r a gi ven v olume th e a i rshi p enve1
0p e
is designed to h ave a mi n i m u m resi st ance and fo r a gi ven cr oss section of
mo de l th e resis t ance i s appreci a bly l ess th an 2 per cen t of th a t of a fla t
p l at e of th e Sa e ar ea p ut n o rma l to th e wi nd Fo r su fli ci en t permanence
of fo rm and ea se of co nst ru c tion mo de ls are ma de soli d and of woo d and
th e resist ance of a sp indl e of grea t en ough st ren gth and sti ffness is a very
l arge pro p o rtio n of th e resist ance of th e m o del F u rther th an thi s it i s
fou nd tha t suc h a Sp indl e afi ect s th e fl ow o ver th e m o de l env e10p e to a
serious ext en t an d in t ro d u c es a Sp u r io u s r es is t an ce u p to 25 per cen t of
th a t of th e envelope AS a c onse qu ence of th e di fli cu lt i es experi en ced a t
th e N atio nal Physi ca l La bora to ry a e tho d of roof suspens ion was devis ed
6
Th e m
o del i s h el d fro m t h e roof of th e wind
an d i s i llu st r a t ed i n Fi g 5
c h ann e l by a sin gl e wi re th e dis tur b an ce fro m whic h i s very sm
all and th e
dra g is t ran sferr ed to th e b al ance by a thi n ro d proj ec ti n g fr o m th e t ail
an d a tt ac h ed by a flexibl e join t to th e ve r ti ca l ar m Th e for ce i s meas u r ed
by w eights i n th e scal e p an as i n th e p rev ious c ase
T h e w eight of th e m
o del pro d uc es a grea t res to ri n g force in i ts pend ulu
ac tion but this is c oun t er ac t ed by making th e b a lan ce un sta bl e so th a t
Th e c o rrec tio n for deflec tio n of th e
suffi c i en t sen siti vity i s obt a ined
Fur t h er th e resist ance of th e
sp in dl e i s eas ily de t er i ned an d app li ed
ca n b e e sti
at ed fr o m st andard c urves as i t s valu e i s a
of t h e resist ance to b e measu red
The metho d h as n ow b een in u se fo r a con si dera bl e perio d and h as
d isp l aced all oth ers as an ultima t e mea ns of es tima tin g th e dra g of bo di es
of low resi stance
m
,
m
.
’
m
’
-
m
,
'
m
,
'
.
.
,
'
’
,
m
m
.
.
,
.
,
,
.
-
,
m
.
.
,
.
,
'
.
m
.
.
,
.
,
,
.
m
.
,
,
m
.
m
.
.
,
.
,
.
APP L IED AER ODYN AMIC S
102
with th e ho riz o n t al mo vemen t s of R it i s necessary to t ake accoun t of th e
inc lina tion of th e su ppo rt in g wi res Th e w eight added a t E meas ures th e
drag excep t for a sm
all co rrec tio n fo r th e i nc lina tio n of th e wir es RS ;
the w e ight added a t N m
easures th e cou p l e a bout R and this p o i n t can be
chosen reasona bly near to th e des ired p l ace without dis tur bing th e lay out
of th e experimen t Th e w eights added a t U and N
us u ally negligibl e Th e co rrec tions fo r
,
.
,
,
-
.
.
mt
57 — Mea s ur c
.
en s of
forces an d
l
cou p e o n a co
m
p l t md l
ee
o e
l
ae rop a ne.
ppara tus and incli na tion of wi re inv ol ve somewh a t l engthy fo rm ul a as
c om
pared with th e aerofoil metho d describ ed ear li er but presen t n o
fundamen ta l difli cu lt ies As an experi men t a l metho d th e proced ure
prese nt s e no rm
ou s advan t age s o ver any othe r and is b ein g m o re ex t en
i c s progresses
s i vely used a s th e sc i ence of aero dy n a m
—
ffi
I t will b e apprec i a t ed o nc e a tt en tio n i s
t
l
t
i
i
C
o
c
i
n
t
S
e
e
S ab y
a
,
.
,
.
.
fro m those on a st ationary aero p l ane
,
,
an d
tha t th e forces and m o men t o n
METH OD S OF MEASUR EMEN T
108
an aerop lane d u rin g a loo p depend appreci a bly on th e angu l ar veloc ity
T h e experimen t to b e described app li es mo re parti c u l ar ly to an aere plan e
for a reaso n gi ven l a t er
B y means of wir es o r any alt ern a ti ve me tho d an axis in th e wind c h anne l
is fi xed a bout whi c h th e aere plan e mo de l can rota t e and a rigi d a rmGFD
i
F
8
r
5
c
o
nnec
t
ed
to
th
e
m
de
l
i
b
ought
th
r
ough
th
e
fl
oo
r
of
th
e
o
s
g
(
)
ch anne l and en ds in a m i rr or a t D Th e angular p osition of th e mo de l a t
an y i nst an t i s th en sh own by th e p osition of th e i mag e of th e la m
p H on
th e scal e K th e ray h avin g been r eflec t ed fromthe m i rro r D Th e arm
GFD i s h el d to th e c h anne l by spri ngs EF and FG and i n th e a b senc e of
wind in th e ch annel will b rin g th e mo de l and th e image o n th e sca l e to a
.
.
,
,
.
.
.
,
,
as
a mean and by adj us t men t of th e m o men t of inerti a of th e
,
WlNO
mt
1h m
58 —
'‘
.
e
easu re
en
o f resi st an ce
deri vati ves
as re
qu i red
for t he t heory
of
s
o lon g th a t
t
em
th
e
ti
e
of
th
e
s
pr
i
n
g
th
e
per
io
d
can
b
e
made
f
f
n
ss
s
d
a
n
sy s
th e ext remes of su ccessi ve os cill ations can b e observed di rec tly on th e scal e
Th e mec h an i ca l arran gem en ts are su c h th at th e dam ng of th e oscil
l a tion in t h e a bs ence of wi nd is as sma ll as p ossibl e an d co nsi dera bl e
Wh e n
su ccess i n th e e li min ation of mec h an i ca l f ri c tion h as b een a tt ai ned
red u ced as m
u c h as p ossibl e th e res i d u a l d a m
p in g is m easu red and us ed
i
n
s
n
i
n
a
c
rrec
tio
I
n
th
e
de
cr
i
p
tio
n
to
follow
th
e
t
r
u
me
t
damp
g wi ll
s
n
o
as
m
.
,
.
.
T h e d ia gram in se t i n Fig 58 will Show why th e fo rc es an d m o m
en t s
o n th e mo de l depend on t h e osc illa tion A narrow fla t p l a t e i s p resu med
to be rot a t i n g a bout a p oin t 0 fr o m whi c h it is d is tan t by a d ist ance I
If th e an gular veloc ity b e q
th en th e veloc ity of th e p la t e n o rma l to th e
c u rren t wi ll b e lqand th e rel a ti ve win d will b e e qu al to lg and in th e o pp osit e
dir ectio n Co mp oundi ng this n o rmal veloc ity with th e wind speed V
.
.
,
,
.
.
APPL IED AER ODYN AMI CS
104
s
1
hows a wi nd a t an inclin a tio n a suc h th a t tan
3
0.
,
thi s will pro d u ce
an d
both a fo r ce and a c ou p l e O pp osin g th e an gular eloci t y If th e a ngl e is
s ma ll th e fo rce o n th e pla te will be p r o p o rtio n a l to th e an gl e an d a ls o to
th e squa re of t he Speed and h ence p ro p o rtiona l to th e pro d u c t of th e
an gular veloc ity and th e fo rw ar d speed
T h e e qu a tion of m otio n of th e m o del i n a wind may th en be expressed as
.
,
,
.
'
B 0+ p V9 + k0= 0
wh ere B is th e mo men t of inerti a pt a cons tan t depen din g o n t h e l engths l
an d area s of th e parts of th e m o del parti c ul a rly th e ta il k a cons t an t
depen din g on th e stiffness of th e Sprin g and 8 th e an gular deflec tion of
th e m o del T h e q
used ea r li er is e qual to 0
Th e solution of e qua tio n (8 9) can b e found in any t rea tise on d i fferen ti a l
equ ations and is
,
,
,
,
.
.
,
pV
k
2
“
l
Si n e
wh ere 00 is th e valu e of 0a t zero ti me and e is a co nst a nt gi vi ng th e p hase
a t zero time For th e presen t it is suffi c i en t to n ot e tha t equ ation (40)
rep resent s a damped o sc illa tion of th e kind i ll ust rat ed in Fig 58 At
e th e valu e of 0 is shown by th e p oin t A and i s a m
Th e
z er o ti m
a x im u m
other end of th e swi ng i s a t B and th e osc illation c on tin u es with decreasin g
amp litu de as th e time in c reases Th e c u rve h as two w ell kn own c h ar ac
t eris t i cs
th e tim
e fro m on e maxi m um to th e next i s a lways th e same as
p li tu des of su ccessi ve oscilla tions Th e c h a nge s of
i s th e r atio of th e a m
th e logarith ms of th e m
axi m um or di nat es are p r o p o rtio na l to t h e di ffer ence s
of th e t i m es a t whi c h th ey o cc ur and th e c ons t an t of p ro p o rtio nality i s
kn own as th e logarith m
i c decrement
In th e ex perimen t th e m
ea sur em
en t of th e logarith i c d ec rem
en t i s
f ac i lit a t ed by th e use of a logarith m
i c sc al e a t K Th e en d s of su ccess i ve
s wi n gs are ob served on t h i s sca l e a n d th e ob se rva tio ns a r e p lott ed a gai n s t
n um
ber of swin gs Th e slo pe of th e lin e so obt ain e d di v i ded by th e ti me
of a swi n g i s th e logarithmi c dec r em
en t requi red an d fr omequ a tio n (40)
i s equ a l to
T hi s expression shows tha t th e dam
p in g i s pro p o rtiona l
,
.
.
.
.
,
-
.
.
,
“
m
.
.
,
.
,
to th e wind Speed and th e experi m
e nt a l result s fully b ea r out t h e p r opert y
i nd i ca t ed
B e fo r e th e v a lu e of n can b e ded u ced it i s n ece ssa ry to d et er m
i ne t h e
m om
en t of in erti a B a nd this i s fac i lita t e d by th e fac t th a t i na ny p racti ca bl e
appara tu s th e va lu e of I: d o es not d epend a pp rec i a bly on th e w i n d fo rces
,
.
,
,
u hge t
m
{2
W ith th e i m
p li fic tio
perio di c tim
e 2
and tha t th e ra tio
mi c decremen t
.
is
very
es
c
s
r a er
a
th an
ns e
17
the
s
q u are of
t h e loga ri t h
qu atio n (40) shows th at
I
;
APPL I ED AERODYNAMI CS
106
top wa tc h th e c oun t er b e in g arran ged to t ransm
it Signa ls to a
outsi de th e c h annel In or der to keep th e speed st eady
it is usu a l to e m
p loy so me fo rm of el ec t ri c indi ca to r un der th e
con t rol of th e o perator of th e el ec t ro m
oto r regula ting switc h es T o r qu e
an d t h r u st are rare ly mea su red sim
ult an eous ly one o r oth er of th e b eams
AF o r AE b e in g look ed as r e q u i red
To make a measur emen t of thrus t
th e sca l e pan a t E is loaded by an ar bit rary am
oun t and t h e wind in th e
ch annel turned o n and set a t i t s re quired valu e Th e a i rscre w moto r i s
and
s
,
.
.
,
.
,
.
.
FI G ti g —Th e
.
.
m
easure
mt
en
of ai r scr ew
t hr us t
a nd
to r
qu e
.
en st art ed an d i t s revolution s i n creased u n til th e t h rus t
th e
weight in th e sca l e pan ; th e r ev olutions ar e kep t co nst an t fo r a su fli
c i en t tim
e to ena bl e readi n gs to b e t aken on a s to p wat c h T h e read in gs
a r e repea t ed fo r th e s ame wind Speed but oth e r lo ad s i n t h e sca l e p an
and fin ally th e scal e pan read in g fo r n o win d an d n o ai rsc r e w rota tio n i s
r ec o r ded
Af t er a su ffi c i en t n u m b er of ob serva tions a t one wind Spe ed th e rang e
m
ay b e ex t ended by t es ts a t oth er wind Speeds inc lu di n g zer o b e fo r e th e
b eam AB i s locked an d the to r qu e measur ed on AF T o r qu e readin gs are
obta in ed in an analogous manner to those of thrus t
th
,
.
,
-
.
,
,
.
.
107
METH OD S OF MEASUR EMEN T
It will be n oti ced tha t in th is experi men t th e i nfl u ence of th e bo dy on
thrust and to rqu e i s co rrec tly r epresen t ed In one ins t ance win gs and
un der ca rr i a ge were he l d in p lace i n the same way as th e bo dy
Th e resi s t ance of th e bo dy in th e a irscre w sli p st ream i s measu red by
re l easin g th e ti e wi res SL and TL an d c onn ec tin g L to the to p of th e b alance
M is di sconn ec t ed fro mthe b alance an d ti ed to th e fl oo r of th e c h ann e l so
as to fix th e m oto r
Fo r a gi ven wind Speed an d a n um ber of speeds of
rot a tio n of th e ai rscre w th e bo dy resis t ance is measured by w eights i n th e
It is found th a t th e in c rease i n the bo dy r esi stance i s
sca l e pan a t E
pro po rtional to the t h rus t o n th e a i rscrew and may b e very co ns i der a bl e
Th e efi ect of th e body on th e thrus t and to r qu e of th e ai rscre w i s re l a ti ve ly
s all
both e ffec ts ar e dea lt with mo r e f u lly in l a t er chap t ers
Th e appara tus is c o nveni en t and acc u ra t e in use and when it can b e
used h as su perseded oth e r ty pes in th e experi
en ts of th e N a tional
Physi ca l La bo ra to ry Fo r sm
all er m
o dels finali ty has n ot been reach ed
and all metho ds so f ar p ro p osed ofi er appreci a bl e d iffi c ulti es In thi s
connec tion th e pro vis ion of a l arge wind ch annel Opens u p a ne w fiel d of
acc ura t e exper i men t o n c o mp l e t e m o de ls i n th a t th e ai rscre w hith er to
o mitted can be represen t ed m i ts c o rrec t runni ng c onditi on
Measure ent of Wi nd Veloci ty and Local Pressur e —Th e pressu re
tube ill us t ra te d i n Fig 40 is us ed as a pri mary standard anemom
e t er
and d uri n g calib ra tio n of a sec o n dary ane o me ter is placed i n th e wind
ch annel i n th e p la ce n o rma lly occu p i ed by a m o de l This seco ndary
anem o me t er co ns is t s of a hol e i n th e si de of th e c h an n e l and th e di ffe r ence
be tw een th e pressur e a t this hol e and th e gen er a l p ressu re i n th e wind
c hanne l buildin g 1s pro po rti onal to th e squ are of th e speed Th e Speci a l
advan tag e of t h is sec on dary st andard i s th a t it a ll o ws fo r th e de termi na
tion of th e wind Speed without obs t ru c ti n g th e fl ow m th e ch annel and
only a person al co n t ac t with t h e subj ec t can im press a fu ll rea lis a tion of
th e e ffec t of th e wi nd shad ows fromsuc h a p i ece of appara tus as an
anem
o me ter tub e A very mar ked wi nd shad ow can b e observed 100
di ame te rs of th e tube a way
For l a bo ra to ry p u rp oses t h e p ressu r e d ifferen ces pro d u ced by both th e
pri mary and seco n dary anemo me ters are mea sured on a sens iti ve ga ug e
of th e sp eci a l type illust ra t ed i n Fig 60 D esigned by P r ofessor Ch a t t ock
and Mr F ry O f Bris t ol th e de tai ls h ave b een impro ved a t th e N a tio n a l
Physi cal La bor a to ry un til th e ga uge i s n ot o n ly acc ura t e but also con
ven i s h t m u se Th e usu al arrangemen t 18 capa ble of r esp o n din g to a differ
en ce of press ure of one t en thousan d th of an inc h of w at er an d h as a tot a l
ran ge of a bout an inc h Fo r l ar ger ran ges of p r essu re a gauge of di ffer en t
pro p o rtio ns i s used o r th e wa ter of th e no rma l ga uge is rep l aced by merc u ry
Th e inst rum
en t d o es n ot need calib ra tio n i t s i ndi ca tions of p res sur e b e in g
ca l c ul a bl e fr o m th e d i mens io ns of th e pa r t s
In p r i nc i p l e th e ga ug e co ns is ts of a U tub e h el d m a fram
e whi ch may
be tilt ed and th e tilt 13 so arran ged as to p r ev en t a ny m
ovem
en t of th e
fl ui d in th e U tube under th e i nfl uence of p r ess u re app li ed a t th e o pen
ends Th e b as e frame i s pro vi ded with thr ee l evellin g sc rews whi c h su pp o rt
it fro m th e obs erva tion ta bl e Th e frame has proj ec tin g u p w ar ds two
.
.
.
.
.
.
'
m
.
m
,
.
,
'
.
,
m
,
.
m
.
,
.
,
.
,
.
.
o
.
.
.
.
,
.
.
,
,
.
,
-
.
.
,
,
APP L IED AER OD YN AM ICS
108
p in dl es en ding i n stee l p oin ts and a th ird p oin t whi ch i s adj us ta bl e i n
h eight by a scre w and wh eel and th e thr ee p oin ts fo rm a su ppo rt fo r th e
u pper frame A st ee l sprin g a t one end and a gui de a t th e oth er are
s u fli ci en t with th e w eight of th e frame to c o m
p l et ely fix th e tiltin g part
in p os itio n Rigi dly a tt ac h ed to this u pper frame i s th e glass wo rk whi c h
essen ti ally fo rms a U tube to fac ilit a t e observa tio n th e usu al ho riz on ta l
lim b is d i vi ded one par t endi n g i nsi de a co ncen t ri c vessel whi c h i s co nnec t ed
to th e oth er part of th e ho riz on t al lim b Abo ve th e cen t ra l vess e l i s a
f u rth er a tt ac h m
en t fo r th e fillin g of th e ga uge Were th e cen t ra l vesse l
co mp l e t ely fill ed with w a t er fl ow fromone en d of th e ga uge t o th e oth er
woul d be p ossibl e wi thout visibl e e ffec t i n th e observ in g mi cro sc o pe shown
e In c i p i en t fl ow i s made apparen t by th e
as a tt ac h ed to th e tilt in g fram
in t ro d uc tion of cas to r oil in th e cen t ral vessel for a dista nce su ffi c ien t to
co ver th e oth erwise o pen end of
th e inner tub e Th e su rf ace of
of th e w a ter an d
separa tio n
S
,
.
.
-
,
.
.
,
.
.
hown by a dep artu re fro m
th e cross wi re of th e i crosc o pe
and is co rrec t ed by a tiltin g of th e
frame In this way th e e ffec ts Of
visc osity and th e w e ttin g of th e
s u r f aces of th e gl ass v ess els a re
is
m
s
,
.
locked by th e c losi n g of a t ap
in th e ho riz on ta l lim b and th e
gaug e th en b eco mes p o rt a bl e
A p oin t of prac ti ca l conv eni
once is th e use of a sa lt w a t er
solutio n of re l a ti ve density 10 7
s t ead of dis t i ll ed w a ter as th e
i
n
go t
p m
n
,
w
cas to r oil in th e cen t ra l vesse l
th en re main s c l ear fo r lon g perio d s A gauge of thi s c ons t ru c tio n
care fully fi ll ed wi ll las t fo r twe l ve mon ths without c l ean in g o r re fillin g A
frac tur e of th e cas to r oil w a t er su rface is follow ed by a tem
p orarily di s
t at hed zero but full acc u racy is rap i dly reco vered T h e zero can b e rese t
by th e l ev e llin g screws a fter such break and u ltimat ely by t ransference
of sa lt w a ter fr o m one lim b of th e U tub e to th e oth er
AS used i n th e wind ch annels of th e N a tion a l Physi cal La bo ra to ry
a r eadin g of a bo u t 600 div isions i s obta ined a t a wind Speed of 40 ft s
a n d th e acc u rac y of readi n g i s one or two d i vis io ns de t er
in ed wholly by
th e fl uc tu a tions of p ressu re Speed s fro m 20 ft s to 60 t h e are read
with all d esi ra bl e a ccuracy on th e sam
e gaug e ; lower speed s ar e rarely
u s ed a nd ga uges of t h e same ty pe but la rger ran g e ar e u sed u p to t h e
high es t ch ann e l Speeds reach ed
Ch at t ock tilt i n g ga ug es h ave a lso been u sed ex t ens i vely for th e measu re
men t of loca l p ressu res on mo dels of ai rcraft and parts of a i rcraft If
is
,
.
-
.
,
,
.
.
,
.
,
-
.
m
-
.
.
.
-
.
,
.
.
.
.
.
APP L IED AER ODYN AMI CS
110
Water Resi stance of Fly i ng B oat Hulls Experimen ts on th e
resi stance of surface craft are made by towin g a m odel o ver still wa t er
Th e gen eral arran gemen t of th e tank c o nsi st s of a t rough som
e 500 to 600
fee t lo ng 80 fee t wi de an d 12 f ee t deep Alo ng th e si des are car e fully l ai d
rails whi ch su pp o rt and gui de a t ravellin g ca rri ag e th e Sp eed of whi c h i s
re gula t ed by th e su pp ly to th e el ec t rom
o t o rs m oun ted a bo ve th e wh ee l s
Th e first 100 to 150 f ee t of th e ru n are re qui red to ac cel era t e to th e fina l
Speed and a ra th er larger am o u n t fo r st e p p i ng th e carri age a t t h e end of
th e run Speed s u p to 20 fe et per sec can b e r each ed and t h e ti me avail
a bl e fo r observa tion i s th en lim
it ed to fift een seconds so th a t all th e m
easu re
men t s are most con veni en tly t a ken a utom
a ti ca lly At lower s peeds t h e
time i s lon ger an d d i r ec t ob ser va tion of some quan titi es c o m es ea sily with i n
th e lim
i ts of p ossibility
Th e w a t er resi stance of a fly i n g bo a t hull i s assoc ia t ed in tim
at ely with
th e p rod uc tio n of w aves and th e law followed i n th e t ests i s kn own as
F ro nde s l a w and st at es th a t th e speed of to wi n g a m o del sho ul d b e l ess
th an th a t of th e full si ze cra ft i n th e prep ortion of th e squ ar e root of th e
rel a ti ve linear di mensions This rul e i s dealt wi th i n grea t er deta il i n th e
ch ap ter o n d ynami ca l Si mil arity wh ere it i s shown th a t o nce t h e l a w i s
sa t i sfi ed th e fo r ces on th e full sca l e are ded u ced f ro m thos e on th e m o de l
by m ulti p lyin g by th e cub e of th e re la ti ve linear d i m
ensio ns
Th e fl yin g b o a t a t res t i s su pp o rt ed wholly by th e re actio n of th e w a ter
an d t h e d i sp l acem
en t i s th en eq
ua l to th e w eight of th e boa t As th e a i r
speed i ncre ases par t of th e w eight i s t aken by th e win gs un til
th e whol e w eight c o mes on to th e win gs an d th e flyi n g bo a t ta kes to th e
ai r
en ts ar e shown di a gra m
ma ti ca lly in Fig 6 1
Th e t esti ng arran gem
Poin ts of a tt ach men t of th e appara tus to th e t ank carri age ar e ind icat ed
by sh aded areas Th e m o del of th e flyin g bo a t hull i s con st rain ed to m o ve
only in a ver ti cal p l ane b u t is oth erwis e free to t ake u p an y an gl e of
inc i dence and ch an ge of h eight under th e ac tion of th e fo rces d u e to motion
Th e measu rin g appara tu s i s a tt ach ed a t A by free joi n t s th e resist ance
b ei ng b a l an ced by a p ull in th e r od AB an d th e a ir lift fr o m
th e win gs b ei n g
repres en t ed by an u p war d p ull i n t h e ro d AD Th e t rimof th e he at can
is
a t P an d th e a ngl e fo r eac h t r im
be ch an ged by th e add itio n of wei gh t z
read o n th e grad u a t ed b ar N whi c h m o ves with th e fl o a t
Th e u pper en d of th e r od AD m o ves in a verti cal gu i de a n d a wi re c o rd
pas sin g o ver p ulleys to a w ei ght 0 gi ves th e f reed omof ver ti cal adju stm
en t
mentio n ed to geth er with th e m
eans of rep r esen t ing th e a i r lift Th e p ull
in th e ro d AB i s t ran sm
it t ed t o a ver ti ca l st eely ar d EFG a n d i s b a l an ced
a in der by th e p ull in th e
i n part by a w eight hun g fr o m G and fo r th e rem
Fr omJ th ere i s a ro d JK Opera ti ng a pen on a rot a t in g dr um
s prin g H J
whilst oth er pens a t L an d M rec o r d time an d d ist ance m
o ved thr ough th e
wa t er Th e reco rd t aken a uto m
a ti ca lly i s suffi c i en t for th e de termina tion
of Speed and resi st an ce
Since t h e m o del i s f r ee t o rot a t e a bout an axi s th rough A th e Ob serva
tions of p ull i n AB an d of lift i n AB a re s u Mci en t i n ad di tion to th e obser
va ti on of i nclination to c omp l et ely defin e th e fo rces of th e m o del a t a ny
T he c on di t ion s of experimen t can be vari ed by ch anges in th e w eigh t s
Speed
Th e
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MET H OD S
OF
MEASU R EMENT
11]
APP L IED AERODYN AM
112
I
CS
a t O and P and th e whol e of th e p ossibiliti es of m otio n fo r th e parti c ular
fl o a t can be i nvestig a t ed
Th e observa tion s i nc lu de a genera l reco rd of th e sh ape of th e w aves
fo rmed th e tendency to th row u p Sp r ay o r green w a t er o r to sub m
erg e
th e bow O ccasio nally m o re el a bo ra t e meas ur em
en ts of w ave form h ave
b een made Flyin g be at s of ce r ta in ty pes bounce o n th e wa t er fr o m p oi n t
to p oi n t in a m
otio n kn own as p o rp oisi n g and by means of su itabl e
arran gements this m otion can b e rep r o d uced i n a m o del
Forces du e to Accelerat ed Flui d Moti on — I n avi a tion it i s u sua l
to as s um
e th a t th e fo rces o n part s of aero p lanes depend o n ly o n th e veloc i
ti es of th e aereplan e li near and angul ar an d are n ot s h eet ed apprec i a bly
by any ac ce l era tions whi c h may o ccu r A littl e thought wi ll Show th a t
this assump tio n can on ly b e
a
ju st ifi ed as an ap p roxi m
tion fo r accel era tion of th e
a i rc ra ft m ea ns a ccel era tion
of fl ui d in i t s neighbo u rhoo d
with a con se qu en t ch an ge
of p r essure di st ributio n and
tot a l for ce on th e m
o del I n
rece n t y ears th e exami na tio n
of th e e ffec ts of acce l era tio n
on aero dynami c fo rces h as
b ecom
e prom
i n en t i n th e
c o nsi dera tion of th e sta bi lity
of a i rshi p s To estim
ate i t s
i mp o rt ance r ecou rse i s ha d
to experimen t s on th e osci l
lat i ons of a bo dy a bout a
s t a t e of s tead y m otio n an d
th e p r inc i p l e may b e illus
t ra t ed fo r a sp h ere
Fig 62
Fl o 62 —Forces d u e t o ac ce le a t ion of fl i d mt i n
s hows an arran g e m
en t whi c h
ca n b e us ed to d i fferen ti a t e b e tw ee n e ffec t s d u e to s t ead y and to u n s t ead y
m otion Th e Sp h ere i s m oun t ed o n a pen d ulumswin gin g a bout th e p oin t
O n an ex t en sio n
A th e Sp h ere itself b ei n g i n so me liqui d su c h as w a t er
of th e pen d uluma t D i s a c ou nt erw eight whi c h b rin gs th e cen t re of mass
of th e pend ulum to A so th a t th e whol e resto rin g c ou p l e i s d u e to th e
s pri n gs a t EF an d F G and t h e eccen t ri c c ou n t e r w e ight C
Th e m omen t Of i n erti a a bout A will b e de not ed by I an d th e osc i ll a
tions will b e su c h th a t 9 is a lways a small an gl e and within th e lim
i ts
0 and cos 0
1 Th e e qu a tion of m otio n may b e w rit t en as
si n 0
,
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,
,
.
.
”
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'
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,
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,
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r
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u
o
o
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,
‘
z :
.
18
W
,
t o—f(e + 1
910)
i s th e
i s th e c ou p l e d u e to th e c oun t er b al ance w eight a t C
wh ere l
Id 5) i s th e
r es t o rin g co u p l e ari si n g f ro m th e Spri n gs a t EF and FG an d f(v
hy dro dyn am
i c c ou p l e Th e lin ear veloc ity of th e cen t r e of th e Sp h ere i s
,
,
.
,
APP L IED AER ODYN AMI CS
114
even kee l an d wh en inc lined as th e resu lt of p i t chin g Advan t a ge i s t ak en
of a th eo rem first p m
by
arr
i
B
ooth
i
n
n
g
l
and
and
by
d
u
n
d
s
E
o
e
i
n
1
9
1
1
H
p
.
a tt ac h ed and is h el d i n an inver ted p os ition by th e wir es which pass o ver
p ull ey s and carry w eights a t th ei r free ends Th e m o del is fi ll ed with w a t er
and a su fli ci en t p r ess ure app li ed to t h e in terior of the envelo pe by con
n ec t i on to a h ead of w a t er
Th e arran gemen t is sh own di agr amm
a ti cally in Fig 68 th e n um ber
of wi res havi n g been c hos en only fo r illust r a tio n and n ot as
an y real riggin g A b eam NO carri es a n um
ber of p ull ey s F E D whi c h
be adj us t ed in
alon g th e b eam so as to vary
,
,
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.
.
,
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,
—Expe i m
e t
F ro OIL
.
r
n
to
d et
m
i
er
ne
t h e nece ssary g as p res s u re i n
,
,
a n on r i i d ai rs hi
g
e
p.
th e ri ggin g wi r es AF B E an d CD Th e t e nsio ns in th es e riggin g wir es
Th e m
o de l being i nfla te d with wa t er
a r e p r o d u ced by w e ight s K H and G
th e pres sure can be v a ri e d by a m
o vemen t of th e r eserv oi r L and can b e
m ea sur ed on th e sca l e M Th e p o i nt s F E a n d D will b e o n th e car of a n
a irsh i p an d th e geo me t ry of th e r iggi ng an d th e loa d s in th e wi res will b e
kn o wn appro xi a t ely fr o mcal c u l a tio n o r gen e ra l experi e nc e O n ce this
po in t h as bee n reach ed a n exp eri m
en t co ns is ts of th e grad u a l lowerin g of
th e r eserv oir L un t i l p uckerin g of th e fa b ri c t akes p la ce a t so me p oin t o r
oth er B y ca re fully a dj us tin g th e p ositio ns of the riggin g wi r es an d th e
ay b e p o ssi bl e t o red u ce th e h e a d of wa t e r
load s to be t a k en by th e mit m
b e fo re p u ck ering again t akes p lace and by a pro cess of t ri a l an d err o r t h e
b es t dis pos itio n of riggin g i s obtained
oi
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MET H OD S OF MEASUR EMEN T
115
Th e rel a tion of th e experi men t to th e full scal e i s found by th e princ i p l es
of Si m
il a rity Th e sh ape of th e enve lope i s fi xed by th e di fferen ce b e tween
th e pressu res d u e to hy drogen and those d u e to ai r T h e i nt ern al pressure
can b e rep res en t ed by th e e ffec t of th e h ead i n a tub e b elow th e enve lope
th e l en gth of th e hy drogen c olu n pro d u ced b ein g an exac tly an alogous
qu an tity to th e l en gth of th e col u m
n of w a t er in th e m
o del experi men t
I n th e m o de l th e sh ape of t h e envelo pe depends on th e d ifference b e tw een
w a t er an d ai r an d th e p r essur es for a gi ven h ead a re 900 tim
es as gr ea t
0 t i mes as g rea t as a t
as t h a t fo r hy d r og en an d ai r a t gr ound l eve l o r 105
fee t Th e law of co mparison st a t es th a t th e str esses i n th e fa b ri c
of th e m
o del envelope will b e e qua l to thos e i n th e a irshi p if th e scal e i s
ft Th e
1050 i s 824 fo r
V9OO al e 3 0 fo r gr ou n d l eve l o r
necess ary i n t erna l pressur e to preven t p u ckerin g of th e a irshi p envelo pe
fa b ri c i s cal c ula t ed fro m th e h ead of hy drogen obt ai ned by scali n g u p th e
h ead of w a t er
Th e me tho d negl ec t s th e w eight of th e fa b ri c but th e erro rs on t h is
accoun t d o n ot appear to b e imp o rt an t
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C HAPT ER I V
DES I GN DATA F ROM THE AEROD YNAM I CS LAB ORA TORI ES
P AR T
I
—Sr aa rc n r
m
a
.
o
mass of dat a rel a t i ng to design pa r ti cu l arly th a t c oll ec t ed un der th e
a usp i ces of th e Advisory Co mmitt ee fo r Aer on auti c s i s very c o nsi dera bl e
and will b e th e ulti mat e reso rt when ne w i n fo rma tio n i s re qui red T h e
r ep o r t s and me m
o ran da h ave b een c oll ec t ed o ve r a perio d of t en y ears
part of whi c h was occ u p i ed by th e Gr ea t War To thi s va lua bl e ma t eri a l
it i s n ow b eco m
in g essenti al to h ave a su mmary an d gui de whi c h in it self
woul d be a serious c omp il ation n ot to b e co mpre ssed in to even a l ar ge
c h a p t er of a gen era l t rea ti se So m
e gen era l lin e of proced u r e was n eces
sa ry th er e fo r e i n p r epa r in g thi s c h ap t e r i n o r der to b r in g it withi n reas o n
a bl e c o mpass and in mak in g ex t rac t s it was thought de si ra bl e i n t h e
fi rst p l ace to gi ve det a il ed d escri p ti ve ma tt er c o verin g th e whol e subj ec t
In sca r ce ly an y ins t an ce h as a r epo r t b een u sed to i t s full
in outlin e
ex t en t and readers will fin d th a t ex t ensio n i n Spec ifi c cases can b e obt a ined
by re feren ce to o rigin al rep o rt s Although det ai l ed re ference i s n ot gi ven
th e i den tity of th e o rigin a l work will a l m
os t a lw ay s b e r ea dily found in
mittee for Aeronautics
th e p ubli shed r eco r d s of th e Adv isory Com
A seco nd main a i m of th e c h ap t er h as b een t h e pro v i sion of en ough
da t a to co ver a ll th e various p roblems whi c h ord in ar ily ari se i n th e aero
dyna m
i c design of a ir cr a ft so tha t as a t ext boo k for stu d ent s th e v olu m
e
a s a whol e i s as c o mp l e t e a s po ssibl e in it self
Th e c hap t er i s d i vi ded in to two part s whi c h c o rres p on d with a n atur al
p hy si cal d i vi sion In t h e fi rs t
e a sur e m
en t s
t he m
St ra ight Flyin g
i n v ol ved are dr a g lift and p it c hi n g m o men t an d h ave o n ly p a ssing refer
e n ce to axe s of in er ti a
Non rec tilin ear fli ght i s however m
ost s uit
a bly approac h ed fro m th e p oin t of v i e w of fo rce s an d m
om
en t s r e l a ti ve
to th e mo v in g bo dy a nd th e sec on d p a r t of t h e chap t er o p en s wi t h a
defini tio n of bo dy axes a n d th e n o m
encl atu re used i n rel atio n to motion
a bout th em The fi rs t p ar t of th e c h ap t er i s n ot rep ea t ed i n ne w fo rmi n
th e seco n d a s th e tra nsfor m
a tions are par ti c ul arly Sim
p l e an d it i s on ly in
p l et e m o del s th a t th ey a re re qu i red I n i t s sec on d p art
t h e case of c o m
this c h a p t er in a dd ition to dea li ng with th e da t a of c irc li ng fl ight gi ves
som
e of th e fun d amen t a l da t a to whi c h th e m ath e m
a ti ca l th e o ry of
s t a bility i s app li ed
—
Wing Forms Th e win gs of a n aer o pl a ne a re d esigned to su pp ort i t s
weight an d th eir qu ality i s mea su re d c hi efly by t h e sm
a lln es s of th e r e
sis tan ce whi c h acco m
pani es t h e lift Th e b es t wi ngs h a ve a r esi stanc e
wh i c h is littl e more th an 4 p er cen t of t h e su pp o rting forc e Al m
o st t h e
TH E
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lie
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APP L IED AERODYN AM IC S
118
call ed th e an gl e of sweep b ack if in p l an Fig 64 (b) an d dih edra l an gl e
if i n el eva tion Fig 64 (c)
When two p l anes of e qu a l c ho rd are co m bi ned th e perpen di c ular
dis t an ce b e tw een th e cho rds i s call ed t h e ga p whi lst th e d i st ance of
th e u pper wi ng ah ead of th e lower i s defined by th e angl e of st agger
Fig 64 ( d) Si mil ar defin itions app ly to a t ri pl ane
Fo r t ail p l an es st ruts e t c th e c ho r d i s t a ken a s th e medi a n li n e of a
sec tion and i n g e n era l t h e cho rd of an ae r ofoil i s th e lo n g es t lin e i n a sec tion
a xi m
um p r oj ec ted a r ea
a n d th e area i t s m
With th ese defin ition s it i s p os sibl e to p ro ceed with th e desc ri p tion of
th e forces o n a wi ng i n m otion th rou gh th e ai r a n d an accou n t of th e
t a bl es a n d d i agra m
s i n whi c h th e res ult s of o bserva tio n are p r esen t ed
—
fi
ti
i
De ni ons (F g
i cs of Wi ngs
In t h e st a n dar d m o de l
Aer ody nam
wi ng th e a ttitu de rel a ti ve to th e wi n d i s fi xed by th e i nclin atio n of
t h e c hor d of a sec tion to th e d i rec tio n of th e rel a ti ve wi n d
Th e a n gl e a i s
Th e fo r ces on th e win g in th e st andard
kn own as th e an gl e of inc i de n ce
at m osp here of a win d
c ha nn e l ar e fi xed by th e
angl e a th e win d speed V
an d th e area of th e m
o del
No m
a tt er wh a t th e rel a
tio n b etween th e angl e
veloc ity and fo rces th e
r can a lw a ys b e co m
l
a
tt
e
wm
o cui r
plet ely r ep re se n t ed by a
fo rce of m agnitu de R Fi g
Q 65 in a defini t e p os itio n
AB
Various alt erna ti ve
me tho ds of exp ressin g thi s p o ssibility h ave c u rren t u se Th e result an t B
ay b e r esol ved in to a lift c o mp on en t L n or ma l t o th e win d d ir ec tio n and
m
a d ra g c o mp on en t D alon g th e wind If y be th e an gl e b e tween AB an d
al t o th e wind d ir ec tio n it will b e seen th a t t h e rel a ti o n b e tw een
th e n o rm
L and D an d R an d y i s
.
,
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”
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”
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,
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.
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"
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,
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,
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'
,
.
,
.
.
.
.
,
L =R
cos
y
,
D = R si n y
I
( )
.
Th e p o sitio n of AB i s oft en de t erm
in ed by th e locatio n of th e p oin t C
whi c h shows th e in t e rsec tio n with th e c ho rd of th e sec tion It i s e qu ally well
de fin ed by a c ou p l e M a bout a p oi n t P a t th e n o se of th e wi ng M b e in g d u e
t o th e res ult a n t fo rce R a c tin g a t a l evera e
Th
e
ig
n
c
ho
n
fo
r
S
i
s
se
g p
ence i n l a t e r wor k
con vem
Th e p oin t P m
ay b e c ho sen a r bit rarily ; in
l
n
e
sm
s
i
s
p
l
a
e
it
us u ally th e ex t reme fo rwar d end of th e c ho r d in bi p l anes
g
t mi d wa y b e tw een th e fo rw a r d e n d s of th e c ho r d s a n d i n t ri p l an es
th e pom
th e fo rwa r d end of th e chor d of th e mi dd l e p l ane
st ep i n r epresen t a tion a ri s es f r o m th e re sult of expe i men t
nex
t
r
s
Th e
It i s found th a t fo r all Si zes of m
o del a n d for all wi n d speed s th e an gl e 31
s nea r ly c o n s t an t so lo n g as a i s n ot c h a n ed an d th a t th e ra tio
to
P
G
g
PQ
i18 s i
al o l ttl e a ffec t ed On th e oth er hand t h e magnit u de of R i s near ly
p r o p or tio na l to th e p l an éarea and to th e squar e of th e speed On th eo re ti ca l
,
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D ESI G N DAT A FR OM AER ODYN AMIC S L AB OR AT ORI ES
119
it i s found th a t th e m
a gnitu de i s a lso p rop o rtiona l to th e den sity
of th e ai r Puttin g th es e qu an titi es i nto m
a th e m
a ti ca l fo rm
shows th a t
groun d s
.
nearly indepen dent of th e Si ze of th e m
o del o r th e wi nd speed
d u rin g the t est Th e qu an titi es are th ere fo re pec uli arly w ell suit ed for
a com
paris on of win g fo rm
s an d th e va ri a tio n of th ei r ch arac t eri s ti c s
with angl e of i nc i dence Th e fi rst qu an tity i s cl ea rly th e same wh eth er C P
and PQ ar e m ea su red in feet o r in met res and i s th er efo re in t erna tiona l
Si milarly th e rad i an as a measu re of an gl e and th e degr ee are i n use a ll
o ver th e c i vili sed wor l d Th e thir d qu an tity can b e m
a de i nt ern a tional
‘
by th e u se of a con s ist en t dy nami cal sy st em of un its
e are
Qu an titi es whi c h h ave n o d i mensions i n mass l ength and ti m
den ot ed by th e c o mm on l ett er is are par ti c ulari sed by su xes an d re ferred
to as coe ffi c i en t s Th e followin g ar e im
p o rt an t p ar ti c ul ar cases as app li ed
by th e
to wings an d are deri ved f ro m th e thr ee al ready mention ed
or d in ary process of resolution cf fo rces an d m
om
en t s
ar e all
.
.
.
,
,
.
.
m
,
,
.
,
C en t re of pressure
L ift
coefli ci ent 5
m
k
of Fig
.
65
.
coefii ci ent
D rag
coefli ci en t
Momen t co e ffi ci en t
de t h e li m
i ts of dyna m
i ca l consi st encv l ead s t o di ffi cu l t i es b e t wee n
t h e p u r e s ci en t i s t and t h e engi n eer Whi ls t bot h agree to t h e fu n da mn t a l ch a rac ter of m
ass as
di fleren t i a te d from
ass and wei gh t or
we h t u sa ge of t h e w or d
p ou n d as a u ni t for bot h m
mon; To au t h or i t ap pears t hat any systemi n whi ch su ch confusi on can occu r
for ce is co m
is defect i v e and i n Eng la n d pa r t of t h e d efec t li es i n t h e a bsence of a legal d efini t i on o f force
f
ngi n ee ri n g
h
u s i n aerona u
whi ch has an v si m
k
d
bl
m
s
T
e
e
re
l
a
t
i
o
n
h
a
a
ro
e
o
l
r
t
o
t
e
w
p
y p
t i es t h e Engli sh spca ki ng races i n vari a bly sp eak of t h e t h rus t of an a irs cr e w i n
nu de a n d
w
f
o f p re su res i n ou n ds e r s u are i n ch or
h
e d iffi cu t y does not
t
h
h
e
o
e
q
f
l
o
u
are
oo
t
T
r
e
s
q
p
p
p
lie here for t h m
ass an d y et t h
e t r i c sy st e m
Fre nc h
es for force and m
h as as p arate nam
logm
mmes per qu are met re i ns tea d of t h e
aer ona u ti ca l engi neer e xp resses ai r
ki
r
s
u
i
n
e
s
re
p
rou ghly eq
l qu a n t i t y m
egad yn es p er sq u are m
of
et re w hi ch i s consi s t en t wi t h hi s sy s t e m
u n i ts
u ch si ni p ler
I t wou ld ap pear t h t t h e c n ce p t io n of
h t as a u ni t of fo e i s so m
as s acce lera ti on t ha t onl
a t i ca lly use t h l t te r
w
t h a n t h at f m
d
If
ey s te m
w
ere
e
s
t
u
e
n
t
s
y
a k e t h e w ei h t of t h e
st a n dar d o f for ce b y S eci f i n
a
d
no w t o m
d
a
r
f
m
ass
i
n
t
o
rese
n
t
a
n
o
t
s
g
p
y g
p
an d i n t rod u ce a n ew u ni t of m
a ss
ber near t o
eas ure m
en t as som
e nu m
g a t t h e p la ce of m
3 22 ti m
es as grea t as ou r
ence o f la ng u ag e
res en t u ni t i t ap ea rs t o t h e a t h or t h a t t h e d i ve
p
p
be t ween sci en ce a nd engi neering wou ld di sap p ea r I n t h is beli ef t h e st a n d i n di ca t ed bo e
ongst t h ose i n cu rren t u se at t eac h in g i ns t i
has been ad opte d t hr ou g hou t t hi s boo k fro ma m
t u t i ch a as be:
e of
slu g
was g i v en
t h e bes t of t hree a l te rna t i ves
Th e r at h er u gly na m
t o t h is u ni t 0 m
ass b y som
e one u n k nown
The st a n dar d d ensi t y of a i r i n aeron au ti ca l
r cu b i c foot
o m
en t s i s 00023 7 Slu
e x er i m
b
f
0
6
5
lb
T
ee t
e
r
cu
i
c
d
n
o
t
0
7
a
n
t
o
o
e
p
g
p
p
n si ona l coeffi ci e nt s so t ha t i n
o bj ec t i ons as far as p oss i bl e f u ll u se has bee n m
ad e of non di m
a ny cases read ers m
m
h
ou t di fli cu lt y i n ap p l y i ng t h e t a b les
t
ay u se t h ei r o wn
w
i
m
t
s
s
t
e
e
y
p
of st a nd ard resu l ts
The
h
c oi ce of un i t s i nsi
e
.
“
,
,
o
.
.
,
,
s
.
e
,
e
,
s
ua
,
a
.
W
o
o
rc
e
a
.
u
,
a
,
.
v
.
.
.
.
,
-
.
e
.
,
APP L IED AER ODYN AMIC S
120
e ult s obt ain ed i n ae ro dynam
i c l a bo r a to ri es ap p ly a l so to a
n on st and a r d a t m os p h er e if th e exp r ess i on s ( 3 ) ar e us e d but th e Speed o f
t est u su ally quot ed app li es o nly to a i r a t 760 m
mHg a nd a t empera ture of
l5 6 C
Fig 66 shows how t h e variou s qu an titi es of (3) are arra nge d in p resenti ng
res u lt s
Th e i n dep en den t vari a bl e of grea t es t o cc u rrence i s
a ngl e of
All
r s
-
,
.
°°
.
.
.
A N G L E O F l fi C I O EN CI
E
a
0
M us t : o r
l N C l O t NC C a
A u G L t or I NC IDENC E s a.
a
—Met h o ds
ll u t a t ing wi ng ch ract ri t ics
i nc i dence but for m any p u rp oses t h e li ft c oe ffi c i en t k, i s u sed as an
i nd ep en dent v a ri a bl e Th e r ea sons for t hi s will app ear a ft er a stu dy o f
th e c h ap t er on t h e P red ic ti on a nd Analy sis of AerOp la n e P erfo rmance
Th e u se ful r ange of an gl e of i nc i d en ce i n t h e fl ight of a n aer o p l an e i s
f ro m—1 t o
an d m o de l exp er i
ent s u su ally exce ed thi s ra ng e a t
b oth e n ds An exa m
p l e i s gi ve n a littl e l at er in wh i c h ob serv a tions w e re
t aken fo r a ll p ossibl e an gl es of i nc i den ce but thi s case is exception al
Pro ( Ni
.
of
i
s r
a
,
e
s
.
,
.
.
m
°
.
,
.
A PP L IED AE RODYN AM IC S
122
—Th e cur v e i s sh own
e scal e as li ft co effi ci en t b u t i s rar ely u sed in thi s f or m
to t h e sam
a lthou gh
s t est e d u n d er st an d ar d
b ers are gi ven i n t a bl es for all win g form
t h e nu m
allness of t h e or d inat es o v er t h e fl yin g r a nge for an y
co n d ition s
Th e sm
reas o n abl e scal e of dr ag at t h e cr iti cal an gl e of li ft i s t h e chi ef reason for
it ed u se of thi s ty p e of di agram
a li m
Fig 66
Centre of Praw n e Coeffi ci ent and Angle of I ncidence — Con
si dera ble v a r i ation i n cu rv es of cen t re of p r essu r e o ccu r i n win g fo r s
but th at ill u st ra t ed i s typ i cal of t h e p r esent day high sp eed wing Th e
cu r v e h as t wo i nfi n i t e b ra n ch es o ccurr in g n ear t o t h e a n gl e of zero lift
For l ar ger angl es of inci d ence
an d t h e ch an ges i n this regio n are gr eat
all er i n am
ou n t an d t h e cur v e h as an a v erage p os ition
t h e ch an ges ar e sm
a bo u t on e thi r d of t h e ch or d b ehi n d t h e l ea d in g e d ge of t h e win gs
Th e
ex act p o sitio n of i nfi nit e cen t re of p ress u re coefli ci en t i s d efin ed by t h e
e
e
r
b
o
p
l
to
ho
d
e
s
r
t
h
c
m
a
a
ll
a n gl e at whi ch t h e result an t force ( R of Fi g 65
e
c
)
e ext en t on t h e d efin iti on of t h e chor d
If
an d th er efo r e d ep en ds to so m
t h e cen t re of p r essu r e m
o ves forw ar d wi th in cr ease of an gl e of inci d ence
t h e t en den cy of t h e wing i s to fu r th er incr eas e t h e angl e a n d i s th erefore
towar ds i nst ability Turn ing u p t h e t ra ilin g ed ge of a win g m
ay rev ers e
t h e t en d ency as wi ll appear i n on e of t h e ill u st ra tions to b e gi v en
Fi g 66 (d)
Mom
ent Ooe aent and Ang le of Inci dence —Th e in fin it e
val u e of cen t re of p r essu r e coeffi ci en t n ear zero lift h as n o Sp eci al signifi cance
ore conv eni ent to u se a m
om
i n flight an d it i s oft en m
en t co effi ci en t
ay b e
ar ked p eculi ariti es o v er t h e flyin g ra n ge but m
Th e cu r ve h as n o m
v ery var i ab l e at t h e criti cal angl e of lift
Lift /Drag and Ang le of Inci dence —Th e ratio of lif t to d rag
Fig 66 (e)
os t i m
p or tan t item
s co nn ect ed with t h e beh a v iour of aer o
i s on e of t h e m
p lane wi ngs an d i n l evel stea dy fli ght i s t h e ra tio of t h e w eight of an
aer o p l an e to t h e resis t an ce of i t s win gs
Th e cu r v e s tar t s f romz er o wh en
ay b e
whi ch m
t h e lift co effi ci en t i s zero an d ra p i d ly r each es a m
a xi m
um
ore slowly to l es s th an h alf th at v al u e
as gr ea t as 20 to 25 an d th en falls m
a d e to use
at m
axi m
umlift coefli ci ent It i s o b v ious th at every effort i s m
Fig
.
66 (b)
Drag
.
M
ei ent and Ang le of I nci dence
.
,
.
.
m
.
,
-
.
,
.
,
-
.
,
.
.
,
.
m
,
.
.
.
.
,
,
.
.
.
,
.
,
,
.
a
wi ng
at
its
bes t
,
i
.
a.
wh ere
L
D
is
a
mxi mum b
a
,
ut
t he
li m
itation of
land i ng sp eed ca n be seen to affect t h e ch oi ce as b elow D en otin g
Sp ee d of fl ight by V an d t h e l an d i n g Sp eed by V1 it will b e seen th a t
co n d iti on of co ns t an t loa d in g re qui res th at
.
,
the
the
V
P s
Equ atio n ( 5)
can
be
as
g d in a m
or e con v eni en t fo rm
arran e
m
ax
wh ere a i s t h e rel ati v e d ensi ty of t h e a t m
osp h ere at t h e p l ace of flight and
o l V will b e r ecogni sed as in d i cat ed ai rs p eed
Th e wh ol e of t h e r ight h an d
s i d e of 6 13 fi x e d by t h e l an di n g sp ee d a n d t h e wi n g f orm
if M be ch osen
( )
a s t h e li ft co efli ci en t for m
a xi m
h i t /drag an d h ence t h e in d i ca te d air Sp eed
um
for gr eat es t effi ci ency 18 fixed
,
-
.
,
,
.
AERODYN AMI CS L AB OR AT ORIES
D ES I G N D AT A FROM
R eferr ing to Figs
l
v a u e of
0 54
an d a
.
an d
66 (a)
it will be fou n d th at It ]
66 (c)
ax i m
um
v al u e of 02 1for m
1
3
1
ti m
es
,
h as a
128
m i mum
ax
Thi s sh ows an in di ca t ed
.
t h e l an d in g sp eed
As app li ed t o an a erop l ane t h e
th eo re
p l et e st ru cture an d not of t h e
u se t h e li ft /drag of t h e co
wings alon e a nd t he nu b er 16 i s u ch red u ced N ear t h e groun d t h e
Sp ee d of
os t effi ci en t fl ight i s w ell below th at of p os sibl e fl ight , but
t h e d i fferen ce b eco es l ess at gr ea t h eight s
For hi gh sp eed fighti n g
sco u t s t h e r a tio of li ft to d r ag for t h e win gs
ay be o nll
in st ea d of t h e
best val u e of 20, and it b eco es i p ort an t t o p rod u ce a wi ng whi ch h as a
h igh v al u e of li ft t o dr ag a t low lift coe ci ent s
Thi s is t h e d is ti ngui sh ing
ch ar ac t eri sti c of a go o d high -s p ee d wing , an d a ppea rs t o be u n a tt a i na bl e
a t t h e sa
e ti
e as a high lift coeffi ci en t
Fig 66 ( f) Lift/Drag and Li ft Coe aent — Th e re ark s o n Fig 66 (e) h av e
in d i ca t ed t h e i p or tance of t h e p resen t cu r ve, an d p arti cul ar att en tion
h as been p ai d t o t h e d ev elo p en t of wi n g fo r s h av i ng a high sp ee d v alue
a i r S ee
p d
of
16
w ould
m
m
m
m
,
of
L
l)
m
m
.
m
.
m
an d as
of
.
.
m
l ift coeffici en t
m
-
.
m m
m
at a
.
m
mm
.
m
.
h igh
a
v alu e
as
p ossibl e a t
a
li ft
u mt h l tt b i g i m
po t
m g t t h mxim
th
lim
bi g of
pl
It will thu b
th t i m
od
p
t h mx imm
li ft /d g of wi g i
p o t t p p ty
t t h m t im
fo m
i t i i m it but o ly it i
i t d with oth p p
efli ci en t
0 9 ti
e c
e
es as
an aer o
n
a
rea
u
as
e
a
‘
an e
s
.
ra
s no
n
a
e
,
e
a
e seen
ern
n
a
r an
os
t in
racti ce
r an
en
er
cc
ro
er
of
it s
ti es
Equ ati on ( 6) su ggests th a t t h e qu antity un d er t h e roo t s ign i s i m
p ort an t
as a n i n d ep en d en t v ar i a bl e an d thi s i s r ecogni sed in cer t a in r ep ort s on
r
as a n n r ns c
er
as
n
,
er
s ass oc a e
r o er
.
,
m
m
i s con
Lift Coefi ci enL—Th e d i agra
pl et e aeroplane, for t h e ch ange fr o t h e cu rv e
co
for win gs a lo ne is al os t sol ely on e of p ositi on of t h e z er o o r di n ate
A
t an g en t fr o t h e new origin sh ows t h e v al u e of t h e axi u lift [drag of t h e
a ero p lan e an d t h e li f t coe ffi ci en t at whi ch it occu rs
Th e d i agr a shows ore
cl ea rly th an any oth er th a t t h e u seful r an g e of flying p os itio ns li es withi n
t h e li it s
a n d 005 for d rag coefli ci en t , an d th a t s all ch an g es o f li ft
coeffi ci en t an d th erefor e of in d i ca te d air Sp ee d p ro d u ce l ar ge ch an g es of
dr ag nea r t h e cr iti ca l a ngl e Th e i n d ica t e d ai r Sp ee d at t h e criti cal angl e
of lift is kn own a s t h e st a lli n g Sp eed , an d h as been u sed in th ese n ot es
”
as i d en ti ca l with
l an di ng sp eed
Th e l a tt er i s how ev er , a lw ays gr ea t er
than t h e for er for reasons of con t rol ov er t h e otion of t h e ae ro p l ane at
t he
o en t of alightin g
66 (g)
Drag M
v eni en t i n i t s r el a ti on t o a
Fig
.
ei er“
:and
m
.
m
m
m mm
m
.
m
.
m
m
.
mm
m
m
,
.
.
P ARTI CUL AR Ca s e s
Effect
0!
Chang e of Secti on (Fig 87
or
and
WI N G F O RM
Tables 2
— Th e sh ap e of t h e
d ar d m
o d el aerofoil i s con v en i en tly gi v en by a t a bl e o f
t h e cc or di n a t es of p oin t s i n it t h e ch o r d b ein g t aken a s a s t an d ar d from
whi ch t o ea sur e an d t h e fr on t en d as ori gin For t wo w ings R A F 15
for hi gh Sp ee d an d R A F 19 for hi gh m
lift coeffi ci en t t h e cc
a xi m
um
Th e l en gth
o r di na t es whi ch d efin e th eir sh ap es ar e gi v en in T a bl e 2 b el ow
eas ur e en t s are gi v en
of t h e cho r d i s taken as unity an d all oth er l in ear m
sec
ti on
-
of
t he
.
t
s an
m
,
,
.
.
.
.
,
.
m
.
,
.
.
A PP L IE D AE R ODYN AM ICS
124
m
i n t er s of it
I t will b e seen
t h e cho rd o f 00 68 , a n d thi s n u
‘
O
'
.
a xi m
th at R A F 15h as a m
um
h eight a bov e
mber is often called t he u pp er su rfac e camber
.
.
.
l
10 0 6 6 42 555
Fro 6 7 —Eflect
.
.
of ch a nge o f
wi ng
sec t i on
.
wi n g secti on Th e oth er win g of t h e t a bl e R A F 19 h as an u pp er
s u r face ca
ber of 0152 or ore th an twi ce th at of R A F 15 and this
for t h e
m
.
,
m
,
.
.
.
.
.
.
,
APP L IE D AERODYN AMI CS
126
l ea d in g e dge thi n
r oo m
,
m
s
i ca
h eets
can
md
be
a e
to fly
st ea
d ily
acr os s
a
.
TAB LE 3
Fone xs
u
.
mM
ox xxr s e x A
Fu
r
Pu
n
.
Cen t re
of
mt
Mo
en
co e ffi c i e nt
T a bl es 4 an d 5ar e r ep r esent ati v e t a bl es of win g ch aract eristi cs in th eir
best form Th e i n t er vals in angle of in ci d ence ar e us ually
with in t er
t
d
d
r
a
l
a
a
f
l
f
e
v
lu
es
t
s
m
o
o
ll
a
n
gl
of
i
i
d
wh
tio
i
t
to
g
a
c
h
ra
es
n
e
n
e
re
t
e
ce
a
p
is varyin g m
ost rap i dly All t h e t erm
s whi ch occur ha v e b een d efin ed
t h e cur v es
an d t h e ch ar ac t eri sti cs of t h e win gs are m
os t eas ily seen fr o m
of Fig 67 whi ch wa s p r od u ce d fro m
b ers i n T a bl es 2—
t h e n um
5
.
,
.
.
.
,
TAB L E 4
m
R A F 16 A
.
Size
L ut
,
mai m
ng
.
of
p lane, 3
D rag
M
.
'
.
o ro n.
Wi nd
x
eie nt
.
sp eed.
40 ft
c6 0
.
.
s.
m
0!
310 111006
D ESI G N DAT A FROM AERODYN AMICS LAB ORATORIES
127
Th e first n oti cea bl e f ea ture of t h e li ft coe ffi ci en t cur v es i s, tha t wh ils t
t he p la t e o nly b egi ns to li ft a t a p os iti v e an gl e of i nci d ence, t h e high Sp ee d
win g R A F 15 lif ts at angl es a bo v e
an d t h e h igh lift win g a t
T h is featur e i s co
on to all si ilar ch an ges of u pp er surface ca b er
Th e sur p r isi ng fact i s w ell est a blish e d th a t a n a erO p lane win g
ay lif t
with t h e win d di rect ed towar ds t h e u pp er surface
.
.
m
m
.
m
m
m
.
.
TAB LE 5
.
R A F 19 Annor o
.
Si
m f pl
o
ane ,
3
.
'
X
.
m
Win d
s
p
d,
ee
wi t
-s .
m
t
b
t l
Mo
Li ft coeffi ci ent
Du g
.
m
e n coe cfen t
a ou
ead i ng
edge .
m mm
b u t t h e v a l u es
at
u
axi
All t h e lif t coeffi ci en t cu r ves sh ow a
for R A F 15an d 0 8 4 for
a re v ery di ffer en t , bein g
for t h e p l at e
Thi s i s p a rtly du e to a p rogr essi v e i ncrease in t h e average
R A F 19
ut
u
r
b
c
e
0
4
5
d
0
n
a
h
o
4
0
81
0
0
0p e of t h e cu r v es t h e v al u es b e in g 003 5
,
,
u
axi
lift co
t o t h e i n cr ease of ran g e of angl e b etween z ero lift an d
effi ci en t
Th e v ery high lift coeffi ci en t o f 0 84 gi v en by R A F 19 app ears
t o be hi gh ly criti ca l an d t h e axi u
i s foll ow ed by a rap i d fa ll so t h at a t
an an gl e of i n ci d ence of 20 d eg ree s t h e di ffer en ce be tw een t h e wings i s
grea tly red uced At sti ll great er angl es t h e effect s of di fferen ces of wi ng
for t en d to di sa pp ear
Th e cur v es gi v ing t h e rati o o f lift to d rag sh ow a d i ffer en t or d er to
u
of 7, R A F 15
ax i
t h e cu r v es for l ift coe ffi ci en t , for t h e pla t e gi v es a
of 166 , an d R A F 19 of 120 It is th erefor e cl ear th a t t h ere i s so e
li ft to dr a g rati o R A F 15i s t h e out co e
sectio n whi ch h as a
a xi
u
a ny e xp eri
of
en t s o n v a r i a ti on of wi n g secti on n on e of whi ch h a s gi v en
a high er ra ti o u n d er s t an d ar d con d iti on s
As i s us ual in t h e ca se of
v ar i a tions n ear a a x i u con dition , it i s p ossibl e t o ch ange t h e section
.
,
.
.
.
.
.
.
m m
m mm
.
,
m mm
.
,
m
.
.
,
.
.
m mm
.
m
.
.
.
m mm
m
m mm
.
.
.
,
.
.
.
.
.
.
m
m
APPL IED AERODYN AMIC S
128
od
m
wi thi n
t ly wi d e li m
its without p ro d ucin g grea t ch anges
era e
wi n g
in
g mas t h e li ft to dr ag cu r v es h as been p l ott ed t h e
cota n gen t of t h e an gl e of in ci d ence as i t b rin gs out a n in t eres tin g p rop er ty
of cam
bered wings For a v al u e of lift to dr ag gi v en by a poi n t on thi s
a l to t h e ch or d an d b oth
cur v e t h e res ult an t force on t h e wi n g i s n orm
R A F 14 an d R A F 19 ha v e t wo s uch p oin t s For v al u es of lift to d rag
whi ch li e below t h e cot an gen t cu r v e t h e res ult an t force li es behin d t h e
a l to t h e ch or d whils t t h e co n v erse hol ds for p oin t s a bo v e t h e c ur v e
n o rm
I t wi ll b e seen th a t t he result an t fo rce on t h e p l a t e i s always b ehin d t h e
e v alu e of
ah ea d of t h e cho r d
al wh er ea s for R A F 15an ex t r em
n o rm
i s shown
When a d escri p ti on of t h e p ress ur e d i strib u tio n r ou n d a win g
i s gi ven it wi ll b e seen th a t thi s forw ar d r es u lt an t i s associ at ed with an
in t ense su cti on ov er t h e forwar d p ar t of t h e u pp er sur fa ce Th e r esu lt an t
a l to t h e wi n d d i r ecti on but i n R A F
i s of course a lway s b ehi n d t h e n o rm
i ni m
umof
Th e valu e of 7 sh own i n Fig 65i s
14 i t s v a l u e h a s a m
th en v ery sm
a gn i
all a nd it will be un d ers too d th at err o rs of a pp r eci a bl e m
tu de wou l d follow fr oman y wan t of kn owl edge of t h e d i recti on of t h e win d
s
rel a ti v e t o t h e w in d ch ann el b al ance a rm
One d egree of d ev i ation w oul d
in t ro d u ce an error of 28 p er cen t in t o t h e dr a g rea d i ng an d even with
easurem
en t s of m
gr ea t care it i s di ffi cu lt t o m
in i m
umdr a g
a k e a b sol u t e m
a d e on t h e
p arati ve exp er i m
en t s m
Com
coefli ci ent to withi n 5 p er cen t
u ch gr ea t er
o del an d with t h e sa m
e a ppa ra tu s h a v e an ac cu racy m
e m
sa m
th an thi s and m
it s in d i ca t ed
ore n ea rly e q u al t o l p er cen t
Withi n t h e li m
ar ka bly co nsi st ent
wi n d ch annel obser v ations are rem
s
Th e cen t re of p res sure coeffi ci en t cu r v es sh ow tha t t h e wing form
en t s
ov em
th at is t h e
R A F 14 an d R A F. 19 h a v e u n s t a bl e m
ov es f orw ar d a s t h e an gl e of in ci d en ce i n cr eas es
cen t r e o f p ress ur e m
Th e p l at e on t h e oth er h an d h a s t h e st a bl e con diti on p re vious ly
r ef err e d t o
Wing Ch aract eristi cs for Angl es of I ncidence outside th e Ordinary FLvi ng
—
e of t h e m
p li cat ed con ditions of m
or e com
oti on
Ra a In d i scu s si n g som
of an a erop l a ne kn owl e d ge i s r eq u i r e d of t h e p r op erti es of wi n gs in
Not on ly i s st ea d y u p si d e d own flyi n g p ossi bl e
ex t r ao r d i n ary a ttit u d es
o ti on occu rs for s h or t p erio d s i n t h e t a il sli d e whi ch i s
but back war d m
eti m
es i n cl u d e d i n a p il o t s t ra i n in g
som
For a flat p l a t e o b serv a ti ons ar e r ecor d ed i n T a bl e 3 for a ra ng e of a ngl es
t h e sy m
a nd f r om
met ry of t h e a erofoil these observati ons
f rom0 t o
0 to
Th e v alu es of t h e li ft coeffi ci en t
a r e s u ffi ci e n t fo r angl es from
li ft t o d rag ra tio an d cen t re o f p ressu re coeffi ci en t ar e sh own in Fig 68 i n
p a r i son with sim
ila r cur v es for R A F 6 wi ng section Th e sh ap e of
com
t h e l att er i s sh own i n t h e fi g ur e a n d t h e d et a il e d d escri p ti on i n t h e h eight
b ers app ly only t o t h e
Th e n u m
o f co n t our s i s gi v en in T a bl e 0 b el o w
a ll ca m
b er of t h e u n d er s u rface i s o f littl e i m
p or t ance
t h e sm
u pp er su r face
od i fi ca tio n kn ow n a s R A F
A m
h as b een
i n t h e p r es en t conn ec ti on
R A F 6 o nly i n t h e fact th a t i n
a ny occa s i on s a n d d i ffe rs fro m
u sed on m
e r t h e u n d er su rface i s fla t
t h e fo r m
ad e it n ecessa ry t o t e s t t h e a er of oil
met ry of t he section m
Th e d issy m
On t h e
sa
m di
e
a ra
,
.
,
.
.
.
.
.
.
.
.
,
.
.
.
,
.
,
.
.
,
.
.
.
,
.
.
,
.
.
.
.
.
.
.
.
,
,
.
.
m
-
.
,
’
.
°
°
,
.
.
.
.
.
.
.
.
,
.
.
.
.
.
.
'
APPL IED AERODYNAMI CS
13 0
mp
i o b etween th e p l ate an d wing section shows a v ery
for t h e v ari ous cur v es a n d i n d i cat es
ilar ity of form
con s i d era bl e d egr ee of sim
t h e sp eci al ch ar a ct er of t h e di fferen ces a t or dinary fl yin g angl es wh i ch h a v e
a ti c s tu dy of t h e effect of v a r i a tion
b een d ev elop ed as t h e r esu lt of syst em
Th e
co
ar s n
,
mm M m
Foru
s
Size 2 5X
"
p
o
on
R AF 6
.
Win d speed
,
.
.
40 ft
.
.
-s .
Cen t re of
mt
Mo
en
coeffi ci ent
+0 3 l l
‘
Wing
ear ly d ays of aer o n auti cs
m
dep endent on Upper Surface ( la ina —In t he
at t h e N a ti on al Physi cal La bo ra t ory a seri es of
Charact eristi cs as
D ESI GN D AT A FROM
m68 —Wi g
P
.
.
n
AER ODYN AMI CS L AB OR AT OR IES
charact er i st i cs a t all
possi ble
angles of i ncid ence.
13 1
APP L I ED AERODYN AMI CS
13 2
p im
ber an d u pp er su rface
en t s o n t h e v ar i a ti on o f u pp er s u rfa ce ca m
sh a p e wa s carr i ed ou t a n d l ai d t h e fo u n d a tion for a r ea soned choi ce of win g
et h o ds of t es t s a n d p ar ti cul ar ly t h e d isco v ery
section
K n owl ed ge of
of an effec t on win g ch ar ac ter i sti cs of s i z e a n d win d Sp ee d ha ve r ed u ce d
th eir value an d oth er exa p l es are now ch osen fr omvar i ous so ewh a t
un connect ed so ur ces No u p to da t e equi val en t of th ese early exp eri en ts
ex is t s but it i s to b e h op e d th a t our N ati on al Ins tit u ti on will u lti a tely
en t s with all t h e r efi nem
un d ert ake su ch exp eri m
en t s of
od ern
etho d s
Un til thi s seri es appears t h e r esul ts d ed u ced fro mt h e ear ly exp er i ents
ay be accep t ed as qu alit ati v ely co rr ect an d althoug h not q u ot ed d ir ectly
h av e been used to gui d e t h e choi ce of exa p l es an d to gi ve weight to t h e
d ed u ctions dr awn fromt h e s tu dy of Sp ecial cases
Aer ofoils h av in g l arge u pp er s urface ca ber are us ed only i n t h e design
o f air scr ews an d on p ages 304 an d 8 05 will be foun d d et ails of t h e sh ap es
of a n u ber of sections and t h e correSpon din g t abl es of t h e aero dyn am
ic
p rop ert i es In ost of t h ese sections t h e un d er s urface was flat Th e
general concl u sion ay be drawn tha t a fall i n t h e value of t h e m
um
a xi
l i ft to drag ratio i s p ro d uced by thi ckenin g a wing to ore th an 7 or 8 per
cen t of i t s cho r d a n d tha t t h e f all is gr ea t wh en t h e t h i ckn es s r ea ch es
Th e exac t sh ap e of t h e u pp er s urface d oes not
20 p er cen t of t h e ch ord
port an t but a seri es of e xp eri en t s at a ca ber
app ear to b e v ery i m
in di ca te d an a dvan t age in havi n g t h e m
u or di nat e
ax i m
r a ti o of
of t h e sectio n i n t h e n ei ghbo ur hoo d of on e thi r d of t h e ch or d fr o
t he
a xi m
um
or d in a t e was fo u n d to h av e
l ea di ng edge Th e p osition of t h e m
ar ke d effect on t h e b rea k down of flow a t t h e cr iti ca l a ngl e of li ft
a
cn t al inf or
ati on it a pp ears th a t th ese
but i n t h e li ght of o d ern exp eri m
di fferences m
ay be l ar g ely r ed u ce d i n a lar ger m
od el t es t ed a t a hi gh er
A v ery si ilar seri es of ch anges t o th ose n e w u n d er r evi ew occurre d
Sp ee d
in t h e t est of an a irscrew section at d i fferen t sp eeds a n d i s illust rat ed an d
d escri bed i n t h e chap t er on Dyn am
i ca l Sim
i l ar ity
Fur th er refer ence t o
t h e efl ect of siz e of m
od el an d t h e Sp eed of t h e wi nd d ur in g t h e t es t i s gi v en
l a t er in thi s ch ap t er
—
L
u
r
f
r
f
il
f
w
r
f
n
A
I t h as b een t h e
r
S
ac
e
o e
Ca be o a
e oo
Changes o
general exp eri ence th at ch an ges of l ow er sur face cam
ber of an aer ofoil
por t ance i n th eir effect on wing charact eristi cs th an ar e those
a r e of l ess i m
of t h e u pp er su r face
Wings rarely ha ve a conv ex lower su rface but for
I n T a bl e 8
secti ons of a ir scr ew s a co n v ex u n d er sur face i s n ot u n us u a l
a n d Fig 69 a re shown t h e effect s of v ari a tion of R A F 6A by a ddi n g a
con v ex low er s u r face t h e or d in a t es of whi ch w ere p r op orti onal t o tho se
of t h e u pp er su rface
Th e ran ge fromR A F 6A to a st rut f ormwa s
co v ere d i n thr ee s t ep s i n whi ch t h e or di na t es of t h e un d er si d e w er e on e
thi r d two thir ds and e qual t o those of t h e u pp er s urf ace I nset in Fi g 69
ar e ill u st rati ons of t h e aer ofoil fo r
I n thi s seri es t h e ch or d was t a k en i n all cases as t h e u n d er si d e of t h e
ina ti on of t h e lift a t
origin a l win g an d t h e t a bl e sh ow s t h e gra d u al eli
ber grows t o th at of
neg a ti v e a ngl es of i n ci den ce as t h e un d er su rface cam
umlift coeffi ci ent is o bserv a bl e
axim
A d istin ct fall i n m
t h e u pp er sur face
with ou t corresp on d ing ch ange of angl e of in ci d ence at whi ch it occu rs
e x er
m
.
m
,
-
-
.
m
,
m
m
m
,
m
m
m
m
m
m
.
,
,
.
,
m
m
.
.
m
m
,
.
m
.
.
,
m
m
m
'
’
-
m
.
m
m
.
m
,
.
'
m
.
.
.
,
.
.
.
.
.
,
.
.
.
.
‘
m
-
,
.
.
.
m
,
-
.
.
APPL IED AERODYN AMI CS
13 4
20 p er
m
cen
.
t
m m
m
m
ight ore th an
in lift to dr ag at a li ft coeffi ci en t of 01
co p ensa t e for t h e s a ll er p ro p o r tio nat e lo ss at lar ger v alues of t h e l ift
it t o t h e a oun t of
ay b e o b ser v ed tha t th ere is a li
It
coeffi ci en t
un der -sur face ca ber whi ch coul d be used with advan tage, and reference
to t h e wing for of R A F 15 sugges ts that t h e a dvan ta ges can be
a tt ai ned by a slight co n v e xity a t t h e l ea d ing ed ge only
of
.
m
m
m
m
.
.
.
.
En
t er o r
VARI ATI ON
m
B or row Ca
or
Aorofoil, 3 x 18
'
Lift w e
m
clen t
e r a or
Wind speed
'
.
,
As s e re
40 ft »
mt
s u rface o f
A
B
C
D
l
i
l
I X
l
.
6A
.
.
eoe
m
I
leadi ng
clent ab ou t
.
.
X
edg e.
.
.
.
.
l
l
en
.
.
l
.
.
A B , C an d D w as t hat of R A F 6A
0, ti c fla t lower su r fac e
Qx o rdi n a t e s of u pp er sur fa ce of B A F 6A
of u p pe r s u rfa ce s o f
lo wer
,
.
Ho
be r
mR A F
.
conv ex .
AERODYN AMI CS L AB OR ATOR IES
D ESI GN D AT A FROM
Changes of Sectton
—
i
n
W
Th e
r
o
l
a
n
e
Ae p
g
ari sing
mth
fro
13 5
Sag of th e Fabri c coveri ng of an
s ha p e of an aere p lan e wing is d et er i ned p r i
ar ily
by a nu ber of ribs ad e carefu lly to t e p l at e, bu t Sp aced so e 12 t o
Th ese r ib s are fixed t o t h e ai n Sp ars ,
15i ns a p ar t on a s all aerOplan e
i s st ret ch ed a lin en fa b ri c i n whi ch a cons i d era bl e tensi on
an d o v er th e
i s p ro d uce d by de p ing with a v arni sh w h i ch con t ract s on dryi n g
On t h e
u pp er surface t h e win g s h ap e i s affected by light for er ribs fr o t h e
l ea ding e dge t o t h e fron t sp ar Fig 1, Chap t er I , shows t h e app earance
of a fini shed w g, whi ls t Fig 70shows t h e contours easure d in a p ar ticul ar
instan ce Fro t h e easure en ts on a win g a o del was ade with t h e
full v ari ati ons of section rep resen te d , an d was tested in a win d channel
m
.
m
.
m
m
e
m
m
.
m
m
m
m
m
m m
.
.
m
m
.
m
.
m
.
.
m
.
.
up p u
Sean ce
Lu
FI G 70
.
.
o
mf
c
l owr a Sean c e
oot
—C
o nto urs o f a
fa br i c -covered wi ng
.
After t h e firs t test t h e dep ressions were fi lled with wax, an d a st an d ar d
ade
section res ult ed on whi ch d u p li ca t e t es t s w er e
of u ni fo r
T abl e 9 gi v es t h e result s of both tests
It i s not n ecessary to p lot t h e res ult s i n or d er t o be a bl e to see that t h e
o d ifyin g t h e aero d yn a i c ch arac
effec t of sag i n t h e fa b ri c of a wi n g i n
Th e high r atio of
t eris t i cs of thi s win g i s s all a t all an gl es of inci den ce
s
u
a
re
i
e
i
s
c
c
o
e
ar
e
i
t
d
g
p
tly
e
l
g
d
l
whi
h
twi
th
t
p
v
o
ly
t
h
e
t
o
a
r
d
u
i
s
r
l f to a
,
u sed i n illus tra tion
—
a
n
r
h
e
ae
r
D
a
T
f
n
a
n
d
H
t
o
d
y
ic
i
c
t
o
Eff
t
s
e
ti
d
g
Aspect Ra o, an
e
a
r
ec
a
a
n
t
o
ec
ra
as
e
h
t
i
ti
fo
l
t
d
by
p
t
tio
pp
i
bl
are
a
f
f
ec
i
ro
n
f
a
a
e
o
e
r
s
cs
ar
c
ac
b er of exp er i en t s i s s all owin g t o t h e fact th at t h e
e x t en t , bu t t h e n u
len gth of a wi ng is fi xed by other con si d era ti ons th an wi n g efii ci ency One
ore co p l ete series of exp eri ent s h a s been u sed t o p rep are
of t h e
m
m
.
.
m
m
m
m
.
m
.
m
m
m
m
m
m
.
APP L I ED AERODYN AMI CS
18 6
Fig 71 i n t h e u pp er di a gramli ft coeffi ci en t i s sh own as d epen d en t on
an gl e of in ci d en ce an d both t h e slep e an d t h e m
a x im
umar e in creased by
a n i n crea se of as p ect ra ti o
Th ese changes get m
o re m
a r ked a t s m
a ll e r
as p ect ra ti os an d l ess m
ar k e d a t hi gh er v a l u es a lth ou gh an effec t can s ti ll
b e foun d wh en t h e win g i s 15 ti m
es as l ong as i t s cho r d
Th e ch an ge s
resu lt in g f ro mch a n ge of asp ect r a tio ar e m
os t st ri kin gly sh own i n t h e
ra ti o o f lif t to dr ag t h e m
v alue of whi ch r i ses fro m10 at a n
a xi m
um
asp ec t r atio of 8 to 15for an asp ect ra tio of 7 an d p rob a bly 20 for an as p ect
ra tio of 15
all an d asp ec t ra tio h as
Th e effect at low lift coeffi ci en t s i s sm
n o app reci abl e in flu ence on t h e choi ce of secti on for a high spe ed wi ng
.
,
,
.
,
.
,
.
,
-
m
u
mm e w
Amm m
sox nsr w
Cou r a
o r an
p
r
u. s
e
e
.
.
.
o r an
'
x 30
.
m
ea r
Wi nd sp eed
,
40 ft
.
.
mwhi h
.
t or
r or
or
a
~
o
.
s
a nni e o r are
s.
mdi fi d
R A F 14
sect i on.
Changes of Wing For
Properti es —Th e win gs of
an d it d oes n ot a pp ear
m mS m
m Sm m F
Ans c ro ft . or U
a s nx o noovs n r o s
Aerofoil,
R A F 14
m
Da
.
h ave Li tt i e Eflect
e
.
m
the Aere dy na i c
on
e e x t en t
op l an es ar e alway s r oun d ed to so m
att ers
Th e d i fference
th at t h e exac t formm
between any r easona bl e roun d in g an d a s q u are ti p accoun ts for an i ncrease
u mvalue of t h e lift to drag ra tio an d an
a xi m
of 2 to 5p er cen t on t h e m
in app reci a bl e ch ange of li ft coeffi ci ent at any angl e
eas ura bl e effec t
A dih edr al angl e l es s th an 10 a pp ears to h av e no m
drag or cent re of p r es su re I t s im
p ort ance ari ses i n a tota lly
on lift
d ifferen t conn ection a dih ed ral angl e b ein g effecti v e in p ro d ucing a corr ec
ti v e rollin g m
om
en t wh en a n aero p l an e i s o v er b ank ed
ilar con cl u sion as to a bsence of effect 18 r eache d for v ari at i ons o f
A si m
m
on
ifi
ca ti on i s n ot v ery com
to
Thi
ty
p
o
d
e of win g m
s w eep b ack u
s
p
c
a er
,
.
.
.
'
°
.
,
,
.
,
APP L IED AERODYN AMICS
188
g of angl e of in ci d ence of 2 to 10
a ra n e
-
°
°
n ot
an
of s p ee d on lift coeffi ci en t
O -2
Fro 72 —Eflcc t
’
is
t h e effec t
i
gl es
mm
t
.
.
r an t ,
Th ere
is
of sp ee d o f te s t
.
all er an d l ar ger
pp reci a bl e ch anges o ccur a t b oth sm
t en dency towar d s an asy m
p toti c v alue at high Spee d s
bu t
a
.
a
,
DESI GN D AT A FRO M
AERODYN AMI CS L AB OR ATORI ES
189
whi ch i s m
or e a pp aren t i n t h e cu rv e sh owin g t h e ratio o f li ft t o dra g
i cal
Th e th eo ry of t h e ch a nge i s di scussed in t h e ch ap t er on Dyn am
i l ar ity wh ere it i s sh own th at t h e co rr ect co m
p arati ve b asi s i s n ot on
Sim
s p eed alone b u t on t h e p ro d u ct of Sp eed an d cho r d
Thus 20 ft s and a
6 in ch ch or d gi v e t h e sam
e cur v es as 40 ft s an d a 8 in ch ch or d
It i s
p rob abl e t h at t h e fu t ur e st an dar d m
o d el will ha v e a 6 in ch chor d an d will
be tes t ed at 11 Sp eed of 60 ft s since t h e r esu lt of a co m
p ar is on with full
scal e shows th a t t h e r es i d u a l co rr ec tion 18 th en with i n t h e li m
it s of err or of
observ a ti on on t h e full scal e For cert ain p ur p oses an d esp eci ally at
m
a xim
ay b e a pp rec i ably
umlift coeffi ci en t t h e range of t est v al u es m
i ncrease d by t h e use of a m
o d el with a chor d of one foo t and a wi n d
s p eed of 10
0 ft s
This will be p ossibl e in t h e l arge wi n d chann el now
bein g erected at t h e N ational Physi cal L aborat ory
.
,
.
,
-
.
-
~
.
.
-
.
.
-
.
.
-
.
,
.
,
,
.
-
.
.
B rp nan s s
’
mT
mi h
a
RI PLAN E S
)
.
i d erin g t h e aero dyna c c aracter isti cs of com
b in ations of
p lan es a n um
ber of n ew v ar i abl es a dditio nal t o those a lread y d iscus sed
et ri cal ar r an g em
en t of t h e p l a n es
h a ve i m
p o rtan ce an d d efin e t h e geo m
os t i m
por t an t is
r el ati v e t o each oth er
Of th ese n ew v ari a bl es t h e m
fi
n
s
a
n
hi
h
d
e
d
as
h
p
p
d
i
d
i
t
b
tw
h
d
w
i
s
e
t
e
e
n
u
l
ar
ce
e
e
n
t
h
e
o
r
s
a
r
e
c
c
e
c
g p
p ort an t fact or i s t h e d ist an ce by
A secon d but l ess im
of t h e p l an es
w h i ch t he u pp er p l an e p r oj ect s ah ea d of t h e lower p lane Gap an d an gl e
o f st agger ar e d efi ne d by t h e i ll u st r a ti on Fi g 64 ( d)
—
i
B
l
a
n
o
f
M
n
l
e
n
T
r
i
l
n
r
i
n
s
e
o
o
a
d
Th e win g sectio n
Co p a o
p
p
p a a
an d t h e
was R A F 15 t h e size of t h e p l anes 8 x 18 with a gap of
For t h e m
onop lane tes ts h av e
a d e at a s p ee d of 4 0 ft s
t est s w ere
en ti on ed an d d e ta ils gi v en in T a bl e 4
Th e corr esp o n d in g
a lrea d y been m
bi p l ane an d t r i p lan e resu lt s are gi v en i n T a bl es 10 an d 11 Th e rel a ti ve
di s p o sition of t h e p lan es i s shown i n t h e sket ch es at t h e foot of Fig 78
a in cur v es of
whils t t h e aero dyn am
i c p rop erti es ar e i llustrate d 1n t h e m
e fi gu r e a s d ep en d en t on angl e o f in ci d en ce
t h e sa m
Th e cu r ves for li ft coeffi ci en t show an a pp reci a bl e fall i n sl op e an d in
—
—
t
e
r
u
h
ight
i
n
o
d
o
p
l
a
n
bi
p
l
t ri p l ane an d ar e
h
an e
er m
on
e
a xi mm e
m
typ i cal rep resen t ati v es of t h e effec t of co m
b ination Whils t t h e lif t
coe ci ent i s r ed u ced at all an gl es t h e tot a l dr a g co effi ci en t i s littl e a flect ed
as a con se qu en ce of w hi ch it will be seen th a t t h e li ft /drag cur v es h av e
or din a t es n early p ro p ortional t o th ose for lift coeffi ci en t Th e less in
i c effi ci ency is a bou t 20 p er cen t for t h e bi p l ane an d 80 p er
aer o d yn a m
a ll er th an th a t
cen t for t h e t ri p l an e
Th e ga p /ch or d ratio of 0 75 i s s m
or e n ea r ly e q u a l t o u n ity an d t h e l a tt er woul d
i n co m
mon u se whi ch i s m
h ave a so m
I n t h e p arti cu lar case n ew d escr ib ed
ewh a t b ett er effi ci en cy
b in ati on of t wo a n d t h ree p l an es h as h ad n o app reci abl e effect
t h e co m
t h e an gl e of n o li ft t o n ear
on t h e pos iti on of t h e cen t r e of p r essu r e fr om
Abo v e t h e criti ca l a ngl e t h e cen t r e of p ress ur e i s
t h e cr iti ca l angl e
fu r th er forw ar d for t h e t ri p l an e t h an for t h e bi p l ane an d t h e l atter i s
forw ar d of th a t for t h e m
bi n ation i s
on O p lan e
Whil s t t h e effec t of com
all it a pp ea rs t o be g en er ally t ru e th a t t h e cen t re
p erh ap s u n u su ally sm
In
cons
,
.
”
,
,
.
,
.
m
m
.
.
.
.
,
'
m
.
"
,
,
.
-
.
,
.
.
.
,
.
m
,
.
'
,
.
.
.
.
,
,
.
.
,
.
,
APP LI ED AERODYN AMI CS
140
TABLE 10
.
m
R A F 15 B rr nw
.
Size
of eac h
l ift
l
a ne , 3 x 18
p
'
coeffi ci ent
'
Ga p
.
Drag
.
.
.
hord
coe ffi ci ent
n.
N o st agger
.
,
Wi nd
s
pe ed ,
40 ft
.
s
.
.
04 45
m
R A F 15Te r
.
Size
An
of eac h
plane a x 18
'
,
'
.
.
.
s
.
No sta gger
Ga p /Chord ,
0 of
In
u se
( degree s )
.
Win d speed
Ce nt re
Li ft
coe ffi ci ent .
Drag
,
40 ft
.
-s .
of
coeffi cie nt .
.
of p r essu r e
ra ng e
is
n ot
v ery
sens
iti v e to ch an ges of
.
Changes
of
Gap i n
a
B i plane R A F 6
.
.
.
.
v
o
er
a
g p
Z ero
the
—
A
Sh eat h
p racti ca l
mp l
co
t
e e
APP L IED AERODYN AMIC S
142
m
u m but i n all case s
o i t d with a d ecrease of t h e a x im
there i s a m
ar k ed a bsen ce of efl ect on t h e drag coeffi ci en t at an gl es b elow
t h e cr iti ca l
Th e cur v es of lift t o drag as d ep en d en t on lif t coefli ci en t
s
mll g
a
a
p s is
ass c a e
,
.
F I G 74
.
-
V a ri a t io n
of
b i p la n e g a p
.
a xi m
at a gap
how a ri se fr o ma m
u m14 0 at a gap /ch or d of
to
p ar is on it i s cer t a in tha t t h e
cho r d of
Alth ough n ot shown for com
mon op l ane woul d show high er val u es for t he same con di tion of test
p ro ba bly in t h e neighbo ur hoo d of 18 5 Since t h e gap i n bi p lanes is
s
,
,
.
D ES I G N D AT A FR OM AER ODY NAMI CS L AB OR AT ORI ES
mi
14s
t i d by st rut s an d wir es t h e b est ra tio of gap t o chor d cann ot b e
ch osen fr o mtes ts on t h e wings a l on e
St r uctura l consi d era tio n s are al so
of s uffi ci en t i p o rt a nce to i m
p ose lim
i t a tions on gap in any p ar ticul a r
a n a ne
,
m
.
TAB LE 12
.
Ca nn o ns
Si ze
o f eac h
or
GAP/CH ORD B a r re, R A F 6 B u
.
'
ane, a x i 8
l
p
'
.
No st agger
o
Co s r rf crs n r
Gap /Ch ord
.
r
m
Wi nd speed
.
Gap /Ch ord
Du
.
.
.
a.
,
40 ft
.
s.
-
APP L I ED AERODYN AMI CS
144
TAB LE l Z
Ca n
Si ze
o ns o r
of e ac h
co nli nucd .
Ga r /Ca c ao B ar re , R A F
.
'
l
a
n
e
, 3
p
'
Cr a
—
X
m
or
No st a gger
.
.
.
6 Bu
m
m
Wi n d speed
Pa sss u aa Co xr rrorxn r
er .
,
4 0 ft
.
-s.
'
.
Gap /Ch ord.
+00 74
04 25
APP L I ED AERODYN AMI CS
146
a ll fo r t h e
h a nges of m
en t coeffi ci ent sh own i n Fig 74 ar e sm
om
wh ol e range but a reference t o t h e t a bl e of cent re of p res su re coeffi ci ents
will show tha t o ver t h e ran ge of flyin g an gl es t h e chan ges are l ess than th ose
u al
om
om
of t h e
en t coeffi ci ent i s v ery closely eq
en t co effi ci en t
As t h e m
t o t h e p rod u ct of t h e lift an d cen t re of p r ess u r e coeffi ci ent s it a pp ears tha t
os t t h e wh ol e of t h e effect s of su p erp os in g p la nes at zero stagger i s
alm
accoun t ed for by a ch an ge in t h e li ft com
po nen t
—
h
i
n
l
Th e win g secti on was a ga in R A F 6
n
f
a
B
C a ges of Sk at er o
pa e
Th e
c
.
,
m
.
,
.
.
FI G 75—Va ri a t i on
.
.
of
bi plan e
s ta
gger
.
.
.
0 and
tio 09 whils t t h e angl es of s tagger were
n d t h e r esu lt s
e n t s a re sho wn a t t h e f oo t of Fig 75(a
a
Th e arran g em
)
As will b e seen from
t h e cu r v es o f Fig 75( a ) t h e effi ci en cy o f
in T a bl e 18
a bi p l an e i s littl e a ffec t e d by i t s st agger but th ere i s a cert a in l oss in t h e
li ft coeffi ci ent by ba ckw ar d an d a ga in by f orwar d
a x im
um
v a l u e of t h e m
p rov e a p ilot s v i ew an d it
Th e la tt er i s u su ally in t ro d u ced t o i m
st agger
will be seen t o be sli ghtly a dvan t ageous Th e effect of st agger on t h e
an d
t h e gap /ch ord
ra
°
,
.
.
.
,
,
,
'
.
,
.
D ESI GN D AT A FR OM
AERODYN AMI CS LAB OR ATORI ES
p os ition of t h e centr e of p ressu r e was not m
easu red
b u t ju d gi ng
t ri p l ane res ults gi ven l at er i s to ov e t h e p osition forwar d on t h e
ch or d for eith er p ositi v e or n ega ti v e s t agg er
m
,
147
m
m
fr o
ean
.
Efi ect of ch angi ng th e Angle between th e ch ords of th e Planes of a
in or v ari a bl es , an d th a t
Bi p lane — Fi g 75( b) sh ows th a t this i s one o f t h e
t o + 08 on li ft coeffi ci en t t h e i nclina tion of t h e
o v er t h e range of
por t an ce in i t s
cho r ds t o each oth er by thr ee d egrees or l ess i s of n o i
A r ep or t by Hun sa ker fr o t h e Massa chu sett s School
e ffec t on effi ci ency
of T echnology s u gg est s tha t t h e cen tre of p r ess ur e
ov e en t s ca n b e
a d e st a bl e by i ncli n in g t h e ch or ds , bu t littl e a tt en ti on app ears to h a v e
b een gi v en t o this p os sibility by d esign ers u p t o t h e p resen t ti e Th ere
i s a p oss ibility th at for high -lift wi ng sections, in cl ina tion of t h e cho r d s ay
a tt a in a bl e, but e vi den ce is
b e us ed t o in crease t h e v alu e of t h e a xi u
m
.
.
m
m m
m
‘
m
.
m
m
.
n ot
m mm
m
l t
wm
p
y et co
'
e e
.
Secfi oa
g Flaps on a Bi plane as a Means of varying th e n
Wh en d es cribing t h e p ro p erti es of aerofoils it was en tion ed t hat t h e con
axi
di ti on of high
u lift co effi ci ent could not be obt ained at t h e sa e
ti e as a high r atio of lift to dr ag at low li ft coeffici en t s , an d, ech ani cal
m
m mm
m
m
m
t b
t th
pp
d f t h o i d tio of wi g
mb Th impl t f t h p p o d
g m t i th
g d fl p t ili g dg t
h f t h wi g
x m
pl
O
p d d i ill t t d i Fig 76 Th bi p l
mo d l
e a n ee
or
diffi culti es ap ar
ea rs o
er e a
e c ns
era
n
n s
e s
es
se
o
o f v ari a bl e ca
er
e
re
ar r an e
en s s
e
e
e o ea c o
a or ra n
e
n s
fi tt i n g of a hin e
ne e a
e
s
us ra e
ro u ce
n
o f t h e chan g es so
e
an e
e
h ad a gap equ al t o i t s chor d an d zero st agger ; each p la n e was
Th e rear p ortion of each win g w as
a n d t h e wi n d sp ee d 40 ft p er secon d
h in ged t o t h e fr on t p orti on t h e d i st ance fro t h e hin ge t o t h e t railin g ed ge
In one p ositi on of t h e flap t h e com
bein g
p l et e secti on
of t h e ch or d
easu r ed fro m
this p os ition an d wh en
was R A F 15 an gles of fl ap ar e m
t h e fla p s are d ep res se d t h e a ngl e i s d efin ed as p ositi ve
b er of
For a n um
10 an d 20 t h e bi p lane was t est ed
an gl es of fla p vi z
for li ft drag an d cen t re of p ress u r e for a r an ge of angl es of in ci d ence
Th e res ults ar e equi v al en t t o t es t s on six d i ffer ent a erofoil sections Th e
u pp er p ar t of Fig 76 show s t h e v ari a tion of lift coefli ci en t with angl e of
in ci den ce an d t h e effec t of d ep ressin g t h e flap i s seen t o be a g en er al
This in crease i s con tin u ed to t h e m
um bu t t h e
a xi m
i n creas e
oun t of t h e cha nge a bove t h e cr iti cal an gl e i s l ess th an t h at bel ow it
a
T h is i s a furth er illus t ration of t h e increase of m
lift coeffi ci en t
ax i
u
whi ch accom
p an i es an in creas e of u pp er su rface cam
ber
Th e cu r v es for lift /dr ag sh ow th at at low li f t coeffi ci ent s t h e sm
aller
bers h ave t h e high er effi ci ency an d th a t thi s p rop erty i s ain t ai n ed
ca m
b elow a lift coeffici en t of
wh en t h e flap s are slightly r a ised A refl ex
cu r va ture of 5 h as p rod u ced a loss of effi ci ency bu t h as h a d a m
ar k e d
R eferen ce to t h e thir d d i agr a
e ffect on t h e cen t r e of p ressur e coeffi ci en t
of Fig 76 show s a p rogressi v e ch an ge i n t h e sl op e of t h e cen t r e of p ressur e
—
a
t
e
fl
a
s
r
se
0
s
d fr o + 2 t o 5 t h e sl op e d ecreases
h
ai
cur v es and
p i
es a s grea t a s for R A F 15 to z er o o v er t h e rang e
fr omb ein g three ti m
on li ft coe fli ci en t
to
T h i s range cov ers all t he o r d in ary
of wi n g i n d i cat ed m
p ort a n t i n
ay b e i m
s t ea dy flight s p ee d s a n d t h e for m
fu tu re flyi ng cra ft es p eci ally as t h e tend ency i s to b ecom
e v ery s t a bl e
,
.
.
.
.
.
m
.
,
.
.
'
.
.
,
.
°
°
.
,
,
.
.
.
,
m
.
,
mm
.
.
m
,
.
°
'
m
,
.
m
.
,
?
.
.
,
.
°
.
APP L I ED AERODYN AMI CS
148
i n a n ose d i v e
t h e o r dinary ran ge
.
O3
02
'
U FI
It m
ay b e feas ibl e to obta in
with a sa fegu ar d a gai ns t one of
ut ra l sta bili ty for
t h e p ossi bl e pos iti ons
ne
04
LIFT COEFFlC IENT
CO EFFIC IEN?
F
m70 —B i pl
.
.
an e
wi t h wi ng
fl ap s
.
m
m
Cri teri on for th e Aerody na i c Advantag es of a Vari able Ca ber Wing
Th e a dv an t a ges d ep en d i n a gn it u d e o n t h e r esi st an ce of t h e p ar t s of t h e
aer o p l a n e oth er th a n t h e win gs , an d al s o o n t h e ch a nges of w e ight w h i c h
acco
p a ny ch a nges of win g area a n d t h e p rov i si on of co n t r ol s for t h e
ov e
e n t of t h e wi n g fl a p s o r oth er
e ch a n i s
for ch an gin g t h e sh a pe
of a wi n g
Th e w eight cons i d era ti ons are i n vari a bly i n O pp osite di r ec ti o ns ,
m
m
m m
m
.
m
.
APPL IED AERODYNAM108
150
that t h e in d ica te d a i rsp ee d for a gi ven v al u e of p
es
l andi n g sp eed Eq u ati on (9) now becom
.
is p
.
es as gr ea t
tim
as
t he
.
forms uit abl e
an d
thi s
Th e
form
u l a i s use d by p l ott in g
cu r
is
a
for t h e
v e ind i ca tin g t h e b est win g
If t h e ru l e be app li ed t o
I
fi
mp
co
ar i s n of
o
for t h e win gs on
various win g
a
base of p
,
sec
tions
.
t h e high es t
.
Fl o 77
.
—Com
p ari
.
so n
t he
v
se era
be t wee n fi x ed
l wings
for
which d et ai ls
an d v a ria b le sec t i on
wi n gs
are
.
gi v en in Fig 76 it will b e foun d th at t h e un b ent wing to R A F 15
section is t h e b est of t h e ser i es for a ny us u al t op s peed of a n aer eplan e
en t th a t R A F 15 i s
This resu lt su pp ort s oth er e v i d en ce for t h e st a t em
one of t h e b est sectio ns for a high sp ee d wi n g of fi xe d form
ent e d on a n d d escribe d i n
win g t h e m
odel e xp eri m
As a v ar i a bl e form
Fig 76 m
h av in g p rop erti es
ay b e r egard ed a s t h e e q u i v a l en t of a fi xe d fo rm
gi v en by t h e e n vel op e t o t h e li ft /drag cu rv e so l ong a s sta bility of fli ght i s
um
a xi m
lift coeffi
n ot u n d er con s i d era ti on
Th e a dv an t ages of a high m
e ti m
e a s a high r a ti o of lift t o
ci en t o f 06 4 5a re th en o bt a in ed a t t h e sa m
d r ag at low li ft coefli ci ent s Th e ori gi na l R A F 15 secti on i s com
p ar ed
an d t h e result s ar e
with t h e v ari a bl e section by m
ea ns of e q u a ti on
illus trat ed i n Fig 77 Th e l an d in g sp eed b eing t aken as u n ity Fig 77
Th e or di n at e i s t h e
sh ows t h e in di cat e d a irs p eed by t h e scal e of a b sci ssa)
It will be seen t h a t t h e fla p s are d i sa dva n t age
li ft /dr a g of t h e wi ngs al on e
.
.
,
.
.
.
.
-
.
.
.
-
.
.
.
.
.
.
.
,
.
.
.
.
DESI GN D ATA FRO M
u p to
ou s
l ti ve sp eed
AERODYN AMI CS L AB OR ATORIES
of
a re a
17 , b u t
t
af er
th a t
are
151
increasingly desi ra bl e
C O E F F IC I E
-
o
-
6
e
I4
a s
04
0 -3
F
at e of
tm
an es i
R1
W
18 re
04
m2
“
.
8
q u i red
,
.
05
—Va ri
e a
a t i on o f
an d a n ot
o
o n
t r i p la n e ga p
o r:
16
18
20
0 3
0 -4
0 -5
0
a s
.
un usu al valu e w oul d be 003
.
Wi th
this v alu e i t a pp ears th a t t h e rati o of lift t o d ra g i s i m
p roved by 5p er cen t
for p
2 an d by 11 p er ce n t for p
3
For a lan d in g Sp eed of 40
.
.
.
.
APP L IED AERODYN AMICS
152
m
m
m
p ee d of 80 p h near t h e groun d or 93 p h at
f eet
3 co rresp on ds with 120 p h n ear t h e gro un d an d 140
h
Si
p
f eet Th e effect of v ar i a bl e sec tio n on t h e to p Sp ee d o f an
at
ay be v alua bl e esp eci ally on lig htly lo a d e d wi n gs
aero p lane
Th ere i s
p o rt an t effect on t h e ra t e of cli b t he u se of fla p s
a s all but l es s i
Th e p ercent ag e loss on t h e ratio o f li ft
r ed u ci n g t h e efli ci ency slightly
to dr ag i s 4 but i n an aerop l an e whi ch cli b s rap i dly t h e p report i o n at e
ore tha n one thir d of thi s n ear t h e
a dd itio n to t h e thr us t wi ll b e littl e
r ea of wi n gs be un ch an ge d fl ap s can a lw ay s be u se d as a
a
ou
d
If
t
h
e
r
n
g
ean s of re d u cin g t h e l an d in g sp ee d ; for t h e exa p l e shown t h e ex t r e e
sa v in g would b e 15p er cen t
No Sta t en— Th e ser i es of
Changes of Gap i a a Trip lane R A F 8
en t s to be d es crib ed corr esp on d s with t h e so
ewh a t s i
e xp er i
ilar s er i es on
bi p l an es t h e ran ge of gap t o ch or d ra ti o bei ng 0 5to
with t h e a d d i tion
p arati v e t est for a ono p l ane Th e n u m
er i ca l r esult s are gi v e n i n
of a co m
T a bl e 14 an d cu r v es h av e b een dr aw n i n Fig 78 fromt h e d at a of t h e t a bl e
p are d with Fig 74 for a bi p l an e fro whi c h it
Th e fig ur e sh ou l d be com
wi ll b e seen th at t h e g eneral ch aract eri s ti cs of t h e cur v es ar e t h e sa e
Fig 78 show s th a t a gap of twi ce t h e ch or d i s app reci a bly r e oved fro m
a n infin i t e gap a n d a l so h ow sens iti v e i s t h e flow of a i r r o u n d one wi ng to
Th e d et a il ed resu lt s a re gi v en ch i efl y for t h e
t h e p r esence of a n oth er
p ur p oses of referen ce as t ri p l a nes ay be i m
p or t an t i n t h e d evelop m
e n t of
v ery l arge aerop l anes Fur th er p ar ti cu l ars can be foun d i n t h e re p o r ts
o f t h e N a tio na l Phy si ca l L a bo ra tory t o t h e Adv i s o ry C o
i ttee for
Aerona uti cs
2 eans
ilar ly , p
m
a s
.
.
m
.
.
.
m
m
.
.
.
.
.
m
m
,
.
,
m
.
m
,
m
m
.
.
-
“
m
m
.
.
m
.
m
,
.
m
m
.
.
.
m
m
.
,
.
,
.
m
.
"
,
m
.
,
m
m
.
.
TAB LE 14
CHAN G ES
Siz e
.
o f each
or
.
G AP/CH O RD Re n o . R A F 6 Ta
.
o sta
er
an e, 3 x 1
l
N
8
g
g
p
L i n : Co s i o s r
'
'
.
mm
.
.
Wi nd
.
Gap ]Ch ord
.
.
s
m
a ra
peed ,
.
40 ft
.
-e.
154
APP L IED AERODYN AMI CS
TAB LE l M
m
Car
a
or
Mi n ued
Pa nacea : Co nf
.
u ci an
+00 72
.
AERODYN AMI CS L AB OR ATORI ES
D ESI GN D AT A FROM
155
m
—
f
T
fi
h
n
s
f
l
n
e
o
o
a
a
Th e seri es of exp er i en t s re la tin g
C a g
SW
a
p
t o t h e chan ges of s t agger of a t ri p lane aga in cl osely foll ows th at of t h e
"
si
il ar work on bi p l an es ; t h e section was R A F 6, each p l ane 8 X
an d t h e s peed of t es t 40 ft s
Th e gap to chor d rati o was un ity , and t h e
°
0 an d
t h e r esult s be in g coll ec t e d i n T a bl e
an gl es of st agger
15an d ill us t rat ed i n Fig 79
Fro t h e curv es of lift to drag rep ro du ced
i t will be seen tha t st agger h as littl e eflect on t h e ax i u value but
t h at forwar d st agger i s
m
.
.
.
-
.
.
.
.
m
m mm
’
,
lightly a dv an t ageous
a xi
Th e increase of m
mumlift coefli ci ent be i t
tween backw ar d an d
f orwar d stagger i s con
s idera ble
Th e chi ef
n ew
in t erest i n t h e
ta bl es i s t h e in cl u s i on
of m
eas urem
en t s of t h e
p os iti on of t h e cent re
of p ressur e B etween
t h e an gl es 0 a nd 14
e ith er forwar d or ba ck
w ar d sta gger p ro d u ces
en t of
a forwar d m
ov em
t h e cen t re o f p ress ur e
t h e am
v arying
oun t
from01 of t h e ch or d
d own t o a bout on e
t enth of thi s For t h e
b ack st agger t h e ch an ge 2
o f po siti on
i s near ly
u n iformand e q u al to
007 o f t h e cho r d ; on e
of t h e r esu lt s of this i s
that t h e m
o v em
en t of
an u pp er wi n g b ack
w ar ds i n or d er t o adj u s t
t h e p os it ion of t h e cen
t r e of gra v ity rel a ti v e
Fro 79 —V ri t i
t gg
of t i p l
to t h e wi n gs i s p a rtly
n ullifi ed
Th e s l op e of t h e cu rv es of cent re of p ress ure ar e i n t h e order
zero s t agger back s t agger a nd fo rw ar d st a gg er t h e d i ffer en ce b etween t h e
ark ed a t or d i n a ry flyin g a n gl es
fi rs t an d las t b ei n g m
Th e t a bl e em
ph a
si ses t h e di ffi cu lti es wh i ch ar e e nco u n t ere d w h en an a ere plane with
od ifi ed aft er t h e fi rst t ri a l fli gh t s
dou btful s ta bility h as t o be m
I n or d er
t o a v oi d su ch ch anges it app ears p r ob a bl e th a t t h e d esign of an a cr e
p l an e in t h e n ear fu tur e will b e b ase d on t es ts of m
o del s of p ar t s and
fina lly ch ecked by a tes t on a com
p l ete m
od el
There i s ev ery reas on t o
be li e v e tha t su ch tes ts fo rmt h e m
ost r eli a bl e gu i d e a va il a bl e
s
.
.
.
°
°
,
.
-
.
.
a
on
a
r
a ne s a
e r.
.
,
,
,
.
.
.
.
APP L IED AERODY N AMI CS
156
Ca
S ize
( d ecrees )
.
m
u s
or
p la ne, 3 x 18
'
of each
Ang le
TAB LE 15
o f sta gge r
Su
m
R A F 0 Ta
s a na ,
.
.
.
as.
Wi nd speed
'
.
Angle
.
,
40 lt
.
of st agger.
.
°
30
1
°
0
Cent re
of
p ressure coeffi ci ent
Angle
( degrees)
mt
Mo
.
coeffi cient .
en
Ang le of st ag ger
of st agger .
.
.
00 19 +0 045 +0004
0 538
0 4 10 —0 227
+0 28 4 2 2
+0 8 00 + 157
0 598
0 777 +0 033
0 4 83
0 582 +0 545
404
0 48 7
0 3 54
0 3 58
0 44 5
0 3 38
0 335
0 3 14
0 408
0 3 18
0 38 4
0 300
0 308 , 0 3 00
0 2 01
0 302
0 3 50
0 28 2
0 290
0 3 20
0 205
0 284 I 0 3 13 1 0 250
0 28 5
0 302
0 33 0
0 280
0 29 3
0 227
0 274
0 28 2
0 3 23
0 28 1
02 78
0 401
O 28 0
0 3 18
04 12
+O OO47 +0 005
-
’
—00 158 —0 0154
—0 0242 002 12
-
-
o
-
00320
—003 8 2
—00465
—0 l 34
‘
‘
-
0 l 53
°
.
.
~
—00307
00344
003 08
0 04 71
0054 7
—00512
00504
0 0750
- 0
0845
0 083 4
00888
0 108 5
—0 144 - 0 155
0 14 8
0 101
—0 104
0 171 1
-
-
‘
.
0 03 28
o 03 09
0 04 79
0 0585
0 08 8 2
—0 0755
0 08 3 2
—0 0905
0 0055
0 107
0 117
0 128
-
-
—0 l 7l
'
-
—
—
.
s.
APP L I ED AERODYN AMI CS
158
TABLE l i t
—30
°
Up p er p lane
Lift
coe ffi ci ent .
—co nti n
d
ue
Sr a oo an
.
.
.
Drag
coe ffi cie nt .
0124
008 9
m
—
Th e n u b ers in T a bl e 17 sh ow th a t t h e u pp er p lane t akes
Tri ph
0 to
t h e a oun t b eing
t h e gr ea t est lift o v er t h e ran ge of an gl es fr o
At t h e s all er angl e t h e
i ddl e p lane
a bout 4 0 p er cen t of t h e tot al
t akes 3 4 p er cen t of t h e t ot al and t h e lower p l ane 25p er cent but at 6
na
.
.
m
m
°
m
m
°
.
.
,
D ESI GN D AT A FRO M
AERODYN AMI CS LAB OR ATORI ES
159
p roporti ons ar e 28 p er cen t an d 3 4 per cent res p ecti v ely
i ddl e p lane i s di sa dv an t ageously
Th e in d i cati on h ere gi v en t h a t t h e
p l aced i s su pp ort ed by t h e co p ari son of t h e a xi umval u es of t h e li ft
i dd l e p l a ne
t o d rag ra ti o whi ch a re 151 for t h e u pp er p l an e 9 5 for t h e
I n all cases t h e ra ti o i s l ess than tha t of t h e
a n d 116 for t h e l ower p l an e
monoplan e val u e of 166
il ar result s an d it
en t s on bi p l an es an d t ri p l a n es all sh ow si
Exp eri m
i s cl ear th a t t h e ai r flow roun d a win g i s ser io u sly o d i fi ed by t h e p resen ce
This
of an oth er wi n g sep ara t ed fro mit by any p rac ti ca bl e dis t ance
sens iti vi ty i s w ell known t o w ork ers i n win d chann el s who fin d it necessa ry
t o a v oi d t h e us e of an y hol d in g app ara t u s oth er th an fine wir es n ear t h e
o d el
u pp er an d low er s urf aces of a wing
a nd
°
12
the
m
.
m
.
.
mm
m
,
,
.
.
m
-
m
,
.
,
m
.
TABLE 17
.
R A F 15Tarn
.
No st agger .
Gap /Ch ord
.
.
Si ze
a
of eac h
m
s.
p lane , 3
'
Wi nd speed
x
,
40 ft
.
s.
-
coe ffi ci ent .
P a Es sUa E Di sr ara u r rox
o n TH E
m
Wm
os
or
A
B I P LA N E
m
Exp eM en t s t o d et er i ne t h e n or al fl u i d p ress ur e on a bo dy
a d e bo th
in flight an d in a win d ch annel , an d p ro vi de
ca n b e
best co p ar i sons b etw een t h e full scal e and
od el
o n e of t h e
Th e exp eri en t s t o b e d escri b ed w er e
ad e
ch aract eri sti cs of wi ngs
wi ngs of a n aerop l an e whi ch di d n ot ha v e t h e stand ar d
od el
on
f or
secti on , but
Th e cen t ral p art s of both win gs w er e of u nifor
o d i fied as i n d i ca t ed i n Fig 80 Th e
a t t h e en d s t h e sh a p e was
m
m
m
.
m
m
m
m
.
m
m
.
.
APP L IED AER ODYN AMIC S
160
es th a t of t h e chor d
h p l ane was 56 tim
t h e s tagger was
+23 an d t h e ra tio of gap to chor d 08 84 Th e sh a p e of t h e wi ng ti p s an d
of t h e
sectio ns are defin ed by t h e co n to urs of Fig 80 t h e u pp er
l en gth of
eac
,
°
.
.
.
Fro 80
.
.
—M d
o
el
fo r p res s u re di st ri b u t i o n
on a
wi ng
.
figur e showi n g t h e sh ap e of t h e t 0p s u rface of each p l ane
fi gu re t h e co ntours of t h e un d er sur face
Th e sh a pe of t h e
i s shown , an d in it a re arked t h e hol es a t whi ch t h e p ress ur es
m
.
,
lo
t h e wer
cen t ra l secti on
a nd
APP L I ED AERODYN AMICS
162
il ar
ewh a t s i m
Som
FI G
re
ar
8 1—I rens u re
’
.
mk
.
s a
pp ly
d is t r i b u t io n
t o t he
on
t he
lower p l ane
u p pe r
w i ng
of a
and
Fig
b i p la ne
.
8 2, b u t
.
with t h e a dd iti on of o bserv a ti ons at an a ngl e of i nci d ence of
Th e
gr eat est su ct i on th en occurs on t he u nd er su rf ace and t h e d i fferences
,
D ESI GN D ATA FRO M
Fro 82
.
—Press
.
ur e
AER ODYN AMI CS
di s t ri bu t i on
on
LAB OR ATOR IES
t h e lowe r wing
of a
b i plan e.
163
APP L I ED AERODYN AMI CS
164
of
s
p r essure
a ll
m
.
mti
f i t be twee n
coe fi c en
t he
t 0p
an d
bo ttomof
m
t he
wi ng
a re
—
Drag fro
h
r
ti
n
s
s
Th e p ressu r e on
t e Ob e va o
eas u r e d by th a t a t a h ol e i n it i s assu
a lly
a s u rf ace
e d t o a ct n or
I n os t ca ses o f a bo d y h a v i ng res is t ance th ere a re t angenti al for ces whi ch
a re no t es ti
a t ed by t h e u s u a l
et h od s o f p r es su r e d et er
i nation An
e xa
p l e o f ea su re en t s o f li ft a nd d rag on an a ero foil by t h e in t egra ti on
whi ch t h e e ffec t o f
o f p ressu re a n d by a d i r ec t p roces s i s gi v e n l a t er , fro
t h e t a ng en ti al co p o nen t s o f fo rce can b e es ti a t e d
en t gi v es only
Acc ep ti ng t h e i d ea t h a t t h e u s u a l
eth o d of
easu r e
t h e p ress u r e nor al t o a s u rf a ce it i s p oss ibl e t o d ev el op a si p l e r u l e
by wh i ch t h e li ft a n d d rag coeffi ci en t s a t a ny secti on ay be fou n d Th e
p roced u re i s i ll u s t ra t ed by t h e d i agra s of Fig 83 I n t h e cen t re i s a
d ra wi ng o f t h e wi n g secti on , an d a tt e n ti on i s co ncent rat e d on t wo p oin t s
Esti
s
on of
m
m
m
m m
Lift
and
m
m
m
m
m
.
.
m
m
m m
m
m
m
m
m
.
,
m
.
.
.
LOWER SURFAC E
w
e r e s u a r ac r
w
m Dm
o
ccn on
UPP ER S URFA CE
er b ei n g on t h e u pp er su rf ace an d t h e l a tt er on
f orm
I f p oin t s on t he wi n g su rface be p roj ec t e d
t h e u n d er su rface of t h e a erofoil
al to a ny lin e an d o r d in a t es eq u a l t o t h e p ress ure at t h e p oin t b e
n orm
ea su r e d off on t h e li ne of p roj ecti on t h e a rea of t h e r esulti n g cu r v e f orm
m
ed
by p oi n t s all o ver t h e su rface i s t h e force coeffi ci en t in t h e d ir ection of
p roj ecti on I n Fig 83 t he p roc ess h as been carri ed ou t for t wo d irections of
p roj ection one of whi ch i s n orm
al t o t h e ch or d of t h e secti on an d t h e o th er
In each i nst ance t h e con t r ib u ti on s of t h e u pp er an d low er su r
al ong it
s
faces are sh own i n sep arat e d i agr am
i ne t h e
Th e areas su ffi ce to det erm
magnitu de and d irection of t h e resu ltan t ai r force
—
i
l
r
C
h
e
r
n
u
r
P
d
t
o
t
h
e
i
Ou t h e u pp er su rface t h e p res su r es
Force e p e
c a
h av e b een sh own as n egati ve at all p oin t s; an d h av e b een u se d to formt h e
l ow est di agram Th e p oi nt a on t h e win g h as been p roj ect ed d own
w ar d s and t h e wi d th of t h e sh ad ed figu re at a i s equ al t o t h e p ressur e
coeffi ci ent
Th e ar ea is t h e force on t h e u pp er sur face in a d ir ec tio n
all d i agr am
al t o t h e ch or d
To t h e l eft i s a sm
of t h e force coefii ci ent
norm
a
and
t he
b,
.
,
,
.
.
'
,
.
.
-
.
,
.
.
.
166
APPL IED AERODYN AMI CS
C as s a va
Pressu res i n lbs per
.
s u r fac e.
m
m
-
q
s
Pa usa
os s o r
.
ft lyfi
.
’
.
.
Wi nd speed
,
50 ft
.
s.
-
AERODYN AMI CS LAB OR AT OR IES
D ESI GN D AT A FROM
m
L
u rn
Win d
( decrees )
DI
NG
sp eed,
Co r
mm
nr s
c
ft
502
.
.
-s .
.
04 53
20
05
0 3 73
e 3 25
0 3 15
0 284
04 09
02 09
t5
0 369
-
°
00383
00751
003 20
00627
003 27
00559
0 110
0 19 1
°
01 63
0 190
0 2 17
0 273
0 169
200
02 54
°
°
Drag Coe ffi ci en t
0
8
12
16
20
24
28
32
40
0 202
0 232
0 28 7
00036
00344
0063 9
0 115
0 253
0 299
°
.
00040
003 3 2
00677
0 103
01 8 8
( No correct i on
md
a
e
for ski n fr i ct i on )
.
167
APP L I ED AERODYN AMIC S
168
Ca
m
s
mm
PR ESS UR E Co x
or
o
xr
.
Lower wing
m
m
—
B
a
l
a
n
F ro t h e fi gures gi v en in T a bl e 19
c
e
t hose
for t h e lift an d dr ag co effi ci ent s a t v arious sections , i t i s p ossi bl e t o
Thi s h as been d o n e
t h e r es ult s to fin d v alu es for t h e whol e win g
su
i n on e of t h e R ep o rt s of t h e N a t i on al Phy si ca l L a bo ra t ory , bu t t h e r es u l t s
are not gi v en h ere b ecau se t h e a ngl es of i nci d ence h av e been chosen a t
wi d e i nt erv a ls , a n d a ore co p l et e seri es for t h e flyi ng r ange e xis t s w hi ch
e co nclus i ons i n gr ea t er d et ai l
shows t h e sa
Th e cur v es of co par iso n
are shown i n Fig 85
, wh er e t h e lift a n d dra g co effi ci en t s , t h e ra tio of l i ft
to dr ag an d t h e cen t re of p ressur e coeffi ci en t s are drawn for angl es o f
°
—
r
both a s ea sur ed i n t h e o r din ary way an d a s
2 to
i n ci d ence f o
eas ur ed by i n t egr a tion of p r essur es
Th e li ft coeffi ci en t cur v es for t h e
two eth ods ar e indi st i ngui sh a bl e within t h e or d er of accu racy of t h e
en t , and are t h e b est j us tifi ca tio n whi ch exi st s for t h e ass u
i on
exp er i
th at t h e n or al p ressure on a su rface can b e eas ured by th a t on a s all
h ol e at t h e p oi n t consi d ere d
Th e cu r v es for dr ag c o effici en t sh o w
ea s u ra bl e d i fferences w h i ch a r e r ep ea te d i n t h e cur v es of lif t to dr a g
Th ese d i fferences a re of su ch agnitu d e a s t o b e reasona bly re gar d e d as
°
d u e to su rface t ractions a lth ou gh t h e cu ri ou s p oin t a pp ears th a t a t — 2
t h e r esu lt an t f orce d u e t o s ki n fr i ctio n i s n eg ati v e
Th er e is no r eas o n
easured
di rect ly
on a
m
.
m
m
m
m
.
.
m
m
m
m
m
m
m
.
m
m
m
m
.
m
.
,
m
.
doubt thi s o bserva ti on eith er on exp eri en t al or th eo reti ca l gr oun ds
a lth ou gh t h e d et a il ed exp l a nati on i s a t p res en t beyo n d t h e p ow ers of
en t a l analy s i s
e xp eri m
Th e cu r v es o f cen t re of p ressu re show an agreem
en t a l m
os t as cl os e as
th at of t h e li ft coeffi ci en t cur v es an d for t h e i m
p o rt an t p u rp os e of t h e
ca l cul a ti on of s t resses i n a win g t h e r esu l t s of p r essu r e d i st r i buti on e x
ri m
e
n
t
s
a
a
b
e
e
e
m
pp
li
d withou t co rrecti on Th ey a dd v ery a t er i ally
y
p
to t h e p ossibi liti es of gu ara n t eein g t h e safety of a d es ign
to
,
.
,
m
.
.
Sr nu r s ,
St r u t s
t
,
s t r u c u re,
wi res
bu t
Wre ns
AND
bl es a re u se d i n a ll t h e m
a i n p a r t s of a n a e r o p l a ne
a lw a y s i n t h e wi nd
Su ch p ar t s as go t o t h e m
a k in
g
a n d ca
a r e no t
Ca nne s
.
APP L I ED AERODYN AMI CS
170
a d e of t u bin g of cir cu l ar sectio n a n d for t h e e x ter n a l fo r m
t t to be m
to be gi v en by a li ght woo d an d fa bri c fai r in g p i ece Th e r esi sta nce of a
ay be o nly 7 p er ce n t or 8
s t r ut m
a de m
thi s way m
er cen t of th a t of t h e
p
tube whi ch i t en closes A som
ilar savin g of resi st an ce a r is es
ewha t s i m
o un t
fro mt h e us e of s t reamli ne wi res i nst ead of wires or cabl es t h e am
bein g a red uction to 15p er cen t or 20 p er cen t of th e resis tance of t h e
c ircu l ar fo rm
s
A co nsi d era bl e d egr ee of fa ir in g i s o b t a in ed by fil lin g
i n t h e sp ace betw een t wo ca bl es so as to m
ake a parall el s i d ed figu re w i th
i circu l ar en ds Th ere i s a p r on oun ce d scal e effect on s t rut form
se m
s w hi ch
i ca l si m
il ar ity b u t it
h as bee n fully d ealt with i n t h e ch ap ter on d yn a m
i s p ossibl e i n a wi n d ch ann el t o sati s fy t h e co n ditions of corres po n d in g
s p ee ds an d so to obt a in r esult s d ir ectly a pp li ca bl e in fl ight
In t h e ser i es
o f s t r ut s to be d escri be d t h e equi v al en t s p eed s reach e d on a s t rut on e i n ch
an d two thi r ds of thi s am
thi ck are 160
o u nt for a st ru t on e a n d a
ha lf inch es th i ck Moreo v er t h e gen er al law of var i a tion i s kn own a n d
show s tha t t h e coeffi ci en t s app ly for a v ery wi d e ran g e of Spee d in clu di n g
s ru s
,
.
.
.
.
-
,
.
.
.
-
.
,
,
.
-
.
,
,
,
Str ata —A
i of s truts of t h e sh a p es shown i n Fig 86 was t es te d
ens io n
l
i n a win d cha nn el t h e d im
be in g t h e sa e for each so t h a t
of f a ir in g for a gi v en ci r cul ar tube
a ny one coul d b e a fo r
Th e sec tio ns
h av e l en gths 21
ber
8 1 8 5i 4 1an d 51 an d are t h e out co m
e of a n um
of p revi ous e xp eri ent s d ir ect ed to fin d form
Th e
s of l eas t res i s t an ce
bers of t h e sec ti ons in Fig 86 h av e been ch osen so as t o i n d i ca t e t h e
n um
ra tio of l en gth of sectio n to i t s b rea d th
Th e dr g of t h e st r ut s h as b een e x resse d i n term
s of a res is t ance
a
p
whi ch t h e ap prOp ri at e ar ea i s t h e p roj ect ed area of th e s t r ut
co effi ci en t m
i n t h e di rection of t h e win d
u l a 18 gi v en i n t h e figur e an d will
Th e fo rm
be seen to gi v e a d rag coeffi ci en t whi ch co m
p ares d irectly with th a t of
a s q u ar e p la t e n orm
al to t h e wi n d for
whi ch k
oo t h
60 A sm
cylin d er h as a co effi ci en t whi ch i s equ a l t o
t h e e xac t val u e de
1
p en d ing on t h e si ze and sp ee d For t h e s t rut s t h e v alue of kn i s ins e t in
t h e sec tio n an d show s a r ange f ro m
0058 for t h e short es t sectio n to 004 1
r
e
for a s ecti on o f fi nen ess r a tio f o ur
Thi s l a tt er v alu e i s seen to be
p
cen t of th a t of t h e l ar ges t ci r cul ar cylin d er whi ch co u l d be en clo sed
Th e v a l u es of t h e drag coeffi ci en t sho u l d not b e us ed for sm
a ll s t r uts
su ch as o ccu r i n m
o d el s or for s t reamline wir es but ar e dir ectly app li ca ble
to win g st r ut s an d un dercarr i age st rut s
It will b e seen fr omt h e v al u es of t h e d rag coeffi ci en t th a t no app reci a bl e
a dvan t a ge ar i ses on thi s accou n t f rom
t h e u se o f a st r ut of finen ess ra tio
an 2 5 b u t on t h e o th er h a n d n o d i sa d v an t a ge 1s in curre d by t h e
t
e
r
th
a
e
r
g
u se o f l on ger s t r u t s
It 13 fo u n d th a t t h e fl ow of ai r r oun d a shor t st rut
i s ex t rem
ely sensiti v e t o s m
all e rr o rs of m
a n u factu re or of settin g alo n g t h e
d i recti on of t h e wi n d an d for t hi s reas on choi ce h as t en ded t o a fi neness
ra ti o of 8 5 or 4 s in ce ext r em
e sensiti vity i s th en a v oi d ed
For a s t ru t of
sec tion No 4 cu r v es ar e gi v en i n Fi g 8 7showin g h ow t h e dr ag an d cro ss wind
force d ep en d on t h e in clin a ti on of t h e p l ane of sym
metry t o t h e wind
Th e d i s po s itio n of t h e s t ru t a n d t h e sig n of t h e for ces are d efi ned by a s m
all
i nse t d i a gra mmFig 87 wh er e t h e forces for i ncli n a ti ons u p to 8 5 are
ser es
m
,
m
m
,
.
,
.
°
,
,
,
.
.
.
.
,
D
.
.
,
.
.
.
-
,
.
°
,
.
,
,
.
-
.
.
.
°
.
,
D ESI GN ; D AT A FR OM
s
hown
.
In
p ite of
S
t he
AERODYNAMI CSIL AB OR AT OR IES
fact tha t co ns i d era bl e
was
D r a g i n lb s
R
L
care
Le n g t h
ll
of
e ercis e
x
171
d in t h e
.
S t r u t i n fe e t
s lu g s
P
Ai r d e n s i t y i n
V
Ve loc i t y i n fe e l p c t
per
.
cu
s ec
b i c ft
.
.
R
STREAM LINE WIRE
to was at 0 angl e of in ci d ence an d wi ll be seen t o be
whi ch i s only 40 p er cen t of th at for t h e st rut wh en
ar k e d ch a n g es occ ur as will be seen
At th ese a ngl es m
curv e of Fig 87 an d t h e d ra g ris es v ery ra p i d ly un til a t
°
,
.
i n clin ed a t 1
f r o mt h e u pp er
,
.
,
APP L IED AER ODYN AMI CS
172
m
es i t s
h
i x teen t i m
in i m
v al u e B etween +15 an d
um
—
2
5
t
h
e
d
g
i
s
u
h
l
o
w
e
r
th
a
n
th
t
a
ra
m
a
t
c
15 to
an d t h e d i fferen c e
+
i s st r o ng evi d en ce of criti cal fl ow su ch as has been observ ed on m
an y o th er
o ccasi ons The sp ec ul a tion ari ses as to wh eth er t he low er dr ag a t 20
co rres p o nds with t h e ty p e of fl ow u p to
an d it i s p ar ti cula r ly i n s u c h
18 5
°
it
reac es s
°
.
°
°
°
.
~
30
-
25
-
20
-
l$
-
lo
“
5
O
F 111 8 7 —Force s
.
.
05
on a n
20
i n clin ed
25
30
st r u t .
s t an ces tha t ou r l ack of su ffi ci en t p ow ers of
um
a ti ca l a na lys is
ath em
of t h e fl u i d m
otion i s so p rom
i nen t
Th e cross wi n d force (or lift if t h e s t rut b e h ori zo n t al) is s hown i n t h e
l ow er di agr a mof Fig 8 7 O ver t h e range 0 to 1 10 t h e st rut be h a v es as
a n or di n ary ae rofoi l a n d gi v es a f orce i n t h e d ir ec tio n exp ec t ed
For a
s h or t er s t rut th ere i s a t en d en cy for thi s p a r t of t h e cur v e t o b eco m
e
r e v erse d in sl o e so th a t t h e f orce on a n i n cli n e d s t r u t i s i n t h e d ir ec ti on
p
m
ci rc
.
-
°
.
°
.
.
APP L IED AERODYN AMIC S
174
It will be seen fr omt h e firs t colu m
n of T a bl e 20 th at t h e res is t an ce of
two circular wires i n con t act i s l ess th an that of a sin gl e wire so lon g as
t h e angl e d oes not excee d
an d th a t u p to 20 t h e shi el di n g i s m
ar ke d
et ers t h e shi el d i ng i s a pp rec ia bl e
Ev en with t h e wi r es sep ara t ed by si x di am
wh en t h e rear wi re is nearly i n t h e win d d i rection rel ati ve to t h e fro n t on e
If t h e gap b etween t h e wires i s fi ll ed to t h e l i nes t o u chin g t h e cyli n ders
a fu rt h er r ed u ctio n of r es is t an ce i s obt a in e d
Ex p resse d as a drag coe ffi
d efined p rev ious ly for st ruts t h e r es i st ance coeffi ci en t for
ci en t o f t h e fo r m
wi res three di am
et ers a p ar t i s
or n early th a t for a l en ti cul ar st rea m
lin e wi re Thi s shows a figu re of
p ar iso n with t h e
gi ve n
for com
i n t h e a bo v e t a bl e for z ero angl e of a tt ack
Th e sh i el d in g of one l enti cular wir e by an oth er i s n ot so grea t as th a t
of t h e cyli n d ers as m
ight b e exp ect e d f romt h e fact th at t h e r esi s t an ce
coe ci ent of t h e s in gl e wir e i s so m
u ch l ess th an th a t of a cylin der U sin g
t h e grea t est d i m
ens i on of t h e sectio n of a l en ti cul ar wi r e as a u ni t by whi c h
easur e t h e sep ara tion of a pai r l eads t o T a bl e 21
to m
°
.
.
.
,
.
.
m
.
.
B au
m
Da
m
or
A
Para
or
D ist a nce bet wee n ce nt res i n t er
mm
Lx
c
a
Wre ns
.
m mi mmdi m i
s of
ax
ens on of
u
o s4
u se
103
res
-
011
7
m
I
e
we
111
2
t abl e shows v ery cl early th at t h e effect of p uttin g one s t ream
li n e
wi re b ehi n d an oth er ay b e to b rea k u p t h e air fl ow su fii ci ent ly to gi v e
Th e las t col u n of t h e
a r es i s t an ce great er th an th at of t h e wi res ap ar t
t a bl e shows t h e p ro po rti ona t e in crease of r esist ance of a sin gle wire du e t o
i nclin at i on a nd i t will be noti ced th at u p to 10 t h e coeffici en t i s not ch an ged
by m
o re th an 5p er cent
Th e
m
m
.
°
,
.
U n dercarr i age st rut s an d wi ng st ruts are frequ en tly set i n an aero p l ane s o
th a t th ei r l engths are m
o re than 20 away fro t h e norm
al to t h e win d
d i rectio n It i s di fi cu lt to gi v e a p recise es tim
a t e of t h e effec t of thi s
i nclination s ince t h e m
e im
p ort ance
etho d of d eali n g with t h e en d s i s of so m
Th e t a bl e below shows how t h e f orces on a wi re an d on a s t rut are affect e d
al p r es en t atio n b ein g coun t ed
by l engthwise incli nation t h e d rag for norm
m
°
.
.
,
of t h e cylin dri cal wi re it was foun d th a t t h e force along th e
wir e was always sm
ore than 6 per cen t of t h e a x im
a ll an d n ev er m
u
drag It wi ll be seen th a t t h e vari ation of dr ag with angl e of in ci d ence
i s rou ghly as sin 3 a for th e wir e an d as si n a for t h e st rut t h e d ifferen ce
In t h e
m
case
.
.
'
.
,
m
D ESI G N D AT A FR OM
m
AER ODYN AMI CS L AB OR AT OR I ES
175
For t h e st rut it i s th erefore a dv i sa bl e to s p ecify
b e i ng v ery ar ked
t he li ft t o d rag ratio , a n d thi s i s gi v en i n t h e l as t colu n of T a bl e 22
m
.
l
a so
.
TAB LE 22
.
A ng le
d eu ce,
a
of
ldu
lnci
m
~
reee)
.
m
Cy ii nd r i l wi re
D ra rat i o
8 111
’
.
.
m
m
ore v ar i abl e fro one aerep la ne to
B ody Resi stance —Th e bo dy i s
It 1s d es igned to carry t he
an oth er th an i s any oth er p ar t of t h e craft
ow
r p la n t
h
e fo re en d, t h e p ilot an d p as seng ers in t h e cen t re an d t h e
t
e
p
t a il organ s at t h e rear So eti es t h e en gi nes ar e ou nted b etween t he
w i n gs and co v ered by a fai r in g It i s essen ti al for full acc uracy
tha t a p re p osed desi gn of bo dy shoul d be sub itte d to exp er i en ta l
d et er in ation on a o del Wh en consi d erin g t h e con t ri butions of each
p l e of a
ai n i t e s of an a ero p l an e t o t h e tot al r esi st ance, an e xa
of t h e
o d el bo d y , co p l et e with engin e, cowl an d tail sur faces will be referred
to In t h e p res en t p aragrap h , how ev er, att en tion i s drawn to t he in creases
p any s uch d efor ati ons of st rea lin e for as t h e
of r esis t a n ce whi ch acco
O p en i ng of a cockp i t an d t h e p ro vi s ion of win d shi el ds
Th e o d el us ed , togeth er with i t s o dificatio ns, i s illus t rate d i n Fig 88
o d el h ad a s qu are section of whi ch t h e ax i u
Th e o r igina l si p l e
l ength of si d e was 2 5 in ch es Th e o v erall l engt h was 24 inch es , an d t h e
test s w ere ade at 40 i t s For t h e p urp oses of co p ari so n with oth er
dr ag coeffi ci ent s of this ch ap t er t h e r ed u ction t o a non- di ensional for
2
i
n
by d i vi d g by pSV h as been a d o p t ed , S in thi s case bein g t h e p roj ecte d
o di fied bo dy i n t h e d ir ection of th e wi n d Thi s area w as
area of t h e un
00484 s qu are foot
m
.
mm
.
m
m
.
m
m
m
m
m
m
.
m
.
m
m
m
m
m
'
m
m
m m
m
.
m mm
.
m
°
.
-
.
.
m
.
m
m
.
.
TAB L E 23
.
p
m
coeiti cle nt .
behi n d t h e p ilo t
9
It
.
.
'
h ea t of wi nd shield cu t back st ill fu rt her
B ot h front and ba ck wi n d shi elds i n p os i ti on
s
h ead
Veloci t y o ver pi lot
'
s
h
ea d
m
V eloci t y i n t ree st r ea
APP L I ED AERODYN AMI CS
176
d rag coeffici en ts as d efined a bo v e a re coll ected i n T abl e 28 togeth er
Th e
,
bl e t h e ch anges of formt o be c o rrelated
“
with a
to t h e
uffi ci en t d escri p tion to
eas ured res i s tan ces
m
s
.
ena
APP L IED AERODYN AMICS
178
fo r ( b)
Th e effec t of t h e wi ngs i s seen t o be a red u ction of body res i s tance ,
an d t h i s wo u l d b e exp l ai n ed by t h e slowi n g of t h e ai r st r ea
du e to t h e
r esi st an ce of t h e wi ngs
I n so e aere plan es t h e ai rs cr ew i s b ehi n d t h e bo dy , an d t h e v ar i a ti o n
o f resi s ta nce with th r u s t w ou l d not b e exp ect e d to be so gr ea t as i n t h e cas e
o f a t ractor
It a pp ears, h ow ev er, th a t t h e for of t h e bo dy j us t in fron t
ar ke d e ffec t on t h e bo dy r es is t ance facto r , an d
of t h e a i rscrew h as a
ay be as grea t a s th a t i n a t rac to r bo dy
t he
An e x a p le
m
.
m
.
m
0
Fro 8 9
.
03
a se o f
body
p ush er bo dy i s illu st ra t ed in Fig
v ari ation was still lin ea r and equal t o
.
R
R0
06
04
res ist an ce
of a
m
m
.
—I ncr e
.
m
m
.
d u e t o s lips t rea
90,
and
it
mf m i
ro
a rscrew.
was foun d
th at
t h e law
of
T
—
0 97 + 1 06 —
fl
z
vn
E
oth er o d el wi th a blu fi er end t h e v ari a ti on of r esist ance fac tor
with th ru st was m
ore th an twi ce th a t i n di cat ed ab ove
—
f
n
d
i
n
a
U
r
c
a
r
r
i
n
l
s
e
n
Th e lan
d in g wh eel s
The Res st a ce o
age a d of Wh ee
a l arg e wi n d
of t h e s m
all er aer e p lan es ar e of a si z e whi ch ca n be t es t ed m
O ver t h e p ossibl e range of sp ee d of t es t t h e resi st ance coeffi ci en t
ch ann el
ay th erefo re b e u sed on t h e full scal e wi th ou t
i s co nst an t and thi s v al u e m
any co rrection
Th e res is t ance of t h e wh eel d ep en ds on t h e sh ap e of t h e
can v as cov ers o v er t h e spok es an d on t h e p r es ence o f t h e st rut s an d a xl e
In
an
'
.
.
.
,
.
D ESI GN D AT A F R OM
AER ODYN AMI CS LAB OR ATOR IES
179
derca rri age Fig 9 1 sh ows a wh ee l whi ch was t es t ed i n th r ee
co n diti ons
A with out any f a b ri c e ver t h e spo kes a n d B an d C with fa b ri c
w veri ngs as sh own in t h e fi gure
Th e drag coe ffi ci en ts for t h e v ari ati on s
a r e gi v en i n T a bl e 24 in t h e u su a l n on d i
with t h e typ i ca l
en si on a l f or
ar ea e u a l t o t h e p roj ec ted a rea o f t h e tyr e
q
of
t he
un
.
.
,
,
.
-
m
m
.
"
25
dvant age of fair ing t h e si d es of t h e wh eel i s seen t o be v ery
c on s i d era bl e t h e coe ffi ci en t for 0 b ein g o n ly on e t h ir d of th a t for A
c on si d er i n g t h e r esi s t an ce of aerop l a ne bo d i es it was sh own th a t v ari a ti ons
p rodu ce d l a rge ch an ges o f resi st an ce an d t h e assem
bly of p art s i n
o f fo rm
t h e su m
a n u n d erca rr i ag e s t ru ct ur e h as a r es is t an ce v ery d i fi er e n t fr o m
a d e 0 11 a n u nd er
An e xp er i m
en t wa s
o f tho se of t h e p art s t a ken s ingly
od el cons i s t in g of on e wh eel
ca rr i age oi t h e typ e sh own in Fi g 9 2 t h e
Th e
a
-
,
.
,
'
.
.
,
m
m
180
APPL IED AERODYN AMI CS
0
APP L I ED AERODYN AMIC S
182
m
bo dy or in t h e fr ee win d The di fierence of p osition h as a ar ked
It shoul d al so be bo rne in m
i n d th a t t he
effect on t h e si ze of a r a di a t or
ay s p oil t h e s t r eam
p resence of a ra d i ator m
lin e formo f a bo dy i n s u ch a
u ch gr ea te r a m
oun t
w ay as to in crease t h e res is t an ce of t h e whol e by a m
th an i t s o wn resist an ce in a fr ee st rea m Th e b es t p os itio n for a ra di a tor
in ed b u t it a pp ea rs t o be ob vious
h as n ot y et been sa tis facto rily d et erm
th a t t h e energy t aken fro mt h e engi ne t o dri ve i t s coolin g m
echani s m
a tion of w eight p er ho rsep ow er of a s t ri c t
s houl d be co un t ed i n t h e es ti m
p arison Th e w eight of t h e coolin g w a ter shoul d a lso be i nclu d ed
com
b radi a t o r of whi ch t h e
In free air t h e drag coeffi ci en t of a honey co m
et er 02 8 i n was foun d to b e 02 6
tub es w ere 4 inch es l on g an d t h e di am
a l to t h e wi n d
t h e area be in g t ak en as th a t of t h e face norm
Th e ho rse
p ow er d iss i p at ed p er un it area was near ly p r o p o rtional both t o t h e
re nce be tw een t h e ai r an d t h e ci rc ula t i n
ff
ra
re
e
s p eed a nd to t h e t em
e
tu
d
i
p
g
wa t er of t h e ra di ato r If t be t h e t em
p era ture d ifference th en t h e cooli n g
i s gi v en app ro xi at ely by
'
aere p lane
.
.
.
.
,
.
.
.
,
.
m
,
.
HP
.
wh ere S
is t he
o
ml f
n r
a
i vo S X 001
.
ace a rea , or
by
H P = tV08 1 X
.
.
if 8 , rep resent s t h e surface of t h e t u bes in con t act with t h e ai r
Ai r cool ed en gi nes d ep en d on t h e h ea t con d u ct e d th r ou gh gills f rom
th e
cyli n d er w all s t o a i r p as s i n g th em It i s a lways n ecessary t o utili se a n
eng i ne cowl eith er to p r ev en t o v erh eatin g or unn ecessary coolin g an d t h e
in t eraction between t h e cylin d ers cowling an d bo dy can only be fou n d
by a d i rect exp eri m
en t
In a r ot ary engi ne t h e l oss of p ow er m
ay b e
15t o 20 p er cen t of t h e tot al but 18 littl e d i fferen t m
flight to th a t on t h e
t est bed Th e cyli nd ers of a ra d i al en gi ne app ear to add v ery m
a t eri all y
t o t h e r es ist ance of an aerOp lane a n d it will be n ecessary to carr y ou t
m
a ny f u r th er t est s b efore a r eli a bl e co ncl u s i on as t o t h e rel a ti ve m
eri t s
o f en gi n es can b e a rr i v e d a t
i cs only
Th e subj ect 18 n ot on e of aero d yn a m
s in ce one of t h e con d iti ons of an ex p er im
en t re quir es t h a t t h e coolin g of
t h e en gi ne sh all be a d eq u at e
—
Th e Re stance of a Com
l
t
e
r
n
e
A
e
l
M
d
e
op a
Th e ex peri m
o el
en t s
p
ost co m
whi ch gi ve t h e m
p l ete analy sis of t h e res is tance of an aerOplane
w ere m
a d e on t h e m
o d el ill u st ra t e d i n Fig 94 at t h e N ational Phys i ca l
mittee whi ch d iscussed t h e res ults of mo del s
L a borat ory for a sp eci al co m
a s a pp li ed to t h e f u ll scal e
Not only was t h e m
o d el t es ted as a com
p l et e
s t r u ctur e b u t t h e r es i s t an ces o f m
ajo r p ar t s both s ep ar a t ely an d m
p l ace
a s p art s of t h e wh ol e w er e al so m
p ari son m
eas u r ed an d a co m
a d e be tw een
t h e s um
s o f t h e res i s t an ces of t h e p ar t s with t h e d i r ec tly m
eas ur ed tota l
I n t h e fi gur e sho wi n g t h e fo r
of t h e m
o d el t h e air screw h as been
sh own a ltho u gh n ot p resen t in t h e t es t s i m
medi at ely und er di scu ssi on
All wi res were al so om
itt ed Th e sp an was 8 7 feet a n d t h e m
o d el was a
on e t e n th sca l e rep ro d u ctio n of t h e B E2c aer O lan e with win gs of R A F 14
p
sec ti on
F orces an d m
om
en t s are exp ress e d thr o u gh out i n lb s an d lb s ft
on t h e m
od el a t a wi n d s p ee d of 4 0 fee t p er seco n d
In d eali n g with s u ch
.
-
.
,
,
,
.
.
,
.
,
.
.
m
,
.
.
.
,
,
,
,
m
.
.
.
,
-
.
.
.
.
.
.
-
.
.
D ESI GN D AT A FRO M
a co
mpl
ex
bo dy
LAB OR A T ORI ES
AERODYN AMI CS
as an aerO pla ne
it
h as
n ot
been fou n d
co n
ve ni en t
to
18 3
u se
tha t n o ty p i cal l ength exi st s on whi ch t o base a form
ul a As an exa m
ple
of t hi s di ffi c ulty r efer ence m
a d e t o a co m
ay be m
mon practi ce of exp ressing
drag coeffi ci ent s in t erm
s of t he win g ar ea wh i ch l ea d s t o t h e resu lt t h a t
.
,
,
s i t ion of
re
e
ss
u
r
p
plo t t i ng ho le s
S i t i on of
we
rc ss
p
lot t i ng h ole s
p
94
.
—Co m
p le te
md l
o
e
aer o
p lan e
.
drag coeffi cien t o f t h e bo dy and und ercarri age
w in gs of di fferen t area to th em
p ris es easure ent s on
Th e seri es of t es ts co m
1
e
e
e
m
o
p
l
t
o
d
h
c
m
e
l
T
( )
e
u
e
2
Th
d
l
with
o
t
t
a
il
p
l
a
l
v
t
o
n
e
e
n
e
a
d
ors
a
( )
t he
.
m
m
m
.
.
is
h
g d by
c an e
APP L I ED AERODYN AMI CS
184
m
m
m
r
r
r
a
a
a
a
e
e
a
r
s
o
r
n
e
e
3
o
d
l
with
ut
t
il
p
l
n
l
v
to
u
d
i
g
h
e
e
o
e
c
T
,
( )
r,
a
n
d
a
n
h
e
wi
g
lo
o
t
d
by
t
t
ot
4
h
e
n
a
n
e
nn
e
e
8
s
ru
s
i
n
o
n
e
cas
e
1
2
s
T
c
c
( )
so a s to p er
it of an esti at e of t h e o d el st rut resi st an ce as w ell as of
t h e lift , dra g an d cen t re of p r es sur e of t h e bi p l ane w gs
.
m
m
T
h
e
to
p
wi
n
g
a
lo
n
e
a
s
a
5
( )
6
e
h
bo
d
y
a
lo
T
e
n
( )
a
h
e
t
a
i
l
p
l
a
n
e
lone
7
T
( )
r
r
a
n
u
n
d
i
g
a
lo
8
h
e
e
e
e
T
c
ar
( )
m
on0plane
m
.
.
.
.
m
.
Th e t a il p l an e an d el evat ors w ere set
p l ane an d kep t th ere for all angl es of
p arall el to t h e chor d of t h e ain
in ci d ence In st ea dy horiz on ta l
fl ight each angl e of inci d en ce corr eSp on ds with a d efini t e ai r sp eed an d with
ent
a sett i ng of t h e tail p l ane a n d el ev a tors whi ch gi v es z er o p it chi ng
om
Ch an ge of t ail settin g d oes not effec t t h e analy sis of resist an ce t o whi ch
.
“
m
,
Alth ough t h e
.
drag coeffi ci en t a s exp l ain ed a bo v e h as d i sa dvan t ages as
a
eans of co m
p ari son of t h e p ar t s of an aer op l ane t h e sa e obj ecti on
d oes not app ly wi th eq u al force t o t h e co m
p l et e o d el I n a d di tion to t h e
fo rces in lbs wi ll th erefor e b e gi v en t h e lift an d dr ag co effi ci en t s in cer tai n
cases t h e s t an d ar d area b ei n g t ak en a s th at of t h e wi n gs
m
m
m
,
.
.
,
.
TAB LE 25
mm
MOD EL
ch or d.
Elev a tors
Co
Ta il plan e pa rallel t o
m
m
1t
)
( lb s )
.
mi
a n
pla ne
L i ft
coeffi ci ent
.
a:
Dr ag ( lb s
)
.
.
.
Wind speed
at
D
confi
dant
,
40
.
.
m
lu m
n con t a in i n g t h e ra tio of li f t
to dr ag show s a m
Thi s wo u l d be
a xi m
u mr ath er gr eat er th an ni n e
easu red d r a g of t h e r esi st an ce of
slightly r ed u ce d by t h e a dd itio n to t h e m
a xi m
um
t h e wi r es whi ch w ere n ot r ep resen t ed
Th e angl e of i n ci d en ce for m
li ft t o d rag i s a bout 6 d egrees a nd t h e li ft co effi ci en t 0 825 whi ls t t h e
umli ft co effi ci ent i s 0 588 ; t h e l east an gl e of gli d e i s th er efore
ax i m
o bt ai n e d at a sp ee d whi ch i s great e r th an t h e s t allin g s p eed i n t h e r a tio
In thi s case a ratio
o f t h e r eci p roca l s q u ar e r oo t s o f t h e lift coeffi ci ent s
i n h or iz o nt a l fl ight with th e e n gi ne r unn in g t h e d rag
o f 18 5 i s o bt a in e d
Co plete
Model (Table
-
Th e
co
.
.
m
,
,
'
.
APP L IED AERODYN AMI CS
186
e Sp eci al co n t r i v an ce t h e a erOp lan e w oul d s till be un s t a bl e
p ro d u ced by so m
as t h e
en t a bo u t t h e cen t r e of gr a v ity d ecr ea ses with i ncr eas e o f an gl e
om
o f in ci d en ce in t h e r an g e 0
to
whi ch co v ers t he com
mon flyi n g
co n di tions
At 10 t h e t ail p lan e i s seen to add a r estor in g
om
en t of
0678 i h fo ot t o t h e o d el aero p lan e with larger v a lu es at gr ea t er an gl es
of inci d ence
— Th e t ail p lane was t est ed as an aer ofo il
Tai l Plane alone (Table
withou t reference to t h e r est of t h e
o d el an d t h e r es u lt s i n T a bl e 27
sh ow by co m
p ari son with s im
il ar fi gur es for t h e com
p l et e odel th a t
an i m
p ort an t d i fference b etw een a freely exp ose d t ail p l an e an d
m
,
°
°
m
m
.
-
.
,
.
m
m
,
TAB LE 27
.
Ta n PLAN E
ALO N E
.
Ele va to rs
Li ft
.
a t zero .
Drag (lbs )
.
p osition b ehin d t h e win gs of an a erop l an e Th e di fi er en ce i n li ft
p l et e o d el an d t h e m
o d el
a t an an gl e of in ci d en ce of 10 b etw een t h e co
without t ail i s 6 3 1 59 2 0 3 9 lb Th e li ft on t h e free t ail p l an e a t an
0 724 lh or n ear ly
an gl e of t a il in ci d ence of 10 d egr ees i s how ev er
d oubl e t h e am
oun t Th e exp l anation of thi s d ifference co m
es f ro
It h as a lr ea dy bee n
a consi d era tio n of t h e a i r fl ow fr omt h e win gs
p oin t ed out th a t t h e wings of an a er op lan e p ro d uce lift by fo rci ng a i r
d own w ard s an d conseq u ently t h e t ail p l an e i s in a d own cu rr en t t h e
in clination of whi ch d ep en ds on t h e angl e of i nci dence of t h e win gs Th e
or e co m
p l et ely in a l a t er exa p l e bu t as a r o ug h
s ubj ect i s d ealt with m
ay be sai d th a t in t h e n eighb our hoo d of t h e t ail p l an e t h e a ngl e
rul e it
of d own wash i e t h e angl e th rou gh whi ch t h e air i s defl ect ed by t h e win gs
i s z ero a t t h e a ngl e of n o li ft a n d in creases h alf as rap i d ly as t h e a n gl e of
inci dence un til t h e criti ca l angl e i s rea ch ed At great er an gles of in ci d en ce
p l e law can be gi ve n
n o si m
Thi s r ul e i s ex em
p lified by t he com
p a ri son now m
a d e s in ce t h e obs er v e d
d i fference of li ft b etween com
p l et e m
od el a n d
o d el withou t t a il (08 9 l b )
ore th a n
i s seen to occu r a t a r ea l a n gl e of t ail i nci d en ce of r a th er m
on e
'
in
m m
.
°
.
.
,
,
m
,
.
.
,
m
m
,
.
.
,
,
.
,
.
.
m
Used
as a sh ort e xp ress i o n
fo r
t a il p la n e
,
an d eleva to rs
.
.
D ESI GN D AT A FRO M
wh ereas
AER ODYN AMIC S
LAB OR ATORI ES
18 7
gl e in d i ca t ed wou l d h a v e been 10 had t h e defl ecti on
t h e a i r st r eamby t h e win gs b een igno r ed
—Th e t a bl e
Model wi th ou t Tail Plane and Undercar ri age (Table
p arati v ely sm
figu r es i s chi efly int erestin g as showin g t h e com
all effect s
om
en t of an aerO lan e
t h e u n d erca rr i age on t h e li ft an d p it chin g m
p
t he
°
an
.
.
TAB LE 28
Mo u nt w rru ou r Tan Pu
Lift ( lbs )
Drag ( lb s )
.
4 x 1)
s s
.
.
mm
.
Us n xac a a
s.
.
I
Thi s f eature can be seen by com
p ari ng t h e corresp on d i ng col u m
n s of
T a bl es 28 an d 26 At an a ngl e of i nci d ence of 12 t h e li ft o f t h e m
od el
ov a l of t h e u n d er carri a g e
wi thou t t a il p l an e was 6 4 7 lbs whil s t t h e r em
p rod uced a ch ange of only 007 lb Th e p er cen t age ch ange on m
om
en t
all as co m
o u n t i s sm
p are d with t he fi gu r es
i s gr ea t er b u t t h e a b sol u t e a m
o d el
p l et e m
n of T a bl e 25for t h e co m
i n t h e l a s t col u m
t h e d i ffer ence b etw een t wo large q u a n ti
a ti on of fo rces f ro m
Th e es ti m
ti es d oes not gi ve high p ercen t a ge a ccur acy
Undercarriag e alone and Co p aris on wi th Undercarri ag e as p art of
—
F
r
t
h
e
e
r
n
n
o
T
bl
e
s
2
9
a
d
u
d
ca rr i age t h e lift a n d
M
d
l
l
a
e
o
e
t
e
Com
p
(
and t h e r es ult s a re gi v en i n T a bl e 29
eas ur e d i n a fr ee st rea m
dra g w ere m
p ari son be tween t h e resu lts i n a free st rea man d i n p ositi on on t h e
Th e com
p l et e m
o d el i s gi v en i n T a bl e 90
com
°
.
.
,
.
,
.
m
.
.
.
TAB L E 29
Us
o nac a aa u o n s
Li ft ( lb s )
the
t
mm
p er cen tage differ ences
o u n t s of i
a bsolute a
Th e
m
.
on
lift
r a n ce
w
a n.
l
.
No
.
i d er a bl e but i n n o case are
a pp reci a bl e err o r woul d ari se
are con s
.
D rag ( lb s )
,
APP L IED AERODYN AMI CS
188
fr o mt h e assum
p tion th a t t h e lif t on an un d ercarri age i s zer o Th e v a l u es
en t
eas ur em
o f t h e d ra g are i n sa ti s fa cto ry a gr eem
en t for t h e accu racy of m
en t to show wh eth er
It wo u l d n eed f u r th er r efin em
en t of ex p er i m
att a in ed
o d el h as any i nfl u ence on t h e r esi s tan ce
t h e p r esen ce of t h e r es t of t h e m
of t h e un d er c rr i age
a
.
.
.
Co st u
m
es
a
TAB LE 30
m
as s
race
Us e r-
mmm
Fe nce s
D i fference
ox
on
m
m
.
e s,
wrr n on r a
nnc
'
Undercarri age
Du
m
as p
m
as cas
Re
m
us
Mo o n
.
nu c s .
Di fference
Und erca rria g e
on
Experi en ts on th e Wings, both as Bi p lane and as Monop lane (Tables
An e xa i nation of t h e drawi n g of t h e
o del will show
31, 32 and
th at t h e cen t ral sectio n of t h e lower wing i s fill ed by t h e bo dy , an d t h e
re
o v al of t h e l att er l eav es a win g st ru cture with a gap ; this was not
en t on t h e win gs
o re
For co p ari son with t h e
clo se d i n t h e exp eri
usu al bi plane observ ations thi s gap in t ro d u ces a s all uncert a in ty, b u t
o di fy t h e ain concl usio ns whi ch will be reach ed B y
cann ot seriously
p ari son with t h e ob ser v atio n on t h e o d el withou t t ail p l ane it will be
co
ic
seen th a t t h e wi n gs con t a in t h e chi ef ch ar act er i s ti cs of t h e aer o d yn a
p rope rti es of a n aerO plane, t h e nex t ost i po r t an t it e s ari s in g fro t h e
t ail p l an e, whi l st t h e b od y and un d ercarr i age are i p ort an t only i n t h e dra g
It i s p oin t ed ou t i n t h e ch ap t er on Dyn a i cal Si il arity th at t h e sca l e
od el s t rut s i s v ery l arge, an d t h e
effec t o n
etho d of d ea l in g with th e
is
t o easur e th ei r r es ist an ce and eli i n a t e t h e effect fro
t h e tot al res i s ta n ce
On t h e f ull scal e t h e corr ec t all owance 18 th en a d e by u s in g cOefli ci en t s
m
-
m
m
m
m
m
m
m
m
.
m
m
m
m m
m
m
m
m m
m
m
m
m
m
d f o mt t
l g m
o d l t ut
Thi p
du
but th t
th y
om
itt d ltog th
th
t i
um
b of t ut
qui d f m h i l
.
.
m
m
.
es s on ar e
of r esist ance obt ai n e r
e s r
s
r oce
s
r e 18
in
a case
a
e ar e
e
er In
e
al s o follow e d for wi r es
e
mo d el whereas a cer a n n er s r s are r e re or ec an ca
r easo ns
Th e a dd e d dr a g d u e t o fo ur e xt ra st ru t s d ecreases as t h e an gl e of
a t ely b ecom
inci d ence i n creases and ulti m
es z ero
Th e a ccuracy of exp eri
en t i s not v ery gr ea t but th ere i s n o (i p ri or i r easo n for d is beli ev in g t h e
ar ka bl e r esu lt
ewh a t r em
Th e effect of t h e s t ru t on t h e ai r fl ow
a bo v e so m
ay be a r ed u ction of th e ir dr a g wh i ch m
o ver t h e win gs m
ore t h an co men
p
sel v es
sa t es for t h e d ra g of t h e st ru t s th em
Th e valu es o f t h e lift to d rag r a ti o for t h e bi p l ane without s t ru t s ca n
p ared with t h e m
on 0pla n e a s t est e d b elow
Th e in fl u ence
n ow b e co m
p ari son an d so ewh a t in t en sifi es t h e
of t h e cen t ral ga p i s p r esen t i n thi s com
Am
a xi m
um
v al u e of t h e ra ti o of
e ff ec t o f t h e a er ofoil s on ea ch o th er
o n0 lan e f a ll s t o n ear ly 15for t h e bi p l ane
o
wh
t
A
s
nea r ly 20 i n t h e m
m
e
a
p
.
,
,
.
m
.
,
.
~
.
m
.
.
.
APP L IED AERODYN AMI CS
190
a d e by subt ractin g t h e r esi s t an ce of t h e bo d y
bett er com
p ari son cou l d be m
a n d st rut s f r o
t h e ob ser v ations on t h e m
o d el Withou t t ail p l an e an d
u n d erca rri ag e but t h e m
a xi m
u v al u e of t h e ra tio of li ft to d ra g is not
grea tly in creased i t s v alu e so d ed u ced bein g 155
m
m
,
.
,
TAB LE 34
BOD Y
Angle
.
m mFm
ALO N E,
on
o
Ru n nxa
.
mt
Mo
o f i ncid ence
en
ab out
TAB LE 35
wi t h ou t fi n ,
m
Co n r a
so n a
m
as s
mmm
Ano n ;
or
ru dd er ,
m
Fo ru
o
en or
m
i es o r
MAIN M
as ,
t e ll ski d
B on y ,
wr
m
s on
.
u
mDm m
sa
'
ou r
B on y
sc ns B ETWEEN
m
Dlfl erencc
od el
.
Nor a —Th e fin
mll
a
:
r
u rn
Inci d ence
( d egrees )
t his s
mw m
D rag
.
0!
u
.
Li ft ( lh aJ
Ang le
Ba
ad d i t i on
on
B od y
D i fference
alone.
on
.
a n d ru dder add
0004 t o t h e dr ag a t
is i nd epe ndent of angle of i n ci d e nce
and
i t h as h ere been
assu
md t h
e
at
.
—'l h e lift
on t h e bo d y an d t h e
B oi v Resi stance (Tables 34, 3 5and
o en t a bo u t t h e cen t re o f grav ity a r e seen t o b e s all as co p ar ed wit h
those on t h e co p l et e od el On t h e o th er h a n d , t h e a dd iti on to t h e dr ag
ay b e a s gr ea t as 26 p er cen t a t t h e an gl es of i n ci d en ce w h i ch co rres o n d
p
with fl ight a t high sp eed s F r o t h e s all t a bl e it a pp ears th a t a bo u t
mm
m
‘
m
m
.
.
.
m
m
m
m
D ESI GN D AT A FRO M
10 p er
u dd er
cen
fin
t
.
of
and
d rag
t ail s ki d
t he
of
AERODYN AMI CS L AB OR ATORI ES
19 1
m
t he
t h e body
i
a r ses
fro
h app end ages
su c
as
i g t h e forces on t h e bo dy with those a dd ed t o t h e m
o d el by t h e
body shows an a pp r eci a bl e e ffect on li ft at sm
all an gl es whi ch i s p r ob a bly
Th e di fi erences
du e to t h e bo dy fi llin g of t h e gap i n t h e low er win g
between t h e two va l u es of d rag sh ow th a t t h e bo dy h as a l ess once t on t h e
p l et e m
o d el th an woul d be estim
a t ed fr om
i t s r esi st a nce i n a free
co m
streamt h e a v era ge d i fference b ein g 18 p er cen t
r
Co
mp
,
.
ar n
,
'
.
'
.
,
Co
mm
so x a
m
TAB LE 3 7
.
ass
Co n n
er :
MO D EL
AN D
Su n
Pa
or
m
s.
'
U nd er
Mcan
Th e di ff ere nce is la rg ely
ca u s ed
by t he ga p i n t h e lower wi ng, d u e t o
re
m
ov a l o f
body
.
m
xp eri m
ad e h a v e sh own a n u
ber of com
p ara ti v ely sm
all
en t s m
d i flerences du e t o p l aci n g t h e win gs bod y t a il p l an e a n d u n d erca rri a ge in t o
thei r corr ect rel a ti v e p ositi on on a n a erOplan e On e l a rge a n d i m
por t a n t
t h e wi n gs on
e ffec t h as a ppeare d i n t h e i nfl u en ce of t h e d ownw a sh from
om
en t t h e res u lts of t h e
Lea v in g t h e la tt er for t h e m
t h e lift of t h e t a il
a n aly s i s of aer e p lane resi st a nce h av e been p r esent e d i n T a bl e 8 7 i n a for
Th e
e
'
,
,
.
.
,
m
APP L IED AERODYN AMIC S
192
w h i ch sh ows tha t i n t h e p resen t st a t e o f kn owl ed ge t h e resistance of
p l et e m
o d el can be estim
at e d t o a high d egr ee of accu racy by the
t h e co m
e
a dd itio n of t h e r esi s t an ces of t h e sep ar a t e p a r t s
Th e p ossi bi lity of som
s
p ort an ce to a d es igner as t h e p roblem
a tio n i s of p r i m
a ry i m
s u ch a pp r o xim
p ro v em
en t ar e th ereby gr ea tly s i m
p li fied
co nn ect ed with p r og r ess i v e i m
I n a dd ition to ill u st ra ting t h e p o ssibility of a dd in g t h e resi stances of
cer ta in p ar t s t o gi v e t h e r esis t ance o f t h e wh ol e T a bl e 3 7 i s of in t erest as
on gst t h e var io us p arts
sh owi ng h ow t h e lift an d dr ag ar e d i s t r ib u t ed am
more than 90 p er cent
O v er t h e flyi ng ran ge of angl es ti e —1 to
ainder
of t h e lift i s du e to t h e wi ngs whi l st t h e great er p art of t h e rem
t h e t ai l p l ane
Th e ra ti o i s n ot grea tly di s tur bed ev en at the
ar i ses fr o m
ore t h an 50 per
s t alli n g angl e
Th e win gs alway s p r ov i d e m
tot al drag in t he m
o del and this p rop ortion will be li ttl e a ffec t ed by
t h e add ition of t h e res i st ance of wi r es
At l arge an gl es of in ci dence
u ch g re at er
b ei ng 75per
t h e p ro p ort io n a l res i st a nce of t h e wi n gs i s m
ce n t o i t h e tot al a t 12 a n d 85p er cen t a t
Of t h e oth er p ar t s t he
bo dy r esist ance i s of gr eat est i m
p ort ance wi th t h e t ail of l eas t im
p o rtance
i n i t s effects on o r di n ary flyi n g
Relati on bet ween Model and Fu ll Scale Th e subj ect of Sca l e effect
was referr ed to a sp eci al s ubco m
mi ttee for
i ttee of t h e Advi sory Co m
e
b er of t h e sam
Aer o n a uti cs i n March 1917 and a r ep ort issu ed i n D ecem
year
As t h e conclus ions reach ed are of grea t i m
p ort ance th ey are
r ep r o d u ced b elow
Car ef ul co ns i d era ti on of all t h e a va il abl e in fo rm
ation l ea d s to t he
followi ng conclus i ons
b
e
r p o se of bi p lan e design m
i
F
or
t
h
e
p
u
d
e
l
a
ro
f
o
i
l
s
m
u
s
t
o
e
()
t est ed as bi p l anes a n d for m
on o p l anes
on0p lan e d es ign a s m
o re cl osely t h e m
od el wi ng t es t e d r ep r esen t s th a t u sed
Th e m
o n t h e fu ll sca l e m
a chin e t h e m
ore reli a bl e will t h e r es ults be
en tio n ed i n p ara gr a p h 7 r em
a in nu
So lo ng as t h e d i fferen ces m
e xp l a i n ed n o high accur acy ca n be obt a in e d in t h e p r e d i cti on
or v er i fi ca ti on of p er form
a nce a t low lift coe ffi ci en t s
a
n
1
1
u
a
r
s
e
s
a
llow
u
t
d
l
t
p
t
wh ere it
D
c
e
m
b
e
m
a
e
f
o
r
e
e
ff
o
n
sca
ec
( )
i s kn own
I n t h e case of st r ut s wir es et c t h e sca l e once t i s
known to be l a rge but th ese p art s ca n b e t est ed un d er co n d iti ons
corr eSp on din g with th ose whi ch obt ain on t h e fu ll scale
a chi ne
n
a
e
iii
s
h
h
e
r
es
c
s
t
i
t
of
v
i
u
p
t
t
p
t
y
T
e
ar
ar
a
e
n
se
ara
e
l
b
e
o
s
s
k
a
)
(
y
a dd e d togeth er t o gi v e t h e res i st ance of t h e co m
p l et e a er O p lane
with good accur acy p ro v i d ed t h e p art s
t h e un d er ca r ri age)
whi ch cons is t of a n um
ber of sep ara t e sm
all p i eces ar e t es t ed as
a co m
p l et e unit
f
r
s
a
e
e
s
n
m
i
v
o
d
l
t
t
o
i
p
o
t
a
t
v
lu
bl
gui
d
M
r
n
a
n
d
a
e
i
a
ae
r
e
a
e
O
l
a
n
( )
p
d esign Wh en e p loy ed for t h e d et er in ation of a b s olute
u s t b e u se d with d i scri
v alu es of res i st ance th ey m
in a ti o n and
a full r eali sa ti on of t h e m
od i fica tio ns whi ch m
a y ar i se o w i n g to
i n t erference an d scal e effect
I n for ecastin g t h e p erfo rm
an ce of a m
achi n e of a kn own ty p e
e tho ds
ca n be em
ploy ed oth er th an t h e a ddi tion of t h e resi stan ce of all el e en tary
,
,
.
.
.
,
.
°
,
.
.
,
.
.
,
.
,
°
.
.
,
.
m
,
.
-
,
,
.
.
.
,
-
.
,
,
.
'
.
,
,
.
,
m
m
,
t
.
,
m
m
.
.
m
,
'
m
'
.
m
m
APP L I ED AERODYN AMIC S
194
m
Th e sec tion of each p l ane was R A F 6A, an d t h e di 0nsi ons were 3 x 18
Th e bi p lan e h a d a gap cho r d ratio of un ity an d zero st agg er
Th e cc or d in a t es of t h e po in t s a t whi ch t h e angl es of d own wash w ere
ea sur ed are d en ot ed by x an d y , an d t h e a x es t o whi ch th ey refer have
th eir or igin i n t h e t railin g ed ge of t h e u pp er win g Th e a xi s of a:is d ir ected
fr o l ea din g edge to t rai lin g edge along t h e chor d of t h e u pp er w ing
eas ur ed a t r ight an gl es to this a x is an d d own w ar ds
whils t y is
For
co n v en i en ce i n u se both a: an d y h av e been gi v en as frac tions of t h e ch ord
Th e t 0p hn o of T a bl e 8 8 shows t h e angl e of in ci d ence d ur ing t h e e xp er i ent ,
whilst t h e secon d gi v es t h e v alu e of t h e lift coefli ci en t correSpon di n g wi th
Th e angl es of d own was h a re gi v en i n t h e body
ea ch a ngl e o f in ci d en ce
o re than
of t h e t a bl e, an d r an g e o v er an ar ea thr ee cho r ds long an d ra th er
Ex t en d ed res u lt s ar e shown in Figs 95an d 97
one ch or d d eep
N ear t h e win gs t h e angl e of d ownwash a pp ears to be v ar i a bl e, and
.
.
.
-
.
m
m
-
.
m
,
.
m
.
m
.
.
.
5
.
20
Pro 06 — Do wn was h
.
.
an d a n le of
g
i nci den ce
.
m
ean i n g of T a bl e
i n t h e fig ure whi ch h as b ee n d r awn to illu st ra t e t h e
8 8 t h e cur v es for p oi n t s n ear t h e win gs h a v e been d otte d as a n in di ca
ti on th a t co ns i d era bl e free d o h as been exerci sed i n d rawi n g th e
F ro t h e ob serv ati ons it i s p ossibl e t o d ed uce t h e an gl e of d own w as h at
J
p oi nt s on t h e ean ch or d of t h e bi p l ane, a n d for five p o in ts A, B , ( , D and
E o f Fig 96 t h e co rr es p on d i ng cu rv es of d ow n w as h h av e be en p re p ared
u si ng t h e li ft coeffi ci en t as a b ase Th e ch oi ce of lift coefli ci en t a s an
i n d ep end en t v ari a bl e i s t o b e p refer red to th at of angl e of in ci d en ce on
general gr ou n ds co nnect e d with t h e o en t u of t h e d own war d o v ing
d d it s rel a ti on t o li ft b u t a s a n e p i ri cal exp ed i ent h as t h e e dv eh
s t rea
t a ge of gi vi n g a n early l in ear r el a tion with t h e angle of d own was h This
lin ear rel ation h ol d s a l ost t o st allin g an gl e a nd a t d i st ances of 2 t o 3
ch o r d s be hi n d t h e wi n gs , i c wh ere t h e t a il u su a lly co
es , i s n ea r ly i n de
p end en t oi wing secti on or wh eth er t h e wings are ar ra nge as on 0p lane,
m
m
m
.
m
.
.
m
,
mm m
m
m
.
bi p l ane or t ri p l an e
Fr omFig 9 6 i t will
m
.
m
,
.
d
m
.
.
be
seen
tha t
the
an
gl e
of
d ownwash
is
gr ea t est
DESI GN D AT A FR OM
AERODYN AMI CS LAB OR ATOR IES
19 5
j us t be hin d t h e win gs and fal ls off ra p i dly to a bout one chor d t o t h e r ear
ore slowly
An exp onenti al cu r v e of v a ri ation h as been
an d a ft erw ar d s m
s ugges t ed of t h e form
,
.
1
9
( )
wh er e so dep ends on t h e angl e of inci d en ce For angl es of
16 an d 20 app r o p r i a t e v al u es of co w ere foun d t o be 2 6 8 10 11 an d 9
ay b e f ou n d
in d i cat ed by equa tion ( 19) m
For so m
e in v es tiga ti on s t h e form
t o be v er y u sef u l
Ov er a l ar ger ran ge of angle of in ci d ence th an is shown in t h e t abl e
a d e at a p arti cular p oin t A of Fig 95 an d t h e angl e
o bs er va ti ons w ere m
o f d own w ash r ecor d e d h as been u sed i n t h e p rep ara ti on of Fig 95
Th e
.
°
°
,
,
,
,
.
.
,
,
.
,
.
Fl o 96
.
—Downwash
.
a nd
lift
.
coeffi ci en t .
general resem
bl ance of t h e cur v e to th at of t h e li ft coefli ci ent curv e for a
ar k ed ev en b eyo n d t h e criti ca l angl e
wi n g secti on i s m
Th e an gl e of z ero
d own was h occurs wh en t h e i nci d ence i s
whi ch i s near to t h e p oin t
—
e
n
e
o
f
n
e
n
c
e
5 to +10 t h e ch a nge
O v er t h e ran g of a gl
i ci d
of z ero lift
of an gl e of d own w as h i s rou ghly e qua l to h al f t h e ch ange of a ngl e of
i n ci d ence
Th e d i stur bin g i nfl u ence of t h e win gs on t h e fl ow of air i s few for con
si dera ble d i s ta nces a bo ve an d bel ow t h eman d i s ill u s t rat ed by Fi g 9 7
Fo ur ch or ds d is tance a bove t h e u pp er win g or b el ow t h e l ow er wi n g an d
t h e d own wash i s 1 or 2 for angl es of
a bout thr ee chor d s b ehin d th em
Th e gr ea tes t a n gl e occu rs i n t h e cen t ral regio n
in ci d en ce of 4 an d
behin d t h e bi p l an e bu t it will be evi den t th a t th ere i s no p ossibility of
a v oi d i ng t h e d o wn wash by any r eason a bl e ch oi ce of t ail p osition
.
°
°
.
.
,
.
°
°
°
,
.
.
APPL IED AERODYN AMICS
196
en t s were m
a d e on a bi p l ane with adj u s ta bl e t railing
Fur ther exp er i m
ed ge or flap s wh i ch show on e or two in teres tin g p oin ts Fig 9 8
The
velocity of t h e a irstr eam
was m
eas ured a t a p oin t on t h e m
ean cho r d and
foun d to be sensibly tha t of t h e un dis tur b ed curren t u n til stallin g angl e
was reached a ft er whi ch a very ra p i d f all occurred
e oth er obs erv a
Som
tions in di ca te a sm
all but m
eas u ra bl e loss of Sp eed below t h e cr iti cal angle
b u t all a gr ee i n showi ng tha t t h e a in in fl u en ce of t h e wi ngs is th a t causing
down wash
,
,
,
,
.
.
.
m
,
.
F
m97 —V
.
.
e rt i cal
di st ri b u t ion
of
down wash
.
m
m
v e of t h e lower figure ar ked origin al o d el shows that
for su ch vari a ti ons of secti on as can be p ro d u ced by t h e u s e of win g flaps
t h e r el a ti on b etween angl e of d own w as h an d lift coeffi ci en t i s not di st u r bed
On t h e oth er h an d t h e re o val of t h e lower flap in t h e wing at a p oint
i
ed i a tel y in fron t of tha t at wh i ch ob servation was
a d e p ro d u ced a
It i s of cou rse p ro babl e
ar k ed red u ction i n t h e an gl e of d ownwash
arked red u ctio n i n loca l lift co effi ci en t occu rred b u t t h e
tha t an equally m
effect on t h e st a bility of t h e aerOp lan e of su ch a ch ange coul d not fail to
be i m
p orta n t Th e ten d ency wo u l d be fo t h e aereplane to be o re stabl e
after t h e low er flap h ad been r em
o ved loc y
Th e
cu r
m
m
m
,
m
.
m
.
,
,
,
m
.
.
A PP L IE D AE RODYN AM ICS
198
m
Varying the Posi fi on cf the Ein —Th e
t ail p l an e and el evators of an aereplane are r equire d p r i ar ily t o ba lan ce
t h e co u p l e on t h e win gs a nd s in ce i n st ea dy fl ight t h e l a tt er d epen ds o n
t h e Sp ee d of fl ight arr ange ent s
u st be a d e for a v ari ati on of t h e co u p l e
ex er t e d by t h e t a il
For s uch
an aau vr es as loo p i ng a n d r a p i d tur ni n g
co u p l es a r e req u ir ed whi ch p r o d u ce t h e necessa r y angul ar ac cel era ti on s
Elcn tou
and
th e Efl cct
of
m
‘
m m
m
,
,
.
-
F
m
m99 —V
.
ar i a t i on of e le v a t or
.
m
a.
v elociti es and in su ch cases t h e el eva tors al one are su fli ci en t ly ra p i d
For st ea dy fl yi n g m
i n acti on
an y a ero p l a n es ar e fi tt ed with a dj u s t a bl e
t a il p l anes so th at t h e aero p l an e can be fl o wn with littl e effor t on t h e co n t rol
n
Th e force on t h e p ilot s h an d h as littl e d ir ect r el a ti on t o t h e
col u m
moment on t h e a erop l ane as ay b e seen fro t h e followi ng exa p l e
o d el u se d was a co m
Th e m
p l ete body a nd t ail un i t an d t h e l att er i s illus
t ra t ed in Fi g 9 9
Th e bod y without un d ercarr i age was a co p y of th a t
u sed i n t h e co m
p l et e odel aero p l ano (see Fig 94) but was to a slightly
an d
,
.
’
m
.
,
m
m
,
.
.
m
,
,
.
.
DE SI G N D AT A FRO M AE ROD Y N AM ICS LAB OR A T OR IE S
di ffer en t
199
I n t h e t a bl es an d figures now use d h ow ever t h e res u lts
l
have bee n co nv er ted to app ly t o a on e t enth scal e m
od el a t a win d s p ee d
of 40 fee t pe r secon d an d ar e th er efore d ir ectly com
p ara bl e with t h e result s
for t h e co m
p l et e m
o del aerop l ane Th e p ositi on of t h e centr e of gravi ty
of t h e aerop lane i s shown in Fig 94
sca e
.
,
,
-
,
.
.
.
TAB LE 39
Angle
mg
Pi te h
m
of in ci d ence of ta i l
pla ne,
dg é lm
t
g n
ty
t
bo lt t he
op
Ang le of i nci dence
of t a i l
.
Wi nd speed
,
40 ft
m
Mo
1
ent “h e
.
-s .
t bi
ri t li y
a
p lane , + l o
°
.
Win d
Mo
p eed,
s
mt
en
0 ‘ele va t o r
t
40 ft
ab o u t
.
.
-
b lu es
( lbs
s
.
0 ! ele vat or
.
a d e for t h e d own w a sh f r om
be en m
a i n p l an es
t he m
en t an d h i ng e m
om
in p rep arin g t h e t a bl es o f p it chi ng m
om
en t
Th e effect
a k e t h e a ngl e of i nci d en ce o f t h e t ail m
of t h e d own wash i s to m
u ch l ess
eth o d of estim
th an th at of t h e wi ngs and t h e m
a ti ng thi s effec t h as a l rea d y
ent s a s t a bul a t ed a re th ose whi ch t en d t o
been d ealt wi th Positi v e m
om
in cr ea s e t h e angl e of a ttack
en t o n t h e aero p l an e an d t h e
om
T a bl e 89 sh ows h ow t h e p it chin g m
No all owance h a s
.
,
.
.
APP L IE D AE R ODY N A M IC S
200
hi nge m
en t on t h e el ev a to rs v ar y with t h e el ev a t or angl e
A p ositi v e
om
a ngl e t en ds t o a d i ve t h e el ev a tors bei n g th en b elow t h e cen tr e line of t h e
t a il p lan e
Th e result s are p lotted i n Fig 99 in a formwhich shows t h e rela ti v e
merits of v ar ious p porti ons of elevator an d tail plane area t he hi nge
o d el ha vi ng been p laced su ccessi v ely at t he p ositions ar ked
in t h e m
Th e hi nge m
en t i s p ro p o r ti ona l to t h e fo rce on t h e
om
A B C in Fi g 99
p ilot s han d an d th erefore an es tim
a te of t h e p itchi n g m
om
en t p ro d u ced
en t i s of di rect app li ca tion in ass ess in g t h e valu e o f
om
for a gi v en hinge m
en t of t a il p lane an d el ev a to rs
An a dvan t age i n
an y p ro p o se d ar r an gem
a xi m
um
light ness of con t rol i ght ha v e been ofiset by a red uction i n t h e m
i nation of t h e cur v es wi ll sh ow
co u p l e whi ch can be a pp li ed but an exam
tha t any sm
all di ffer ences i n t h is r esp ect a r e fa v o u r a bl e to t h e light con t ro l
T h is can be seen m
os t si m
p ly i n t h e fi gure for zero an gl e of i nci d ence t o
a , = 0 wh er e t h e narr ow el ev a to rs d enoted by C
i
v
e
a
t h e t ail p lan e
g
p itchin g m
om
en t 5
0 per cen t gr eat er than t h e wi de el evators for t h e sam
e
om
en t whils t t h e b rea k d o wn of fl ow i n d i ca te d by t h e su dd en r is e
h in ge m
i n en t i n th e la t t er without l ea d ing to a lar ge r
of t h e cur ves is m
ore p rom
p itchi ng om
en t
Ex cep t for t h e sm
all lack of sym
m
et ry i n tr o d uce d b y
om
en t an d p itchi ng
om
en t wo ul d beco
e zer o
t h e bo dy t h e hinge m
together for an an gl e of i nci dence of th e t ai l p lane of
This con dition is
with a large angl e of in ci dence on t h e t ai l p l ane
app reci a bly d ep ar te d fro m
en t ha v i ng a r ela ti v ely lar ge v alu e wh en t h e p itch in g
om
t h e h inge m
momen t is zero The lack of sy etry of t h e cur ves is now well mar ke d
altho u gh t h e tot al ran ge of efl ect i ven ess is n ot grea tly re d u ced an d t h e
f eatur es rel at i ng to ligh tn ess of cont rol are not fun dam
ent ally affecte d b y
t h e ch ange of an gl e of in ci dence of t h e t ai l p lan e
it s of effecti veness of t h e el ev ato r
I n o r d er t o show i n det a il t h e li m
a te d it is necessar y
co n t rol an d h ow t h e forces on t h e p ilot s han d are es t i m
to carry out t h e seri es of cal cula tions in d i cat ed mChap ter IL T a bl e 40
s of t h e ca l cula tion for t h e el ev at o rs
l
r
es
i
v
t
h
e
u
t
d
C
pp
es
ar
a
li
k
e
as
ed
g
to t h e aero plane of wh i ch Fig 94 rep resen ts t h e co m
p l ete m
o d el
.
,
.
m
,
.
,
.
m
,
.
’
,
m
.
,
.
,
,
.
,
m
m
.
,
m
,
mm
.
,
'
,
.
’
,
m
.
Co s
de
A
i nci
e of
nce of
( decrees )
m
-
en
Angle of
Inci de nce of
“ 1p ”
( degrees )
.
TAB LE 40
Foru
m
0
s on a n
An
m
.
(
m
o
m
an )
n.
.
0002
gl e of d ownwash gi ven i n col u m
n 2 of T a bl e 40 was d ed u ced
t h e ob ser v a ti ons on t h e co m
fro m
p let e m
o d el t h e t est s of t h e m
o d el without
Th e
an
,
APP L IE D AE R OD Y NAMI CS
202
m
p la t e of di am
et er e qu a l to th a t of t h e en velOp e
Th e m
e tho d s of
eas ur e
ment su ita bl e for bod i es of high resistance coefi ci ent fail t o gi ve suffi ci en t
o d el s an d it wa s o nly i n t h e l at es t st ag es of d ev elo p
acc u racy for env eIOp e m
e err o rs of i m
p or tan ce w er e di sco vere d an d st ep s t ak en t o
en t th a t som
i n futur e
I n a win d ch a nn el of us u a l fo r
t h e ai r i s slowly
a v oi d th e
accel era t e d i n t h e cent ra l p ort i on as it p asses alo n g t h e t r un k an d th ere i s
t h e in t egra l efi ect of t h is sm
a sm
all dro p of s t a ti c p ressur e
all d ep ar t ure
fr omuni for ity i s of i m
p ort an ce i n t h e a ir shi p en velo p e an d n egli gibl e
for t h e a ero p l an e
o d el At t h e p resen t ti m
e t h e error i s es t i
a te d a n d
but st ep s are b ein g t aken so to o di fy t h e wi n d
a corr ecti on a pp li e d
.
m
,
m
m
.
,
'
m
m
m
.
,
m
N9 5 5
h annel as t o eli m
in a t e i t Wh en exa m
in ing t h e res ult s of t h e exp er im
e n ts
ma de at d i fferent wi n d Speeds app reci able changes of resist ance coeffici en t
e d o u bt th en ar ises a s t o t h e co rrect v al u e t o be
ar e ob serv ed a n d so m
I n t h e case of ci rcul ar wir es it was foun d th a t
a pp li ed on t h e full scal e
t h e d r ag coe ci ent v ari ed a b ov e and b elow a roughly co ns ta n t valu e a n d
o d el a n d full scal e i nd i ca t es a so m
p ari son b etw een t h e m
il ar
ewh a t s i m
com
Th e ra nge of scal e i s how e v er so v e ry gr ea t th a t t h e
e ffect for a i rs hi p s
en t s i n a l a bor a to ry
i nt erv en i n g ga p cannot b e cov ered by a ny eXp eri m
an d it i s p rob a bl e th a t t h e l aw s of scal e effect on en v el op e fo r s will firs t
a th em
a ti ca l solution of t h e
be sa ti sfact orily en unci at ed as t h e r esu lt of a m
o tio n of a v i sco us fl ui d
e q u a ti on s of m
c
.
,
,
,
m
.
,
.
,
,
m
.
,
D ES I G N DA T A FR OM A E RODYNAM IC S L A B ORA T ORIES
208
A ser i es of fiv e m
o d els shows for env el ope form
s h ow t h e drag co
effi ci en t s v ary with t h e fin en ess ra ti o or l ength to di am
A si m
il ar
et er r atio
seri es of t es t s for s t rut for
s h as alr ea dy been gi v en i n whi ch t h e dr a g
s
coefi ci en t o n p roj ect e d area was r oughly 0 042
On t h e en v elo p e form
t he co effi ci ent i s a pp reci a bly l ess an d m
ay f all t o h alf t h e v al u e j u s t quot ed
Th e for s t es t ed w er e soli ds of r ev olution of whi ch t h e fr on t p ar t wa s
elli p soi d al
i n all cas es t h e
umdi a eter was a de to o ccur at
a xi
one thi r d of t h e t ot al l en th fr o m
t h e nose
Th e sha p es of t h e longi t u
g
di na l secti ons are shown i n Fi g 100 an d h av e n u bers a tt ach ed to th e
whi ch a re eq
ual to th eir fin eness ra tio Th e observ ations a de are re
It i s po in t ed out i n t h e
co r d e d i n T a bl e 41 and n ee d a littl e exp lana tio n
i cal si m
o del n or t h e
i l ar ity th at neith er t h e s ize of t h e m
ch a p t er on dy na m
spee d of t h e win d h as a fun d am
en ta l ch arac t er i n t h e Sp ecifica tio n of
r es i st a n ce coeffi ci en t s bu t th a t t h e p ro d u ct of t h e two i s t h e d et er i n i n g
n of
vari a bl e In acco r d ance wi t h tha t chap t er th er efo re t h e firs t colum
Ta bl e 4 1 shows t h e p ro d u ct of t h e win d sp ee d i n feet p er secon d and t h e
e t er of t h e m
d iam
o del in feet Fu r th er two drag coeffi ci en t s den ot ed
res pecti v ely by
er gi vin g
an d 0 h av e been u se d for eac h m
o del t h e form
h
r
a
a
n
d
t
e
a di rect co m
i
o
n
with
ot
r
d
t
a
t
h
e
b
as
i
s
of
p
r
oj
ec
t
e
d
a
e
ar
s
h
a
n
e
o
p
l atter a coeffi ci en t of sp eci al utility in a i rshi p d es i gn w h i ch i s closely r el a t ed
t o t h e gr oss lif t
m
.
,
.
m
.
mm
,
m
m
m
-
.
.
,
m
m
.
.
m
,
.
,
,
.
,
,
,
.
TABLE 4 1
B u rsaAi
.
m C mm
s
on
c
mmE
A
'
rs o r
sr
No 4
.
Th e
coe
m
ci ent s
kn
an d
D
C
ra
ar e
drag
m
Fe
No
.
.
d efined by t h e equations
z
=
g kp pv
:
s vxno n
Cp
2
V
"
.
1
2
2
l
u
m
e
v
ol
)
(
wh ere d i s t h e m
u mdi am
a xi
et er o f t h e env eIOp e
.
m
.
No 8
.
.
APP L IE D A E RODY N A MI CS
204
m
m
m
m
x i na ti on of th e col u ns of T a ble 4 1 show s so e cu r ious ch anges
o f coeffi ci en t whi ch are p er ha p s
ore rea d ily a p preciat ed fromFig 101
where th e v alues of kn are p lotted on a b ase of Vd For t h e longest odel
t h e cur v e first s ho ws a fa ll to a m
u mfollow ed by a r ise to i t s in itia l
ini m
t he
v al u e For t h e o d el of fi neness ra tio
in i u occu rs la ter and
it is p ossi bl e tha t t h e thr ee sho rt od els all h av e ini a out si de t h e range
of t he d ia gra
It is cl ear ly i p oss ible t o p r o d uce th ese cur v es wi th any
d eg ree of cer tai nty In Cha p ter II it was d ed uce d tha t for a r igi d ai rs hi p
t h e full sca l e tr ia ls gi v e t o 0 a v alue of 00 16 an d for a non rig i d
An
e a
m
.
.
m
m
.
m mm
mm
,
m
m
.
,
,
.
.
-
-
,
,
Fro l Ol
.
—R
.
eei et an ce of ai rshi
p en velop e
,
md l
o e s.
Th ese figures co nt ain t h e allowan ce for cars and rigging and d o not in d i cat e
t h e fi gu r e of 00 18 gi ven a bo v e for t h e en v elo pe
any
ar ked d ep ar tur e from
p ar iso n i s v ery ro u gh but accurate fu ll scal e exp eri m
alone
ent s
Th e com
i lar to those on m
o d els h a v e
of a na tur e s im
t
d
e
b
e
a
e
m
t
o
y
It wi ll be n oti ced fr o mT a bl e 4 1 th a t w hilst t h e dra g coeffi ci en t calcu
l a te d on a xi m
ump roj ec t e d ar ea fa lls with d ecrease of fin eness ra ti o t he
p ar es t h e form
s on u n it gr oss lift i s l ess v ar ia bl e
coeffi c i en t C whi ch com
p ortan ce of t h e second
an d h as i t s l eas t v al u es for t h e lon ger m
o d els
Th e im
dr ag coe ffi cient C i s th en seen t o be cons i d era ble as an ai d t o t h e choi ce
of en velope for
p l ete m
od el illus tr a ted
0011191
0“ new of a Non-rig id Ain h i —A co m
m
,
.
-
,
.
'
m
,
.
m
.
m
,
AE RODYNAMI CS
A PP L IE D
206
TAB LE 4 2
.
R ssrsr axox
Di a
Dr ag
or
m Am
Nos -
orn
mt
e er of env elope,
i ns
su rv
.
A ng le
Wind
.
s peed ,
o f i ncide nce
40 ft
( degrees )
.
.
g,
.
Descri p t i on of hlod el
.
0102
0109
0132
01 70
0066
0073
0092
01 27
0187
0052
0068
0078
01 15
01 71
0051
0035
0021
00 16
00 12
0054
0036
0001
01 29
0101
00 16
00 18
00 11
0022
0054
0021
00 16
00 11
0032
0090
0074
002 1
00 16
00 12
0048
0 102
01 11
0134
0171
0220
00 18
t a bl e shows t h e drag on t h e m
od el an d i t s p ar t s
5d
1
of 40 ft s t h e m
axim
umd iam
et er of t h e m
od el
in
—
e
r
o vi n g p ar ts su ces s
bein g
i nch es Th ows a f gi v e t h e r es u lt of r em
—
m
e
m
o
e
r
s
d l whils t ow f j refer to t h e r es i stan ces of
t h e co p l t e
si v ely fr o m
th a t i s with t h e ai r
At an angl e of in ci d ence of
t h e p ar ts sep ara t ely
metry of i t s envelop e t h e to t al
shi p t r a v elli ng a long t h e a x i s of sym
es th at of t h e en v elo p e alon e
Fr o t h e
r esi st an ce i s n ea r ly three tim
fu rth er figur es i n t he colum
n it will be seen tha t t h e difl erence i s al ost
t h e r u dd er
t h e car
eq u ally d is t r ibute d b etw een t h e r i ggin g ca bl es
p l an e ( i ) a nd t h e el ev at o r p l ane
o d el is
Th e resi st ance of t h e whol e m
very closely e q u a l t o th at es tim
at e d by t h e a d ditio n of p ar t s t h e fi gur es
easu r ed a n d
as foun d by a d di tion
bein g 0102 as m
Th e agr ee en t
b etw een dir ect ob serv ation and co m
p ut ation fro m
p arts is l ess sa t i sfact o ry
p arin g with t h e
a t l ar ge an gl es of in ci d ence a n ob ser v ed figur e of 0 8 00 co m
mu ch lower figure of 02 20 Th e d i fference is prob ably connected with
ore arked
t h e in fl u ence of t h e ru dd er an d el ev a t or fi ns in p ro d u cin g a m
li n e f or mth an t h e in cli n ed env el op e al one
d ev i ation fr o mst rea m
—
Pi
i
M
m
n
if
n
d
c
h
n
t
i
i
r
L
t
a
hi
of t h e
o e t on a R g d Ai rs p
D ag
Th e form
g
o d el to 5
l
x
i
u
t
s
a
e
a
d
a
m
m
ai rs hi p i s sh own i n Fig 103 a n d t h e m
h
c
h
a
m
0
i nch es
d i am
F orces are gi ven in lli s on t h e m
o del at 40 ft s
et er of
en t s ar e gi v en i n lb s ft Ap ar t fr om
whil s t m
om
a ny scal e efi ect a pp li ca
a d e by i ncreas in g t h e forces i n p ro p or tion t o t h e s qu are
ti on t o full scal e i s m
o f t h e p r o d u ct of t h e scal e an d Sp eed whils t for m
om
en t s t h e s quar e of t h e
ai ns but t h e thi r d p ow er of t h e scal e i s re q ui red
At a
sp ee d st ill r em
0 V b ei n g t h e v eloc ity i n f eet p er secon d an d d
valu e of Vd eq u al to 5
et er i n f ee t t h e p artiti on of t h e r es i s t a nce was m
t h e di am
easur ed as i n
T a bl e 48
m of t h
Each colu n
lbs at a wi n d
e
-
.
.
.
,
o
.
,
.
m
m
,
.
'
m
,
.
,
m
.
.
,
.
.
,
.
.
.
'
-
.
,
,
.
,
,
,
.
-
D E SI G N D A T A FROM A E RODYNAM ICS LAB OR A T ORIE S
207
It
ti on
w as n oti cea bl e th a t t h e v ari a
C
for
of r esi s ta n ce coeffi ci en t
mpl t mod l with p d f t
m h l m k d th t h t
th
v l p
l
o ffi i
gi g f m00 19 5 t 00210 f
t h e co
was
the
ran
e e
ess
uc
en
n
S
e
e
a one,
e o e
c e
t
of
c en
t
or a
o
ro
es
a
an
e
ar
o
ee
g of Vd of 15 t o 50 w h i lst for t h e
en v el op e t h e ch an ge was 0009 6 t o
ran e
,
00 13 1
.
TAB LE 43
m md l
Co p le te
En v e lop e
Fi ne and
V alue
o
m» m
d c
"
C
di ctent
.
00207
00 13 1
00014
o e
a lone
con t rols
0003 8
00024
T a ble 44 are coll ected t h e re
o d el
su lt s of ob serv ations on t h e m
a irs hi p for a ran ge of angl e of in ci
t h e lift an d
d ence —20 to
om
en t as well as t h e drag
p itchin g m
In
°
en t on t h e
valu e of t h e p itchin g m
om
en v elo pe a l one h as b een a dd ed
A
furth er t a bl e sh ows t h e var i a tion of
p itch in g m
o en t du e to t h e u se of t h e
e le va to rs an d t h e sali en t f ea tures of
t h e two ta bl es are ill u s t rat ed i n Fig
104 ( a ) an d ( b)
An gl e of in ci d ence has t h e u su al
ean i n g a p os iti v e v a l u e
con v en ti on a l m
i n di ca tin g tha t th e nose of t h e airshi p
oti on is h or i z ont al
i s u p w h i ls t t h e m
A posi ti v e i n clin a ti on o f t h e el ev a t ors
i ncreases th ei r l ocal angl e of i nci d ence
a n d cl ea rly t en ds to p ut t h e n o se o f
t h e a ir shi p d own
T a bl e 44 i n d i cat es a m
ar k ed i n
c rease of r esi s ta n ce d u e t o a n i nclin a
tion of 10 of t h e a xis of t h e a i rshi p
ewh a t
t o t h e rel a ti v e wi n d bu t a som
more remarka bl e fact i s t h e magn itu d e
ay be 2 5 ti m
o f t h e lift whi ch m
es as
grea t as t h e d ra g a t t h e sa m
e angl e of
i n ci d en ce
en t
om
Th e col u m
n of p it chi n g m
mon to all typ es
s h ow s a fea tu r e co m
all a ngl s of i nci d ence
en t a t s m
om
of a i rshi p i n t h e a b s ence of a righti ng m
e
I t d oes not foll ow th a t t h e a i rs hi p i s th erefor e u n s t a bl e si n ce th er e i s a
.
m
,
.
.
,
.
.
°
,
,
.
.
,
mm t d
mth t f t h
fur th er
a
AERODYNAMI CS
A PP L I E D
208
p i t chi n g
o
pp recia bly fro
en
o
a
ue
e e
xis t ence or oth erwi se of a
TAB LE 44
Du
e,
mP m Mm
t
M imm
di m
e
rrc
r
ax
o
a
u
Moo n
.
.
or
Win d speed
,
a
B ra
er ,
or e
;
en
n
.
m
40 “A l
.
Li ft
Drag
mll
i ns
r
.
o r: A
o
e e r,
m ov i t
ighti g m
om t
d ist ri buti on of w eight
to t h e
gl es of in ci dence t h e in di ca tion of T a bl e 44 i s tha t t h e ele
vat or fins a nd el eva to rs neut rali se only on e thi r d of t h e cou p l e on t he
en v el op e alon e b u t at grea t er angl es wh ere t h e fin i s in l ess d is tur b ed a ir
ore t h an 85per cent i s neu tralised A p osition of equilibriu mwhi ch is
s t a bl e wo u ld e xis t a t an i nclin ation of a bout 8 5 to t h e rel a ti v e wi n d
At
s
a
an
-
m
,
,
,
.
.
°
.
TAB LE 45
.
PITGHI XO
M
t
ou n r on A
B ra
mA m M
ru
-ft
b
e
L
(
.
Angle
( doc )
.
.
at
O D EL no t
40 ft
.
o f eleva t or
( degrees)
.
70
m
EL B V ATOI S
.
A PP L I E D AE RODYNA M IC S
210
o d el as
Fig 104 (a) sh ows t h e p i t ch i ng m
om
p l et e m
en t on t h e co m
a ll ang l es of
depen d en t on a ngl e of i nci dence Th e ra p i d ch an ge at sm
i nci dence i s f ollowe d by a fallin g off to a m
umat 10 an d a fur th e r
a x im
Th e low er d ia gramFig 104 (b) sho ws how t h e cou p l e whi c h
fall at
ca n be a pp li e d by t h e el e v a tors com
p ares with th at on t h e a irshi p I t
a pp ear s th a t at an angl e of 20 t h e m
a xi m
um
momen t can j us t be overcome
by t h e el evat o rs an d tha t a gus t whi ch li ft s t h e nose to 10 will requi re
an el e va t o r an gl e of th a t a m
ou n t to neut ra lise i t s effect It i s qu it e
p ossibl e th a t m
ost a ir s hi p s are u ns t a bl e to so m
e sli ght degree but ar e all
e di ffi
con t r olla bl e at low sp ee d s with ease a n d a t high sp ee ds with so m
culty
en t of fin s of a rea re quis it e to p rod u ce a r ightin g
Th e at t ac hm
moment at sm
a ll angl es of in ci d ence i s see n to p resen t a p r obl emof a
seri ou s engin eer in g ch arac t er
e
a n d t h e t en d en cy i s th erefore to so m
sa cr i fi ce of aer o dy na m
i c a dvant age
Pressu re Distribu ti on rou nd an Ai rsh i p EnveIOpa —A dr awi ng of t h e
.
°
.
,
,
.
°
°
,
.
,
.
,
.
od l i
m
ar k ed t h e p o sitions of t h e p in t s
gi v en in Fig 1058 on whi ch a re m
o
easu r e d
ewh a t grea t er p reci sio n i s gi v en
Som
a t whi ch p ress ur es w ere m
n of whi ch s how s t h e p r essu r es for t h e con d iti on
by T a bl e 4 6 t h e l ast col u m
Oth er figu r es a n d
in whi ch t h e axi s of t h e env el op e was a l ong t h e win d
d i agram
s show t h e p ressu re d i s t r i buti o n wh en t h e a xi s of t h e ai rshi p i s
Th e p ro d u ct of t h e
i ncli ne d to t h e r el a ti ve wi n d at an gl es of 10 a nd
et er i n f eet was 15 whils t t h e
win d Sp ee d i n f eet p er secon d a n d t h e d i am
2
p ressu res h av e b een d i v i d ed by pV t o p ro v i d e a s ui t abl e p ressure cccii i
c i en t
With t he a xi s al ong t he wi n d Fig 105A sh ows a p ressure coeffi ci en t
of h alf a t t h e n os e whi ch fall s v ery ra p i d ly t o a n ega ti v e v a lu e a s h o r t
di st an ce f u r th er b ack Th e p ressu re coeffi ci en t d oes not ri se to a pos iti v e
os t co m
v al u e till t h e t ail region i s a lm
p l et ely t ra versed and i t s g rea t es t
v alu e a t t h e t ail i s o nly 10 p er cen t of th at at t h e nose It i s of so e
i n t erest a n d i m
p or t an ce t o know th at t h e regi on of high p ressu re a t t h e
nose ca n b e in ves tig a t e d on t h e hy p oth es i s of a n i n vi sci d fl ui d whi ch th ere
e
s
.
,
,
.
°
,
.
.
,
,
.
,
.
.
m
D E SI G N D A T A FR OM AE RODYN AM ICS LAB OR A T ORIE S
gi v es
21]
tisfac tory res u lts as t o p ress ur e di s t rib u ti on Th e s ti ffenin g of t h e
n ose m
en ti on ed in an ear li er ch a p t er ca n t h e refore be p r ov e d o n d
n ori
p
rea so n i n g
Wh en t h e a xi s of t he envel ope i s in clin ed t o t h e win d lac k of sym
metry
in t rod u ces com
p l exity into t h e o bs erv a ti on s an d r ep resenta tions B y
ro llin g t h e m
od el a bo ut i t s a xis ea ch o f t h e p ressure h ol es is b rou ght in t o
fer en ce with t h e hol e on t h e
p os iti ons rep resen ta ti v e of t h e wh ol e ci rcu m
win d war d s i d e t h e an gl e h as been d enot ed by
an d t h e sym
et ry of
m
o d el sh o ws th a t o bse rv a ti on s a t 0 a n d 180 w ou l d be t h e sa m
t he m
e
Th e
3
F romt h e la tter it will be
r es ult s a r e s h own i n T a bl e 4 6 a n d i n Fig 1051
sa
.
'
.
,
.
°
°
.
.
(
0
.
3 0°
F
m1 3 »
.
0
1
~
Press u re d is t ri b u t i on
on a n
i ncli ned
a i rs h i
d
m
p
o el
.
al t o t h e
t h a t t h e p ressure ro u n d t h e env el op e a t any sectio n norm
a xis i s v ery v ar ia bl e a pos iti v e p res su re on t h e win d war d si d e of t h e n os e
s for a n
gi vi n g p l ace t o a large negati ve pres su r e a t t h e b ack Th e di agram
ost st r i ki n g formowin g t o th ei r
i n clina ti on of 80 sh ow t h e effec t s i n m
a gnit u d e
—For ty p i ca l ob ser va tions on kite b all oons t h e r ea der
Kite B au
i s r e ferr e d t o t h e secti on i n Ch ap t er II wh ere i n t h e co urse of d iscussion
p l et e accou n t was gi ven of t h e
o f t h e co n dition s of equilib ri u ma co m
o bse rva ti ons o n a m
od el
s ee n
,
.
m
°
.
m
.
.
,
A PP L IE D AE RODYN AM ICS
12
Pa nss
Va lu es
H ole No
.
of
m
p ress u re
TAB LE 46
on A
as a
.
mm
Mo u nt A
.
fract i on of p V
‘
.
ar
i
.
Incli nati on
,
C H A P TE R
IV
TH E A ERO D YNA M I CS L AB ORA TO RI ES
D ES I GN DATA FROM
—
P A R T II B on x AxE s
.
A ND
NO N
-
RE C TI L I N E AB
F LI G H T
m
oll ecting t h e o re com
p l ex data of fli ght it is a dv isa bl e for ease of
p ari son and u s e th at r es ult s he r eferr ed to som
e st a n d ar d sys t e
of
com
a d e owin g to t h e n ec essity for co m
Th e ch oi ce i s n ot eas ily m
ax es
p rom
men d e d
i se but recen tly t h e R oya l Aeron auti ca l Soci ety h as r eco m
p l et e sys t emof n ot a ti on an d sy m
bols for general a d op tio n Th e
a com
d et ails ar e gi v en i n A Glossary of Aeron au ti ca l t er s and wi ll be
Th e a x es p r ep osed d i ner f rom
foll ow e d i n t h e ch ap t ers of thi s b oo k
oth ers on whi ch aeronauti cal d at a h as b een based an d so m
e li ttl e car e i s
om
en t s
necessa ry i n a tt achi n g t h e co rr ec t sign s to t h e v ari ous fo rces an d m
It h app ens th at v ery si m
p l e chan ges only ar e r e ui red for t h e grea t bul k
of t h e a vaila bl e da ta
—
Fig
p l et e ai rcraft lS co m
1
6
Th e ori gi n of t h e a x es of a com
0)
monly
Axes (
ta ken at i t s cen t re of gr av ity an d d enot ed by G Th e reason for thi s
oti on of t h e cen t re o f gr a v ity
i cal th eo r emth a t t h e m
t h e d yn am
a ri ses fr o
i n ed by t h e resu lt an t for ce whi ls t t h e rot a ti on of a
of a bo d y 18 d et erm
bo dy d ep en ds only on t h e res u ltan t cou p l e a b ou t an axi s thro ugh t h e
Thi s th eo rem1s not t ru e for any o th er p ossibl e origin
cen t r e o f grav ity
an y p ur p o ses
F romG t h e longitu di nal a xi s GX go es forw ard and for m
al a x i s GZ
may be rou gh ly i d entifi ed wi th t h e a irscrew axis Th e norm
metry an d i s d ownwar ds whi lst t h e la t eral a x i s
li es i n t h e p l an e of sym
al to t h e oth er t wo a x es a n d t ow a r d s t h e p ilot s r ight h an d
GY i s n orm
o ve wit h
Th e a x es ar e cons i d er ed t o b e fi x e d i n t h e aer o p l ane an d to m
it so th at t h e p osition of any gi v en p art such as a win g ti p a lways h as t h e
otion This w oul d not b e t ru e if wi n d
e co or d i n at es th r o u ghout a m
sa m
cu lt i es w oul d th en occu r i n t h e ca l cula ti on of
a x es w ere ch osen an d di
For m
any p u rp oses t h e axi s GX m
oti on a s s p inni n g
ay b e
su ch a m
cho sen a r bit rarily whi l st i n oth er i ns t an ces it i s co n v eni en tly t a ken as
on e o f thg p ri nci p al a xes of i n er ti a
i rcra ft it i s not alway s p ossibl e t o rel at e t h e
I n d ea li n g with p a rt s of a
result s i niti ally to a x es s u it a bl e for t h e a i rcra ft si n ce t h e l a tt er m
a y no t
th en be d efin ed It i s consequ ently necessary t o consi d er t h e co n v ers ion of
result s f r om
on e set of b o dy a xes t o anoth er
So fa r a s i s p ossibl e t h e a xes
of sep arat e p art s a re t aken t o conformwith tho se of t h e co m
p l et e a i r craft
—
i
n
Any p ossibl e p o sition of a bo d y r el a ti v e
Angl es rel ati ve to th e W d
ea ns o f t h e angul ar p ositio ns of t h e ax es
t o t h e wi n d ca n b e d efin ed by m
Two a n gl es th ose o f p it ch a n d y aw a re req u i r ed an d are d enoted t eep ee
bols or an d B Th ey a r e sp eci fied as follo ws fi rs t p l ac e
t i vely by t h e sy m
IN
c
m
.
,
m
.
”
,
'
.
,
q
.
.
.
m
.
,
.
.
,
,
.
,
’
.
m
,
-
,
.
.
,
.
,
.
.
,
.
.
,
,
.
,
,
2 14
D ESI GN
D AT A FROM AE R ODYN AM ICS LAB OR A T ORI E S
215
xi s of X alon g t h e win d secon d rot a t e t h e bod y a bout t h e axis of
Z t hrou gh an an gl e B a n d fin ally rot at e t h e bod y a bout t h e new positi on
o f t h e a xis o f Y t h ro u gh an a n gl e or
Th e p ositi v e sign i s a tta ch e d t o a n
a n gl e i f t h e r o ta ti on o f t h e bo dy i s from
GX t o GY GY t o GZ or GZ t o
GX
Thi s i s a con v eni en t con v enti on whi ch is also app li ed t o elevat or
a n gl es fla p sett i n gs an d r u dd er m
o v em
ent s
With such a con v en tion it i s
fo u n d th a t confus i on of sign s i s ea sily av oi d ed
An gl es ar e gi v en t h e na m
es roll p i t ch or y aw for rot a tio ns a bo u t t h e
a x es of X Y a n d Z r es p ecti v ely
It sh oul d be noti ced th at an an gul ar
en t a b out t h e ori gi na l p osition o f t h e a xi s o f X d oes no t ch an g e
di s p l acem
t h e a ttitu d e of t h e bo d y rel a ti v e t o t h e win d
Forces along th e Ar ea —Th e res u ltan t fo rce on a bo d y is co m
p l et ely
s p ec ifi e d by i t s como n en t s a l ong t h e thr ee bo d y a x es
u
t
e
d
p
s
iti
v
C
o
n
e
o
p
wh e n act i ng fr omG towar ds X Y an d Z ( Fig
they a re d enot ed by
mx Y an d 1712 an d Sp oken of as longitu dinal force
force and
the
a
,
.
,
.
,
.
,
.
.
,
.
,
.
.
,
m
.
,
,
,
FI G 106
.
m
.
St a nd ard
-
a xes .
ay n ot b e
p
t t he m
ass of an a i r cra ft an d m
i cs ] d at a i s b eing o bt ai n ed ; t h e for i s
t h e aero dyna m
otion
co n v eni en t wh en app lyi ng t h e eq u a tion s of m
p l et ely
Mo ents abou t th e An a — Th e resu lt an t cou p l e on a bod y i s com
p on en ts a bou t t h e thr ee b o dy ax es Coun t ed p os iti v e
S p eci fi ed by i t s co
GZ to GK an d fr o m
GY to GZ fr om
w h ere th ey t en d t o turn t h e bo dy fr o m
bol s L M an d N an d are known as
G X t o GY th ey are den ot e d by t h e sy m
om
om
o en t
en t p it chin g m
en t an d ya wi n g m
r ollin g m
—
Th e co m
r
p on en t an gul ar v elociti es
l
u
A
x
Ang u a Veloci ti es abo t the
es
b ol s p q
a nd
kn own as r ollin g p it chin g an d y awing a r e d en ot ed by t h e sy m
ov e t h e bo dy so as t o i ncr ease t h e
r an d ar e p ositi v e wh en th ey t en d t o
c o rr es p on di n g an gl es
a gn itu d e of t h e
Th e forces an d co u p l es on a bo dy d ep en d on t h e m
r e l a ti v e wi n d V t h e i ncli n a ti ons a an d B a n d t h e an gula r v elociti es p q
o d el i s st a tionary r el a ti v e to t he
In a win d ch annel wh ere t h e m
and r
ost of t h e observ ations hi th er to
an d r ar e ea ch z ero a nd
ch ann el wa lls p q
m
r e resen s
m
,
.
m
.
,
,
,
,
m
,
,
m
.
,
.
,
,
,
.
,
,
,
m
A PP L IE D AE RODYN AM IC S
216
md
how t h e forces an d cou p l es as d ep en d en t on V at an d 3 only To
od el i s us u ally gi v en a sim
fin d t h e v ari ations du e to p q
a nd r t h e m
pl e
oscill atory m
otion an d t h e cou p l es ar e th en ded uced fr o mt h e rat e of
d am
p in g At t h e p r esent tim
u ch of t h e d at a i s b ased on a com
bina tion
em
en t an d ca l cul a tion a n d d iscu ssi on of t h e m
of exp er i m
etho ds i s d efer red
to t h e next ch ap t er Exam
p l es of r es ult s a re gi v en i n t h e chap t ers
on Aeri al Man aau v r es an d t h e Eq u atio ns of Motion an d St a bili ty
In
t h e p resen t section t h e res ults ref erre d t o are obt ai ned with p q
an d r
zer o
a
e s
.
,
,
,
.
,
.
.
,
.
u i valent Meth ods of representi ng 3 Gi ven Set of Obser vati ons
Eq
Fig 107 shows th ree etho ds of rep resenting t h e force an d cou p l e on
m
’
.
.
Directi on
a
Direction,’
.
m—M
1
.
ct h ods of represent i n a
g
gi ven
set of obscrva t i o
m
.
m
l ateral a x i s i s n ot sp eci fically in v ol v ed owing to t h e sy m
et rv
ed but i t s i n t ersecti on with t h e p l an e of sy m
assu m
metry at A an d B is
r equi r ed
An aerofoil i s su pp osed to b e p l aced in a unifor curr en t of ai r
a t a n an gl e of in ci d en ce or
Th e s im es t
etho d o f sh owi ng t h e a ero
d y n am
i c efi ect i s th at of Fig 107 (a ) wh er e t h e result ant force i s d ra wn in
o d el
etho d how ev er r eq u i res a d ra wi ng
p ositi on rel ati v e t o t h e m
thi s m
an d is th er efore not s uit ed for t a bul ar p r esen t atio n
Fig 107 (b) sho ws t he
win g
.
Th e
mm
,
.
.
m
'
.
,
,
.
.
APP L IE D AER ODYN AMI CS
218
m
m
m
Yawi ng Mo ent on a Model of a Flyi ng B oat B u
A dr awin g of t h e o d el
is shown in Fi g 108, t og eth er with t wo s a ll in set di a gra s of t h e pos itio ns
of t h e a x es Exp eri en t s were a d e to det er ine t h e lon gitu di n al a n d
n or al fo rces an d t h e p it chi n g
o en t for v arious angl es of p i t ch a b u t
wi t h t h e an gl e of y aw z er o , an d a lso to d et er in e t h e l ongitu di n a l a n d
l a t era l f orces an d t h e yawin g o en t for v ar i ous angl es of y aw 3 b u t with
t h e an gl e of p it ch z ero
Th e rea d i n gs are gi v en i n T a bl es 1 an d 2 a n d
cu r v es fr o
t h e ar e sh own i n Fig 109
m
.
m m
m
m
mm
mm
m
.
m
m
‘
,
.
.
.
TAB LE I
Fos cas
m
-
AN D
Mou rners
on A
Wi nd
mB
Fu
s eed ,
p
.
e
40 ft
.
.
oar
Hu
m( P
.
rrcs
)
.
s.
ml f
Nor
Fe nce s
a ox a
.
Lo ngi t u d i nal fo rce
( lb s )
( lb w
.
.
mB
Fu
Wi nd s p eed
A ngle of y a w 3
( d eg rees )
o rce 1
112
TAB LE 2
mMm
m
o
a
,
e
40 i ts
x
m
o ar
.
.
-8 .
L ate ral for ce
( lb s )
.
H u nt ( Yaw)
.
Y
m
mm t N
Y awi n g
( lb e
o
en
.
Fig 109 sh ows th a t t h e norm
Z an d t h e p it chin g m
om
en t M
al for ce m
ou n t s th an t h e l on gitu d i n al f orce
u ch gr ea t er p ro p or tion at e am
ch ang e by m
mx wh en t h e angl e of p it ch is changed an d th at t h e l at eral force m
Y an d
en t N sh ow a si
ya wi ng m
om
il ar f eature as t h e a ngl e of y aw i s ch anged
.
m
,
.
DESI GN D A T A FRO M
AERODYN AMI CS
A N G L E OF P I T C H
A N G LE or
Fro
.
109
-
.
Forces
a nd
mm t
o
Y AW
en s on a
md
L AB OR AT OR IES
219
Of
3
[
o e l of a
ny in g b oat h u ll
-
.
la t t er case Fig 109 (b) it m
ay b e n o ti ced th at t h e l ongit u d i n al
es zero a t an angl e of y aw of
force m
ent was
x b ecom
Th e r olli ng m
om
consi d ered t o be t oo sm
all to be w orthy of m
easu rem
en t
In t he
,
.
,
.
AER ODYN AMIC S
A PP L IED
220
gl e of p it ch it is ob vious th at th ere will be a di agra mi n
whi ch t h e angl e of y aw is v ari ed Th e nu m
ber of i nstan ces i n wh i ch
measuremen ts ha ve been m
a d e fo r l ar g e v ari a ti ons o f both a: a n d B i s v e r y
all an d p a rti a l res ults h a v e th erefore b een used ev en wh ere t h e m
sm
or e
co m
p l ete ob serv ations woul d h av e b een d ir ectly app li ca bl e It o nly n eed s
to be p oin t ed ou t tha t t h e si x qu antiti es X Y Z L M N ar e n eed ed for
all an gl es a B for all a n gu la r v eloc iti es
a
n
d
l
of t h e
s
tt
g
or
l
f
a
s
e
i
n
r
p q
el e v a to rs ru dd er a n d a il er o n s for it to be r ea lise d th a t it i s n ot p oss ibl e
t o cov er t h e wh ol e fi el d of aerona uti cal r esearch in gen era l form For
a t ely
t h i s reason it i s exp ect ed th at sp ec ifi c t ests on ai rcra ft will ulti m
be m
ad e by co nst ru cti ng fir m
i cs l a bo ra tori es
s an d th a t t h e aero d yn a m
en t
will d ev elo p t h e new t est s requi red and gi v e t h e l ea d t o d ev elo p m
For
h
eac
an
.
.
,
,
,
,
,
,
.
,
,
,
,
.
,
.
TAB LE 3
Fe nc es
u rn
Mou rners
B od y wit hout an
on as
.
An t on
iu
s] : B o n r
m
“ (I ru dder .
force i lin i
.
“ fl “
Lat eral force
(M )
ru dd er
‘
l o ng i t u d i na l
$ 3 3
.
.
B od y wit h an and
( dd fil i t
of
Long i t u d lnal
( Yaw)
1
0sec
TAB LE 4
.
Er r er
or
Wind
A118 “
01
B od y wi t h fi n and
( rud de r i t
Re na t
sp eed ,
r u dd er
a
( Yaw)
.
40 i t s
B ed
wi th an and
ru dd er i t
ru dder
APP L IE D AERODYN AMIC S
222
m
of t h e body of
an d h en ce t h e p ositio ns of eq u ilib r iu
s h ow n by
Ta bl e 4 a t —12 for a r u dd er an gl e of 10 an d a t —25 for a r u dder an gl e
of 20 m
us t b e d u e to t h e coun terac tin g effec t of t h e fi x ed fin It will
thus be seen t h a t t h e li ght ness of t h e ru dd er of an aerop lan e d ep en ds on
t h e a rea of t h e fi x ed fin
Th e b es t r es ult will cl ear ly be o bt ain ed if t h e
fi n j u s t co un t eract s t h e effect of t h e bo d y
Th e exp eri m
en t to fi n d t hi s
cond iti on coul d be p erfo rm
ed by m
om
en t on t h e
eas u rin g t h e y a wi n g m
bo d y an d fin wi th ru dd er i n p lace but not a ttach e d a t t h e h i nge I t
woul d not b e s uffi ci en t t o m
o v e t h e ru dder s ince t h e forces on
er ely r em
t h e fi n woul d th er eby b e a fi ect ed
Th e p ossibiliti es of t h is line of i n qui ry
h av e not been seri ously in v es ti ga t ed
°
°
°
°
.
.
.
.
,
'
.
.
Th e Efl ect of th e Presence of th e B ody and Tai l Plane and OI Sh ap e of
Pi n and Ru dder on th e Effect i veness of th e Latten —E t hi s exp eri en t t h e
m
m
m112 —M d
F
o e l ae ro p la ne
.
co
mp l t
e e
t a il u n i t
.
m
was set at zero an gl e, an d ca nnot th er efore b e di fferenti at e d fr o
Th e b asi s of co p ari son h as b een a d e t h e l a t eral fo rce p er un it
t h e fin
I t i s fo u n d tha t t h e coeffi ci en t
a rea d i v i d ed by t h e s qu a r e of t h e wi n d sp ee d
so d efi n e d d ep en d s n ot only on t h e s h a p e of t h e v er ti ca l s ur face, but a ls o
Th e dra wi n g
on t h e p resen ce o f t h e b od y a n d t h e t ai l p l an e a n d el eva to rs
of t h e o d el u sed i s s h own i n Figs 112 a nd 113 , t h e l att er gi v in g t o an
en l arg ed s cal e t h e sh ap es of t h e fi ns a tt ach e d i n t h e seco n d s eri es of ex
e
n
t
s
e
r
i
p
Th e exp eri e n t s r eco r d ed i n T a bl e 5 a pp ly t o t h e o d el as i llus t r a t ed
by t h e full lin es o f Fig 112, th a t i s wi thout t h e fi n arked AI Th e t es t
l ea d in g to t h e secon d colu n of T a bl e 5was a d e with r u dd er alone h el d
i n t h e win d , a n d will b e fou n d to show great er v alu es of t h e l at eral fo rce
o d el A ran ge o f angl e
coeffi c i en t th an wh e n i n p ositi on as p ar t o f t h e
o n i n st ea dy st ra ight flyi n g a nd t h e
of p it ch o f 10 d eg rees i s n ot un co
with it a l on g t h e wi n d
bo d y w as t es t e d with t h e a xi s of X u p war ds
r
u dd er
bod y wi t h
m
.
m
.
m
m
.
.
.
m
.
m
mm
m
m
m
m
.
.
.
AERODYN AMICS
D ESI GN DAT A FROM
with it p itched do wn war ds
p osition
both with and with out
and
in
L AB OR AT ORIES
the
228
l v a tors
e e
.
TAB LE
m
E
L a t eral forces
on
t he
cr or
B o nx
ru dde r of
u
p
Eu
5
.
v a r o ns o n
mR
a
Fi g 112 i n lbs di vi ded by
wi n d speed i n fee t per sec
a rea
.
.
u p e es .
in
s
qft
.
.
a nd
by
q
u are of
s
.
e levators .
0000104
0000205
00003 15
000042 1
0000528
000005
0
0000104
0000170
000024 9
00003 3 0
0000057
0000114
0000186
0000265
00003 5
0
0000071
0000155
000024 7
000033 7
0000433
Consi der in g fir
s t t h e coeffi ci en t s for t h e
e l ev ato rs
I n all
t h e v al u e i s
.
ml l3 —V
F
.
.
a ri a t i o ns o f
fin
1
1
0000063
0000083
0000l 83
0000133
00002l6
0000274
0000303 I 00003 70
00004 8 2
0000402
md
0000065
0000143
0000226
00003 06
00003 9 7
l with t ail p l an e and
l ess than th at for t h e free
a n d ru d
o
de r
e
a rea .
e i n di ca tion of a grea t er sh i el d i n g by t h e b od y
dd er a nd th ere i s so m
Thi s f ea tur e i s
w h en t h e n os e i s u p th an wh en it i s eith er l e vel or d own
—
r
t
e
0
r
d
ily
f
r
Fig
1
1
4
a
wh
e
e
h
c
u
r
v
es
f
a
n
d
5
o
n
o
ore rea
m
see
p itch ar e
( )
s een t o li e b e l ow th ose of t h e ru dd er a l on e b u t a bo v e t h e cu r v e for an an gl e
Fig 114 (b) sh ows i n thi s in st ance t h e effect of t h e p r esen ce
o f p itch o f
o f t h e el eva t o rs ; as or d i n a t e i s p l o tte d t h e l a t e ra l fo rce c o e ffi ci en t with
ila r co effi ci en t with ou t t ail p lane Th e
t a il p l an e on an a bsci ssa of t h e s i m
sel v es a bou t a s t ra ight lin e whi ch sh ows a
p oin ts are seen to gr ou p th em
ru
,
m
.
°
.
°
,
,
.
,
,
,
,
.
APPLIED AE RODYN AMI CS
224
loss of 14 per cent du e to t h e p r esence of t h e ta il p lane A fur th er reductio n
t h e i n t r od u ction of t h e m
p let e
m
ay b e e xp ect e d fr o m
ai n p l an es i n a co m
a ir craft du e to t h e slowin g u p of t h e ai r wh en gli d in g
On t h e oth er h an d
t h e in fl u en ce of t h e airscrew sli p st ream ay be to in crease t h e v a l u e
materi ally un til t h e final resul t an t effect is great er than that on t he fr ee
r u dd er
Th e t est s on t h e efiect of sh a p e w er e carr i ed out on t h e sam
e bo d y
but without tai l p lan e a nd elevators an d t he r es ults are gi v en i n T a bl e 6
Th e fins w ere d i v i d ed in to two grou p s A l to A 6 an d B 1 t o B 5 o f
.
.
m
.
,
.
'
,
,
.
,
O N OS
,
,
00 005
'
00 004
0 0 003
00003
00 002
0 0 00 !
0 000 1
00W !
00002
( Wi th ou t
00003
00004
To i lp lc n e
.
00 00:
)
m
0-0004
(
00 003
00003
0 0002
00002
oo;
0 000 :
Z
6
4
A rca ne
or
Ya w
.
Fro l l4
.
(
2
0
Au
)
d egr e e s
—Efi
eet of va ri a t i ons of
'
.
c u : or
fin
5
6
4
Ya w
( negr o
an d ru dder area .
whi ch A 1 an d B 1 w ere i d entical in size an d sha p e In t h e A ser i es t he
f orm
s of t h e v er ti cal sur face w ere roughly s im
ilar t h e a in change b ein g
e of s iz e
on
Fi g 114 (0) i n di ca t es li ttl e ch ange i n t h e l a t era l f orce co effi ci en t
un til t he area has been u ch r ed u ced Seri es B on t h e oth er han d shows a
marked loss of effi ci ency du e to red uction of t h e h eight of t he fi n (Fig
114
an d both result s ar e co nsi s t en t with an d are p r o b a bly exp l a i n ed by
a r ed u cti on i n t h e sp ee d of t h e a i r i n t h e i m
medi at e n eighbour hood of t he
bo dy Exp eri m
en t s on t h e fl ow of fl ui d r ou n d s t rea m
li n e fo r s h a v e shown
ay b e m
ar ked o v er a lay er of ai r of a pp recia bl e
th at thi s sl owi ng of t h e air m
t h i ckn ess
.
,
.
.
m
.
m
,
,
.
m
.
.
APPL IED AERODYN AMI CS
226
l v t
d oes n ot need a sep ara t e figure I t shou l d be not ed th a t
en t i s p os iti v e an d th er efor e t en d s t o in crease a d evi a ti on
om
t h e y awin g m
metri cal position The effect of t h e lateral force which
fromt h e sy m
a pp ears wh en a n a i rshi p i s ya w ed t en d s on t h e o th er h a n d t o a red u ctio n
u l a t e a t h eo ry of m
oti on b efore a
of t h e a ngl e a n d it i s n eces sary t o f orm
sa ti sfact ory b a l an ce b etw een t h e t wo t en d enci es i s o bt a i n ed
—
Th e fi rs t ill u s t ra ti on h ere gi ven of t h e
n
Fl
a
Ai l erons and Wi g
d et erm
i nati on of t h e three com
pon en t fo rces an d com
p onent m
om
en t s
ple m
o d el aerofoil A
i n whi ch or a n d 3 ar e both v ari ed relat es t o a s im
l at er t a bl e whi ch i s an ext ension shows t h e effec t of wi ng fl ap s Th e
model was an aerofoil 18 i ns long and 3 i ns ch or d with s qu are ends ;
en t s with fl a p s two r ect a ngul a r p orti ons
i ns lon g
for t h e exp er im
a n d 11 6 in s wi d e w ere a tt ach e d by hi nges so th a t th e ir an gl es coul d
a i n su rface
be a djust ed in d ep end en tly of th at of t h e m
e e a ors an d
.
,
.
,
m
.
.
.
.
.
.
.
.
As s oron
.
R AF
.
.
.
m
Mu r
0, 3
TAB LE 8
m
ou ss x
u rs o n
ls
Mo u nt
.
m
mF
Wm
en u s,
u
a
.
wr
u
s on ar. ro
rs
n -s r s nn o r
B ot h flap s
40
s
m
s
Foru
.
n s r ca n s ee
m m
s ar
'
.
at
La te ral
ml mm
t
m
Z
L
N or
a
o
( lb s )
-
i
en
( lb s- ft )
o
.
.
M
-i
l
b
a
t
(
)
.
b
l
s
ft )
(
.
00257
°
Angle of y aw 0
Angle
of
°
y a w 10
8 shows th a t , t h e angl e of y a w h a v in g b een set at t h e v alu es
°
°
eas u re
10 an d 20 i n each seri es of
ent s, t h e an gl e of p it ch was
°
°
v ari ed d u r i n g t h e exp eri en t by s t ep s of 4 fro — 8 t o
Th e
et ry 006 i n a b ov e t h e
o r igi n of t h e a x es was a p oi n t i n t h e p l an e of sy
i ns b ehin d t h e l ea d i n g e d ge
ch or d a n d
With t h e a xi s of X i n t h e
°
d i recti on of t h e win d t h e aerofoil a d e an angl e of in ci dence of 4 wh en
t h e angle of y a w was zer o : i c t h e a ngl e arc of Fig 107 c was
With t h e
T a bl e
m
m
m
.
.
.
m
m
mm
.
.
.
D AT A FRO M
D ESI G N
AERODYN AMIC S
L AB OR ATORI ES
227
gl e of y aw zero it follows fromsym
met ry t h at t h e la t eral force and t h e
a tt er wh a t t h e a ngl e of p it ch
om
en t s are all z ero no m
rol lin g an d y awin g m
e an d a
Th e l ongitu d ina l force on an aerofoil a pp ears for t h e fi rs t tim
a n egati v e v al u e at an a ngl e
co ns i d er a ti on o f t h e t a bl e shows th a t f rom
8 it r ises to a grea t er p ositi v e v alu e a t
an d th en a ga in
of p itc h of
beco m
Th e norm
al
es n ega ti v e a s t h e criti ca l angl e of a tt a ck i s exceed ed
for ce —m
om
en t
Z has t h e gen er al ch ar act er i sti cs of lift whi ls t t h e p it ch in g m
di ffers fr o mt h e q u an titi es p rev iously gi v en o nly by bein g referred to a
n ew a xi s
an
.
,
-
°
.
,
.
Anae ron. wr
Flaps
a t :iz i
mWm F
o
°
0 ( ri gh t hand fla p down ,
u rs
a nd
.
left -h an d fla p
up
)
.
—00 140
—0064 1 —00027
00685 4 00068
—00684
00 165
00698
00244
00955
00 197
01043 +00 15
2
00 152
- 00664
00030
00724 +00076
—00730 00 180
0073 2
00266
00209
00767
00723 4 0009 7
00654 —00 146
—0003 6
+00076
00713
00 182
-
—
—
—
Angle of ya
Angle
of
w —
10
°
—
—
—
°
ya w 0
00268
004 15
00267
00 134 +00 111
—00654 —00 125
—0063 2 —00040
00671 +00072
0063 2
00 175
00260
0023 3
+0033 5 +00 132
00628 - 00 118
00603
00621 +00060
0053 1
00 154
00233
+cocce 1 00222
+00550 +00 151
—
—
Angle
of
a
w
y
°
10
—
—
At an angl e of y aw of 10 all t h e forces a nd cou p l es ha v e valu e, bu t
n ot all ar e l ar ge
Th e l at eral force Y i s n ot i p ort an t as co p ared
with longitu di nal force, whils t t h e y a wi ng o en t N is s all co p ar ed
with t h e p it ching o ent On t h e oth er h an d , t he rolli n g o ent L
°
‘
.
mm
.
m
mm
m
m
m m
mm
APP L I ED AERODYN AMI CS
228
es v ery i m
becom
po rtan t at la rge angl es of inci dence This m
ay be ascr ibed
to t h e criti cal fl ow occu rr ing m
ore readi ly on t h e win g whi ch is d own wi n d
.
00 2
-
o 04
-
O 06
-
O 08
-
0 02
-
O 04
-
0 -06
-
C OB
-
O IO
.
'
°
-
'
'
'
20
-
10
AN G L !
0
or Y AW
a
e r ees
d
( g
olli ng a nd
than
-
)
y a win g
20
-
10
to
0
A N G L E or v aw
mm t
o
en s
d ue t o t h e
tha t facin g in t o t h e wind
littl e less force wh en t h e an gl e of y aw is
du e t o t h e y a w
with
.
.
—R
Fl o 115
.
lo
20
IO
.
( d e gr ees )
u se of ailerons
.
.
m
Th e r e ar ks
Th e resu lts show
tha t
APP L IED AERODYN AMI CS
23 0
gl e of in ci dence corresp o n ds with a s t ea dy fl ight sp eed l arge an gl es b ein g
e im
p ro v em
associ at ed wi t h low Sp ee ds it wi ll be seen th at so m
en t co u l d
an
,
,
F
m116
.
.
-
Tho ba la n ci ng
of a i le
m
ns .
bt ained o v er t h e range 4 to 12 by t h e u se of a sp ri ng with a cons ta n t
p u ll acti ng on t h e ail eron l ev er
Th ere i s of course no r eason why thi s ty p e of balance shoul d not be
app li ed to el ev ato rs an d ru dd ers a s w ell as t o a il ero n s a n d m
an y in s t an ces
of su ch u se e x i st
Owing t o l ack of Opp ortun ity for m
a k in g m
eas u r em
en ts
be
°
°
o
.
,
,
,
.
D ESI GN D AT A FROM AERODYN AMI CS
L AB OR ATOR IES
23 1
fic accu r acy littl e is known a s to t h e v al u e of t h e d egr ees of b al ance
ti
i
i
o btai n d
di ca t i on gi v en i s th a t p lot s d isli ke a cl ose
Th e cl ear es t m
e
a p p rox1m
a t 10n to ba la
e i n o r di nary fl ight
o f sc en
,
m
.
.
O -O O I
I
AN G L E
m111 —M m t
F
For ces
or
.
o
FL A P Neg r oe s )
e n s on
.
.
b a lanced ai le ro ns
.
exp eri
and
ar
i
n
r
e
s
e
a
c
a
n
B
2
ll
d
l
t
h
e
E
th
th
t
d
ib
d
P
tI
e
o
f
o
r
m
e
t o a sm
e
t
h
o
f
a
re
s
e
o
f
t
h
r
o
n
r
o
lt
h
v
b
d
t
h
e
p
o
p
ti
qu
i
n
b u t t h e r es u s a e een i ncrease
s
i
n
h
e
T
r
e
m
a
a
e
c
r
r
c
o
e
b
e
m
li n ea r di m
io
to
d
i
tly
o
p
bl
w
g
en s n s e t c so as
e
r
a
e
o
e
m
f
t
h
r
o
a
s
o
o
fi
n
n
a
d
ih
d
l
gl
th
er
e
Ph
t
p
h
d
l
e nor w as
e r a an
g
h ad no
y
men ts refer
a
,
.
,
.
,
.
APP L I ED AERODYN AMI CS
23 2
R eport No 111 Advis ory Com
mitt ee for Aerona u tics The axis
o f X was t a k en t o li e along t h e win d a t an angl e of i nci d ence of 6 an d an
an gl e of y a w of
en t s w ere m
a de for l arge v ar i atio ns of angl e
Exp er im
o f y aw an d sm
all v ari a ti on s of angl e o f p itch
ited i n scep e
Althou gh li m
t h e r esu lt s ar e t h e only on es av a il a bl e on t h e su bj ect of fli ght at large angl es
of y aw an d r ep r esent one of t h e li m
i t s of knowl edge
App li ca tion i s s till
furth er fromcom
p l et en ess
h
s own
in
.
,
.
°
,
.
.
.
TAB LE 10
.
Fo ac ss
mM
on
11
Co
sca le .
ot h
{
Anne
oi
aw
|
J
Th e r esu lt s
grap hi ca lly for
mpl t
4 0 ft
e e
.
fo rce
tu x ( lb s )
Co
mm M
.
Mod el B E 2 AerO plane
.
Y ( th
m
ml
Nor
mm t
Rolli ng
a
fo rce
.
Anaor nax s
ou nt.
a
Angle of i n ci den ce =angle of p i tc h
-s .
Lonni t u dlna1
'
xr s on A
e
o
en
L
)
mm
mm t
Ya wln
Pi t chi ng
o
en
en
o
M ( lb s
N ( lbs
b v ati ons ar e gi ven in T a bl e 10 an d are sh own
z er o a ngl e of p it ch i n Fig
118
Th e si x cur v es th r ee for
for ces an d three for m
om
ent s a re ra p i dly d i v i d ed in t o t wo gro u p s accor d in g
t o wh eth er th ey a r e sy m
met ri cal or assy met ri cal with respect to t h e
v erti cal at zero y aw I n t h e sym
met ri cal gr ou p are longit u di nal force
n orm
a l fo rce a n d p it chi n g m
om
en t whil s t i n t h e asy m
metri cal group are
l at era l force rollin g m
om
en t
It i s for this reason
en t an d p it ch i ng m
om
th a t cert ain m
o ti ons a re sp o k en of as l on gitu di nal or sym
metri cal and
oth ers a s l a t era l or a s m et ri cs ] an d corr es p o n din g wi th t h e d i st in ction
y
i s t h e sep ara ti on o f two m
a i n ty p es of st a bility
Up t o a ngl es of y aw of + 20 it a pp ea rs th a t l on git u d i nal force a n d
of
the
o ser
,
.
,
.
,
,
.
,
,
.
,
,
.
°
APP LIED AERODYN AMI CS
234
R AF 6
It was b en t a bout two lin es near t h e cen tre, t h e d etails bein g
shown i n Fig 119
Th e origin of axes was t ak en as 007 in a bo v e t h e chor d
an d
i ns b ehin d t h e l ea di n g ed ge, whi ls t t h e axis of X was p arall el t o
t h e chor d
In thi s case, th er efore, angl e of p itch an d an gl e of inci d en ce
.
.
.
.
.
.
.
.
.
Fl a 119
.
m
—Model aerofoil wi t h di h
.
ed r a l a n le
g
.
ean in g
e
Man y observ ations w er e m
hav e t h e sam
th e
a d e a n d fr o m
has been ex t ract ed T a bl e 11 whi ch gi v es for on e d ih edra l an gl e an d sev era l
pon en t forces an d m
om
Fig
en ts
an gl es of p it ch an d y aw t h e thr ee co m
om
en t o nly for v ar i a tions of y aw
120 on t h e oth er han d shows r olli n g m
p it ch and di h edral angl e
.
m
,
,
.
,
,
,
.
Fo ae as
AN D
Mou rners
.
o n an
Angle of p i tc h
,
Annoron
na
.
m
e a
Wi nd speed
-
An gle
of
i
t
c
h
,
p
,
D
m
nnau
40 feet per
.
sec.
Ar
m
an or
D ESI GN D AT A FROM
AERODYN AMIC S
TAB LE l l
Angle
of
—o
c
23 5
L AB OR ATORI ES
n lrn u ed .
'
°
p i t ch , 10
Angle of p i t ch ,
+00208
Angle of pi tc h,
0
00017
00034
0004 7
004 15
00424
00456
00503
00542
00573
005
99
00064
00077
0009 3
00 109
Th e v ari atio n
of longitu d inal force with di h ed ra l an gl e p res ent s no p oin t
of im
p ort an ce excep t at high angl es of i nci d ence where as usua l t h e fl ow
ore regu l ar an d i s
Lat eral force i s how ever m
shows erra ti c f eatur es
r oughly p r op o r tion al b oth t o t h e di h e dr al angl e an d t h e an gl e of y aw and
I t s v alu e i s n ev er so
i n d ep en d en t of t h e angl e of i n ci dence u p to
ar k ed i m
gr eat as to gi v e m
p ort an ce in cons i d erin g t h e m
Y any m
otions of
No rm
p ort ance ev en at l arge
an aero p l ane
a l fo r ce sh ows n o ch an g es of i m
en t i s not s t ri k in gly affect ed by
om
an gl es of in ci d en ce whil s t p it chin g m
t h e d ih edral a ngl e ex cep t a t t h e criti cal angl e of in ci d en ce
ost in t eres tin g p rop erty of t h e dihedral angl e i s t h e p ro d u ction
Th e m
of a rollin g m
om
en t n ear ly p r op o r ti ona l to it self an d n early i n d ep en d en t
of angl e of i nci d ence u ntil t h e criti cal v al u e i s app roach ed Thi s i s m
os t
rea d ily a pp r eci a t ed fr omFig 120 o r d i nat es of whi ch sh ow t h e r ol lin g
o del at a win d sp eed of 40 ft s Th ere i s
momen t in lbs feet on t he m
a sm
a ll rollin g co u p l e for n o di h edral angl e a t a ngl es of in ci d ence u p t o
10 and a consi d era bl e cou p l e at 15 an d
At v ery l arge angl es
e r ev ers ed a n d is not t h e
t h e e ffect of t h e di h edra l angl e h as b ec o m
,
,
.
,
,
.
,
.
.
.
,
'
-
.
.
°
°
o
.
APP L IED AERODYN AMIC S
23 6
00 6
04 0
c
m
:
o -c c
0-0 7
o -os
0 -0 5
0-0 4
0 02
m120 —R lli g mm t
F
.
o
n
o
en s
d ue t o
a
d i h ed ral
a ngle on a n acroi oi l.
APP L IED AERODYN AMI CS
23 8
Th e
s
ion
fo rm
ulae gi v en in
thereby annplifi ed
With t h e values gi ven
is
an d
3
( )
4
( )
s
easu re d
uffice t o con v er t forces m
.
t erm
s Of X0
,
Y0 Z
,
0
,
,
the
ex
p res s i ons
X Y Z
for
,
,
,
L,
,
an d
N in
L0, RIO and NO ar e
II XO
X
Y =1
2Xo
Z lsxo
l
o4
mY
mY
‘
z
»
o
m
o
z
L
M
11110
"t i
1
1
1
2 0
M 2310
N
1
4
1
3 0
e
mM
o
a
" zzo
” 3 20
No
" sNo
" sNo
"i
—
m
be
M
h
n
i
i
b
t
r
i
i
h
r
f
f
h
e
r
n
w
i
n
o
igin al a x es
a
o
f
t
o
h
e
o
D
e
c
t
o
I
C
e
O
u
t
C
a
g
g
g
( )
an d Z oGo of Fig 122, an d t h e origi n i s to b e t ran sferre d
KOGO,
.
Fl o 122
.
,
-
L
Th e
—C h
ange of ori i n of
g
G, t h e cc or di na t es
3
n
d
e
a
n
th
,
y
to
being 2
.
of
t he
m
= L0 + Yo
forces are not a ffect ed by
t he
.
la tt er
z
—
body
axes.
l ti v e to
re a
mZ
o
.
y
.
the
D ESI GN D AT A FROM
For
ml
u ae
AERODYN AMI CS L AB OR ATOR I ES
uati ons
for Sp eci al Use wi th th e Eq
of
Moti on
and
23 9
Stabi li ty
.
otion i n general fo rm
of m
d o not con ta in t h e an gl es a an d
t h e com
p onent s of v elocity
B ex p li citly but obt ain t h e equi v al en t s fr o m
Th e result an t v elocity bei ng d enot ed by V
al ong t h e cc o r di na t e a x es
p onen ts al ong t h e ax es of X Y and Z by u v an d w it will be
an d t h e co m
4
n
d
a
a
th
t
8
seen fro m
)
)
(
(
Th e equ ati ons
,
-
.
,
u
an d
t he
=V
cos a cos
i p rocal
r ec
fi
,
v
-
V sin
w = V sin
fi
,
,
oz cos
B
8
( )
l tions are
re a
w
m
=
,
—si n
o
"
‘
1
m
d
n
o
t
u
l
8
n
9
i
s
fi
c
a
it
d
if
t to p ass fro t h e u se of t h e
( )
( )
vari a bl es V a and B to u , v an d w
St a bility as covered by t h e th eo ry of s a ll oscilla tions app ro xi at es
t o t h e v alu e of fo rces an d cou p l es i n t h e neighbour hoo d of a con di tion of
by us in g a li nea r law of v ari ati on with each of t h e v ar i abl es
eq u ili b riu
Ma th e a tically t h e p osition i s th at any one of t he quantities X
is ass u ed to b e of t h e for
By
eans of
'
m
.
,
m
m
m
m
.
m
’
w, p ,
v,
q)
,
r
.
t i values of u
r whi ch will b e d en ot ed
gi ve a cond ition of eq u i librium For t h e us ua l
h eavi er th an air craft it i s ass u ed th at X can b e exp an d ed
and cer a n
m
.
-
-
et c
.
,
i t
res s an ce
m
i d
for
m
d eri v ati v es an d d enot ed by
m
t us ually app ea r in t er s of V a: and p
i t i s co nv eni ent to ded uce t he d eri vati v es fr omt h e o rigin al cu rv es an d thi s
an d r ar e z er o) by m
ea ns of t h e
i s p oss i bl e ( for t h e cases i n whi ch p q
u l ae below
s tan dar d fo r
X,
"
etc
.
As t h e aer o dyn a
ll d
ar e ca e
in t he
c
a a
,
,
,
m
,
cOS ar si n
cos a cos
B
fl
v and
w constan t
( 12)
APP L IED AERODYN AMI CS
240
t
u a n d v cons an
fi
sin a cos
o
2V
_ 2V co s a
q
W
a
cos
with sim
il ar rela tions
é
=2
xu =
+V
fl
.
F
a
z
for t h e
cos a cos
m
a“
Fx — V
o
u
é
+V
sin e:
a“
5
3 7
a
2
—
33
1
64
—V cos
ther qu antiti es
,
so
mfi
th at
— cos
B
a s
az sin
B
V
1
2"
v
1
V
“
1
V
1
—2 smfi
—2
Ji m
m
e
or cos
—cos fi
X
fi V2 +
GE,
cos a
'
cos
fl
6a
—
sm
a
m
s
t
C HAPT ER V
AERI AL MANOE U VRES AND THE EQUATI ONS OF MOTION
’
o ditions of st ea dy flight of a ir craft h av e b een d ealt with in consi der
wh ere t h e eq u ati ons u se d w ere sim
p l e b eca use
a bl e d eta il i n Chap t er II
p li city of t h e p robl em Wh en m
oti ons su ch as 100p in g sp inn in g
of t h e si m
an d tur ni n g ar e be in g in v es tigat ed or ev en t h e di st u r ban ces of s t ea d y
motion a change of meth od i s found t o be d esirable Th e equati ons
of m
p l es t or t h e m
os t
otio n n ow i n t r o d u ced a r e app li ca bl e to t h e si m
s y et p r op osed
p l ex p robl em
Ev i den ce of an e xp er im
en ta l char act e r
com
u l at ed
an d a pp ara tus n ow exis ts whi ch en a bl es a n
has b een accum
en t s t o b e m
ade
ov em
ber of r ecor ds
aer i a l m
Th e n um
analysis of
taken is not y et great bu t i s su ffi ci en tly im
p ort an t to i n t rod u ce
oti on of an aer o p l an e d ur in g
t h e subj ect of t h e cal cul ati on of t h e m
Aft er a b ri ef d es cri p tion of th ese reco rds t h e
aeri al m
anoeu vr es
ula t e t h e e qu ati ons of m
oti on an d to a pp ly
ch a p t er p r ocee ds to f orm
e of t h e ob serv ed m
th emt o an in v estigation of som
oti ons of aerepla n es
in flight
—
i
n
In aki ng a loo p t h e first o p era tion i s to d i v e t h e aerOplan e
Loop g
—
r
e
l
s
A
n
n
c
a
a
i
s
ee
8
0
100 m
o
d
to
g
i
p
d
i
d
i
t
d
p
d
of
r
r
a
n
ee
h
i
s
u
s
ua
l
y
n
e
i
p
bered that t h e
su ffi ci en t but at co ns i d era bl e h eight s it sho u l d b e r em
em
Sin ce t h e air forces d ep en d on
rea l sp eed i s gr ea t er th an t h e a irsp eed
in di cat ed a irsp ee d an d t h e kin eti c en ergy on r eal sp eed th rough t h e ai r it
will be ob vi ous tha t t h e rul e whi ch fi x es t h e a irsp eed is f av ou ra ble to loop in g
Hav ing reach ed a s uffi ci en t Sp eed j n t h e d i v e t h e
at co nsi d era bl e h eight s
n i s p ull ed s t ea dily b ack as far as it wi ll go an d thi s woul d
con t rol colum
p l etion of a loop Th e p ilot how ever swit ch es
be su fi ci en t for t h e com
ak es u se of hi s el evator to com
e out
off hi s en gi ne wh en u p si d e d own an d m
of t h e di v e gen tly Not un t i l t h e ai rspeed is th at s uit abl e for clim
bing i s
t h e en gin e restar t ed
I n loo p ing aer o p l an es whi ch hav e a rot ary en gin e it ay be necessary
to u se cons i dera bl e r u dder to coun t eract gyroscop i c co u p l es Th e efi ect
of t h e airscrew i s felt i n all aerop l anes an d unl ess t h e ru dd er i s used t h e
loo p i s i m rfect in t h e sens e th at t h e wings d o not keep l evel
an y m
inor v ari ations an d un ti l
00p i ng i s subj ect to m
Th e op er ation of 1
otion i s known it
t h e p ilot s u s e of t h e el eva t or an d engi ne d ur ing t h e m
etho ds of ca l cul ation i n st r i ctly co m
p ar ati v e
i s not p ossibl e to a pp ly t h e m
form A full a ccoun t of t h e ca l cul ati on i s gi v en a li ttl e l a t er in t h e ch ap t er
and fro m
it h av e been extract ed t h e p arti cula rs whi ch w oul d b e exp ect ed
fr omin str u m
ent s used i n flight
Th e ins t rum
en t s w er e s u pp osed to con
s is t o f a r ecor d i n g s p eed m
et er a n d a recor d i n g a ccel er om
e t er
B oth h a v e
THE
c n
.
,
.
,
,
.
,
.
,
.
,
.
m
.
,
.
.
.
.
,
.
,
.
,
.
,
,
,
.
m
.
.
m
,
.
,
'
.
.
.
.
MANOEUVR ES
AERIAL
AND EQ UATION S
OF MOTION
24 3
ay p er h a p s b e r eca ll ed h er e th a t
been ref err ed to in Ch ap t er III but it m
t h e l a tt er gi v es a m
ea sur e of t h e ai r fo r ces on t h e aerop l an e
Th e accel er o
meter is a sm
ov es with t h e aerop lan e ; t h e
all p i ece of a pp ara t u s whi ch m
movin g p ar t of it wh i ch gi v es t h e reco rd has acting on it t h e force of
gra v ity an d any forces du e to t h e accel erations of i t s s u pp ort It is
th erefore a ass whi ch t akes all t h e forces on t h e aereplan e proportionally
ex cep t those d u e to t h e ai r
o v em
en t of th is sm
all m
ass
Th e di fferential m
an d t h e l ar ge m
ass of t h e aero p l an e d ep en ds o nly on t h e ai r f orce alo n g i t s
axi s
et ers wo u l d be r equi r e d wi th
p l et e rea di n gs thr ee accel erom
For com
th ei r ax es m
utu ally p erp endi cu l ar In p ra cti ce only on e h as b een u sed
with i t s ax is app ro xi m
a t ely i n t h e d i r ection of lift in st ead y fl ight an d t h e
accel era tion m
easure of t h e
easu r ed in unit s of 9 h as b een t ak en as a m
incr ease of win g loa di ng
Speed and Loading Records i n a Loon —Fi g 123 sh ows a r ecor d of t h e
t ru e sp eed of
d ur i n g a l oop with a corr esp on din g di agr am
.
,
.
'
,
,
m
.
.
.
.
,
,
.
.
,
L OO P
LO O P
ml23
F
.
.
-
Speed
a nd
force
on
wi ngs d u ri ng
a
loop
( observed )
.
e sca l e for t h e two cur v es i s t h e sam
force on t h e win gs Th e ti m
e
an d co rres p on di n g p oin t s on t h e d i a gra m
s h a v e b een m
a r ked for ea se of
ref eren ce
Th e p reli m
inary di ve from0 to 1 t akes n early h alf a in u t e
e t h e force on t h e wi n gs was r ed u ced b eca us e of t h e
d urin g whi ch tim
i nclination of t h e p ath
At 1 t h e p ilot began to p ull b ack t h e s ti ck with
an in cr ease i n t h e fo rce on t h e wi n gs to 8 5
ti m
es i t s u su a l v al u e wi t h i n 5
fo
r t he
.
m
.
.
,
,
APP LI ED AER ODYN AMI CS
244
Whi lst t h is force was bein g d evelop ed t h e sp eed h ad scarcely
bing to t h e to p of t h e
ch anged
B etw een 2 an d 3 t h e aerop l ane was clim
loo p with a rap i d fall of force on t h e wings From3 to 4 t h e r eco v erin g
d i v e was t akin g p l ace with a sm
a ll i ncrease of force on t h e win gs after
ed
whi ch 4 t o 5 t h e aerOplan e was fl att ened out an d l ev el fl ight res um
an d at 5in t h e Sp eed
Th e t wo d ep ressions j us t before 4 on t h e force di agram
di agramp rob ably corr es p on d with swit ching t h e en gine off and on
Th e ca l culat ed sp eed an d l oa din g d u r in g a loo p are shown in Fig 124 t h e
or
6
secs
.
.
.
,
.
,
,
,
.
,
.
,
CALcu u r ro )
10 MI N S
F
m124 —Sp
.
.
eed and
force
on
wi ngs d uri ng
a loop
(calcu lat ed )
.
im
p o rtan t p er io d of t en secon ds bei ng shown by t h e full lines Th e d ott ed
e x t ens i ons d ep en d v ery gr ea tly on wh at t h e p ilot d o es with t h e co nt r ols
an d a r e of littl e i m
p ort an ce in t h e com
p ar ison Th e m
ai n f eatur es of t h e
observ ed r ecords are seen t o b e rep eat ed t h e general di fferen ces in di catin g
t h e adv an tage of an in t elligen t u se of t h e el ev ator i n r ed ucin g t h e p ea ks
an o eu vre as sum
of stress o v er th at of t h e rigi d m
It
ed i n t h e cal cu l a tions
wi ll b e f o un d wh en consi d er i ng t h e cal culation th at an u se of t h e el ev at or
y
can be r ea d ily in clu d ed an d t h e corr es p on din g effect s on t h e loo
n
d
a
p
s t resses in v es tigat ed i n d et a il
.
,
.
,
.
.
APP L IED AERODYNAMIC S
246
with
gi n e off As flyin g sp eed i s lost t h e ru dd er i s p ut har d ov er
i n t h e d ir ec ti on i n whi ch t h e p il ot wi sh es t o sp in
So lo n g as t h e con t rol s
p ar ti cu l arly so l ong as t h e col u m
n i s b ac k t h e a erop l a n e will
ar e h el d
To r eco v er t h e ru dd er i s p u t cen t ral an d t h e el ev a t or
co nti nu e t o sp in
e ith er cen t ra l or s lightly f orw ar d
t h e sp inn i ng ceases an d l ea v es t h e
whi ch it i s flatt en ed ou t
aero p l a n e in a n ose d i v e fr o m
en t a lly an d t h eor et i
Sp innin g h as b een stu di ed car efully both exper im
It p ro vi d es a si m
ple m
cally
ea ns of v erti cal d es cen t to a p ilot who i s n ot
an oeu vr e i s n ot
e gi ddy
Th ere is evi d ence t o show th a t t h e m
a p t t o b eco m
00p in g whi ch
u n i v ersa lly con si d er ed as com
fort a bl e, i n sh arp cont rast t o 1
h as far l ess efi ect on t h e feelin gs of t h e a v erage p ilot
Force and Speed Records i n a Spi n —At
Fig 126 t h e
eed f e ll
e
s
h
k
T
was
a
ti
b
g
p
ull
d
b
p
70 m
h
n
d
t
h
e
s
ac
k
c
e
i
n
e
p
the
en
.
.
,
,
.
,
,
.
.
.
'
.
,
.
.
.
.
.
M I NS
.
S PIN
Fl o l 26
.
.
Speed
-
an d
force
on
wi ngs d u ring
s p i nni n
g
.
l t h e aereplane h a d st all e d a nd was p u tt i ng i t s nose d own
ra p i dly
This la tt er p oin t is shown by t h e red ucti on of ai r force on t h e
win gs Th e angl e of in ci d en ce con tinu ed to in crease alth ou gh t h e Sp eed
was ri si n g an d a t 2 t h e sp in was f u lly d ev elop ed
u su ally in clin ed at an an gl e of 70 —
80 to t h e h ori zon t a l an d i s r ot ati n g
a bout t h e v erti ca l once i n ev ery 2 or 2
t
ti
qui
s
e
cs
r
t
e
T
h
e
o
a
o
n
i
s
n
o
t
5
r egu l ar as will b e see n fro m
both t h e v el ocity and force di agram
s b u t h as a
an d a t
.
.
,
.
°
°
,
.
,
,
AERI AL
MANOEUVR ES AND EQU AT ION S OF MOT ION
247
ut ation su p er p osed on t h e av erage speed Th ere i s no reaso n to su pp ose
that t h e p eri od of n uta tion i s t h e p eri od of sp in
At 3 t h e ru dd er was cen t rali sed an d t h e sti ck p ut sligh tly f orw ar d an d
a lm
os t i m
med iat ely flattenin g ou t b egan as shown by t he i ncreased force
at 4
ai n d er of t h e hi st ory i s th at of a di v e t h e fl a tt enin g out
Th e rem
h a vin g b een accel era t ed som
ewh at a t 5
It h as b een shown by exp eri m
en t s on m
od els th a t s t alli ng of an aer o p l ane
a n d th a t
a utom
a ti ca lly l ea ds t o sp i nni n g
t he m
ain f eat u re of t h e
p h eno m
en o n i s cal cu l abl e q u it e si m
p ly
—Th e r ecord of p ositi on of an aerop lane shown i n Fig
Roll ( Fig
a s econ d a ere p lan e
127 was t aken by cin em
era fro m
Moun t ed
a ca m
i n t h e rear cockp it t h e cam
era was p oin t e d o v er t h e t ai l of t h e ca m
era
aerep la n e towar d s th a t p hotograp h ed
era a ero p l an e was fl own
Th e cam
carefully i n a s t ra igh t lin e but t h e cam
er a was f ree t o p it ch an d t o
rot a t e a b out a v er ti ca l a x i s
For this r eason t h e p i ctur es ar e
not a lw ays i n t h e cen t r e of t h e
film In d i scu ssin g t h e p ho t o
gr ap hs whi ch w ere t aken at
i nt er v als of a b ou t
sec
i t is
ill u m
inatin g t o u se at t h e sam
e
e a v elocity a n d fo r ce r ecor d
ti m
a
t
Fig
ltho
gh
it
d
o
u
e
s
n
o
(
app ly to t h e sam
e a erop la n e
Ph ot ograp h 1 shows t h e
aer o p l ane fl yin g s t ea dily and on
an
k eel so m
ev en
e di st an ce
a bo v e t h e ca m
era
Th e Sp ee d
i s
woul d be a bout 90
I3
I4
j us t befo re 1on t h e Sp eed ch ar t
Fro 127 —Ph t ogra p hi c reco ds of ro lli ng
Th e seco n d p h ot ogra p h show s
t h e b egin nin g of t h e roll whi ch i s acco m
p an i e d by an i ncrease in t h e angl e
of in ci d en ce
Th e l att er p oin t is sh own by t h e in creased l en gth of p r o
ec
t
i
o
n
of t h e bo d y as w ell as by p eak 1 i n t h e force di agram B oth roll
j
a n d p it ch are in cr eased i n t h e n ex t i n t er v al with a corr espo n d in g f all of
At fou r t h e ba nk i s n ea rly 90 a n d t h e p it ch i s slightly r ed u ced
sp eed
Th e v ertica l b a nk i s th erefor e rea ch ed i n a littl e m
ore th an a secon d
O nce o ver t h e v erti cal t h e angl e of in ci d en ce (or p it ch) i s ra p i dly red u ced
an d as t h e sp ee d i s fallin g ra p i d ly t h e t ot a l a i r f or ce on t h e win gs f alls un til
t h e aer o p la ne i s u p si d e d own a ft er ra th er m
or e th an i i secs
At a bou t
thi s p eri o d t h e force d i a gra msh ows a n ega ti v e ai r f orce on t h e win gs a nd
u n l ess s t rapp e d i n t h e p il ot wo u l d h a v e l eft hi s sea t
T h i s negati ve a i r
for ce d o es n ot a lway s occu r d uri ng a roll a nd i s a v oi d ed by m
ai n t a i n in g t h e
angl e of i n ci d ence a t a h igh v a l u e for a l on ger ti m
e
Th e p il ot t en d s t o
fa ll with a n accel era ti on eq u a l t o g but i f a d own wa r d a i r force occu r s on
t h e win gs of t h e a ere p la n e it t en ds t o fa ll fa st er th an t h e p ilot an d th ere
fo re m
ai n t ain s t h e p r es su r e o n h i s sea t
or e u su a l con d iti on i n a
T his m
r oll i nv o l v es as a con sequ e nce a v ery r a p i d fa ll wh en t h e a erop l an e i s u p si d e
n
.
.
,
.
,
.
,
.
.
.
.
,
.
,
.
.
,
.
,
.
.
.
.
.
.
r
o
.
.
,
.
.
‘
,
°
.
.
.
,
.
,
.
,
.
,
,
.
.
APP L I ED AER ODYN AMI CS
24 8
m
m
d own Th e ost noti cea bl e feat u r e of t h e re ai n i ng p hotograp hs i s t h e
fac t th a t t h e p ilo t i s hol din g u p t h e n ose of t h e aer op l ane by t h e ru dd er
a
ano eu vr e acco m
p ani ed by vigorous si d e sli pp i ng As t h e an gl e of
inci d en ce i s n ow n or a l t h e sp eed p i cks u p ag ai n d uri ng t he rec o v ery
of an ev en k eel
Th e m
anoeu v r es a ft er 3 Fig 128 a re tho se conn ec t e d
with flatt enin g out and occur subsequ ently t o t h e roll Th e co p l ete r oll
t ak es rath er l ess th an four secon ds for co p l eti on
Th e r oll m
ay b e carr i e d ou t eith er with or wi th ou t t h e engi n e an d ex cep t
for sp ee d t h e m
an o eu vres are t h e sa
e as for a sp i n Le t h e s ti ck i s p u ll e d
back an d t h e ru dder p u t h ar d o ver Th e angl e i s nev er red uced t o th a t
a n d thi s i s t h e essen ti a l aero dy na
i c di flerence fr omsp inn i ng
.
m
,
m
.
,
.
.
,
m
,
m
,
.
.
m
,
.
,
m
.
'
.
RO LL
m
O S
us
°
.
RO LL
m
p hotograp hs sh ow th a t th ese sim
p l e i nst ru ctions are su pp l e ent ed
by oth ers at t h e p il ot s d iscretion a nd th at t h e aero d yna i cs of t h e m
otion
Th e
m
’
,
d ealin g with t h e m
or e com
p l ex m
otions of
a ir craft i t i s fo u n d t o b e a dv a nt ag eou s t o foll ow som
p re
e d efi n it e an d com
h en si ve sch e e whi ch will co v er t h e gr ea t er p ar t of t h e p r obl e s li kely to
occur
oti on a re
Sy st e s of ax es an d t h e corr eSp cn di ng eq u ati ons of m
t o b e foun d i n a dv ance d b ooks on dyn a i cs a n d fromth ese are sel ect ed
t h e p arti cul ar for s r el a ti ng to r igi d bo d i es
An aer op l ane ca n o v e fr eely i n or e d i r ecti on s th an any oth er v ehi cl e
i t can m
ov e u p w ar d s forw a r d s an d si d ew ay s as w ell as r oll p it ch a n d t u rn
Th e gen erali ty of t h e p o ssibl e
i nence t h e v alu e t o
oti ons b r in gs in to p ro
q
E
uati ons of
.
Moti on
.
m
m
m
~
1n
m
m
m
m
,
m
,
.
m
,
.
APP L IED AERODYN AMI CS
25
0
to th ei r resp ecti ve ax es i s t h e s t at em
en t of t h e p r obl em
co m
p l et e As a n
ex am
p l e cons i d er t h e fli ght of an aerep lane : t h e forces and cou pl es on i t
d ep en d on t h e v elociti es lin ear an d an gul ar thr ough t he ai r an d h ence two
set s of a xes are h er e r equi r ed one i n t h e a ero p l a ne a nd t h e oth er 1n t h e a i r
Th e w eight of t h e aer ep lan e b rin gs i n fo rces du e to t h e earth a nd h en ce
I n t h e rare cas es in whi ch t h e r ot ation of t h e earth i s con
ea r th a x es
w ould b e
si der ed a f our th set of ax es fi xe d rel ati v e to t h e st ellar sys t em
int ro d uced and so on
en t of a p robl em
p rio r to t h e app li cation of ath em
Th e st at em
ati cal
an aly si s r eq u i r es a k nowl ed ge of t h e f or ces an d cou p l es a cti ng on a bo d y
for all p ositions v elociti es accel era ti ons et c rela ti v e to ev ery o th er
en ta l an d h as som
bo dy concerned T h is d at a i s us ually exp erim
e d egr ee
of app ro xi m
a tion whi ch i s ro u ghly known
B y accep tin g a low er d egree
ore set s of axes m
ay b e eli m
i nat e d fro m
t h e p robl em
of p r ecisi on on e or m
a th em
a ti cs
with a corr es p on di ng si m
p li fication of t h e m
This st ep i s t h e
j usti ficat i on for igno rin g t h e effect of t h e ear th s r ot ation in t h e u sual
otion of ai rcra ft
es t i m
ation of t h e m
A f u rth er si m
p li ficatio n is int ro d uced by t h e negl ect of t h e vari ations
of gra vit ational a ttr action wi th h eight an d with p o siti on on t h e earth s
su rf ace t h e conse qu en ce of whi ch i s th a t t h e cc o r di na t es of t h e cen t r e of
otio
ra v ity of an aerop lane d o n ot app ear i n t h e equ a tio ns of m
n of a ircra ft
g
in still ai r
Th e angular cc or di nat es app ear on acco u n t of t h e v arying
p onent s of t h e w eigh t along t h e a xes as t h e ai rcra ft roll s p it ch es an d
co m
turns I n cons i d erin g gust s an d th eir effects it will b e foun d n eces sar y to
in t r odu ce lin ear co o r din at es eith er exp li citly or im
p li citly
otion r el ati v e to t he ai r d ep en d m
arkedly
Th e fo rces on ai rcraft du e to m
on t h e h eight a bov e t h e earth an d of r ecen t y ea rs co nsi d era bl e i m
po rt an ce
h as a tt ach ed to t h e f act
Th e v er ti ca l co or di n a t e how ev er r ar ely
app ea rs di rectly t h e effect of h eight b ei ng r ep res ent e d by a cha ng e i n t h e
d ensity p and h er e again t h e app ro x im
ation oft en su ffi ces tha t p i s co ns t an t
d u r in g t h e m
otions consi dered Ap art fr o thi s reserv ation t h e air forces
otion an d a dvan t ag e i s t ak en
on an a i r cra ft d ep en d o nly on t h e rel a ti v e m
of thi s fact to u se a Sp eci al sys t em
of ax es
At t h e ins t an t at whi ch t h e
motion i s b eing consi dered t h e body axes of t h e aircraft have a cert a in
p ositi on rel ati v e to t h e ai r and t h e air a x es ar e t aken to m
om
en t a r ily
coinci d e with th em Th e r at e of sep ara tion of t h e t wo set s of a x es th en
oti on
p ro vi d es t h e n ecessary p ar ti cul ars of t h e rel ati ve m
ajori t y of t h e kn own
otion whi ch co v er t h e m
Th e equ a tions of m
ro
p
bl em
s r equi r e t h e u se of t h ree set s of ax es as follow s
B od y a x es
For conv eni ence t h e
( 1) Ax es fix ed i n t h e ai rcraft
o rigi n of th es e i s t aken a t t h e cen tre of gra vity an d t h e di rection s ar e
made to coinci d e with t h e p rin ci pal axes of in erti a Th e l att er poi n t i s
er
p ort an t than t h e for m
fa r l ess i m
r ax es
n st an t an eou sly coin ci d en t with
r
i
x
d
i
I
x
i
n
t
h
e
a
A
2
A
s
fi
e
e
( )
os t cases t h e ai r i s su pp os ed still rel a ti v e to t h e
t h e bo dy a x es
In m
e a r th
Ear th axes
( 3 ) Ax es fi xed i n t h e ear th
Th e an gu l ar rel ati ons b etw een t h e a xes d e fined i n ( l ) a n d (2) h av e
.
,
,
,
,
.
,
.
,
m
.
,
,
,
.
,
,
.
.
,
.
’
.
’
-
,
-
.
,
.
-
.
,
‘
-
.
,
,
m
,
.
,
.
,
.
.
'
"
.
.
,
.
.
.
.
.
.
.
.
,
MAN OEU V R ES
AERI AL
AND EQU AT ION S
OF MOT ION
251
ngl es of p itch an d
as
Ch
p
t
p
g
a
r
e
a
a
e
[
V
(
di rection cos in es an d t h e co m
p o nen t v eloci ti es
m
u
v an d w
Th e corr esp on di n g r ela ti ons b etween ( 1) an d ( 3 ) are
1 t h e aere plane i s p ut
r eq u i re d
t h e angl es b ein g denot ed by 0 st an d 5
in t o t h e p os iti on d efin ed by th ese angl es by first p l acing t h e body and
ear th a x es in to coinci d en ce an d th en
alr ea
dy been r eferr ed
w
a
s
s
a
l
o
by
e
n
a
;
y
to
of
.
,
.
,
,
:
,
,
r
: a bo u t
ot ating t h e ai rcraft through an angl e (1
the
Z
a
x is of
0 a bo u t t h e n ew p os i
tion of t h e Y a xi a of
an d
5
bout
t
e
h
a
9
fin ally
()
0
tion
of
pos i
of t h e X ax is
the
n ew
a ircr aft .
Th e angl es st , 0 an d d ar e sp oken of as angl es of y aw, p i t ch an d roll t e
sp ect i v ely , an d t h e d o u bl e use of t h e exp ressions
a n gl e of y aw
and
an gl e
of p it ch
shoul d be n ot ed
Confusion of u se i s n ot seri ously i ncurr ed since
t h e an gl es or an d 3 do not o ccur i n t h e equati ons of
otion b ut are
r ep r es en t ed by t h e co
p onen t v elociti es of t h e resu ltan t r elati v e win d
Tha t i s, t h e qu an titi es V or an d 3 of t h e aero dyn a i c easure en ts are
co n v er t ed in to u , v and w befo re
a th e ati cal an aly si s i s a pp li ed
With these exp lana tions t h e equa tions of otion of a rigi d bo dy as
app li e d to a i rcraft ar e wr itt en d own an d d escrib ed i n d et ail
'
m
mm m
.
m
,
m m
,
m
.
.
m{ + wq
m{ 6 +
=
=
mgw+ p — q
Z
} m
’
1
i
ur
v
’
u
h1
he
it s
“
h
P 3
+
f hi
=M
“
h1 = p A — F
—
r
E
q
h2 = q
B — r D —p F
m m
mm
m
m m m
m
m
mm
m
m
q
m
m
m
m
I n th ese equ ations
i s t h e as s of t h e a ir craft whilst A, B , C, D , E
and F ar e t h e
o ents an d p ro du ct s of i nerti a All are exp eri ent al
an d d ep en d on a kn owl ed g e of t h e d is t ributio n of
a tt er thr ou gh ou t t h e
'
'
'
’
'
aircra ft
Th e q u an titi es
x , Y , Z , L , M an d N a r e t h e f orces
and co u p l es on t h e a i rcra ft f ro
all s our ces, an d on e of t h e fi rs t op era ti on s
i s t o di vi d e th e in to t h e p a rt s whi ch d ep en d on t h e ear th an d those whi ch
ari se f ro
otion rel ati v e to t h e a ir The re ai n in g qu antiti es, u , v, w,
n d r d efin e t h e
a
n
o
e
otio
f
t
h
b
o
d
y
a
x
l
ti
v
to
r
a
x
es
T
h
e
i
a
es
t
e
r
e
a
e
h
p
equ a ti ons ar e t h e gen er al seri es a pp li ca bl e to a rigi d bo d y , an d only t h e
descr i p ti on i s li it ed t o ai rcra ft
Th e q u a ntiti es
F are fa ili ar i n dyna i cs a n d do not
A
,
.
m
'
.
,
.
,
,
.
.
.
APP L IED AERODYN AMIC S
252
mm
m
d fur th er a tt ention excep t to n ot e th at D an d F are zero fr o sy
et r y
It h as a lready been shown how t h e p ar t s of X —N whi ch d ep en d on m
oti o n
rela ti v e t o t h e ai r are
eas ur e d i n a wi n d ch annel i n t er s of a v an d w
n ee
p
q
an d
'
m
to d t m
i
emi
'
m
,
.
,
1
r
e er
n e t h e com
It now r a ns
p onen t s of gra vita tiona l att ractio n
p onen t p ar ts of t h e w eight along t h e
A littl e t h ou ght will show tha t t h e com
s o f t h e d ir ec ti on cosin es of t h e
bo dy a xes are r ea dily exp ress ed 1n t erm
downwar dly d ir ec te d v erti cal r el a ti v e t o t h e a x es Rot ation a bout a
v erti cal axi s th rou gh an an gl e h as no effec t on th ese d i r ec ti on cosin es
ill
t
t
d
s
a
u
an d t h e o nly angl es whi ch n ee d b e cons i d ere d a re 0 a n d c
s
r
a
e
}
Th e ea rth axes are GXO GYo an d 020 an d b efore rot ati on t h e
i n Fig 129
GY a nd GZ ar e su pp osed to coin ci d e with t h e
a x es GX
,
.
.
.
,
,
.
.
,
,
Z
Pro l 29
.
.
-
I ncli nat i ons
a
o f a n ae rop lan e
t o t he
ea rt
h
.
R ot atio n th ro ugh an angl e 8 a bout G
YO b rin gs X0 to X and Z 0 to Z 1
a sub seq u ent rot a ti on th rough a n angl e 5
6 a bout C X b ri n gs Yo to Y a n d Z ,
Th e
t o Z an d t h e b ody a x es ar e n ow i n t h e p ositi on d efin ed by 8 an d (6
d i rection cosin es of OZ O rel at i ve t o t h e bo d y a x es ar e
,
.
,
cos
n,
712
n3
=
cos
X GZ O
Y OZ O
Z GZ O
—sin
0
cos 0 si n
cos 0 cos
m
es t h e co rres p o n d in g di rec tio n
om
p o nen t s of t h e w eight ar e y ti m
T
i
bol s 111 112 an d 913 h a v e o ft en b een u se d t o d en ot e t h e
s
n
os
e
h e sym
c
p l e of cal cu l ation fromt h e
Th e firs t ex am
lon ger exp ressi ons gi v en in
otion will b e th a t of t h e 100p i ng of a n aereplane and co n
e qu ati ons o f m
an d
t he
c
,
.
,
Th e
re la t i v e
e xp er i
mt
N o n angu la r
t he d epen de n ce o f X
n o t y et s u ffi c i e nt t o cove r a wi de ra nge o f ca lcu la t i o n
en a l
t o t h e ai r i s
k n ow led ge
of
'
'
.
ve loc i t i o i
APP L IED AER ODY NAMI CS
25
4
Fi g 13 2
.
and
is
h wn
S o
as
dep en den t only
on
t he
res
ult an t v el ocity of t h e
—
0 01
-
03
“
02
Fl o 13 1- Nor1n a l force
~
p l ane an d h er e
only be j u s t ifi e d a s
aero
,
on an aere p la ne
re la t i ve
wi n d
.
ful stu d ent will see th a t t h e rep resen ta tion can
s t an ces
good a pp roxim
a ti on i n t h e s p eci al c ircu m
a care
a
d u e t o i n clina t i on t o t he
AER I AL MANOEUVR ES AND EQU ATION S
OF MOTION
mm
m
m
255
The ch i ef it e s i n p i t c hin g o en t a re illus t ra t ed by t h e cur ves of Figs 13 3
er rela t es to v ar i a ti on w ith a ngl e of in ci d ence,
and 13 4 of whi ch t h e for
Si nce
and t h e la tt er to v a ri a tio n w ith t h e a n gula r v elocity of p itc hin g
t h e t ail 9 si p l e
the cou p l e du e t o p itchi n g ar ises al ost wh olly fro
If l be
appro x i
a ti on all ows for t h e ch an ge of fo rce du e to p itchi n g
,
.
m
m
m
.
.
,
m
.
v u oc n v
m13 2 —Ai
F
.
.
rscre w
t h ru s t
fy
s ec
an
d
m
.
aero plane ve loci t y .
dis tan ce fr o t h e centre of grav ity of t he aero p la ne t o t h e cen t re of
pressure of t h e ta il t h e equation
M I Z H= M
can be seen t o exp ress t h e a bo v e i d ea tha t t h e tai l 19 t h e o nly p ar t of t h e
aer o p lane whi ch i s efi ect i v e i n p ro d u cin g chan es du e to p it chin g
g
With t h e aero dyna i c dat a i n t he for gi v en equations (7) are con
t he
,
H
'
m
m
v
z
—
-
+g
M,
cos
2
v
'
,
r2
—wq
uti
.
fi
~
M,
+
v
B
Z
m
K
h
2
V
q
B
+
Y
"
3i
V
APP L IED AERODYN AMIC S
256
Th ese eq u a ti ons Sh ow t h e chan ges of u w a nd qwith tim
e for a ny
gi v en con di ti on s of m
oti on and ena bl e t h e Jeep t o b e ca l culat ed fr o m
t h e ini ti al con di ti ons by a st ep to st ep p r ocess
Th e in iti a l con di tio n s
,
,
.
Fl o l 34
‘
.
mt b
.
—Pi t chi ng m
om
en t
o f an aero la n e .
p
m
gi ve a leap and so e f u rth er exp eri en ce or t ri a l
U su a lly
a n d err o r i s n ecessar y b efor e thi s can b e d one sa t i sfactor ily
e consi d era bl e a ltit u d e b u t t h e cal c ula ti o ns
l oop in g t a kes p lace only at som
osp h ere of s t an da r d d ens ity
e an a tm
n ow gi v en assu m
us
h
d u e t o pi t chi n g
e ch osen su c
as
to
,
.
,
.
,
APP L IED AERODYN AMI CS
258
im
ilar p rocess will now be foll owe d in
v al u es of t Equa ti on ( 10) m
ay be w r itt en as
A
the
s
evaluation of w for s
mll
a
.
an d
fr om
Fig
for
l v t
.
13 1it is
fo un d th at
15 t h e
°
e e a ors a t
in t h e neighbo u r hoo d
)
+ 0 07
— 4 77w
°
,
er i ca l v a l u es
In sertin g n um
,
60
-
q
e ua
tio n (20) beco m
es
4 77w + 168 4
°
v alue of q
p re viously obta in ed
i n t egral of (22) is
Th e
Ao
Sin ce
w
V
= —00 6
"
" 77‘
in iti al v alu e of w i s
so th at
an d
e
,
t he
at
£0
Valu es
of
73 36
w an d i h are
a
e
used
a n d an
(23 )
ti m
e t = o it f oll ows tha t
,
v al u e of
40 8
4
my b
quation
t he
w
a nd
q
2
( 2)
408
180
V ==
a nd
V = 180
wh en
°
w
a nd
v al u e of Z , is gi ven by
—1
Z
of
th en foun d to
A in (23 ) is
1022 6
4
t he
be
9“
7 "
h own i n T a bl e 1
s
.
TAB LE 1
.
Of t h e v a ri ou s
is
li m
it a ti on s i m
p ose d by ( 14) t h e one of grea tes t i m
p ort a nce
th a t rel a ting t o t h e const ancy
tha t this Shoul d
n ot
Ta bl e 1 t h en sho ws
method is start ed
.
veni en ce
.
be
Across t h e
g
an d r e fer e nce
p u sh ed furth er th
mit
li
Th e
a
of
;
of
tim
e of
an t h e
0 10 see
to Fig
v l
a ue
befo re
.
13 3
for
t he
will ind ica t e
9
1
V
:
t p
t o -s t e p
work m
ay b e arranged as i n Ta bl e 2 for co n
e a r bit ra rily
hea d of t h e t a bl e ar e i n t erv als of ti m
.
s e
-
MANOEU VRES AND EQU ATI ON S OF MOT ION
AER IAL
259
hosen ; as t h e ca lcu l a ti on p roceed s and t he t ren d o f t h e r esu lts i s seen
e of m
it i s us u ally p ossibl e t o u se i n t erv als of ti m
u ch great e r m
agnitu d e
tha n those sh own in T a bl e 2 For t = 0 a n u m
b er o f qu antiti e s s u ch as
V w q0 ar e gi v en as t h e in iti al d a t a o f t h e p ro bl emw h ils t oth ers li ke
c
.
mx
,
,
,
,
m
m
T
h
t
et c are d ed u ce d f ro
e c ur v es of Figs 13 0 134
A
p
a
r ison
co
"
s
v
as
be tween t h e exp res si ons i n t h e t a ble and tho se in eq u ations
an d
ollow e d
f
T
h
e a dd iti onal e qu a tio n for
e
will
i
n
d
i
ca
t
t
h
e
e
tho
d
1
1
( )
,
.
fin di ng
,
es fr o m
V com
-
.
m
.
2
V
y
b
im
p l e differenti atio n
S
.
u
?
?
w
mt
g
an d a rr an e
en
of
t er
m
s
.
TAB LE 2
.
0
1 c os
4 4
-
eo n )
-
m
1539
-
-
14 53
-
I
m
=0 05a re t aken fr o T a bl e 1, an d u si n g
Th e fun d a en ta l fi gu res for t =
an d ( 11
t h e t h e necessary cal cu l a tio ns in di ca t e d by eq u ati on s
)
a re
2, in,
Th e n e ces sa ry bas i s
a d e t o gi v e t h e ins t a n t an eo u s v alu es o f 1
f or st ep -t o step ca l culati o n i s th en co p l e t e, an d any d ifferen ces of d et ai l
m
m
-
°
m
q
m
.
APP L IED AERODYN AMI CS
260
m
woul d p ro ba bly be ad e t o suit t h e h a bits of an i n d i v i d u al cal cu l ato r
p ti on whi ch was m
n wa s
a d e i n p r oceeding t o t h e n ext col u m
Th e assu m
t
e
u es o v er
e
u
a
r
th at t h e v alu es of 12
w
q
l
to
a
v
g
v
a
l
a
t
e
re
e
e
h
a
q
p l e consi der t h e v alue o f
t h e in t er va l o f tim
e 0 to 0 10 secs
As an ex am
w
At t = 0 05 w 770 and in t h e in t er v al of 0 10
w ; at
Addi ng thi s to t h e v alu e of w at
see t h e cha n ge of w i s t ak en as 7 7
10 as t a b u la t ed
p ar i son
1 0 gi v es —3 1 as t h e v alu e of w at
A com
of t h e v al u es of w a q and 0 as ca l cula ted i n this way with th ose of
ati on s of ( 14) had n ot
T abl e 1 will Show th at t h e m
a th em
a ti ca l app r oxi m
inary s tag es of ca l cu l ati on for t =0 15 are
led t o l arg e errors
Th e p reli m
Shown an d t h e p r oced ur e follow ed will n ow be cl ear
.
q
,
,
.
'
,
,
.
.
:
,
,
,
q
.
’
.
.
,
TAB LE 3
.
l ulati ons were carri ed ou t for a com
p l ete 100p and T abl e 3
es
At t h e
sh ow s t h e v ar i a tio n of t h e qu an titi es co ncern e d a t ch osen ti m
beginn in g of t h e 100 p t h e angl e of i nci den ce i s Sh own as 0 4 d egree
whils t l ess than h alf a secon d l at er it h as ris en t o 11 0 d egrees Th e
e f ro m
t h e v alu e of 1112
leadi n g on t h e win gs can be cal cul at ed at any tim
b ers for V an d w an d Fi g 13 1 Th e
co r resp on d in g with t h e t a bul at ed n um
maxi mumis 52 times t h e weight of t h e aerop lan e but owi ng to t he fact
th a t t h e l oa d on t h e t ai l i s d own war d thi s d oes n ot rep resent t h e l oad on
t h e wi n gs whi ch i s th en a b out 10 p er cen t g reat er
Th e Sh ap e of t h e loo p can b e obt ai n ed by i n t egra ti on at t h e end of t h e
cal cul ations since t h e h ori zo n t al cc or d i na t e i s
Th e
ca c
,
.
-
,
.
.
,
.
.
,
-
cos
whi lst
.
t he
v erti cal
co
0+ w sin 0) dt
or di n ate i s
[
(
u
sn
—w eoe
o
m
di
.
.
APPL IED AB ROD YNAMI CS
262
foun d n ecessary to app ly ru dder t o coun t eract t h e gyrosco p i c effect of t h e
a in t ai n an ev en keel
air screw an d so m
Failur e t o co plete a Loop —Th e ca l cula ti ons j us t m
ed an
ad e ass um
e sm
all res er v e
ini tial sp eed of 180 ft s in a d i v e at
an d in d i ca t e d som
of en ergy at t h e t op of t h e loop A red u cti on of t h e sp eed to 140 ft s
an d l ev el fl ight b efo r e p u lli n g o v er t h e con t rol col u m
n l ea ds
with t h e
sam
e a ss u m
p tions as t o t h e a ereplane to a fai lur e to com
p l et e t h e loop
m
.
.
.
-
.
.
.
-
.
,
.
,
TABLE 4
I
(
m1
Resu lt ant
( ne w
1
.
velocit
.
I ncli nati on of ai rs cre w
“ l“
“
W
y V
.
Angle of i nci dence
( degrees )
.
Th e fi gur es i n T a bl e 4 a re of consi d era bl e in t eres t a s showi n g on e of
ay t e p o ra rily beco
e unco n t roll a bl e
t h e w ay s in whi ch a n a erop l ane
owing t o loss of flying Sp ee d Up to t h e en d of four seco n ds t h e cours e
a t eri al for co
oti o n p resen t s littl e
of t h e
en t
t h e aero p lan e i s th en
ovi n g v er tically u p war ds at t h e low speed of 60 ft 41 an d i s turnin g
u ch
Th e energy i s in su ffi ci en t t o carry t h e a erop lan e
ov er back w ar d s
fur th er b u t a t 5 secon ds t h e aerOp lan e is 20 d egrees ov er t h e v er ti ca l
m m
m
m
m
m
m
m
.
.
m
.
.
,
with
mll
p iti v
gl of i i d
but a sp eed of only
ft s
I n t h e n ext h al f seco n d t h e a er op lan e begins to fall an d a t t h e en d of
is
secs i s s ti ll l os in g s p ee d an d h as a larg e n ega ti v e an gl e of att ack i s
flyi n g on i t s b ack with t h e p ilot s u pp ort ed fr o hi s belt Owin g to t h e
ust per
low Sp ee d t h e cont rol s are p racti cally in op era ti v e and t h e p ilot m
force w ait u n til t h e a ero p l an e reco vers Sp eed b efore h e can r es u m
e n or a l
fl ight If t h e aerop l ane i s uns t a bl e i n nor m
e di ffi
al s t rai ght fl igh t som
ay b e exp eri en ced i n p assi ng fr o m
a s t ea dy s t a t e of u p si d e d ow n
c ulty
al a ttit u d e
flyin g t o on e i n a norm
Th e d et a il ed ca l cu l a ti ons fromwhi ch T a bl es 3 a n d 4 h av e b een com
t h e a uth or i s
p il e d w ere m
3 M Ca v e B ro wn e Ca v e t o wh o
a d e by Mi ss 1
in d ebt ed for ass ist an ce on thi s and oth er occasi on s
i ng and t he Spi ral Glide —Th e equ ation s
Steady Moti ons i ncludi ng h
oti on gi v en i n ( 1) a n d (2) t ak e sp ci a l for
otion i s st ea dy
of m
s if t h e m
e
a s
a
e an
os
nc
e
28
en ce,
-
.
.
-
,
m
.
,
.
,
.
m
,
.
m
~
.
.
,
-
.
m
-
,
m
.
m
.
OF MOTION
AERI AL MAN OEU VR ES AND EQ UAT ION S
q
263
Not o nly are t h e q u an titi es 12 13 iv p an d 1 eq u al to zero b u t th ere i s a
rel at ion b etw een t h e qua n titi es
an e
a
n
s
e
r
n
ro
fo
a
p
l
d
7
t
h
c
es
o
n
ae
A
p q
a lon g i t s ax es d ep en d on t h e in clin a tions of t h e a er op lan e rel ati v e to t h e
v er ti ca l it will be evi d ent th at th ey ca n only rem
a in co ns t an t if t h e res ult an t
r ota tio n i s a lso a bout t h e v erti ca l
Thi s rota tion i s d enot ed by £2 an d
l oo k ing d own on t h e a i rcraft t h e p ositi ve di rection 1s clockwi se
Th e d ir ection cosi nes of t h e bo dy a x es rel ati v e t o t h e v erti cal w ere
foun d an d r ecor d ed 1n
th emt h e com
p onen t an gul ar v el ociti es
an d from
a bout t h e bo d y a x es are
—0 sin 0
,
,
,
,
"
,
.
,
,
.
,
.
0
cos
n
cos
With t h e p ro d uct s of iner ti a D
st ea d y m
o ti on ar e
—
=X
wq vr =
an d
g
-
0 sin
0cos
5
9
6
9
F e q u al
sin
o
to
t he
z er
e
q u ati ons of
0
(3 10)
1
3
( 10)
3
( 1p )
1
3
( g)
3
1
7
( )
ar
— uq Z + g cos O cos q
S
—
—
=
rq
L
E
C
B
) pq
(
2
2
—
=
=
A
r
M
r
O
E
(
)
p
)
(p
B
(
ap
In equ a ti ons
X Y Z L M an d N refer on ly to forces and cou p l es
du e to r el ati v e
an d r gi v en
otio n thr ou gh t h e a i r If t h e v alu es of p q
by (30) are use d i n
t h e so ewh a t di fferen t fo r s b el ow are obt ain ed :
vp
m
:
,
m
.
X
¢
sin
-
—v sin 0
g
s
c
o
g
,
O
d sin
.
m
—(A
( 22 {
,
m
Q cos 0( w si n
Q(
,
,
,
Ocos d cos <1
S+E(sin 2 9
C) s
cos o sin g
t{
( 3 2a )
¢ ( 3 2s)
( 32 )
1
3
2
( 1)
(3 2g)
3
( 27)
—(B
u a tions for stea d y rec tili nea r sy
oti on
Th e eq
et ri ca l m
fr om(32) by p uttin g n 0 95 0 th ey th en b ecom
e
Q
fl
mm
m
a re o
l3 2
bt ai ned
,
Y ==
0
=O
L=
M ==O
an d
=0
N=
grea t si m
p li city of formi s v ery noti cea bl e Th e soluti ons of (3 3 )
os t
f orm
e d t h e subj ect m
a tt er of Ch ap t er II an d co v er m
any of t h e m
s i n fl ying
im
p ort an t p robl em
ore gen era l
e d iscuss i on of t h e m
So m
e qu a ti ons ( 3 2) wi ll now b e gi v en
t h e p r ocess follow ed will be t h e ded u c
tion of t h e p arti cula r fr omt h e genera l cas e Thi s m
eth o d is n ot alw ay s
a dv an ta geous
metri cal
but i s n ot u n sui ta bl e for t h e d i scu ssion of asym
motions
Equa tions (3 2) con t ai n si x rel ations be tween t h e tw el v e qu an titi es
“ v w 9 ¢ a
X Y Z L M N an d cer t ain COD St fl n t S Of t h e a i r cra ft
Th ere ar e only four cont rols t o an aero p l an e an d th ree to an airshi p con
s i stin g o f t h e en gin e el ev at or ru dd er a n d a il ero ns for t h e fo r
er an d t h e
and
the
.
-
,
.
.
,
.
r
)
,
,
r
,
,
,
,
,
,
.
,
,
m
,
APPL IED AERODYNAMICS
264
thr ee of th ese for t h e latt er In t h e bes t of ci rc um
st a nces th erefo re
four of t h e quantiti es X Y Z L M an d N are in depen d ently v aria bl e
but all ar e f un ctions of u v w 0 s an d O whi ch ar e d et er m
i nabl e i n a
win d channel or by other m
eth ods of ob t ai nin g aer o d yna m
ic dat a
ay th en be l oo k ed on as si x e qua tions between t h e
Equ a ti ons (32) m
qua n titi es u v w 8
0 of whi ch four ar e in d ep en d en tly v ar i a bl e i n an
aero p la n e a n d th r ee in an ai r sh i p
It has alrea dy b een shown i n t h e case of sym
metrical st raight fli ght
t hat t h e el evator d et erm
in es t h e an gl e of in ci d ence whi lst t h e en gin e
i nes by
co n t rol s h eet s t h e a ngl e of d escen t
Th e aer o p lan e th en d et erm
oti ons t h e n ew
i t s accel erations t h e Sp eed of fl ight
For t h e lat eral m
co nsi d erati ons show t ha t t h e rat e of tur ni n g an d angl e of ba n k can b e
v ar i ed at wi ll but tha t t h e rat e of si d e sli pp i ng is t h en d et erm
ined by t h e
p re p ortions of t h e aer o p la n e
It foll ows fromt h e e q u a ti ons of m
otion t ha t within t h e lim
its of
ay ch oose t h e sp eed of fli ght t h e ra t e of cli m
b
his cont r ols a p ilot m
t h e rat e of t u rni n g an d t h e an gl e of bank but t h e angl e of inci d ence an d
r at e of s i d e sli pp ing ar e t h en fi x ed for hi m A v ery u s ua l co n di t ion obser v e d
d ur ing a t u rn i s tha t si de sli pp ing s ha ll be zero an d t h e an gl e of bank
ultan eously consi d ered as an in d epend ent v ar i a bl e
cann ot b e s im
b er of cases of lat eral m
ot ion will now be cons i d er ed i n relat i o n
A n um
first
only
.
,
,
,
,
,
,
,
,
,
,
,
.
,
,
.
,
,
,
,
,
.
,
'
.
.
,
.
,
,
,
,
,
.
,
.
Hori zontal Ci rcle wi th ou t Bi de Bur
sli pp in g i s occu rr ing i s sh or tly s t a t e d as
Tur ni ng i n
th at
i de
no s
a
m
a
—Th e con dition
th a t of ho ri zont al fl ight i s l ess di rect If k be t h e h ei ght
gr oun d t h e r esolution of v elociti es l ea ds to t h e eq u ation
i1 =a sin 0—v cos 0 81n ¢ — w cos 000s ¢
bu t
.
,
an d
for t h e
con
.
m
d iti ons i p ose d (3 5) beco m
es
u sin
mPIifi
0 = w 008 0 008
3
6
( )
¢
various exp ressi ons can be obt ai ned by a car e fu l
Th e axis of X will be t ak en as
choi ce of t h e p os ition of t h e b o dy a x es
hor izont al an d th erefore alon g t h e di recti on of fli ght ; this is eq u i v al ent
mx
to
e qu a l t o t h e lift
n =v
t h e ai rscrew thr us t an d will be foun d t o be zer o
di ffers fro t h e drag
Th e si x e qu ations of otion now b eco e
Si
the
ca ti on of
.
,
m
a
.
,
m
m
.
,
X =0
.
—B )
V0
VQ
i ¢ cos ¢ = L
s n
Q E
Q
2E
12
mt h
e
gt cos (6
N
3
( 77)
w a nt o f sy m
met ry of t h e aero p lane whi ch arises
Y will not be s t ri ctly
u se of ai l er o ns an d ru dd er t h e l a t era l fo rce
O wi n g
fr o
s in
t o t h e sli ght
,
m
APPL IED AERODYN AMI CS
266
Th e an gl e of ban k for Y =Oi s i d en ti cal wi th tha t gi v en by ( 3 8) for h ori zon t al
t u rnin g with ou t si de sli pp in g whils t t h e n or al ai r force i s
m
,
—m
Z =m
0
08
g
08 8 0
¢
e grea t er
0 ay b ecom
I t a pp ears th at t h e an gl e of t h e sp ira l with Y
ore
an d great er un til t h e a xis of X i s in clin ed t o t h e ho ri zo n ta l at 80 or m
Th e following
an d t h e ra di u s of t h e ci r cl e of turn i n g i s on ly a few f ee t
t abl e i n dicat es som
e of t h e p oss ibiliti es of st ea d y Sp i ra l fli ght
m
°
,
.
TAB L E 5
.
Ang le
of
b a nk
Rad i us
Resu lt ant ang u lar
n
v eloclt
( ax
a
m
of
of
plan
B
.
i s an
T a bl e 5app li es t o an a erop l an e a t an a n gl e of in ci d ence of
an gl e w ell a bo v e t h e criti cal an d i s d ed u ce d fr om
o b ser v a tions i n fli ght
ar k a bl e
oti on of win gs a t l ar ge an gl es of i n ci d en ce prod u ces r em
Th e m
effec ts an d it wi ll b e seen fr om
o d el tha t t h e r ot a ti on
an exp er im
en t on a m
otion
a bout t h e a xi s of d escen t i s n ecessa ry i n o r d er t o p ro d u ce a st ea d y m
whi ch is st a ble
.
.
,
,
.
m
m
m
od el
Aer oplane duri n g Co plex Man oeuvres — A co p l et e
aero p lane
oun t e d in a wi n d ch ann el a s sh own i n Fi g 13 6 was f ou n d to
rot at e a bou t an axi s a l ong t h e wi n d with a d efin it e sp ee d of r ot a ti on for
each an gl e of in ci d en ce a n d wi n d Sp eed
Th e analysi s of t h e exp eri en t
i s of v ery grea t i p or t a nce as it show s t h e p ossi bility of buil di ng u p t h e
on
an
m
.
m
m
.
tota l fo rce or cou p l e froma cons i dera tion of t h e p art s
If t h e ax is of X be i d en tifi ed with t h e a xis of rot ati on t h e v ar i ous
otio n
cons t r a in ts i n t r od u ced by t h e app ara t u s re d u ce t h e si x eq u a ti on s of m
to one (3 1p ) Sin ce q
p l e for m
i s zer o thi s equation t a k es t h e v ery s im
0 an d on e of t h e soluti ons for equilib rium
o d el
L
i s th at for whi ch t h e m
is n ot r ota ti ng
At sm
all a n gl es of i nci d en ce thi s con diti on i s s t a bl e an d
rot a tio n i s ra p i d ly s to pp ed sh ou l d it b e p ro d u ced by any m
eans
Abo v e
t h e criti cal an gl e of in ci d ence t h e con di ti on of n o rot at i on i s u ns t a bl e an d
dir ecti on p r o d u ces a n a ccel eratin g
a n acci d en t a l di s tur b an ce i n e ith er
o del i n conti nu ou s r ot ati on
cou p l e u n t i l a s t ea d y s t at e i s reach e d with t h e m
Figs 13 7 a n d 138 rel at e t o t h e m
od el with i t s ru dd er a n d a il er ons i n
t h e sym
metri cal p osition t he d irection of r ota tion bei ng d etermin ed by
acci d en t al d i stur b an ce
Th e Sp eed of rot a ti on was t a k en by sto p w a tch
a n d t h e fi rs t ex p er i m
en t co nsi s t ed of a m
en t of t h e sp ee d of r ot a tion
ea su r em
a t v a r i o us win d sp ee ds
As w as t o be exp ect e d 011 th eoreti cal grou n ds t h e
,
.
,
.
,
,
,
.
,
.
,
.
.
,
.
.
-
,
,
m”
F
(I
—Mo de l ae ro plane ar ranged
to
s
ho w
a u t o rot a t i o n.
Dig it iz ed by
G o ogle
AERIAL
MANO EUVRES AND
EQU AT ION S
OF MOT ION
267
p d of rotation was foun d to be p rop ortional t o t he win d Sp eed ( Fig
Th e secon d exp er i ent co vere d t h e v a ri ation of rot ati onal sp eed with
s ee
m
Fi o 13 8
.
.
.
—A11t o rota t i on of m
od e l ae roplan e as
d epen d en t
on
wi n d
s peed .
h ange of angle of in ci d ence a nd it will be n oti ced tha t i ncrea se of t h e
l atter l ead s to fast er sp i nni ng at l east u p t o angl es of
Th e a na lyti ca l
p rocess n ow to be d escribed i f carri ed ou t over t h e whol e range of possibl e
it ed range
a ngl es of i nci d ence sh ows th a t t h e sp i nni ng i s confin ed t o a li m
c
,
,
.
,
.
AND EQUATI ONS
AE RI AL MA N OE U V R ES
di st a nce fro
an
mt h
e a
gl e of inci d ence
Si nce
3
is co nst an t
xis of rot ation to
to
du e
l
a on
an an
l
mt
an e e
en
gular v elocity
ay
g t h e wi ngs it m
be
as
14
Fi g 189
.
b rin gs
I4O
at
—
C l
a eu la t i on of
.
p os i t ion
for t h e
Th e cou p l e
es
on
ti
mtio
a
the
n
of
t he
b ei ng
t
e
h
y
,
by
t he
m f ( 4 3)
for
u se of
o
i nd efini t e t em
p orarily
°
say ,
will
in to
°
26
be shown
m
t h e di fferen ce 81
c
om
p l et e aerofoil
t he
be
v ari a bl e
h
an d
a
t h e cou p l e du e
h al f sp an yo
of
g d to
c an e
yp
v
by
s peed of a u t oro t a t i o n o f a n aerofoil.
c
can
g
r
,
is
inst ea d
of
one
y
g
,
en
t he
(43 )
mo
re s
an d
to
th en
S
y d y
Th e
h
c an e of
oughly equ al to
i
s
p
l eft
269
OF M O T ION
uita bl e
for
es
it th en b ecom
i nt egr ati on
)
?
s
v alue of
ca n
be foun d
by t he
A PP L IED AE RODYN AM IC S
an d
u v e AB CD i s continu ed un til t h e ar ea b etw een it an d AB
T h i s occurs at t h e or din at e ED whi ch th en rep resen t s t h e valu e
t he
zero
.
c r
,
bo th
yo
V
an d
mo
dm
itt
known ,
an d
h ence p
is
d ed u ce d
fro
mt h
mthod
e ra
tio
of
so
of cal cu l ation will be gi v en l at er but t h e
e d a bo v e a r e th ou ght t o b e j us ti fied by t h e s i m
pli city of t h e
err o rs a
cal cu l a ti ons a n d t h e consequ en t eas e with whi ch t h e p hysi ca l i d eas can b e
a te m
t ra ced i n t h e u lti m
otion On one win g t h e an gl e of in ci d ence is seen
to b e in crease d t o a bou t 3 7 at t h e ti p whi ls t on t h e oth er it i s redu ced
to
Fig 139 b efore st ea dy rotation i s reach ed Fu r th er t h e Sp inni n g
A
t
are
is
re acc u r a e
e
,
.
°
,
,
.
FI G l 4 l
.
.
.
—C m
p
o
ar is on of
t he
,
o b ser ved and ca lcu la t ed s peeds o f a u t or ot a t i on of an
ae r ofoi l.
to d ep en d on t h e evi d ence of an in t ersection of t h e li ft cur v e an d i t s
im
age a co nd iti on whi ch woul d n ot h a v e occur r ed h ad t h e an gl e of i n ci dence
b een ch osen as
en t s agrees wi th
Qualit ati v ely th erefore t h e th eory of a dd ition of el em
p ar iso n can be a d e s in ce t h e a er ofoil
o b ser v a ti on
Th e q u an tit a ti v e com
a w i n d ch an nel
t o whi ch t h e li ft cu r v e of Fig 13 9 a pp li es was t est ed m
an d t h e ob ser v e d a n d ca l cu lat e d cu r v es of r ot a tio na l sp eed a r e r ep ro d u ced
in Fig 141 Th e aer ofoil was 18 ins l on g with a chor d of 3 i ns a n d t h e
sp ee d of t est 30 f ee t p er sec
en t b etw een t h e cal cu l a t ed an d o bserv ed v alu es of t h e s p eed
Th e agreem
of r ot a tion i s clo se p er h ap s cl oser th an wou l d b e exp ect ed i n v i ew of t h e
a tio ns i n t h e cal cul ati on an d m
ay b e tak en a s s t rong su pp o r t
a pp r oxi m
ent th eory
for t h e el em
Th e ext ra p ower gi v en in t h e cal cu lati on of aer o
is
seen
,
,
m
,
.
.
.
,
.
.
.
,
,
.
.
,
AE R IAL MA NOE U V R E S
AND
E QU A T IONS
OF
MOTION
271
p lane m
oti on i s ext rem
ely grea t an d will en a bl e fu tu r e i n v estiga t ors t o
p roceed t o a n aly se i n d eta il t h e m
oti ons of Sp i nn i ng rolli ng a n d ra p i d
en t s
turn i n g with out reference t o com
p l e x e xp eri m
a d e o n t h e c h ee t of
F u rthe r o bserv a ti ons i n t he win d ch an n el w er e m
ch ang es of win d s p eed a n d of a sp ec t r a ti o
p l et e
As i n t h e ca se of t h e com
od e l ae rop l an e t h e Sp ee d o f rot a ti on was foun d t o b e p rop or ti on al t o t h e
wi n d sp ee d
R eference t o (44 ) will sh ow th a t t h e int egral d ep en d s only
y
? and h ence for aer ofoil s of grea t er l ength it woul d be
on t h e va l u e of
,
,
.
'
“
m
.
,
.
e
p t d th a t t h e ra t e of t h e st ea dy sp in w ou l d b e p rop orti on at ely l ess
Th e o b ser ved and ca l cu la t e d res ults ar e gi v en i n T abl e 6
ex ec e
.
.
TAB LE 6
.
As p ect
Angle
o f i n ci d en ce ,
Angle
of i nci den ce ,
mu
O b se r ed rat e o f spi n
Calcu lat ed
v
fl
rate of
m
m
°
17
°
22
m
It will
m
b e n oti ce d th a t t h e agree en t i s far l ess co p l et e th an was
t h e cas e for v a ri ati on of an gl e of in ci d ence
It i s p ossi bl e th at t h e ti p
effect s whi ch h av e b een ignor e d ar e p r o d u cin g
eas ur a bl e ch an g es i n thi s
case a n d for a high er d egree of accur acy reso rt shoul d b e h ad t o ob serv a
tions of p ressur e dis t r ib u tion on an aerofoil It is to be exp ect ed th at
future exp er i ent s will thr ow furth er li ght on t h e p ossibi liti es of t h e
el e en t th eor y an d p rob a bly l ea d t o gr ea t er accur acy O f ca l cula tion
More Accu rate Develop ent of th e Math e ati cs of th e Aer otoi l Ele ent
Th eory - Any el e en t t h eory can only b e an app r oxi ati on t o t h e
t ruth , an d for t hi s r eason so ewh a t di ffer en t exp ressi ons
ay b e eq u ally
jus tifi abl e On t h e oth er han d all su ch th eori es assu e th at t h e for ces
on an el e en t ar e d et er
ined by t h e loca l r el ati v e win d and are sens ibly
in d ep en d en t of ch an ges of v elocity r oun d n eighbou r in g el e en t s Fur th er
it i s n ot u s ua l t o a ke a ny a pp li ca ti ons t o s all ar eas of a bo dy but only
to st ri p s of aerofoils p a rall el t o a p l an e of sy
et ry an d t o t a ke t h e a:
co-or d i n at e of thi s st r i p a s th a t of i t s cen t r e of p r essu r e
Th e l as t assu p
ti on a y be regar de d as a con veni en t eth od of t akin g a w eight ed ean
of t h e v ari a ti ons ov er a s t ri p a n d n ot i n t ri n si ca lly
ore sou n d th an t h e
t aki n g of ar eas s a ll i n b oth di recti ons an d su
ing t h e r esults
Usu a lly , t h e aerofoils to whi ch cal culati on i s a pp li ed li e either i n t h e
et ry or n ea rly n or al to it an d cons is t of t h e fi n a n d r u dd er ,
pl a ne of sy
tail p lan e an d el ev at or an d ain p lan es Of th ese, t h e l as t p rov i d es t h e
ore co p l ex p robl e on acco u nt of t h e d ih edral an gl e, a n d si n ce t h e
t rea t ent co v ers t h e subj ect a p ai r of wi ngs h as b een ch osen fo r illus t ra ti on
of t h e
etho d of cal cu la ti o n
.
m
,
m
m
.
m
.
m
m
,
m
m
m
m
m
m
m
.
m
,
m
m
m
m
m
m
m
m
m
m m
m
m
.
m
m
m
,
.
,
.
,
,
m
m
,
.
m
m
,
.
m
.
AE RIA L MA N OE U V R ES AND E QUA T I ON S OF M OT IO N
273
quantiti es a and V su di ce t o d et erm
i n e t h e lift an d dr a g on
a n el em
p refera bly on e in whi ch th e p ress ur e
a st an dar d t es t
en t f ro m
i n ed
d is tributi on Ov er a si ilar aer ofoil was d et erm
ed resoluti on of fo rces l ea d s
Using Fi g 142 (b) as r ep resen tin g t h e ass um
t o t h e fo rce and m
om
en t e qua tions
Th e t w o
m
,
.
.
,
d
(
l
l
kn sin a:
kn cos a
0
7”
d
kn
008 or
z
m
d
lh
l
dL 1
—xl d
dMl
m)
a
p
y,
2
V cd
y1
i
l
2 177“
1
—y 1d
l
8
)p
2
V cd
1
en t of t h e e l em
om
p l ete t h e s t at em
en t th eo ry an d will
be seen t o assum
e tha t t h e r esu lt a n t force li es i n a p lan e p ar a ll el t o X I GZ
P
s e qu a tions (4 5
m
a
t
t
v
i
t
b
e
In cert ai n p robl em
h
e
m
os
co
n
e
n
e
n
)
y
f ormof a pp li cati on but in gen eral it will be necessary to resol ve t h e
p on en ts a bou t t h e origi nal ax es before in t egration can be effect ed
co m
Th e n eces sa ry relat ions for thi s p ur p ose are gi v en
Forces and Mo ents relat ed to St andard Ar ea —I t m
ay be n oti ce d
tha t th e angl es of r ot ation “ x an d I corres p on d closely with th ose of 0
t as illust rat ed in Fig 129 A p ositi ve d ih edr al angl e on t h e right
an d s
how ev er corresp on ds with a negati v e d Th e d i recti on cosi n es
of t h e di sp laced a x es rel a ti v e t o t he origi n al are
Equa tions (48 )
c
,
,
,
.
m
.
“
.
,
.
.
,
t;
m
l
lg
1
,
a
m
K OK ,
cos
cos
YGX , = 0
cos
Z GX ,
ce s
X GYI
008
YGYI
cos
Z GY ,
cos
X OZ
Z GZ
cos
m
— si n
= 008
a x sin
I
‘
P
cos a x si n
si n a ; cos
si n
I
‘
‘
I
‘
I
= cos az‘ cos l
ex p ressions with t h e sign of
"
I
for t h e ri ght han d wing an d si i lar
for t h e l eft -h an d win g
If x, y an d 2 be t h e co-o rdi na t es of
-
si n a
x
z :
l
=
Z
s
Y
G
co
,
3
n, s
cos a )
‘
I
h
.
m= l
a
,
11
1
In
a s
im
ilar way
31
1
24?
= 13az
’
u,
,
”1
lx
73
+
P
l ti ve to t h e stan dar d axes
re a
my +
,
1113
W
" 23
” all
” 33
ll u
m
1
“
2
M + 7121”
1
3“
7733 0
fl aw
vi ng
" if
z
n gf
a
n ar
lap
lap
n
i
mq
mq
a lw
g d
c an e
,
A PP L IE D AERODYNAM IC S
l ti ons gi v en by
ina ti o n
(50) a n d (51) su ffi ce for t he d et er m
o f t an o q
an d V as gi v en by equ ations
an d th ence t h e
(46) an d
el em
en tary forces a n d cou p l es fro m
exp eri m
en t an d equa tions
Th e
final st ep i s t h e r esolution fr o m
t h e di s p laced to t h e st an dar d a x es whi c h
i s co v ered by t h e followin g eq u a tions
Th e
re a
,
dx
M
ll dx l
m
d
JY
mdx
dZ
" 1dX 1
l
dL
(
mdL
M
(I
l
N
1
2
4
1
3
l
"
1
d
1
s
fl s
" 161
111
a
i
” adz 1
i
7n
l
mdz
m
s
1
n
1
s
1
i
mt
m
xp ressi ons i n (52) now all a pp ly t o t h e sa e a x es t h e el e
b e su m ed by in t egration t h e el em
en t of l en gth b ein g
As t h e
my
a
e
m
en s
,
1
wh ere 12 m
re
a
a
n
d
n
2
2
cen t r es o f p r essure
"t i
t h e di sect i on
,
m
lads?
63
11
dy
(59-0 )
n zdz
oi
c s n es of
t h e li ne
j oin ing s uccess i ve
m
.
—
i
n
E
U
o
f
th
e
G
e
n
e
r
a
l
ua
t
o
s
Two exa p l es will be
Exa ples of t h e se
q
gi v en one d eali ng with t h e p robl e of a uto rota ti on di scussed ear li er, a nd
t h e oth er with t h e p r op erti es co nnect ed with a d ih e d ral angl e
t d escri be d ear li er i n t h e ch ap t er
1 Au to tafi o —In t h e experi
an d r w ere all zero
i t was arranged tha t t h e q uan ti t i es x, z, I , v, to,
otion was a r ot ati on a bout t h e ax is of X an d t he
Th e only p oss ibl e
p ort ance D enoting t h e win d
cou p l e L was th erefore t h e only on e of i
v elocity by n o and u s ing e qu ati ons (4 5) t o (52) l ea ds to e x ero, an d
m
mm
,
m m
.
.
q
‘
m
m
.
,
.
t
I;
lg
I3
,
2
a0
y r,
“ o 008 (10
w;
no s
Ther efore
Py
ao + y
p
“O
m
1,
"
= 10 + p
an d
Bi ll “ 0
008 ao
Wh ere
,
= u 02 + p 2y 2
2
V
Finally fro m(52)
0
0
‘
I1
1
= u o 008
=y
171
7 009 “0
P1
al
l
0
331
“I
m= o
m= l
cos a o
=0
= Sin
-z
t he
va l u es o f dL l
= ta n
(54 )
s
5
5
( )
p
.
a
and
2
8 90 “
d N1 i s
m
p
obtain ed t h e rela ti on
4
k
( L
Eq u a tio n ( 56)
as a s
mll qu
a
an
re
tity
,
d u ces
i
.
c.
cos
to
k,, s in a)
p
l
an e e
if t h e
m t of
lin ear
"
d(
n)
3
s ec
6
5
( )
qu a ti on (4 4) if y be consi d ered
v eloc ity of t h e wi ng ti p d u e t o rota ti on
en
e
.
275
AE RIAL MA NO E U V R E S A ND E QUAT IONS OF M OT ION
mll o mp
d with t h e t ransl a t i onal v elocity
o bt ain ed by in t egra ti on as
is
s
c
a
ar e
8 (k,
cos
,
kl,
y
.
si n
y)
.
v alu e
Th e
.
sec?“
d(
m
is
( 57)
)
2
1
8 60 1
—
L
of
i n p on t h e t wo
t h e differen ce of t h e v al u es of k cos pa l kn s
el e
en t s of t h e wi ngs o f t h e a erofoil wh ere i t h as t h e sa
e nu
er i ca l
val u e bu t opp osit e si gn
2 Th e Efi ect of a Di h edral Angle du ri ng Si de Sli p pi nx — Th e si p l est
cas e will b e t a k en a n d t h e origi n chos en on t h e cen t ra l ch or d at t h e cen t r e of
p r essure Th e win gs will be ass u ed t o b e s t ra ight an d of uni for ch ord ,
a n d t o b e b en t a b out t h e cen t ral cho r d
ath e ati ca l co n d iti ons
Th e
8
igni fi es
m
s
.
.
.
.
m
.
m m
.
m m
m
m
.
.
a re
= ao
'
u l
]
00
0
to
0
(l l
U1
1
11
3
It sh ou l d be noti ce d th a t t h e co or di na t es a re i n this case t aken with
resp ect t o d i sp l ace d a x es as thi s i s con v eni en t i n t h e p res en t illus t ra tion
713 a re gi v en by
Th e d ir ection cosin es ll
an d
are
I a re
i n d ep end ent of 311 an d t h e foll owing fur th er r el ati ons are obt ain ed
-
,
.
‘
,
ul
= u 1 = u o co s
'
-
eo
u 0 si n a o si n
P
l
-
vo cos
-
si n
=
=
0
1
P
t an
(
V
2
91
0
1
-
n o sin
or,
(
‘
I
=O
J
no s i n
F
cos
‘
i
“ 0 003 d o
no si n ( 10 s i n
no cos
T1
i
‘
no sin no cos
I
"
no cos
1)
no sin
‘
I
V are seen from( 60) an d (61) t o b e in d ep en d en t of y ,
f
n
w
s
a
it
th
ollo
th
t
48
e
( )
B oth
or a nd
(61)
F rom
.
z
— k
( t
;
SID a
kl)
003 “ I
o
z
pcv dy l
1
d
?
)
(kl Bill or;
th ese exp ressi ons
v ari a ti on al on g t h e wings
k
in
dL
.
cos «0
)
.
zl
an d
2
kn COS a 1 pcV dy 1
a
n
d
kn
my
a
be
fun ctions
of
y
,,
o
wing
to
Si nce
dL l
s in d o cos l
‘
dN ,
( 63)
m
val u e can be obt a in ed fr om( 62) for t h e right h an d wi ng A si il ar
exp r ess i o n h ol d s for t h e l eft h an d win g i f t h e sign of 1 b e ch anged
Th e
im
p ort an t q u antiti es co an d I only app ear exp li citly i n t an or and V9 an d
eas u red i n a wi n d ch annel
V rep resen ts t h e q u antity us u ally m
t he
-
.
‘
-
.
‘
.
AE RI AL MA N O E U VR ES A ND E QU A T ION S OF M O T IO N
277
I t h as been seen i n Ch ap t er I V
Calcu lati on of Rotary Deri vati ves
th a t t h e ra t es of v ari ation of forces an d cou pl es with v ar i ati ons of u , w
i ne d in a win d ch ann el whil s t v ari ations with p ,
a n d v are easi ly d et er
p ly obt ain ed Th e n u b er of ob serv at ions in t h e
an d r are l ess si
l a tter case is so ewh at s all an d as a consequ en ce t h e el e en t th eo ry
h as been fr eely used i n cal cul atin g t h e rot ary d er i va ti v es re quired for
I t 13 us u al to consi d er s , p , an d r as s a ll qu an titi es
a ero p l ane s t a bility
r i v ati v es th en b ein g fu n cti ons of no and
a nd to n egl ect s q u ar es t h e d e
o
o r of V an d d o
'
It i s n ow con v eni en t t o exp ress t h e valu es of a and V in t er s of u ,
’
an d r i n st ea d of t h e co rres p on d i n g v ari a bl es for t h e d is p l ac ed
v , w , p
F r o t h e equa tion s d ev el op ed earli er it wi ll b e seen th at
a x es
-
.
m
m
m
m
q
.
m
,
.
m
,
m
q
'
.
m
.
,
m
.
q
'
m
,
.
“1
“
l
'
3
9
m(
)
e q u a tions
ry
with t wo si m
il ar
val u es o f n o 111 an d w, ar e
'
v
1
for
‘
m
i
l
:
‘
“
'
w
i
"1
Us in g
c l an d 1
01.
+ 173!
a s
)
n ot atio n
2
7
( )
3
4
hort er
,
,
,
= ao( 1 + a 1p + a 2q
+ aor)
bo( 1
01
bi ?
bgq bar )
ul
a
$5
m
0
w h ere
bo
co
1
115
0
oz
n ay
0100
With thi s
oz
ao
and
W
§B
l
i
(
si n d o 005 %
i (01
z
Vo
a0
2p (
2
a ,ao
290
x
ocd y l
+ cos
‘
‘
l
“
[
ao
( 1300
" 25
5
babe
1133:
c c
o o
§
2ko
,
o
2
(
C3
lq (03
as
G3
)”
l}
8V
babe”
on e
(
03002
)
of t h e qu antiti es p
k°
+
aw
.
q
l
7
—
5
3]
i}:(
g
i
) $l
V
kf
k +
l
V
) g ( 32 5
3”
,
Q
x
-
(75)
ca f
l
- -
a?
k;
'
s
“
n
i g eq u a tion s are gi ven i n (48 )
a n n
i n:
o
“2
2
01
100
'
m ly
— 1y
mm
mm
m
l (02
‘
g
cos d o
mi
" 13?
"
“o
re
n ow
(02
z
bi bo
'
s i n cro
an d
t he
'
v
o
)
be u sed t o r ep res en t generally
=
=
m
d
x
(
l pV
an d
'
C1
2
bsbo
2
3 20 0
If w
71210
t tion
111
or;
n lw
no a
=
or
'
,
“ 20 0
m
m
" all
lv
'
112
babe = 1
23
coco
loz
"113
”13
m
m
ll u
'
lzu
'
lou
'
do
0 100
01
?
00
i
0
to
“
6
7
( )
or r ,
the
A PP L IE D AE ROD YN AM IC S
D en ot e by
and
by
the
2ko
e
xp r ession
the
v,
xp ression
'
e
,
2k,,
-
a(
l
Z (
k
1 o
'
kL
'
gz
Vo
- -
6
- -
-
)
:
do
—
V0
‘0
5
to red u ce equ ations (77) and (78 ) t o
m
1 msi n
[ Noa h/10
2
W
1
1)
(
PVoCdll l i v
( dx l )
u
Appli cati on t o L,” L,, Np,
Assu
md
e
N, for
)
003 d o
a p air of
Str ai gh t
Wi ngs
.
o di ti on s z
an
dt h
b ve
e a o
cos
bo
= u o si n
do
a l ao
a
x
a
w
s
i
n
+
o
x
3;
a za o
oc
x
y s i n or ,
Vocos aro
no
an d
y
cos az
,
y
sin a ,
z
Vo s in ao cos d o
1 co s a:
T t;
‘
o
th ese
;
= Vo sin
0
0
coco
cos or
y
Vo8 111 d o
or
=0
bgbo
cos on
a
=
V
os a o
c
o
x
no
0
al so
a
1
00
—
1
t an
x
=
= — y sin
bi bo
m
— wo s in
0
co
Usin g
si n “ 0
c n
4
7
( )
3
V51
and
)
008 010
“c
03
co
s
s
o
p res si ons
ex
an d e
qu ati ons (75)
)
s in a o cos aao
do:
-
a?
do:
a
(
co
av
(JV
2
00
“1
1
cl
no s i n ozo cos oro
)
2
bl bo
ol coz
2
babe
6 3 002
an d
gs
( 76)
f
Vo
/w—
ig
sir
a
AND
A E RI AL MA N OE U V R ES
Si nce
t ake t h e for
m
t he
u l as for dL
form
E QU A T IO N S OF M O T IO N
and
8
1
( )
s
I
I
I
and
gi v en by
an d ( IN.
.
279
f ro m(79 )
and
i
.
33
a
:
c
(85)
=
ei
e
wo
a
s
es 1
l
3i
s
{is d
'
en s
as
6
0
+
,
ue
If t h e vari ations of li ft and dr ag towar ds t h e win g ti p s be ignored t h e
in tegrals take si m
p l e form Calli ng t h e l ength of each wing l t he valu es
are for co nst an t chor d
.
,
,
,
W
e
{
l
o
as
L.
— § lflpc
-
{
l
(
(
+ a
W)
95
'
.
o
e
sp
— gzapc
2k,
110
(
+ k
'
M
Q )
(
a xi m
Num
er i cal v al u es can be ob t a in e d for th e co n d iti on of m
umlif t
o f t h e win gs i n illus t ration of 88
e
i
n
su
n
s
as
to
wi
g
b
g
med of
T
h
e
( )
ch or d 6 ft an d l en gth 20 ft
t h e v elocity of 15
0 feet p er s ec wi ll b e t ak en
as along t h e ax is of X
ic
App roxi m
at e v al u es for t h e aer od y n a m
qua n titi es inv ol ved are
.
.
.
8k]
8ku
.
dai
da
k, _ 02
an d
—
=
=
0
N
1
1
5
0
0
0
L
4
,
an d l ea d to
It
01
,
ko
an d
-
0 01
N, =—200 (92)
ti on with rap i d turning th a t v al u es of p i n ex cess
of 05 w ere obt ained and it n ow app ea rs tha t a roll ing cou p l e of m
or e th an
e by t h e a il ero n s if t h e con ditio ns
10 000 1b s ft wou l d n ee d t o b e ov ercom
u ch l arg er
of (92) app li ed Th e angl e of i nci d ence i n fl ight is h ow ev er m
an d t h e Sp ee d low er both of whi ch l ea d t o l ow er v a lu es of t h e tot al cou p l e
In t h e case of t h e t a il p l ane of an aerop l a ne t h e effect of d own wash
It i s t h e val u es of t h e ai r vel ocities at t he ae rofoil
shou l d be inclu ded
was
in
seen
conn ec
,
-
,
.
.
.
,
,
.
,
.
CH A PT E R VI
I G ENERA L T H EO RY
.
th eory of t h e Op era ti on of air scr ews h as b een m
a d e t h e subj ect of
i t s b ro a d outli n es i s w ell est a blis h ed
m
an y sp eci a l exp er i m
en t s an d
otion fr omfirs t p rin ci p l es i s far bey on d our
Cal cul a tion of t h e fl ui d m
p resen t p ow ers and t h e hyp oth eses use d are j u stifi a bl e only on exp eri
mental grounds Wh ilst frankly emp iri cal t h e ma in p rincip l es foll ow
li nes i n di cat ed by som
p l e th eo ri es of fl u i d m
otio n an d mthi s
ewh at si
co nnection t h e cal cul a ted m
oti on of an i n v i sci d fl u i d m
ost n early a pp roach es
th a t of a real fl u i d Th e di scon tin uous m
otion i nd i ca te d by a jet of fl u i d
r esem
bl es t h e m
otion i n t h e s t r ea of ai r fr oman ai rscr ew an d W E
Fr ou de h as form
ulat ed a th eory of p ro p uls io n on t h e analogy
In t hi s
at ed fr o
t he m
th eory t h e thr us t on an ai rscr ew i s es tim
om
t
en tum
era
e
n
ed
g
r
e
secon d i n t h e sli p s t r ea
p
An oth er th eory n ot n ecessari ly u n conn ecte d with t h e fo r er was al so
p r op osed by F rou d e an d dev elop ed by D rzewi ecki an d oth ers Th e bla d es
of t h e ai rscrew ar e regar d ed as aer ofoils t h e forces on whi ch d ep en d on
th ei r m
otion r elati v e to t h e air i n t h e sa e way as t h e forces on t h e wings
en tar y l en gth s beh av e a s
of an aerop l an e It i s assum
ed tha t t h e elem
i lar ity of t h e n eighbo ur in g el em
though unaffect ed by t h e di ssi m
en t s an d
t h e fo rces acti n g on th em
are d ed uced fr o m
wi n d ch ann el exp er im
en t s on
t h e lift an d d rag of aerofoil s
os t su ccessful th eory of airscrew desi gn com
bines t h e two
Th e m
main i deas indi cat ed abov e
In sp it e of im
p erfections t h e st u dy of t h e m
otion of an in vis ci d i n
com
p ressibl e da i d form
s a goo d i n tr o d u ction to exp eri m
ent a l wor k as it
draws a tt en tion to som
e sali en t f ea tur es n ot oth er wi se eas ily app reci at ed
I n co nn ec ti on with t h e es t i m
om
generat ed
ati o n of thr us t by t h e m
en tum
W E Fr ou d e in t rod u ced in to a irscrew th eory t h e i d ea of an actu ator
ech ani s m
i s p os t u l a t ed bu t a t a cer t a in di s c AB C Fi g 143 it i s
No m
p r es u m
ed th a t a p ressur e d i fi erence m
ay b e gi v en to fl ui d p assin g thr ou gh it
Th e fl ui d a t an i nfi ni t e d ist an ce b oth b efore and b ehind t h e di sc h as
a uni form
v elocity i n t h e d ir ecti on of t h e axi s of t h e a ctu at or At infi n ity
t h e fl ui d h as t h e
ex ce p t i n t h e sli p st r eam wh er e t h e v elo city i s
v elocity V
Th e only ex t ern al forces acti ng ou t h e fl u i d occur at t h e
a ctua tor di sc an d t h e s i m
p l e formof B ernoulli s equ ati on d ev elo p ed i n
t h e ch ap t er on fl ui d m
ay b e a pp li e d sep ara t ely to t h e two p ar t s
oti on m
of st ream
li nes whi ch are sep arat ed by t h e ac tu ator di sc
Ta n
m
,
,
.
m
.
,
'
,
m
.
,
.
m
m
.
.
m
.
,
,
.
m
,
.
,
-
.
.
,
.
,
.
.
.
,
,
.
,
,
'
.
,
,
.
,
.,
'
,
.
28 1
,
A PP L IE D AE RODYN AM IC S
oti on o f an i nv i sci d fl ui d i n a la t er ch ap t er
Wh en dealing with t h e m
i t i s sh own th a t p r ess u re i n p arall el s tream
s i s un iforma n d i f thi s th eo rem
be app li ed t o t h e hy p oth e ti cal flo w illus tr a t e d i n Fig 143 i t will b e seen
tha t t h e p ressure o v er t h e b o u n d ary DEGF t en ds t o b eco m
e unifo rm
wh en t h e b ou n d ary i s v ery l ar ge Th e con t i nu ou s p r essu re a t t h e bou n d ary
of t h e sli p s trea mi s associ a t ed wi th di s con tin u ou s v elocity
Th e t o t al f orce on t h e bl ock DEFG i s du e p ar tly t o p r es s u r e an d p ar tly
es
to m
es z er o wh en t h e p r essu r e b eco m
om
en t u man d t h e fi rs t p ar t becom
un i formo v er t h e su rfa ce Th e excess m
om
en t u mp er sec: l ea vi n g t h e
bl ock i s t h e i ncrease of v elocity i n t h e sli p s t reamo v er th a t w ell in fr on t
ulti p li ed by t h e area of t h e sli p s t reami ts v el oci ty
of t h e actu ator m
f
t
h
fl
ui
d
I
e
th
a n d t h e d ensit
ru s t T a pp li ed by t h e a ct u a t or
o
f
t
h
e
y
,
,
.
.
.
,
.
,
,
.
AC TUATO R
Fro 143
.
.
al a nced by a force b etw een t h e d i sc an d t h e bl ock DEFG a n d t h e l a tt er
b
i1s t o b
e i n e q u ilib r iu mt h e foll owi n g equ ati on for m
om
en t u m
:
—
T=P m
V
V
1
a
V
i m
(
)
mus t be sati sfied
Mak in g u se of B ernou lli s eq u a ti on anoth er exp ressi on m
ay b e oh
ed for T whi ch by co m
t am
p ari son wi th ( l a) l ead s t o t h e i d eas m
en ti on ed
i n t h e O p en in g p aragr ap h s of t h i s ch ap t er
For any s t r eam
li ne not p a ssi ng th rou gh t h e actu at or di sc B ernoulli s
eq u a tio n gi v es
2=
2
V
V
+
+
1
p
1
p
l
p
p
l
wh ere p , a nd V, ar e t h e p r essu r e an d v elocity of t h e fl ui d at any p oin t
o f a s t r ea m
lin e Thi s eq u a tion a pp li es t o t h e wh ol e r egi on in fr ont o f
t h e act u at or a n d t o t h e fl u i d b ehi n d ou t si d e t h e sli p s t ream I nsi d e t h e
s
,
‘
“
.
a
.
’
,
.
’
a
.
.
mt h
p ress u re b ei ng p 2 an d t h e v el ocity V2 t h e equ ation corre
Sp on di n
t
o
2
a
i
s
(
)
g
z
B
8
a
v
V
2
( )
ip z
iP
P
P
If p be eli m
ina t ed betw een (2a ) an d ( 3 a ) an i m
p o r ta n t exp ressi on for
t h e p ress ur e di ffer en ce on t h e t wo s i d es of t h e a ct u a t or i s o bt ai n e d a s
s
li p
A IR S CR E WS
st r ea
,
e
,
an
-
uo
ao
,
(P3
"
"
M
P1)
”
"?
V1 )
Vi
4
a
( )
)
Co n tin ui ty of ar ea of t h e st rea mi n p ass in g thr ou gh t h e actu a t or d i sc
ed t h e v alu e of V 2 will gr a d u ally a pp roa ch th a t of V, as
be in g p r es um
t h e p oin t s 1 an d 2 on t h e s t rea m
li n e a pp r oa ch t h e d i sc On t h e d i sc bo th
es
v elociti es will be t h e sam
e an d e q u al t o V0 a n d eq u a ti on ( 4 a ) b ecom
,
.
,
5
a
( )
(Pg
ight h an d si d e of ( 50 ) i s consta nt for all st ream
lin es i nsi d e t h e
s li p s tr eaman d h en ce t h e p ress u re d i ffer en ce on t h e t w o si d es o f t h e
ac tu a t o r i s uni form
o v er t h e wh ol e d isc
A seco n d e q u a tio n for t h e thr us t T o bt ain e d fr o mthi s unifo rm
p ressur e i s
—
fi
n
V
t)
e
ce
a
Th e qua n ti ty of fl ui d p assin g th ro u gh t h e a ctu a to r d i sc b ein g t h e sa m
e as
th a t i n t h e sli p s t r ea mi t foll ows th a t r ozvo i s e q u al t o r i w vw a nd
usi ng thi s rel ati on with ( 1a) an d ( 6a) show s th a t
Th e
r
-
.
,
.
,
,
.
m
a
7
( )
m
ea n
Th e v alu e of V0 o v er t h e a ct u a t or d i sc i s seen fr o ( 7a ) t o be a
of t h e v elocity of t h e u n di s tur b ed st r ea
a n d t h e v eloc ity i n t h e sli p
s t r ea
a ft er i t h as r ea ch ed a un ifor
v al u e
F or t h e p ur p os es o f exp eri en t a l ch ec k i t i s cl ear th a t no
easu r e
en t s far fro
t h e a i rs crew wi ll b e sa t i s fact ory owin g t o t h e b r ea k i n g u p
of t h e sli p s trea du e to vi sc osity an d t h e p osi tio n of l east d i a et er of
s li p s t r ea
i s us u ally ta ken as s u ffi ci en tly rep resen ta ti v e of p ara ll el s t rea
l i n es B y a odi ficat i on of equ a ti ons (40 ) an d (60) diffi cul ty i n an exp eri
en t al ch ec k ca n b e a v oi d ed
A rearr a ng e en t of t er s i n (4a ) an d (60 )
l ea d s t o t h e eq u a ti on
m
m
m
m
m
.
m
m
m
m
m
,
m
m
.
,
m
m
.
m
-
=
Pz
a
V
P
i sf
P1
V
iP L
z
eas ur ed
q u antity p + § pV h app ens to b e v ery easily m
It is
th erefore p ossibl e t o choo se t h e p oi n t s 1 a nd 2 i n any conv eni en t p l ace
on e i n fr on t a n d on e be hi n d t h e a i rscrew
Eq u a tio n (80 ) i s gi v en as a pp li ed t o t h e whol e a irs cr ew as though
e
r
s
t
h
7
o
r
e
r
t
h
s
M
w
o
t
t
o
v
w
ol
d
i
igo
ou
ly
V
V
r
h
c
e
e
e
ere
a
n
n
s
c
2
1
p p 13
en t ary a nnu l u s an d t h e
e q u a ti on sh o u l d b e d ev elo p ed to a pp ly t o a n el em
and
the
2
.
,
.
.
e
.
.
xp ressi on
P
L
b
eco m
es
2
m
,
in
2an dr
'
T
is
th en obt ained by in t egr ation
.
With
od i fica ti on ( 8a) a pp li es with co n si d era bl e accu racy to t h e r ea l fl ow
thi s m
of a i r th rou gh a n a i rscrew
en t whi ch was
Ha d t h e ac tu a to r gi v en t o t h e fl ui d a p r essur e i n cr em
o
.
AIRSCREW8
F ro 144
.
’ ’
.
.
var ia t i on a lo ng a n ai rsc re w
—Co m
pa ri son b
FI G l45
.
l h ru st
e t ween
t wo
mt h
e
b la de ( experi
od s o f
t h r us t
m
mt
en a l
eas u re
)
.
mt
en
.
2
as
h
ee
t t l h d
b n di v i d ed by p V D b efore p l ottin g Th e r easo n for
t his choi ce of v ari a bl es is n ot of i p or t ance h ere a nd will b e d ealt with
a t a l a t e r s t ag e
Th e cur v es of Fig 144 sh ow t h e v ari a ti on o f th rus t al ong
o a
ea
s
m
.
.
.
AERODYN AMI CS
A PP L IE D
l
e
t
e
t
h
e
d
w on t h e b as is of eq ua t ion (Ba ) whils t t h e ar ea com
by
p
li n e of zero or d inat e is p ro p o rti ona l t o t h e t o ta l thr ust It will be noti ced
t ha t t h e inner p ar t of t h e ai rscr ew o pposes a r esi s tance to t h e ai r fl ow an d
tha t by far t h e gr eat er p rep ort i on of t h e thm
at i s d ev el op ed on t h e ou t e r
half of t h e bla d e Th e to tal thr us t as shown by t h e ar ea of t h e cur v es
t he
i
a r scr e
,
.
,
.
d
1
un ity
)
m
.
m
X
ecreases as
i
wo u l d b eco
n creases , and
e zer
o
for
V
-
e
~
nD
qu al
to
n ea r
ly
m
m
m
For co p ar i so n with t h e t o ta l thr us t as ca l cula ted fr o e q u a ti o n ( 8 0 )
a n d Fi
w
a
s
a
e
a
i
u
r
t
of
t
l
th
u
t
d
by
d
re
t
a
r
s
1
4
4
a
eas
e
n
t
h
e
c
e
t
o
g
eth o d a n d led to t h e cu r v e of Fig 145 Th e po in ts
ar k e d i n t h e fi gu r e
ar e t h e r es ul t o f t h e exp eri
I t will be n oti ced t h a t
en ts j us t d escr ib ed
t h e a gr ee en t be tw een t h e two
etho d s 18 goo d , wi th a t en d en cy for t h e
p oin ts t o li e a li ttl e b el ow t h e cur v e Th e agree en t i s a l os t as grea t
a s t h e acc ur acy of o b ser v a tio n , a n d t h e co nclus io n
ay b e dr a w n th a t i n
a pp li ca ti on s o f fl ui d th eory t o a ir screw s a r eas o n a bl e a pp li ca ti on of B er
n o ulli s th eo re
wi ll l ea d t o goo d r esu l ts L a t er i n t h e ch ap t er i t will b e
s hown th a t thi s th eo r e
ca rr i e d t h r o ugh i n d e t a il en a bl es a d es i gn er t o
cal cula t e s u ch c ur v es a s tho se o f Fig 144 , a n d th a t t h e a gr ee
en t with t h e
o b ser v a ti ons i s ag ai n sa t i s fact ory
u ch i nfor ation on t h e a i r
H av ing shown tha t t h e tot al h ea d gi v es
fl ow r oun d e u ai rscr ew, it i s p r op osed t o ex t en d t h e consi d era ti on of t h e
fl ow t o t h e di fl eren t p robl e of t h e di st ributi on of v elocity b efore an d
b ehin d an a irscr ew d isc R ep la ci ng t h e p itot tub e by an ane o et er ,
r ep et iti on o
ea n s o f
t h e p revi ou s exp er i en t s p r o v i d es an a d e u a t e
ea sur i n g t h e v el ocity an d d ir ec tio n of t h e ai r n ea r t h e a i rs cr ew
m
m
.
m
m
m
.
.
m
.
m
m
.
m
’
m
m
.
m
.
m
.
m
m
'
.
mm
m
q
m
.
m
.
m
—Exp eri m
en t s
of a ir n ear an ai rscrew h av e b ee n
ca rri ed ou t a t t h e N
a n d fr o m
a co ns i d era tio n of t h e r esu lt s o bt a in e d
Figs 146 and 147 h a v e b een p r od u ced Whil st th ey gi v e t h e g en era l
i d ea of fl ow to whi ch it i s now d esire d t o d raw a tt enti on it sh o ul d b e
mentioned th at t he cur v es shown ar e fair ed and th erefore for t h e p ur p oses
o f d ev elo p in g or ch ecki n g a n ew th eo r y of a i rscrew s l ess reli a bl e th an t h e
or igin al ob ser v a tion s
It will r ea dily b e u n d erst oo d th a t m
en t s o f v el ocity a n d
easu r em
d i rec ti on o f t h e a ir fl ow cann ot b e m
med i at e n eighbour h oo d
ad e i n t h e i m
o f t h e a i r scr ew d i s c a n d any v a l u es g i v en i n t h e fi gur es as r el a tin g t o t h e
a irs crew d is c ar e t h e r esu l t of in t er p ol a ti on an d are c o rr es p on di n gly
ay b e t a ken as corr ec t
u ncer ta in Qu alit a ti v ely h ow ev er t h e fi gu r es m
r ep r esen t a tio n s of o b ser v a ti on
w h ils t qu an tita ti v ely th ey are r oughly
co rr ect
Ea ch fi gu re h as b een s ub d i v i d ed in to Figs ( a) (b) and (c) whi ch h a v e
t h e foll owi ng f eatur es
m
e d i at e n eigh
a gr amshow s t h e
d
i
t
am
li
n t h e im
e
s
r
n es
a
T
h
e
i
( )
bo u rh ood of t h e ai rscrew t h e lin ear scal e b ei ng exp r essed 111 t er s
s t r eam
e t er o f t h e a i rscr ew
of t h e d i am
li nes
On each of t h e
ar e n u
b ers rep resen tin g t h e v elocity of t h e a i r at sev era l p oi n t s
w h ils t a t a few of th ese p oin t s t h e a n gl e o f t h e Sp i ra l follow ed by
Ai r
a n
on
t h e flow
.
.
,
,
,
.
,
.
,
,
,
.
.
,
,
.
m
,
m
‘‘
”
.
,
AIR SCREWS
m
i s i n d i ca t ed by fur th er n u b ers
Th e v el ocity i s d en o t e d
by V, an d t h e an gl e of t h e Sp ir al by gt
Th e d is t rib u tion of v eloci ty at v ar iou s ra dii i s shown i nl t h ese
d i agra s Each of t h e cur v es corresp on ds with a section of (a)
p arall el t o t h e airscrew di sc, an d t h e p osi tion of t h e sec ti on i s
indica t ed by t h e n u ber att ach ed to t h e cur v e Th e rad ii are
e xp r essed a s fracti ons of t h e di a
e t er of t h e ai rscrew
If t h e
o vi ng rela ti v e t o air at infini ty t h e v elocity scal e
a irscr ew b e n ot
i s ar bit r ar y , as it d ep en ds on t h e r ev oluti ons of t h e ai rs cr ew only
Wh ere t h e airscrew is o vin g with v elocity V r el ati v e to t h e d is t an t
ai r thi s 18 a conv en i en t
easur e for oth er v elociti es conn ec t ed with
t he
oti on of t h e a i r th r ough t h e ai r screw
Each of t h e s t r ea li n es of (a) i s a Sp ir a l , with t h e an gl e of t h e
s p ir a l v ar i a bl e fro
p oin t t o po in t Th e rel a tion b etw een t h e angl e
of t h e Sp i ra l an d t h e r a d i u s i s s hown i n
eac h cur v e as b efore
corr es p on d i n g with a di fferen t sec ti on of (a )
the
ai r
.
.
m
.
m
m
m
m
m
m
m
m
.
.
.
.
.
.
li m
its of accuracy a ttain ed t h e figur es gi v e a com
p l et e accoun t of t h e
moti on of t h e air ov er t h e most i mp ortant region and t h e t wo group s of
fig u r es h av e b een ch osen to r ep resen t wi dely difl er en t con di tion s of ru nn in g
I n Fig 146 t h e a irscr ew was st a ti onary r el ati v e to d i s t an t ai r an d i t s efli
ci s n ey th er ef ore zero
a xi m
um
I n Fig 147 t h e con dition was th a t of m
effi ci en cy an d was o bta i ned by s uit a bly choos in g t h e r a tio of t h e f orw ar d
Sp eed to t h e r ev oluti ons
Th e figur es are st r i kin gly di ffer en t ; for t h e s t ati on ary airscrew t h e
s t ream
li nes con v erg e rap i dly 1n fr on t of t h e ai rscrew di sc an d for som
e
littl e d is t ance behin d T
h ey ar e n ear ly p arallel at a di s tan ce b ehi n d t h e
d is c equ al to half t h e airscr ew di am
et er
For t h e m
o v ing airs crew t h e
most n oti ceabl e fea ture 18 t h e bulgi ng of t h e s treamli nes j u st b ehi n d t h e
ai rsc r ew dis c an d near t h e a xis
Outsi d e t h e cen tr al r egion t h e s tr eam
li nes are nearly p arall el to t h e ai rscrew axi s but show a slight con v ergen ce
towar ds t h e r ear
t he
,
.
.
,
.
.
,
.
,
.
.
.
.
Ha d t h e v alu e of Y b een i ncr eased fr om
075to 2 0 t h e airscrew wo u l d
-
-
nD
h a v e b een runni n g as a wi n dm
lin es ar e m
or e
i ll
Th e co rr esp on di ng st ream
clo sely r el a t ed t o t h e m
o v in g ai rscr ew th an to t h e s ta ti on ary on e t h e only
si m
p l e ch ange fr om
Fig 147 b ei n ga sli gh t d i v er gence of t h e s tream
s b ehi n d
ai rscrew
lin es t en d s t o p ersis t
Th e bulge on t h e m
n er s t r eam
.
,
.
.
.
Stati onary Airscrew, Fi g 146
n
b
a
h
c
a
e
t
r
a
T
r
a
e
di i in t h e
u
v
v
lo
ity
h
w
v
y
p
i
d
h
g
c
r
r
es
o
f
e
s
o
a
e
c
( )
n eighb our ho od of 08 t o 0 5
Th ese rap i d ch an ges d efin e t h e edg e
D
of t h e sli p st rea , s o far as i t can be d efined Wh en t h e str ea li nes
h av e b eco e rou ghly p arall el at 0 5D ( Fig 146 a) i t will be n oti ced
t h a t t h e great er p ar t of t h e fl ow occurs wi th i n a r adi us of 0 4 D, an d
thi s r ep resent s a v ery consi d era bl e r ed u ctio n of area b elow th a t of
t h e a i rscr ew d isc an d a con seq u en t con si d era bl e er ca se of av erag e
v el oc ity b etween t h e a i rscrew d isc an d t h e in i u secti on Th e
fi gur e sh ow s t h e v eloci ty at t h e d i sc t o b e r o u gh ly 70 p er cen t o f
.
m
m
.
'
.
.
m
°
.
m
m mm
.
.
AIR SCR EWS
lu es ar e of t h e o rd er of 10 or
At t h e ai rscr ew d i sc t he
in terp ol a t ed cur v e show s angl es of 10 at t h e cen tre f alli ng t o
3 or 4 j us t i nsi d e t h e bl a d e ti p
If t h e d ed u cti ons fromt h e fi gure be co m
p ared with those f ro t h e
°
va
°
,
°
°
m
.
DlSTANC E ALO NG AXIS O F A I RS C REW
m
e s s es b a c
NUMB ERS ATTAC HED r o cuav r s
ARE ousn nc s s ALONG AXIS O F Alas cs ew
SPIRAL O F
S LI P S TREAM
Fro l 4 7
.
th eoreti ca l
.
—Flow of ai r near a m
ovi ng
a i rs cre w.
lys is gi v en ea rli er i t wi ll b e seen th a t t h e i d eas of t rans
otion of a i r
la t i onal an d r ot a tiona l i nfl ow are app li ca bl e to t h e av erag e m
F u r th er th er e 18 a r egion of r oughly p arall el m
otion
r oun d an a ir scr ew
som
e m
ay b e
o d era t e d i s t an ce behin d t h e ai rscr ew i n whi ch it m
at
an a
,
.
,
U
AERODYN AMICS
A PP L IE D
u pp o sed th a t t h e p ress ur e di st ribution a dd s n othi ng to t h e thr us t a
en t u m
p r ess u r e an d m
om
by t h e us e of (8a)
ca l cu l a t ed fr om
Movi ng Air screw (Fi g
a
o
n
v
l
oc
ity
d
o
es
n
t
c
h
e
ra
p
i
d
l
y
with
h
e
ra
d
i
u
s
a
l
g
b
t
ar
t
e
T
h
e
e
( )
g
i s n ot cl ear ly d efined
ost
r adii a n d t h e e d ge of t h e sli p s t rea m
Th e m
m
ar k ed ch an ges of v el ocity occu r at t h e cen t r e an d j us t be hin d
of
S
p
ee
d
i
s
th
r
v
y
r
d
Thi
e
er
t h e ai rscr ew boss
Th e d m
e
m
a
k
e
s
p
p ar t of an a ir scr ew a dd s v ery little to t h e t otal thr us t or to r q u e
p ort an t Th e v elocity i s u ni ty w ell ah ead
a n d i s r e la ti v ely uni m
of t h e a i rscrew an d h as a dd ed t o it an a m
o un t n ev er e xcee di ng
Alon g each s t ream
lin e r o u ghly h alf t h e increm
en t
7 p er cen t
of s p ee d i s sh own a s h av in g occ u rr e d b efo r e t h e ai r crosses t h e ai r
Thi s co n d ition oi t h e wor kin g of an airscrew is o f grea t
screw d i sc
p r ac ti ca l i m
p ortance and t h e accu racy of di rect ob ser va tio n i s
b ett er th an for t h e s ta tionary ai rscrew Th e con t raction o f t h e
all but t h e in cr em
en t of m
om
s t rea m
i s sm
en tu m
is not i nconsi d er
a bl e
r ew t h e twi s t i s s hown by t h e ob ser v a ti on s t o
r
a
s
f
o
t
of
i
0
I
n
r
n
t
h
e
c
( )
u ch
all
b e sm
Ev en b ehi n d t h e ai rscrew di s c t h e an gl es ar e v ery m
a n d d o n ot an yw h ere
all er th an for t h e st a ti on ary a i rscrew
sm
exceed
s
.
.
.
,
.
.
,
.
,
,
.
.
,
.
,
.
.
,
II Ma r u s x ar roa r
'
.
.
m
Tu s onv
or
ru n
Arnscns w
Th e ex p er i en t a l wo rk j us t d escrib ed was n ecessary i n o r d er to outli n e
p ti ons on whi ch a th eo ry of t h e ai rscr ew shoul d
cl ea rly t h e b as i c assu
a d e t o e xp er i
en t on ly for t h e
r es t
In t h e t h eo ry it self a pp ea l i s
d et er in ati on of one nu b er , whi ch i s t h e ra ti o of t h e v elocity a dd e d a t
t h e ai rscrew d i sc to th a t a dd e d b etw ee n t h e p ara ll el p a r t of t h e sli p s t r ea
an d t h e p ar all el s t rea s i n f r on t
a d e th a t
Th e assu p ti on i s u su ally
thi s n u b er i s const an t , i e d oes n ot d ep end on t h e rad i u s a n a ssu p ti on
‘
e
s
wh i ch i s only j u sti fied by t h utility of t h e re u lti n g e q u a ti ons
In t he
ea r li er s t ag es i n o r d e r t o b ri ng i n t o p ro
i nence i t s ac tu al ch arac t er thi s
p ti on will n ot b e a d e
ass u
Th e airscr ew s t rea i s ill u st ra te d i n Fig 148 to sh ow t h e no encla t u re
u sed Th e h alf d i a et er of t h e a irscrew 18 d en oted by ro whil s t t h e h a lf
d i a et er of t h e sli p s t rea at i t s in i u section 18 11,0 R a d ii eas u red
i ni u
sec tio n by r ]
a t t h e a ir scre w di s c a r e d en o t e d by r a n d a t t h e
Th e a x i al v eloc ity of t h e ai r a t t h e a irs cr ew d isc 13 V( 1+ a l ) an d a t t h e
ini u section
V b ei n g t h e v el ocity i n fr on t of t h e ai rscre w
”
i n fl ow an d out fl ow
a t a n i n fi n it e d i st an ce ; a , an d b , are t h e
fact ors of t r ansl a tion al v elocity
t h e n ex t d i agr a , whi ch
Th e r ot a ti on al v elocity i s b ett er s een fr o
al so i n t ro d u ces t h e i d ea of t h e a pp li ca ti on of t h e aerofoil an d i t s kn own
ch ara ct eri s ti cs
Ea ch el e en t i s consi d ere d as th ou gh i n d ep en d en t of i t s
e a ssu
p ti on as t o t h e asp ect r a tio of
n eigh bo u r , a n d thi s i nv ol v es so
m
m
m
m
.
m
.
m
m
m
m
,
m
m
m
m
m
.
.
,
m
m
.
,
.
.
m mm
m
.
m
,
m mm
.
m
m
.
m mm
“
m
.
m
.
La te r e x
ye t u nd c ve
1
as
ri
g
mt
c n s a re
i
ng
ro
v
i
d
p
m
d a t a fo r
m
m
a
m
ore
e ral ass u
e
n
g
mpt i
on ,
bu t
a
p p li cat i on is
AIR SCREWS
the
ofoil on wh i ch t he b asi c d at a w ere obt ai n ed
a i r s cr ew bl a d e
Th e v alu e t aken i s
kn owl ed ge i s n ot y et reach ed
aer
an d
the
.
h p of t h e
sin ce rea l
s a e
.
Fig 149 r ep resent s an el em
en t of a n a i rscrew bl ad e a t a r a d ius r
Th e
t ransl ati onal v elocity r el ati v e to ai r a consi d era bl e d ist ance aw ay i s V
t o b ei n g t h e a n gul ar v eloc ity of t h e a i r
a n d t h e ro ta ti ona l v eloc ity cu r
.
.
,
,
D I R ECTIO N O F
RELATIV E WIND
V( I
t tional in fl ow factor T h ese t wo v elociti es
t
d efin e t h e an gl e a
di r ec ti on of t he r el a ti v e win d an d sin ce t h e
a kes a kn own an l e with t h e a ir screw di sc t h e
en t m
chor d of t h e el em
g
an d wr
(1
as
b e in g
)
an
,
the
t he
ro a
.
,
AIRSCREWS
m
Th e el e en t s of th r u st and t or q u e can n ow b e w r itt en d own
o f a i r fl o wing t hr ough t h e a nn ul u s of t h e s li p s t rea
i s 21rp V( l
t h e v elocity a dd ed fr o r es t i s b,V, an d th er efor e t h e thr us t i s
m
m
dT
Usin g
e
quations (4)
2vrp ( l
5
( )
an d
to transfor
m(7) l
ea
.
Th e
m
ass
d s to
at es
if t he o m
en tu m
an d aerofoil th eo ri es are t o l ea d to i d en ti ca l es t i m
H ence
t h is thr ust shoul d be t h e sam
e a s th a t gi v en by
m
and
JR
) XV rdr
21rp ( 1
y)
008
(9)
"
a,
1
s of a , a n d a,
t h i s equa tion every t ermi s by hy p oth esi s known i n term
a n d eq u a tio n 9 i s th er efo re o n e rel a tion b e tw een a , a n d 0 2
seco n d
A
( )
r e l a tio n m
ay be obt a in ed from
t h e e qu ality of t h e exp r essions for to r qu e
Th e el e en t of tor qu e i s r ea d ily seen to b e
In
,
,
.
m
and
1
1
( )
mki
a
an d
.
dQ
n
g
5
( )
the
o
c rr esp
my b
a
b,) b2Vw r , dr l
9d ?p ( l
o n d in g
a
a ss
um
p ti on
Azbz
02
e
used to t ransform( 10)
dQ
t o (4 )
th at
o
to
27TP( I +
o
m
f dr
U nli ke e qua tio n (9) for t h e el e en tary th r u st whi ch cont ai ns r o nly
for el em
en tar y to r qu e i n v ol v es b oth r a n d r , an d t h e r el a
e qua tion
tion wh i ch i s gi ven by ( 6) d o es not l en d it self to si m
p l e s ubs titution
,
,
,
,
in
Equa tin g ( 12)
an d
as
A,
a nd
gi v es a secon d
2
( )
si n
(,t
y)
A,
l tion
re a
b et ween
'
a , a n d 02 as
21rp ( l
bein g kn own const ants equ atio ns ( 9) an d ( 18) a r e su fii ci ent t o
d et erm
in e both a , an d a , in t erm
s of aer ofo i l ch arac t er i s ti cs
Transfor ati on of Eq
for
u ati ons ( 9) and ( 13 ) to
ore Conveni ent For
Calculation —Fr o m
t h e geo m
et ry of t h e a ir fl ow it follow s th a t
V
t an
( 1 + a 2) w r
and th a t t h e res u lt an t v eloc ity i s
,
m
m
.
.
en t of for ce
l m
ments t o have th e for
Th e
e e
(1
, dB ,
m
an
6
9
)
is kn own fromg enera l wi n d ch annel
p odr
a , wr sec
-
(l
a
g
z
w
m
see
“
a f( )
.
a
m
AERODYN AMI CS
A PP L IE D
wh ere c i s t h e su mof t h e chor d s
n t of
b
olut
o
f
i
a
i
t
h
e
a
s
e
c
e
fi
c
e
s
j( )
known th a t
il el em
en t s
t he
of
aerofo
ult an t force
res
a t ra
In t h e
.
diu s
r , an d
mw y
sa
e
a
is
it
1
( 7)
s
t ransform
i n g ( 9) by
Th e algeb rai c wor k i n
p l e, an d l ea ds to
im
1
5
s
1
i
7
( )
a,
whi ls t ( 18) b ecom
es
— 3
I
an d
u se of
d
5
2
:
72
o
f( )
c sec
a
qsee p sin (95
t
y)
os t co n v eni en t to ass um
ol v e i n a ny p ar ti cul ar cas e it i s m
e
su ccess i v e v alu es for a
Si n ce ¢ + a is kn own fr omt h e g eom
e t ry
in e
of t h e a ir screw thi s fi x es 95 an d e q u a tio ns ( 18) an d ( 19) th en d et erm
To
s
,
.
,
Fi nally , e
a , a nd a 2.
m
q
m
f o mwi d h
of
q
u at i on
1
4
( )
S as su ed
Exa ple of th e Calcu lati on
V
gi v es t h e correct v alu e of —
for t h e v alu es
m
.
—Th e forces on an aerofoil as t a k en
r
en t s ar e m
n
c ann el exper i m
os t co m
only gi v en as lift a n d d ra g
I n t h e p r esen t n ot at i on
co effi ci en t s k, a n d
of a ,
-
k, =
k,,
f( )
a
T ake r =3 8 6
i ns
.
,
09 8
cos
s in
}
m}
4
sm
¢ + a cos ¢
in
k. 09 8 96
e
c=2 ><
y
y
9 65 i n s
m
.
,
i
.
s
-03 5
,
575,
.
m
p r oceed t o fill t h e t abl e b elow fro kn own d a ta
Th e t a bul ation st ar t s fr omcolu n ( 1) with ar bit r ar i ly ch osen v alu es
of a an d mthis illust ration a v ery wi d e rang e of a h as b een t aken Si nce
=22 l colu n (2) follows i m
a +q
S=
medi at ely Th e lift coeffi ci en t k, i s
t ak en fromwi n d ch ann el ob serva ti ons on a suit a bl e a er ofoil for t h e gi v en
(1
m
,
°
-
v alu es of
th e
u se
a,
m
.
.
£2
;
o mt i
t he
.
ra
ti o
lu m
n ( 4)
of co
is
s
im
il arly obt ain ed
an d
mi i
olum
t rig n o e r cal t a bl es l ea d s t o colum
n
foll ow a s ar ith m
t h e firs t
eti ca l p r ocesses f r om
of
,
Th e
re
by
a n n
g
ns
four c
olum
ns a n d
eq u atio n
Th e v al u es fou n d for a , show v ery gr ea t v a ri a tion s but d i scus sion of
t h e r esu lt s i s d eferr ed u n til a , h as b een ev alu a t ed
—
A
Assu m
i
n
n
1
Th e assu m i on whi ch h as recei v ed
t
o
s
(
as t o
p
2 a d
2
most a ttenti on hitherto h as b een th at A2= O an d eq u ati on ( 19) th en sh ow s
th a t a , i s z ero T h i s i s equi val en t t o ass u m
i n g n o r o t a tion al i n fl ow
p ti ons now app ea r to b e b ett er
a n d o t h er a s s u m
A2 p l ays t h e sa m
e p a r t i n rel a ti on t o t or q u e th a t A, d oe s t o t h e thr u s t
c
m
,
.
.
,
.
.
,
A IRSCR E W S
a n
A,
s o
it w oul d
d
A2
and
mp
e
ar
so
p ossibl e to carry em
p i r i ci smone s tag e furth er an d ch oo se
th at both t h e th r u st and tor qu e agree d with exp erim
en t at
be
ticu l ar v alu e of
33
.
T hi s wo u l d l ea d to
mo 11:11t
o
b v io
A m
re o
but not to fun dam
en t ally di fferen t i d eas
e qu ally p r ob a bl e assu m
p tion i s th a t t h e air at t h e airscr ew di sc
a n add ed v el oc ity i n t he di rection Opp o si t e to dB i n whi ch case
t i o ns ,
.
l ul a
ca c
re (
us
is
an d
gi v en
,
kL
—0 l 70 —3
°
+0 0l0
'
22 1
'
l
+03
cos
7)
—
é k
p s ln
w
—o l 73
+06 3 2
'
u
6
:
a 7
e
W
—00 196
2
—00 20
73 0
019 5
+00 0 5
+01 76
19 15
+0069 !
08 4 3
+00 74 0
+0108
lS l
’
04 25
+0 4 0l
'
+04 67
+09 54
150
71
A
mlt ip li d b y
0560
l3 5
0 210
+06 3 6
059 5
80
0 124
+0 58 1
°
+06 4 0
12 1
+ 106 0
13 0
—l 10
°
u y of thi s assu m
p tion i s n ot l ess th an th a t r el ating t o c l
a d e th a t a ,
p tion i s m
Th e ra di al v el ocity i s s till ign or ed an d t h e assum
or e corr ec t
a n d a z are co ns t an t a cr os s t h e bl a d e whi ch wi ll p rob a bly b e m
ai n s as b ef or e b u t
for n arr ow th an for wi d e bl ad es
Eq u atio n ( 18) rem
e qu atio n ( 14 ) b ecom
es
Th e
acc r ac
.
,
,
,
.
01
5
t a n (5
Y)
A IRSCRE WS
of i nci den ce u p to
w ar ds m
ore
or e a n d m
m
l ti v to
as a
ra
p i d ly
m
li
mit
,
as
t he
.
_
n
El
f
i
th
Th e u seful wor k
o
e
e
e
n
t
c e cy
ai r a t in fin ity , i s VdT, whi ls t t h e p ow er
e
E
re a
efli ci en cy
is
ai rscre
.
mov b
b i g m u
w
ac k
es
d on e e n
eas r e d
exp en d ed i s wd Q
Th e
,
.
th en
V
dT
w dQ
Sub stit u ti n g
fromequations ( 1)
an d
2
( )
o v er t s (26) to
c n
V
om
bin in g t hi s wi t h ( 14) l ea ds t o
an d c
Wi n d m
i ll
+ 23 8
06 08
e s t at i c
.
t es t
con di t i on
might h b
ho i g VdT
m my b
er e
id
e sa
as
to
m
g of effi ci ency an d t h e
as a m
r eason for c
os n
eas u r e oi w or k d on e
Effi ci ency i s a
t h e f oll owi n g exam
r el a ti v e t er
e seen fr o m
p l e : Im
as
a
agin e an
aero p la n e fl yi ng t h rough t h e a i r ag ains t a wi n d h av in g a sp eed e qu a l t o
i t s own
R ela ti v e to t h e gr oun d t h e aerop l ane i s st a tionary but t h e
p et rol consu m
p tion i s j u st as great as if th er e w er e n o win d As a m
ean s
of t ransp ort o v er t h e gr ou n d t h e aereplan e h as n o effi ci ency in t h e abo v e
i ns tance On t h e oth er h an d if i t turns roun d and fli es with t h e win d
t h e aer ep lan e wou l d b e sai d to b e an efli ci en t m
ean s of t ran sp o r t an d y et
i n n eith er case d o es t h e aerop l ane d o any u seful wor k i n t h e sense of s torin g
en er gy unl ess it h as h app en ed to clim
b It i s o b v i ous t h at n o useful
d efinition of effi ci ency ca n d ep en d on t h e st rength of t h e wi n d an d wh a t
A
word
v eloc i t y ,
th e
eani n
'
.
,
.
,
.
.
,
,
.
,
AERODYN AMI CS
A PP L IE D
ea n t by t h e e fli ci en c
usu ally m
i
w
v
l
o
f
t
h
r
i
s
s
s
a n i ns t r u
a
rs
e
c
i
t
a
u
a
e
e
y
men t for t h e p u rp ose of m
o vi n g t h e r es t of t h e aer ep lan e throu gh t h e a i r
Th e concep ti on of effi ci en cy i s n ot si m
p l e an d w ell r ep ay s sp eci al a tt en ti on
d u ri ng a st u dy of aerody nam
i cs
ay b e cal cul a t ed t h e v alu es of effi ci en cy 1
F r omequ ati on (28) m
)
co rresp on di ng with T a bl es 1 an d 2
Th e v a lu es ar e gi v en in T a bl e 3
In i n t erp retin g T a bl e 3 it i s con v eni en t to r efer to Fig 150 whi ch
show s t h e a i rs cr ew ch ara ct eri s ti cs of t h e el em
p ari son with th ose
en t i n co m
is
.
.
.
.
.
F
of
t he
on a n
m150—C mp
.
l
.
mt
e e
o
y
en ar
a ri s on of cha racte ris t i cs
a er
ofoi l
gl e of i n ci d ence of
.
the
Th e
h
e le
mt
e n s of aerofoi
t i ti cs
c ar a c er s
foil
aero
of
,
an d
ar e s
,
l a nd ai rs crew
.
hown
as
d ep end en t
u v es show Y and
t he
c r
efli ci en cy
lift coeffi ci en t and l— for t h e aerofoil
E
l
At an an gl e of i nci d ence of —10 t h e thr u st an d tor qu e are both
n ega ti v e an d T a bl e 3 show s t h e effi ci en cy to b e p ositi v e
Th e ai rscr ew
ill t h e w ork out p ut i s wdQ and n ot VdT an d
i s w orki n g as a wi n d m
2
r
e
p
r
e
e
n
t
h
i
p
r
o
a
l
of
t
e
e
fli
i
e
n
c
o
f
t
h
e
wi
n
d
m
i
ll
t
e
v
a
lu
e
e
6
s
t
e
r
c
h
c
h
s
c
;
( )
y
of Ta bl e 8 r ep r esen t s a r eal effi ci en cy of 4 2 p er cen t
At an
a n gl e of i n c i d en ce of
t h e p oin t of z ero t or q u e occu rs and t h e
ill i s zer o corr esp on d in g with an infi ni t e v al u e in
efli ci en cy a s a wi n d m
T a bl e 3 As t h e an gl e of i nci d ence i ncreases t h e t or qu e b ecom
es p o siti v e
whil st t h e thru st r em
a in s n ega ti v e an d t h e efli ci en cy i s n eg a ti v e
At
al
t h e thr u s t b ecom
es p ositi v e a n d t h e a irs cr ew b egi n s i t s no rm
functions a s a p r op elli ng agen t t h e effi ci en cy b ei n g zer o a t thi s p o in t
for t h e air screw
iii
an d
"
.
°
.
,
,
,
.
,
.
,
.
,
,
AIR S CR E WS
i i g
p i d ly to 08 3 at an angl e of i nci d ence of a bout
At
g r ea t er an gl es of inci d en ce t h e effi ci ency fa ll s to z ero wh en t h e ai rscr ew
i s n ot
ovin g r el ati v e to d i s ta n t air I f t h e airscrew b e o v ed b ackw ar d s
V dT i s n ega ti v e and t h e effi ci en cy i s n ega ti v e but this co n d iti on i s
u ni
o
r
t
a
t
d
e
t
a
il
d
tu
d
y
o
f
i
t
i
s
i
v
n
n
a
n
d
n
o
e
e
s
p
g
lift
a
e
Th e g en eral si il arity of t h e effi ci ency an d
b
n
oti
c
ed
cur v es
y
bu t
r s n
ra
m
m
and s
O th er
i
s
7
,
.
m
t h im
po t
ugges ts
e
ra
n
e
g
m
.
e
m
fi g
of high l —
r an ce
i
iE
ra
tio
.
Thi s
is
seen
to
be
a
l p ro p erty of ai rscrew el em
en t s by r ef erence to e qu a tion
thi ngs b eing equ al equ ati on (28) shows ax i umefi ci ency wh en
(
mm
,
l eas t
,
i
.
s.
wh en
is
m
m
grea tes t
.
m
Relati ve I p or tance of Inflow Fasten —I t is now p o ssibl e to ak e a
u a n tit ati v e exa ina tion of t h e i p or t an ce of t h e infl ow fac tors a , an d
a z, an d for t h i s p ur p o se T a bl e 4 h as b een p rep ared
Th e fir st colu n
co n t ai ns t h e an gl e of in ci d en ce of t h e bl a d e el e en t , whils t t h e r e a i n i n g
q
m
m
ol u
c
ns s
( 1) both
is
how
TAB LE
the
v alu es of
l
wr
an d 77 on
the
p
.
m
t hy p oth eses th a t
used ; (2) th a t n eith er i s used and (8) tha t only
p or t an t
Th e g eneral co nclusio n i s reach ed th a t a , i s v ery im
ay be ignor ed i n
any cal cul atio ns wi thout serious error
,
m
E
4 —
'
m
,
.
mCmmm A
Em
B
mm
I s r now F a
r ron u rn E
rr s c r o r
,
m
.
se ar a e
a ; an d a 2 ar e
used
but th at az
a,
m
m
o us o n r
ou
or
D V AN C E r
:
or 3
u
r n
mv
;
ow
'
s nr.
At t h e angl e of no thr u st ,
t h e th r ee hy p oth eses d i ffer by v ery
s all an d un i
p or t an t a oun t s but at an angl e of i nci d ence of
whi ch
woul d corr esp on d with t h e b es t cli bin g rat e of an aer op l ane, t h e differen ce
m
m
m
,
m
Y for t h e assum
p tion of no in fl ow an d th at
—
m
full i nfl w i s ore th an
p
20 p er cen t
If V b e fi x ed by t h e con d itions of flight t h e th eo ry of no
i nflow w oul d i n d i ca t e a l ow er sp ee d of r ot a tion for a gi v en th ru st th an d oes
.
for
AIR SCR EWS
s een
mll i om
p i o
mll 5my b g t
to
be
s
a
t an
s
a
n c
ar s n
e as
a
.
rea
ii
"
f
with uni ty if
i
as
s
,
l arge
.
i n t he
1
20 an d t an 91
an d
p art s of
an
i hi p a irscrew whi ch are im
p ort an t Hence for t h e
ci r cum
s t an ces of gr a t es t p racti ca l i m
r t an ce we m
p
o
ay u se 29 as in di catin g
( )
e
a goo d a pp ro xi
ati o n ; o v er t h e w ork in g ran g e a l d o es n ot ex cee d 0 8
a n d an error of 5 er cen t in c m
ak es a n err or of 1 p er cen t i n t h e es ti
p
l
mated efli ci ency At aximumeffi ci ency t h e approxi ma ti on i s v ery
much closer Inst ea d of (18) a n ew app roximat e express ion for a ; for t h e
o r di n ary d es ign of ai rscr ew s i s
D
e fli ci en t aer ep lane or a r s
.
m
,
m
.
.
.
.
A]
a]
—
1 + al
21r
0
°
°
r
k° cos d
'
i
cosec
af
t
m
it
Poi nts of no Tor u e, no Th ru st , and no Lift —Fr o
e q u ation
w ill b e seen th at t h e tor qu e of t h e ele en t will b e z ero if dB si n
0,
6
b
e
se
t
h
e
n
f
n
o
r
e
u
d
o
d
iti
o
o
to
qu
a n d if t h e v al u e of dR fro
1
n
c
,
( )
r ed u ces t o
0 wh en fl a ) s i n ( t
y)
q
m
m
q
l
a e.
wh en k,
wh en
In
a si
il
m
ar
way
ay
it m
sin c
}
]
p oin t
of n o
cos
6
5
0
— cot ¢
;
D
.
be foun d
tha t
11
91
1611
Th e
kn
1
2
1
:
lift occurs of course wh en
,
,
d in ar y p ra cti ce 41 i s p ositi v e at t h e angl e of n o li ft an d t h e p ositi ons
f ou n d from(82) a nd (8 3 ) are n ot far r e o v ed fro t h e no li ft p ositi on
It
for (8 2) an d +2 for
p le l
For t h e el e en t o f t h e p re v i ous exa m
In
or
m
m
m
,
~
.
“
kD
wh en t h e solution i s obt ai ned t h e angl es of i nci d en ce bei ng
n o to r qu e
n o lift
n o thr u s t
Thi s r esu l t m
p ort an t secti ons of air
ay be t a k en as ty p i ca l of t h e i m
screw bl a d es
en ts to ob tai n Thr u st and Torq
ue
I ntegr ati on for a Nu ber of Elem
en t can
for an [th eore —Th e p rocess carr i ed ou t i n d et a il for a n el em
b e rep ea t ed for oth er ra di i an d t h e tot a l th rus t an d t or qu e obt ain ed
ay b e coll ect ed as
Th e exp r ession s m
3
8
( )
,
m
.
m
.
m
a
i
i
n/
“
v
m
=
1
Q
<
f
n
w
e
sec
z
c. cos
it
c, si n
a
sa
a
sin
d
r
a
k,
cos
d
r
¢)
0
2
c
p
0
)
a,
2
7 3
01 7
s ee
2
sa
.
APP L IE D AE RODY NAM IC S
fro mt h e aerofoi l si d e
,
and
T
21rp ( l
s
s
a
w
o}
!
1
m
2 p (1
fromconsi d erati ons of
wh ere
1
d efin es
th e
o i d er ing
In
c ns
mom tum
en
a I f dr
)
r l of
)
a,
ii
i
-
Vai n
"
r dr
,
1
)
91
A!
r I dr 1
en t it
gl e el em
a si n
3
?
h as
b een
sh
own th at
02
my
a
be
It h as b een shown th a t (8 5) an d (38 )
2
ay m
os t
can b e m
a d e to a gree by s ui t a bl e choi ce of a l a n d (38) m
ay b e un kn own equa tio n
b e used d ur i n g i n t egr a tio n to fin d T
AS as m
i
s
s
e
u
d to cal cul ate Q
36
( )
t aken
as z er
o but th a t
,
i s fini t e
.
,
.
,
.
Deter
th a t for
m
i ti
na on of
Ap —I f
v ariou s v alu es of i ll be chosen it is ob v i ou s
so m
e p r t i cul ar on e t h e cal cul at ed th ru s t at a gi v en a dv an ce
a
It
p er rev ol u t i on w i ll a gr ee with t h e observ ed thr us t on t h e a irscrew
ay be su pp o se d th a t this h as b een d o ne i n a p ar ti cul ar case see Fi
(
g
V
a n d th a t for a v al u e of
o f 06 4 5 t h e bes t v a l u e of a
l l ha s b een foun d t o
m
.
.
ED
be 0 8 5
.
U si ng thi s v alu e of A, for
0562 an d
0 726 fu rth er v alu es
A IR S CR E WS
a y b e co m
to t a l th rus t ar e cal cul a bl e an d m
p ar ed with ob serv ation
ay b e com
en t s m
p ar ed by t h e m
etho d us ed by
Cu r v es for t h e bl ad e el em
eas ur in g t h e t h r u s t on t h e el em
en t s
D r St an ton an d Mi ss Marsh all in m
an d t h e r es u lt of t h e co m
p ar is on i s
o f an a ir screw bla d e ( see p age
os t co m
1 T hi s i s t h e m
pl et e ch eck of t h e in fl ow th eory
s hown i n Fig 15
whi ch h as y et been m
a d e Gen erally t h e agr ee en t b etw een cal cu la ti on a
nd
er ous ass u m
ob serv ation i s v ery goo d in vi ew of t h e n um
p ti ons i n t h e th eory
It will be r ealiz ed t h at i n t h e ch eck as app li ed a bo ve any err o rs i n
of
.
.
m
.
.
.
,
.
,
o ur
an d
k nowl ed g e of t h e
will
in the
s fi ect
sa
m
e
h t
i
dT
r g
t h e sections
of
wi ll app ear as a tt ribut ed to i nfl ow
t h e v al u e of A, any loss of effi ci ency at t h e ti p will a
pp ear
Page h as shown , how ev er , th a t for a
way
o d era t e
m
.
ran ge of a irscr ew d esi gn and
for
u h v alu es of
s c
35
—
,
as ar e
used
in
p racti ce
oughly cons t ant Th e b es t v al u e i s y et to be d et erm
ine d but i s
Th e com
p ari son gi v en i n Fig 151
a pp ar en tly i n t h e n eigh bour hoo d of
en d lo ss
t h e th ru s t o b ser v ed
show e d t h e p r esen ce of an a pp r ecia bl e
n ear t h e ti p bei n g l ess th an th a t ca l cul at e d un til a r ed u ctio n o f lift s e
ad e
At a li ttl e o v er 9 5 p er cen t of t h e ra d i u s t h e
e fli ci en t h a d b een m
lift coeffi ci en t was app arently r ed u ced to h alf t h e v alu e it w oul d h av e
t h e ti p
h a d if far fr o m
It wi ll be seen th at on p r esen t assu m
p tions t h e v alu e of t h e to r qu e i s
i ned wh en A, i s known
p l et ely d eterm
p ared with exp eri
Wh en com
com
men t t h e cal cul ated values of t h e t or que are i n goo d agr eemen t wi th
o b ser v a ti on t h e a v er ag e d i ffer en ce b e in g of t h e or d er o f 2 or 8 p er cen t
ar y of Conclu si ons on th e Math e ati cal Th eory —As a r es ult
Su
en t a l exam
of a com
bin ed th eo reti ca l a nd exp eri m
i na ti on of a i rscr ew p er
ay b e n egl ec t ed a n d th a t
an ce it i s con cl u d ed t h a t rot a tion a l i n fl ow m
form
may b e u sed for t h e t ransl ati ona l i n flow factor
a n a v er ag e v alu e o f
Th ere i s a ti p l oss whi ch is t aken t o b e i n a pp r eci a bl e a t 8 5 p er cen t
A,
o f t h e r a d i u s 100 p er cen t a t t h e ti p an d 40 p er cen t a t
of t h e
ma xi mumra d ius Th e val u es of th ese losses althou gh ad mitt edly not
o f hi gh p er cen t ag e accur acy ar e of t h e n a t u re of co rr ectio ns an d t h e fi n al
en t wi th p racti ce
c a l cu l a tions o f th r u s t a nd t or qu e are i n g oo d a gr eem
A, i s
r
.
,
.
"
,
.
.
.
.
m
m
m
,
.
.
.
,
.
.
.
,
.
.
,
,
,
.
III
.
APPL I CATI ONS
or
THE
Ma
m m
s nx
cxn
Ta s oav
—
e th o d o f cal cu l a tio n for t h e p erf orm
I n d ev el op i ng t h e m
ance of
q an d n ecessary
a n ai rs cr ew O pp or t un i ty will b e t a k en t o collec t t h e form
d a t a F ollowi n g t h e p rev ious p a r t o f thi s ch ap t er i t will b e u nn eces sary
u las i n u se a s th ey m
ay b e obt a in e d fr o m
t o p r o v e any of t h e form
equ atio n s
6
n
d
8
s
a
8
7
by
i
m
p
l
e
t
ra
n
s
f
tio
wh
th
y
d
i
o
r
m
a
n
s
er
e
e
ffer
( )
( )
s th er e sh o wn
fromt h e f orm
Th e firs t st ep w ill b e t o coll ec t a r ep r ese n t a ti v e set of aerofoil sec ti ons
s u it a bl e for ai rscrew d esign tog eth er with t a bl es of th ei r ch a r a ct eri sti cs
Th e r es ult s chosen w er e obt ai n ed i n a wi n d ch ann el a t a high v alu e of
ay b e u sed without scal e co rr ectio n
Th e sh ap es of si x a erofoil
1
71 an d
Airscrew
.
.
,
.
,
,
m
.
.
AI RSCREWS
TAB LE 6
.
-
Ar:ao r orr s
.
SUI TA BL E r o a
Ab solu te lift
No 3
N0 2
.
.
.
.
Ai as c as w D ss
m
s
.
coeffi ci ent .
No 4
.
.
No 5
.
.
No 6
.
.
+ 0025?
+0 OOI 2
°
+00 125
o 3 l4
°
No 4
si on al or
the
i
a rscre
b olut e
a s
w
.
Th e
No 6
.
No 6
i l ar p r oced ur e wi ll be follow ed for
un i ts an d a sim
ty p i cal l ength of an ai rscrew i s al m
ost alway s t aken
,
X
APP L IED AERODYN AMI CS
as
d i am
et er a n d t h e wi d th of t h e ch or d of a ny sec ti o n will b e e xp re s s e d
fra cti on of D Si ilar ly t h e ra diu s of t h e secti on will be gi v en a s a
its
as a
m
me diu
,
.
fr action of
An
a
the
e
x tre
pp li catio n
ra
the
of
i
w p er r ev oluti on
s u ch th a t
k, ,
a
s
.
P
2
of
.
m
X
ws
a rscre
fraction of
as a
,
i ts
i
kt p n D
di a
)
mt
7
e er
or
th e
; a
a
dvance
17
nD
s
s
m
su i
.
er
2
1 + 02
1T
i
“1
l
+ a,
.
D 2?
°
11
Th e
27
'
s
77
kQ
”D
V
1T
1
( X
EA
;
)
01
t an
v alu e of A, will b e t aken to b e
TAB L E 7
.
Th e
gi v i n g
mof t h bl d
m f t h wi d th
p l an for
t he
su
o
e
a
s of
e
I n t h i s e xa
es
m
pl
e 0is
of
the
the
t he
the
‘
m
2
of
t h rust co effi ci e n t
Th e equ ati ons alr ea dy d ev elop ed ar e easily co n v ert e d to a for
s of t h e g en erali s ed
a bl e for t h e ca l cul a tion of k, an d kg i n t er
a n d t h e fi v e eq u atio ns r equi r ed ar e
'
the
m
i
e
m
dyn a i cal si il ari ty s ugges t s
i
for
T
tor qu e co effici en t kq d efin ed by
,
of
p rinci p l es
foll owing v ari abl es as sui t a bl e
a rscre
i
s,
su
i
w
a rscr e
two bl ad es
mf t h
o
is
d efin ed by T a bl e 7
for
e ch ords of
,
v arious v al u es of
t wo b lad es
.
t
,
AIRSCREWS
m
S i nce D i s not s p eci fi cally d efin ed , t h e sh ap e a pp li es t o all si il ar a i r screw s
I n a ddi ti on to t h e bl a d e wi d th s, t h e p ar ti cu l ars of t h e sec ti on s at v arious
v a
lu es
;
of
1
n c
gi v en
a re
Th e
.
.
n
olu m
i n t h e firs t
Fi g 152
those of
a s
i
2
i d ence of eac h section
for
c
t he
,
n of
l ast colu m
whi ch
th e
a er
ofoil
Nos
t h e t a bl e
is
a
m
s
.
b ei n g
.
hows
the
the
an
mxi mum
a
sa
m
e
gl e of
.
Th e sh ap e of t h e bl ad e i s not co p l et ely d efi n ed un til t h e i n clin ation
gi v en Thi s
o f t h e ch or d of each sectio n to t h e scr ew di sc h as b een
a n gl e, d en ot ed by 91
e
n
s
n
t
h
f
r
e
s
d
p
d
o
e
d
uti
e
o
wh
i
c
h
h
i
w
to
a
r
t
e
s
c
re
i
s
0
b e d es igned
u forw ar d Sp eed of an ai rcra ft ,
In g en era l t h e
a xi
t h e Sp eed of r o t a ti on of t h e engin e an d t h e a ir scr ew d i a et er are fi x e d
b y i n d ep en d en t cons i d era tio ns if t h e di a et er i s o p en to choi ce, a suit a bl e
v a lu e can b e fix ed fr o
g en eral kn owl ed ge by t h e use of a ch art su ch as
.
m mm
m
,
.
m
,
m
t hat
p ag e 8 19
on
q
n ot
a
e a
mk
’
nD
known
firs t set
calc
V
whi ch
is
e
m
my b
mo t
the
a
md k
n
e
s
own
1+ 0 1 V
°
11
is
s
iD
7
°
gi v en i t
n ow
fi xed i n
e seen
thi s
from
way i s
n ot
s
uffi ci en t to
t h e v al u es of a , an d
as
a,
v eni en t m
etho d of p ro ced u r e i s t o
of cal cu l a tion s with app ro xi a t e valu es and to r ep ea t
if grea t er accu racy i s d es i red Ins t ea d of t h e v alu e o f
an d
ass u
gu ess a v alu e for
ti o n
as
s
ul atio ns
the
v alu e of
Th e
mof
t er
d efin e t i n
a re
.
'
con
o
at s
m
mp
e s
.
d
ee
fl ight i t
of
,
mtio
i n t h e firs t a pp ro xi
u pp osed th at
the
d esign
a
re
o v eni en t to
is
c n
i n t h e illus t ra
n, a n d
qui r es th at
mxi mum
at
a
ci enc
Th e
in
e
y
m
p reli in a ry
l ul ation s m
ay
ca c
be
md
a
e
with
a,
= 0 an d
Wi th th ese con di tio ns t h e calcul ati on
qu ation
g = 0 88 p r oceed s as i n T abl e 8
9
gl ectin g P
ne
1;
ti on
at
bit rari ly cho sen v alu es
of
for t h e
sec
r
°
fi
1
Th e firs t
ai
Y
t
D
.
,
71
v alu e of
t a bl es of
lu m
n of T a bl e
co
i
a n d s n ce
1+ 0 1 V
_
7r
nD
°
. .
o t i
8
2
;
c n a ns
thi s l ea d s
-
1
m2
t a n (b i n colu
t rigon o etri ca l
m
.
n
.
5
i
s
3
«
functi on s
,
This is i n
ar
ra
p i d ly by
u se o f
S by
obt ained fr o mt an q
a n d t h e an gl e a i s chosen
.
o d an ce with
a cc r
the
ear
li er
a na
to
44
( )
t he
as
the
u se
8
°
of
wh en
ly sis whi ch
howed th at t he m
u meffi ci en cy of a sectio n o ccurred wh en t h e
axi m
lift
um Th e choi ce of a as 8 wh en
axi m
ra ti o of t h e a erof o il w as a m
s
°
(i t
i
¢
.
.
ig
°
l 5 8 fi x es t h e
the
°
v alu e of 95
s o f t h e bl a d e angl e t o t h e ai rscrew d i sc
i
0
(
1
e
re
s
s
n
t
h
e
r
o
a
n
e
i
i
g
v
lu
obt
i
d
f
xp
io
a
a
r
e
f
o
n
a
e
s
a n
¢o
m
re
,
.
m
AI R SCR EWS
Nu m
b ers can
Th e v alu e of
be
mT
d ed u ced fr o
thr u s t
foot
e
r
p
pV
D
'
a
bl e 9
for
p
m
co
i o with Fig
ar s n
.
151
.
ru n
_
’
v al u es
and
mi l
l ul ated by
ca c
m
eans of
9
4
( )
an d
D
p lott ed again s t
gi v e cur v es
those of Fig 151 Th e cen t ral p ar t of t h e ai rscr ew h as
p or t an ce
b een ign or ed as of littl e i m
U si ng e q u a ti on (46) i n t h e formshown t h e va l u e of
+ a ,) was
2
an d t h e v a lu e of t h e in t egr a l obt ai n ed g r ap hi ca lly
p lott ed on a b ase of
v ery
si
ar
to
.
.
.
,
,
t he
r es
ults b ei n g set out in
t h e t abl e
below
.
TAB LE 10
.
I f t he
i t will
v al u es of k,
b e foun d
th a t
ar e
k,
p lott ed
b eco m
es
on a
z er
b asi s of
an d
o wh en y
t h e cur v e p ro d u ced ,
an d
nD
thi s
nu
mb
er i s
tio of p i tch to d i am
et er for t h e a i rscr ew i n q u es ti on
Th e p it ch
h ere d efined i s call ed t h e exp eri m
ea n p it ch
a n d i s t h e a d v an ce
en t a l m
er
re
s
i
s
z er o
u
v
ol
tio
n
f
i
w
wh
t
h
r
u
t
e
o
t
n
t
h
h
e
a
rs
re
e
c
p
Torq
ue —
Th e ca l cu l a ti on of t or qu e foll ows fr o meq u a ti on (4 7) as
b el ow
the
ra
.
”
,
.
.
.
TAB LE 11
.
“N
Tab le 8
M
ab le
.
1
f
it
8
.
l
h ii ifm
ts
d ol xi
2 and 5
t'
ir
.
'
i
fl
AERODYN AMI CS
A PP L IE D
Th e
to
a
nu
mb
v al u e
an d
of
T a bl e 1] corr esp on d
in
23:
of
D
r es ult s of
t he
T abl e
ers
wi th
t a bl e was
Th e
l l ti ons
s u ch
ca cu a
X
a
r
e
e
a
n
s
w
e
p
lott
d
g
i
t
11
nD
t d
p
re ea e
as
are
s
p rep ar ed by r ea di ng ofi v alu es of a ,( 1
V
)
a,
,
an d a
pp ly
oth er v alu es of g
r
for
hown
F r omt h e curv es
'
was
those in T abl e 8
in
so
ol u m
n
c
6
p lott ed T abl e
t an (95 y ) at
h
c osen
,
of
12
v alu es
nD
TAB LE 12
.
ot an a»
n
Th e
a
b sci ssa
n
,
um
b ers
T a bl e
w er e p l ott ed
12
u v es for each v al u e of
an d c r
the
a reas of
in
cu r
qu ation
ee
T
a
bl
e
s
(
of e
£5
1
Th e efli ci ency
of
t he
.
s een
.
w h ol e ai rscrew i s
-
mQ
2
i
1
211
°
v
)
111
s.
.
kq
bta i n ed fr omT a bl es 10 an d 18 an d eq u ati on
th a t a high effi ci ency of 08 25i s fou n d a n d thi s i s p ar tly
ev a l u es of 1)
W11
1b e
Th e
b
a
t
u
(
1
n
n
It
.
et er g a v e t h e v al u es of t h e i n t egr a l
v es obt ain ed by p l ani m
th emt h e cal cu l ati on for kQ was easily com
an d f r om
p l e t ed
a
th
o rdi nat es with
drawn thr ough th e p oint s
TAB LE 13
an d
as
a re o
,
AI RSCR EWS
du e
e
to
the
ffi ci ency
fact
that
t h e sa
at
mt
all e e
mv
e
l
en s
lu e oi
a
h av e b een ch osen to gi v e th ei r
m i mm
ax
u
V
nD
m mm
_
Y
_ for t h e s t a t e of
u effi ci ency
ax i
nD
a n d r ep ea t i n g t h e calcula tio ns , t h e effec t of v ari a tion of p it ch co ul d h a v e
b ee n ob t ai ned Inst ea d of r ep ea ti n g t h e cal cu lati ons, a n exp eri en t
d es cri bed in a r ep o r t of t h e A eri ca n Ad vi sory C o
i tt ee on Aer on a uti cs
b e us e d to illus t ra t e t h e effec t of v ari a ti on of p it ch - di a et er ratio
Th e r ep o r t , by D r D u ra n d , con t a i n s a sy st e ati c seri es o f t es t s on 4 8 a i r
differen t v al u es of
0 4
.
mi 53
.
v
-
Efl ect
.
o
05
7zD
.
m
m
.
F
m
m
m
m
.
of va r ia t i o ns of
i
p t ch dia
mt
e e r ra t i o o f an a irscrew.
w s of v ar i ou s p l a n form
s a n d p it ch es a n d t h e res u lt s shown a re ty p i ca l
o f t h e whol e
For d et a ils t h e ori gin a l w ork shoul d b e con su lt ed
en t r eferr ed to w er e
Th e thr ee scr ews us ed i n t h e p arti cul a r exp er i m
e
of t h e sam
e di am
et er a n d h a d t h e sam
e aero foil s ec ti on s a t t h e sam
r a dii
Th e g enera l sh ap es of t h e secti ons w ere n ot gr ea tly d i ffer en t fr o m
an ce o f a n ai rsc re w
those j us t referr ed to i n t h e cal cul atio n of t h e p erfo rm
an d ill us t ra t e d i n Fig 15
2 t h e low er su r f aces b ein g fl a t excep t n ea r t h e
cen t r e of t h e ai rscr ew
Th e ch or d s of t h e s ecti on s w er e i ncli n ed a t 8 t o
t h e s ur fac e of a h eli x a n d t h e p it ch of t h is h eli x was 0 5D 0 7D an d 0 9 D
i n t h e thr ee a i rscr ews used i n p ro d u ci ng t h e r es ult s p l ott ed in Fig 158
scr e
,
.
.
,
,
.
.
,
°
.
°
°
°
,
,
.
Th e
p er i
ex
mt l m
ean
en a
o w er e 0 69D 0 87D
t h e h eli ca l p it ch es
z er
'
,
‘
,
.
p it ch es
a nd
,
109 D,
the
an d
v al u es of
do
n ot
b ea r
Y
7‘
‘
wh en
a ny s
t he
th ru s t
im
p l e r el a ti on
.
is
to
A IRSCRE WS
m
m
till un b ro k en and of p r acti cally i t s m
i ni umd i a et er Th e v elocity of
t h e ai r both t ransl a ti onal an d r ot a tiona l a t t h e r ea r ai rs crew can b e
a pp r oxi m
a t ely ca l cu la t e d by t h e u s e of eq u a ti on s ( 4 5
a
nd
a
n d an
)
e x am
f
a
b
w
p l e of t h e m
eth od whi ch m
e
o
llow
e
d
will
n
o
b
e
g
v
e
n
i
y
Th e fo r wa r d ai rscr ew will b e t a ken to b e th a t wo rked o u t i n thi s ch ap t er
—
o n p ages 806 t o 8 1
a
a
0 a nd of whi ch d et i ls are gi v en i n T bl es 8 18
Th e fi rs t Op era tion p ec uli ar to t an d ema i rscr ew s i s t h e cal cul a ti on of
t h e d etails of t h e sli p s t ream
F r omt h e v al u es
fr om
t h e forw ar d ai rscr ew
o f a 1( 1+ a l
a l cula t ed witho u t
a
t
h
i
s
gi
v
T
bl
e
v
lu
of
c
e
n
i
n
a
e
9
e
I
c
+
l
)
)
(
d i ffi culty si nce
s
f
=
5
s
5
1
a
o
s
1
o
e
«
+
.
,)
( )
(
mred
T akin g
p l e t h e followi n g t a bl e show s t h e req
0 6 as exa m
s
.
,
,
.
.
,
.
,
-
°
t p
s e s
mt h
,
l ul ation of t h e rad i u s of t h e sli p s tr eam
e ca c
TAB LE 14
.
0 8 72
m
fr omT a bl e 9
.
15
4
b ase of
on a
,
Th e
mugg t
mi d o di
2
p lott ed i n Fig
ca se
04 00
Th e firs t two r ow s of T a bl e 14 ar e obtain ed
t h e s econ d s in ce
r ow i s easi ly o bt ai n ed fr o
are
-
-
08 14
0 8 92
fou r
0 84 8
0 83 8
-
a for
s
t h e th ir d
fi gu res i n r ow
a nd
d by equ ation
es e
nate m
eth o d o f
i n t egral r eq u i r ed was obtai n ed by t h e
r
fin di n g t h e area of a d i agraman d t h e resu lt i s sh own 1n t h e low er p a r t
e v a l u e of t h e s q u ar e of t h e ra d i u s of t h e s li p
o f Fig 15
4 Th e ext r em
es th a t of t h e a i rs crew an d t h e r a di u s o f t h e
i s seen to b e 0 87 ti m
s t r ea m
a
es as gr ea t a s t h e ti p r a d iu s
Thi s v al u e m
s li p st r eam0 93 ti m
y be
p ared with t h e d i r ect ob ser v a ti ons illust rat ed in Fig 147
co m
—
m
i
l
i
r
t
o
l
l
a
Frome q u a tion (47) t h e r el ati on
i
n
i
oc
S
t
e
ty
S p
Ro at na Ve
Th e
-
,
.
.
,
.
.
.
.
ii ; (sil
5
2
( )
I)
b t ain ed by d iffer en ti a ti on
F r ome q u a tion ( 12) a secon d rel ation for t h e sam
e q u an tity i s obt a i n ed
Thi s la tt er exp r essi on i s
s of t h e ou t fl ow fac t or bz
i n t erm
is
O
.
_
_
_
g
.
{
213
2
Zr
“
1}
APPL IED AE R ODYN AM IC S
om
bi nation of (52) and (53) l ea ds to
an d a c
(1
an d a ll
the
t a bula t ed
Th e
r
qu antiti es
re
qui red
)
01
l ula ti on
for t h e
ca c
Of
bz
h av e a lr ea dy been
.
otational
ai r
v el ocity
m
is ba
0 l
.
2
( )
i
F10 l 5L
.
In
0
—C
olcu la t i o n of
thi s exp ression
V
t an
i s t he
t he size
will
V
A1
v elocity b etw een un di stu r b ed
01
f
be
air
of
t h e sli p s t rea
ogniz ed
r ec
an d
t he
s
as
mf
o
the
a n ai rsc re w
.
a
dd ed t rans l a tional
li p st ream and
,
t he
factor
om
p onen t r ot ationa l v el ocity whi ch woul d follow
c
51
fromt h e assu m
p tion th a t t h e di r ectio n of t h e resu ltan t f orce at t h e blade
a i n i n g f act or i s d u e to
i s a lso t h e di r ectio n of a dd ed v elocity
Th e rem
t h e ch ang e fr o m
et er t o s li p s t r eam
diam
a i rs crew d i am
et er
Th e followi ng t a bl e sh ows t h e v a l u es of b2 and t h e a ngle of t h e Sp iral
i n t h e s li p s t r ea m
ca l cu l a t e d from( 5
4) a n d
a n d t h e l a tt er can be
c om
p a r e d d i r ec tly with Fig 147 for o b ser va ti on s on a n a i rscrew
.
.
‘
.
A IRS CR E WS
TAB LE NS
.
0 602
—o ooeo
-
—00 166
4
029 7
0 4 12
-
4 00903
oo 17c
-
0 3 24
-
-
oi 1095
m
of
Th e ca lcu lations for a seco n d a i rscr ew wor kin g 1n t h e sli p st rea
'
t h e fi rs t can n ow h e p r oceed ed with a l ost as b efo re
I f a { an d a 2
e
eani n g as b efore,
a pp ly to t h e secon d ai rscr ew whi ls t V h as t h e sa
t h en t h e whol e of t h e p rev ious equ a tions can b e us ed with t h e followin g
m
Ins tea d of a l
'
use
a1
X1
1 bg
i t d of a 2 u se
an d n s ea
mm
.
'
a2
v alu es of a , and bz b ein g t aken wi th t h e co rr esp on d in g v alu es of r ]
T a bl e 14 Th e am
biguity of sign corr esp on d s with
a s obt a in ed fr o
e an d i n O pp os it e d i r ectio ns resp ecti v ely
r ot a ti on s i n t h e sam
If t h e r ear ai rscrew r uns i n t h e O pp osit e di r ecti on t o t h e fron t one
es
t h e ex i st ence of bz t end s to i ncr ease t h e e ci ency s i nce ( 28) n ow b ecom
the
m
.
m
t an
an
.
,
,
t
f
m
d as bz i s n egati v e t h e n u
Th e t ran sla ti onal i n fl ow
t h e fa c t or
l ti v e
x1
i n to
to the
re a
the
ai r
is
t
i
s
i
n
cre
d
5
as
e
7
( )
red u ces t h e efli ci en cy by
er a or of
d enom
in ator but
,
n ow
high er
the
as
t he
v alu e
p d
S ee
of
.
t he
of
i n t ro d u ction
the
i
of
w
r ea r a rs cre
18
i ncreased
owin g to t h e la rg er v al u es of 45
e loss of e fli ci en cy occu r s in t h e u se of
I n g en eral i t app ears th a t som
t a n d ema i rscr ews
i ned exp eri m
en t a lly an d
Th e su bj ect h as b een exa m
en t s i s q u ot ed b el ow b ecaus e of i t s b earin gs on t h e
on e o f t h e exp eri m
p resen t cal cu l a tion s
Th e a i rscrews w ere us ed on a l arg e a erOp lan e an d ea ch a b so r b ed 3 50
horsep ow er a t about 1100
r ot a ti on b ei ng i n O pp o sit e d i r ecti ons
et er of t h e fr on t ai rscrew w as 13 feet a n d th a t of t h e r ear a i rscr ew
Th e d i am
12 f eet
ax i m
u mSp eed of t h e aerOp lane i n l ev el fli ght was a b out
Th e m
100 m
Mo d els of t h e ai rscr ews w er e m
a d e a n d t es t ed i n a
wi n d
p h
ch ann el an d f rom
t h e result s obt ai n ed Fig 155h as b een p rep a r ed
Cur v es for th ru s t coefli ci en t an d effi ci ency a r e sh o wn for both a i r screw s
I n t h e case of t h e fron t ai rscr ew t h e cu rv es w er e n ot app r eci a bly alt er ed
by t h e ru nni ng of th a t a t t h e rear An exam
i n a ti on o f t h e fi gu r e will
s h ow th a t t h e r a ti o of p it ch to d i am
et er of t h e r ea r screw i s 0 8 6 whil s t
.
.
,
.
,
.
,
.
.
.
.
,
.
.
.
.
,
AIR SCR EWS
m
m
m
m
Th e v elocity in t h e sli p s tr ea
of t h e fro n t ai rscr ew i s n o t unif or
,
a ki n g t h e assu
p tion
a n d t h e v a l u e a s gi v en i n T a bl e 16 i s obt a in ed by
tha t t h e thr us t coeffi ci en t of t h e r ear a i rscr ew wh en wo rkin g i n t an d e
h as t h e
mi
sa
mv
e
lu e
a
wh en w orkin g
as
al n e,
o
if
v alu e
th e
of
Z
D
is the
b eing t h e av erag e v elocity of t h e a irscrew
r e la ti v e
Th e ca l cula ti on in v ol v es t h e v ari ation of en gin e p ow er
p loy abl e ar e gi v en i n t h e ch ap t er
etho d s em
w ith Sp eed an d d et ai ls of t h e m
e d a t i s sa ti sfi ed
In t h e p resen t ins t an ce t h e obj ec t ai m
o n P r ed i ction
ai rscr ew s h as b een d ev elo p ed an d t h e
w h en t h e d et ail e d th eory of t an d em
sa
e
n
t h e two
to t h e ai r
m
cas es,
'
V
.
,
.
r es
ult ill ust rat ed
I t will
.
be
n
oti ced fr omFig 155th a t for valu es
.
of
X
nD
of 062 t h e effi ci ency of t h e com
bination i s gr eat er th an th at of
t wo i n d ep en d en t air screw s li k e t h e for w ar d one
axi m
umSp eed
At t h e m
arr ang em
en t of ai r scr ew s
o f an aer Op lan e t h e loss of effi ci en cy on t h e t an d em
in
excess
.
is
n ot
gi v in g
v ery grea t si nce
is
,
m imm fli
ax
e
u
ci en cy
u ally ch osen
us
At
lm
bin g
ci
a
li ttl e l arger th an
v
say a t
t he
v alu e
0 4 , t h e effi ci ency
fif)
of t h e rear airscr ew i s 82 p er cen t of t h e forw ar d ai rscrew an d t h e co
b i n a ti on h as an efli ci ency 9 1 p er cen t of th a t of t h e fron t air scr ew a lone
whi ch was desi gn ed with out res tri ction as t o di a et er I t ay b e con
en t of ai r scr ew s
clu d ed th er efore th a t t h e lo sses i n a t and emarran g em
may be very small at t h e aximumSp eed of flight and th at they will
e g rea t er an d gr ea t er a s t h e m
a xi m
be com
umra t e of cli m
b an d t h e r es erv e
h orsep ow er for cli bi ng i n crease I t will h owev er b e t h e usu al case
th a t t an d e ai rscr ew s are only n eed ed on t h e ae r op l anes whi ch h av e l eas t
r es erv e ho rsep ow er i c wh ere t h e l osses ar e l east
.
,
"
.
m
.
,
m
,
m
m
,
Tu n Er as er
or r u n
.
,
,
.
Pa s s s n cn
or r u n
O F AN
m
m
m
.
,
,
.
.
m
,
Ann op na u s
on
m
A
mP
a
E RF O RMAN CE
s cns w
Th e n u b er of t es t s whi ch r ela t e t o t h e effec t of t h e p r esence of an
a er O p la ne on i t s a i rscr ew a re n o t v ery n u
P ar ti al exp eri en t s
erou s
o n a co
bi na ti on of od el a irscrew an d bo dy are or e n u er ous chi efl y
beca use t h e effec t of t h e a i rscr ew sli p s t rea i n i ncreasin g t h e bo dy r e
s i s t en ce i s v ery gr ea t
Thi s i ncrease of r esi stance i s d eal t with el sewh ere
i n d i scus sin g t h e es ti a ti on of r es ist a n ce for t h e aerop l ane a s a wh ol e
a n d i n d et ail
All t h e a v a il a bl e exp eri en t s show a consi st en t effec t of
b o dy on air screw whi ch i s r oughly equi v al en t to a s a ll i ncr ease of e
ci en cy an d an i n cr ease of exp eri
en t al
ea n p it ch
On e exa p l e h as
b een chosen, and t h e resu lt s are illustrat ed i n Fig 156 This ex a p l e i s
ty p i cal of su ch effect s as aris e fr o a n acell e closely surro u n di ng t h e
en gin e, a n d app ly p a r ti cul a rly to a t r act or a i rs cr ew
Wh er e t h e fr on t of
t h e b ody of a t ract or a er o p l a n o i s d esigned t o t a k e a w a t er cool ed engi n e
t h e r esult s woul d a lso a pp ly , but i t
ight b e a n ti ci p a t ed th a t t h e l arge
bo dy req u ir ed for a rot ary or ra di al en gin e w oul d h av e ore app r eci a bl e
e ffect s
m
m
m
m
m
m
.
m
.
.
,
m
m m
m
m
m
m
.
.
.
.
m
.
-
m
m
APP L IED AERODYN AMI CS
ffec t s of t h e nacell e of a p u sh er a ero p l ane a r e of t h e sam
e gen era l
ch aract er as for a t ra c t or ; bo t h t h e t h ru s t an d t or q u e coefli ci ent s a r e
i ncreased by t h e p resence of t h e nacell e and t h e efli ci en cy an d p it ch a re
i ncr eased Th e a m
o un t s a r e on t h e whol e r a th er gr ea t er th an th os e
s h own i n Fig 15
6
Fig 156 shows t h e thr u st coefi ci en t an d effi ci en cy of a fou r bla ded
t ractor a i rscrew wh en tes t ed al on e wh en tes ted i n fr on t of a bo dy an d
wh en t est ed in fr on t of a com
p l ete aer op lan e Th e ob serv a ti ons w er e
ta ken on a m
o d el i n a wi n d ch annel Th e cr oss secti on of t h e bo d y a
short di s t an ce b ehin d t h e ai rscr ew h a d an a r ea of 7 er cen t of th a t of
p
t h e ai rscrew di sc
Th e thr us t co effi ci en t i s i ncr eased by t h e bo dy ov er t h e
Th e
e
,
.
.
.
’
.
,
,
.
-
.
.
.
an a
u m ffi
m im
ax
e
i
y
c enc
is
mou
n
t whi ch
m
littl e affect ed but
,
m
D
pe im t l m
V
cr ea ses as
cr ea ses
.
Th e
n
th e
ex
r
en a
ean
p itc h
0 04
.
o
F10 150
.
.
—Eff
0 2
e ct o f
04
t h e b od y
0
a nd
w ings
o f a n ae ro p la ne o n
t h e t h ru st
of a n a irs cre w .
i ncreas ed by n ea rly 3 p er cen t Th e a dditi on of t h e wi ngs and g enera l
s t ru ctu r e o f t h e aer op l an e b r in gs t h e t o t a l effec t on t h e a i rscr ew to an
i ncrease of 1 p er cent on effi ci ency an d 5p er cen t on p itch
a xi m
um
On a p ar ti cu l ar p u sh er n acell e of gr ea t er r ela t i ve b od y area t h e m
a n d t h e exp er i
en t a l m
ean p i t ch by
effi ci ency wa s ra i sed by 3 p er cen t
9 p er cen t
In t h e p r esen t st a t e of kn owl ed g e i t wi ll p rob a bly b e su fli c1en t t o
a d e on an ai rscr ew alon e can b e a pp li ed to th e
e th a t cal cula tio ns m
ass u m
is
.
.
.
.
.
m
,
.
a i rscrew i n p l ace o n an aer o p l an e by ch an gi n g t h e scal e of
V
by 5p er
-
nD
t
cen
.
i ncr easing t h e or di nat es of t h e t h ru s t co effi ci en t an d efli ci ency curv es
by 2 p er cen t Th ese ch anges ar e s all , an d gr ea t a ccu racy i s th erefore
n ot r e qui r ed i n t h e p r acti cal app li cations of a irscrew d esign
an d
.
m
.
AIR SCREWS
Ap r no x
m
ar ros s r o
Ara s c a nw Cn a nx cr n
m
s r l cs
B efo re p r oceedi n g t o t h e d etai l ed d esign of a n a i rscr ew i t i s n ec essa ry
t o k n o w t h e g en era l p rop or ti ons of t h e bla d es , and t h e s ecti on s t o b e used
T h es e a re a t t h e ch oi ce of any d es ign er , who will ade p t s tan d ard s of hi s
it ed th a t rou gh g en era liz a
o wn , but t h e choi ce for g ood d esign i s s o li
.
m
,
a d e for a ll ai rscrew s
ti ons ca n b e m
Th e p lan f ormo f t h e bl ad es i s
p erh ap s t h e qu an tity whi ch v ari es m
os t i n any d esign an d i n co nn ec tio n
u l a a n d cur v es i s gi v en a dra win g of t h e
a t e f orm
wi th t h e app roxi m
p l an fo rmto whi ch th ey m
or e p a r ti cul ar ly re fer ( see Fig
.
,
.
thi s sh ap e of bl a d e H C Wa tt s h as gi v en a nom
og r a m
conn ecti ng
t h e a i rscr ew di am
et er of t h e m
ost effi ci en t ai rscr ew s with t h e h orsep ow er
s p eed of t ran s l a tion an d ra t e o f r ot a ti on
For
.
.
,
.
AIR SCREWS
P
D
an
9
two oth ers d enot ed by To ari d Q0
mb
To i s a n u
.
uh th at T le
er s c
o ,
e v alu e of
t , an d
im
ilarly QJc,
for t h e sam
I
3
1
?
fin ed on p 306
a re t h e u su al a b solut e t h r us t an d t or qu e coeffi ci en t s a s d e
p l e a fu r th er not e i s r eq u ir ed ; i t can
To a pp ly t h e curv es to t h e ex am
7
3
wh en
kQ
d
$
an d s
.
0
03
02
Fro 150
.
be
ra
D
ti o
o
e
the
i
y
a e
ax
u
nP
e
is
a
e
y
c en c
.
v al u e
y
b ut
a o
p lane ai rscrew
ccu r a t a
ti ng t h e effi ci ency
c enc
of an aer o
o
on
.
09
a nda rd ai rscre w char acte rfii t ics .
ca c
.
v alu e of X
md t o
m i mu mffi i
ax
—St
05
m 159 by l ul
f 0 7 t h m i mmffi i
d ed uced fro Fig
e ffi c en c
is
.
of
V
7
1?
u rs
occ
0 65
.
as
high
will oft en b e
e
For a p it ch
as
a
d iam
et er
= 0 6 whil s t
a b out l
°
”
In
mwh
so
at
.
ord er to
P
,
k eep t h e
p ossibl e t h e
a
v erag e
m i mump
ax
for
s
ee
d
t gr ea t er th an th a t gi vi ng
or e
mp
v en 0 75at m
u
ax i m
Y
S ee
d
.
APP L IED AERODYN AMICS
Con tin ui ng t h e
pl
m
e,
exa
i t i s th en foun d tha t
10 5
v
a?
an d
th erefor e P
15fee t
,
P
an d
Calcu lati on of To and
08 0, t h e th r u s t i s fou n d fro
—0 7
—
P
15
f)
18
say
—Th e effi ci ency h av i ng been foun d to
mt h e horsepower availa ble Sin ce
be
.
400 x 0 80 x
thr ust
1000 lb s
120 X 88
0 0023 7 x 176
2
x 13
.
3
Th ese fi gur es will dep en d on t h e air d ensity both T an d k, _b ein g
a ffec t e d
Th e h orsep ow er a v ail a bl e for a gi v en th ro ttl e p ositi on et c v a r i es
o re r a p i d ly th an d en s ity a nd h en ce t h e th ru s t v a ri es ra p i d ly wi th
r a th er m
d en sity
It , i n v ol v es t h e ra ti o of h orsep ow er t o d en sity an d i s not th er e
t ly alt er e d
Grou n d con ditions o f d ensit y an d h ors ep ow er m
a
y
"
e
a
a t e exp r e ss i ons for T0 a n d Q
th erefor e alway s b e u sed i n t h pp roxi m
V
0 7 i s seen t o be 0 63 5 a n d
F r omFig 159 t h e v al u e of T ole, a t —
,
.
,
.
,
,
,
.
.
.
.
h en e
nP
63 5
2——
T
00808
400 X
t or q u e
F rom
Fig
0 805
i
the
md
a
p l anes
a er o
v alu e of o qa t
0 805
e of
all
is
rea
d
O ff
61 5
.
00 13 ]
v al u es
0 7 an d
,
’
of
an d
y
i zD
Q0 i n thi s way
a r e r ea
ati ons
th ese app rox i m
in
h
the
,
t i ti cs
c ara c er s
d ily d ed u ced fromFig
an a
lysi ng
the
.
of
159
p erform
an ce
.
of
.
I V Fonos s
.
at
3
1
3
x
0 0023 7
D
w
.
2100
vi ng d et erm
i ned P P To
a rs cr e
Use i s
159 t h e
H ence Q0
.
Ha
.
2100 lbs -ft
6 2 8 x 1000
k“
as
,
Ara s cns w
O N AN
m
m
wa res
ran
rs n o r
mm A
ov
xra nnv r n a o u o n
e
Ara
Mo d ifications of for ul ae a lr ea dy d ev elop ed will b e cons i d ered i n
otion of t h e a ir scr ew r el a ti v e to t h e ai r u n
or d er t o co v er n on a xi a l
d i s tu r b ed by i t s p resence It i s necessary to i n t ro d u ce a sy st e of a x es
as be low
Th e axi s of X will b e t aken along t h e ai rscrew a xi s and i n r el a tion t o
Fi g 160 i s d i rect ed in t o t h e p ap er Th e v elocity of t h e ai rscrew p erp en
-
m
.
.
,
.
.
AIRSCREWS
dicu lar
to
the X
a
xi s i s
v, an d
t he
m
a
x is of Y i s ch osen
m
so as
to i nclu d e
m
Th e on ly n ew ass u p tion to b e ad e i s tha t t h e co p on en t of 0 a l ong
t h e a i rscrew bla d e i s without app reci abl e effec t on t h e fo rce on it
otion i s
Th e v elocity of t h e el e en t AB d u e to r ot a tion an d l at era l
a d e u p o f t h e cons t an t p ar t our an d a v ari a bl e p ar t —11 cos 0, an d co
p a r is o n with Fig 149 sugg es t s t h e w ritin g of t h e r esu lt an t v elocity no r al
t o t h e X ax is i n t h e for
m
m
m
.
0
cos
sa
to
m i
mk
e as n
a
0 now
m
.
takes
t h e case for
t he
whi ch
as
e an
p l ace
of
o u p t o the
t h e v alu e of
ez
to
furt her p roced ur e
Th e
a 2.
p oi n t
0 2,
i
.
m
m
is the
whi ch i t was necessary
un til t h e com
p l eti on of
at
e.
F111 160
.
ot ation al in fl ow p reviously i nclu d ed wi ll
u nim
po rtan t i n t h e present connection )
es
Equ a tio n ( 14 ) beco m
h
T
( e
as
.
r
be
ignored h ere
.
Thi s equation
my b
a
1_
e
w ri tt en
t an 41
as
l
,
1
(l l
v
«
of
V
"
v
gi v en v alu es of 0 t h e corr esp ondi ng v alu es
ca l cu l at ed an d are gi v en i n T a bl e 1
For
For t h e
p ur p oses of ill us t rati on
,
18
taken
as
of a ,
08 40,
so
h av e b een
th a t
ults
res
m1
AIRSCREWS
t a ken from
T a bl e 18 an d t h e 3 r d and 4 t h colu m
ns
a r e th en r ea d fr o m
Fig 161 Thi s fi gur e show s kn and kn as d ep en d en t
o n a n gl e of i nci d en ce a n d a g r ees wi th t h e v a lu es of T a bl e 1
an d ( 4 7) ar e d ed u ced t h e exp ress i ons
C o lu
ns
an d
2 a re
m
.
.
g
PV
and
)
see
z
cr
a s . cos
s
d
'
z
aco
dr
fro mt h e v alu es
in
T a bl es ( 18)
.
“
q
l
,
a nd
Ti kt
r
1
9
t
h
e
( )
k. si n
m
k,
a
r
a;
1
6
( )
¢ol (62)
005
igh t h an d si des of th ese
-
ANG L E O F i N C ID EN C E
( D EG R E ES )
Fro 16 1
.
.
ex
c
p ressions
are eva
om
p ara ti v e v alu es
an d
lu ated
of
s
10 5r esp ec ti v ely
to
c
formt h e
d
dr
1
1
18
of T a bl e
1
s
cr
dr
for t h e
a
xi al
19
Th e
.
motio
n
~
er s
.
t a bl e shows th at du e t o a n i nclin ation of
t h e thr us t on t h e
bl a d e el em
en t v a ri es from
74 p er cen t to 126 p er cen t of i t s v al u e wh en
en t ary t or qu e r ang es b etw een 79 p er cen t a n d
movi ng axi ally Th e elem
123 p er cen t of i t s a xia l v a l u e
Th ere a re seen to b e a pp reci a bl e fl u ct u ations of th ru s t an d tor qu e on
i n ed
e ach bla d e d ur i n g a r ev olu tion of t h e a i rscrew w h i ch n eed to b e ex am
Th e
,
.
.
.
.
.
.
f u rth er
1 19
(
.
r
2
p
t
l
mt
r e resen s an e e
y force acti ng on
en ar
h bla d e of
eac
a
APP L IED AERODYN AMI CS
t wo- bl a d ed
w norm
al to
i
a rscr e
mt
bo tto mthi s
t he
b lad e at
th e
ele
to p i t
en ar
t h e dir ection
é
i s 1 290pV dr
dr ,
°
’
°
é
i s 0 828s
y force
n
th e
blad e
On t h e
.
whilst
t or qu e on
Th e
.
motio
of
the
at
O
ppo s i n g
two bl ad es
i s t h en
on
m
m
1 06p V 0dr as ag a i ns t t h e v alu
for a xi al
oti on Si i la r
res ult s follow for oth er po sitio ns , a nd for a i rs crew s wi th t wo or four b la d es
t h e v ari a tion of tor q u e with 0 i s seen to b e v ery s all
’
e 1 05
r
c
d
V
p
2
°
‘
m
low er bl ad e
On t h e
a
the
fo rce
0 828p V
'
g
’
.
t
t he
ac s in
dr
.
xi s O f Y whils t on t h e u pp er bla d e t h e force i s 1 290p V
°
the
g
’
,
di rection of
dr i n t h e O pp os i t e
—0 23p V ’cdr
on t h e p ai r o f
di rectio n Th ere i s th erefore a force dY
bla des ; thi s i s t h e sa e effect as w oul d be p du ced by a fin i n t h e p l ace
of t h e airscrew an d lyin g along t h e a xi s of X an d Z Su ch a fi n woul d O p p ose
a r esi st an ce to t h e n on a x i al m
oti on
m
m
.
°
,
.
-
Th e
th ru st on
bla d e i s 3
mp
'
17p V
the
.
low er bla d e el em
en t i s 1 8 1
pV
°
g
’
dr, t h e r esult an t
§
3
dn
an d on
th e
u p per
thr u st on t h e two blad es bei n g 2 49pV
'
’
ed r
d with 2 4 6pV cdr i n t h e axi a l m
o tion As for tor qu e i t a pp ears
th a t t h e eflect of l at eral m
oti o n on thr u st at any i nst an t i s v ery sm
al l
for two a n d four bl a d ed ai rscrew s
On t h e low er bla d e t h e thr us t gi v es a cou p l e a b out t h e axi s of Y of
ar e
as co
3
°
,
.
'
.
whils t
on
t he
u pp er bl a d e t h e cou p le i s
3 17pV
°
~
3
£
r dn
Th e
ult an t cou p l e i s th en
Th e low er bl a d e as illus t ra t ed in
Fig 160 wo u l d th en t en d to en ter t h e p ap er at a gr eat er rat e th an t h e
cen t r e
oti on for
Th e v a l u es of t h e di fl erences b etw een a xi a l an d n on axi a l m
t h e el em
en t of a si ngl e bl a d e are gi v en b elow as t h e result of calcul a ti on
fr omt he followin g form
ul ae
r es
,
.
,
.
'
-
0 525
pV
'
0rdr
)
05
25pv
cr dr
l 23p vs
r
z
‘
8M
2
‘
cos
0
si n
0
1 23pV cdr
z
°
m
m
e two bl a d es for t h e ai rscrew an d t h e di ffer ences
Th es e for ul ae as sum
fro a xi al oti on are u sed i nst ea d of t h e ac tu al forces d u rin g l a t era l
’
moti on ; 0 525pV crdr and 12 3s 0111 are t h e el e ent ary t orqu e an d
t h ru st on eac h bl ad e d u ri ng a xi al m
o tion
m
°
,
m
;
.
AIR SCREWS
m v lu gi v t t h
f 8T SQ 8Z
d 8N
v i ti
mll omp d wi th t h v
foot of T abl e 20 show th a t t h e a v erage
a r a on s o
an
as a r es ult o f n on axi al m
oti on are v ery
s
a
as c
e a erag e thr u st a n d tor q u e on t h e el em
en t
ar e
Th e la t era l force 8 Y i s a bo ut 4 p er cen t o f t h e thr u s t i n thi s ex am
p le
whi ls t t h e p it ch in g cou p l e SM is a bout 3 2 p er cent of t h e t or que T h ese
mean figur es app ly to any nu m
ber of bl a d es For v ar i a ti on s on two
bla d es d ur i ng rot ation t h e l ast si x colu m
ns of T a bl e ( 20) sh oul d b e in v ert ed
an d t h e fi g u r es a dd ed t o tho se th ere gi v en
For th r us t an d t or qu e t h e
e ffect is t o l eav e sm
e app li es to t h e
all di ffer en ces a t a ll an gl es
Th e sam
a l force SZ an d t h e y a win g cou p l e SN
n o rm
For t h e l a tera l force 8Y a nd
t h e p i t chi n g m
om
en t 8M t h e effec t i s t o d oubl e t h e fi gu r es a pp roxi
p ar e with d ou bl e thr ust an d tor q u e
a t ely an d th ese th en co m
Th e
ea n
es
a
,
en a
e
-
,
.
.
,
.
.
.
.
.
.
m
.
,
TAB L E 20
.
8Y
’
V
d
p
0 a n d 360
20
34 0
40
3 20
60
300
80
280
100
260
120
24 0
220
14 0
200
[ 60
AL
GZ
__
’
d
V
p
r
BN
p VW
r
’
V
a dr
p
'
—002
04 74
0 174
~
—0 700
—1000
v eragi ng of 8Y an d 814 wo u l d b e
ns di sp la ced by 90 i n 0 wo u l d th en
a pp r eci a bly be tt er s i n ce four colu m
b e add ed to p rov i d e t h e r esu lt an t
ay easily occur i n t h e n o r
a l r ang e of
An angl e of 10 as h ere u se d m
en t of v el ocity b ein g th en
hori zon t al fl ight of an aerop l an e t h e d i sp l acem
i n t h e v erti cal p l an e Th e neces sa ry ch a ng es of n ot a ti on b etw een Y a nd
Z M an d N can readily b e a d e For l at er a l st a bili ty t h e p r esent not ation
os t con v eni en t
is m
—
Ele ent to Ai rscrew
Th e r ep etition of t h e p re
I ntegrati on fro
For
a
four - bla d ed
i
w
t he
a rs cre
a
°
,
m
.
°
,
m
m m
y
m
d
ti
um
b of l
t
v lu of
f
p ov i d
yf d t m
i tio of t h to q
th u t l t l f
w
th t f
l mt m
o t of t h ff t f
xi l
im
p o t t d tt ti wi ll b d i t d to t h v lu ti
ym
mt y f t h figu f T bl 20 d th i g l
Th
gg t t h pp li bili ty f i m
pl f m
ul
l g
th
.
.
,
.
.
i g
ce d n
l ula
ca c
ons
or a n
er
e e
en s an
d a t a n ecessa r or e er na n
et c
on an a ir scr e
It h as b een seen a or an e e en
r an an
a
en on
otio n ar e u n
r es
of Y an d M
e s
e
o
e r
es s
e a
o s
ca
a pp ea ra nce s u
a ngl e of y aw d oes not exceed
all
the
.
u e,
r
e
es
a
r
r
s
,
a er a
es
orce,
.
,
m
.
s
e e
e
rec e
a
o
e
ec s o
or
e
n on -a
e e a
an
ae, so
e r
on
a
a
on
en era
as
e
AIRSCREWS
With th ese v alu es equ ation ( 69) l ea ds to
}
( 71)
A
eri cal f act or sh ou l d b e h al v e d b efor e co m
bl a d e t h e n um
p ar is on
wi th colu m
T able 20 Th e si m
p l e exp r ession
gi v es results i n
n 4
goo d agreem
en t wi th tho se of T a bl e 20
r
r
r scr ewm
6
5
r
n
f
f
n
x
es
s
o
o
t
h
e
a
i
Fr o m
d
th
e
e
p
i
M
o
s
t
6
9
a
a
b
e
w
r
i
t
e
n
a
(
)
)
(
y
For each
,
.
,
.
“
Th e
a
d ep ends
k,
,
i 0
41
7
v erage v alu e of M
on
the
with X
wr
a
l tion
half
is
cos
)
¢
kn
i $50
S ll
0
dkn
”
8 111
da
cos
40
960
}
mxi mum
the
re
is
re a
960
a
.
m
v a lu e
Th e
of
M
vp V
v ol u tion ch i efly b ecause of t h e v ari ation of
dvance p er
Th e
.
309
,
n ot
so si
m
e as
to
be
ob v i ously d ed u cibl e
s ch an g e 1n Opp osit e di rec ti ons
fro m(7 s in ce t h e im
p o r tan t t erm
T rea ti n g t h e tor qu e equa tion (63) ma si m
ila r way t o tha t follow ed
th r us t will gi v e t h e la t era l force
.
1
2
7V
1
V
008
8
w’
.
V
.
z
8 00
for
¢o
00 z 8 8 02
t
a
s
n
i
n 0
0
08 0
k
0
5
0
0
0
0
0
l)
0
0)
2
V 8 00 ¢o
I
d
——i ‘ si n 00
5
cos ¢O
k; cos $5
i
n
s
k
n
0
9
t an
i
0}
”
wr
'
v
dh
8in
.
dc
dkn
d,
With
the
v alu es gi v en
in
o
c nn ec
l ea d s to
1
s
cr
tion wi th p i tchi ng
dQ
8
mom t
0 22 cos 0
7
6;
en
2
¢o
60
cos 5
8
7
( 1
,
qu a tion (73 )
e
4
7
( )
eri cal f actor s houl d b e h a l v ed . Com
i gl e bla d e t h e n um
p ar ed
with colum
n 5 of T a bl e 20 t h e r es ult s wi ll a gai n b e foun d to b e i n
en t
goo d agreem
Th e v a lu e of t h e l a t er al for ce Y on t h e whol e ai rscrew i s
a nd
for
a s n
.
Y—
fiv
m
n
-
v
008
2
0
(
kn
1
00 8
¢o
4 »
4
da
an d
the
a
v erag e va l u e i s again h alf
the
mxi mum
a
.
m
m
m
m
m
w
ati on of Ia teral Force on an l nclined Ai ri
Experi cntal Deter
—Th e exp eri en ts whi ch led to t h e cur v es of Fi g 162 w ere obtained on
a Sp eci a l b a l an ce i n on e of t h e win d cha nn els of t h e N a ti ona l Phy s i ca l
Th e a i rs crew was 2 f ee t i n d i a ete r , b u t t h e result s h a v e
L a bo ra tory
b een exp ressed i n a for whi ch i s i n d ep en d en t of t h e si z e of t h e airscrew
i n acco r d ance w ith t h e p rin ci p l es of dy na i ca l si il arity
Th e o r di n a t es of t h e cur v es of Fig 162 ar e t h e v alu es of t h e l a teral f o rce
’
2
on t h e a i rscr ew d i vi d e d by p V D ex cep t for on e c ur v e whi ch shows t h e
2 2
thr us t di v i d ed by p V D t o on e tenth i t s t ru e scal e Th e n u b er of degrees
t he
s h own t o t h e l eft of each of t h e cur v es i n di ca t es
o
ai rscr ew a x i s was i ncli n ed t o t h e d ir ec tion of r ela ti v e
.
.
m
m m
m
.
.
.
.
m
-
.
m
=O
.
4 66
-
o 2 ee
.
06
fo un d t o b e
o rdi na t es for t h e d ifferen t a ngl es of y aw
n ea rly p rop or ti on a l t o t h e ra tio of l a t er a l v elocity t o
v elocity i
to 31
Th e
v a lu es
,
at
.
s
.
{
high v alu es of
1ncr ea ses
AS
t ak e
a
g rea tly
Th e ch ang e of
$5
,
an d
i n all
ax
as
nD
e
u
l at er al force co effi ci en t with
ca ses
the
ra
ti o
of
$3
1
i l
ax a
mll
is s
a
l a t eral force to thrust
V
1ncreases
mpl of t h mg
t m i mmp d
a n exa
be
the
of
e
S
ee
a
n
.
itu d e
of
t he
l a t era l for ce
for
fl yi n g Sp eeds
A IR S CRE WS
l)
= 9 ft V = 160 ft
.
s.
-
1
0
9
(
Th e l a t er al fo rce 1s 48 lbs
At t h e Sp ee d of cli bi ng
m
ft
Th e la t era l force i s 28 lbs
.
.
,
.
and
,
s
.
mp
.
X = 0 75and angle of y aw = 10
.
fi
the
thr us t
655lbs
.
V
68
(
an d
°
‘
D
t h e t h ru s t SI 5lbs
.
m
V TB S Sr a s s s s s 11s A s cs s w B LAD ES
ore i m
p ortan t s tresses i n an ai rscrew bl a d e are du e to ben din g
Th e m
u n d er t h e co m
bin ed acti on of ai r forces an d cen t r ifu gal forces an d t h e
d i rect effect s of cen t r ifugal force in p r od u cin g t ens ion B oth typ es of
s t r ess ar e d ea lt wi th by s t r ai ghtfor w ar d a pp li ca tions of t h e en gin eer s
th eo ry of t h e strengt h of bea s Recen tly a tt en tio n h as b een p ai d to
t orsi ona l st res ses an d to t h e twis t i ng of t h e bl a d es but t h e cal cu la ti on s
r eq
uire ore el a borat e th eo ri es of st ress Th e p rogress a d e althou gh
co ns i d era bl e h as n ot y et h a d an y app r eci a bl e effec t on d es ign an d t h e
im
p ort ance of t orsi onal stresses i s not y et acc ura t ely esti at ed A fu r th er
ser i es of cal cul a tio n s d ea l s with t h e r es onan ce of t h e n a t u r al p er io d s of a n
a i r scr ew bl a d e with p er io d s of d i stur b an ce an d on e g en era l th eo re
of
im
p ort ance h as been d ed u ced It s t at es th at t h e natural fr equ ency of
v i b r a tion of an ai rscr ew bl a d e m
u s t be hi gh er th an i t s p er io d of r ot a ti on
a n d th a t a s a co nseq u ence r es o n an ce can o n ly o ccur f r om
ca us es not con
nec t ed with i t s own r ot a tion
Th e cal culation of st r esses du e to ben d i n g an d cen t rifuga l force will
e d et a il but torsi on a n d r eso n an ce wi ll n ot b e fur th er
be d ealt with i n so m
t rea t ed AS a general rul e i t ay be sai d th at t h e ev i d ence i n re l a ti on
al
to a ir screws of n orm
d esign i s th a t t h e t wi stin g i s n ot d efi nit ely
dis cernibl e i n t h e aero dy n a i cs b u t a pp ear s occasiona lly i n t h e sp littin g
of t h e bla d es Th e flexur e of t h e bla d e u n d er t h e i nfl u ence of t hr us t
i s s uffi ci en t to i n t ro d u ce an app r eci a bl e co u p l e as t h e r esu lt of t h e
d efl ec ti on an d cent rifugal force
—Th e bl a d e of an airscrew i s
B endi ng Mo ents du e to Ai r For
twi sted a n d t h e ai r forces acti n g on it a t v ariou s ra d ii h av e r es u lt an t s
lyin g i n d i fferen t p la nes As each secti on i s chosen of aerofoil for on e
en t s of in erti a of t h e s ec tio n i s sm
a ll a s co m
of t h e m
om
p ared with t h e oth er
an d i t i s s u ffi ci en t t o con s i d er t h e b en d i n g whi ch o ccurs a b ou t a n ax i s of
inerti a through t h e cen t re of ar ea of a sec ti on an d p arall el t o t h e ch or d
Th e r esol u tio n of t h e ai r forces p r esen ts no p ar ti cu l ar di fli cu lt y an d t h e
d et ail s a r e gi v en bel ow All t h e ai r forces on el e en t s b etw een t h e ti p
of a bl ad e an d t h e secti on ch osen for ca l cu l ation ent er i n to t h e b en din g
momen t and it i s necessary t o h av e a d istingui shin g notation for di flerent
For thi s p ur p os e d a sh es h a v e b een a dded t o l ett ers to sign i fy
sec ti on s
u se i n co nn ecti on with t h e b ase el em
om
en t for whi ch t h e
en t i s b ein g
.
.
’
m
.
m
,
,
.
,
m
m
,
,
.
m
,
.
,
.
,
.
,
m
m
,
.
m
.
m
m
,
.
,
.
m
.
'
,
m
mf omt h
.
u l ae r equi re d foll ow i n m
for m
os t con v eni en t for
r
p ressions for th ru st and tor qu e as th ese a d m
i t of r ea dy a dditi on
Th e
,
e ex
for t h e
AIRS CR E WS
F or t h e
fou n d
2r
p ar ti cu l ar v alu es of
p o d u ce d fr omT abl e 9
re r
D
hosen
c
Th e
.
,
the
whol e of colum
n 2 will
valu e of ¢o for
g
:
0880i s
be
gi v en in
T a bl e 8 as
d egr ees an d t h e oth er v alu es w ere t aken fro mt h e
si
ilar ta bl es n ot rep rod u ced Si m
ilar rem
ar ks app ly to 1
> an d
t
1
q y
a s sh own at t h e foot of t h e ta bl e an d t h e l as t col u
n of T a bl e 21 i s
obta in ed fromt rigonom
et ri ca l t a bl es
m
,
m
.
,
.
TAB LE 22
.
m
2 x 8 x 5t o vs
ele ent al
e uat i on ( 79
“1,
q
0 3 24
-
01 12
0 602
04 20
04 160
0 880
l m
en t s of t h e i n t egr a l of 79 are ca l cu la t ed a s i n T a bl e 22 fr om
( )
v alu es ex t rac t ed fromT a bl e 21 Th e p r ocesses are si m
p l e an d call for n o
men t Th e valu es i n column 6 of Ta bl e 22 are p l ott ed as
Sp eci al co m
Th e
e e
.
o r di nat es
the
by
in Fi g 163
v alu es
th e
.
of
t he
m
id o di
-
r
,
with
i n t egr al
t
2
5
-
at
as a
t he
mtho d
u v mk
na e
e
b sci ssae
Th e
v ari ou s v alu es
Th e
.
.
a r ea s of
m
an d
of
v alu es whi ch
,
th ese cu rv es gi v e
p
w ere o bta i ned
r e r esen
t
M
.
pl
l3
fr
2’
V
i n Fig 168
hown in t h e c r e ar ed
Si nce
om
en t fact or
all t h e cu r v e t en d s
t h e a i r forces on t h e bl a d e near t h e cen t r e are sm
t o b ecom
e s t ra ight as t h e r a d i u s d ecr eas es an d for p racti ca l p ur p os es
Th e v alu es
ay b e ex t rap ol a t ed i n accor d a nce wi th thi s ob serv a tio n
m
ar e S
“
.
.
.
of
t he
i nt egral d enot ed by
,
eg)
P
h wn
ar e s o
in
T abl e
24 ,
but before
A PPL IED AERODYNA M I CS
u se can
t he
md
mom t
be
ar ea ,
a
ate
th emt o cal cu l at e s t resses i t i s necessary to es ti m
of i n er ti a an d d is ta nce t o ou t s i d e fi b res for each o f t h e
e of
en
0 5
0 1
Z
FI G 163
.
.
08
0 7
03
r
0
C alcu la t ion of ben d ing st res ses due t o t h rust
—
.
ofoil secti ons used Th e v a lu es are gi v en i n T a bl e 28 i n t erm
s of
ch or d so a s t o b e app li ca bl e t o a i rs crew s o f d ifferen t bl a d e wi d th s
a er
.
t he
.
h
mmmm t f
md
i
ti (
’
cg
Area o f sect
h e]
.
Mi ni
ion
o
u
en
’
Di st ance
o
of
Iner a a x s assu e
p aralle l t o ch o rd )
.
.
u sed t o d en ot e t h e chor d to di stingui sh i t fr o mc
wh i ch i s t h e su mof t h e ch or d s of all t h e bl a d es If t h e n u m
b er of bl a d es
i s t wo t h e v al u e of M gi v en by ( 79) sh ou l d b e h al v ed whil s t for f ou r bl a d es
o n e qu ar t er of t h e v alu e shoul d b e t ak en
Usi n g t h e o r d i n ary engin eer s
exp ressio n t h e m
axi m
umstress du e t o bendin g i s
T a bl e
In
23 ,
c, i s
,
.
,
,
’
.
,
f
-
"
M
y
i
'
mc
s s nws
x
m
om
en t o f i n erti a an d y i s t h e di s t an ce to t h e ex t r e e
wh ere I i s t h e m
fib re
U sin g ( 80) i n conj u nction with ( 79) an d d en otin g th e in t egral of
.
l ea d s to
WGHZ
QT,
b ei ng
ci en t of
It ,
as
the
t he
i
coe ffi c en
in
su
Ta bl e 24
l
co u
mf
n
.
h o rd s
c
‘
D
m
i n col u n 4 o f T a bl e 28 an d k, t h e coefli
eani n g
Th e 0 of eq u a ti on (8 1) h a s i t s u su al
o f t h e bl a d es
Ev a l u ati on of ( 8 1) l ea d s t o
o f c,
m3
t he
o
t
P
“
m
,
.
.
TAB LE 24
.
[M
‘
°
m
g
ft
Co
~
res alvc
Ts
an
lb s- ver se In
Th e
21
v al u es of :an d
1
3
(%
is t h e
F
or
d i na t e
of
{5
of
t he
m
T a bl e
cu r
ve
24
in
are
Fig
.
m[l
'
.
m
o st r u |
M
“
’
t aken d i rectly fromT a bl e
163
th e
at
p re p er v al u e
7
.
of
m
k!
an d
i s d ed uced fro
T a bl e 23 Th e fifth col u n of T abl e 24 foll ow s
I} I
fro t h e figures i n t h e p r ev iou s col u n an d e q u a tion
B efor e t h e
res u lt s ca n b e i n t er p r et ed n u
eri ca lly i t i s f u r th er n ece s s a ry t o kn ow
-s
=
=
=
n
f
a
n
d
co
lu
n
s
a
n
d
a
r
c
a
l
u
l
a
t
d
f
r
o
0
0
23
a
d
V
1
4
t
6
e
c
e
7
o
7
7
s
p
e
s
t
1
o
r
t
h
e
p
p
r
ti
n
f
ti
h
o
s
e
n
e
t
n
il
e
s
t
ress
d
u
e
0
F
r
s
n
h
0
e
o
o
o
s
e
o
c
c
(
t o b en d in g i s t wo thi r d s of t h e co p r essi v e s t r ess
t h e ti p i nw ar d s for t h e fi rs t qu ar t er
Th e s t resses i ncr ease ra p i d ly fr o
or e sl owly , t h e high es t v al u e sh own b ei n g 2000 lbs
of t h e bl ad e an d th en
p er s q u ar e i n ch for t h e secti on neares t t h e cen t re for whi ch ca l cu l a ti on s
.
m
m
m
'
.
m
m
-
m
,
m
.
.
.
h a v e b een m
ad e
It i s i m
p or t an t to not e th at t h e st ress i n t h e airscrew h as b een cal cu l at ed
.
witho u t
fi xi n g i t s
d i am
et er
.
Si nce i n t h e cal cul a tions
,
by hy p oth esi s t h e ch oi ce of V 1s equi v al en t to a
st res s d ep en d s on eith er V or n D
Th e l att er
of d i fferen t d i am
et er s i s p rop or ti on a l to t h e ti p
,
.
hown X
s
,
’
D
ch oi ce of 11D an d t h e
qu an tity for a i rscrew s
sp eed a n d h en ce t h e
11
,
,
o lusio n i s rea ch ed th at for t h e sam
e ti p Sp ee d an d v a lu e o f
c nc
i s fi x ed
V
73
11
t he
t
s ress
A IRS CRE W S
2T
D
of
2
)
as a
b sci ssa
fi n d i ng
r
a eas,
.
Th e
i n t egr al w a s o bt ai n e d by
t h e v alu e
of
the
t he
mi d o d i
-
i n t egra l b ei n g z er o a t
the
r
ti p
t
na e
of
mth
e
th e
o
d
bl a d es
0 00 1
o
0
5
21
0 7
(0 V
FI G 164
.
wh e re
of
2r
D
1
.
.
-
Ca lcu la t i on
F romt h e cu rv e
T a bl e 25w ere rea d
of ce nt r i fu ga l s t res ses .
111t eg ra l
for t h e
t he
v a l u es
l
i n co u
m6
n
o ff
.
2'
( D)
(D)
b‘ s
w
42 lbs p er
.
I Valu e
ol
lnt es ral
u a t ion
q
'
St rcss lbs
of e
ubi c foot ( waln ut)
c
.
.
m
d i rect st ress du e t o cen t ri fu gal force can be cal c u l a t ed fr o eq u a ti on
d
t
h
e
r
e
s
o
f
a
5
n
fi
e
2
h
gu
T
bl
e
s
t
of
o
t
il
a
6
T
re
s
s
i
8
s
c
u
rs
e
e
s
n
e a nd i s
)
(
a dd iti v e to t h e s t r es s ca l cul a t e d a n d sh o wn i n colu
n 7 of T a bl e 24
bi ned st ress i s 2300 lb s per s qi a a nd 3 000 lbs p er sqi n i s
The com
This v al u e w oul d be
not rega rd e d as a n ex cess i v e v a lu e fo r w a l n u t
ewh a t high er v al u e o f n D
reach ed for a so m
t he
m
.
.
.
.
,
.
.
.
.
,
.
.
APPL IED AEROD YNAM I CS
m
B endi ng Mo ents du e to Eccentri city 0! B lade Secti ons and Cen tri f ug a l
—I t will b e seen sh ortly t h a t as a r es u lt of cent rifuga l force t h e
For
b en di ng o en t s ar i si ng fro s all ecc ent ri city o f t h e ai rscrew sec t i o n s
fro
t h e a i rscr ew d i sc a re of a pp r eci a bl e
agnitu d e
Th e eccen t ri c i t i es
m
mm
m
mm
m
i d er ed will b e of co m
p ara bl e si ze with tho se p rod u ced by t h e d efl ec t i o n
o f t h e bl a d e u n d er t h e a cti on of th ru st
Th e cal cu l a ti ons a r e som
ew h a t
p l ex and will b e i llust ra t ed by a d i rec t exam
p l e whi ch a ssu m
com
es t h e
v al u es of t h e eccen t ri citi es Th e m
o re p racti ca l p robl em
i n v ol v es p r oc es s e s
of t r i a l an d error for com
p l et e su ccess
2
As t h e a rea of t h e sec tion o f a bl a d e a t ra di u s r i s i k (c ) t h e ce n t r i
fug al f or ce o bt ai ned fromeq u a tion (8 6) 18
.
co ns
.
,
.
.
’
'
3
9
'
”
)
24
i
t
5
h
ers)
d
e
C ons i d er n ow t h e cou p l es ac ti ng du e to cen t rifugal force if froms o m
p a ir of fi xed a xes t h e cc or d ina t es of t h e cent r es of ar ea of each s ec ti on
a ny on e o f t h es e
b e gi v en a s a:a n d y t h e perp en d i cul ar d i s t a nce p f rom
c ent r es of ar ea 0 11 t o t h e a xi s o f l eas t i n er ti a o f a n oth er i s
—
s
i
n
s
x
co
x
y
a
a
)
(
(
y
>
-
,
,
'
'
an
d t he
r esu
712
M
P
C
—
__
P
p
o
en
at
the
sec
tion d enot ed by d ash es i s
1
°
2
V D3
8 p
Th e
e
mm t
lt a n t
,
f ormof ( 89)
h as
been ch osen
for
co n
v eni ence
mp
o f co
i o
wi t h
l
l t d
ar s n
q u atio n
2
Gi v en
from( 89)
a: a nd
a nd
y
as
f u ncti on s of
(g)
d at a p r ev i ou sly gi v en
the
As
a n e xa
mp l
e
the
v al u es
of
011 a 12 ft 6 i n s
.
.
i;
ca n
be
ca cu a e
.
TAB LE 26
t
v a lu e of MC,
.
h av e b ee n t a ken
e t e r ai r scr ew
di a m
as
t he
t he
o ne-
hu n d r ed th
t i ity
eccen r c
du e t o
A IRSCREWS
d e s ign a n d d efl ecti on u n d er lo a d w ou l d b e
i n s a t t h e ti p of t h e bl a d es
ay ea s ily occ u r i n p racti ce
E cc en t ri citi es o f g rea t er a ou n t
Th e v a l u e
of
as
e en t a k e n as z ero e v e rywh er e
h
b
T
a
bl
e
h
w
t
d
t
y
s
2
h
a
6
o
s
e
a
n
ss
a
r
ec
e
y
fo r t h e ca l c u l a ti o n o f o en t s fr o e q u a ti on
Th e d et a il s a r e gi v en
b e l ow i n T a bl e 27
m
m
m
mm
.
.
.
.
.
TAB L E 27
.
miii 3 (i
C‘” 9 “
i nta zr
—o 4 c x
0 105
e a se
e
c
-
~
t
1
1 se)
,0
-
.
1
0 9 15
0-3 62
0 94 9
-
Th e
v a lu es gi v en i n
l
co u
m6
n
of
T a bl e
2
g
Fig 165with ( )
i
.
g
as a
b sci ssa
.
For each
27
a re
p l ott e d
v a l u e of 2
a s or
d i n a t es
in
a
( 12)
th ere i s a sep arat e
v e t h e ar ea of whi ch i s req ui red F ou n d i n t h e u su a l way th ese ar ea s
i n t eg ra l cu rv e of Fig 165
a re p l ott e d t o gi v e t h e
To s how t h e r esu lts i n com
p a ri son with t h ose for b en d i n g d u e t o thr u s t
cu r
.
,
.
hown in T a bl e
as s
ta b u l a t e d
in
24 t h e
T a bl e 28
valu e
Mo,
of
.
vr
.
TAB LE 28
.
pV
2
3
D
.
h as
b een
l ul a t ed
ca c
a nd
AIR SCR EWS
Fo a n o
m
s s on
Arascas ws
s u oos sr s n B Y
m m
S
In t h e
th at 1 i s
$
fou n d
1
)
a con
m
xos s o r
s au
Dv
mm
ca n
rv
en t
d et ail ed t reat m
t he
cou r se o f
u a
Co xs
’
v eni en t v ari abl e
w th eory i t
of ai rscr e
It
.
m
h as
l o b een
a s
seen
b een
h as
th at
t he
d en sity of t h e a i r an d of t h e at eri al of t h e a i rscr ew are i m
p ort an t I n
d i scu ssi ng t h e forces on aerofoil s i t was sh own th a t both t h e v i scosi ty an d
ela s ti city of t h e ai r are p o ssibl e v ari a bl es whil s t con si d er a ti on of t h e
el as ti city of t h e ti m
b er occu rs as an it emi n t h e cal cula tion of d efl ections
a n d s t resses
It m
ay th en b e co nsi d er ed
i n su m
mary th at t h e vari abl es worth
con si d er a ti on ar e
V z —t h e forw ar d v el ocity of t h e ai rscrew
n a
th e r ot a ti on a l Sp eed
D E t h e d i am
e t er
E
t h e ai r d en sity
p
.
,
.
,
,
.
.
.
.
w
d en sity of t h e m
at eri a l of t h e a i rs cr ew
the
9
a a
EE
.
v elocity of sou n d i n ai r as rep resen ti n g i t s el asti city
You ng s o d ul u s for t h e at er i a l of t h e a irscrew
the
m
m
'
.
qu an titi es thru st t orq u e effi ci en cy
d ep end on a fu nction of fi ve v a ri a bl es of whi ch
All t h e
,
,
.
,
t
s r ess
,
t i
an d
th en
s ra n
,
VD
v
(T
F
’
,
,
my b
a
e
t a ken
a nd
is t h e
ta ken i n
is
ty p i ca l
as
mo t
h
choo si n g
s
s
.
.
’
ar
g
p
gu m
en t
V
,
b
’
is
of
gr ea t i m
p ort ance
a
600
if thi s v i bl
ar a
e 13
.
ign ored
be
a
d i ng
.
Th e fi rs t
.
E
ance
t i ti c v ari a bl e of ai rs crew p erform
If care
o d el aerofoil and win d sp eed
a su ffi ci en t ly l arg e m
my
v
ft
a
v
1 w
c arac er s
2
-
V
700 ft
or
ig
nor e
d
.
Y b ecom
p or t an t
es i m
.
bu t
-s
Th e
.
om
p l et e
a
gu m
en t
ar
m
failur e
c
1
i
-
,
ly
on
at
occu rs a t
m
f si p ly
t t
s a es
P 9
ti p
1100
th a t
t h e ra ti o o f t h e d ensity of t h e
a t eri a l of t h e a i rscrew t o th a t of t h e
a i r a ffect s t h e p erfor
an ce
Sin ce th r u s t d ep en d s p ri a rily on p an d
m
t i fu gal
cen r
t h e t wo
5
,
is
h ed
reac
ly
in
it
be
is
s
ob viou s th a t
im
p ly
g d
re ar
to
l t d
re a e
i
f
mm t
o
en s
be
for ces from
an d
t
con s an
t
.
A si
mil
ar
1
3
”,
ad e
d ensity a n d el asti city of t h e m
a t er i al s of whi ch ai rscr ew s a r e m
rar ely i n t ro du ce d i nt o t h e fo r
a t e r i al
u l a e o f p racti c e
Wh ere t h e m
Th e
ar e
on
ca u ses can on
l ion
c on c u s
force
w
m
.
p
m
.
APP L IE D AERODYN AMI CS
m
i s wood t h e ch oi ce h as b ee n b e tw een w a l nu t a n d
ah og any , a n d n eith er
'
t h e d en sity n or el as ti city a re a pp r eci a bly a t t h e ch oi ce of t h e d es ign e r
So e p rog ress h as b een a d e with e t a l a i rscrew s , a n d t h e s t res s es cau si ng
g r ea t es t di fii cu lt y are th ose l ea d i ng t o b u ck li ng of t h e th i n sh eet s u sed I n
ou n t it i s
o rd er t o r ed u ce t h e w eight o f a
e t a l a i rscrew t o a r eas on a bl e a
o b v i o u s th a t h oll ow con s t ru cti on
u s t be u sed an d th a t si ila rity of d esign
ca nno t cov er b o t h w oo d a n d
e t a l a ir s cr ew s
So e v ery Sp ec i a l a t er i a l s
su ch a s
h av e b een u sed i n a few cases a nd s i n ce t h e bl a d es
icar t a
are s o li d a nd h o
en t s fro
si
ila rity ight b e a pp li ed
og en eo u s , t h e a r gu
with t er s d ep en d i n g on d en sity a n d el as ti city
Mi ca rt a i s a p r e
a t eri al )
p ara t i on o f cott on fa b ri c t r ea t ed with ce en ti n g
o n fo r
s of e x p r essi on u s ed are
Th e co
m
m
m
m
m
mm m
m
m
m
m
m
T q
or
Effi
sa
m
l
foll ow s t h e
e v a u e of
u l)
m
mm
m m
.
,
pn D
3
p n D F2
y
F3
r ess
= na 3D 2F4
the
t t
s a e
t
m t th
s r ess
en
a
t
d ep en d s
'
(
5
ue
i
—
'l
3
3
( )
111
(
for
on
im
ilar ai rscrews work i ng
the
m
.
at
s
ti p sp eed
i s oth erwi se i nd ep end en t of t h e d i a et er
Th e
F , a n d F2 a re u su ally gi v e n u n d er t h e d escri p ti on o f
to rqu e coeffi ci en t s r es p ecti vely
an d
.
’
c en c
St
t he
m
m
m
m
.
.
Th ru s t
Fr om(94)
.
.
of
mi
t he
i
w
a rs cr e
,
l v al u es o f
a b sol u t e t h ru s t an d
nu
er ca
CHAPTER
M OTI ON
FL UI D
EX PE RI ME N TA L I nw
Tn s o
MATI C AL
s r na n o n s o s
m
s s es
V II
m Mo
AE RO D Y NA MI CS
R E MARK S
r ro w
A ND
a
on
Ma
m
a
no n v x a n xcs
F e nce s on aerep lanes and p art s of a er eplan es ar e consequ ences of m
oti on
a th em
ati cal knowl ed g e
th rough a v is cou s fl u i d t h e ai r and if ou r m
w er e su ffi ci en tly a dv an ced it wou l d be p ossi bl e t o cal cu l at e fr omfi rs t
p ri nci p l es t h e li ft an d d rag of a new wi ng form No su ccess h as y et b een
fr o m
t h e si m
p l est ass um
p ti ons
a tt ai n ed i n t h e analy si s of su ch a p ro bl em
ad e t o d ir ect exp eri m
en t s
a n d reco urs e i s at p resen t m
Th e vis cosity of
p ort an t i n i t s effect on m
oti on and as t h e effect d ep end s
ai r i s alw ay s i m
on t h e si ze of t h e obj ect it wi ll b e n ecessa ry t o d is cu ss t h e con d itions
ay
b e r ep r es ent ed by m
u n d er whi ch a i r craft m
o d el s Th e r el ati on
be tw een fl ui d m
il ar obj ect s i s so i m
p or t an t th a t a
o ti ons r ou n d s i m
Dy nam
i cal Si m
il ar ity
se p ara t e ch ap t er i s d ev ot ed t o it u n d er t h e h ea d
os t a er ody nam
It will be fou n d th a t for m
i cs connect ed with aere plan e
ay b e r egar d e d a s a n i n com
oti on ai r m
p r es sibl e fl ui d
ari d ai rshi p m
a t eri al on fl u i d m
oti on w h i ch th row s
Th e p r esen t ch ap t er con t ai ns m
It al so cov ers i n b ri ef r ésu m
s om
e li ght on t h e r es i st an ce of b od i es
é t he
it ati on s but
exi sti n g m
ati cal th eo ri es i n di catin g th ei r u ses a n d li m
ath em
a d e to d ev el op t h e th eo ri es of fl u i d m
oti on b ey on d t h e
p t is m
no a tt em
ea r li es t s t ag es as th ey can b e fou n d i n t h e s t a n d ar d w or ks on h y d r o
d yn am
i cs For exp eri m
ent al r easons t h e p h o t og rap h s sh ow n will r ef er t o
w at er It will b e fou n d th at a si m
p l e law wi ll ena bl e u s t o p as s from
moti on i n one fl u i d t o moti on i n any oth er and t h e anal ogy b etween
w at er a n d ai r i s ill u st rat ed by a st ri king ex am
p l e u n d er t h e t rea t m
en t of
il ar m
oti ons
si m
otio ns with wh i ch aer on auti cs i s
Whils t i t i s t ru e th a t t h e fl ui d m
d i rec tly concern ed a r e u nkn own i n d et ail th er e a r e n ev er th el ess som
e
t h e d i scu ss i on o f
oth ers whi ch can b e ca l cul a t e d wi th gr ea t a ccu r a cy
whi ch l ead s t o t h e i d ea s whi ch exp l a i n fail u re t o cal cu l at e i n t h e general
o ti on an d w h en t h e m
Fig 166 r ep r esen ts a cal cu l a bl e m
a th em
case
a ti ca l
th eory i s d ev el op ed l a t er i n t h e ch ap t er it i s carr i ed t o t h e s t age at whi ch
Fi g 166 is su b st an ti ally r ep rod u ced Th e p h ot ogr ap h was p r od u ced by a
meth od du e t o Professor Hel e Sh aw who ki ndly p roffered t h e loan of h is
a pp ara t u s for t h e p u rp ose o f t a ki ng t h e o rigi n a l p h ot og rap h s of whi ch
—
6
F i gs 166 171 17 178 are r ep ro d u cti ons
Th e app ara t u s cons i s t s of t wo su b s t an ti al p l a t es of gl ass sep arat ed f rom
each o th er by ca rd boa r d on e or two h u n d r ed th s of a n i n ch thi ck
In Fig
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FL U ID MOT ION
m
m
Unsteady [ Od o —Th e root i d eas un d er lyi ng t h e u ns t ea d y oti on of
—
n
6
a fl u i d ar e far l ess si
p l e th an th ose for s tea d y otio
Figs 1 7 170 all
r e f er t o t h e sa
oti on , a n d y et th er e i s li ttl e ev i d en t conn ecti on b etw een
e
t h e fi gur es
An a tt e p t will n ow b e a d e t o t ra ce a co nn ecti on , an d w e
s t a r t with t h e d efi nitio ns s ugg es t e d by t h e i llus t ra ti ons
otion t h e p osition of each strea lin e
St rea Li n ea — In an un s tea dy
d ep en ds on t h e ti e I n all cases wi th whi ch we ar e concern ed i n aer o
d yn a i cs t h e p osi ti on of t h e s tr ea lin es i n t h e regi on of di s tur b ed flow
r e p ea t s a t d efin it e in t erv als , i e t h e flow i s p er i od i c
Th e p erio d i n Fig 167
c an b e seen to ex t en d ov er 13 or 14 p i c t u res
In p r o d u ci n g Fig 168 t h e
fl ow w as r ec or d ed by t h e oti on of s all oil dr 0p s, a n d n o l ess th an eighty
p erio ds w ere o b serv ed Th e cin e a t ogra p h p i ctu re for t h e b egin ni ng of
ea ch p er i od w as sel ec t e d an d p r oj ect ed on a screen whils t t h e li n es of fl ow
m
m
mm
m
m
m
.
m
m
m
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m
m
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m
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m
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m
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Fro 168
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FLO W
—I
n st a nt aneou s
dist r i bu t i on
of ve loci t y
in
an edd y
a rk e d a n d Fig 168 i s t h e r es ult of t h e s u p erp os itio n of 80 p i ctur es
w ere m
en t been p erfec t n on e of t h e li n es so p l ott e d
H a d t h e a cc ur acy of t h e exp er im
w ould h av e crossed each oth er As i t i s t h e crossi ngs d o n ot confu se t h e
fi gur e un til t h e edd i es h av e b rok en u p a pp r eci a bly
If ne w one p roceed s t o j oi n u p t h e lin es so th a t th ey becom
e co n tin u ou s
li nes St rea m
acr oss t h e p i ct u r e t h e r esu lt i s t h e p rod u c tion o f s t r eam
li n es h av e t h e p rop erty th a t a t t h e i ns t an t con si d ered t h e fl ui d i s ev ery
wh er e m
o v in g a l on g th em
Fig 169 sh ow s t h e general ru n of t h e s t r eaml i nes a t in t er va ls of one
t en th of a com
p l et e p eri od O nly fi ve di ag ram
s a re sh own si nce t h e
a i n i n g fi v e are o bt a in ed by r ev ers i n g t h e o th ers a b ou t t h e d i r ec ti on of
rem
moti on ; Fig 169 (j ) wou ld b e li ke Fig 169 (a) tu rned u p si d e d ow n and so on
Mos t of t h e s t rea mlin es foll ow a s i n u o u s p a th acros s t h e fi el d but occasi on
a lly be n d b ack u p on th em
s el v es ( Fig 169
Two p art s m
ay th en a pp ro ac h
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F L UID MOTION
h own an d so on Th ese closed st ream
s r ep r esen t v ort ex m
oti on and as
ewh a t ra i d ly d i ssi p a t ed
t h e v o r ti ces t r av el d ow n s t r eam
th ey ar e som
p
Fig 168 sh ows th a t t h e v elocity in si d e t h e v er t ex i s sm
all co m
p ared
wit h t h at of t h e free s t ream
Path s of Parti cl es — Fig _
170 sh ow s t h e p a th s follow ed by i n d i v i d u a l
p articl es across t h e fiel d of v i ew Unli ke s t reamlin es
p a th s of
p ar ticl es cr oss fr eq u en tly Som
e of t h e p ar ti cl es w er e n ot p i ck ed u p
by t h e ca m
era u n til well i n t h e fi el d of v i ew
I n on e ca se ( t h e l ow es t of
Fig 170) a p arti cl e h ad en t ered a v ort ex an d for four com
p l ete t u rns t ra v ell ed
s l owly aga i n s t t h e m
a i n s t r ea mwhi ch it t h en j oi n ed
Th e u pp er p ar t of
Fig 170 sh ow s a seri es of p ath s v aryi ng froma 100p t o a cu Sp for p arti cl es
a ll of w h i ch h a d p asse d clo se t o t h e cy li n d er
s
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Mo t i on of pa rt icles
FI G l 7o
.
-
.
of
fl ui d
in
a n ed d
y
To
p r od u ce th ese cu rv es it was only necessary t o exp ose t h e p l at e i n a
era d u r i ng t h e p a ssa g e of a st r ongly ill u m
ca m
ina ted oi l dr op a cross t h e
fi el d
Si nce o bser v a ti on of all oi l d r op s across t h e fi el d gi v es b oth s t r eam
lin es and p aths of p a r ti cle s one set of p i ct u r es u s t be d ed u cibl e
t h e oth er
fr o
B efor e p a th s of p ar ti cl es ca n b e obta i n ed by cal cu l a ti on
fr om
t h e st rea mlin es of Fig 169 t h e v el oc ity a t each p oin t of t h e s t rea m
li n es m
us t b e d ed u ced D r aw a line AB across Fig 169 as ind i cat ed ; t h e
q u an tity of fl ui d fl owi n g b etw een each of t h e st reamli n es b e i ng know n
t h e n um
ber rep r esen ti ng thi s q u a n tity ca n b e p l ott ed a ga i n st d i stances o f
t h e s t r ea
A Th e s le p e of t h e cu rv e so obt a in ed i s t h e v elo city
li nes from
a t r ight angl es t o AB
Si nce t h e resu lt an t v el ocity i s a l ong t h e s t r eam
li ne t h e com
p onent th en l ead s t o t h e ca l cu l a ti on of t h e r esu lt ant v el ocity
Th e cal cu l a ti on i s s im
ay n eed t o b e r ep ea t ed so m
p l e bu t m
any ti
es as
to b e l a bori ou s i n a ny sp eci fied i ns t an ce of fl u i d m
oti on
For t h e p r esen t
we only n eed t o see th a t Fig 169 gi v es n ot only t h e st ream
li nes bu t t h e
v el ociti es al ong th em
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APP L IED AERODYN AMI CS
F romFig 169 we can n ow cal cula te t h e p ath Of a p ar ticl e Star ti n g
a t C for i ns t a nce i n Fig 169 (a ) a sh or t li ne h as been dr aw n p arall el to t h e
n ear es t s tr ea m
lin e
This lin e rep resen t s t h e m
o v em
en t o f t h e p ar ti cl e i n
e in t er v a l b e tw een s u ccess i v e p i ct u r es
t h e tim
In t h e n ex t p i ctur e t h e
p oin t D has been chosen as t h e end of t h e firs t an d a noth er short li n e
d rawn and so on t h e whol e l ea di ng to t h e lin e CG of Fig 169
Fur th er
a pp li ca tion of t h e p r ocess wou l d com
p l et e t h e l oop Th e lin e CG i s i llus
t ra t i ve only since t h e v elocity a long each of t h e st r ea m
li nes was not
ca l cu la t ed ; it i s s uffi ci en t to show t h e conn ec tio n betw een t h e li n es O f
Fig 169 obt ain ed exp erim
en t a lly an d tho se Of Fig 170 al so d ed u ce d from
t h e sam
e exp er im
en t
Th ere are two s tan d ar d m
a ti ca l m
etho d s of p resen ting fl ui d
a th em
motion whi ch corresp ond with t h e difi er ences betw een streamli nes
an d
p a ths Of p ar ti cl es
en t lin e
Filam
s h av e b een so ca ll ed s in ce th ey ar e t h e
ent Li nea — Fila m
ins tan tan eou s formt ak en by a filam
en t O f fl u i d whi ch crosses t h e fi el d O f
di s tur b ed fl ow Th ey ar e t h e lin es sh own in Fi g 167 Th e col ou r in g m
a tt er
O f Fi g 167 was i n t ro d u ce d th rough sm
a ll hol es i n t h e s i d e of t h e cy li nd er
Th e whit e lin es th er efore rep resen t t h e for
t a k en by t h e lin e joini n g all
p ar ti cl es whi ch h av e a t any ti m
e p as sed by t h e sur fa ce of t h e cy lin d er
Th ey coul d b e d ed u ced fr omt h e p a th s O f p ar ti cl es by i sol a ti n g a ll t h e
p a th s p as sin g through one p oin t m
ar k in g on ea ch p a t h t h e p oin t co rr e
sp e n di ng wi th a gi v en ti m
e an d j oi ni ng t h e p oi n t s
o ti on it i s i m
I n exp eri m
p or t an t t o b ear
en t a l in v es tig a ti ons of fl u i d m
in m
i n d t h e p r op er ti es of fi l am
a tt er i s
en t li n es wh en g en era l col ou r i n g m
us ed Th e u se Of Oil drOp s p resen t s a far m
or e s uit a bl e lin e of exp eri m
en t a l
r es ea rch wh er e a tt em
a ti ca l
a th em
p t is m
a d e t o r el a t e exp eri m
en t a l an d m
methods
Al though eddyi ng m
oti on i s v ery com
mon i n fl u i ds it i s not t h e
un i v ersal con d ition i n a l arge m
p l es will b e gi v en of a com
ass
Two exam
p ar ison b etw een s t ea dy free fl ow an d t h e flow ill u st rat ed by P rof H el e
f flo w
etho d O
Sh a w 8 exp eri m
en t s
Th e qu es ti on will a ri se d o es t h e m
b etw een p l at e gl ass su r faces in di ca t e t h e only ty p e O f st ea d y flow Th ere
i s of cou rse n o ob v io u s reason why i t sh oul d
p le Of
As a fu rth er ex am
P r of H ol e Sh aw s m
eth o d o f illus t r a ti n g fl u i d m
otion t h e cas e Of a s t ru t
secti on will b e con si d er e d (Fig 171 O pp os it e p
It will b e n oti ced
th at t h e s t ream
s w er e q ui t e g en tly d i s t u r b ed by t h e p r es en ce of t h e
O b s t ru cti on
If we con si d er t h e fl ui d m
li n es
ov i ng b etw een t h e s t ream
s whi ch a re
a n d t h e si d e o f t h e m
od el i t will b e n oti ced th a t t h e s t r eam
wi d es t a h ea d of t h e m
od el gr a d u a lly n arrow to t h e cen t r e O f t h e s t ru t an d
th en agai n exp an d Th e fact th a t t h e col ou red b an ds keep th ei r p ositi on
a t all ti m
es m
eans th a t t h e sam
e am
ou n t of fl u i d p assi n g b e tw een a ny
p oi n t O f a st r eamli n e an d t h e s t ru t m
u s t al so p ass i nsi d e a ll o th er p oint s
o n t h e sam
e s t ream
li ne a nd b eca u se of t h e cons t ri ction t h e v elocity wi ll
b e gr ea t es t wh er e t h e s t r eam
i s n arrow es t a n d vi ce versd
It i s i n t eres ti ng t o com
p are Fig 171with an o th er figu re ill u st ra ti ng t h e
flo w o f w a t er r ou n d a s t ru t O f t h e sec ti on u sed fo r Fig 171 t h e flow n o t b ei ng
confined by p ara ll e l gl a s s p l a t es
li nes i n Fig 172 a re sh own as
Th e s t rea m
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'
‘“
”
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m
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’
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APP L IED AERODYN AMI CS
850
ov ed f r om st r eam
blu nt an d far r em
li ne to p rodu ce eddyi ng m
o ti on
Th e i nfl u en ce a t wor k t o p rod u ce thi s r esult i s t h e v i scous d r a g of t h e
w a t er o v er t h e su r face of t h e t wo sh eet s of p l a t e gl as s It i s o b v io u s
without p roof th a t thi s vi scou s d ra g will b e gr ea t er t h e cl oser t h e su rfaces
a r e to ea ch oth er a n d th a t on m
each oth er thi s essen ti al
o v i n g th em
far fr om
co ns t ra in t i s r ed u ced
I t i s n ot equ ally O b v i ou s th a t an i ncr ease of v el ocity
O f t h e fl ui d b etw een t h e p l at es h as t h e effec t o f red u ci ng t h e co ns t r a i n t
b u t on t h e p rin ci p l es of dy nam
il arity t h e law i s d efi ni t e an d ad
i cal s i m
v an t ag e i s t aken of thi s f act i n p r o d u cin g Figs 176 an d 177 whi ch show
d i ffer en t m
oti ons for t h e sam
e O b st acl e
Th e p hotogra p h s t ak en by P rof essor H al e Sh aw s m
eth od Sh ow t h e
fl ow roun d a narrow r ect angl e p l aced across t h e streami n a p ara ll el si d ed
ch ann el
a d e su ch th a t a t low
Th e thi ckness of t h e w a t er fi lmwas m
v elociti es it was only j us t p ossibl e to p r odu ceFi g 176 whi ch show s s tr eam
s
b ehi n d t h e r ect angl e whi ch ar e sym
met ri cal with th ose in front Wit hou t
ch a ngi ng t h e app ar a tu s i n an
O f t h e fl u i d wa s v ery
v
l
ity
w
t
h
e
o
c
a
e
y
y
g reatly 1ncreased an d Fig 177 p r o d u ced I n fron t of t h e ob s tacl e car eful
e x am
in a ti on of t h e figur es 18 n ecessary i n or d er to d et ect d i fferen ces b et ween
Figs 176 a nd 177 b u t a t t h e b ack t h e ch ang e 18 O b v i ou s Th e fi rs t p oin ts
a t whi ch t h e di fi er en ce i s cl early m
ar k ed a re t h e fr on t corn ers O f t h e r ec t
a ngl e
Th e fl u i d i s m
ovi ng p as t t h e corn ers wi th s u ch high v eloc ity th a t
t h e con s t ra i n t of t h e gl ass p l a t es i s i n su ffi ci en t t o su pp ress t h e effect s of
in er ti a Th e fl ui d d oes not n ow cl ose i n b ehi n d t h e O b st acl e as b efore
a n d a n a pp r o ach t o
d ea d w a t er i s ev i d en t Th er e i s a w an t of d efi n iti on
i n t h e st ream
e m
s t o t h e r ea r whi ch seem
s t o i n d i ca t e so m
i xi ng O f t h e
cl ear a n d colour ed fl ui d s,but th er e 18 n o ev i d en ce O f e ddyin g
We are t hus
led t o consi d er thr ee d i sti nc t s t ag es O f fl ui d m
oti on
ar
1
e so gr ea t th a t
t
ea
d
otio
wh
t
o
v
i
sc
o
s
ity
S
m
n
e
r
e
t
h
f
r
s
d
u
e
o
e
c
e
( )
y
th ose d u e to i ner ti a a re i napp reci a bl e
n
d
n
r
a
2
e
a
o
s
o
s
e
S
t
d
y
otio
wh
f
v
i
ity
i
ti a ar e bo th
m
n
e
n
r
u
t
c
d
t
h
e
o
ce
s
e
( )
a pp r eci a bl e ; an d
e
o
o
n
e
n
t
h
e
r
t
a
t
d
y
ti
p
ibly
t
d
y
ti
wh
i
i
3
n
s
a
s
m
e
n
s
ea
n
a
U
m
o
o
n
d
os
( )
forces a r e l arge com
p ared with th ose du e to v i scosity
Th e ex t rem
e cas e O f ( 3 ) i s rep r es en t e d by t h e con v en tio n a l i n v i sci d fl ui d
a th em
of m
It i s
a ti cal th e ory wh er e t h e for ces d u e t o v i scos ity a r e z er o
n o t a littl e s u r p r i s i ng t o fi n d th a t t h e cal cu l a t ed s t re am
li nes for t h e s t ea dy
moti on of an i nv i sci d fl u i d ar e so nearly li ke those o bta i ned i n P rofessor
th em I t
H el e Sh a w 8 exp eri m
en t s a s to b e scarcely d i s ti ng u i sh a bl e f rom
n eed e d a m
a th em
a ti ca l a na ly si s by Si r Georg e St ok es t o sh ow th a t t h e v ery
e ca l cu l a ti on
Th e
d i ffer en t p hysi ca l cond iti on s sh ou l d l ea d t o t h e sa m
c om
mon cal cu la ti on ill u st rat es t h e i mp or t an t i d ea th a t ma th emati cal
method s d evel op ed for one p u rp ose may h av e a pp li ca ti on s i n a t otally
a th em
a ti ca l
d i ffer en t p hy si ca l sen se a n d t h e stu d en t of a dv anced m
p hy si cs fi n d s hi m
p or t an t tool a pp li c
self i n t h e p osses s i on O f a n i m
a bl e i n m
Thi s i s p erh ap s t h e chi ef a dva n tag e t o b e
any d i r e cti on s
O bta i n e d fr omt h e s t u d y of t h e m
ot i on o f a co n v en ti on a l i n v i s ci d fl u i d
B efo re con si d er i n g t h e th eory one fu r t h er ill u s t rati on fr omexp eri en t
will b e gi v en
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.
.
.
.
.
,
'
.
.
,
.
‘
.
.
.
,
,
.
.
-
’
.
.
,
,
.
,
.
,
m
.
F111 l 76
—Vi
flow rou nd s ect i on
Sh aw)
Low s pee d
scous
.
.
.
—Vi
Fi o 171
.
—Vi
.
.
flo w roun d s ect i on of flat p la to:
( ll ele
Sha w )
H i gh s peed
scous
—Visc
F lo N th
.
fla t pla t e ( H ele
s cous
.
Fro . l78
Of
flow
o us
.
rou nd
flow
wing
ro u nd
sect i on
( Hole Sha w )
-
wi ng s ect i on ( free
fl u i d)
.
.
D git z ed b y G o o le
g
l
Dig it iz ed by G oo e
g
APP L IED AE ROD YN AMI CS
8 52
bl ad es i n w a te r w ou l d be a vi ol a ti on O f th is ass um
p t i on A
cav it a ti on a r i ses fr omt h e p r es en ce of po i n t s O f v e ry low p res su r e
i
i s cl ea r th a t e v en in a hy p oth eti cal fl ui d no soluti on ca n b e acc ept e d
fo r w hi ch t h e p res s u r e a t any p o in t i s r e q u ir ed t o b e en orm
ou s an d
n ega ti v e
An i ns ta n ce o f t h is occ urs i n r el a ti on t o o n e o f t h e se n ti on s
for th e m
o ti on o f a n i n v i sci d fl u i d rou n d a p l a n e s u rf ace
Assu m
i n g co nti n u i ty an d in com
p r essib i li ty for t h e fl u i d i t i s ob v i o u s
ll
r
O
e
e
r
p p
.
f
.
’
.
.
,
F
m180
.
.
-
Flu id
mt i
o ns
o
de ve lo ped fr o
th a t t h e v el ocity O f ou t fl ow across
ca lli n g t h e v el ocity v w e h a v e
m
d s in ks in
so u rces a n
the
i l
c rc e
CPG
an
will
fl u id
in d
be
.
un i formand
,
m==m
2
( 1)
0
1
( )
so
th a t
th e
a ll er
es sm
v el ocity b ecom
f omt h
mll
th di t
i d i om
p
ibl fl i d w h t v th
i
d v l ity t
p
i
t
t
li
f
m
y
t mly u f l i p ti l hy d d y m
i
a nd s
a
er a s
e
s a n ce
r
c
i ncr ea ses
oti on o f any i n v i sc
For t h e m
nc
r es s
e
u
a e er
er e
a rel a ti on b e tw ee n t h e p res sur e a n
a
e oc
an
o n o a s r ea
ne
Th e eq u a ti on p rov ed l a t er i s e x r e e
se u
n
ra c ca
ro
na
cs
a n d i s on e p ar ti cu l ar fo rm
o f B ern ou lli s e q u a ti on
It s t at es th a t
s ou rce
.
,
,
,
s
.
,
’
.
1+9
)
mli
n es in
cons
t
2
( )
.
it v olu m
e of t h e fl u i d
We h a ve seen th a t
Fig 180 (a ) are ra di al li n es and fr om( I n) i t ap pear s th a t
wh er e p i s e q u a l t o t h e
t h e s t r ea
m
192
ass o f u n
.
.
,
F L UID MOTI O N
858
a t ely b eco m
lti m
es zero wh en r b eco m
es v ery g r ea t
thi s i s t r u e for all
t h e s t ream
lines i m
p ar ti ally If in (2) t h e v alue of c b e p ut equ al to zero
w h en r i s v ery gr ea t i t will b e seen th a t t h e
co ns t
on t h e r igh t h an d
s i d e i s t h e p r essur e of t h e st ream
a lo n g way fr o m
t h e sou rce and s in ce
t h i s i s t h e sam
e for all s t ream
lin es it follow s th a t ( 2) gi v es a r el a ti on
b e tw een p and v for any p oin t wh at ev er i n t h e fl ui d
Th e sam
e p r o p os itio n
i s t ru e for all m
otions of fri ctionless in com
p ressibl e fl u i d s i f t h e cons t
d o es n ot v ar y fr o m
to t h e n ex t Most p robl em
on e s t r ea m
s com
e withi n
thi s d efiniti on Equ ation (2) i s only t ru e for an in v isci d incom
p ressibl e
fl u i d a n d cann ot b e a pp li ed with co m
p l et e accur acy to any fl ui d h avi ng
v i scosity
Str ea Fu ncti on — I t h as b een shown th a t t h e total qu an tity O f fl ui d
mo vin g across t h e circl e CPG i s m Th e same q u an tity Obv i ous ly fl ows
a cr o ss an y boun d ary whi ch en cl os es t h e so ur ce
It i s co nv eni en t t o h av e
a n exp res sio n for t h e q u an tity of fl ui d whi ch goes acr oss p ar t Of a
b oun d ary
Th e
s t r ea mfun ctio n
whi ch gi v es thi s i s usu ally rep r o
1
s
a r th a t t h e sa m
u an tity of fl u i d fl ow s acro ss a ny
e
It
i
l
q
s en t ed by 1
c
e
1
l i ne j oin in g tw o s t reamlin es and t h e ch ange O f fromon e s t reamlin e
t o an oth er i s th erefor e a lways t h e sam
e no m
a tt er wh a t t h e p a th t ak en
I t foll ow s fromt h i s th a t alon g a st r eamli n e
co ns t
a r ked th a t t h e only assum
I n arr i v in g at thi s co n clusion i t will h e r em
p
tions m
a d e ar e th a t t h e fl u i d fi ll s t h e whol e Sp ac e an d i s in co m
p ress ibl e
I t n eed not be in v i sci d
I n t h e p ar ti cu l ar case of t h e source of Fi g 180(a ) it i s i m
medi at ely ob vi ous
v u
.
”
,
.
,
.
“
.
.
.
,
m
.
.
.
.
.
,
.
,
.
,
.
.
.
th a t
th e
mou
a
n
t of fl u i d fl owing
across
lin e
th e
CP i s
e
qu al t o 1mand
,
2”
i t i s u su a l to wr i t e
v alu e O f .p whi ch co rr es p on ds wi th a sou rce of s tren gth mt h e
If
b e su it a bly chosen , t h e d i ag ram
n ega ti v e si gn b ei n g conv en ti onal
ay b e d i v i d ed u p by equ al an gl es su ch th a t
a lo n g 0G
of Fig 180(a ) m
=
a n d so on
i
a
n
n
i
n
e
m
i
ee
n
OP
lo
g
l
ght
h
v
b
ll
d
a
n
a
ca
e
A
z
1
o
0
e
0
g
y
fi
1 a s i n all cal cul a ti ons i t i s only t h e d iffer en ces b e tw een t h e
th a t O f z ero 11
p ort an ce
v al u es Of ewhi ch a re Of i m
Fig 180(b) sh ow s t h e d rawin g of st reamli n es fo r a co b in ation Of s i p l e
Two sets of r a d i a l li nes sim
ilar to Fi g 180 (a ) ar e d rawn
so u rce an d s in k
eq u a l
For t h e cas e sh ov
an d th ese p rod u ce a seri es of in t ersec tio n s
I f th e
an gl es r ep r esen t e qu al qu an titi es O f flow for b oth sour ce an d s i n k
s tr engt s h ad b een u n eq u al t h e angl es w oul d h av e b een p re p or ti on ed so
h
a s to gi v e equ al fl ow i e t h e lin es are li n es of const an t
differin g by
one lin e to i t s su cces so r
ou n t fr om
equ al am
If li n es be dr aw n fromO t o 0, th rough t h e p oin t s O f in t ersecti on Of t h e
s t ream
li n es 1n t h e way OAO, an d OB O; ar e d rawn t h e lin es so O bt ai n ed
a r e t h e s t rea
lin es for a sour ce an d si nk of e qu al st rength L in es d rawn
thr ou gh th e p oi n ts of in t ersecti on a long t h e o th er d i agon al s Of t h e ele
en t ary qu ad r il at er al s woul d
i
v
t
li
q
l
ou
e
t
h
e
s
r
e
a
m
e
s
f
o
r
t
o
e
a
s
rc
es
n
w
u
g
for t h e
m
.
,
,
.
.
,
,
.
,
.
.
m
m
.
,
.
,
.
m
,
,
.
,
,
.
.
.
,
m
,
.
m
.
2
A
APP L IED AERODYN AMI CS
3 54
h er e b een m
ad e th a t t h e efl ec t o f a si n k on t h e
motion i s i nd ep end en t of t h e exi stence of t h e sou rce and vice car ed Th e
p ti on i s l egiti m
a t e for an i n v i sc i d fl ui d but d oes n o t alw ay s h ol d for
a ssu m
o ti ons O f fl u i d s ; it i s p r o v ed with ou t di fli cu lt y th a t a ny
t h e v i scou s
a
ed t og e t h er
b
a
b er of sep ara t e p ossibl e i nvi sci d fl ui d m
o ti ons m
e
dd
n um
y
ore co m
ak e a m
p l e x p ossibl e m
otion
to m
Addi ti on of Two Valu es of ill — Th e con stru cti on gi v en i n Fi g 180(b) ca n
a
b
e
s m
en t th a t two sep ar a t e sy s t em
b e seen t o foll ow f romt h e st a t em
y
a dd ed t og eth er to p ro d u ce a r es ult a n t n ew sy st em Th e gr ou p o f r a d i a l
lin es r oun d 0 18 nu m
b ered 1n accord ance with t h e sch em
e of Fig 180 (a ) a nd
For t h e si n k a set O f nu m
b ers ar e
for a sou r ce
rep resen t s v a lu es of
b ers
a rr an g ed rou n d 01 t h e s in k b ei ng i nd i ca ted by t h e f ac t t h a t t h e n um
i ncrease wh en t rav ellin g r ou n d t h e ci rcl e 1n t h e O pp o site way t o th a t for
: t h e v a l u e for t h e sou rce
in creas ing n u m
b ers ro u n d t h e sou rce I f we ca ll 91
1
for t h e com
bi na
t h e v alu e for t h e si n k t h e a dd iti on gi v es
and
ti on or
Th e
a ssu
mp tio
'
h as
n
.
,
m
,
.
.
.
.
.
,
.
,
.
‘
,
,
mli
As a s t rea
$
n e is
i n d i ca t ed
by
1
b ein g cons t a n t
1
1
111
con s
112
t
,
we
my w it
a
r
e
5
a)
(
.
by gi v ing t h e cons t v a ri ou s v al u es t h e n ew s t reamlines ca n be
d raw n As a n exam
p l e t a ke const = 3 l and consi d er t h e p oin t A of
t h e sou rcethr o u gh t h i s p o i n t i s
1i g 18 0 ( b) ; t h e li n e from
an d fr om
l l l = 26
or
t h e Si n k lfiz
H e n ce
l
bo th A a n d E are on a s t reamli n e o f t h e n ew sy s t em Th e a dv an t a g e
e th o d li es i n t h e ea s e with whi ch it ca n b e e x t end ed a n d t o o n e
of t h e m
su ch e xt ens i on i t i s p re p ose d t o ca ll i m
med i at e a tten tion
A s t ea d y s t r ea mo f fl u i d will b e s u p er p osed on t h e sou rce a n d s i n k
o f Fig 18 0 ( b)
Th e s t rea mli n es for t h i s ar e e q u i d i s t an t s tr a igh t li n es a n d
b
e
a
n
ar
a
e
t
e
k
o
t
h
y
will
t
e
p
ll
l
0
I
ca n eas ily b e s h ow n th a t t h e cu r v es
0
t
1
p l e sou r cea n d
O f Fig 180 ( b) ar e ci rcl es but thi s w oul d O n ly b e t ru e for a s i m
s i nk a n d n ot for a ca s e p r esen tly t o b e d i scu ss e d
etho d o f p r oce d u r e
Th e m
i s n o t confi n ed t o su ch a s i m
p l e sou rce and si n k I f p arall el li nes b e d ra wn
on a sh ee t O f t r a ci n g p a p er w hi ch 18 th en p l a ce d o v er t h e li n es fr om
s ou rce
t o si n k a s et O f i n t ersec ti ng l i n es will a ga i n b e f orm
ed o f whi ch t h e d i ag on a l s
ay b e d ra w n t o form
t h e n ew sy st e m
Fig 180(c)
m
; t h e r es ul t 18 i n d i ca t e d m
an o v al sh ap e d s t r eam
Th e result 1s i nt eres ti ng
i dd l e of t h e
in th e m
I n si d e th ere ar e s t r ea mli n es p ass in g
fig u r e sep arat es i t i n to t wo p ar t s
from
s ou rce t o si n k a n d ou t si d e s t r ea m
s p a ss i ng f rom
a gr ea t d i s t an ce o n
on e s i d e t o a gr ea t d i s t an ce on t h e o th er
As t h e fl u i d i s fri cti onl es s t h e
ay b e r ep l aced by a soli d O b s t r u ct i on with out d i s tu r bi ng t h e s t r eam
ov a l m
eth o d of sou r ces a nd si nk s m
a
th
li n es a nd t h e m
b
e u sed t o d ev e l o p
n
e
y
fo rm
s o f ob s t acl es a n d t h e corr es p on d i n g flow O f a n i nv i sci d fl ui d r ou n d
th em
B y t h e a dd iti on of t h e v elo citi es of t h e fl u i d d u e t o sou rce s in k a n d
t ransl a ti on sep a ra t ely by t h e p ar all el ogr amo f v el ociti es t h e r esu ltan t
v el oc ity o f t h e fl u i d a t any p oi n t rou n d t h e o val can b e o btai n ed Th e
a nd
.
”
.
.
,
,
1
.
,
.
,
.
.
.
.
.
.
.
,
.
.
,
.
.
.
,
.
,
,
.
,
,
.
APP L IED AERODY NAMIC S
356
m
th a t t he figu re ight hear as m
u ch r esem
blance as p ossible
e
t o t h e p hot ograp h s sh ow n by P r of esso r H el e Sh aw
Th e resul t is som
wh a t s tri k ing
—
i
u ati ons of Moti on of an Invi scid Flu d
R ead ers ar e r eferr ed
Th e Eq
i cs for a fu ll tr ea tm
en t of t h e subj ec t as
t o t h e t ex t books on Hy d ro d ynam
p ressibl e fl ui d s and t h e effect s of gra vi ty and a tt en tion wi ll
a pp li ed t o co m
i ted t o t h e cases ou tlin ed in t h e p r ev iou s n o t es
b e lim
otion i n t h e p la ne of t h e
Su pp ose th a t Fi g 182 r ep resen t s a st ea d y m
a ll el em
p a p er I sol a t e a sm
li nes an d consi d er t h e
en t b etw een two st r eam
forces actin g on i t wh i ch ar e to b e su ch t h a t i t wi ll not ch ange i t s p osi tion
with ti m
e a lth ou gh fi ll ed with n ew fl ui d
en t ar y
Th e force on t h e el em
block i s d u e to p ressur es Ov er i t s four faces an d t h e d i ffer ence b etween
w er e
fi ll ed i n
so
-
.
.
-
,
.
.
.
,
.
the
mom tum t
o
t to m
k i
bl oc
f ace AD an d
I f th e
BC
v e t h e res u lt an t of th ese t wo m
u s t b e zero
—
i
f
M
ot on
I f p b e t h e p r essur e on AD th a t
th e Di recti on o
s no
For ces i n
the
.
.
.
on
B 0 wi ll b e p
be
v ari abl e
z ero,
i g by
en er n
en
f
g
—ds
,
an d a
long
th e
faces AB
an d
DC t h e
ult an t O f t h e u niformp r essu r e p o v er
an d t h e t ot al fo rce ag ain s t t h e arr ow i s th er efor e
.
Th e
r es
p ressur e wi ll
all
t he
faces
4
( )
is
if we n egl ect qu an titi es of r el ati v ely high er o rd er Th e m
ass O f fl ui d
e i s t h e sam
e a n d i s e qu a l t o p vdn wh er e p
p assin g AD and B C p er unit tim
en tum
om
en t er i ng i s
Th e m
i s t h e d ensi ty Of t h e fl ui d an d 0 i t s v elocity
.
,
.
th en pv dn
’
,
an d
(
th a t l ea vi ng i s pvdn
v
an d
the
d i fferen ce i s
F L U ID MOTION
8 57
i n t h e d ir ecti on of t h e arr ow an d th erefor e exertin g
di r eact i on on t h e el e en t
Th e force eq u ati on i s
an c
l is
m
Eq u a tion (6) i s
easi
force 1n t h e Opp osite
made u p of (4) and
,
.
ly in t egrat ed
v ery im
p or tant
otion of r ea l fl ui ds
Eq u a tion ( 7) i s
m
ml t
Forces Nor
p a th
ar
of
o
t h e cent ri fu ga l
,
w ar ds
3?
a
is p
dad s ,
moti
?
an d
oft en app li es
app r oxi
mM ly
e
to
di us of t h e
mmo vi ng out
ra
of p ressure p r od u cin g thi s force i s
motio
n at r
ight angl es to
the
d i rection
of t h e fl u i d i s
m(7) f
or
g
,
an d
8
( )
+
r
b ecom
es
an
0
71
6
d eali ng wi th sources and sink s equa tion ( 9) was assum
ed to h ol d
1s n ow seen th a t t h e as su
p tion was j us tifi ed si nce r is infini t e an d
i s z ero
If
,
(7)
th e Directi on of Moti on —I f r be t h e
for ce n ecessary t o keep t h e block fr o
0
an
o t
c ns
.
h ence t h e eq u ation of
Sub stitut e fro
In
an d it
“
du ds , wh i l st t h e di ffer en ce
an d
on
gi v es
W
p
t he
an d
a
a
long
h
eac
of
t
s r ea
mli
,
n es.
ra
di u s
,
.
of s r ea
he
n es
i nfin i t e
:
2
ti on ( 9) show s th a t
mus t be z ero i e th e v elocity mus t be uniformfromstreamto st ream
Equ ation (7) th en show s th a t p i s cons t an t
Th e con v ers e i s of course
t ru e th at uni formp ressur e m
v el ocity an d strai ght str eam
ean s un ifo rm
th e
t
th e
m
mli
,
,
q
e ua
.
.
.
,
m
m
co pari son of Pressur ec i n a SOurce and Si nk Syst e
wi th th ooe on
a Model i n Ai —Th e ca l cula tio ns an d exp er i en t s t o whi ch r efer ence
will n ow b e a d e ar e d u e to G Fuhr ann working i n t h e Get t i n gen
Uni v ersity La bora tory
Th e g eneral li nes of th e cal cu l ation s follow those
o ut lin ed , bu t t h e sour ce and sin k sys te
od el s,
i s not si p l e Th e
i nst ead of b ein g long cyli n d ers as i n t h e cases wor ked ou t i n p r ev iou s
p ages , w er e soli d s of r ev ol u tions b u t t h e t ransfor ations on t his accou n t
ar e ex t r e
ely si
p l e Th e co p l ex sour ces and sin k s are obtai ned by
m
m
.
.
m m
m
,
.
m
m
m
m
m
.
m
APP L IED AERODYN AMI CS
8
35
en tary si m
i n t egration fr oma nu m
b er of el em
p l e sour ces an d sim
p l e sink s
ad e to t h e
an d p r esen t littl e d i ffi cu l ty
Fo r d eta il s r efer en ce shou l d be m
en ti oned
o r igin a l r ep or t or to t h e p a p er by T ayl or a l rea d y m
ann con t a in s t h e an a ly si s an d exp eri
Th e o rigi nal p a p er by Fuhr m
men tal work r el a ti ng to si x mod els of t h e sh ap e taken by airshi p envelop es
h ad both p oin t ed
e of th es e sh a p es h a d p o i n t ed t ai l s whil s t on e o f th em
Som
h ea d an d t ail Th e in v estigati on w as carr i ed ou t in r el a tion to t h e
d ev el op m
and t h e m
od el m
os t
en t of t h e w ell known P a rs ev a l a i rshi p
li ke t h e env elo p e of th a t ty p e of di r igibl e i s ch osen for t h e p ur pose of
illu st ra ti on Star ting with v a riou s sou rces and s in ks t h e flow was calcu
l a t ed by m
ila r to thos e l ea d in g to Fig 180 but need in g t h e
etho d s s i m
,
.
.
.
,
.
-
,
.
.
E10 18 3
.
.
,
—Ca lcu la ted flow of i nvisc id fl ui d round an a i rsh i p en ve lope
.
m
pp li ca tion of t h e in t egr al ca l cu l u s for th ei r si p l es t exp ress wn Th e
ty p e of sou rce ch osen for t h e od el i n q u es ti on i s i ll u st ra ted by t h e sket ch
a bo v e Fig 18 3
Th e si n k b egi n s at 0 ge t s s t r ong er gr ad u ally to D an d
th en w eaker t o B ; a t thi s la tt er p oin t t h e source begi ns an d g rows i n
s t rength to A wh en it ceases a b r u p tly
p l ex sou r ce and si nk s o d efin ed a re r ep r od u ced in Fig 183 t h e
Th e co m
u p pe r h al f of whi ch sh ow s t h e s t rea m
li n es d u e t o t h e sy st e
Th e res em
s the
bl an ce t o ci rcul a r arcs i s sligh t Su p er p osing on th ese s tream
ann fou n d t h e b all oo n sh a p ed
a pp r o p ri a t e t ransl a ti on al v el oc ity F u h r
li nes p as t it Th ese s t reamlin es
bo d y i n d i ca t ed togeth er with t h e s t r ea m
od el h as a ro un d ed h ea d
Th e m
a r e sh ow n i n t h e l ow er h a lf o f Fig 18 3
a li ttl e d i s t a n ce i n fron t o f t h e sou rce h ea d A an d a p oi n t ed t a il t h e t i p
of whi ch coin ci d e s wit h t h e ti p C of t h e s i n k
Hav i ng obt a i n ed a bo dy o f a d esi red ch aract er Fuh r ann p roceed ed
a
m
.
.
.
,
.
,
m
.
,
.
m
.
-
.
,
.
.
,
.
,
m
APP L IED AERODYN AMI CS
360
mod
l bo th are of cons i d era bl e i m
p or tance i n t h e m
ea su red t ot a l
r esis t an ce
Fr omt h e an alogy with fla t boar ds tow ed with t h e su rfa ces
i n t h e d ir ection of m
o ti on so t h a t t h e n o rm
al p r es sur es cann o t e xer t a
re ta r di n g in fl u en ce t h e t a n g en ti a l d r ag i s g ener ally r ef er r ed t o a s
s ki n
fr i cti on
It will be seen th a t app reci abl e error 50 p er cen t or 60 p er
cen t
w ou l d r esu lt i f t h e p r essu r e di st rib u ti on w er e t ak en t o b e th at of an
in vi sci d fl u i d
od el s i n all w er e t es t ed in t h e ai r ch ann el a t Get t ingen an d t h e
Si x m
res ult s a r e s um
marised i n t h e followi ng t a ble
of
the
e ;
.
,
,
”
.
.
.
,
,
.
-
TAB LE 2
,
mb
Nu
er of
—Tnx
mR
F or
ns n
sr as c s
ar
m8
x11:
,
F aron on
or
mm E
A
s
r
xv xw r s s .
Fracti on of res i st a nce
md
Relat i ve tot al
o e l.
m
( for
res i s t ance
)
.
general concl u si on whi ch m
igh t h av e b een d rawn i s t h a t for form
s
ore d ep end en t on form
of r ev ol u tio n of a ir s hi p sh a p e t h e res i s t an ces ar e m
r es i s t an ce th an on s k in f r i cti on
T h i s con cl u sion shou l d be accep t ed with
r eserv e i n t h e light of m
o r e r ecen t exp er i m
en t s
Th e exp er im
en t s r efer red t o a bo v e w er e all carr i ed ou t at on e sp eed
en t s w er e
Measur em
ad e of t h e t o t a l r es is t an ce a t m
any sp eed s but th ere
are no correSp on din g r eco r d s of p r essu r e m
en t s
eas urem
A ser i es of t es t s
on a m
od el o f a n a i rshi p en v eIO e h as b een carr i ed ou t a t t h e N P L a t
p
a nu
b er of sp eed s with t h e foll owin g r esu l ts
Th e
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.
m
.
.
,
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m
TAB LE 3
.
.
—Vanu r xon
or
F0 “
Ras t er /
m mS mc
os
a
ar
rrox
'
.
.
wrrn Sr s s n
F r omt h e l as t r ow of T a bl e 8 it wi ll b e seen th a t th e f ormr esi st an ces
a r e fa r sm
all er f rac ti ons of t h e tot al a t all sp ee d s t h an tho se gi v en i n
T abl e 2 F urth er exam
i n a tion of t h e origi n a l figur es show s th a t t h e
measu remen ts of t ot al resi stance at t h e N P L are v ery much t h e
sa m
e in m
agn it u d e a s th ose a t Get t i ngen
No su gg es ti on i s h ere p ut
forw ar d t o a cco u n t for t h e d i ffer ence t h e exp eri m
en t s a t v a r i ou s Sp eed s
h av i ng an i n t eres t a p ar t fro mthi s
I t will b e n oti ced th a t both t h e
,
.
.
.
,
.
.
.
FL U ID MOT IO N
3 61
formr esistance an d t h e ski n fr i cti on v ary with sp eed an d in t h e
p ar ti cu l ar ill u s t ra tion t h e v ari ation of t h e p r essur e i s t h e gr ea t er Thi s
p tion som
e ti m
es m
a d e th a t t h e
ev i d en ce i s d i r ectly aga in st an assum
p r essu r e on a bo dy v ari es a s t h e s qu are of t h e sp eed whi l st t h e skin
fri c ti on increas es as som
e p ow er of t h e sp eed a pp r eci a bly l ess th an t wo
Th er e i s cer t a inly no th eo reti cal j u s tifi ca ti on for su ch an a ssum
p ti on
an y p r acti cal r een t s coul d b e p ro d u ced to
a s will b e seen l a t er an d m
p ti on
show th a t exp er im
en t al ev i d en ce i s ag ai ns t su ch a ssum
On e oth er ill u st ra ti on of t h e v ar i a ti on of p r essur e di st r ib u ti on wi th
Sp eed m
ay b e m
A si x in ch sp h er e i n a win d of 40 ft s
en tio ned h er e
os t wholly on t h e p ressu r e ov er i t s sur face
h as a r esi st ance d ep en d en t alm
but this r esi st ance i s ex tr em
ely sensi ti v e to ch an g es of sp eed ; t h e cur i ou s
r esu lt 18 obt a in ed th a t for cer t ain condi ti ons a r ed u ced r es i s t an ce aecom
cov er i ng
a
n
i
n
n
fl
1
8
r
c
e
d
a
p
d
o
p
o
d
i
g
p
o
u
by
e
s
n
i
n
c
rea
s
e
o
f
S
ee
A
c
rres
e
ec
t
d
p
with san d t h e sm
oo th s ur f ace fo rm
ed by v arni sh on woo d
At a bout t h e
Sp eed m
ay b e d ecreas ed t o l ess th an h alf by su ch
en tio n ed t h e r es is t an ce m
t ou gh eni ng
Th e gen eral a sp ect s of t h e subj ect ar e d ealt with u n d er t h e
h ea di ng of Dyn am
i cal Sim
i l ar ity For t h e p r es en t it i s o nly d esi red t o
draw a tt en tion to t h e fac t th a t t h e law of r es i s tan ce p r o p or tion al to sq u are
of sp eed i s n ot accur a t ely t ru e for eith er t h e p r essur e di s tr ibution on a bo d y
i n a fl ui d or for t h e sk i n fr i cti on on it
Th e d ep ar tur es are n ot u su ally so
gr ea t t h at t h e 112 law i s seri ously at fa u l t i f ca r e i s t ak en i n app li ca ti on
en t will a pp ea r shor tly wh en t h e
A fu ll er ex p l ana ti on o f thi s st a t em
con di ti o ns un d er whi ch t h e a2 law m
ay b e t ak en to app ly with su ffi ci en t
a ccuracy for g en er a l p ur p o s es will b e d i scu s sed
otion s alr ea dy di s
Cy cli c Moti on i n an I nvi s ci d Fl u i d — I n t h e fl ui d m
cu s sed t h e flow h as b een obt a in ed fr o m
bin a tion o f a m
oti on of
a com
t ransl ation an d t h e efllu x an d i nfl u x fr om
a so u r ce and s i nk sy s t em Th e
i niti al assu m
p ti on s inv ol v e as cons equ en ces
a
Fi
n
it
e
s
l
i
pp
i
n
g
f
t
h
e
fl
u
i
d
o
v
e
r
t
h
e
bou
n
d
y
w
a
ll
s
ar
o
( )
n
n
b
No
s
ult
a
n
t
fo
r
o
n
t
h
bo
d
y
i
n
a
d
i
r
ec
ti
o
r
e
c
e
e
;
y
( )
c
u i d p r es sur es
li
a
bility
to
p
o
d
u
g
a
ti
v
fl
e
r
c
e
n
A
e
( )
No th eory h a s y et b een p re p osed and fromt h e n atur e of an inv i sci d fl ui d
it wou l d app ear th at n o th eory cou l d ex is t whi ch av oi d s t h e fin it e
sli pp i n g o v er t h e boun d ary
It app ears t o b e fim
dam
en t ally im
p o ssibl e
to rep resen t t h e m
otion of a r eal fl ui d accura t ely by an y th eo ry r el ati ng
to an i nvi sci d fl ui d It i s not how ev er im
med i at ely ob vious th a t su ch
s in
th eori es cann ot gi v e a goo d app ro xi m
a tio n to t h e t r uth an d as cla im
thi s di rection h av e o ft en b een m
a d e fur th er s tu d y i s n ecessary b ef or e any
O p in ion ca n b e fo rm
ed as to t h e m
erit s o f any p ar ti cul a r solutio n
Th e di fli cu lt i es (b) an d (0) can b e a v oi d ed by i n t r o d u cing Sp eci al
ns
ng
cy cli c
as su m
t
i
o
t
w
o
s
t
a
n
d
ar
d
tho
d
r
d
v
lo
p
d
o
n
e
i
n
v
ol
v
i
m
e
s
a
e
e
e
e
p
motion an d t h e oth er di scontin uous moti on
L eav i ng t h e seco n d of th ese for t h e m
en t a tt en tio n w ill b e d ir ect e d
om
to t h e case of cy cli c m
p l e cy cli c
A s im
otio n of an in v i sci d fl ui d
moti on can p erh ap s b est be d escrib ed in r eference t o a simp l e sour ce In
t h e s im
p l e sou rce t h e s t reamlin es w ere ra d i al an d t h e v elocity ou tw ar d s
v ari ed in v ers ely a s t h e ra di u s In a sim
p l e cy cli c oti on t h e st reamli nes
,
.
.
,
'
,
.
-
-
.
.
,
.
,
.
.
.
.
.
.
,
.
,
.
.
.
,
.
.
,
,
,
,
.
,
,
”
“
.
,
.
.
.
m
APP L IED AERODYN AMI CS
3 62
o
t i i l t h e v elocity i n each ci rcl e b ein g in v ersely p rop or ti on al
t o t h e ra d i u s
F romt h e conn ec ti on b etw een p ressu re an d v elocity i t will be seen th a t
t h e s u rfaces of u n i formp r essur e in a cy cli c m
otion du e to
oti on an d in m
a si m
p l e sou rce a re t h e sam
e
As i n t h e case of sou r ces an d s in k s com
p l ex cycli c m
o ti ons co u l d be
p r od u ced by a ddi ng t ogeth er any num
ber of sim
p l e cy cli c m
oti on s: Cy cli c
a nd n on cy cli c m
o ti on s m
ay al so b e a dd ed
Cons i d er t h e effec t of su p er p osin g a cy cli c m
oti on on t o t h e fl ow of an
i nv isci d fl u i d rou n d a bod y say a cylin d er p l aced acr oss t h e s tr eam be for e
t h e cy cli c m
oti on i s a dd ed t h e s t reamlin es are th ose ind i ca t ed in Fig 166
a dd t h e cy cli c m
oti on as i n Fig 185
Th e an gl es AOP DOP B OO an d COO h av i ng b een ch osen e qual t h e
sym
metry of Fig 166 shows tha t t h e v elociti es th ere will be equal
for t h e u pp er an d low er p ar t s of t h e cyli n d er
Th ese v elociti es are
i nd i ca t ed by sh or t li nes on th e ci rcl e t h e arrow h ead in di catin g t h e
di r ection of flow Si nce t h e
p r essure i n an i nv is ci d fl u i d
i s p erp endi cu l a r to t h e sur
fa ce i t ca n easily be seen th a t
t h e p res su r es a ll be i ng e q u al
an d sym
metri cally d isposed
Q h a v e no r esu lt an t Su pe rpose
o tio n whi ch h as i t s
a cy cli c m
cen t r e at O an d whi ch add s
a v el oc ity a t t h e s u r fa ce r e
p r es en t ed by t h e li nes j u s t
out si d e t h e ci r cl e AB CD On
h
h
t
e cyli n d er
u
pp
l
f
t
e
r
h
a
f
P
Fro 18 5 Cy li fl w r nd cy li nd er
e
o t1on a dd s t o t h e
t h e cy ch c m
v elocity and a dd s e q u ally a t A an d B B el ow t h e v elocity i s r ed u ced or
p ossibly rev ersed b u t t h e resu lt ant h as t h e sam
e v alu e a t C an d D
Fromt h e rel a ti on b etw een p ressu re an d v elocity gi v en i n eq u ati on (2)
t h e d ed u ction i s i m
n
a
n
d
a
ed i a t ely m
a d e th a t
l
ss
th
a
n
d
ar
e
e
p
p
pb
a n d a si m
p l e app li ca ti on o f t h e p arall el og r amo f forces th en sh ow s th a t a
r esulta n t fo rce act s on t h e cyl i n d er u p w a r d s
ewha t
Th e r esu lt i s som
cu r io u s an d m
ay b e s u m
mari zed as follows if a cyli nd er i s mov ed i n a
s t ra ight li n e th rou gh a n i n v i sci d fl u i d whi ch h as i m
p os ed u pon it a cy cli c
motion concen t ri c with t h e cylin d er th ere will b e a force acting on t h e
oti on
cylin d er a t r ight ang l es t o t h e p a t h b u t n o r esi s t an ce t o t h e m
I f t h e b o dy h a d b een a w i ng for mit a pp ea rs th a t t h e r esu lta n t force
wou l d n ot th en h av e b een a t r ight angl es t o t h e l in e of m
o ti on a n d th er e
w oul d h av e b een a resi s t an ce com
p on en t
a ny a n d Jou k owsky i n R u ss i a h a v e d ev el op ed t h e
K u tt a i n Germ
math em
a ti cs of cy cli c m
oti on i n r el a ti on t o a erofoil s t o a g r ea t ex t en t
St ar ti ng fr oma ci r cu lar arc K u tt a cal cu l a t es t h e li ft a nd d ra g for v a ri ou s
angl es of i nci d ence an d co m
p ar es t h e r esu lts with th ose obta in ed in a
wi n d t u nnel B efore gi v ing t h e fi gu res it i s d esi ra bl e t o out li ne t h e b as i s of
are c ncen r c c rc es,
.
.
,
-
.
,
.
.
.
,
,
,
.
.
“
-
,
.
,
,
.
,
.
-
.
.
c
c
o
’
ou
.
.
,
.
,
m
m
,
,”
.
,
,
.
,
,
,
.
.
,
,
.
APP L I ED AERODYN AMIC S
3 64
of
g i v e p ressur e at
o f v el ocity
n e at
any an
gl e of i nci d ence wh at ev er
lim
i ted
for
a
Mu
s eu
range
.
TAB LE 4
.
K 1rrr a
’
s
Tu
H on
I ncli nat i on of
chord o.
ored llft
per u nlt aroa.
L lllentln l
rn a
CO U PAB I SO X
or
mm
C
r r n u rn
mF
ou l s .
m
Calcu la te d lift
p er
u nlt a
.
(W
M
.
en ts referr ed
ta bl e of fi gur es by Kutta i s gi v en abo v e Th e exp er im
t o w er e p rob a bly not v er y accura t e an d t h e d isagreem
en t of th e cal cula t ed
a n d o b s erv ed v al u es of li ft an d dr ag i s n ot so gr ea t as to d i scr ed it t h e
ay b e n ot iced th a t t h e cal cul a te d dr ag h as b een com
th eory I t m
p ar ed
with t h e excess of t h e ob serv ed dr ag a bo v e i t s m
i ni m
umv alu e and so
t hr ows n o light on th e econom
i cal f ormof a win g Th e th eory cannot
i n i t s exi s ti n g f orm
i n d i ca t e ev en t h e p o ssib i li ty of t h e w ell kn own criti ca l
angle of an aerO lan e wi n
s
u s ti fy t h e a ssum
n
s
I
t
p
o
ibl
to
j
p
tio
s
e
i
n
t
s
o
p
g
mad e and th e r esu lt i s a somewha t com
p l ex an d n ot v ery accur at e
em
p ir i cal form
ul a
ea n in g
Di sconti nu ons Flu i d Moti on
p l es t illus trati on of th e m
Th e si m
of d i sco n tin u o u s m
otion 18 th a t p resen t ed by a j et 1ssu i ng into air froman
ori fice i n t h e si d e of a t an k of w at er If t h e or ifi ce i s r ou n d and h as a sh ar p
ed g e t h e w a t er f orm
s a sm
e d i st an ce a ft er
ooth gl ass lik e s urf ace for som
i ssu i ng Aft er a littl e tim
e t h e co lum
n
b reaks i n to dr 0p s and L or d
R ayl eigh h as sh own th at thi s i s du e to su r f ace t ensi on ; fur th er i f t h e jet
i ssu es h ori zon t ally t h e cen t re li ne i s curv ed du e t o t h e acti on of gra vity
whil st if v er ti ca l an in crea se of v elocity ta k es p l ace whi ch r ed u ces t h e
s ecti on of t h e col u m
n
N egl ectin g t h e eflect s of gra vi ty and sur face t ension a horiz on t al jet
w ould con t inu e through t h e ai r with a f r ee su rface along whi ch t h e
p r essur e was co nst an t an d eq u al to th a t of t h e a t m
osp h ere Th e
method of di scon tin u ou s motion i s essenti ally i d entified with t h e mathe
mat i cal analysis relat ing t o cons tan t p ressur e fr ee su r faces Th e
e xa m
os t
p l es actually worked ou t app ly to an in v i sci d fl ui d an d a lm
excl u si v ely t o t wo d i m
b st a t es th a t t h e fir s t exam
pl e
en s i ona l flow
L am
was du e to H el m
holtz an d it app ears th at t h e m
etho d of cal cula ti on w as
made r egu l ar an d v ery g eneral by K ir chh off and Lor d R ayl eigh Th e
mai n resu lts h av e b een coll ected in R ep or t No 19 of t h e Advisory Com
mi t t ee for Aeronau ti cs by Si r George Gr eenh ill and si nce th at tim
e ex
a d e t o cur v ed b arr i er s
t ensi ons h a ve b een m
Th e
.
,
.
,
.
.
,
.
.
-
.
-
.
,
,
,
.
'
,
.
,
-
.
.
,
.
.
,
.
FL U ID MOTION
3 65
e thod s
It i s n ot p ro p osed to a t t em
p t any d escri p ti on of th e sp eci al m
of solution b u t to di scu ss som
e of t h e r esu lt s
Th e firs t p robl em
ex am
in ed
by Sir Geo rge Gr eenhill is th e m
otion of t h e fl ui d i n a jet b efor e an d af t er
im
p ingi ng on an in cli ned flat sur f ace Th e j et com
ing fromI Fig 188
im
p in ges on t h e p l ate AA and sp li t s i n to two j ets t h e sep ar a t e ho rns of
es u p to t h e
whi ch are con tinu ed to J an d J On e s t reamlin e IB com
b arr i er at a st agn a ti on p oin t B and th en t ra v el s along t h e b arr i er A A i n
t h e two d ir ecti on s tow ar d s J an d J
Finit e sli pp ing i s h er e in v ol v ed an d
a tio n to r eality
t h e analy s is m
u s t th erefore b e l ook ed on as an a pp ro x im
e th a t t h e
on ly
In t h e ca se of j ets i t app ears to be j u s ti fi a bl e to assum
all co m
otion an d p r es su res 18 v ery sm
p ared
effec t of v i sco sity on t h e fl ui d m
en tuman d so far a s
wi th tha t ari sin g fr o mth e u su al res olution s of m
om
otion of j ets w orked
ex p er im
en ta l ev i d ence exi s t s it sugg es t s th a t t h e m
ou t i n thi s w ay i s a sa ti s fac t ory i n d i ca ti on of t h e
otion of a r eal fl ui d
u ch l es s d en s e fl ui d su ch as ai r
su ch as w a t e r wh en i ssu i n g in t o an oth er m
.
,
.
,
.
,
'
,
'
.
'
,
’
.
,
.
,
m
,
.
,
Fro 188.
.
—
Db
6
0 n t i nu ous
mt i
on of a
o
jet
of
fl u id
.
Fr omI to J fromI to J a nd fromA to J A to J t h e flu i d i s bou n d ed
by fr ee sur f aces al ong whi ch t h e p ressur e i s cons tant F romeq u ation (2)
thi s wi ll b e seen to i m
p ly t h e con di tion th a t t h e v elocity i s const an t ;
furt h er i f t h e f ree su rf ace ex t en ds to gr ea t di st ances fr omt h e b arri er t h e
v elocity all a l ong it m
u s t b e th a t of t h e fl ui d a t su ch g rea t d i s t an ces
os t alw ay s in v ol v e t h e assum
oti ons alm
p tion
Solutions of di scon t i nu ou s m
th a t t h e v el ocity al on g t h e fr ee su r f aces i s th a t of t h e st reamb efor e dis
t u r b an ce by t h e b arr i er
Fi g 168 al r ea d y r eferr ed t o show s b eh i n d a cylin d er a r eg ion of a l o st
it s of whi ch i n t h e d ir ection of t h e str eamare v ery
s t a gn an t fl u i d t h e lim
sh ar p ly d efin ed an d it i s cl ear th a t i n r ea l fl u i d s i n a dd ition to t h e p erio d i
Di rec t exp eri
ci ty th er e i s in d i catio n of t h e ex i s t en ce of a fr ee sur f ace
en t s show th a t in si d e su ch a r egio n t h e p res sur e i s oft en v ery uni fo r m
but app reci a bly b elow th a t of t h e fl ui d far fr omt h e o d el
in g a fr ee su rf ace encl osin g st agnan t fl ui d ex t en di n g far b ack
Assum
f romt h e m
od el t h e whol e d et ail s of t h e p r essur e p osi tion of cen t re o f
p ressur e an d sh ap e of st ream
lin es for an in clin ed p l at e h av e b een work ed
'
,
.
,
,
.
m
.
.
,
,
,
m
,
.
,
m
,
,
,
.
APP L IED AERODY NAMI CS
366
out I n ad d iti on to fin it e sli pp in g a t t h e m
o d el th ere i s now a lso fin i t e
s li pp in g o v er t h e b ou n d a ry O f t h e s ta gn a n t fl ui d an d o bj ec tions on t h e score
n o ta bly by L or d K el v in
o f s t a bility h a v e b ee n r a i se d
Th e followin g
s um
mary O f t h e p os ition i s gi v en by Lamb
As t o t h e p ra cti cal v a l u e O f this th eo ry O p i ni ons h a v e d iffer ed
On e
i s th a t t h e u n li m
as s of
O b v i o u s cr iti ci sm
it ed m
d ea d w a t er foll owin g
p li es an infini t e ki ne ti c ener gy ; but this only m
ean s th a t t h e
t h e d i sk i m
typ e O f m
p l et ely es ta blish ed i n a fin it e
o ti on i n q u es ti on cou l d n o t b e co m
ti m
r es t a lth ou gh i t m
ight ( con cei va bly) b e app roxim
a t ed t o a s m
e fr om
y p
t ot i cally
An o th er obj ecti on i s th a t su r f a ces O f d isco n ti nui ty b etw een
fl ui d s of com
p a ra bl e d ensity ar e as a ru l e highly u ns t a bl e I t h as b een
u rged h ow ev er by L or d R ayl eigh th a t thi s ins t a bili ty m
ay no t s erio u s ly
otio n for som
e d i st ance fr o m
a ffec t t h e ch ar a ct er of t h e m
t h e p l ace O f or igi n
o f t h e s u rfac es 1n q u es ti o n
a i n t a i ns th a t t h e typ es of m
L or d K el v in on t h e o th er h and m
o tion
h ere con t em
p l a t ed with su rfaces of d is con ti n u ity h av e no r esem
bl a nce
t o anythi ng wh i ch occu rs i n act u al fl u i d s ; a nd t h a t t h e only l egitim
a te
a pp li ca ti on of t h e m
eth od s O f v on H elm
h oltz an d K i rchh off is t o t h e cas e
of fr ee su r f aces a s of a j et
en t a l hy d r od yna
i cs an d si nce t h e
Wi th t h e a dva n ce O f exp eri m
t h e p ositi on t a ken by L or d K el v i n h as
a dv en t O f a v i a ti on p ar ti cu l ar ly
ent al s u pp or t
o n e i ns t a nce O f t h e d i ff er en ce
r ecei v ed co ns i d er a bl e ex p er im
b etw een t h e p res su re of a i r on a flat p l a t e and t h e p r essur e as cal cu l at ed
ak e a n exp er im
p ossibl e t o m
I t i s cl early i m
en t on a fla t
i s gi v en b el ow
en t a l r esu lt s ar e n ot
s ur fa ce of no thi ck ness and for th a t r ea son t h e ex p er im
p a ra bl e wi th t h e cal cu l a tio ns : i n a dd iti on t h e con d iti ons were
s t ri ctly co m
ens ion al flow
p ti on o f t wo d im
N ev er
n ot s u ch a s t o f u lly j u s tify t h e a ss u m
en t an d calcul a tio n
port an ce b e tw een exp er im
t h eles s t h e d i scr ep anci es O f im
en t a l si d e but to t h e
ar e no t t o b e exp l a i n ed by errors o n t h e exp erim
i niti a l a ssu m
p ti ons m
a d e a s t h e b as i s o f t h e ca l cu l a tio n s
en t s w e r e carri ed ou t i n a n a i r ch ann el at t h e N a tion al
Th e exp eri m
Physi ca l La bo rat ory and a re d escri b ed 1n one of t h e R ep or ts O f t h e Ad
v i sory Com
mitt ee for Aeronau ti cs Th e a bscissae rep res ent ing poin ts a t
t h e l e ad i ng ed g e O f t h e
whi ch p res su r es w er e O b serv ed are m
eas u r ed f r o m
p l an e a s f racti ons o f i t s wi d t h Th e scal e O f p r essu r es i s s u ch th a t t h e
e x cess p ressu re a t B o v er th a t a t i nfi n i ty w ou l d j u s t p rod u ce t h e v el ocity
I t a pp ears t o b e v ery cl os ely t ru e wh eth er
v i n t h e a b se n ce O f f ri c ti on
li n e
t h e fl u i d b e v i scou s or i n v i sci d th a t t h e d rop O f p r essu re in t h e s t ream
z
whi ch com
Th ere ar e oth er r eas o ns whi ch
es to a s t agna ti on p oi n t i s lpv
2
o ti ons for ch oosi n g pe as a b as i s
will a pp ear i n t h e d i scu s si on o f si m
il ar m
for a p ressu re sca l e
i s fo u n d o n t h e un d ersi d e of
en t t h e p r es su re of +i
I n t h e e xp er i m
t h e i ncli n e d p l a n e v ery n ea r t o t h e l ea d i ng ed g e ; this i s sh o wn a t B i n
Fig 189 T ra v elli ng on t h e l ow er s ur face tow ar ds t h e t ra ili ng ed ge t h e
o re slowly u n til i t ch an g es s ign
p ressu re a t fi rs t f all s r ap i d ly a n d th en m
e d ge
e O f t h e u pp er su rface i s
t
ra
n
e
e
a
h
n
t
h
e
T
h
e
t
b
i
g
ili
g
wh
o
l
r
c
o
u
s
e
f
r
j
t h e t ra ilin g ed g e to t h e l ea d in g
u n d e r r ed u ced p r es su r e t h e v a ri a ti on f r om
e d g e be i n g in d i ca t ed by t h e cu r v e EFGH K A
.
,
,
.
.
-
,
.
.
,
,
.
,
,
,
,
r
m
.
,
.
,
.
,
,
-
.
,
.
,
.
,
,
.
.
,
’
,
.
,
,
.
”
,
.
,
.
.
,
.
APP L IED AERODYN AMI CS
8 68
m
H K A gi v es a eas ur e of t h e for ce
Th e a r ea i ns i d e t h e cu rv e AB C
°
At an in cli na ti on of 10 i t app ea rs
on t h e p l at e d u e to fl ui d p r ess ur es
th a t ore th an two-tt hir d s of t h e force du e to p r essu r e i s n ega ti v e a n d
i s du e to t h e u pp er s urface Th e sa e hol d s for a er ep lan e wi ng sections
to p erh ap s a grea t er d egr ee , t h e n eg ati v e p r essu r e a t H so eti es ex ceed in g
th ree ti es t h a t shown Fig 189
Fi g 189 sh ows for t h e sa e p o siti on of a p lan e t h e p ress ur es ca l cul at ed
oti on of a fl ui d On t h e un d er sur f ace t h e
a s d u e to t h e d i sco n tin uou s
?
t h e sa e p l ac e as t h e exp eri
v alu e of you at B 13 r each ed v ery u ch
T ra v elli ng b ackw ar d s on t h e un d er sur face t h e p ress ur es
en t al v alu e
fall t o z er o at t h e t ra ili n g ed ge but are app reci a bly gr ea t er th an thos e of
t h e e xp eri en t al r es ul t s
On t h e u pp er su rf ace th ere i s n o nega ti v e
p ressu r e a t a ny p oi n t Th e t ot al f orce i s again p rop or tiona l to t h e ar ea
i nsi d e t h e cu r v e AB C
H K A and i s cl early uch l ess than t h e area
en t a l d et er
in at i on Th e
o f t h e corr eSp on di n g cu r v e for t h e exp er i
d egr ee of a pp roxi ation i s o b vi o u sly v ery unsa ti sfac to ry i n sev eral res p ec ts,
2
t h e only a gr ee en t b ei n g at t h e has poi n t
For t h e sak e of co p ar i son t h e p r essu re d i s tr i bution co rresp on d i n g
wi th t h e s o u rce an d s i n k hy p oth es i s i s ill us t r a t ed i n Fi g 189
As b efor e,
‘
n
n
r
e
a
d
a
e
o
n
h
e
n
t
d ge A
t v lli g
u d er si d e, t h e lpv
s t ar ti ng a t t h e l ea di n g
p oi n t at B o ccu rs i n u ch t h e sa e p l ace as b efore, but f ro th i s p oin t t h e
p ressu re fall s rap i dly a nd b eco es n eg ati v e j us t behi n d t h e cen tr e of t h e
p l an e ; p r oceed in g fur th er, t h e p ressur e con tin u es to fall or e an d or e
ra p i dly u n til i t b eco
es i n fini t ely gr ea t at t h e t r ai li n g ed g e
Ex actly t h e
t h e tra ili n g ed ge
sa
e v ar i a tio ns of p ressur e are ob ser v ed on return in g fr o
t o t h e l ea d i ng ed g e ai d t h e u pp er su r face as h av e b een d es cri b ed 1n p assi n g
i n t h e r ev ers e di r ection on t h e low er su r face
t h e gr ap hi ca l const ru ction
Th e tot al area i s now z er o , t h e conv ention
b ei n g th a t wh en t rav elli ng roun d t h e cur v e AB CD
EFGHK A areas
t o t h e l eft h an d sh all b e coun t ed p osi ti v e an d ar eas t o t h e r ight h an d
I t 13 cl ear, how ev er th a t th e o en t on t h e aerofoil 18 n ot z ero
n eg ati v e
a n d t h e cen t r e of p r essu r e i s th er efor e a n i nfini te di st an ce aw ay
t h e cou p l e
t en d s to i ncrease t h e angl e of i nci d ence, and fur th er analysi s show s th a t t h e
cou p l e d o es n ot v an i sh u n til t h e p l a t e i s b ro a d si d e on t o t h e s t r ea
It wi ll b e n oti ced th a t t h e ed ges of t h e p l a t e ar e p osi ti ons of i n t ense
n eg at i v e p r essu r e su ch a s we h a v e seen n o r ea l fl u i d i s a bl e to with st an d
ary co v ers i n ess en ti al s all t h e co n ven tion al
This b ri ef s u
a th e
at i cal th eo r i es o f t h e
otion of in vi sci d i nco p r essibl e fl ui d s, and will
i t 18 ho p ed , h av e shown how far t h e th eo ri es fall sh or t of b ei ng sa t is fac to ry
os t of t h e p r obl e s r el a t i n g t o aero n au ti cs
en t i n
s ub stit u t es for exp eri
m
.
m
.
m
m
.
m
m
.
mm
.
.
m
m m
m
.
.
m
,
.
.
m
,
m
m
m
m
m
.
,
.
m
m
.
m
m
m
m
m
.
m
m
.
.
m
.
.
.
mm
,
,
m
.
mm
,
m
m
m m
'
Mon oxs
111
m
m
Vrscou s
—OO, Fig
m
m
.
.
.
s
190, i s a fla t su rfac e o v er whi ch a
v ery vi scou s fl ui d , su ch as gly cerin e i s fl owi n g as t h e r esult of p ress ur e
F
B y d i r ec t ob serv ation t h e
app li ed acro ss t h e fl ui d a t AB
v elo city i s kn own to b e z er o all al on g 001, a n d to gr a d u ally i n cr eas e as
t h e fla t s ur face 1ncr eas es
If t h e v elocity 13 p r op or t ional
t h e d i st an ce fr o
D efini ti o n of Vi scosi ty
.
,
;
,
.
m
.
.
.
FL U ID MOT ION
to y t h e d efinition of v i scos ity
n
u
t
i
v
e
by
q
t
o
h
a
e
e
i
n
g
thi s equ ati on
t
s t a es
th at
8 69
t h e fo rce
t he
on
s
u r face
00, i s
m
v eloci ty of t h e fl ui d at a d i stance 3; fro t h e
s ur face an d
Ar ea r ep r es en t s t h e ex t en t of t h e s ur fa ce of 00, on whi ch
t h e fo rce i s m
eas ured
pt i s t h e coefli ci en t of v iscosity
If t h e fl ui d v elocity i s n ot p rop or tion al to y but h as a f or su ch
as th a t show n by t h e d ott ed lin e of Fi g 190 t h e fo rce on t h e s ur face i s
Q
2
Area X t c
In exac tly t h e sa e way t h e for ce actin g on a
In
is the
0
,
m
.
.
)
(By
,
m
s u rface
flu id
e
uf
s r ac e
q u i v al en t to
su ch
the
DD ,
as
mt
t t
en
s a e
F1o l 9o
.
i s Ar ea X 11 x
.
th a t
i
v
(ay
Th e
DD |
t h e forces d u e
d efiniti on
to v i scos ity
—Lam
inar m
ot i o n of a viscous fl u id
.
ar e
is
o
r
p
.
m
p or tional t o t h e ra t e at whi ch n eighbour ing p ar t s of t h e fl ui d ar e ov i n g
p ast each oth er
—
D
i
n
i
e
t
e
r
a
t
o
n
o
f
f
t
h
Experi en tal
e
I
oti on of a v i scous fl u i d as
p
d efi ned ab ov e b e e xa i n ed i n t h e cas e of a ci rcular p i p e p ressu re b ein g
a pp li ed a t t h e two en ds it i s foun d th a t un d er cer t a in ci rcu s t an ces t h e
th eor eti cal co nsi d er a tions More
o tio n can b e cal cul a t ed i n d et a il fr o m
o v er t h e p r ed i ctions of th eo ry ar e accu ra t ely bo rn e out by d ir ect exp er i en t
O nly t h e r esult of t h e ath e a ti cal cal cul at ion will b e gi v en as i t i s d esi red
to d raw a tten ti on to t h e r esult s ra th er th an to t h e etho d of cal cul ati on
Th e q ua n tity of fl u i d fl owi ng p er secon d thr o u gh a p i p e of l ength l i s
fou n d th eoreti cally to b e
m
m
.
m
m
m
..
m
,
,
.
m m
,
v ol
m
1
31—1
12a;
1
"
.
e
r
s
ec
.
p
d
4
m
.
,
.
2
.
et er an d p l an d
th e p r essu r es a t th e en d s of t h e
wh er e d i s t h e d i am
es th a t p i s co ns tan t a n d th at t h e m
l eng th l Th e cal cul ation assu m
oti on
sa tis fies t h e co n d iti on of no sli pp in g a t t h e s i d es of t h e tub e
en t i s carr i ed out i n cap ill ary tub es of
Wh en t h e corr esp ondi n g exp er i m
et ers an d di fferen t l en gths it i s f ou n d th a t t h e law of v ar i a
d i fferen t d i am
tion gi v en by (8) i s sati sfied v ery accur a t ely L am
b s t a t es th a t Poi seu ille s
t h e tim
e of efilu x of a gi v en v olum
en t s sh ow ed th a t
exp er im
e of w a t er i s
.
.
.
,
’
.
2
1
3
APP L IED AERODYN AMI CS
8 70
l ength of t h e t u b e i n v ersely as t h e d i ff erence of p r essu r e a t
a
or
m
e
r
e
a
m
e
t
e
r
o
f
h
u
o
r
d
i
v
ly
f
th
p
ow
d
i
t
F
ul
s
t
h
e
a
s
n
s
a
n
d
n
er
e
e
t h e t wo
s
a
a
m
f
a
n
i
s
i
c
c
n
i
n
th
gi
v
p
ti
l
d
t
i
g
whi
h
t
th
t
l
o
t
e
m
s
f
er
e
a
n
o
m
ca
a
n
e
s
a
r
c
e
8
p
( )
i n in g s tan d ard v a lu es for any flu 1d
a lw ay s a d op t ed i n d e t erm
s
1
s
1
n
t
h
e
o
n
o
r
n
c
e
s
k
t
h
i
d
i
t
d
by
it
ily
th
a
t
i
f
i
ti
ee
n
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8
i
a
a
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s
n
c
A
p p
( )
di rectly
as
the
,
.
,
.
.
e qu al
of t h e
the
t o (p ,
the
or
cross- sec
ti on
Al so t h e
.
a
t ot a l fo rce an d v t h e v el ocity
v ol
Su b stituti ng for p ,
h av e
mwhi
we
p r essu r e d rop a nd t h e
V 0 l p er 8 ec
v eloc1t y
v erag e
,
the
If
.
.
19
ar ea
F
18
th en h av e
z
77d
'
3
v
s ec.
er
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of
—
(P1 l
F
we
p r o d u ct
‘
l
1
o
l
d
v
a
n
p2
.
er
s
ec
.
p
i n ( 8) t h e
v alu es gi v en by
h i t a pp ears th a t t h e force i s p r op or ti onal to t h e coeffi ci en t
of v i sco sity i s t h e v el oci t y v and to t h e l ength of t h e t u be 1 Th e v ari a tion
otion
of force a s t h e fi rs t p ow er o f 0 a pp ears t o b e ch arac t eri sti c of t h e m
of v ery v iscous fl u i d s
If t h e exp erim
p t ed i n a l arge t u b e at high Sp eed s t h e r es is t
en t i s a tt em
a t ely a s t h e s qu are of t h e s p eed a nd it i s
a nce i s f ou n d t o v a ry a pp roxi m
th en cl ear th a t eq u a ti on ( 8) d oes n ot h ol d Th e exp l ana ti on of t h e d i fl erence
oti ons w a s fi rs t gi v en by P r of essor O s bo rn e
o f high sp ee d an d low Sp eed m
R ey n o l d s who illus t ra t ed h i s res u lt s by exp er im
en t s i n gl as s t u b e s
Wa t er
a t ank w as a ll ow ed t o flow sl owly th rough t h e tub e i n t o whi ch w as
fr om
i tt ed a s t reak of colou r ; so l ong a s t h e sp eed was k ep t b elow a
al so a d m
cer t ain v a l u e t h e colou r b an d w a s cl ear an d d i s ti n ct i n t h e cen t r e of t h e
tub e As t h e sp eed was rai s ed gra d u ally th ere cam
e a tim
e a t w hi ch
more or l ess su ddenly t h e col ou r b roke u p i nto a conf u sed mass and b ecame
mi xed wi th t h e gen eral body o f t h e w a ter Thi s i n d i ca t ed th e pr od u cti on
of edd i es a n d P r of es sor O s b or n e R ey n ol d s h a d s how n why t h e law o f
moti on as cal cul at ed h ad fail ed
en t fu r th er
i t was shown th a t t h e law of
Car ryi ng t h e exp er im
b reak d o wn cou l d be fo rm
ula t e d th a t i s h av i ng observ ed t h e b reak
d own i n on e case b r eak d own co u l d b e p r ed i ct ed for oth er t u b es a nd for
e fl u i d a t d i ffe r en t t em
p era tu r es D eno ti ng t h e
o th er fl ui d s or for t h e sa m
mass of uni t v ol u me o f t h e fl ui d by p O sborn e R eyn ol d s fou n d th a t b reak
d own o f t h e s t ea dy flow alway s occu rr ed wh en
fro
c
.
,
,
.
,
’
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-
-
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,
,
,
,
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,
,
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,
.
,
,
,
,
.
,
,
CH APT ER VI II
D YNAMI CAL S I M I LARI T Y AND SCALE E FFECTS
m
m
—
Th e i d ea of sim
il arity as app li ed t o so li d obj ec t s
Geo et ri cal M ari ta
in ed by i t s sca l e b u t if
Th e actua l s iz e of a bo dy i s d eterm
i s fa ili ar
by su ch a r ed u ction as occurs i n ta ki ng a p ho tograp h i t i s p oss i bl e t o
ilar If one
make t wo bod i es app ear ali ke t h e ori ginals are sai d t o be si m
of t h e bod i es i s an a ircra ft or a st ea m
s hi p a nd t h e oth er a s m
all s ca l e
all er bo dy i s d escri bed as a m
o d el
r ep ro d u c ti on of it t h e s m
ilarity ex t en ds t h e ab o v e s im
p l e i d ea to co v er t h e m
oti on
Dyna i cal Si m
of si m
il arly sh a p ed bo d i es Not only d oes t h e th eo ry co v er si m
il ar m
o ti ons
bu t also t h e sim
i lar m
oti ons of fl u i ds
o f aerop l a n es a n d oth er a i rcra ft
It ay app ear t o b e u sel ess t o a tt em
p t t o d efine si il arity o f fl u i d m
o ti ons
otion i s in ca l cu l a bl e bu t t h i s i s not t h e case
i n those cas es wh er e t h e m
It i s i n fac t p ossibl e t o p red i ct si m
il arity of m
otion t o lay d own t h e
l aw s with consi d era bl e p rec isi on an d t o v er i fy th emby di r ect ob serv atio n
e of t h e
Th e p res en t ch ap t er d ea ls wi th t h e th eory i t s a pp li ca tion an d so m
more st ri ki ng and im rt ant exp eri men t al v er ifications
ple the m
oti on of t h e li nks of Pea u celli er
A co n v eni en t ar bit rary exa m
en tal i d ea s r ela ti n g to
cell s l ea d s to a r ea d y a pp rec i a tio n of t h e fun d am
A Pea u celli er cell cons is t s of t h e sys t emof li n ks ill us
il ar m
otio ns
s im
t rat ed in Fig 19 1 Th e fou r lin ks CD DP PE an d EC ar e equ al an d
AD a nd AE are equa l an d are hi nged t o
freely j oin t ed t o each o th er
CDEP a t D an d E an d t o a fi xed base a t A Th e li n k B C i s hi nged t o
e fi xe d b ase a t B
otio n
Th e o nly p o ssi bl e
CDEP a t C a n d t o t h e s am
of P i s p erp en d i cu l ar t o AB F Th e i m
po rt an t p oin t for p r es en t p ur pos es
i s th a t for any gi v en p o siti on of P t h e p ositi ons of D C a n d F ar e fi xed
ech an i sm
by t h e li nks of t h e m
Co ns i d er n ow t h e m
otion of a seco n d cell whi ch i s L ti m
es gr ea t er t ha n
tha t of Fig 191 an d d enot e t h e n ew p oin t s of t h e lin k w or k by t h e sa e
l et t ers with d ash es Th e l ength AN will b eco m
e
Put P
an d t h e sh a p e of t h e li nk w or k w ill
i n s u ch a p os ition th a t
ilar t o th a t of Fig 19 1 A l i i t ed class of s im
ilar m
oti ons m
ay now
b e s im
es t h e two cel ls h a v e
b e d efin ed for t h e cell s as b ei ng su ch th a t at all ti m
il ar sh a p es
sim
otions i s obtained by co ns i d ering
An ex t ens io n of t h e i d ea of si il ar m
il ar p ositions to occur at d i fferen t tim
Im
es
a gin e two ci n em
a
t h e s im
o ti on s of t h e cells t h e i m
p l oy ed t o p hotograp h t h e m
ag es
cam
eras t o be em
e si ze of p i ctur e
b ei ng r ed uced so as t o gi v e t h e sa m
oti on
Make one m
ov e t h e co rr esp on d in g cin em
twi ce as f as t a s t h e oth er an d m
a ca
era
twi ce as f as t Th e p i ctur es t ak en will b e exact ly t h e sam
e for both cells
oti ons w ill ag a in b e ca lled s i m
ilar m
oti on s
We a re th u s led t o
an d t h e m
.
,
.
-
m
,
.
.
m
m
,
.
,
.
,
,
,
.
m
,
.
,
,
.
.
.
,
,
.
m
.
.
.
,
.
.
m
,
.
'
.
.
.
m
,
.
m
.
,
.
.
m
,
.
3 72
SI MILARITY AND SCAL E EFFECT S
DY N AMI CAL
8 73
o i d er a scal e o f ti m
e T a s w el l as a scal e o f l ength L
ilar
All s im
motions ar e red ucibl e to a st and ard motion by ch anges of t h e scales of
l eng th and t i m
e
If t h e li nks be gi v en m
ass it will b e n ecessary t o a pp ly fo rce a t t h e
p oi n t P i n o r d er to m
oti on of a ny p r ed et erm
in ed ch arac t er and
a in t ai n m
t h is fo rce d ep en d i ng as i t d oes on t h e m
ass o f ea ch l i n k m
ay b e d i fferen t
for si m
ila r m
otions Th e stu dy of t h e forces p r o d u ci ng m
otion is known as
dyn am
i cs an d dyn am
ical s im
il arity i s t h e di scu ssion of t h e con d itions
un d er whi ch t h e e x t erna l forces act i n g can p ro d u ce si m
il ar m
otions
p l e an ext ernal force can b e p rod u ced
Still r et ai n ing t h e cell as ex am
by a sp ri ng stretch ed b etw een t h e p oi n t s P an d F Th e force i n this Sp rin g
d ep en ds on t h e p osi ti on of P an d th erefo re on t h e m
oti on of t h e cell I t
may b e ima gined that a sp ring can b e p rod u ced h av in g any law of force
c ns
.
,
,
,
.
,
,
,
.
"
,
.
,
.
.
,
Fro 19 1
.
.
—P
ea u ce llie t cell.
functi on of ex t ensi on an d if t w o su it a bl e Sp ri n gs w ere u sed i n t h e
o tion s coul d b e
ilar f ree m
il ar cell s i t wou l d th en foll ow th a t si m
si m
ass i n t h e two cases
p rod u ced no m
a tt er wh a t t h e di s t r i buti on of m
as a
,
.
,
Parti cu lar CIass
Si
of
m
il
ar
Moti ons
thi s p o i n t t h e genera l th eo r emwhi ch i s i nt rac ta bl e i s l eft for an
im
p or ta n t p ar ti cul ar cl as s of m
p li fied as b elow In t h e cell
o ti ons e x em
ay s till b e su pp osed to b e qui t e a r bi
ass m
o f Fig 19 1 t h e di s t r ib u tio n of m
a d e whi ch r eq u i r es
a r es t ri cti on i s m
il a r m
ech an i sm
t rary b u t in t h e si m
a ss sh a ll b e M ti m
es a s gr ea t a s
th a t a t each of t h e si m
ilar p oin t s t h e m
th a t for t h e cell of Fig 19 1
en t of t h e secon d cell
il ar m
F or s i m
o tio ns of cell a ny p ar ti cu l ar el em
mov es in th e same d irecti on as t h e corr es p on d in g el emen t of t h e first It
I t s v eloc ity i s th er efo re
e T ti m
es as gr ea t
mo ves L ti mes as far i n a tim
At
,
,
.
.
,
.
.
.
.
ti m
es
as
p rod u ci n g
of
t he
g rea t
motio
forces
,
n
on
a nd
is
its
q
e ua
th e
o
l
acce era
es a s
tion 12 ti m
’
T
l to t h e p rod u ct
c rres
m
en s
is
t
l
.
acce era
ass
mt
p ond in g el e
a
r
e
g
f
?
fi
ll/
Si nce t h e
,
t he
tio
,
ti on
Thi s
ra
force
ra
tio
for t h e
APP L IED AERODYN AMI CS
3 74
li m
it ed as sum
p tion as to di st rib u ti on o f m
ass i s con s t an t for all el em
e nt s
a nd m
us t al so app ly t o t h e wh ol e m
ech ani sm Th e f orce a pp li ed a t P
,
'
.
w ill th er efo re b e
HIJ
l—
—
-
3
T
tim
es th a t a t P if t h e r es ulti ng m
otio ns are sim
ilar
Th e co ns t r ai n t s i n a fl ui d
t h e Peau celli er cell , b u t th ey
o ti on o f ea ch el e
en t
Th e
m
.
d ifferen t fr o th ose du e to t h e lin ks o f
n ev er th el ess a rise f r om
t h e s ta t e of
o ti on
u s t b e co ns i d er ed in s t ea d of t h e
oti o n
m
of a ny o n e p oin t a n d t h e force on it d u e t o
p ressur e vi scosity gr av ity et c
us t
be
a t ed
es ti m
If t h e fl u i d b e i nco m
p ressibl e
as s of corr espo n d in g el em
en t s wi ll b e
the
p r o por ti onal t o th e d ensity and v olu e
e t he
otion
Cons i d er as an exa
of
si
il ar cyli nd ers th r ough w a ter an accoun t
of whi ch w as gi v en i n a p re v iou s ch a p t er
Th e cyli nd ers bein g v ery l ong i t m
ay
U b e as s um
ed th a t t h e fl ow i n all sec ti ons i s
t h e sam
e an d t h e equ a tio ns o f m
oti on for
t h e block AB CD Fig 192 co nfin ed to t w o
F m
19 2
d im
p r essi bl e
en si on s
Th e fl u i d b e i n g in com
otion
a n d without a f r ee sur f ace gr a v ity will h a v e n o i n fl u en ce on t h e
an d t h e fo r ces on AB CD will b e d u e t o e flec t s on t h e fa ces of t h e bloc k
a l an d t an gen ti a l p r essur es d u e t o t h e
Th ese ay b e d i v i d ed in t o norm
ac ti on of i n ert i a a n d sh ear o f t h e v i scou s fl u i d
F r omany t ex t boo k on Hy d ro d yn am
i cs i t wi ll be foun d th a t t h e app ro
r
i
a
t
e
q
tio
of
ti
o
n
o
f
t
h
e
bl
o
c
k
ar
e
u
e
a
n
s
m
o
p
are
m m
,
,
m
.
,
,
,
m
m
m
.
.
,
m
m
m
m
m
.
,
.
,
,
.
.
,
,
.
m
.
,
,
'
m
.
’
.
-
m
P
“
5
5
e
+
a
e
By
It i s t h e sol u ti on of th ese e q u a tion s for t h e corr ect co nd iti ons on t h e
n d a ry o f t h e cyli n d er w h i ch w o ul d gi v e t h e d eta il s of t h e edd i es s hown
ou
b
ma p rev i ou s ch ap ter With su ch a sol u tion t h e p res en t d i scu ssi on i s n ot
con c e rn ed a n d it i s o n ly t h e g en era l b ear i n g o f e qu a ti ons 1 whi ch i s o f
( )
i nt eres t Eq u a ti on s ( 1) a re th ree r el a ti ons f r omwh i ch t o find t h e q u a n ti
ti es u v a n d p a t all p oi n t s d efined by a:and
If by an y sp eci al hy p o
th esi s u a nd c b e kn own th en p i s d et er m
i ned by eith er t h e firs t or t h e
s eco n d e q u a ti on of
C on si d era ti on of t h e firs t eq u a ti on i s all th a t i s
r eq
mr ed i n di scu ssing s im
i l ar m
o tio n s
D efin e a secon d m
oti on by d ash es t o o bta i n
z
u
w
7
a
a
1
n
5
3:
.
,
.
,
,
.
j
.
'
D1
As
a
pp li ed
to
a s
ex
’
)
( W)
2
im
il a r an d si m
il arly sit u a t ed block th ere will
be
t
cer a i n
APP L IED AERODYN AMI CS
3 76
i l ar m
be tw een corr esp on d ing p oin t s in sim
otions t h e increm
en t s o f
‘
p r essure dp v ary a s p U
altho u gh
e l en gth
as will b e
Thi s case h as been d ev elop ed at som
t h e law of corr eSp on di ng s p eed s can be foun d v ery ra p i dly
shown
withou t Sp ecifi c r efer ence to t h e equ ati ons of otion It h as been sh o w n
en t al b asi s why a law of corr eSpon di ng sp ee ds i s requi r ed i n
o n a fun d am
t h e case of cylin d ers i n a v iscous fl ui d an d th a t t h e p r su res th en calc u
l a t ed as actin g in si m
i l ar otion s ob ey a cer t a in d efin i t e law of co nn ecti o n
i
Two m
ot ons of v isco
ay be ex p r essed i n w or d s as f ollow s
Th e r es ul t m
and
.
,
m
,
m
.
—Appli
.
.
m
,
.
LOG | o —Q
%
20
Fro 193
,
ca ti on of
t he laws
of s i
m
il
ar it y
to th e
resi st ance of cy linders
m
.
fl ui ds will be si i l ar i f t h e si z e of t h e ob stacl e an d i t s v el ocity are so rel a ted
to
t he
v isc si
o ty tha t
UD
-
-
is
v
o
c ns t a n
t
Th e
.
p ressur es
a t all si
mil
ar
p oin t s
t h e t wo fl u i d s will th en v ary a s
’
Si nce t h e p r ess ur es v ary as p U at all p oin t s of t h e fl ui d , i nclu d in g t hose
' ’
on t h e cyli n d er , t h e tot a l r esis t ance will v ar y as p U D , an d it follows th at
of
R
pU D
’
v
ar i es only wh en
a
[
2 v ar i es
v
.
mi whi h it
m th wi p
Th e law i s n ow s t a t ed i n a for
S
i t t e d to exp eri en t al ch eck
m
m
‘
.
n
oo
c
r es
can
ro
d ily b e su b
ran ge of cyli n d ers
v ery
vi d e a
rea
DYN AMICAL
of
di am
et ers
di fleren t
'
i d erabl e
cons
SIMIL ARITY AND SC ALE EFFECT S
th ey
and
,
g of Sp eed
r an e
can
t es t ed
be
Two out of
.
in
1
wi n d ch ann el ov er
a
thr ee qu an titi es
th e
377
a
in
th en i n d ep en d en tly v ari a bl e an d t h e r esi st ance of a w ir e
in in d i am
et er
t est ed a t a sp eed of 50 ft s can b e co m
p ared wi th th a t of a wi re 0 5i n
i n d i am
e t er a t 10 ft s
Th e exp eri m
en t h as b een m
ad e t h e d i am
et er of t h e cylin d ers v a ryi ng
i ns a n d t h e wi n d s p ee d s f r o m
f r o m0002 i n to
10 to 50 ft s
Th e
n um
b er of o bs erv a tio ns was r o u gh ly 100 an d t h e r esu l t i s shown i n Fi g 193
,
.
-
.
.
.
-
.
.
,
-
.
.
.
,
.
I n st ea d of HEas a v ari a bl e
V
u ve
i ng
c r
is
o r d inat e wi th
th en
,
mo
III2
log
t h e v a l u e of log
r e ea s
ily
b i
as a sc ssa
-
d
r ea
is
Th e
.
.
.
,
to gi v e a
h as b een
r es
used
,
as
ult of p lottin g
narr
t h e resul t
2
65
5
1
2
as
ow b an d of p oin t s whi ch
e t ers
i nclu d es all ob ser v a ti ons for wi res of th i r t een di fl erent di am
ilar m
otio ns i s
Th e ra th er sur p ri si n g r esu lt of t h e consi d eration of s im
th a t i t i s p ossi bl e to say th at t h e r es i sta n ce of on e bo d y i s cal cula bl e fr om
en t al tho u gh
th a t of a sim
i l ar b o dy if du e p r ecautions are tak en in exp eri m
n eith er r es i s t an ce i s ca l cul a bl e f r om
fi rs t p r in ci p l es
Th e im
p o r t ance of
o d el s wi ll b e a pp r eci at ed
t h e p r i nci p l e as a pp li ed t o ai r cr aft an d th ei r m
V
'
.
,
.
Furth er
.
.
m
nstrati ons of
m
th e Law of Corresp onding Speeds for Inco p ressible
Vi scous Flu i ds
m
xp er i en t s t o tho se on cyli n d ers i s gi v en in t h e
P hilosop hical Tr ans acti ons of the Roy al Soci ety i n a p a p er by St an to n a nd
P ann ell Th ese exp er i en t s cons titut e p erh aps t h e os t co nv i nci n g
p tion th a t in a ny
e v i d en ce y et a v a il a bl e of t h e s u ffi ci ency of t h e a ss um
ilarity to fl u i d
i cal si m
otion
a ppli ca tions of t h e p r in ci p l es of d y n a
p ort an ce
ev en wh en t u r b u l en t v a n d p are t h e on ly p hy si cal co n st a n t s of i m
Th e p i p es w er e ad e of s ooth d rawn b rass an d v ar i ed fr o m
in
t o 4 i n ch es
B oth wa t er an d air w er e u sed as fl ui d s an d t h e Sp eed r a ng e
w a s excep tion ally gr ea t co v erin g fro
1 ft s t o 200 ft s a t or d i n ary
osp h eri c t em
p era tur e an d p ressu re Th e v alu e of v for air i s approxi
at m
a t ely 12 ti m
es th a t for w a t er
A
p arall el
se t
of e
m
.
m
,
m
m
m
m
m
,
.
,
m
.
,
m
,
.
,
.
-
.
.
-
.
.
.
Th e
u v e co nn ecti ng f ri ction
c r
on
the
w alls of
the
D
U
—
p i p es with
;
s
i
p lott ed as for Fi g 193 wi t h a r esult of a v ery si m
il ar ch aract er as to t h e
ea n li n e
Th e exp er i en t s co v ered n ot
Sp rea d i n g of t h e p oi n t s a bout a m
only t h e fr i ction a l r esi st a n ce but a l so t h e d i s t ributio n of v elo city a cr o ss
wa s
.
m
,
.
the
pipe
,
an d s
how ed th at
th e
fl ow
a t all
p oin t s i s
V
a
at
function of
Th e
v
o rigi n al p ap er sh oul d b e consult ed by tho se es p eci ally i nt erest ed i n t h e
th eory of s i m
i l ar otion s
en t a l w ork a s t r i ki n g O p ti ca l illu s t r a tion of
In t h e co u rs e of exp eri m
il arity of fl ui d m
oti on h as b een fou nd Work i ng wi th w at er E G Ed en
si m
m
.
.
Fu rt h er pa r t i cu lars
a re
gi ven i n R a M No 102, Ad vi so ry Co
.
.
.
,
mmi t t
ee
.
.
for Aer ona u t i cs
.
APP L IED AERODYN AMI CS
3 78
ob ser v ed t ha t t h e flow r ou n d a sm
a ll in clin e d p l a t e ch an g ed i t s ty p e as t h e
Sp e ed of flow in cr eas ed
I n on e cas e t h e m
otio n i ll u s t ra t ed i n Fig
194 was p rod u ced ; t h e
col ou r ed fl u i d f or m
e d a co n t i n u ous s p ir a l s h ea th
an d t h e m
otion was
a pp ar en tly s t ea d y
I n t h e oth er cas e t h e m
oti on led t o t h e p r od u ctio n
o f Fig 19 5 a n d t h e flow w a s p er i o d i c
Th e flow Fig 194 i s fr oml ef t
to right t h e p la t e be i ng a t t h e e x t rem
e l eft of t h e p i ct u r e
Th e s t rea m
w as r en d ere d v i s ibl e by u si n g a soluti on of N es t lés m
il k i n w a t er and t h e
whit e st rea k Show s t h e w ay i n whi ch thi s col our i n g m
a t er i al en t er e d t h e
r egio n u n d er o b ser v a ti on
a tt er Sp rea d a nd
At t h e p l a t e t h e colour i ng m
l eft t h e cor n er s i n t w o co nti nu ou s sh ee ts wi n d i ng i n w ar d s Th e for mo f
th es e sh ea ths ca n be r ea lis e d fr om
t h e p h ot o gr a p h
For Fig 195t he flow i s i n t h e sam
e d i r ec ti on as b efor e a nd t h e p l a t e
more r ea d ily vi sibl e I nst ea d of t h e fl ui d l ea v ing i n a corkscrew shea th
the m
oti on b eca m
e p eri od i c a n d l oo p s w er e f or m
ed a t i n t er v als a nd su c
cee ded ea ch o th er d ow n s t r ea m Th e ob s er v a ti on of thi s ch ang e o f ty p e of
flo w seem
ed t o f or m
a co n v eni en t m
ea ns o f t es ti n g t h e su it a bili ty of t h e
law of s i m
il a rity th ought t o b e p r e p er t o t h e exp er i m
en t
To t es t for thi s
a sm
a ll a i r ch a nn el was m
a d e a n d i n i t t h e flow o f a i r w as m
a d e v i s i bl e
by t ob acco sm
ok e ca r e fully coo l e d b efor e u se
Th e effec t s of t h e h ea t
f ro mt h e el ect ri c a re n ecessa ry to p rod u ce en o u gh light for p hotogr a p hy
w as fou n d t o b e grea t er th a n for w a t er a nd equ a l st ea d i n ess of fl ow w a s
d i ffi cul t to m
ai nt a i n
I n Sp it e o f t h es e d i fficu lti es i t w a s i m
med i a t ely foun d th at t h e same
ty p es o f flow coul d b e p ro d u ced 1n ai r as ha v e been d ep i cted 1n Figs 194
a n d 195 Va ria ti ons o f t h e s i z e of p l a t e w ere t r i e d a n d in v ol v ed ch an es of
g
Sp ee d t o p ro d u ce t h e sa m
e ty p es o f fl o w
Two p h ot ogra p h s for ai r a re
s h o wn i n Fi gs 19 6 a n d 1
p ar ed with Fi gs 194 a n d
97 a n d sh ou l d b e com
o ke
195 fo r w a t e r
e d i rec ti on as b efor e a n d t h e sm
Th e flow i s i n t h e sa m
e
t
a
n
d
p
l
a
t
e
r
e
e
as
ily
s
h
h
a
th
i
s
n
o
t
so
p
er
f
ec
tly
i
6
a
ee
n
e
s
e
f
1
9
o
T
F
j
g
d e fin ed a s i n w at er but i t s ch aract er i s unm
i s t a ka bly t h e sa m
e as th a t
o f Fig
otio n fou n d i n
194
Fi g 197 foll ows t h e h igh speed ty p e of m
w at er and p h o tog ra p h ed i n Fi g 195
en t s
a ke t h e ch eck on si m
eas u r em
To m
il ari ty still m
o re co m
p l et e m
w ere t ake n o f t h e a i r a n d wa t er v el ociti es a t whi ch t h e flow ch ange d i t s
ty p e for all t h e si zes o f p l a t e t es t e d T a k i ng th ree p l at es 1 i h i i n
a n d 2 i n s qu a r e all i n w a t er it w a s fou n d th a t t h e s p eed s a t whi ch t h e
fl ow ch a nge d w er e r ou ghly i n t h e r a ti os
1 1 r esp ecti v ely
Usi ng a
p l at e 1; i ns s qu are i n t h e ai r ch ann el t h e s p eed of t h e ai r wh en t h e fl ow
—
e
w
f
e
r
a
a s o u n d t o b e 6 or 7 ti m
with i ih p l a t e
ch a ng d ty p e
es th at of w a t
!
Thi s i s i n acco r d ance with t h e law of si m
il arity whi ch st a t es th a t 2
.
.
“
,
.
.
,
.
,
.
,
-
,
.
’
,
.
.
.
.
,
,
.
,
-
.
.
,
,
.
,
.
.
.
.
.
.
,
,
.
.
.
,
.
-
.
.
.
.
,
.
,
.
,
,
.
,
.
.
.
.
.
ai n s
ho u l d b e co n st an t ; i f for i n s t an ce t h e fl ui d 18 no t ch anged v r em
If bo th I a n d v ar e ch ange d
co ns t a n t a n d u sh ou l d v ary in v ers ely a s I
es cl ea r ly
by d o u bli ng t h e scal e o f t h e m
od el a n d i n crea s i ng v 12 o r 14 ti m
en t s w e r e n o t so e xactly
u st be 6 or 7 ti m
es a s gr ea t
Th e exp eri m
0 m
but it i s cl ear th a t
carr i ed o u t th a t gr ea t a ccu racy cou l d b e o bt a i n ed
gr ea t accu r acy was n ot need ed t o es ta bli sh t h e g eneral law of si m
il ar ity
V
S
,
.
,
.
,
.
DYN AMIC AL
Th e Pri nci ple
SIMILARITY AND SC ALE EFFECT S
Di
of
mi
ens ons
as
app li ed to Si
m
il
Moti ons
ar
3 79
All
s d ep en d i ng on m
a ss M l ength
d yna i ca l e q u a tions are m
ad e u p o f t er m
s sep ar a t ed by t h e Sign of a dd iti on
e T a n d are su ch th a t a ll t erm
L an d ti m
e
d i ensions i n M L and T
or o f s u bt racti on h a v e t h e sa m
s of i m
p l es of som
ili ar t er m
p or t ance i n aer on au ti cs
e fa
As ex am
a
a d e t o t h e t a bl e be low
b
e
r efer ence m
y
m
-
.
,
,
“‘
m
m
m
.
,
.
TAB L E I
Angu lar
ve loc i t y
Angu la r
accelera t ion
Li nea r
.
a cceler a t ion
Force
Ki ne
mt i
a
c viscos it y
m
In or d er to b e a bl e t o app ly t h e p r inci p l e of d i ensi ons, i t i s necessa ry
p rod u cing
to know on exp er i en t al gr o u n d s wh a t quan titi es ar e in v ol v ed
otion U si ng t h e cyli nd er an i nco p ress ibl e v i sco u s fl ui d as an
a gi v en
e xa p l e, w e say t h a t as a resu lt of exp eri
en t
Th e res i s tan ce of t h e cyli n d er d epen d s on i t s Si ze, t h e v eloc ity r el a ti v e
to d i s t an t flu i d , on t h e d ensity of t h e fl u i d an d on i t s v i sco sity , an d so far
a s i s kn own on n o t h in g else
Th e l as t p ro vi so i s i p o rt an t , as a f ailure 1n
app li ca tio n of t h e p ri n ci p l es of d yna
i ca l si i l arity ay l ea d to t h e
d isco v ery of anoth er vari a bl e of i p o r tance
a th e a ti cally t h e s t a t e
Exp ressed
en t i s e qui v al en t to
m
m
m
m
.
m m
m
m
m
m
m
m
.
m
m
m
whi ch
m
is
o i t
c ns s en
m
t wi th
an
m
.
As t h e d i ension s of R andf u st b e t h e sa
o f j i s s ubj ec t t o cer t ai n
sh ow th a t t h e fo r
e xa
i ne t h e exp ressio n
m
m
unr es t ri ct ed
m
li ttl e cons i d era ti on will
res t r i ct i ons
For i n sta nce
e, a
,
.
i n t er p re ta tion
ML
M
L
T
d i ens ions of R ar e i whil s t those of ___ ar e F
w p
T
v
I
i fi
ensi on s of t h e two s i d es of ( 10) a r e i n co ns i s t en t
an d t h e d i
2 a
l
e
p
m
’
Th e
of
'
.
M
“
IT
’
2
n
.
6
APP L I ED AERODYN AMIC S
3 80
It i s not howev er
e qu atio n b e t h e sam
e
,
,
s
ufii ci en t th a t t h e di
m
en s
ions of
the
mof
ter
s
an
t h e eq u a ti on
a 2
=
It
pl v
m
1
1
( )
m
m
m
h as t h e corr ec t d i ens io ns, but cl ear ly
ak es no us e o f t h e co n d ition
th a t t h e fl ui d i s v i sco u s, and t h e f or i s t oo r est ri ct ed for v ali d a pp li ca tion
It will n ow be app r eci at ed th a t t h e co rr ect for of (9) i s th a t whi ch h as t h e
b i nation of t h e
cor r ec t d i
ensi on s a n d i s a l so t h e l eas t r es tr i ct ed co
m
qu an titi es whi ch m
a tt er
may be fou n d as follows
Th e r equir ed fo rm
e a s a p ar ti cular case of ( 9) th a t
Ass um
R
a n ew e qu a tion su b sti tut e for R
a n d t o fo r m
ensio n s
exp r essi ng th ei r d i m
.
m
.
,
,
M
Lb
1
1 1:
Equ a te t h e
d im
ens i ons sep ara t ely
th erefore a
1
.
For
the
L
r
L
p
,
'
,
l
,
v, an
d
v
qu antiti es
d
1
T
M we h a v e
M
“
q u a tio n i s
Lt
L
an d
e
f
For
.
M
an d
Il
z f’
1
2
( )
-
a
with a ==l thi s l ea ds to
b
Th e
e
qu ati on
for
T
20
d
4
is
T
2
T
d
c
d
e
2
F r o mequ ations ( 16) a nd ( 17) ar e th en obta i ned t h e rel a ti on s
a nd
an d
w ith
qu a ti on ( 12)
e
a
0
2
b
d
beco
d
m
es
R
cl
Th e
the
valu e of
d is
i l
l
di m
ensi ons of
u n d et er m
i ned
mi
ar e exa
ne
d
,
,
and
for
4
th e
o
for
be
fo u n d to
reas n
th ey wi ll
thi s will
be
be
seen
z er
o
.
if
It
s of t h e sa m
l o cl ear th a t any nu m
b er of t erm
but with d i ffer en t
e f orm
v alu es of d m
ight be a dded an d t h e su mwou l d still sati sfy t h e p rinci p l e
of di ensions All p o ssibl e com
b i nations ar e i n clu d ed i n t h e exp ression
is
V
a s
m
.
w
wh ere F i s an und et erm
i n ed fu nction
m
m
m
e)
s
m m
i la r o ti ons
et h od of d ea ling w i t h s i
u ch o f t h e
Th is for u la a nd
o f d i e n s i o ns a re d u e t o Lord Ra y le i gh , t o wh o
a grea t i n d ebt edn es s is a
s cient i fic w or kers i n aer onau t i cs
m
.
m
APP L I ED AERODYN AMI CS
3 82
oun d i n t h e u n d is tur bed fl ui d i t d oes not gi v e any i n d i ca ti on of h ow
r es i s t an ce v ar i es with v elocity
e im
portan ce i n aeronauti c s
Th e knowl ed ge of t h i s l a tt er poin t i s o f so m
otio n for an in vi sci d co m
p ressi b l e
an d a sol u tion of t h e e qu a ti on of m
fl u i d will b e gi ven in o rd er to in d ica t e t h e li m
it s within whi ch ai r m
ay sa fe l y
p ressibl e
be r egar d ed a s in com
In d ev elo p i ng B ern ou lli s equ a ti on wh en d ealin g wi th i n v i sci d fl u i d
motion t h e equ ation
dp + pvdv = 0
2
5
( )
b etw een p ressu re d ensity and v elocity was ob t ained an d in tegra t ed on t h e
p ti on tha t p was a co ns tan t Th e fl u i d now cons i d ered be in g com
ass u
p ressibl e th ere i s a rel ati on be tween p and p whi ch dep en d s on t h e law o f
ing a d i a ba ti c flow t h e r ela t ion i s
Assu m
exp a ns i on
of s
,
.
,
.
’
m
,
.
.
,
o
.
e s t an d a r d p o in t i n t h e s t ream
wh ere p a an d po r efer to so m
wh ere v i s
u n i fo rm
an d eq u a l t o v0 an d y i s a consta n t for t h e
Dif
ti
ti
as
e
f
re
n
a
ng
g
i n (26) a nd s u bs tit u ti ng for dp i n ( 25) l ea ds to
.
2
1 P0
7
and
of
t h e co ns tan t i s
2
d i t b d fl ui d
2
an d
s ur
.
e
.
i
z
v
t
co ns a n
v lu at ed by p uttin g v
00
t he
the
e a
e
t
27
0
qu al to
s
o
,
un
is
(P
f
2
74
Equ a tio n (27)
q u are of
wh en p
beco m
es
po
.
Th e
v al u e
v elocity of soun d i n t h e
P
P0
i
and s n ce
l tion fro m(28)
is
a new re a
(
2_
po
)r
1
-
2
1
§
v
-
a
e
’
z
; fi
”
m
Th e grea tes t
positi v e p ressu re d i fference on a o vin g bod y occu rs
s t a gna ti on p oi n t
at a
i a wh ere
Mak in g v = 0 i n (29 ) a n d
exp an d i ng by t he bino i al th eo re m
”
m
,
.
.
8
or si nce
°
a
‘
DYN AMI CAL
3
9
If
be
SIMI LAR ITY AND SCALE EFF ECT S
fp
mll t h i
mly mov i g t mi
l p e im t
F
e
a
s
n cr ease o
t g
ressu r e a t a s a n a
38 3
tion p oi n t ov er th a t
n
un ifor
s r ea
s
an d thi s v al u e i s u s u ally fou n d
or ai r a t
or d i nary t em
p era t u r es t h e
i n wi n d ch a nn e ex r en s
v elocity of so un d i s a bo u t 1080 ft s a n d t h e v el ocity of t h e fast es t
a erop l an e i s l es s th a n one qu ar t er o f thi s
Th e secon d t er mof ( 8 2) i s
th en n ot m
ore th an 1 5 p er cen t of t h e firs t As t h e gr ea t es t su cti on on
eri cally th ree or four tim
es th a t of t h e g rea t es t
a n aer op l ane wi n g i s n u m
p os iti v e i ncre m
p ress ibility m
en t t h e effec t of com
ay l oca lly b e a li t tl e
mo re mar ked bu t t o t h e ord er of accur acy y et reach ed air i s subst an ti ally
i nco m
p ress i bl e for t h e m
otio n ro u n d wi n gs
e eq u a tio n show s th a t for a i rscr ew s t h e ti p s of t h e bl a d es
Th e sam
ay reach sp eed s of 700 or 800 ft s
t h e effect o f com
p ressi
o f whi ch m
bi li ty m
p or tan t At s till high er v el ociti es i t app ears
ay b e exp ec t ed to b e i m
t h a t a r a d i ca l ch ange of ty p e of flow occurs and wh en t h e ti p sp eed
etho d s of d es ign
al m
ex cee d s th a t of h al f t h e v el ocity o f sou n d n orm
en t e d by t erm
s d ep en d i n g on co m
p ressibility
n ee d t o be s u pp l em
—
i
M
o
t
i
o
n
s
as
a
ff
t
a
ti
o
n
An aerep lan e l a s u pp o r t ed
i
l
r
e
c
t
e
d
b
G
r
a
v
i
S
a
y
oti on
a ga i ns t t h e action of gr a vi ty an d h en ce g i s a fac t or on whi ch m
d ep en d s Ignori ng vi scosity and co m
p ressibility t em
p ora rily t h e o ti on
will b e see n t o d ep en d on t h e a ttit u d e of t h e a e rep lan e i t s s i z e i t s v el ocity
en
Th e p ri nc i p l e of di m
on t h e d ens ity of t h e fl u i d a n d on t h e v alu e o f g
s io ns th en l ea d s to t h e e qu a tio n
of
t he
-
.
-
.
.
,
.
.
.
,
.
,
.
-
.
,
.
,
m
.
.
.
m
,
,
.
,
,
,
.
R = p l%
For
s t ea
two si m
il ar
o p l an es to h av e
aer
m
tb
1
us
e
mmotio
m F t
sa
z
d ily t h e i niti al v alu es of 3
1
the
t h e sa
9
an d t h e la w
ns
e
e
or
.
wh en not flyi ng
t i l p u rp oses
erres r a
v ery nearly co n s tan t
of co rr es p o n d ing Sp eed s say s th a t
t h e s p ee d o f t h e l a rg er a erop l ane
all er
u st b e gr ea t er th an th at of t h e s m
i n t h e p r opor ti on of t h e sq u a r e roo t s of th ei r scal es
Thi s ay b e r ecognised
as t h e Fr ou de s la w whi ch i s a pp li ed i n conn ecti on with N av a l Archit ec tu re
Th e i n fl uen ce of grav ity i s th ere felt i n t h e p ressu r es p ro d u ced a t t h e b ase
of w av es owi ng t o t h e w eight of t h e w a t er
—
E
f
f
i
l
f
m
v
i
Th e
Co bi ned
ects o
Vi scosi ty Co pr essi b i ty and Gra ty
p ri nci p l e of d i m
en si on s n ow l ea d s to t h e eq u a tio n
i
s
g
,
m
.
m
’
.
m
.
.
.
2
11
’
v
an d a
not
law
of
p ond in g sp eeds i s
corres
no
a
’
lg
)
l onger a pp li ca bl e
p oss ibl e mone fl ui d an d with t err est ri al con d iti ons t o
.
mk
It
a e
is
l
c ear
ly
2
?5 £
whi h
h cons t an t for two si m
il ar bo d i es ; It 18 only 1n th ose cases for
c
only one or t wo of t h e argum
en t s a re gr ea tly p red o m
i na n t th a t t h e p r in
ci p les of d yn am
i ca l si m
il arity l ea d t o eq u a tions of p racti ca l i m
p or t ance
—
l
m
il
m
t
i
P
r
o
i
r
St a c
b e s and S
a i ty of Stru ct u res
Th e r u l es d ev el op ed for
i l arity ca n be a pp li ed t o s tati ca l p robl em
d yna m
i ca l si m
s a nd on e or two
eac
1
.
APP L IED AERODYN AMIC S
3 84
m
i n t erest i n aerona uti cs
So e i d ea of t h e rel a ti on b e t w ee n
t h e s t rength s of s i i lar s t r u ct u r es can b e obt a i ned quit e r ea d i ly
Consi d er fi rs t t h e st r esses i n s i i l ar s t r u ctur es wh en th ey a re d u e t o
ay e ith er b e
a d e of t h e
Th e p ar t s
t h e w eight of t h e s t r u ctur e i t sel f
a t eri a l s , but t o t h e sa
If of d iffe r e n t
e dr awin gs
sa e or of di ffer en t
ed t o re t a i n
a t er i als t h e d ens iti es of co rr esp on d i n g p ar t s will b e assu
Si nce t h e d ens ity a p p e a r s
a co ns t an t p r ep o r tio n thr oughout t h e s t r u ctur e
3
b
ru
r
e
e
r
e
a
n
t
h
e
c
p
t
ly
w
ight
p
n
t
d
by
l
d
if
s
t
t
u
n
r
b e e es
e
e
ca
se ar a e , t h e
p g
I f f,
n ot r ed u n d an t it i s k no wn th a t t h e s t ress d ep en d s only on p , l an d g
is
ens io ns an d for
r ep r es en t s t r ess, t h e e qu ati on of co rrect di
cases ar e of
m
.
m
m
m
m
.
m
m
.
.
m
m
.
,
m
m
.
3
5
( )
a t eri al s s t r ess i s p r o p o r ti o n a l
Thi s equ ati on shows th a t for t h e sa m
e m
to t h e scal e of st ru ct u re an d for thi s con d ition of lo a d in g l arge s tr u ct ur es
all ones
It i s in accor d an ce with (8 5) th a t i t i s fo u n d
a re w eak er th a n s m
or e di ffi cult to b uild b ri dges as t h e sp an i ncreas es
t o be m
or e a n d m
e con d iti on of lo a d in g i s th a t in whi ch t h e w ei gh t o f
Th e oth er ex t rem
po r t an t and t h e st r es ses ar e alm
t h e st ru ctur e i s uni m
os t wh olly du e t o a
bo l
lo a d i ng not d ep en d en t on t h e s iz e of t h e s t ru ct u r e I f w be t h e sy m
il ar s t ru ct u res t h e
rep r es en ti n g a n ex t ernal a pp li ed lo a d fact or b etw ee n si m
p ri nci p l e of d i m
ens i ons show s th a t
,
,
.
.
,
.
,
l l oa ds i ncrease as P Le as t h e cross secti ons of t h e si m
ila r
memb ers eq u ation (86) shows th at t h e s tress is i ndep endent of t h e si ze
3
h
t
h
e
s
r
r
r
n
t u ctu e how ev e i cr eases as 1 if t h e sa e m
T e w eight of
at er i al s
are used
I n an aere plan e t h e co n d i tio ns of load in g ar e n ear ly th ose requi r ed by
If t h e loadi n g of t h e wi n gs i n p oun d s per s qu are foot i s const an t t h e
t ot al w eight t o b e carri ed vari es as t h e s qu are of t h e linea r d im
en sio n s
Of thi s to t a l w eight it app ears th a t t h e p ro p o r tion du e to st ru cture v ari es
all es t aerop l ane t o 3 8
fr o m
a bout 26 p er cen t for t h e sm
t
t
f
r
er
c
e
n
o
he
p
l arg es t p resen t day a erep lan e Th e ch ang e of lin ear di m
ens io n s co rre
Sp en di n g with th ese fi gur es i s l to 4 b u t it sh oul d n ot b e fo r gott en th at
i l ar i ty ar e app r eci ably d ep ar t ed fr om In bui l d ing
t h e p rinci p l es of si m
or e a tten ti on t o d et ails b ecause
a l ar g e aero p l an e i t i s p o ssibl e to gi v e m
of th ei r rel ati v ely l arg er s iz e an d b eca u se t h e scan t lin gs ar e th en not so
f requen tly d et erm
in ed by t h e li m
it ation s of m
an uf actur in g p r o cesses
all aerop l an es h a v e b ee n u sed for fi ghtin g p ur p oses wh er e th ey
Sin ce sm
are s ubj ec t e d t o h igh er s t r esses th a n l arg er aero p l an es a l ow er fa cto r of
all
sa fety h as b een allowe d for t h e l a tt er
ar gi n of sa f ety for s m
Th e m
p resen t day aereplanes wo uld b e alm
ost twi ce a s gr ea t as for t h e l arger
ilar d u ti es
on es if b oth w er e used on s i m
ulti
In t h e case o f en gin es t h e p ow er i s f re q u en tly i ncr eas ed by t h e m
p li ca t io n of u n its an d n ot by an i ncr ease of t h e di m
en sion s of each p art
ay b e t wo for 80 to 60 hors ep ow er 12 for 800
Th e nu m
b er of cyli n d ers m
h orsep ow er t o 500 h orsep ow er a n d for s till high er p ow ers t h e wh ol e engi ne
If t he
t
ex ern a
-
.
,
m
,
,
,
,
.
,
.
.
.
-
~
.
,
.
,
.
,
.
-
.
.
,
,
APP L IED AEROD YN AMI CS
3 86
ly be fou nd fr omt h e a bo v e
shows th a t
easi
fig u res t o be 4 4 2,
Fig
a nd
193 t h e n
.
R b ein g t h e r esi s tance of a p i ece of tub e of l en gth equ al to i ts d i a m
e t er
Th e r es i s t an ce o f t h e whol e tube i s th en
144 x 0 595x 00023 7 x
ft
At
to
.
x 100
t h e res i stan ce will b e differen t
mti
000175, a n d t h e k in e
R
is 4 3 2, and th at of
a
c
vi scosity to
i s 0 592
1?
5
-
-
d ens ity i s th er e eq
u al
Th e
.
.
Th e
0000201
.
valu e
of
Fi na lly t h e r es is tan ce i s 2 60lbs
.
2
3 54 lbs
2
-
m
p tionh
.
log
f
i
.
a d e th at r esi s tan ce v ar i es a s
b een m
R
t h e s qu ar e of t h e Sp eed an d t h e f ac t th a t a 2 h as cha ng ed 18 an i nd i ca t i on
In thi s cal cu l a tion
no assu
as
'
,
pl
of d ep ar t u r e from
t h e s qua re law and st ri ct
t)
si
m
m
m
il
i ty
ar
.
Th e v al u e
of
B
T
5?
fr o
0 59 5 to O 592 as a r esult of ch an gi n g t h e h ei g h t
fr o 1000ft t o 10, 000 ft Mo s t of t h e ch ange i n res i stan ce 18 du e to ch an g e
i n ai r d ens ity It ight h av e h ap pen ed th a t t h e cur v e of Fi g 193 h ad b ee n
h as
ly ch anged
on
m
.
.
.
.
h ori z on t al s tra ight lin e
a
,
woul d
h av e ch anged
n ot
an d i n
th a t
a t all, a n d
th e
case
motio
res is a n ce c effi c en
t
o
i
t 3}
w
?
e
v al u es of 3 woul d h a v e
n s a t all
ay th en r eg ar d t h e v ar i a ti ons o f t h e o r di na t es of Fi
b een s i m
ila r We m
g 193
sim
il ar ity It d oes n ot follow th a t si m
easur es o f d ep ar tu r e from
as m
il ar
.
.
.
fl ow n ecessarily occu rs wh en
!
be i ng th at if
is
o t
c ns an
v
If such curv es
as
motio might b
p i io w my
ns
rec s
a
n
es
In
5
“
e
s
a
cu ar
ar
e
a r cr a
a
th a t of Fig
.
193
do
es
se
.
gi v en attitu d e of
p d
s ee
v
e
n
i
g
,
and
by
i s wo
rt h
not
a
lu e
,
cases ou r
e o
e
an
;
3
9
t he
flu i d
the
m
u h th at
s c
of n early s i m
il ar otions an d thi s
ns b etw een m
o d el s of a ircraf t and t h e
e re a
r u na e
or aer onauti cs m
ost of t h e forces for
ai rcr aft or p ar t v ar y n ear ly a s t h e s quare of t h e
ea s
,
,
.
Th e law
of
i t
r es s a nce
e.
p i l a tt enti on i n i t s b ear in g
s ec a
t cond i ti on
kn owl edge i s
ly of i m
p or t ance as a co rrection
.
.
o
c rr ec
,
l ack of
1
11
i s on
i
t he
v ary grea tly wi th
m
my
mu t b md f t h i d
l tio
pp li p ti l ly to t h
ft th ml v
i
Fo t
t ly f
us e
c
a
vl
on
e
d escrib ed as n early si ilar an d with a cer ta in l oss of
say th at t h e resis t an ce o f t h e cy lin d ers d o es n ot d epen d
a
e
mv
sa
.
e
pp r eci a bly
mu h
t
h as t h e
on
the
p resen t p oi n t
.
B oth
mod l
e
DYN AMI CAL
an d a r cra
i
of
.
l o
is a s
mov
ft
to be
e
SIMILARITY AND SCALE EFFECT S
in t h e
t
cons a n
t
mmdium
t h erefore
follows th a t
o t
sa
it
e
e
an d
,
of i s c ns an
t
,
v
is
3 87
o t
c n s an
t
.
If
qu a tio n ( 3 7)
an d e
th en s how s tha t R i s co ns tan t Thi s m
ea ns th a t s i m
i l arity of flow can
o nly b e ex p ec t e d on th eo r eti cal gr oun d s i f t h e fo r ce on t h e m
o d el i s a s
r ea t a s th a t on t h e a ir cra ft
St a t ed in this way it i s o b v i ou s th at t h e
g
la w of co rresp o n d i n g sp eed s as a pp li ed to aero dy n am
i cs i s u sel ess for
c om
ay b e p ossi bl e to d oubl e t h e s iz e for
p l et e air craft For p art s i t m
wi n d ch ann el t es t s an d so get t h e ex act equi v al en t of a d oubl e win d sp eed
T hi s i s t h e ease for wir es an d strut s and t h e law of corresp on d in g sp eed s
V
.
.
.
,
,
,
.
,
For
ai rcra
ft
as a
wh ol e
i n v estiga t e t h e na tur e
an d
for
of F , o v er
wi ngs
th e
p ar ti cular it i s
in
n ecessar
l
whol e r ange of l b etween
i
V
mod
y
to
l
e an d
f ull s cal e if cer tai n ty i s to exi st and if t h e changes ar e gr ea t t h e assi s t ance
w hi ch m
o dels gi v e i n d esign i s corr esp on d in gly red uced si nce r es ul ts are
s ubj ec t to a scal e co rrec tion
—
n
n
Aeropla e Wi g s
Th e scal e effec t on aereplan e wi ngs h as r ecei v ed
or e att en tion than th a t of any oth er p ar t of a ir craft for whi ch t h e ran ge
,
,
,
,
.
m
}
9
of
ca nn
ot
be
m
o v er ed wi thout fli ght tes t s
c
It
.
h as
b een foun d p ossi bl e
fl igh t t o easure t h e p r essur e d is t ribution r oun d a wi ng o v er a wi d e
For t h e p ur p oses of co p ar ison a co m
p l et e o d el stru cture
r ang e of sp eed s
w as set u p i n a wi n d ch ann el an d t h e p r essur e d is tr ibution ob serv ed at
Th e full sca l e exp er i en t s are
or e d i ffi cul t to
co rres p o n di ng p o i n ts
o d el an d t h e accuracy is rel ati v ely l ess It i s
ca rry o u t th an thos e on t h e
how ev er gr ea t enough t o w arran t a d irect co p ari son su ch as is gi v en i n
Th e a b scissae of t h e d ia gr a s r ep r esen t t h e p osi tions of t h e
Fi g 198
p oin t s at w h i ch t h e p ressur es w er e eas ur ed whi ls t t h e v alu es of t h e l att er
’
l
d i vi d ed by p v ar e t h e o r d in a tes i n each case Th e p oin ts locat ed on t h e
Th e
u pp er sur face will b e cl ear fro t h e ar kin g on each d iagra
e ob ser v ed a ngl es of i nci d en ce for t h e low er
cur v es rep resen t t h e extrem
an d u pp er wi n gs of a bi p la ne t h e co n ti n u ou s cu r v es b ei n g obt ain ed on t h e
full scal e and t h e d ots on t h e m
od el
arked tha t n o h es it a ti on
Th e genera l si il ar ity of t h e cu rv es i s so
o d el wi ng i s nearly
w ill b e f el t i n sayi ng th a t t h e fl ow of ai r r oun d a m
il ar to th at roun d an aereplan e wing
si
A close exa ina tion of t h e di a ra s d i scloses a d ifference on t h e low er
g
su rface of t h e u pp er wi ng whi ch i s sy st e a ti c a n d gr ea t er th an t h e acc i
d ent al errors of ob ser v a tion It i s di fii cu lt to i agine any reason why thi s
di fi erence shoul d app ear on on e wi ng an d n ot on t h e oth er an d n o sat i s
fact ory exp l ana ti on of t h e di fferen ce h as b een gi v en I t m
ust be conclu d ed
fr o t h e evi d ence av ail a bl e th at t h e m
od el r ep r esen t s t h e full scal e wi th
en t s si n ce i t i s not p o ss ibl e
an accuracy as gr ea t a s th a t of t h e ex p er i
It follow s fr o thi s th a t
t o gi v e any qu an tit ati v e v alu e t o t h e di fference
un til a high er d egree of accuracy is r each ed on t h e full scal e t h e ch ar act er
en t s
i st i cs of aereplan e wi n gs can b e d et er i n ed co p l et ely by exp er i m
o d els
on m
in
m
.
.
m
.
m
m
-
,
m
,
m
m m
.
m
m
.
,
,
m
.
.
,
m
m
m
.
m
m
.
m
.
m
O
'
o
,
m
.
m
,
.
m
.
m
m
APP L IED AERODYN AMI CS
3 88
s o f p re
p ossibl e fr o md i a gr a m
s s ur e d i s t r i buti on a l o n e t o
d eterm
i ne t h e lift a nd d rag of a wi ng An i nd ep en d en t m
ea su re
en t i s
n ecessarv be for e reso l u ti o n o f fo rces ca n b e e ffect ed a n d o n t h e full sca l e
It is
not
m
.
,
F
m198 —C mp
.
.
a r i s o n of
o
w i ng ch ara cte r is t ics on t h e
md l
o
e
d fu ll scales
an
.
eas u r em
en t i nv ol v es e ith er a m
eas u re o f a n gl e of in ci d en ce of gli d in g
thi s m
a n gl e or of th r us t
Of th es e t h e d et erm
i na ti on o f gli d in g angl e with air
screw s t e p p ed gi v es p r o m
i se of earli es t r esult s of su fli ei en t accu racy For
,
.
.
APP L IED AERODYN AMI CS
3 90
Ju d gi ng f romth ese r esults alone it m
ight be e xp ect ed th a t for effi c i en t
o d el t es ts woul d b e v ery accura t e bu t th at a t v ery high an d v ery
flight t h e m
a gn itu d e w ould b e n eces
low sp ee ds of fl ight scal e facto rs of a pp r eci abl e m
om
en t all th a t can b e sa i d i s tha t full scal e exp er i
sary
At t h e p r esent m
ments have not sh own any ob v ious err ors ev en at t h e ex treme sp eed s
ore th an or di nary t esti ng app ea rs to b e r equir ed if t h e correc
et hi n g m
Som
tions ar e t o b e ev alu a t ed an d for t h e p resen t win d ch annel t est s a t = 3 0
ay b e appli ed to full sca l e
6 cho r d an d a win d Sp eed of 60 ft s ) m
wi thout any of factor
,
,
-
.
.
,
,
"
-
.
.
Vari ati on
of
th e
Maxi
mmLift C
.
Model Rang e of 112
n ei ghbour hoo d of t h e
u
axi
oefi ci ent
u
i n th e
m mm
m
.
v ari atio n of lift coeffici en t i n t h e
v ar i es v ery greatly f ro one win g section to anoth er For t h e for shown
in Fig 198 t h e ch an ges ar e app reci abl e but n ot v ery st r iki n g i n char ac t er
Chan gin g to a u ch thi ck er secti on su ch as i s u sed i n a ir screws t h e effec t
of ch ange of sp eed is ark ed an d shows th a t t h e fl ow i s v ery cr iti ca l i n
axi
umlift coefli ci ent Fig 199 shows a goo d
t h e n eighbour hoo d of t h e m
p l e of thi s criti cal flow Th e section i s shown in t h e top l eft han d
e xa
co rner of t h e fi gu re a n d t h e v alu e of of i s t h e p ro d u ct of t h e wi n d v elocity
axi
umdi ension of t h e section i n feet Wi t h
i n f eet p er seco n d an d t h e m
a t an angl e o f
axi m
u of
111 5t h e cur v e for lift co effi ci en t reach es a
i nci d ence of
an d a ft er a f all to
agai n r i ses so
ewh a t i rregul ar ly
at an an gle o f i nci d ence o f 4 0 d egr ees
to
At t h e oth er ex t re e o f
u
ax i
14 5 t h e firs t
h as a v alu e of
at
follow ed by a
of i s
a t 15 an d a v ery sh ar p r is e to
at
fall to
For gr ea ter an gl es
of in ci d ence t h e v alu e of t h e li ft coeffi ci en t f all s t o
at
an d agr ees
for t h e l as t 10 d egr ees of thi s ran ge wi th t h e v al u e for of= 5 In t er ed i a t e
ed i a t e v alu es of e l an d i t app ears p r ob a bl e
cu r v es ar e o bt ai n ed for i n t erm
ewh a t grea t er v alu e of c l th an 14 5 t h e firs t
th at at a som
ini u woul d
axi
d isapp ear l eaving a si ngl e m
um
of n early 08 Th e drag cur v es show
l ess stri ki ng b u t q u ite co nsi d era bl e ch an ges with ch an ge of 111
en t fr o m
th e an gl e of
Th e cu r v es for all v alu es of cl ar e i n goo d agr eem
n o hi t u p to 6 or 8 d egr ees an d for t h e high er v alu es of of t h e regi o n o f
a pp r eci abl e ch ang e i s res t ri ct ed t o a b ou t
If t h e exp er i en t s h ad bee n
3 0 it a pp ea rs p rob a bl e th at sub st an ti al i n d ep end en ce o f c l
carr i e d t o of
wo u l d h a v e b een a tt ai ned It i s to thi s s t age th at m
o d el exp eri en t s
Th er e
shoul d i f p o ssibl e b e ca r ri e d b efore a pp li ca tion to full scal e i s m
ad e
i s of co u rse n o cer t ai n ty th a t b etw een t h e l ar g es t c l for t h e od el an d th a t
for t h e aer o p l an e so e d i fferen t ty p e of criti ca l fl ow ay n ot exi s t
Th er e
i s how ev er com
p l et e a b sen ce of a ny ev i d ence of f u r th er criti cal fl ow an d
u ch evi d en ce t en din g i n t h e rev erse di r ecti on
Th ere are n o exp er i en t s on aero p l ane b od i es or on ai rshi p s an d th eir
mo d el s whi ch in di cat e any i nstability of fl ow co p ara bl e wi th th a t shown
for an aerofoil i n Fig 199
I n all cases th ere i s a t end ency to low er dr a g
c o effici en t s as 112 i ncr eases t h e p r op or tio n a t e ch an ges b ein g gr ea t es t for
T a bl e 3 sh ows t h ree ty p i cal result s ; i n t h e firs t
t h e airshi p en v el op es
n i s t h e sp eed o f t es t whil s t i n t h e o th ers ar e fi gur es showi n g t h e
col u m
ch a n g e o f dr a g coe
ci en t with ch an g e of sp ee d o r wh a t i s t h e s a
e thi n g
so l on g as t h e m
od el i s u n ch a n g ed with ch an g e 0v
Th e fir s t od el w as
Th e
m
m
.
.
.
m
,
m
.
.
-
.
m m
,
,
m
,
m mm
,
m
m
.
m
m
.
°
m
m mm
.
,
m
,
,
.
,
.
m
,
m
,
.
,
,
m
,
,
,
m
m
m
.
.
,
,
m
.
.
m
.
,
m
.
,
,
,
,
.
m
m
DY N AMI CAL
SI MIL ARIT Y AND SC ALE EFFECT S
39 1
m
om
p ar abl e i n s ize wi th an aerop l ane bo dy but i t s sh ap e was on e of u ch
l ower r esist an ce for a gi v en cross secti on Th e ch ange of drag coeffi ci en t
o ver t h e ran ge sh own i s a bo u t 8 p er cen t
p ar ison wi th ac t u al ai rshi p s
Com
i s di ffi cult for l ack of in form
a ti on , b u t i t i s cl ear th a t t hi s ra t e of ch an e i s
g
c
,
-
.
.
o i u ed u p to t h e v1sui t a bl e for airshi ps an d i t i s p rob a bl e th a t t h e
w h ch
a ni f es t a ti on of ch an g e of ty p e of flow f r om
r a t e of cha nge i s a local m
i
As ap ph ed
i t i s i m ssi ble t o dra w r eli a bl e d ed u ctio ns for ex t r a p ol a t i on
t o aere pla n e b od i es h owev er t h e r an g e of s t cov ered i s so gr ea t th a t t h e
n ot c n t n
,
m
o
.
,
APP L IED AERODYN AMIC S
3 92
light ex t rap ol a tion r eq u i red m
a d e witho u t d an ger
This con
ay be m
ns whi ch show t h a t wh e n
el u s i on i s s t r en gth en ed by t h e l as t two col u m
r iggi n g w i n d screen s e t c
are a dd e d t o a f a ired bo d y t h e d r a g coe fli c i en t
p ti on th a t t h e d ra g co effi
ch a n ges l ess r a p i d ly with vi a n d t h e u s u a l a s su m
c i e n t of a n aero p l a n e b o d y i s in d e p en d en t of vl i s s uffi ci en tly ac cu ra t e fo r
p r esen t day d esign
s
.
,
,
.
,
,
,
-
.
TAB L E 3
.
—Sc u
.
.
n
En
a cr o n
Ra t i o
of
Axa o r aa s s B o ni
ns ar
mAm
8 it 1
011
8
.
Str a ta —I n d escr ib i n g t h e
thi ck ening of t h e secti on led
the
ae ro
.
Th e Red st an ce of
hown th a t
.
.
Mod e l of
s
s
drag coe ffi ci e nt s at var io us speed s to t h e
dr ag coe ffi ci e nt at 60 ft -s
.
was
Mo o n
an i e
p ro p er ti es of aerofo il s i t
to a criti cal typ e of flow
0 24
0 16
0 04
Fro
.
M
.
—S
ca le effect o n
t he
res i st an ce of a s t ru t
R
res is t an ce i n lbs
l = s a ller di e ns i o n of cross
le ngt h of s t ru t i n fee t
L
s pe ed i n fee t
1
:
m
m
.
.
-
sect i on
i n fee t
.
.
.
t i
gl es of i nci d ence A fu rt h er ch ange of aerofoi l sec ti on l ea d s
e e x t re m
en t sh ow s t h a t t h e flow i s a p t t o b ecom
e ly
t o a s t ru t a n d e xp eri m
Ev en w h en
cr iti ca l
es p ec i ally wh en t h e s t ru t i s i ncl i n e d t o t h e wi n d
at cer a n a n
.
,
,
.
APP L IED AERODYN AMI CS
3 94
e nt a s ta ti c t es t h as b een carr i ed out a t s p ee ds
l ow er sp ee ds One experi m
0 ft s
I n t h e neigh bou r hoo d of t h e v elocity o f so u n d t h e ty e
u p to 115
p
was eli m
i n a t ed an d t h e m
o f flow chan g ed ra p i dly so th a t t h e sli p s t r ea m
ai n
out fl ow cen t rifugal Th e noi se p ro d uced was v ery gr ea t an d di sco m
fo r t
e
It i s cl ear th a t no cer t ai n ty i n d es ign a t p resen t
felt i n a sho r t tim
e x i st s for ti p s p eed s i n e xcess of 800 ft s
—
su m
éof t h e a pp li ca tio ns of t h e p r i n
Thi s r é
ar y of Cond u si
Su
i cal si m
il arity wi ll h a v e i n d i ca t ed a fi el d of r esearch of whi ch
ci ples of dyn am
o nly t h e frin g es h av e y et been tou ch ed So fa r as r esear ch has gon e
t h e r es ult 18 to gi v e su pp ort t o a r eason a bl e a pp li ca ti on of t h e r es ul ts o f
ent s
T h i s co n cl u sion 1s i m
p ort an t s in ce m
o d el r es ul t s are
mod el exp eri m
more rea di ly an d rap i dly obtai ned than corresp on d ing quan titi es o n t h e
fu ll sca l e an d t h e p rogress of t h e sc i ence of aeron a uti cs h as b een an d w ill
bi nation of exp e r i m
con ti nu e to b e assi s te d gr ea tly by a j u d i ciou s co m
en t s
o d el an d f u ll scal es
on b o th t h e m
,
.
.
-
,
.
,
.
.
m
m
m
.
-
.
.
.
,
.
.
CHA PT ER I X
TH E P RED I CTI ON AN D ANAL YSI S OF AEROPLANE PERF ORMAN CE
TH E
P ERF O RMAN CE
Aa a or na x s s
or
an ce
a s a pp li ed t o ae r Op lan es i s u se d
term p erform
as
an
e xp ress i on t o d en ot e t h e grea t es t sp ee d a t whi ch an a erop l an e can fl y
a n d t h e gr ea t es t rat e a t whi ch it can cli m
b As fli ght ta kes p l ace in t h e
a i r t h e st ru ctur e of whi ch i s v ar i a bl e fr om
day to day t h e exp ression
o nly rece i v es p reci sion if t h e p erform
an ce i s d efin e d r el a ti v e t o som
e
s p ecifi ed se t of at m
osp h eri c con d itions As ae plan es h av e reach ed
h eights of n early
f ee t t h e st ra tumi s of consi d era bl e thi ckness an d
i n B rita in aero n au ti ca l exp eri m
en t s an d ca l cul a tions a re r ef err ed t o a
os p h ere whi ch i s d efin ed i n T a bl es 1 a n d 2
s t a nd a r d a t m
Tu n
.
,
,
.
m
,
,
.
The pr a w n
'
1s
in
mlt i pl
u
60
7
f
es o
m
mf m
.
o
ercu ry , a nd
t he d en si t y
u g p er
2
d
0
00
8
7
1
0
cubi c
ma h i
Ane
i
( it )
e
.
fl
.
gh t
APP L I ED AERODYN AMI CS
3 96
TAB L E 2
.
The p r
m
u re i s
in
mlt i p l
f
es o
u
760
—As s ao mH m
om
.
mm f m
.
o
er cu ry , a nd
t he de ns i ty
]
0
00023 7 d ay p er
Ab so lu te
cu bi c
fl
.
St a nd a r d
l OOO
‘
t a bl es show t h e q u an titi es of i m
p ort ance i n t h e s t an d ar d a t m
o
s p h ere with t h e a ddi ti on of a qu an tity call ed
a ner oi d h eight
Th e
et er i n a n aer op l ane t h e
t erma ri ses f romt h e u se of a n aneroi d b arom
d i vi sio ns o n whi ch are gi v en i n tho u san d s of f ee t an d fra cti ons of t h e
main d i vi si ons As a meas u re of h eight t h e inst ru ment i s d efecti v e an d
i t will b e n oti ced f romt h e t a bl e th a t a n a ner oi d h eight of
f ee t
corres p on d s with a r ea l h eight of
feet i n a st a n d ar d a t m
osp h ere
et er i s r eg a r d ed sol ely
I n aerona u ti ca l w or k of p r eci si on t h e aneroi d b arom
as a p r essu r e i n d i ca t or a n d t h e r ea d i n gs o f a n eroi d h eight as t a ken a re
ea ns of T a bl e 2 b efo re a ny us e i s m
a d e of
co n v ert ed i n to p r ess u r e by m
Th e t e r m a n eroi d h eight
t h e r es u lt s
i s usef u l a s a r ou gh gu i de t o
et er
t h e p os iti on of a n aerop l ane a n d for thi s reason t h e an er oi d b aro m
h as nev er b een d i s p l a ced by a n i ns t ru m
en t i n whi ch t h e s ca l e i s ca lib ra t ed
i n p res su res d i rec tly
n of T a bl e 1 s h ow s for a s t an d ar d a t m
os p h er e t h e real
Th e fi rs t col u m
h eight of a p oi n t a b ov e t h e earth (s ea l ev el) whil s t t h e o th ers sh ow r el a ti v e
p r ess u r e r el ati v e d ens ity a n d t em
p er a t u re both Centigr a d e an d a b sol u te
Th e
"
.
,
.
,
.
,
,
.
,
.
,
,
,
.
APP L IED AER ODY N AMIC S
3 98
m
m
Sp eci a l care i n r egul at i ng t h e p e t rol co nsu p tion t o t h e a t os p h e r i c
co nditio ns i s re qui re d ; wi th ou t r egul a tio n t h e p et rol -ai r
i x tu re t en d s
t o beco e too r i ch as t h e h eight i ncr eases , wi th a co ns equ en t loss of en gi n e
p ower, an d an i ncreased p et ro l cons u p ti on Th e followi n g figur es wi ll
show how i
p o rt an t i s t h e regul a ti on o f t h e p et rol fl ow
In a p ar ti cul ar aereplane t h e ti e to cli b t o
f eet wi th u n
co n t r oll ed p et rol w as 25
ins , an d thi s was r ed u ced to 2l 5 i ns by
s u i t a bl e a dj us t
en t
Th e i n cr ease of Sp ee d wa s f ro
84
p h to 9 1
an d altho u gh t h i s i s p r ob a bly an ex t r e e cas e, i t i s cl ear th a t t h e
of altitu d e con t rol beco es ess enti al for a ny acc ur a t e
u se of so e fo r
easur e en t s of aer o p l an e p er fo r an ce
Th e rev oluti on coun t er an d t h e
a i rs p ee d i n d i ca to r affo r d t h e p ilot a
eans of a dj us ti n g t h e p et r ol - ai r
i x ture to i t s b es t con dition
Th e p r ed i ction an d r ed u ction of aero p l an e p erfo r ance p roceeds on
t h e ass u p tion th a t all p reca utions h a v e been t a ken i n t h e adj u s t en t
of t h e p et rol su pp ly t o t h e engi ne, an d th a t d ur in g a seri es of t ri als t h e
p re v al ence of u p curren t s will obey t h e law of a v era ges , so th a t t h e ean
will not co nt ain any err ors whi ch ay h av e occurred i n sin gl e t ri als
Th e qu estion of t h e ca li b ra tion of i ns t ru en ts i s n ot d ealt wi th h ere,
but i n t h e section d ea lin g with etho ds of eas ure en ts of t h e q u an ti t i es
in v ol v ed in t h e stu dy of aero d yna i cs
m
m
m
m
m
m
m
m
.
m
m m
m
m
.
m m
°
.
m
m
m
m
.
.
.
.
.
.
m
.
m
—
When
m
m
m
m
m
m
m
m m
.
.
ubj ect of p red i ction i s consi dered i n full deta il t aki n g
a cco u n t of all t h e kn o wn d a t a i t i s fo u n d to n ee d co ns i d era bl e kn owl ed e
g
an d exp eri ence befo re t h e b es t r es ult s are obt a in ed
A first app roxi at i on
to t h e final r esu lt can how ev er be m
a d e with v ery littl e difi cu lt
y an d
t h i s ch ap t er begi ns with t h e m
a t eri a l a n d b as i s of ra p i d p redi ctio n an d
p roceed s to t h e m
ore accurat e m
etho d s i n l a t er p ar agra p hs
—
i
in atio n of n u m
i
i
bers of m
o d ern aero p l an es
Rap d Pred ct on An ex am
will i ndi ca t e to an o bser v er th a t t h e differences i n formand co ns t ru ct ion
are n ot su ch a s to m
ask t h e gr ea t gen eral r esem
bl an ces Aero p l an e bodi es
a n d un d erca rr i ages p r es en t p er ha p s t h e gr ea t es t in di v i d ua l ch arac t eri s ti cs
but a firs t generali sation i s th at all aerOplan es h av e sensi bly t h e sa e
il ar d ra wi n gs bu t of differen t sca l e
ex t erna l form AerOp lan es to s i
an d t h e s i
would be d escr ibed as of t h e sam
e fo r
il arity i s ex t en d e d to
t h e ai rscrew
a t wo bl a d e d a irscr ew t o on e wi th
Ev en t h e ch ang e fr o
four bl a d es i s a seco nd a ry ch aract eri sti c i n rap i d p r edi ction
Th e a xi umho ri z on t a l sp ee d of whi ch a n a ero p l an e i s ca p abl e i t s
umrat e of cli m
b and i t s cei li n g are all shown l a t er to d ep en d
a xi
on ly on t h e ra tio of h orsep ow er to tot a l w eight an d t h e wi n g l oa di n g so
l ong as t h e ext ernal fo rmof t h e aero p l ane i s cons t ant Th e generali sa tion
eth o d of p rep ari n g ch ar t s o f p erfo r
s ugg es t s a m
as t o ex t ern al fo rm
an ce
a n d s u ch ch a r t s are gi v en i n Fig s 20
2 204
—
Fig
Th e o r d i n a t e of Fig 202 i s t h e m
a xi
um
Maxi u Speed (
s p ee d of a n aero p l an e i n
whils t t h e a b sci ssa i s t h e s tan d ar d h o rs e
p ow er p er 1000 lbs gr o ss lo a d of a er o p l an e Th e s t an d ar d horsep ow er
a xi m
i s th a t on t h e b ench a t t h e m
umrev olutions for con t inuou s r un ni n g
t he
s
,
m
,
.
,
,
,
,
.
.
m
,
m
m
m
m
.
,
.
mm
mm
-
.
,
”
,
,
,
.
mm
-
.
.
.
.
.
m
m
,
.
.
PR ED ICT IO N
AND AN AL Y SI S
FOR
AEROP L AN ES
3 99
APP L IED AERODY NAMI CS
4 00
f am
i ly of cu rv es r el a t i ng Sp eed a n d p ow e r i s sh own ea ch cu r v e of
ily co rresp on d i ng wi th a d efini t ely ch osen h eight Th e cu rves
t h e fa m
m
ay b e u se d d i r ec tly i f t h e win g lo a di n g i s 7 lbs p er s qf oot ; fo r any
oth er wi n g l oa d i n g t h e fo rm
u l a on t h e fi gu r e s houl d b e u s ed
A
,
.
.
.
.
m
Exa p le 1- Acrop la ne wei gh in g 2100 lbs , h p 220 Fi n d t h e p rob a b le t o p s p eed
ft and
ft , assu in g t h at t h e en gine
00 ft ,
at t h e grou nd , 65
ft ,
a t each of t h es e h e i gh ts
Th e wi ng load i ng i s t o b e 7 lb s p er
all ou t
ay be ru n
s
foot
105
h p p er 1000 l
m
.
m
.
,
.
q
.
.
.
and
.
.
.
.
.
.
.
fro
.
mFig
.
.
202 i t i s fou nd t h at
.
Top s peed
At grou nd
ft
ft
ft
ft
124
123
121
117
103
.
.
.
.
mp h
.
.
Thi s e xam
p l e ill u st ra t es t h e gen er a l law th a t t h e t 0p Sp ee d of a er o p l anes
with non su p erch arg ed en gin es f all s off a s t h e altitu d e i n creas es s lowly
or e r a p i d ly a s t h e cei lin g i s a p p r o ac h e d
for low altit u d es bu t m
ore a n d m
,
-
,
m
pl
,
.
m
2 —Th e s a e aer e p lan e w i ll be t a k en t o h av e in creased we igh t an d h ors e
t
s
i
i
n
i
n
w
er
h
e
l
oa
d
n
b
e
0
b
w
l
1
foot i ns tead of 7 lbs p er s
o
ft , b u t t h e
,
g
g
p
g
p er s
h ors epower p er 1000 lbs as b efor e
Exa
e
.
q
.
.
l
By t he
ru e o n
.
q
.
.
.
.
105
Fig 202 fi nd
.
t e.
a
cup
88
.
10
7
O n Fi g 202 read o ff t h e s peeds for 88 h p p er 1000 l
we igh t
G rou nd
Speed for 88 h p p e r
Speed for 105h p p er
1000 lbs a nd 10 lb s
1000 lbs a n d 7 lbs
r
e
s
r
f
t
ft
e
s
p
p
ft
ft
ft
ft
.
.
.
q
.
.
.
.
.
.
.
.
.
q
.
.
. .
.
.
.
.
To get t h e real s pee d for 105h p pe r 1000 l
.
by
Th e r es u lt s
i s t h e in crease
of
t op
s
ar e
.
gi ven i n t he las t
peed near
t h e gr ou
nd
.
mlt iply t h
colu
u
m
n of
d u e to
an
mp h
.
e
140
138
13 6
13 0
105
.
figu r es in t h e se co nd
t h e t ab
le
,
a nd
t he po in t
in cre as e i n load i ng
.
of
.
.
co lu
Maxi
mmR t
m
.
b ( Fig
m
n
i n t erfi i
Th e p e nalt y
for t h is i ncr eas e i n t op s peed i s a n in cr eas e i n la nd i ng s peed i n t h e p ro por t io n of
Th ere are also losses i n ra t e of cli b a n d in c e i li ng
si c of near ly 20 p er cent
.
m
to
.
Th e or d i na t e of t h e fi g u r e i s t he
b i n feet p er i n u t e whilst t h e a b sci ssa i s s till t h e s t an d ard
ra t e of cli
h orsep ow er p er 1000 lbs gr oss w eight Th e s a e a erop l an es as w e r e u sed
for Ex a p l es 1 a nd 2 will a gai n b e co n si d ered
u
m
m
m
a e of
Cli
m
.
m
,
.
.
.
m
m
Fin d t h e r a t e of cli b o f a n aer opla ne w eigh i ng 2100 lbs w i t h a n cngi
Exa p le 3 —
h orsep ower of 220, t h e load in g of t h e w i ngs be i ng 7 lbs p er s foot
Th e s t and ard h p per 1000 lbs i s 105, a nd fr o Fig 203 t h e fo llo w i n g r a t e s o f cl i b
ar e r ea d o ti
Ra t e o f cli b
1530 ft
in
1120
8 90
.
.
.
.
m
m
q
.
.
.
.
.
o
m
.
m
APP L IED AERODYN AMI CS
402
l
r u e on
Fi g 203 is
.
a
o
li
e
d
b
e
l
w
pp
( 2)
("
”ul h
t
l
w h i st ?
.
Grou nd
ft
ft
ft
ft
26
36
.
.
.
( 4)
( 2)
1°
106
nu
m
b
ers
Ra te
( 4)
mf m
of cli
b
and H z
ro
.
1350
100
93
89
'
7
Th e
( 5)
In
7
31
43
54
72
90
45
60
83
.
( 8)
g
/
w
(lb
.
38
a
.
ffec t of i ncreas i ng t h e l oadi ng i n t h e r ati o 10 to 7 i s seen t o be a
r e d u cti on i n t h e ra t e o f cli m
b of n ear ly 200 ft p er m
i n u t e an d a r e d u c tio n
o f t h e ce ili n g of a bo u t 3000 ft
Th e four ex a m
p l es illus t rat e a general ru le i n m
od ern high S pee d
e
.
,
.
-
Flo 2O3 A
.
.
Th e cu rve a p pli es a t
An a pproxi a t e for
m
IC C
12 0
so
S t a nd a r d N P IOOO lb s
.
—C
e i lin g a nd
h orsepow er chart for
loadi n g of 7
u la wh i ch a p p li es t o
m
/
ra pid
.
r edi ct i on.
p
a
all loa di ngs
m
(i )
is
i
At
W
cei ling ,
wt i n lbs
.
.
,
0
)
rela t i ve
00 10
d ens i t y, w
load ing i n lhn /i t }
.
th a t hi gh sp eed is m
or e econ om
i cally p ro d u ced with h es
wi n g l oa d in g th an with light lo a d in g whil s t r a p i d clim
b an d hi gh cei li n g
or e eas ily a tt ai n e d wi th t h e li ght lo a d i n g
ar e m
Th e r easo ns fo r thi s
a s t u d y of t h e a erod yn a m
i cs o f t h e aerop l an e whi ch show s
a pp ea r fr om
aerOp lan es,
.
,
.
,
PR ED ICT I O N AND
AN AL Y SIS
AER OP L ANES
FOR
408
th at t h e an gl e of i nci d en ce at t 0p Sp ee d i s usu ally m
u ch b elow th a t gi v i n g
b est li ft /drag for t h e wi n gs so th at an i ncr ease of lo a di n g l ead s to a bett er
a ngl e of inci d ence at a gi v en Sp eed
For cli b i ng t h e an gl e of in ci d en ce
i s us u a lly th at for b est lift /dr ag for t h e whol e aer op l an e a n d t h e horse
p ow er exp en d ed i n forw ar d otion (n ot in clim
bin g) i s p ro p o rtional to
To su pp ort t h e ae ro p l an e thi s Sp ee d of fl ight m
ust
t h e Sp ee d of fl ight
b e i ncr eased i n t h e p rO p ort i on of t h e s qu are root of t h e i ncr eas ed loa d in g
to i t s original v alu e It i s not p ossibl e i n cli m
bi ng t o choose a b ett er angl e
o f i n ci d en ce
for th e Aeropl e of Exa p le L —In es ti m
a tin g t h e
Roug h Outline Desi
ance t h e d a t a used h as b een v ery li m
a t e p erfo rm
i t ed an d no
a pp ro x i m
i n di ca tio n h as b een gi v en of t h e u ses to whi ch su ch an aere p lane coul d
u ch of t h e t ot al w eight of 2100 lbs i s r equi re d for t h e
H ow m
b e p ut
u ch fu el wi ll b e
How m
en gi ne an d t h e s t ru cture of t h e a ero p l an e ?
il es
00 m
Wh at sp ar e l oa d will th ere b e 2
re q u i r ed for a j our n ey of 5
—
i
Th e p er cen t age whi ch t h e s t r u ctur e w eight b ears
Stru cture We gh t
to t h e gross w eight of an a ero p l an e v ari es fr om27 to 82 as t h e aereplan e
s to on e of
a gr o ss lo a d of 15
lb
s
rows i n si z e fro
0
0
l
b
T
h
e
g
all er a er o p l an es us u ally h a v e a f acto r of saf ety great er th an t h e l ar g e
sm
o nes an d so for equ al fac tor of safety t h e di fference in t h e st ru ctur e w eight s
wou l d be gr ea t er tha n th a t quot ed a bo v e For r ough g en eral p u rp oses
ay b e t a k en as 8 0 p er cen t o f t h e gross w eight
t h e s t ru ctu re w eight m
—
i
i
w eigh t p er s t and ar d
e
W
t
Th e re p r esen t a ti v e fi gu re i s
Eng n
e gh
h orsep ower a nd for non su p erch ar g ed m
oto rs t h e fi gu r e v a ri es f rom
a b out 2 0 lbs p er h p for a ra d i a l a i r cool ed engi n e to 8 0 lbs p er h p
For l arg e p ow er w a t er cool ed en gin es
for a light w a t er cool ed en gi n e
all er p ow ere d en gi n es m
ay b e eith er ai r cool ed
a re t h e r ul e whil st t h e sm
As a g en er al fi gu r e 8 lbs p er h p sho u l d be t a ken a s
o r w a t er cool e d
ore r ep r esen t a ti v e v al u e
t he m
Wei gh t of Petrol and Oi l An ai r cool ed non rot ary engi n e or a w a t er
es a pp roxi m
5l b of p et rol an d oil p er b ra ke
a t ely 0 5
cool ed en gi n e cons u m
horse p ow er h ou r wh en t h e engi n e i s all out
Th e co nsu m
p ti on of p et rol v ari es wi th t h e h eight a t whi ch fli ght t akes
p l ace r oughly i n p r e p or tio n to t h e rel a ti v e d ensity a Th e g eneral figu re
for fu el consu m
p tion i s th en
0 550 lb p er s t an d
h p h ou r
—
l
5
Es t i m
at es of w eigh t ava ilab le for net load can n ow b e m
ad e
e
Exa m
p
m
,
.
,
m
,
.
,
r
.
m
.
m
m
,
.
.
‘
m
.
.
,
.
,
.
.
-
,
.
.
-
.
.
-
.
.
-
.
,
-
,
-
.
.
.
.
.
-
-
-
.
.
-
.
.
.
.
.
.
.
.
Tot al w eigh t of aerop lan e
St ru ct ure 2100 x 0 30
Engi ne 220 x 3
Fu el for 500 i les , c 4 hr s
4 x 05
5 >< 0 74 x 220
m
.
2100 lbs
at a
h igh t
e
.
66o lba
of
ft
.
.
360 l
450 lbs
.
.
Out of thi s 4 50 lbs t h e p ilot an d p asseng er w eigh 180 each on t h e
a v era g e l ea v i n g a bout 90 lb s o f u seful lo a d in a two Bea t e r a er eplan e or
270 lbs of us eful lo a d i n a si n gl e sea t er aerop l an e
i n a ti on o f t h e p ossib i liti es of a d esign
I n thi s way a p reli m
i nary ex am
a d e b efo re e n t eri n g in to grea t d et ail
t o s ui t an en gi n e can b e m
.
,
-
.
,
-
.
.
.
APP L IED AER ODYN AMIC S
404
MO RE
ACC U RA TE
METH O D
o r P RE D I C TI N G
AERO P L AN E
P E RF O RMA N C E
etho d of p redi ctin g ae ro p l an e
uccee d in g p ara gra p hs a m
p erform
p l e At t h e
an ce will b e d es cri be d an d i llus t ra t ed by a n ex a m
en t a l d a t a to w h i ch res o rt i s
p resen t ti m
knowl e d g e of t h e fun d am
e
a k es
n ecessary b efo r e cal cu l a tio n s are b egu n h as n o t t h e accu r acy whi ch m
full cal cula tion a dvan t ageous Si m
p lifyin g ass um
p tions will be in t r od uce d
a t a v ery early s t ag e but i t will b e p oss i bl e for a ny on e wi shin g t o ca rry
out t h e p rocesses to th ei r logi ca l co nclu si ons to p i ck u p t h e thr ea ds an d
p lifyin g ass u y
An ot h er r eason for t h e u se of s im
eth od
el a bo ra t e t h e m
tio ns i s t h e p oss ibili ty th ereby Op en e d u p of rev ers in g t h e p rocess an d
a nce t ri a l
It ap p ears i n t h e conclus io n
an aly sin g t h e res ult s of a p erfo rm
an ce i s su fli ci en t ly
th a t t h e n um
b er of a in f actors in a erop l ane p erform
all for efl ec t i v e a n aly si s of aero p l an e t ri a ls with a pp ea l o nly t o g en era l
sm
o d el o f t h e aero p l a n e
kn owl ed ge an d not to p arti cu l ar t es ts on a m
s of i m
p ort an ce i n t h e d es ign of an ae r o
a tin g t h e v a riou s it em
In es ti m
an ce it i s co n v eni en t to gr o u p th em
p l ane as th ey affec t achi ev ed p erfo rm
u n d er fou r h ea d s
atio n of t h e res i s t a nce of t h e aero p l an e as a gli d er
es ti m
e
T
h
a
( )
without air screw
a ti o n of a i rscr ew ch arac t eri s ti cs
h
e es ti
b
T
( )
r with Sp ee d of rot a tion
n
v
i
tio
n
of
e
n
gi
e
p
ow
e
r
a
a
h
e
T
0
( )
v
ri a tio n of en gin e p ow er with h eight
a
h
e
T
d
( )
It i s t h e co nnection of th ese four qu an titi es wh en actin g togeth er w h i c h
an ce
I n t h e e xa p l e
i s now referre d to as p redi cti on of a er o p l ane p erform
s ( a) t o (d) are a r bit ra rily chosen a n d d o no t co ns ti tu t e
cho sen t h e i t em
I t i s p rob a bl e th a t t h e bes t d esign for a gi v en en g in e
an effo r t a t d esi gn
will only be at t ai ned as t h e r es u lt of r ep etitions of t h e p r ocess now d ev el o p e d
b er of r ep etitions b ein g d ep en d en t on t h e s ki ll of t h e d esign er
t h e n um
o d el ex p er i m
Of t h e fo u r it em
s ( a ) an d ( b) are u s u ally bas ed on m
e n ts
of wh i ch ty p i cal resu lt s h av e a pp ea re d i n oth er p arts of t h e boo k Th e
thi r d it e i s o bt ain ed f ro mb ench t es t s of t h e engin e whils t t h e f o ur th
h as h ith erto been obt ain ed by t h e analy sis of aero p lan e tr i als wi th s u p p o r t
f romb ench t est s i n h igh l ev el t es t hous es
It h as b een s h ow n th at t h e r esis t an ce of an aerep lan e m
a y b e v er y
t h e ai rscrew an d for a
a pp r eci a bly d ep en d en t on t h e sli p st reamfr o
b
s i ngle seat er a er o p l an e of high p ow er t h e i ncrease d r es is t an ce d u r i n g C lim
ay b e thr ee ti m
es as gr ea t as th a t w h e n
of t h e p art s i n t h e sli p s t rea mm
gli d i ng One of t h e firs t consi d era tions i n d ev elo p i n g t h e fo r ul a e o f
eth o d of d ea li n g with sli p s t r eam
p red i cti on rel at es to t h e m
e ffect s
o d el s o f a i rs crew s a n d b od i es a t t h e N a tio n al P h y s i ca l
Exp eri en t s on m
utu al i n t erf e r en ce
L a bo ra to ry h av e show n cer t a i n consi s t en t effect s o f m
en t a l m
ea n
Th e e ffec t of t h e p resence o f a bo d y i s to i n creas e t h e e xp eri m
p it ch an d efli ci ency of a n a irscrew whils t t h e effec t of t h e a i rsc re w s li p
i s t o i n creas e t h e r esi st a n ce o f t h e bo d y a n d t ail v ery a p p reci a b l y
s t r ea m
Th e fi rs t po in t h as bee n d ea l t with un d er Airscrew s an d t h e l a t ter w h e n
d eali n g wi th t es t s on bo d i es It i s co n v en i en t to ex t rac t h ere a t y p i ca l
In t h e
s
,
.
,
.
,
m
.
m
.
'
,
.
,
m
.
.
-
.
.
m
.
,
.
,
.
,
,
m
.
,
-
.
m
,
,
m
.
m
-
.
.
,
.
.
APP L IED AERODYN AMI CS
406
wh ere
b
a an d
ar e
o t
t
c n s an s, an d
t hr us t
k, is t h e
ffi ci en t d efin ed
coe
by
us u ally l ess th an unity a pp aren tly owi n g to t h e Sh i el di n g of
I t s v a lu e i s seen t o b e 0 85i n equa ti on
t h e b ody by t h e a i rs crew bo ss
b is m
ore v ari a bl e an d t h e
a n d thi s i s a us u a l v a lu e for a t rac t or scout
t es ts on v ar i ous co m
binations of bo dy an d a irscrew m
us t be exam
i n ed i n
ad e
an y p a r ti cu lar cas e if t h e b es t ch oi ce i s t o be
U sin g t h e vari ou s exp ress ions d ev elop ed e quation (2) b eco m
es
is
a
.
.
,
m
.
,
Equ ati on ( 6) will n ow be
kg , an d kn by di v i di n g
Th e
valu e of
c i ent ly
~
p SV
2
is
t he
m
a
for
’
,
wh ere
S is
d ep en di n g on
t h e wi ng area
.
oun t of t h e loa d
on acc
on
t he
illus t ra ti on of m
eth o d as s u fii
With th ese ch anges e quation ( 6)
th e
.
ig
w
)
2
?
Th e fa c t or
k;
i
p
an ex r ess on
p SV
t i tly equ al to
b ecom
es
]
to
not s r c
pp roxi ation i s used in
a ccu ra t e for p res en t p ur p o ses
t a il but
,
W
o v ert ed
thr ough by
c n
S
ba
rn) , i n e qu a tio n ( 7)
all an
gl es of i nci d ence an d it
thr us t coeffi ci en t d efin ed by
,
w i ll now
is
be
o veni en t
c n
ogn ised
rec
to
t
as a co ns a n
i n t ro d u ce
a
t
fictiti ou s
r
D
Th e
u ve
ti ng thi s o v erall th ru st coefli ci ent as a f u n ctio n o f
a dvan ce p er rev oluti on d iffers fr o m
th a t of t h e ai rscrew i n t h e scal e of
i t s or di n a t es
a t e t h e v al u e of t h e m
To esti m
ulti p lyin g factor for t h e
n ew scal e t h e foll owi n g a pp ro xi m
a t e v alu es m
ay b e u se d
p
c r
r e resen
.
S
5
i
Ij
and
t he
b
,
( kn);
9
( )
o ffi ci en t o f It , i n ( 8) i s
Th e n ew o r di na t e o f th r us t i s th en
6 p er cent l ess th a n th a t of t h e r eal th r u s t
As t h e e ffect of t h e bo d y i s t o
i ncrease t h e a i rscr ew th r us t it will b e s ee n t h a t t h e fi ctiti ou s thr us t co
e ffi ci ent i s withi n 5 er cen t of th a t of t h e a i rs cr ew alone o v er t h e u seful
p
c e
.
.
,
.
m
m
Th e t er
c co),
a
( kp h
o
y be r eg a rd e d a s a fi ctiti ou s d rag coefli ci ent
for t h e aerop l a n e a s a gli d er
Th e co r r ect ex p ression for t h e gli d er d ra g
cocfli ci e n t b ei ng ( kn o f kp , t h e d e a r tu re o f t h e coe ffi ci en t
fro
a
( h
l
p
u ni ty i s a easu re o f t h e d i fferen ce b etw een t h e fi ctiti ou s an d r ea l v al u es
o f t h e d ra g co e ffi ci e n t
F ro t h e n u eri ca l e xa p l e qu ot e d i t will be
f ou nd th a t t h e d ifference i s 0 p er cen t o f t h e i ni u d ra g coefii ci ent
.
m
'
.
"
'
m
m
.
m
m mm
m
AND AN AL Y SIS
PR ED I CTION
FOR
AEROP LAN ES
407
whol e aero p l an e It h as been p reviou sly rem
ar ked th at this
d i ffer en ce ari ses fr o mt h e shi el d i ng of t h e bo dy by t h e airscr ew boss
a n d i n any p ar ti cul ar cas e t h e effect coul d b e esti m
at e d wi th fa ir accuracy
i f r equi r ed as a r efin em
en t i n p r ed i ctio n
Th e equ a tio n for forces whi ch corres p on d s with (7) i s
of
th e
.
.
,
.
T
w h ere T
a nd
'
D
'
my b
Ve
W
D
( 10)
V
g d d p r o vi si onally as t h e thru s t of t h e air
s crew an d t h e drag of t h e a er op l ane es ti m
a t ed sep ara t ely
Sin ce D d ep en d s o nly on t h e ai r Sp eed of t h e aer o p l an e i t i s p ossibl e
1
0
r
a
n
o
f
a
a
to d ed uce fr om
e
l
tio
i
m
p
l
e
n
a
tu
r
e
b
tw
th
u
t
i
b
s
e
n
n
ee
r
s
a
d
c
l
m
( )
if fl yin g exp er im
en t s b e
a d e at t h e sam
e air sp ee d b u t at d i fferen t thr ottl e
s
o
Th e r el a ti on i s
p itions
W
V
en t in ra t e of clim
wh ere 8 V i s t he i ncrem
b co rresp on d in g with a n i ncrease
a
e r e ar e
.
'
,
m
,
.
,
8T
thr us t
of
E
?
i
n
ce
S
’
.
an d
V
8 V,
are
m
eas
ur ed d ur in g p erform
ance
,
qu ation ( 11) can be used in t h e rev erse or d er to d ed u ce 8 1 froma t ri al
Th e t rea t m
p l etes t h e sp eci al
en t of sli p s t reamgi v en a bo v e com
p ti ons ; a t v ar i ou s p la ces assu p ti ons h av e b een i n di ca te d w h i ch
a ss u m
en t a l d a ta
may becom
o re accu r a t e
Th e m
e l ess a cc ura t e t h an t h e exp er i m
a lg eb ra i c wo r k whi ch woul d th en be requi r e d p r esen t s no ser ious di ffi culty
"
e
.
m
.
.
Detai ls
of a
Predi cti Cn Calcu lation
ed d a t a co rresp o n di n g r ough ly
C al cul a tions will b e m
a d e on a ss u m
with a high Sp ee d m
o d ern aero p l an e ; a lthough t h e act u al n um
b ers ar e
g en erally r ep resent ati v e of an aerop l an e th ey h av e b een t aken fr om
v ar io us sour ces on accou n t of co m
p l et eness and n ot on accoun t of sp eci al
qualiti es as an effi ci en t co m
bi n ation i n an a ero p l an e
-
,
.
D ra g an d li ft co effi ci ents of
s hi el d in g of ai rs cr ew boss
1
( )
the
aer
o p l an e as a gli d er
,
o
t d
c rrec e
for
.
2
( )
( For
t he
Th e
Thr u st
gen era l
ana
lysi s
ii D
o
c rr ec
ti on
for
md
g
ho
3
i
E
( )
p tu
d t m
a
e
n
e
p h er e
ne
era
is
£
i
3
”
s
o i i
t of
c eff c en s
,
h as
the
i
a rscr e
b een p referr ed
a re eas
;
w as d ep end en t on
if P
and
D
ily con v ert ed fr omone to
li p s treamfa cto r i n di ca t ed
-
i n (8) i s
s
be kn own
t he
oth er )
.
u pp o sed to h a v e
.
re
gi
4
ne
E
n
( )
s
t o r qu e
v ar i a bl es Y
b een
an
an d
used
.
rse
p ow er as d ep en d en t
v olut i ons at st an d ar d d ensi ty
on re
.
horsep ow er
as
d ep en den t
on
hei ght
.
A
s t an ar
d d
a
tm
o
APP L IED AERODYN AMI CS
408
b rake horsep ow er of t h e en gi ne u n d er s tan dar d con di ti ons w ill be
d enote d by St d
whils t t h e fa ctor exp ress i ng v ari a ti o n w i th
os p h ere t h e b r a ke
h eight will be f(h) At any h eight i n t h e st an d ar d a t m
ho rsep ow er a t gi v en r ev olutions will be
The
“
.
.
=j (h)
/
V
n
e
m m
m
of
m
217
d ed u ced
.
.
.
m
m
tor q u e Q
,
,
an d
t o rqu e
xp r ession
550
is
.
P t a Aow cr e
nr vo u
on A6 A FRA CYION
or h i t t xe c n
r cu
e nu t n u s
o r d in ary d efini ti on
t he
St d B H P
h
fl )
St d
.
.
It shoul d
be
n
oti ced from( 13 ) th a t
the
v al u e
of
is
i n d ep en d en t of
APP L IED AER ODYN AMI CS
4 10
D
feet
l ea ds to
,
an d
p itch P
the
f ee t
,
kg
th ese d a t a equ a tion ( 13 )
For
.
—
fl
a
l ti v e d ens ity a i s u nity i n a st an d ar d a t m
os p h ere a t a h e igh t
of a bout 800 t es t t h is v al u e ha v in g been chos en to conformwith t h e
s ta n d a r ds of t h e Aer o d yn am
i cs l a bora tori es throu ghout t h e wo rl d an d
et eorologi ca l con diti ons thr ou gh out t h e y ear
with t h e av erage m
Th e followin g t a bl e i s co m
p il ed from
Fi gs 204 an d 205an d equa ti on
Th e
re a
,
,
,
.
.
TAB LE 3
E p
.
m
.
I
.
I
.
sooo a
226 0
223 4
2200
216 5
Fro
be
c an
mt h
r ea
d
8 2a x
94 0x
1000 x
11 10 x
'
°
e
v al u es of
off an d
th e
m
00 124 5
00 1380
00 1510
10
4
10
10
10
kQ
v alue of V
i aooo n
I
.
u v es
c r
m
of
00 1125
00 124 5
00 13 60
00 1505
Fig
.
204 t h e
.
.
It
.
.
.
m
va
31
a?
1400
13 5
0
1300
1250
16115
13 75
9 10
00 92
0 6 11
p er
0035
01 96
2
cc
.
.
a?
.
165
14 3
1075
.
00 1080
00 1105
00 13 15
00 14 60
Gro und
B p
en
v alu es of
,
TAB LE 4
se
.
l ulat ed l ea d in g to T able 4
ca c
m
ooo n
1
00 1215
00 1
00 14 75
t he
an d
l
.
0 660
07 5
0
00 92
170
148 5
118
175
156
135
89 5
04 30
I
07 68
0 652
0520
.
vn
.
p er eec
.
179
1615
14 15
108 5
T a bl e 4 show s t h e r el a ti on be tw een t h e en gin e revoluti ons a nd t h e
forw ar d Sp eed of a irscrew for all altitu d es t h e en gi ne bei ng all out
Th e corr esp on di n g
Th e r el ati ons h i p i s Sh own d i a gra m
m
a ti ca lly i n Fig 206
”
,
.
.
.
l ti on b etw een
re a
in
Fig
206
.
Th e
Y
nP
an d
t he
f orw ar d Sp eed
of
the
i
w
a rscr e
is
l o sh o wn
a s
.
fall
of re
v oluti ons with h eight whi ch
is
o
b serv ed
in
l evel flight s
Thro u gh ou t t h e t h eo ret ica l pa rt of t h e book t h e u n i t s u sed h a ve been t he foo t and seco nd
wi t h fo rces
eas u red i n
Th e u n i t o f ass is t h en conv eni ent ly t a ke n as t ha t i n a
”
body we igh i ng g lbs , an has bee n ca lled t h e
Slu g
Co
on la ngu a ge h as o t h e r u ni t s ,
8
i n , an d ro t a t i o n i n re volu
of fl i gh t
b in fee t per
in i les
r h ou r , rat e o f cli
t ou s pe r
i n u te
are t h e
on langu age, ea r ly ad opt i o n
[ resu lt s are re u i red in t h e co
oft en lea ds t o a sa v i n
g of la bou r
m
m
.
m
m
m
q
.
.
.
mm
m
m
m
.
PR ED ICTIO N
AND AN AL Y SIS
FOR
AEROP L AN ES
m
d ed u cibl e fr o th ese ob serv ations and t h e p ro p erti es of
as b elow
Th e exp ressio n for lift co effi ci en t i n t er s of w eight i s
is
the
4 11
o p lane
aer
m
W
1
loa d in g B will
be
"I
and
in t he
exa
mp l
e
the
8
S P E ED
mm
il —C l
F
a
s g
w
v
t aken as
per
Conv ert i ng t o
co
m
m
on
b e t ween forward
rev olu t i on as a fracti on
u nit s
an d
.
s
qu are foot
.
(f /s )
engi ne sp eed . a nd ad va nce
a c u la te d rela t i ons
.
7 lbs p er
0
t h e p i t ch
.
p ar ti cu l a r va lu es for
th e
p l ane l ea ds
a er o
to
13 72
i5
-
“
Th e
qu an tity
W
’
a
is
( Va
im
p or t a n t
nn
and
2
h as
b ee n
ca
ll ed i n d i ca t ed
ai r
APP L IED AERODY NAMIC S
4 12
Equ a ti on ( 16) sh o ws t h a t k,
a n d n ot on t h e t ru e Sp ee d
Fig 207 sh ows t h e v al u e of dra g
s
p eed
.
d ep en ds
on
t h e i n d i ca t e d
ai r s
p ee d
.
.
coe
ffi ci en t
for
a
parti cu la r
aero
p la n e
L OFT CO EF FIC IENT
Fro 207
.
—A
.
ero p la n e
gli der
charac t er i s ti cs us ed
m
in
ex a
mpl
e of
p redi c t i o n
.
gli d er as d ep en d en t on lift coe ci en t an d h ence with ( 16) l ea ds t o a
knowl ed g e of d ra g coeffi ci en t for a ny v al u e of t h e i n di ca t ed a ir sp eed
Th e e qui v al en t o f e q u a ti on ( 10) a s a pp li ed to t h e r el a tion be t ween
t hr us t dr a g an d lift coeffi ci en t s 18
,
.
,
APP L IED AERODYN AMI CS
4 14
I ndlcat ed
c
i vi li
Fro
an
m 14
eo
.
m
Fro
.
for
ai rscre w.
and
5
.
m
Fr o col 5
and curve for
ai rscrew
lm
m
o
d 7
d
Fro
.
eo
an
an
e ua t i on
q
.
01 65
0
m
m
To cal cul at e t h e t 0p Sp eed u se i s ad e of a gra p h i cal etho d o f
wh en t h e engi ne ho rs ep ow er i s th a t required by t h e aer o d yn am
i cs
,
fin d i n g
.
TAB LE 6
.
m
I “) fro
l
e
H orse
ol ely fro mt h e engin e ar e p lott ed i n Fig 208 fr o mt h e n u m
b ers of T a bl e 6
p l e t h e n ecessary d a t a b ein g c o n ta in ed
Th e n eces sary cal cul ati on s ar e s i m
i n Figs 205(a) an d 205(b)
i cs a lone
aero d y n a
Th e cur v e PQ of Fig 208 i s th a t obt ai n ed f ro
Th e sep ara t e sho rt cur v es m
ar k e d wi t h t h e
a n d a pp li es a t all h eight s
h eight are f ro mt h e en gi n e d a ta An i n t ers ection in di ca t es b a l ance
b etw een p ow er a va il a bl e a n d p ow er r equi r ed for l ev el fli ght At
ft
thi s gi v es t h e
t h e b al an ce occu rs w h en a t r p m 1227 Si n ce a
s
.
.
,
m
m
.
.
.
,
.
.
.
r
.
m
p
.
.
as
.
14 27
.
.
.
.
.
PR EDICT IO N
AND AN AL Y SIS FOR AEROP L AN ES
f ur th er un i qu e rel a tion in d ep en d en t o f
ns 1 an d 6 of T a bl e 5
thr ottl e i s giv en by colu m
A
.
F
m208 —C l
.
the
.
a cula te d rela t i on
o
f (R
4 15
p osi ti on of t h e en gin e
For any v a lu e of o ‘ r p
the
.
.
m
.
1000
P
M
)
bet ween h o rsep ower and rev olu t i o ns for st ead y h ori zo n t al fl igh t
.
lue of
Full p a rti cu l ars of t op
i s kn own (see Fig
s p ee d of aerep la n e are n ow obt ai n ed fr o
t h e in t ersections of Fig 208 an d
va
m
.
.
APP L IED AERODYNAMI CS
4 16
t h e a bo v e rela ti on be tw ee n
are co
ll ec ted i n T a bl e 7
i n d i ca ted ai r Spee d an d rev olutions
.
60
"
a
H eig h t
E
a
t
a
mp h
"
80
?v w
.
as )
lv
’
’
r
.
.
129
117
1055
95
83 5
( 5015
1
69 5
( 66)
F1g
Th e res ul t s
.
209,
whi ch sh o ws
t he
l i
re a t o n
p
m
.
.
m
e
13 1
11
l
be t w e en
.
( 1230)
a ir s
t
ee
d
a
n
a
d
p
APP L IED AERODYN AMIC S
4 18
TAB LE 8
.
1
1
fl
-
1 V
ll c
fi
’
h
}!m
dglim 8
l
001
l
nus
00760
B est lndlcete d
G rou nd
20
4 0
F 1o
.
21l
.
w
o
m
)
e n
—De ta i led
so
res u lt s o f
pe rfor
m
a nce ca lcu la t i ons.
l
r
5
n
.
AND AN ALY SI S
PRED ICT ION
FOR
AEROP L AN ES
4 19
m
Th e w ell-k nown ch aract er i sti cs of v ari a ti on of p er fo r ance with h eight
axi
u r at e of cli b d ecr eases r ap i d ly
Th e
are shown i n t h i s t a bl e
with h eight fro 18 15 ft i n n ear t h e gr o u n d to z ero a t a littl e ore
th a n
f eet Th e b es t a i r sp eed an d a irscrew re v olutions bo th fall
0 6 as t h e h eight in cr eases
Th e r esult s of t h e cal cul a ti ons of to p s p ee d a nd rat e of cli b are
As t h e
coll ec t ed i n Fig 211, an d illus t ra t e ty p i cal p er for an ce cu rv es
d at a w ere n ot r ep r esen t ati ve of an y Sp eci al aerOp lan e it i s n ot p ossibl e
to ake a d et ail e d co p ari so n wi th any p ar ticul ar t ri al s, but withi n t h e
li it s of general co p ari son t h e accur acy of t h e etho d of ca l cul ati on
i s a p ly grea t
m
.
-
.
m
m mm
m
m
.
-
.
’
m
.
m
.
m
m
m
m
m
.
m
.
T H E ORY or THE R ED UCTI O N o r TE E O B SE RVATI O NS or AEROPL AN E
P ERFORMA N CE FRO M AN ACTUAL TO A STAND ARD ATMOSPH ERE
p robl emi s to fin d how to a dj us t O bserv a tions un d er n on st an d ar d
con d iti ons so th a t t h e r es u lt s will rep res en t tho se whi ch woul d h a v e b een
os p h ere Genera l
O b t a i n e d h ad t h e t es t b een ca rr i e d out in a s t an d ar d a t
th eor eti cal l aw s go v ern t h e aer o dyn am
i cs of t h e p robl eman d a rel ati on
b e tw een t h e p ow er r e quir ed by t h e a i rscr ew an d th a t av ail a bl e f ro mt h e
u st b e sa t i sfie d
e n gi ne
os t aero nau ti cal p r obl e s t h e assu m
p ti on i s m
a d e th a t o v er
As i n m
t h e r an ge of Sp eeds p ossibl e i n flight t h e r esi s t a n ces of t h e aerO plane for
a gi v en a n gl e O f in ci d ence an d a dva nce p er r e volution of t h e a irscr ew
v a ry as t h e s qu ar e of t h e Sp ee d With t h e p ossibl e ex cep tion of ai rscr ews
h a vi ng hi gh ti p Sp eed s t h e assum
p ti on h as grea t p racti ca l an d th eoreti cal
s an cti on
eth od co ns i d er t h e fo r ces actin g on a n aer o p l an e
To d ev el op t h e
wh en flyin g st ea di ly Th e w eight i s a fo rce whi ch bo th in i ts di rection
i s in d ep en d en t of t h e m
oti on th r ough t h e a ir
a gn itu d e
an d
Th e
us t be e qu al an d O pp osit e to t h e w eight if t h e flight
res ul t an t ai r for ce
a gnitu d e an d d i r ec tio n are fi x ed sol ely by m
o ti on
i s st ea d y but t h e
Fig 212 h el p s tow ar d s t h e m
a th e
a ti ca l exp r essi on
rela ti v e to t h e ai r
r el a tin g t h e w eight an d res ult an t ai r f or ce
ed p ara ll el t o t h e w i ng chor d for co n v en i ence i s fi xe d
A lin e assu m
met ry of t h e aer op lane Th e di rec ti on Of
ar bit rarily i n t h e p l an e of sy m
otion ak es an angl e a with thi s d a t u lin e and t h e v elocity i s V Th e
i l arity of e xt ern al for i s k ep t an d t h e
a i rscrew r ev oluti on s a re 11 a n d i f s i
di m
en t a lly th a t
ens io n of t h e a erop l a n e d efi n ed by 1 i t i s k n own ex p er i m
R and y t h e result an t force a n d i t s angul ar p ositi on ar e d ep en d en t on
As was shown in di scu ssi ng d yn am
i cal
a V 71 l an d t h e d ensity of t h e ai r
il arity a li it to t h e for of p er i ssi bl e fu nctions Of co nnecti on i s
si
Th e
-
m
.
,
m
m
.
,
.
m
.
m
,
.
,
m
m
,
,
.
.
m
.
’
.
,
,
m m
m
m
,
.
m
,
.
,
,
,
m
,
,
,
Th e
,
m
m
m
d f om t
.
v ar i a bl e 1w i ll be d ep art e
a
r
once a nd
will
be
2
two v ari a bl es
S for 1
2
,
an d
D
for l
.
Th e
qu an tity
p l aced by
re
mt
us
be kep t
MBRODYN AMICS
APP L IED
420
mon
o t t bu t otherwise t h e use Of t h e t wo l ea d s to exp ress io ns of co m
formm
Th e f u ncti ona l r el a ti on s req u i r e d ar e
or e rea d ily th a n 1
c ns an
,
.
1
$5
B
1
7
‘
)
a
t h e fir s t
gi v i n g t h e m
a gnitu d e O f R
an d
m
y
t he
Th e co n diti on s of s t ea d y oti on are seen
0 an d eq u a tio ns ( 20) a n d (21) beco e
m
,
W
o d
i t s d i r ection
s ec n
fro m
Fig
.
.
212 t o be R==
W a nd
1
S p\
'
2
.
Th ese eq u ati ons
a re of gr ea t i n t erest
the
l oa d ing
—
0 a)
t he
er
p
en t a l form
ula e of r ed u ction a n d
t i t h e f u n d am
It wi ll be n oti ced th at t h e im
p or t an t vari a bl es a r e
con a n
.
unit
a rea ,
E
t he
,
ai r S
gl e of i nci d ence of t h e wi ngs
p eed
(
et er
ti on as a f racti on o f d i a m
5
—
As t h e a ngl e o f cli m
b
h
i
Level Fl g t
,
an
.
,
a
.
,
V, t h e
01
a nd
t he
a
an
gl e
Of
dvance p er
c
li
mb
re vo lu
7
,
e
q u a ti on (23 )
sh o
ws th a t
Y
nD
3
is
.
a
fu ncti on
is
z ero,
of a o n
.
8 is
ly
.
e
qu al
to
a,
an d
Equ a ti on (22) t h en
APP L IED AERODYN AMI CS
422
For kn own
a r ela
v a lu es
tion b etw een
of
11,
a
an d
p
,
V
—
nD
t
h
e
p
D
an d
e
q u ali ty
of
In t h e
.
the
ea r
two v alu es
Q gi v es
of
ly p ar t of thi s c h a p t e r
,
wh en d eali ng with p red i cti on t h e d et ail ed i n terp re ta ti on of thi s r el a tio n
i nd ep en d en t of a Th eo r eti cally t h e
was gi v en D bei n g co ns t a n t a n d
p resen t equ a ti ons are m
ore exact th an thos e use d befor e b u t th ey a re
os t con v eni en t fo rm Equa tin g t h e two v a l u es of Q
not y et i n th ei r m
l ea ds to
.
,
,
.
a
w
n
550
D
1
”
7
D
\
0 S)
1
O
S
‘
0
118 5
Th e n ex t s t op i s t o u se
W, a n d eq u a ti on (27) b eco
e
2
5
?
¢
)
55
2 V) ( D
V)
”
2
I
)
2
2
8
P
If
x(
c,
(28)
l oa d i ng
the
qua ti on (22)
to
m
5
du
ces
t o t he
s
ub s titut e
’
71
s
t erm
for p V S i n
2
2
“
11l)
°
1
r
e
p
s
qua re foot
.
8
d en ote d by
be
,
q u an tity begi nn in g with
for t h e
im
p or t a n t
Le
,
3
1
7
w,
e
an d
qu ation
l ti on
re a
4
W5
1) ,
)
£5
lysis h as b een t o in t rod u ce a v ari a bl e whi ch
c o n t a i ns as a fa ct or t h e h ors ep ow er p er uni t w eight a q u an tity w ell k n own
a ry i m
to b e of p ri m
por t ance i n t h e es tim
a ti on of t h e p erfo rm
a n ce of an
aero p l a ne
bi nati on of e qu ati ons (22) an d (29) show s th a t t h e a ngl e of
A co m
i nci d ence an d a dv an ce p er r ev ol u tio n of t h e ai rscrew a re fi xe d for all
Th e
res
ult
a
es
W
I)
2
1
—
[0
73
5
8
6
w ritt en
be
)
1
3
re
P
3
)
2
:
5)
11D
t he
of
a na
,
.
op lanes of
are
a
is
kn o wn
f u ncti on
a
In
.
of
funct i on
m
the
aer
sa
l ev el flight it
the
Of
t
e ex ern a
a
dvance
M
“
,”
1
l formi f
h as
been
seen
v olutio n
r
r
e
e
p
Th e
an
gl e a
is
q u an titi es
t he
th a t
an d
the
it
n ow
ly u s ed
rare
an
an d
1
3
p
gl e of in ci d en ce
follow s th at
111 re
d ucti on
,
bu t
.
5
18
g
P
X
3
1
of i m
p ortan ce Th e p ower P as u sed h as b een t h e act u al p ow er an d 15
e qu al to f P5 wh ere P,5 i s t h e s t a n d ar d h orsep ow er an d f t h e p ow er f acto r
wh i ch all ows for ch ang es of p ress ur e a n d t em
p er atur e fr o mt h e st an d ar d
con diti on
is
.
.
.
,
PR EDICT IO N
AND AN ALY SI S
FOR
AEROP L AN ES
423
ill ust ra ti ng t h e rel ation b etw een t h e qu antiti es of i m
po rt
a n ce i n l ev e l fl ight 1s sh ow n ( Fig
Th e uni t s a re f eet an d secs
wh ere n o t o th er wi se Sp eci fied For i n t ern ation a l co m
p ari so ns p woul d
b e b ett er tha n a as t h e d i m
ens io n s o f t h e q u a n titi es a r e th en zer o a n d
i ca l un i ts
e for a ny co ns i s t en t s et of d yn am
co nse qu en tly t h e sa m
For cli m
bin g fli ght t h e for
a dOp t ed n ee ds d ev elo p m
s i n ce
en t ;
P
w
V
p a d
d e term
in e bo th a an d
i t foll ows fromeq
n
u at 1on (23 )
m
A figu r e
.
.
.
.
,
m
W
tha t th ey also
fix
m
,
O— a
li m
b
Th e
v al u e
—Fu nda m
en ta l cu rv es o f aer Op lane p
m
,
the
an
gl e
.
of c
.
of
i
i
?
]
is
e
qu al
20
Fl o
to
si n
0,
a nd
.
fl it
h ence a n e qua tio n for
0
o r,
mlti p lyi
u
n
Eq u a tio n ( 22)
g by
s
p
h ows th a t
0 11
:
t of cli m
bm
ay
t he
VF
e r for
ra e
(
“,
a
an ce.
be
w ritt en as
)
nD
bo th si d es
,
is
a
f u ncti on of
a. a n
d
and
h ence it
APP L IED AERODYN AMICS
4 24
ME
follows fro m(3 1) th at
b v e of
seen a o
Th e
Fig
in
.
213 ,
whi ch
lim
b i s to
be
to a d ep en d ent v ari a bl e
mki g
mo t i mp
n
a
an
an d
a
.
a
n
a c
t
r a e of c
o
W li mbi
m
now conn ec s
both l ev el
a ls
f u nction
the
g t es t
i
ser es
w as
or a s
li m
b i ng fl ight s
mxi mumo v
F
om
pl t
c n
a
ert s
J;
o d of
e e r ec r
or a c
of
cu r
v es
of
whi ch
hown
o dition th a t
Th e
.
V
ar e s
P
v ari a bl es
c
p l an e
the
c n
fro ma n in d ep en d en t
p l an e p er fo r
aero
m
an ce
i d ep en d en t v ar i a b l es
as n
t h e fi gur e
illust ra tes
or a n
.
.
,
of
th e For
the
can
.
,
Apnfi cafi on
,
t h e t wo
t cases
Th e g en era l th eoremh as i m
p o rt an t a pp li ca ti ons i n whi ch all
v a ri a bl es are u sed For t h e red u ction of p erform
a n ce si m
p lifi ca ti ons
be m
a d e s i n ce i n t h e p rocess W 10 an d D ar e co ns t a n t
s
t
.
011 a n a er o
d n ee d to be co nsi d ere d
i nfi ni t e
o f a an d
”
ults obt ai n ed fro
an d
t
is
an d
,
r es
V
ml
n ae of
Redu cfi an to
a
Parficnla
asc
O bserv a tions on a h igh sp eed scout t aken i n fli ght are shown in T abl e 10
-
TAB L E 10
.
m
ia h i
e gh t .
Ane
feet
.
—ll ) C i mb
l
mg
Te
.
crat u rc,
c
.
l I nd id ote d
ca
sp ee
( 2) Level S p eeds
n
.
I ndi cat d
e
ai r speed
an
p n1
.
RN
“
APP L I ED AERODYN AMI CS
m
oo
EN GlNE REVOLUTIO NS
Fro
Fro
.
215
.
.
—V
214
.
S t an dard h orsep ower
-
R P
.
.
M
a nd rev olu ti ons.
P RO PO RTIO N A L P R ES S UR E
ar i a ti on o f
h ors ep ower wi t h p ressure
and
.
te
mp
erat u re
.
AND AN AL YSI S
PREDI CT ION
FOR
AEROP LAN ES
4 27
ob ser ved a n d 4 th en foll ows fromFig 215 Th e rel a ti v e
d en sity a w as cal cul at ed fr o mcol u m
n s 2 an d 3 by us e of e qu a ti on
a n d t h e l as t col u m
n f oll ow s fr om
n 6 an d t h e o b ser v a tio n s o f r evolu
colu m
tions
Fu rth er calculation l ea ds to t h e r equi red fun d am
en t a l
d at a of
r e d u ctio n
Colu
m3 w
n
as
,
.
.
.
.
00768
lum
t h e thi r d i s
n s of T a bl e 12 a r e o bs er v ation s ;
mt h e secon d and Fig 214 an d t h e four th an d fi fth are calcu
Th e firs t
o bt a i ned fro
two
lo o
14 00
90
100 0
co
.
,
0 02
FI G
.
216
.
—S t
a n dar
d
cu rv es of
p erfor
m
a n ce red u ct i on.
l a t ed u sin g t h e fi gures i n T a bl es 10 an d 11 Th e res ult s a r e p lott ed in
a xi m
Fig 216 an d are n ow st a n d ar d r ed u cti ons of m
um
Sp ee d
a n ce i n a s t an d ar d a t m
os p h ere t h e p rocess i s
To fin d t h e p erf or m
Fr omt h e d efin i tion o f a st a n d ar d a t m
osp h ere an d
r e v ers ed a s follows
p era ture as
t h e law of v ari a tion of h o rsep ow er wi th p ress ur e an d t em
gi v en i n T a bl e 1 t h e ca l cul a ti on p ro ceed s as for T a bl e 11 e x cep t for t h e
l as t col um
n
.
.
.
,
.
,
.
APP L IED AE RODYN AMIC S
428
TAB LE 13
.
F romt h e st an d ar d cu rv es of Fig
num
b ers
.
216
are
th en o btai ned
the
followi n g
L eo /J
m
m
Th e fi na l fi g u res for p erfor an ce i n a s t an d a r d a t osp h er e ar e o bt a in ed
by fi n d in g th a t sol u tio n of T a bl es 18 an d 14 whi ch i s cons i st en t with fu ll
p ow er of t h e en gi n e Th e cal cul a tion i s si p l e, an d at
ft i s fou n d
by ass u in g v alues of 110 an d 113 for a W an d cal cul ating t h e v a lu es of
r p
a n d P,
r p
1605, P8
c ‘v = 110,
24 8
.
.
m
m
m
.
.
.
.
0
&V
= 113
.
.
m
.
F8 = 265
,
Th ese fi gur es are rea d ily obt ain ed by cal cul a ti on fr omn u m
b er s alrea dy
t a b u l a ted Th e two v alu es of r p man d P s ar e th en p lott ed in Fig 2 14
Th e in t ers ection with t h e r eal ho rse p ow er
an d j oin ed by a st ra ight li ne
cur v e o ccu rs wh er e t h e r ev olutio ns a r e 163 5 an d t h e r ea l Sp eed i n m h
p
130 m
h
r
n
B
a
e
e
t
h
e
r
p
titio
of
p
o
l
ces
s
i s 163 5 X 0 0795==
t
h
e
fi
n
a
p
y
osp h ere i s fo u n d
p erfor ance d u rin g l ev el fl ight i n a stan d ar d a t m
see
T a bl e 15
‘
.
.
.
.
.
.
,
m
.
'
.
.
.
,
.
TAB L E I 5
.
m
t
32m
i 2
Maxi
n
m
rue
1
!
W
.
.
APP L IE D AER ODYN AMI CS
4 30
u v es of Fig 216 V
a re sh ow n i n T a bl e 18
c r
a nd
.
.
V.,
ar e
th en
r ea
di ly
l ul a t e d
ca c
Th e
.
r es
ul ts
.
TAB LE 18
m
mm
St a ndar d hei gh t
Ti
Rate of cli b
( tu - l )
mt
e
.
mb
o cli
(
.
m
m
.
ai r sp eed
p i n)
Th e thi r d colu n of t h e a bo v e t a bl e i s obtai ned by
t h e nu b ers i n t h e seco n d col u n a n d p lotti n g
m
of
h eight
6c
,
.
.
m
$
p l otti ng
g i t
a a ns
1
H
i nt egral
Th e
.
Engi ne
I ndi ca t ed
o
,
re vo lu t i o ns
.
t a king t h e r eci p r o cals
aga ins t t h e s t a n d ar d
bt ain ed by
t he
any o f
mtho d gi v t h v l of t u p to y h ight H
—
i
t
Th ob
vatio
u d f t h ill
th
R
d
m
k
m
R
x m
pl w
t k di tly f om p ilot
po t I om p t
t ti
p ti l ly f t h i d i t d i p d
di g th
ly i how th t
im
p o v m t of ob v tio woul d l d to th b tt
lt
O th
oth h d i t i k ow both p ti lly d th o ti lly th t t h b t
t g
t f li m
bi
tly ff t d by md t h g of i p d d
t th
by pp i bly i
o
t h p i m y f to i
t
s an
d ar d
cu ar
e
r
er
ra e o
e
r
e
or
s
,
ar
ac
r
a ue
an
a r-s ee
ca e
rac
n,
a
s no
ra
ca
an
ec e
ere
o
se
r
e a na
er
e
e re
n err r .
or
ns
.
er a e c an
r ec a
a
s re
n s,
ea
n
r ea
r ea
.
ns
’
a
r
e
ser
e
rec
en
n
n
s no
c
a
s er a
en
an
e
e uc o
e
er e
e
ve e a
ar
es
s,
on
ar s
e
ra
e
e
e r eS
s s s
er resu
s
ca
a
es
,
us
ec s
s
n
.
a r S
a
e
e
es
ee
an
m
p ro ced ur e follow ed i s v ery g eneral i n ch aract er an d ay be a pp li ed
to a ny horsep ow er facto r whi ch d ep end s on p ress ur e an d d en sit y no
It 18 shown l at er t h e ch ap t er th at fl yi n g e xp eli
a tt er wh a t t h e law
ay b e so co n d u ct ed th a t a ch eck on t h e law of va ri a ti o n wi th
en t s
h eight 13 obt ai ned fr o t h e t ri als th e sel v es t h e essen ti al ob ser v a tions m
ber of fl ight s n ear t h e gr o u n d with t h e engin e all out t h e
elu d i n g a nu
a xi m
um
s p eed l evel to
ax i
u
b
ra t e of cli
co n d itio ns ra n gi n g fr o
en t s ca n o nly gi v e t h e p ow er f acto r for t h e p a r ti c u l ar
As t h e fl ight exp eri m
r el atio n b etw een p an d t whi ch h a pp ens to exi st it i s s till n ecess ary t o
s t an d ar d co n d iti o ns b u t
a pp ea l to b en ch t es t s for t h e corr ectio ns f ro
a i n v a ri a ti on
n ot for t h e m
eth o d o f r ed u cti on o f B riti sh p erfo r
ance t r i a l s h as
Th e st an d ar d m
u p to t h e p resen t d at e b een b ased on t h e assum
p tion th a t t h e en gine
horsep ow er d ep en ds o nly on t h e d ensity Qu es tions are now be in g r a i sed
p tion an d t h e law of d ep en d en ce of
as to t h e st ri ct v ali d ity of thi s ass u
p ow er on p r essu re and t em
p era t u r e 18 b ei ng exa i ned by eans o f s p e c i a ll y
e d i fl er en ces fr o
en t s
Th e e xt r em
or e e l a b o ra te
t he
con d u ct e d ex p eri m
p ti o n d o n ot a pp ear to b e v ery great an d affec t co m
p arati v e res ults
a ssu m
osp h ere d i ffers gr ea tly f r o
o
o nly wh en t h e actu a l a t m
t h e s t an d ar d a t m
Th e
,
m
m m
m
.
m
m
mm
m
,
,
”
“
m mm
m
,
m
.
,
.
.
,
m
m
m m
m
m
,
,
.
m
,
PR EDICTIO N AND
AN AL Y SIS
AEROP L AN ES
FOR
43 1
p h ere It a pp ears th a t a st age h as been r each ed a t whi ch t h e d iffer ences
co
e withi n t h e l i
it s of easu r e ent an d t h e r ath er o re co p l ex law
will th en be n eed ed
If t h e horsep ow er d ep en ds on t h e at osp h eri c d ensity only t h e
red u ction of ob ser vatio ns i s s i
p li fied for t h e h eight i n t h e s t an d ar d
at
os p h ere is th en fi x ed by t h e d ensity a l on e an d all obs er v a tions of sp ee d
a n d rev olutio ns a pp ly a t thi s s t an d ar d h eight i rresp ec ti v e of t h e r ea l
h eight at t h e ti e of o bser va tion For l evel s p eeds o nly t h e l at 2n d 3 rd
an d 5
t h col u n s of T a bl e 11 are r eq u i re d
F ro t h e v alu es of a an d
T a bl e 1 t h e v alu es of t h e s t an d ar d h eight are obta in ed an d us i ng th ese as
a bsci ssae t h e i n di ca t ed ai r s p eed s an d t h e rev olutio ns of t h e en gi n e ar e
p lotted Thi s 18 now t h e r ed u ced cu rv e an d at e v en h eights t h e st an d ar d
v alu es of ai r sp eed an d rev olutions ar e r ea d fr o t h e cur v e
For cli bs t h e fir s t si x colu ns of T a bl e 16 ar e r eq u ired an d t h e rea l
r a t e of cli
b i s th en p lott ed a ga ins t t h e s t an d ar d h eight as d eter i n e d by
a
Th e re ai ni n g p rocesses follow as for l ev el fli ght s
B y wh a t ev er
eans t h e ca l cul a tio ns ar e carr i ed out t h e res ult s of t h e
r ed u ctio n of perfo r m
a nce to a s t an d ar d serv es t h e p ur p os e o f co m
p a ri son
betw een v ariou s aerop l an es an d en gin es i n a formwhi ch i s esp eci ally
suit a bl e wh en th ei r d uti es a re bein g ass i gn ed
For so m
e p ur p os es su ch a s t h e ca l cula tio n of t h e p erfo r
ance of a
w eight carr yi n g aero p l an e or a long d is tance m
achi n e i n whi ch t h e w eight
of p e t rol consu ed i s i po r t an t t h e s t an d ar d r ed u ction i s a pp reci a bly
l es s useful th an t h e i n t er m
e d i a t e s t a ge rep r es en t ed by T a bl es 12 an d 17
o r p re fera bly by cur v es obt a in ed fr o
th e an d t h e lo a d i n g to gi v e t h e
fo rmof Ih g 218 Th e lo adi n g 10 was 8 5 lb s p er squ ar e foot
s
m
m m
.
m
m
m
m
m
m
.
m
m
,
,
,
m
.
.
,
,
,
.
m
,
m
m
m
'
.
m
.
m
,
m
.
,
m
.
,
m
-
m
-
,
m m
,
'
.
En
.
m
pl
es of
,
th e Use
m
Ot
andard
.
.
,
Ourvss of the Tn e sh own in Fi e 213
.
Meri t —Th e fir s t p oi n t to b e
n o ti ce d i s th a t t h e c ur v es
Aerody na i c
i ne d by t h e a er o dyn a i cs of t h e a ero p l an e a n d a i r
a r e es senti ally d et er
Thi s wi ll h a v e b een ap p r e
scr e w , an d do n ot d ep en d on t h e en gi n e u sed
t h e f ac t th a t a Sp eci a l cal cu l a tio n wa s n ecess ary t o en s ur e
c is t ed f r o
th at t h e en gi ne was gi v in g full p ow er i n any p arti cular co n di tion of
m
m
m
fl ight
.
“W (
.
/
V N e again st
" 3
f
y
W
ens ion al coefli ci en t s whi ch for t h e aer o p l an e an d ai rs cr ew p l ay t h e
n on d i
e p a r t a s t h e fa m
ili ar lift an d dr ag coeffi ci en ts for wi n g fo rm
U si ng
sa m
s
e i th e r a or p t wo se t s o f cu rv es for d i fferen t a erop l a n es m
ay be s u p er p o se d
p are d di rectly
a n d th ei r ch a ract eri s ti cs co m
I f for a gi v en v a l u e o f
v ari a bl es
Th e
-
‘
m
'
,
.
,
.
on e aer
o p l an e gi v es great er v alu es of
J;
V
and
/i
I LA
th an
i c d esign of t h e fo rm
oth er t h e aero dyn am
er i s t h e b ett er
In thi s
a r k e d th a t t h e m
eas u re o f p ow e r i s t h e t or q u e
co nn ec tion it sh oul d b e rem
et er on t h e engi n e t es t b ed a n d th a t t h e en gi n e i s u s e d a s an
om
d y n am
ed i ary s t a n d ar d
i n t e rm
It i s u n fort u na t ely not a thor oughly goo d i nt er
an
,
.
,
.
APP L IED AERODYN AMI C S
43 2
y an d t h e a cc uracy of t h e cu rv es i s u su ally li m
i ted t o th a t o f a
All a ero p l a nes gi v e cu r v es
kn owl ed ge of t h e en gi n e h ors ep ow er i n fli ght
t h e d i ffer ences bein g s i m
of t h e sa m
ilar in p ro
e g en era l ch a ra ct er
ou
n t to tho se b etw een t h e li ft an d d ra g c ur v es o f g ood
at e a
r
t
i
o
n
o
p
win g sectio n s
—
h
n
w
ith
ou
t
o
f
M
u
s
e
u
E
n
i
n
C
a
e
e
Sin ce t h e aer o
Ch ang e of
g
g
d yna m
i cs oi t h e a ero p l ane i s n ot chan ge d by t h e ch an ge of en gi n e it
medi ately app li ca ble Th e only
foll ow s th a t t h e st an d ar d cu r v es a re i m
e ffec t of t h e ch a nge i s to in t r o d u ce a n ew engi ne cur v e to rep l ace t h e ol d
o ne i n o r d er to sa ti sfy t h e co n di tion th a t t h e en gi n e i s fully O p en ed u p
a xi m
d ur in g l ev el flights or m
umcli m
b
—
Aga i n t h e aer o d yn am
i cs is no t ch a n ge d
Change of Wei gh t carri ed
As an exam
p l e cons i d er t h e
an d t h e c ur v es a r e a pp li ca bl e as th ey s t an d
2000 lbs an d a l oa d i ng
e ffec t o f ch a ngin g t h e w e ight of an aere p lan e fr o m
mdi
e
ar
,
.
m
,
m
.
,
.
.
.
,
.
,
.
2 4 -0
220
BRAKE
14 0
1
3 00
Fro
.
217
.
IS O O
14 00
—B ala nce oi
1700
16 00
1
8 00
19 00
EN6 1NE S PEED R P M
.
.
2 00 0 2 1
00
.
ai red an d
h orsep ower
i s cha nged
h orsep ower
t he gross i
2 200
a vai la ble
whe n
.
2
l
b
s
a
foot
to
w
ight
of
a
n
d
lo
d
i
g
of
a
e
5
0
0
a
n
1
0
l
b
s
r
e
q
p
_
ft
sqfoot t h e h ei ght be in g
Th e v al u e of a at a h eight of
ft i n a st an d ar d a t m
os p h ere is
0 74 0 an d t h e ho rsep ow er fa ctor will be ta ken as f
Th e e n gine
cu r v e of s t a n d ar d ho rsep ow er i s shown in Fig 217
To b egin t h e cal cu l a ti on two v a l u es of st an d ar d ho rsepo w er P,
ed
a n d t h e c u r v e of Fig
217 show s th a t 160 an d 220 are
a r e ass u m
reas on a bl e v alu es
Grea t er accuracy woul d be a tt a in e d by t a k i n g th ree
v alu es
T a k in g one loa di n g as ex am
p l e t h e p roced u r e i s as follow s
of
8 lbs p er
.
.
s
.
.
.
.
,
.
,
.
.
,
,
.
.
,
.
.
,
)
1
( )
P,
220,
f
m
.
I;
227
s t a n d a r d cu r v es
2
r
o
t
h
e
F
( )
o f 22 7 a s a bsc issa e t h e or d i na t es
,
f ro mt he d a ta gi v en
of Fig
to get
.
213
rea
d
o ff,
.
for t h e
a bo
v e v alue
APP L IED AER ODYN AMI CS
484
Ss r x
m
r ro n os
AE RO P L AN E
m
fo m
Ars s can w Er r xcxs n o
AN D
s
p lan e p er r an ce no
a er ep lan e an d a irscr ew h as bee n
s ep ar a ti on of t h e efli ci enc i es of t h e
p t ed an d t h e an alysi s h as been b ased on v ery st ro ng th eoret i cal
a tt em
groun d Th e p ro p osa l now b efo re u s i s t h e rev ersa l of t h e p rocess follow ed
i n t h e d et ai l e d p redi ction of a ero p l an e p erfor ance an d i n or d er to p roceed
general kn owl ed ge In t h e
at all i t i s n ec essary to i n t r o d u ce d a t a f r o
ch ap t er on Ai rs cr ew s it w a s p oi n t ed out th a t all t h e ch ar ac t er i sti cs of ai r
a t ely by a ser i es of st an d a r d cu r v es
s crews can be ex p r essed a pp ro xi
a pp li ca bl e to a ll
Th e i n di v i d u a l ch ar act er i sti cs of each a irscrew ca n be
rep res en t e d by f ou r cons t a n t s a n d t h e an aly si s shows how th ese co ns ta nt s
t ri als i n flight Th e d et er i n a tio n of th ese four
ay be d et er i ned f ro
cons t an t s a ls o l ea d s to t h e d es i re d s ep ara tio n of aerop l an e a n d a i rscrew
p rev ious
In t h e
re
d u cti on
an d a n a
lysi s of
aero
,
m
m
.
,
.
m
.
m
m
e fli ci en ci es
m
m
,
.
.
Th e p ri nci p l es i n v ol v e d h a v e b een d ealt wi th in t h e earli er sec tio n on
d et ailed p red i ction wh ere t h e f u n d a en t al e qu ations w ere d ev elo p ed
ed i at ely with an app li ca tio n to an
Th e a n alysi s will th er efo re begin i
aere p lan e
u
w
ro
p
l
a
n
e
ho
se
n
f
o
r
i
l
l
s
t
ra
tio
n
a
s
a
two
s
e
a
t
er
ae ro p l an e wi th
c
Th e ae
w a ter-cool ed en gi ne Th e choi ce was a d e b ecaus e t h e fl ight ob ser v a t i ons
o re co p l et e th an us u al Th e o b ser va ti ons re d u ce d t o
a v ai l a bl e w er e
osp h ere are gi v en i n T a bl e 19 b elow whil st t h e st an d ar d
a s t an d ar d a t
en g i n e ho rs ep ow er as d et er i n e d on t h e be n ch will be foun d i n a l a te r
m
m
m
.
.
m
m
m
m
.
.
m
,
t a bl e
.
TAB LE 19
Le vel flig h ts
.
.
(
Th ese le vel fli gh ts w ere
md
a e
wi t h t h rot t led
engi ne .
mmm
o
a
.
PR EDI CTIO N
AND AN AL Y SIS
F OR
v l ti ons o f t h e a i rsc rew w ere l ess th an th ose of
t h e g e a r in g ra ti o bei n g 0 6 to 1
Fur th er pa rti cul ars a re
Th e
48 5
AEROP LAN ES
t he
re o u
en
gin e
,
.
Gross
wei ght of aereplan e
Air screw
di a
meter
It wi ll be fou n d th a t it i s p oss ibl e t o d e d u ce f romt h e d a ta gi v en
h
r
h
e
o
p
it
h
f
i
ew
1
T
c
t
e
a
rs
c
( )
a
a
r
r
e
2
h
v
i
tio
f
gi
n
p
ow
with
h
ight
e
n
e
e
n
o
e
T
( )
3
h
e
e
f
fi
c
i
e
n
c
y
o
f
t
h
e
a
i
r
s
c
re
w
T
)
(
s i s t an ce of t h e a erop l an e a p a rt f ro m
4
i
t
s
a
rs
cre
e
r
i
w
h
T
e
( )
.
.
.
.
d e d uced fr o
F lo
218
.
v es
f ollow s
of
cur
i
g
.
—8 t a
Fig
Fr o
.
ca n
mt h
be
f s
a e r o p l an e
,
h e p it ch of t h
t q u e coeffi ci en t of t h e a irs crew as shown by t h e st and ar d
e or
n da rd ai rscre w cu rves an
218
th e
m
.
and
th e
be i ng k nown
,
i
mb
f o mb
nu
l l t d
ca cu a e
the
ers
r
b ench t es ts
i n T a bl e 19
h t est s of
enc
v l tions
en g n e r e o u
t h e v alu e of
5as
7
713
7
d in th e
an d
h own
s
a nalysi s of aerop la ne
an c e
p ow er of t h e en gin e as
a n d equ a ti on ( 18) t h e v alu e of
on
the
t he
en gi n e
Th e
an d
a r s cre
.
gea ri n g
in
m
pe rfor
t he
s
p eed
of
w di a m
e ter
i
T a bl e 20 i s easily calcu la t ed
U s in g e qu ati on ( 18) a n d p utti n g i n t h e n u m
eri ca l v alu es
St d B H P
.
.
.
.
the
of
t he
e
.
m
xa p l e
APP L IE D AE R ODY N A M IC S
486
an d
t he
v alu es gi v en i n t h e l as t col u
ula
T abl e 20 sh ows th a t
thi s f orm
a re
m of T
n
obt ained
,
fr omt h e
on e
a
,
TAB LE
za
.
mm
x
20 are
mxi muml v l
f li m
b Th p
f romt h e t es t for m
u mrat e o c
a xi m
w ere ex t ract ed fr omcol u m
n s 1 6 an d
20 —En
the
at
.
bl e
'
s
.
7
s
e
ar
a nd
t i cu l ars
oth er
t he
in
T a bl e
21
20
.
m m
s
ca c
p ee d
e e
of Tabl e
w rrn Exa
m
l ula te d fr o
sa m
e h eight two v alu es of
a
“
a
.
o ur
.
00201
00304
0023 2
0029 7
00408
TAB LE 21
.
Lev el nig h t
For
h
eac
of
r ow
d X i s ob ta ined
-
nD
t he
i
a rscre
h own
cur v es
s
w if
us e
t a bl e f(h)
This
be
l tio n
re a
md
c a
,
1
)
,
P
a
e
v
lu
of
e
h
T
D
is
o t
is
c ns a n
is
t
,
s ufii ci en t
and a
to
l tion b etw een
re a
m
d et er in e
t he
p i t ch
k0
of
t h e s t an d ar d cu r v es of Fig 218
As
Air screws t h e o r d in at es and a b sci ssae of th ese
but t h e sh ap e i s d et er i ned wh en t h e p itch
a
h p t er on
u n det er m
in ed
P
1s kn ow n
in t h e
ar e
.
t he
.
f o un d
e of
.
m
.
as
fo llow s
.
.
APP L IEDl as nonv
4 88
of T a bl e 20 by
i s rea
t he
u se o f
m
d f ro
t he
t
s an
t he
mm
cs
p itch di am
et er
d a r d cu r v es
for
ra
i
ti o
ws
a rs cr e
RELATIV E DENS ITY O
l
a rea
,
for t h e
dy
v al u es
"
F
of
m219 —C l
.
3;
a cu la t ed var ia t i on of
.
in
ol u m
n 3
c
b etw een th ose for
,
the
P
h ors ep o wer wi t h h eigh t fr o
p ar ti cul ar v alu es
l 2
°
a nd
P
-
14
.
mb
o serva t i ons
i n fli ght
.
b ei ng in terp ola ted
n 6 foll ows by d i v is ion
C olu m
D
D
t h e nu m
b ers i n colu m
n 5by th ose i n colu m
n 4
.
l 000
4 10
09 9 5
09 88
09 82
358
33 0
‘
09 78
of
‘
PR E DIC T IO N AND A N AL Y S IS
m
s on AE ROP L AN ES
439
Th e va lu es of Qofl h) i n col u n 5are th en p l ott ed i n Fig 219 with a as
a b as e
Th e p oin t s li e on a s t raight lin e whi ch i n t ers ect s t h e o r d in at e a t
a
1 a t t h e v alu e 43
Sin ce f( h) i s th en un ity thi s v al u e d et er i nes Q0
for t h e airscr ew , col u n 7 of T a bl e 23 i s obt ai n ed by d i v i s i on an d sh ow s
t h e var i a tion of en gi n e p ow er wi th h eight
Th e la w of v ar i a ti on as thu s d ed uce d e p i ri cally
ay be exp r esse d as
.
m
.
m
.
,
.
o
m
m
3
01 5
m
h ows th at t h e b rake ho rsep ow er falls off app reci a bly o re
th an t h e rel a ti v e d ensity
I n t h e co urse of t h e cal c ul a ti on of f(h) it h as bee n sh own th a t
an d s
.
o
09 63
09 03
m
03
05
-
Deter i nati on of th e Aerop lane Drag
—
T
r
To d et er i n e t h e a erOp lan e
Facto ,
‘
s
facto r
m
To
u se
,
is
m
md
a
e
of
q
e ua
ti on
d r ag
th ru s t
and
two v al u e s of
coe
{
ffi ci en t
J
P
for t h e
p eed b ei n g e x t ract ed fr omt h e o b serv a tion s so th a t t h e d rag
ay b e e li
in a t e d as i n d i ca t ed i n p rod u ci ng eq u a ti on ( 1l ) Th e
coeffi ci en t
sa
e ai r S
m
m
,
.
Engi n e t h rot t led
.
446
APP L IED AERODYN AMIC S
lift coefli ci en t kn i s n ow an i m
p o rt an t v aria bl e an d gi vi ng t h e p ar ti c ul a r
v al u es of t h e exam
p l e to t h e qu an titi es of e qu a tion
s h o w s th a t
,
,
,
With thi s form
ula and t h e ra t es of cli m
b gi v en in T a bl e
Y ca n be cal cu l at ed
and
k,
\
.
Th e res ults are
m
F ro mt h e n u b ers in T abl e 24
of
k, i n
or
,
£
1
my b
d er th a t v alu es of
a
gi v en i n T a bl e 24
x t ract ed
e
v a lu es
e
for
of
.
l evel fl ight i s p lotted on
for
3
19 t h e
v alu es of
a
b as e
t he
air
L I FT COEFFIC I EN T
between ob ser va tio ns
m
m
Th e
.
o di tion r equire d
c n
is
th at
m
m
V
r l ev el fl ights sh all be t a ken a t t h e sa
v alu es of 5 fr o t h e cur v e fo
7
71
bin g Cons t a n t ai r sp eed eans constan t kL F r o
ai r Sp eed as for cli
Fi g 220 T a bl e 25 i s co p il ed p ar t of t h e da t a b ein g t aken
-
mT
.
fr o
,
a
bl e 24
.
m
m
,
.
TAB LE 26
mul
Th e for
fr o
mq u
e
a
.
ti on
a
.
whi ch l ea ds to t h e thr ust coeffi ci en t f act o r
ay b e w ritt en as
an d m
(
Tc he
l
l
a
?
,
To, i s
e
APP L IED AER ODYN AMIC S
442
a n d To = 7 5
6
Th e oth er v alu es of h. y i eld To
ns i s t ency
, a n d t h e co
o f t h e red u cti on i s seen t o be on ly
o d era t e An ex a ination of e q u a tion
d
0
s
s
4
t
h
a
a
n
how
why
e
wh
di
fl
ere
n
o
d
p
d
b
g
s
ll
er
ces
n
i
e
e
n
s
e
i
n
c
h
T
,
( )
o
s a ll er as t h e ra t e of cli
b di i ni sh es In ean in g t h e ob se r v a tions , due
w eight i s gi v en to t h e r el ati v e accuracy if t h e n u era tors an d d en o i n a tors
of t h e fractions for To be adde d b efore d i v i sion Th e res u lt i n t h e p res ent
ins t ance i s to gi ve
m
,
°
m
.
m
m
'
m
m m
m
m
.
.
To :
o
4
4
( )
o
In
t es ts ca rr i ed ou t with a v i ew to a pp lyin g t h e p resen t lin e of an alysi s
t h e e v i d en ce of gli d es wo u l d be i nclu d ed an d t h e accur acy of r ed u c ti on
a pp reci a bly i n crea se d
Aerop lane Drag
in ed equ a tio n (4 0) i s a
To h avin g been d et erm
,
.
.
-
,
L :n
Fro
.
22l
.
—A
crop la ne
mk
cocrrlcna
gli der dr ag
as
03
-
L
d ed u ced by ana ly si s
of
per for
m
ance
'
t n ala
.
l ti o n b etw een t h e d rag coeffi ci en t kn an d kn own qu an titi es The
ca l cula ti on i s gi v en i n T a bl e 26
usi ng fi gu res fro mT a bl e 24 a a
b as i s
Colu m
n 1 i s t a ken from
T a bl e 24 a n d colum
it by
n 2 i s d ed u ced fr o m
re a
.
s
,
.
,
PR EDICTIO N
AND AN AL Y SIS
FOR
AER OP L AN ES
44 3
n 3 th en foll ow s
of on e of t h e st an d ar d a i rscr ew cu r v es Fi g 218 ; col u m
f r ome quation
n s ar e a l s o ta ken from
Th e fo ur th a n d s i x th colu m
n s 3 a nd 4
T a bl e 24 whil s t t h e fifth colu m
col u m
n i s d ed u c e d f r o m
Th e cur v e sho wi ng kl, a s d ep en d en t on k, i s gi v en i n Fig 221 t og eth er
with t h e cur v e whi ch was p rev i o usly use d i n t h e exa m
p l e of p r ed i ction
u ch
For v alu es of t h e lift coe fii ci ent b elow
t h e ca l cula t e d p oi n t s fall m
b elow t h e cur v e d rawn as p ro b a bl e A di scu ssi on of thi s r esul t i s gi v en
a li ttl e l a t er ;
p l e of an aly si s t h e d rag as d ed uced wi ll b e
as a n e x a m
fo un d to rep r esen t t h e observ ations
p l et e as
Air screw Efl i ci ency —Th e a naly si s i s p racti cally co m
a l r ea d y gi v en bu t as t h e a i rscre w effi ci en cy i s on e of t h e qu an titi es use d
i n d escri bi ng t h e p erfor m
ance of an a i rscr ew i t s v a l u e will b e cal cu l a t ed
Th e for m
u la i n con v eni en t t er m
s is
u se
.
,
.
.
,
.
.
.
.
.
,
.
‘
1
or,
i n t he
x
e a
mp l
7
mt h
t
e s an
if
re
V
T ole,
e
1
F ro
P 99
1 29
V
nP
d ar d ai rs crew cur v es
qui red )
is
eas
Tokv
ily obtain ed
o q
t he
as
efli ci ency at
various
Valu es
of
T a bl e 27
in
.
TAB LE 27
.
a
m mumi
per
cent
.
w effi ci ency i s seen t o b e 70 5p er cen t
—
l
on th e Ana y si s
Th e an aly si s shoul d b e r egar d ed as a
t en t a ti v e p rocess whi ch will b eco m
o r e p r eci se if regu lar exp erim
en t s
e
Th e st an d ar d ai r
be
a d e to obt ai n d a t a with t h e r e qui sit e accu ra cy
i n or m
o d i fication but i t i s o b v i ous th at a fur th er
ay n ee d m
s cr ew cu r v es m
s t e p coul d be t a ken whi ch r ep l a ces th em
F rom
i n a p ar ti cu l a r i ns t an ce
t h e d ra wi ngs of t h e a irscrew t h e for mof t h e s t an d ar d cu r v e coul d b e
It i s not
eth od s o utli n e d i n t h e ch a p t er on Ai rs cr ew s
c a l cul a t ed by t h e m
th en n ecessary th a t t h e cal cula ti ons of efli ci ency thru st or to r qu e as m
ad e
f r omdra wi ngs sh a ll be reli ed on for a bsol u t e v a lu es of t h e fou r ai rscrew
i ned as now outli ned but only for t h e g en er al sh a p e of
co n st an t s d et erm
t h e a i rscrew cu r v es
B oth t h e d r ag of t h e aer o p l an e an d t h e effi ci ency of t h e a i rscr ew as
Th e a xi
Re arks
m
a rscr e
.
m
m
.
,
.
.
,
,
.
APP L IED AER ODYN AMI CS
444
d ed u ced by ana lysis are l ess th an those u sed i n p r edi ction i n a n ea r li e r
p ar t of t h e ch ap t er an d t h e di fferences are utu ally correcti v e Th e
act u a l v alu es d ep en d p ri
a rily on To an d for thi s p ur p ose l ar g e d i ffer en ces
of r a t e o f cli
b are requi red if acc u racy i s to be att ain ed Thi s o bj ec t
ca n b e a chi ev e d by a n u m
b er of j u di cio u s ly ch osen gli d es
m
m
,
m
.
,
m
—
.
m
.
Th e Sh ap e of t h e Drag Coeffi ci ent Li ft Coe ci ent Cu rve at S a ll
—
Valu es of th e Li ft M
elani
Th e difl erence b etw een t h e res ul t o f
a na ly s is a n d th a t of di r ect ob ser v a tion on a
o d el i s i n t h e exa p l e
so st ri ki n
t
h
a
r
t
t
h
n
T
h
e
fu
th
e
tt
tio
d
v
ot
d
o
p
oi
t
o
d
l
t
r
a
e
n
n
i
s
e
e
e
e
g
?
c u r v e as u s ed i n p r ed i ct i on , Fig 207 sh ow s a
in i u for kl, a t a bo u t
'
m
m mm
n
m
m
,
”
.
,
.
.
m
ar s o n / e s
THREE Essent i a l s
GINE
I 000
.
50
4 0
3 0
80
60
90
I IO
IOO
I NDICATED AIRSP£ ED
m222
F
.
.
g rea t i ncrease i n v alu e o ccu rs u p to kg
It is
ak e a v ery d i rect ex am
i na tion for t h e con st an cy of kn o v er a
p ossi bl e to m
li m
i t e d range of km
whi ch i s i n d ep en d en t of t h e st a n d a r d cu r v es for ai rscr ew s
It h as been shown i n e qu a tion (20) th a t t h e d ra g co effi ci en t of an a erOp lan e
an d n o
kn:
:
.
is
d ep en d en t
d ep en d ence
by
on
on a
a nd
is
and
a.
.
V
Si
not
mil
a
ii D
ar
ly
o n ly
t he
,
and
th ru s t
t he
n ew
li m
it ation
mo v
re
coefli ci en t of an a er Op la n e
pp r eci a bly d ep en d ent
on
es
t he
i s fi x ed
It th en follow s th a t
a.
t d rag coefli ci en t in vol v es const an t a dvan ce p er r ev ol u ti on for t h e
ai rs cr ew
Adv a n t ag e i s t a k en of thi s r el a ti on i n p lotti n g Fig 222
Th e
or d i n at es a r e t h e v a l u es of a t r p mfor t h e en gin e a n d t h e a b sci ssa e ar e
cons t a n
.
.
.
.
.
,
.
APP L IED AERODYN AMIC S
446
v luti on of t h e a irscrew) t h e thr us t i s i n v ersely p ro p o rtiona l to t h e lif t
a n d k3 =0 10 th ere i s a 5
0 per cen t i n crea s e
B etw ee n
coeffi ci en t
i n f orce a n d if t h e bl a d e i s li a bl e t o twis t un d er loa d t h e r es u l t will b e a
t h e ass um
p tion t h a t
en t al p it ch an d a d ep ar ture fr o m
ch an ge i n ex p er i
a n a irscrew i s sens ibly r igi d
It ay th en b e th a t fa ilure to o bt a in a st an d ar d ty p e of cur v e as a
At
res ult of an a ly s i s i s a n i n di ca ti on of twi s t i n g of t h e a irs crew bl a d es
en t s w h i c h w ill
a ny ra t e t h e r es ult h a s been t o s u gg es t fur th er e x p er i
It will b e a pp reci a t ed th a t t h e sources of e r ro r
re o v e t h e u ncer t a i n ty
n ow d i s cussed d o n ot app ear i n t h e t es t of an aero p l ane whi ch i s gli di n g
d own wi th t h e a ir screw s to pp ed Th e an aly sis of s uch exp eri en t s m
ay
a tio n a s t o t h e co ns t an cy of kn a t
b e ex p ect ed to fur nis h d efin it e i nf or m
high Sp ee d s Flyin g exp eri en t s will th en gi v e i nform
a ti on as to t h e e ff ec ts
p ressibility an d th e a dv an t a ges of research i n t h is
o f twis tin g an d com
d i rec tion d o n ot n eed fur th er e p h asis
re o
°
.
.
m
,
m
m
.
m
,
.
.
”
m
.
.
m
m
,
.
C HAPT ER X
M OTI ONS OF A I RCRAF T
THE S TAB IL I TY 01? TH E
P ART I
.
m
m
St abili ty
General lntrcdu cti on to th e l roble s oovered by th e ter
Th e earli er ch a p t ers o f thi s b oo k h a v e b een chi efly occu p i ed by consi d er a
ti ons of t h e st ea dy oti on s of a i rcraft Thi s i s a fir s t r equi sit e Th e
th eory of st a bility i s t h e st u d y of t h e oti ons of an aerop l ane a bout a
s t e a d y st a t e of fl ight wh en l eft to i t s own d ev i ces , eith er with con t rols
‘
m
m
.
.
.
h el d or a b an doned
Figs 223 an d 224 Show obser v ati ons on two aerOplanes i n fligh t t h e
e w er e p hotogr a p hi ca lly r ecor d ed
S p ee d s of whi ch a s d ep en d en t on ti m
On e aerop l an e wa s st a bl e an d t h e oth er un st a bl e an d t h e d iffer ences i n
p ort an ce Th e flight s o ccurr ed i n
ar ka bl e a n d of gr ea t i m
r e co r d are rem
goo d or di na ry flyi n g w ea th er an d n o ser i ou s err or will ar i se in su pposin g
th a t t h e ai r was still
—
A Speci a l clut ch was p r o vi d ed by
eans
St able Aer oplane ( Fig
t h e r eco r d begin s with t h e
of whi ch t h e cont r ol colum
n coul d b e lock e d
an d t h e look j us t p ut i n to o p era tion
AS
rop lan e flyin g at 62
a er op l a ne b in g s t a bl e co m
h
t
h
e
e
n
c
d
m
e
t h e s t ea dy Sp ee d was th en 73 m
p
e
ark i t p assed to 83 m
to d i v e an d ga in sp eed O vershootin g t he m
h
p
b efo re agai n turni ng u p ward s ; th ere IS a v ery ob viou s dyin g d own of t h e
in ut es t h e m
otion woul d h av e b eco m
e s t ea d y
o scill a ti on an d i n a few m
p t h e aero p l ane con t r oll ed itself
Th e rec or d Show s th at a ft er a big bu m
ore th an two m
il es with o u t any Sign of d anger
for m
Unstable Aeroplane Th e nex t reco r d Fig 224 is v ery di fl er en t an d was
n o t so ea sily obt a in ed s in ce no p ilot car es to let an u n s t a bl e aer o p l an e
No p os iti v e lock was p ro vi d ed b u t by gen tly n ursi ng
a tt en d to it self
otio n it w as fo u n d p ossibl e t o get t o a s t ea dy flyin g Sp eed with t he
the m
O n ce th ere t h e p ilot h el d i t as long as h e
n a ga i ns t a st 0p
co nt rol col u m
in ut e Aft er a few
ca re d to an d t h e cl ock sa i d th a t t h i s was l ess th an a m
secon d s t h e no se of t h e a erO plan e b ega n t o go u p l oss of sp eed res ult ed an d
s
e r a p i d ly t h e aerop l an e b e an t o g a th er
t
s
n
o
Dropp m
I
s t alli ng occu rr e d
g
g
p h t h e p ilot agai n took
Sp ee d a n d get i nt o a v er ti ca l d i v e b u t at 80 m
e d o r d in ary fl ight
Th e a er o p l an e I n t hi s con dition i s
co n t rol an d res u m
to p h ea vy
A st all e d a er o p l an e h as b een Show n Ch ap V to b e li a bl e to Sp in an d
e i n effecti v e
N ear t h e groun d an acci d en t al st allin g
t h e ail er on s b eco m
may be di sas t rous Th e i mp ortance of a stu dy of s ta bility Shou l d n eed
no f u r th er su pp o rt th a n i s gi v en by t h e a bo v e illus t ratio n
.
,
.
.
,
.
,
m
.
.
'
.
.
.
.
,
,
.
,
.
,
.
.
.
'
.
-
,
,
.
,
.
,
.
.
,
,
.
.
,
.
.
.
.
,
.
.
,
,
.
.
.
44 7
APP L IED AER ODY N AMI CS
44 8
m
In all p rob a bili ty difli cu lti es i n resp ect t o s ta bility li i ted t h e d ura t i on
of t h e early fl ights of San tos Du
ont , Far an, B l eriot , et c It a y be
sa i d th a t t h e co n t r ols w er e i
p erfec t before t h e Wright B r os in t ro d u ced
th eir Syst e of win g w arp in g in conj un ction with ru dd er ac tion , an d t h at
thi s d efici en cy i n con tr ol woul d be su ci en t to accoun t for t he p a r ti al
fail u r es of t h e early a vi ators Although thi s obj ection ay hol d goo d it
i s ob v io u s th at a
achi n e whi ch is tot ally d ep en d en t on t h e skill of t he
m
m
m
m
m
-
m
F10 28
.
p il ot for i t s sa fety
p ilot s assi st ance
I S n ot so
.
m
.
—The u ncont
.
r olled
oo
d
a
s
g
mt i
o
on e
m
.
,
on of a st a b le aeroplane.
whi ch
can r
ight i tself wi th out
’
.
t he
m
ay b e d efined
A st a bl e aero p lane
Defini ti on of a Stable Aerop lane
ay h a v e got ei t h er
any p o sition i n t h e ai r i nto whi ch it
as on e whi ch , fr o
’
a s t h e r es ult of gu st s or t h e p il ot s u se o f t h e co n t rol s , s h all r ec o v er i t s
ac hi n e to
co rr ec t fl yi ng p o sitio n a n d Sp ee d wh en t h e p ilot l ea v es t h e
ch oose i t s o w n cou rse, with fi x ed or fr ee con t rol s, acco rdi n g t o t h e ch ar acter
m
m
-
.
m
bi lity
ed to allow a n aer o p l an e
Su ffi ci en t h eight a bo v e t h e gr ou n d i s p r es u m
to reach a st ea d y flyin g st ate if i t is a bl e t o d o so Th e m
ore ra p i dly
ore sta bl e i t ay be sai d
t h e a erOplan e r ecov ers i t s flyin g pos ition t h e m
ay r e t u rn t o i t s
If a p ilot i s n ecessary i n or d er tha t a n aero p l ane m
t o be
n o rm
a l fli ght p os i t i on th en t h e aer o p lane it self cann ot be sa i d to be st a ble
of
t he
st a
.
.
.
,
m
APP L IED AER ODYN AMI CS
4 50
lthough t h e t ermm
ay b e a pp li ed to t h e co m
binati on of aer ep lane and
p ilot
A sub di visi on of st ability i s d es ira bl e t h e t erm
s
an d
i nh eren t
a ti c
a u to m
b ein g alrea dy i n u se An aero p lan e i s sai d t o be in h eren tly
st a bl e
if wh en t h e co nt r ol s are p l aced i n th eir norm
al fl yin g p osi ti on
whil st t h e aero p l ane is i n any p osition and fl ying at any Speed t h e result
a l fl yin g p osi tion an d Sp eed
i s to b ring t h e achin e t o i t s n orm
Auto
matic st abili ty is used t o d escri be stabi lity obt ained by a echani cal
dev i ce whi ch o p erat es t h e co nt rols wh en t h e aerop l ane i s n ot i n i t s co rr ect
flyin g a ttitu d e
Although t h e subj ect of st a bi lity
ay be thus s ub d i v i d e d it will be
foun d tha t t h e m
etho ds use d for p r o d u cin g i n h eren t s t a bility thr ow light
on t h e re qui r em
a ti c s t a bility d ev i ces
en t s for a u t o m
B efore a d esi gn er
p l et ely satis fact ory p osition h e m
u st ha v e inform
a t ion whi ch
i s i n a co m
will ena bl e hi to fin d t h e m
otion of an aero p l ane un d er any co ncei vabl e
set of ci rc u s tan ces
Th e sam
e in for a tio n wh i ch en a bl es hi
to cal cula te
t h e in h eren t sta b ility of an aer Op lane i s als o tha t whi ch h e uses to d esign
eflect i v e con t r ols an d t h e sa e as th at r equi r ed for any effecti v e d ev elop
a t i c st a bility d ev i ces
ment of aut om
A d esi gn er cannot for et ell t h e d et ail ed na tur e of t h e gus t s whi ch his
aero p lane will ha v e to enco u nt er an d th er efore cann ot an ti ci p at e t he
achi ne
In t h i s res p ect h e i s only i n t h e usual
co ns equ en ces t o t h e fl yi ng m
p osi tion of t h e en gineer who uses his knowl edge to t h e b est of his abi lity
itting his lim
i t at ions p ro vi d es for u nforeseen co n tingen ci es by
an d a d m
us ing a fact o r of safety
ay be
M eet 01 Gu
Th e aero p l an e us ed as an i n ch cat i on of wh a t m
n
e x ec t ed of an i nh er en tly st a bl e
i
ac
hi
n
e
h
a
d
t
h
e
a
dv
a
n
of
fl
yi
g
t
a
e
n
g
p
parati v ely st ill air It i s not n ecessary d ur i ng calula ti ons t o p resu e
c om
For ins tance t h e at h em
at i cal
s till ai r an d n egl ect t h e exi st en ce of gus ts
for t h e effect s of Si d e sli pp in g of t h e aer o p lane
t reat m
en t in clu d es a term
Exactly t h e sa e ter a pp li es if t h e aerep lane co n t i n u es on i t s course but
h
A
s
d
ar
r ecei v es a
a
n
w
n
u
t
f
o
S
i
d
h
d
u
t
u
p
w
d
i
d
are
s
i
m
i
s
r
e
e
e
a
a
n
t
g
g
a th em
pli
ati cs an d ev en for gust s of a co m
larly co nt e p lat ed by t h e m
for exam
i n i ng t h e eflect s on t h e oti on of an
cat ed na tur e t h e
ech ani s m
aereplan e 13 p ro vi d ed
al m
at h em
at i ca l t rea t m
en t of st a bili ty
I B efore enterin g on t h e form
eas urem
en t will b e gi v en
an d a
a fu r th er ill us t rat ion of full scal e m
o d els wi ll be d escri bed with th eir m
otions an d th eir pec uli ari ti es
ser i es of
of const r u ction Th e seri es of o d els correSpondS exact ly with t h e out
a th e ati cal analysi s
st an di n g f ea tures of t h e m
—
i
e
An aerOp lane h as
any
ann
The Produ ct on of an Unst abl M
types of inst a bili ty one of t h e m
ore i nt erest i ng bei n g ill us trat ed i n Fi g
whi ch in ci d en tally shows that an aero p l ane ay be st a bl e for so e con
Th e reco r ds w er e taken by t he
di t i ons of fli gi t an d uns ta bl e for oth ers
equi val en t of a p in hol e ca era carr i ed by t h e aer o p lan e an d di r ected
to war ds t h e su n I n o r d er t o reco r d t h e p itchin g osci llat ions t h e pilot
t h e su n by obs er v in g t h e s ha d ow of t he
a rran ged to fl y d i rectly away fro m
wi n g st ruts on t h e lower wing Th e p ilot st ar t ed t h e p redo inant
a
.
,
.
,
m
,
m
.
“
m
.
,
.
m
m
m
m
.
m
'
,
.
.
.
m
,
,
.
m
-
.
.
m m
m
m
m
,
m
m
.
.
,
'
m
.
m
-
.
m
m
.
.
a
m
,
-
m
m
m
.
.
.
.
m
451
ST AB IL ITY
oscill ations by p utting
o
the
lo
of
t he
M
m
n se
.
cc
o
(
FT
s
.
De g r
o p l an e u p
aer
or
an d
d own
th en
.
lo o
ees
MP H
)
m
M
s
Fr
90
.
M PH
.
M
m
M
s
4 0 0 0 Fr
m225
F
.
.
Thc
-
u n cont rolle
d
mt i
o
on
t he
b an d oning t h e cont rol colum
n
o f t h e fi gu r e
The u pp er d i agr a
a
70
h ow i ng t ha t
e
.
M PH
s t a b i li t y
depen d s
.
l
gles is shown by t h e si d
shows th a t a t a Sp eed of 100
h
a
n
d
I
p
A
m
.
.
l
s peed o f fl igh t
o n of a n ae ro p an e , s
.
m
sca e of an
m
.
.
.
APP L IE D AER ODYN AMIC S
4 52
h eight of
ft t h e aero p lane was st a bl e D ur i n g t h e p erio d a
b t h e aere p lan e was l eft to
t h e p il o t di d hi s bes t to fly l ev el whi ls t for
pe tit or t o t h e p il ot At t h e
i t s o wn d e v i ces an d p r o v ed to b e a goo d co m
en d of
b t h e p ilot r es u e d con t rol p u t t h e nose d o wn and a ba n d on e d
eas ur e of t h e
whi ch gi v es a m
t h e colum
n to get t h e o sci ll a tio n di a gr a
h
t
40
0
0
ee
o
t
a
f
t
n
e
of
h
e
At a Sp ee d of 90 m
s t a bi lity of t h e aer o p lan e
p
lower di agr a s of Fig 225shows an osci ll ation whi ch di es d own for t h e first
es s t ea d y
Th e st a bility was v ery s all for
few per io ds an d th en beco m
h
w
as
t h e con ditio ns of t h e fl ight an d a r ed ucti on of s p eed to 70 m
p
Two reco rds of t h e l atter
su ffi ci ent to p r o d u ce an i ncreasi ng oscill a tio n
or e ra p i d ly in cr eas in g reco r d b ein g t ak en whi l s t t h e a er o
are sho wn t h e m
p l ane was cli m
bin g slightly
oti ons o bserv ed ar e cal cu l a bl e an d t h e obj ect of thi s ch a p ter i s
Th e m
etho d
a th em
to i ndi cat e t h e m
Th e m
a ti cal th eo ry for t h e aero p la ne as
but h as sin ce been
now used was fir s t gi v en by P r ofesso r G H B ry an
bine d with d ata obt ain ed by sp eci al exp eri m
en t s
Th e p r esen t li m
ita
co
tions i n app li cation are i m
p os ed by t h e am
oun t of t h e exp eri m
en t a l d a ta
ath em
a ti ca l di ffi culti es whi ch ar e n ot serious
an d not by t h e
Th e reco r ds d escr i bed h av e b een con cern ed eith er with t h e v ari ation
of Sp eed of t h e aerOplan e or of i t s angl e to t h e gr o u n d ti e wi th t h e longi
otion Th ere are no corres p ondi n g fi gur es ext an t for t h e l ateral
t u di nal
motions an d t he d escri p tion of these will be deferred un til t h e flyi ng
mo d els are descri bed i n det ail
.
.
,
‘
m
m
.
m
,
.
.
.
.
m
.
.
.
,
.
.
.
~
,
.
,
.
m
.
,
.
.
m
m
.
,
,
.
.
.
.
F L YI N G MOD ELS
I NSTAB I LI TY
—Th e sp eci al fea tur e of t h e
Model sh owing Co plet e St abi li ty ( Fig
mo d el i s that i n a room20 feet hi gh an d with a cl ear hori zo nt al t ra vel of
8 0 f eet i t i s not p ossibl e so to l a unch it th a t it will not be fl yin g co rrectly
b efore it r each es t h e groun d Th e m
o d el m
ay be dro pp ed u ps i d e d own
wi th one wing d own or with i t s t ail d own but although it will d o di fferent
manoe u vres in recov eri ng fr omt h e various l a unchin gs i ts final attitu de is
e
always t h e sa m
o d el i s a b n o rm
a l beca us e t h e s t a b i lity
Th e a pp earan ce of t h e littl e m
a d e v ery grea t
R eco v ery froma di v e or Sp i n w h en ass i st ed
h as been m
a
n
ee
f u lly by t h e p ilot m
d
5
0
0
f
ee
t
to
1
0
0
0
f
t
o
p
l
ee
o
n
a
n
aer
a
n
e and
y
a ltho u gh t h e m
odel i s v er y sm
a ll it m
us t be m
a d e v ery s t a bl e if i ts
ch aract er is ti cs are to be ex hi bit e d i n t h e confin es of a l ar ge l ec tu r e h all
To I LL USTRATE
m
STAB I LI TY
A ND
.
,
,
.
,
,
.
.
,
whi ch St abili U of th e Model depends —
In
a i n p l an es a n d t h e t ail
a ho r i zon t al p l an e th ere ar e two s urf aces , t h e
p l an e whi ch tog eth er accou n t for lo ngitu d ina l st a bili ty Th e an gl e of
inci d ence of t h e ain p l a nes is g reat er th an th at of t h e ta il and t h e cen t re
of gra vity of t h e o d el li es one t h i r d of t h e wi d th of t h e ai n p l an e fro
i t s l ea di ng e dge
I n t h e v erti ca l p l ane are two fin s ; t h e r ear fin ta kes t h e p l ace of t h e
u s u al fi n an d r u dd er , but t h e fo r w ar d fin i s n ot re p r esen t ed i n aerO lan es
p
by an a ct u al su rface It wi ll be foun d th at a di h ed ra l angl e on t h e wings
i s eq
ui v alen t i n so e res p ects to thi s l arge forw ard fin
.
Dist ing uishing Features
m
m
,
on
m
.
.
m
.
-
.
m
.
.
m
ST AB IL ITY
m
All t h e ch an g es of s t a bi lity whi ch occ u r ca n b e acco un t e d for i n t er s
o f t h e fo ur s u rf a ces of thi s v ery s t a bl e
o d el Th e ch anges an d effect s
wi ll be referr ed to i n d et ail i n t h e su ccee d in g p aragrap hs
A fl yi n g od el
ay be co
p l et ely st a bl e with only one v i sibl e surface
t he
ai n p l an e
od el i s sh own in Fig 227 It has how ever
Su ch a
p r o p erti es w h ich i n t ro d u ce t h e e ui va l en t s o f t h e fo u r s ur f aces
Th e s i p l est ex p l a n a ti on of s t a bility a pp li es t o a n i d eal o d el i n whi ch
t he
a i n p l a n es p ro d u ce a fo rce whi ch a lw ay s p a sse s th rough t h e cen t re
m
m
m
m
m m
m
.
.
.
,
q
m
.
.
,
m
FI G 228
.
,
.
.
I n a ny ac tu a l m
o d el
gra v ity of t h e aero p l a n e m
o d el cen t r e of p r ess ur e
p li cat e t h e th eory but Fig 228 ay be t aken t o
c h an g es e xi s t whi ch co m
od el i n sy m
metri cal flight
r ep resen t t he essen ti al s of a n i d eal m
a gi n e t h e m
ple im
o d el t o b e h el d wi th i t s m
ain p l an e
I n t h e firs t exam
om
en t of r el ease it will begin to
h ori z on t al j us t b efo re rel ease At t h e m
fa ll an d a li ttl e l a t er will exp eri ence a win d r esi st ance un d er both t h e
main p l ane and t he tail plans Two t hings happ en t h e resi stance t ends
t o s t 0p t h e f allin g a n d t h e f orce F 2 on t h e t a il p l an e actin g at a consi d er
o d el d own
G t en ds to p ut t h e n ose of t h e m
a bl e d i s t an ce fr om
o d el i s h el d with t h e m
otion i f t h e m
a i n p l a ne
Now cons i d er t h e m
v ert i ca l j us t before rel ea se Th er e will b e n o force on t h e m
a i n p l an e d u e
to t h e fall but as t h e t ai l p l an e i s i n clin ed to t h e d i r ecti on of
motion it will exp eri ence a force F2 ten d ing t o p ut t he nose of t h e
mo d el u p The mo d el cann ot th en stay in eith er of t h e a ttitu d es
illus t rat ed Had th er e n ot b een an u p war d l ongitu d ina l d ih edr a l
ai n p l an e an d t a il p l ane th er e woul d h av e
a n gl e b etw een t h e m
been no r esto ri n g cou p l e i n t h e l as t illus t ra tio n an d it wi ll be
seen th a t t h e p rin ci p l e of t h e u p w ar d l on gitu d i n a l d ih e d r a l a n gl e
It i s f ur th er cl ear th at t h e m
o d el
i s fun d am
en t a l to s t a bility
cann o t st ay i n an y a ttitu d e whi ch p r o d u ces a f or ce on t h e t a il
us t li e a l ong t h e t ail p l ane
a n d ulti m
a t ely t h e s t ea d y m
oti on m
a i n p l a nes i s fi x ed t h e an gl e o f
a n d si nce t h e an gl e t o t h e m
u s t be a , wh en t h e st ea d y s t a t e o f
in ci d ence of t h e la tt er m
moti on has been r each ed
en t et c
Fromt h e p ri n ci p l es of for ce eas urem
i t i s kn own Fm229
th a t t h e d i r ecti on of t h e r esu lt an t force on an aerof oil d ep en d s
o n ly on i t s angl e of i nci d ence an d as t h e force t o be coun t er act ed m
u st be
o d el thi s resu lt an t force m
us t b e v ert i ca l in t h e
t h e w eight o f t h e m
s t ea d y m
otio n
Thi s l ea d s di rectly t o t h e th eoremth a t t h e angl e of gli d e
i s e qu al t o t h e an gl e wh ose t a ngen t i s t h e d ra g/lift o f t h e aerofoil
i ned
Alth ough t h e di r ecti on of t h e result an t force on an a erofoil i s d e t erm
agnitu d e i s not a n d i ncr eas es as t h e
so l ely by t h e an gl e of in ci d ence t h e m
of
.
,
,
.
m
.
.
.
.
.
.
.
,
.
.
,
.
,
,
,
m
.
,
.
,
.
,
,
.
.
,
.
APP L IED AER ODYN AMI CS
a gni tu d e of t h e r es ulta n t
quare of t h e Sp eed In a stea dy sta t e t h e m
f orce m
us t be eq ual t o t h e w eight of t h e m
od el an d t h e Sp ee d i n t h e gli d e
will in crease un ti l t his st a t e is reach ed Th e sch e e of o p erations i s n ow
p l ete an d is
com
h
e an gl e of i n ci d en ce of t h e m
d
i
tio
n
of
p
l
rm
n
t
a
i
n
a
n
es
a
h
a
e
e
t
e
T
( )
by t h e u p war d setting of t h e t ail p lan e angl e
t
s
e
n
f
a
e
o
fi
o
qu
n
of
a
h
gl
e
gli
d
e
i
x
d
s
n
s
c
e
b
A
a
c
e
e
( )
( )
d
e
i
s
fi
x
a
a
n
b
h
e
e
l
e
qu
of
t
v
o
ity
of
gli
d
d
e
n
ce
c
c
s
a
s
co
n
e
A
( )
( )
( )
s
.
m
,
.
,
-
.
.
.
m
d ep arture fr o t h e steady state of flight gi v en by (a) ( b) and (0) in t ro
d u ces a force on t h e ta il to correct for t h e d is tur ban ce
—
No assum
p t ions ha v e been a de as to t h e size
Decree of se wn
o f t h e ta il p la n e n ecessary for st ability
nor of t h e u p war d tail se ttin g
It
In t h e i d ea l od el any siz e an d angl e are su fi ci en t to ensure st a bility
a ll tai l t h e f orces woul d be s a ll an d
i s how ev er cl ear th at wi t h a v ery sm
su ch a
o d el wou l d h av e
t h e co rr ec ti n g d i ve et c co rres pon di ngly slow
If t h e tail be lar ge an d at a co nsi d era bl e angl e t o t h e
a ll st a bility
sm
o d el will switch r oun d qui ckly as a result of a distur ba nce
ain p lane t h e
ay ha v e a
It wi ll be seen th en tha t s ta bility m
an d will be v ery s t a bl e
wi de range of v alues d ep en din g on t h e d isp ositi on of t h e t ail
,
,
m
.
m
,
.
m
,
,
.
,
m
.
,
,
m
.
m
.
,
,
.
—Fig
m
227 shows a s t a bl e od el without a v i sibl e t ail p l an e
I n th e
a i n p lan es was s u pp o se d to act
cas e j u st d is c ussed t h e force F ; on t h e
throu gh t h e cen tre of gravity at all angl es of inci dence Thi s i s equi val en t
t o n o ch an g e of cen t re of p ress ur e on t h e win gs, a case w h i ch d oes not
od el of Fig 227 i s su ch th a t wh en t h e an gl e o f
o ft en occ ur
Th e
i nci d ence falls below i t s n or al v al u e th e ai r p ressure act s a h ea d of t h e
Th e co u p l e, d u e t o thi s u p war d ai r
cen t re o f gra vity , a nd vi ce versa
force thr ou gh t h e cen tr e of p ress ur e an d t h e d own war d force of w eigh t
thr ough t h e cen tre of gr a vity , t en ds t o r es t ore t h e or igin al angl e of
Angle
.
m
.
.
m
.
m
.
.
m
m
in ci d ence Th e s all m
i ca odel has an equi v al en t u p w ar d t ail settin g
to m
os t cam
ber ed p l an es for whi ch t h e
co n t ra d is tin ction
an gl e i n
ewha t l arg e
ui va l en t an gl e is n egati v e an d som
T ail p lanes ar e th er efore
eq
n ecessary t o b alan ce thi s n ega ti ve an gl e before th ey can begin t o act as
r ea l s t a bilis in g sur f aces
Th e u n s t a bl e aerop lan e for whi ch t h e reco r d i s
gi v en in Fig 224 h a d eith er insuffi ci en t t ai l ar ea or t oo sm
a ll a t a i l
.
-
,
-
.
.
.
qui v l t t il set t ing angl e of a n aerop l an e is not easily
r ecogni sa bl e for oth er reas on s t h a n tho s e ar i sin g f ro m
ch an g es of th e cent r e
of p r es s u r e
T ail p l an es ar e u su ally not flat s urfaces but hav e a p l an e
o f sym
metry fr omwhi ch an gl es ar e measur ed Th e li ft on su ch a
t ai l p l an e i s zero wh en t h e wi n d blows al ong t h e p l an e of sym
metry Th e
a i n p l an es on t h e oth er b a n d d o n ot cease t o lift un til th e ch or d i s i n clin ed
e su ch an gl e a s
d ownw ar ds at som
I f t h e p l a n e of sym
metry of t h e
t ai l p lane i s p arall el to t h e chor ds of t h e wi ngs th ere i s no geom
et r i ca l
di h edr a l an gl e but aero dyn am
i cally t h e an gl e i s
A co m
p li ca tion of a different n atur e a ri ses fr omt h e fact tha t t h e t a il
p l an e i s in t h e d own wash of t h e ai n p l an es
Th e
a en
e
a
-
,
,
.
m
.
.
,
,
m
.
ST AB IL ITY
mm
Althou gh all t h e a bo v e con si d era ti ons ar e v ery i
rt an t th ey d o n ot
t ra v erse t h e corr ectn ess of t h e p rin ci p l es outlin ed by t h e i d eal o d el
Lateral st abi lity
Su pp ose t h e v ery st a bl e o d el to be h el d p ri or t o
rel ea se, by one wi n g ti p s o t h at t h e
ain p l an e i s v er ti ca l
At t h e o en t
of release th ere wi ll b e a d ir ect fa ll whi ch will s hor tly p r o d uce win d for ces
on t h e fi ns, but n ot on t h e
a i n p lan e or t ail p l an e
On t h e fr on t fin t h e
force F3 Fi g 23 0, in a ddi tion to ret ar di n g t h e fall , t en ds to roll t h e aereplane
so as to b rin g A r ou n d towar ds t h e h ori z ont al
Th e air for ce F4 on t h e t ail
fin t en ds to p ut t h e n ose of t h e aeropl an e d own t o a d i v e an d so get s t h e
a xis in to t h e d irection of
otio n
B oth actions con ti n u e with t h e resu lt
th at t h e ai n p lan es and t ail p lane are a ffect ed by t h e air forces an d t h e
longitu dina l st a bility i s call ed int o
p lay I t i s not un til t h e aero p l an e
i s on an ev en k eel t h at t h e fin s cease
to gi v e r est orin g cou p l es
Any
.
—
m
m
m
mm
.
,
.
m
,
,
.
.
.
m
m
.
,
.
m
fu r th er a dj ust en ts
by t h e di scus sion
.
ar e
of
th en co v ered
longi tu din al
t bili ty a lr ea d y gi v en
L at eral st a bility in v ol v es rollin g
yawi n g an d si d e Sli pp in g of t h e
a er o p lane
all of w h i ch di sa pp ear
i n st ea d y fl ight
i ca m
od el
Th e m
Fig 227 has rolli n g an d yawin g
moment s, du e to centre of pressur e
ch an ges wh en si d e sli pp in g occurs
Th e equi v alen t fin s ar e v ery sm
a ll
an u f act ur e l ea d t o
an d t h e s t a bili ty so sli ght tha t sm
a ll in accuraci es of m
oti on
c ur v ed p ath s an d err ati c m
o d el i s nev er p r esen t i n an
Th e l ar ge cent ra l fin of t h e v ery s t a bl e m
ore
aer o p l an e as it i s fo u n d th a t a d ih ed ra l an gl e b etw een t h e win gs i s a m
con v eni en t e qui va l en t
ely w ell an d whi ch h as no
o d el whi ch fli es ext rem
Fi g 23 1 sh ows a
fr on t fin Th e dih edr a l an gl e b etw een t h e wings is n ot great each of th em
b ein g incli ned by a b out 5 to t h e li n e joinin g t h e ti p s Th e p ro p erti es of a
l a t eral di h edra l an gl e h av e been referr ed t o in Cha ps IV an d V
en
Unstable Models —Two ca ses of u ns t a bl e a ero p l an es ha v e b een m
Th e t ail p l an e
o d el s
t i on ed an d both i ns t abili ti es can b e rep rod u ce d in m
a ll whils t t h e b al an cin g
of t h e m
o d el Sh own i n Fig 23 2 will b e seen t o be sm
a ll
w eight whi ch brin gs t h e cen t re of grav ity in to t h e corr ect p lace is sm
o d el for
o
en t of i n er ti a of t h e m
a n d w ell fo rw ar d so p uttin g u
m
m
h
t
e
p
p it chi n g m
oti ons
oti on of Fig 224 t h e t ai l p l an e wou l d be set
To r ep ro d u ce t h e ty p e m
d own a t t h e b ack t o m
o d el
ak e a slight n eg a ti v e t a i l s ettin g an gl e an d t h e m
l au n ch ed at a hi gh sp eed It wo u l d ri se at first an d lose sp eed aft er whi ch
t h e n os e w ou ld fall an d a d i v e en su e ; with su fli ci en t h eight t h e m
o d el
w ou l d go over on to i t s back an d excep t for t h e l a t er al d ih edral angl e
w ould s tay th er e
e fr oma r olli n g o v er of t h e
Th e r ighti ng woul d com
mo d el and t h e p rocess would rep eat itself until t h e groun d was reached
s a
.
,
,
.
.
.
.
,
.
,
m
.
.
,
.
°
.
.
.
.
~
.
,
,
.
,
.
.
-
,
.
,
.
,
.
ST AB I L ITY
457
m
d es cri p ti on of s ta bl e a nd unst a bl e oti on j u st co nclu d ed app li es t o t h e
s ta bl e a n d u ns t a bl e
oti ons of an aero p l ane fl yin g un d er p ow er
Fr o mt he short d es cri p ti ons gi ven it will h av e b een o bserv e d th a t t h e
si
p le otions of p it chin g fall i ng ch ange of sp ee d ar e i n t errel a t ed in t h e
lon gitu dinal oti ons whi ls t t h e la t eral
oti ons i n v ol v e si d es li pp in g
rolli n g an d y a win g
Th e o bj ect o f a m
a th em
a ti ca l th eo ry o f s t a bility i s
to sh ow exactly h ow th ese oti ons ar e rela t ed
m
m m
m
.
,
m
,
,
m
.
,
.
MATH EMATI CAL Tn a oa r
or
STAB I LI TY
Th e
th eory will be t a ken in t he ord er of longit u d in a l s ta bili ty la te ral
st a bili ty a n d s t a bility wh e n t h e t w o m
oti ons a fl ect ea ch o th er
,
'
,
.
LO NG I TU D I NAL STA B I LI TY
Th e
p lan e
mti wi th whi
f ym
mt y f
o
o
on s
s
e r
o
an
h l ongitu d in al st a bility d ea ls all o ccur In
a i rcr a ft
Chang es o f v el ocity occu r a l ong
c
.
t he
t he
HORlZ ONTM UNE
.
FIG 234
.
.
X an d Z whi ls t p it ch i n g i s a bout t h e a x is o f Y Ax es fi xed i n
t h e bo dy ( Fig 284 ) a re u sed a lth ou gh t h e t r ea t en t i s n ot a pp reci a bly
p l er th an with fi x e d a xes ex cep t as a li n k with t he genera l ca se
si
a
x es
of
m
.
Th e
e
n
f
m
ti
n
s
o
o
ti
o
o
q
ua
Th e gr o u p
m
,
.
.
,
a re
u a t i ons s ho w n i n 1) has
f eq
val i d a p li ca t i on on l
p
i f gy r os co p i c
e
at c
a f i os i a na lys i s assu
t o t h e ro t a t i n g a i rs cre w a re i gn ore d ; t h e con di t i o ns o f t he
e a i rcra ft , a n d t ha t t h e s t e ad
t
etr
occ
rs
i
n
h
ea r
u
o
t
t
l
t ha t co p le t e s y
i
o
n
i
s
r
e
c
i
i
n
y
y
t
s t a ke n u
et r
i
s
o
i
n
t
er
i
T
h
l
a
a n d i n t he p la ne o f s y
p
p
y
o t i on o f t he a i rs c re w re la t i ve t o t he a Irc ra ft
A po i n t o f a di ffe re n t k i n d co n ce rns t he
os t log i ca lly be d e a lt w i t h b y t he i n t rod u ct i on of a fou rt h e u a t i o n
a n d w o u ld
o
m ym
mm
m
mm
m
mm
.
m
m
m
.
,
q
m
( la )
q
o
e n t o f i ne rt i a o f t he a i rs cre w, Q. i s t he ae ro d n a
i c t o r u e , a n d ( L ie t he
w h e re I i s t h e
y
All p re se nt t re a t e nt s of ae re p la ne s t a b i li t y a ke t he ass u p
t o r u e i n t he e ngi n e s h a ft
t i on , e i t he r e x p li c i t ly o r i p li ci t ly , t h a t I i s ze ro
Ma t he a t i ca lly t his i s i ndefens i ble as an e u i valent o f ( l a ) , b u t t h e ass u p t i on is
q
m
.
m
m
.
q
m
m
m
AERODY NAMI CS
A PP L IED
4 58
fo rces m
x
'
Th e
and
p a rtly on a i r fo rces
th rou gh t h e a i r
mZ d
'
e
M t he
.
,
p en d p artly on gr a v i t a ti ona l a tt rac tio n and
p itchin g m
otion
om
en t d ep en d s only on m
,
Gravitati onal Attracti ons —Th e
on ly fo rce d u e t o g ra v ity , a n d t h e
.
Z
a re
— g sin
m —G
i
s
w ei ght o f t h e a i rcra ft m
g
pon en t s a lon g t h e a x es of X
co m
0 an d g
c03
t he
,
,
a nd
0
al
force an d
lly t h e lon gitu di na l fo rce norm
ad e
om
en t d ep en d on u
p itch i ng m
u st be m
w and q An excep ti on m
for light er th an a i r cra f t a t thi s p oi n t and t h e a na lysi s co nfin ed to t he
Th e exp r essions for X Z an d M a re
a er o p lan e
Air Ro
a
en era
,
.
,
-
,
,
-
,
.
,
X =fx lu w
Z = f2 ( u w
M
w
9
0
mt
Ru t at e
on
Su b stituti ng for
m
)
u ati o
ol u ot i on as
th e Eq
X , Z and M i n ( 1) l ea d s to t h e
ot
f
0
3
t
'
q
q
l w
‘
"
9
008
3
1
(
ml
,
appli sd
e
to
au
A
p ana
q u a ti ons
w: Q)
8 + fz < u t w, )
f. ( u . w. )
q
q
genera l ca se w hi ch w oul d cov er l oop in g th es e e q u a ti ons ca nnot
a ry t o reso r t to
For su ch sol u ti ons i t h as b e en cu sto m
be sol v e d exa ctly
p l e of whi ch h as been gi v en in Ch ap t er V
s t ep t o st ep i n t eg ra ti on an ex am
Th e pa rti cula r p ro bl emd ea lt with und er s ta bili ty s t a r t s w ith a s t ea dy
i n es t h e conse qu en ces of sm
all d i st u r b an ces
moti on and exam
oti on e q u a ti ons
If
w 0 b e t h e v a l u es o f u w an d 8 i n t h e st ea d y m
e
( 4 ) b ecom
111 0 +fx (u mw
0
8
)
g
w
0
0
8 0 + fa( u
0
)
9
0
w
)
fu
mu st be zero i n a ny st ea dy l ongitu di nal
Sin ce q 0 i t foll ow s th a t q
motion 0 being con st an t
Th e thi rd e qu a ti on o f ( 5) show s th a t t h e p it chin g m
en t in t he
om
oti on m
s t ea dy m
u st be zero Th e fi rst t wo e q u a tions exp re ss t h e fa ct th at
t h e r esult an t ai r fo rce on t h e a erop lan e m
u st be equ al a n d o pposit e t o t h e
w eight of t he a ere plane Th er e i s no d iffi culty in sa t i sfying e q u a tions
a n d t h e p ro bl em
s r e l a ti ng t o t h em
h a v e b een dea lt with i n Ch a p t e r I I
In t h e
,
,
’
.
-
-
.
,
.
,
e,
,
,
0
"
0
0
m
as
c’
e,
e,
oo
,
,
.
.
.
.
.
m
he less sa t is fa ct ory i n t h e p rese n t s t a t e o f kn ow led ge
Th e da p i n g o f an y
rot a t i ona l d i s t u r ban ce o f a n a i rs crew i s ra p i d , w h i ls t ch a n e s o f forwa rd a
we d of an
g
ae ro p la ne a re s low a n d a re t h e onl
t
es of a
e t o wh i ch t h e a i rs cr e w
c
h
a
re
c
e
i
i
a
a
n
b
l
u
d
n
g
y
pp
g
h as t o res po n d
Th e e x t ra e ua t i on o f ot i on d oe s n ot lea d t o a n y se ri ou s ch a n ge o f e t h od , b u t i t a d ds
t o t h e co p le xi t y o f t h e a ri t h e t i ca l p roce sse s , a n d t h e s i
li fi ca t i on w h i ch res u lt s fr o t he
=
t
as s u
o
i
n
I
O a p pea rs t o be ore v a lua b le t han t h a t of t e e xt ra a ccu ra cy of re t a i n i ng i t
p
A li t t le la t e r i n t h e ch a p t e r i s gi ve n a n u e r i ca l i n v es t i ga t i on o f t h e val i d i t y of t he
as s u
p t i on . b u t i t i s a lwa ys o pe n t o a s t u de n t t o r e cas t t he e u a t i ons o f s t a b i li t y s o a s t o
us e t h e v a r i a b les u , w,
a n d 1: i ns te a d of confi n i n g a t te n t i on t o u , w an d
only
n e ve rt
.
m
m
.
m
m
m
q
q
m
m
m
m
m
m
.
q
q
.
APP L IE D AERODYN AMI CS
4 60
li m
i na ti on
u a t i on
eq
d t he
s t a bility
an
of a ny
e
,
Th e
to di v i d e t h rough
“
is
3
wi se t h
i nst ea d of
e
BI
E
B
E
D1
:
of
Coe fii ci en t
of
3
X
coeifi ci encow
Coefii ei en t
con
of
d iti ons
qu an titi es A1
,
“
fi
M, i n
in
p ow ers of Ai s
x,
,
of
an d
,
is
in
o r d er
to
w r it t en ins t ea d
i na n t o t h e r
w d et erm
'
a ne
3Z
O
A
,
"0
E
Su bs t i t u t ing t h e
for
4
x,
Z:
Zw
M,
M“,
,
an d
t he
r esu
“
t bility
’
co
lts
BIG
+
uo
+Z q
3
.
213
X,
Z
n
M
3
M”
.
M,
co s
0
6“
sin
0
M
0
“,
He a th , an d a re th a t t h e fi v e
Q— A D sh all all b e
—
o
s
iti
v
e
D 1 a n d Al B l Cl C
F I
p
s a
a re
gi ven by
-
l
7
B
— a°
a,
4 3
-
X.
r
008 1
M
ead s t o
l
3
i
n
1
4
1
(
)
)
(
14 8
mpl
X
.
va u es of
is
,
“
l bll
,
'
l
15
X
—0 l 59
r
q
1
X
+
'
= 122 4
A,
ot on
hi ev ed
ac
1
5
My
X
1
=—0 W
M =
'
ily
51,
,
X
,
,
X.
eas
1
z,
Zw
B 1 Cl ,
u,
mi
,
,
of
t he
M is B
.
Thi s is efi ect ed i f y
.
in st ead
M“
a nd
s
M I
Co effi ci en t
l
A, i
.
s
a
Th e
t he
1
2
( )
'
LIa
0,
?
A
an d
xp ansio n of
Coeffi ci en t
A,
B
l ea d s to
e as
sa
gi v en b el ow
a
v erywh ere by
e
0
)
high est p ow er of
t he
dm
Th e
a re
of
coelfi ci ent
an d
u.
M,
.
w
u,
weA
Xw
z,
A
z
M
v ari a bl es
the
of
t wo
,
B1
C,
A ‘B q
et ely s t a b le .
620 ,
Cl
,
C,
9 80
Al D ‘
z
.
D,
8 4 20
24 6
STAB IL ITY
m
So e fu rth er p a rti cula rs
bi u a drat i c e u at i on i n A.
q
‘
q
moti
t he
on are
obt ained by sol v i ng
t he
.
qu a tion
fac t o rs
Th e
h as t h e
of
4 61
A4
e
14
(A
°
9 SOA
62 0A2
BA3
216
0
0 075 t 01 700
0
'
I
m
m
m
‘
m
All t h e roo t s are co p l ex
A p ai r of co p l ex r oo t s in di ca t es a n oscil
la ti on Th e r ea l p ar t of a co p l ex r oo t gi v es t h e d a p in g fact o r, an d t h e
i agi nary p art h as i t s n u er i ca l v alu e e qu al t o 2" d i v i d ed by t h e p erio d i c
ti e of t h e os cilla tion I n t h e a bo v e case t h e fi rs t p ai r of factors i n di cat es
p i ng fact or of TM ,
a n o scil la tio n with a pe rio d of 257 sees an d a d a
w h ils t t h e second pai r of co p l ex f acto rs corresp ond s with a p erio d of
8 70 sees an d a d a p in g f acto r of 00 75
Th e
p i ng fa ctor i s oft en i ll u st r at ed by co p uti n g t h e
eani n g of d a
ti e t aken for t h e a p litu d e of a d i stu r b ance to di e to h alf a gnitu d e If
'
.
m
m
.
m
m
m
m
m
m
.
be
Ta ki ng
m
m
.
.
‘
h
u le
‘
u
will
m
.
.
u
m
half t h e i nitial v alu e a , wh en
1
logarith m
s
»
,
—Al t
t
to h alf
a
mplitu d
e
A1
illus t ration t h e m
o re rap i d o scilla ti on d i es d own t o h alf v alu e
in l ess th an 115t h seco n d whils t t h e sl ow er osci lla ti on re q u i res
seco n d s
It will be r eadily und ers tood fro mthi s i llust ra ti on th a t a ft er a seco n d
o r so o n ly t h e s low osci lla tio n wi ll h a v e a n a pp r eci a ble r esi d u e
Th e
blan ce t o t h e curv e sh own i n Fig 223 of t h e o scillatio n s of an
r ese m
p ari so n
a e ro p lane will be r ecogn i s ed wi thout d et a il e d com
In t he
,
.
.
.
,
,
.
m
A
t ha t t h e
mt i
o
sc as w
m
Is s a
m mm L
cr
as a
md
t mh fi
be
on of a n aer op la ne
is no
of t h e exa
C or r es pond i ng wi t h t h e d a ta
d y na i c t or u e a nd engi ne t or
m q
q
e of
a
a
uc
mpl
t he i
Exa
m
s
mp
ort a nce of
by t h e i ne rt i a of t h e a i rs cre w
t wo foll owi ng e ua t io ns for aero
ect e d
t he
e a re
as r
q
.
ue
Q
w ow
Q
8 75
.
a nd
o
,
a
00018
2
.
14 6n
'
q
1224 lead s t o t h e va lu e a
So l v i ng t he e u at i on, Q,
Q” for u
u for u i n e u a t i on s ( 15
a ) a nd ( 15
n for n a nd u .
6) a nd sep ara t i ng
Su bst i t u t i ng n ,
t hos e for st e ad y
ot i on fro
ot i on co n vert s
t h e pa rt s co rres pond i ng wi t h dis t u r b ed
e u a t i on l a i nto
m
q
q
m
m
’
or
wit h
u,
a rson.
1s
-
n
dn
an
For
.
m+
a
mh d
et
u.
o
0 256u
550n ==
’
of so lu t i on , see
'
App endix t o t h is
cha pte r .
APP L IED AER ODY N AMIC S
4 62
mt b k w
”
w
b t t hi
f
f
d f th f m
w
f
ll mt i
di t b d
wh t h
d th
lt
mi d t
d i ( l5
d)
B efore a ny s olu t i on of ( 15
d)
q
ca n
b e ob ta i ned a
no
e
us
n as a fu nct i o n o f n a nd t
m
.
es a d efi ni t e
e
s ass u
e or
u
I n e u a t i on ( 15) a va lu e o u as ou n o
,
e
or st ead y
rela t i on b et ween a and a
or a
o ons
er
s ur e
Th e valu e
.
“
s o fou nd
o see w het h er a ny fu nda
th e
ay b e u se
n
an
e r esu
exa
ne
ent a l assu
i s no w
A solu t i on of
p t i ons on w h ich i t was based are v i olated
m
m
m
.
.
i
a
excep t
i n t he
case
wh en t h e
wh ere A,
m
q
t h com m
ci ll t i
q
s
i
,
for
os
l
l
a nd followi ng t h e usua ru le h
e
x
,
p
A
a nd t h e t wo roots ar e
l
e
e
n
t
ar
roo
t
”
y
p
e u a t i on ( we) is re p laced b y
fre u ently co
e
a
solu t i on
on ,
“
e
c eas
n
5
n ‘6
6
is
( 151)
c ease s }
n
A
+ 559
,
+
—
“ : 008 ( H
l
5
sis6 4 p
i }: is wri t t en for A, a nd A—i t
F or a n
co ns id ered t o get h er
.
74 t
-
,
m( a
a
m
u se
k
wh er e
m
nt of i t s i ni t i a l va lu e i n la s t h an
r
ce
u
t
o
l
d
e
ce
d
s
e
Th e firs t t er of ( Me) a nd ( 15
i
r
0)
p
at
a lu e of t h e second t e r
occu
rs
a
u
v
e
x
i
I n t h e case of ( 15
t
h
one s econd
1)
b eco es u n i por ta nt i n ab ou t a
u ,, a nd li ke t h e first te r
t =0 18 s ec a nd i s 0 125
m mm
m m
.
°
°
.
Had t h e i nert ia
wou ld h a ve b een
of
t he
a i rscr ew
neglec t ed
been
=JE§
I ns t ead
of
w hi ch t h e
.
m
q
u a t i on
ore accu ra t e e
m( 15d)
t h e relat i on ob t a i ned fr o
E
( 1543) gi v es
a ft e r
oases
A
6
9
5
,
mm l
m m
m
m
1 s ec
.
m
q
st e a d of
a
e
h
b
e
u
s
d
i
n
5
1
t
o
n
a
i
a
u
e
A
l
re
a
b
e
i
a
t
e
t
h
t
i
f
)
ed
a nd i t i s se en i
(
,
,
y
y
o
i
i
s t o p e rsi st aft er
t
e
o
t
n
h
f
o
t
e
n
c
er
0
If
5
s
a
a
l
l
c
o
r
e
d
t
h
s
w
i
5
6
i
if
1
p
p
( ) A,
A
a
o
f
o
t
o
n
s
n
ae
r
o
l
a
n
e
i
s
i
t
r
t
a
n
o
re
i
o
t
h
e
A
e
x
c
a
n
d
i
n
eed 0 69 ,
p
,
1 sec , , cannot
p
I n s u ch cas es t h e assu p t i on i s jus t ifi ed t ha t
u ch less
s t u r ba nce of t he
t
d
i
e
h
e
t
o
f
n
n
d
e
e
d
a
t
a
t
i
n
s
s
i
ll
s
u
b
n
e
o
n
s
a
o
i
s
nd f rward
r ev olu t i
p
y
p ed
s t eady
ot i on
s b ot
a da
ot i on sh own b y ( 15
9 ) i n volve
In t h e cas e of a n os cilla t i on t h e
p ing
Th e d a p i ng fact o r corres p ond i ng Wi t h ( 153 ) is
fa cto r and a ph as e d i fference
m
.
m
n
wh ils t t h e p has e d i fi erence is
mpl
e
I
.
\ 7h
) a nd
( 155
p
proxi
mt
a e
for
ml
u a
-
( 15k )
( 1511)
11
k
'
559
h
-
a
2b du
O
_
c
7
m
h
—I
m
“
Appli ed t o Exa
m
m
m
.
.
wh ils t t h e
m m
m
.
.
gi ve
k
0 O75u a nd y
°
240
°
gi ves
0040“
a nd
y
0
m
APPL IED AERODY N AMIC S
4 68
i n creases t h e angle of in ci d ence fu r th er d ecreases t h e lift a nd a ccen tu a t es
t h e fall
At t h e h igh er sp eed s t h e d am
p in g of t he ra p i d o scilla ti on is grea t a nd
i n la t er ch a p t ers it i s sh own tha t t h e m
otion rep resen t s (a s a m
a in f ea t u re)
t h e a dju st m
en t of an gl e o f in ci d en ce t o t h e n ew co n d i tio ns
Th e slow o sci lla ti on in t h i s i ns t an ce d o es not b eco m
e un s t a bl e b u t i s
n o t a lway s v i go ro u sly d am
p ed a t 60 ft s t h e dam
p in g fac to r i s on ly 00 8 1
A m
od i fi ca tio n of a ero p lan e su ch as i s o bt ai n ed by m
o v in g t h e cen t r e of
g ra v ity b ackw ar d s will p ro d u ce a ch ang e of si gn of t hi s d a p in g fa c to r
an d an i n crea si n g p hugoi d osci ll a tion i s t h e result
At h igh s p ee d s t h e p eri o d of t h e p hugoi d oscilla ti on beco m
es g r ea ter
a n d ulti m
at ely t h e o sci lla ti on gi v es p la ce to t wo su bsi d en ces
In a l ess
st a bl e aerop la n e t h e o sc illa tio n m
a y ch an ge t o a sub si d en ce and a d i v e r ge n ce
] w hi ch ca se t h e aer o p la ne woul d b eh a v e in t h e m
a nn er ill u s tra t ed i n
,
,
.
,
.
,
.
o
.
.
m
,
.
,
.
,
iii g ? 24
.
.
All t h e
in
b serv ed ch arac teris ti cs of aerOp lane st a bility are rep resen te d
ca l cula tio n s s i m
i la r to those a bo v e Many d et ails req u i re t o b e fi ll ed
o
.
HORIZONTAL
DIRECTION OF
MOTIONO F G
.
b efore t h e cal cu la ti ons b ecom
e wh olly r ep r e se n t a ti v e of t h e di s tu r bed
motion of an aerop lane The de ta i ls are d ea lt with in t h e d eter i na tion of
t h e r esi st a nce d eri v a ti v e s
Cli bi ng and Gliding Fligh t —Th e effec t o f cutti ng off t h e en gi ne or
o f o p eni n g o u t i s t o a lt er t h e a i rs cr ew race eff ec t s on t h e ta il of an ae ro
oti on
p lan e Th e efl ec t s on t h e s t ead y m
ay be consi d era bl e so th a t each
co n d iti on of en gi ne m
u st be t rea t e d a s a new p robl em Th e d eri v a ti ves
Th e effect of cli m
a re a lso ch a n g e d
bi n g i s t o red u ce t h e st a bili t y of an
a ero p l a n e a t t h e sa m
e sp ee d o f fl i ht if w e m
d
ubt
l
u
p
tio
k
e
a
t
e
h
o
f
u
a
s
s
m
n
g
th a t t h e ch an ges of t h e d eri v a ti v es du e t o t h e a i rscrew a re un i m
p ort ant
Th ere i s no t in t h e a nalysis so far gi ven a ny e xp ress i on fo r t h e ih
c li n a t i on of t h e p ath o f t h e cen t r e o f g ra v ity G
R eferrin g t o Fig
i s seen th a t t h e angl e of p it ch a i s i n v ol v e d a s w ell as t h e in clin a ti on of t he
a xi s o f X to t h e h or i zo n t a l
Th e a n gl e of ascen t 9 i s
a
s
0 or i n t erm
of t h e qu a n titi es m
o r e co m
monly u sed in t he th eory of sta bi lity
in
m
.
m
.
m
'
.
,
.
.
.
,
,
,
.
.
.
,
.
—
=
d
l
ta
t an
“
1
ST AB I L ITY
4 69
l ev el fli ght 6 i s zer o an d t h e v a lu e
i n ci d ence of t h e a in p lan es by a con st an t
In
m
of
,
f romt h e
0 di ff ers
an
gl e of
.
b i ng flying l ev el or gli di n g
Wh eth er cli m
,
,
the
gl e of p it ch
an
°
,
1e
.
.
t en
s
d
,
m
lm
ost in d epe n d en t of t h e incli nati on of t h e p ath ; i t i s a rked ly a
80 fo r l ev el fl ight or
fun cti on of Sp eed Th e cu r v e i n Fig 28 8 m
a rk ed
is
a
.
.
an
gl e of p i t ch
c
it
h
p
,
f
i
t an
"
IS
it
mo t s ti f
s
to rily d escrib ed
s ac
a
an
as
gl e
of
.
V ariati on
l ong i t udinal St abili ty with Heig h t and wi th Loadi ng
Wh en d i scu ssi n g aerOplan e p erfor an ce , Le t h e st ea dy oti on o f an ae ro
p l an e , i t was show n th a t t h e a erodyna i cs of otion n ea r t h e grou n d coul d
b e rela t ed t o t h e
otion for di fferen t h eight s an d load i ngs i f cer t ai n
fu n ctions w ere chosen a s fun d a ent a l v a ri a bl es In p ar ti cu la r it was
of
m
m
s
h own th a t si m
ila r st ea d y
ke p t
w ere
o t
c n s an
t
moti
for t h e
sa
m
m m
.
m
on s
.
.
foll ow ed if
m
e or
for
im
ilarly sh ap ed
s
aer ep la n es
v
(i
.
e
u se d for t h e wi ng loa ding t o av oi d t h e d ou bl e u se of w in t h e sa m
fo rm
ula ) It was foun d to be unn ecessa ry to cons i d er t h e v ari ati on of
e n gi n e p ow e r with sp ee d of rot a tio n an d h eight ex cep t wh en i t w as d es i red
um
r a t e of clim
to sa tis fy t h e co nd ition of m
b
um
a xi m
ax i m
Sp ee d or m
In o r d er t o d ev elo p t h e co rr espo n d i n g m
etho d for s t a bili ty it i s
n ecessa ry t o e x a m
i ne m
o re closely t h e formt a ken by t h e res i st ance d eri v a
en t s on an a ero p la ne w ere
ti v as In equ a ti on ( 8) t h e forces an d m
om
e xp resse d i n t h e f orm
is
n ow
,
.
,
.
.
=fx( u
X
’
w: 9)
fun ction of u w an d q No assu m
p tion was m
a d e th a t
fo r a gi v en d ensity a ttit u d e an d a dv ance p er r ev ol u ti on t h e f orces an d
mo m
en t s w ere p ropo r tio nal to t h e s q u a r e o f t h e sp eed
If a pp ea l he m
i ca l si il arity i t will b e
a d e t o t h e p r i nci p l e of d y n am
s of exp ressi o n for X i s
fou n d th a t one of t h e p o ssibl e form
wi th
known
n a
.
,
,
,
,
.
mx
p
m
z
l v
wh ere p i s t h e d en sity of t he flu i d V i s t h e r esult an t v elocity of t h e aerop lan e
an d l i s a ty p i ca l l en gth whi ch for a gi v en aer op l an e i s co n s t an t
,
,
.
Th e
t he
an
ar
en t s
gu m
1
i if
$ 3i
3,
,
-
a re
of
t he
na
tu
re o f a n
gl e o f i nci d ence o f t h e aerop lan e as a whol e
gl e
of an
bla d es
.
of
i nci d ence
.
an d
"
7
1
d efi nes
t he
an
gl e
gl
es
qp
m
2
9 is
l
V
‘
re
of a
a
t l
u re of
eas
l h
g
res e n s oca c an es
t ta ck
of
t he
i
a rscre
w
APP L IED AERODYN AMIC S
4 70
Si nce V i s t h e
resu
lt a n t v e locity
dv
aw
,
u
av
V
alga - const
_
_
-const
n
.
w
V
.
P roce e d i n g now to fi n d on e of t h e d eri v ati v es by d i fferen ti a tio n
are cons tan t l ea ds t o
with r esp ect to u whil st w an d q
,
of
X
,
2
1
p
X“ — _ 2v
ma
i
7
5
n
n
—
w BF
lg2
V
10
a
v
X“
or
If
d u ri ng ch ang es of
n ow ,
p
mk
a
a
mi
Th e
ne
m
a nd
e
,
o t
e
c ns
p arti a l d ifferenti al
.
,
i
3
h v
d
g g 73
an d
t
coeffic en s
mv l
mwhi
e
cons
n -
v
a ue
t er
7
a
a
sa
for v ari a tions
h d oes n ot
c
qu ation ( 17) shows
t
.
,
an d
61
an
a
e
ex
t he
a
a
v
v
d er t h e restr i ct ed con d itions Th e o utst an d ing
o b v i ou sly sa ti sfy t h e con d ition of co n s t a n cy i s
un
.
nl
a
thi s m
u s t b e e xa m
in e d fu rth er ; i t wi ll
co m
pl ex m
an n er th an t h e oth e r q u an titi es
Th e a i rs crew to r q u e m
ay b e e xp r essed a s
an d
be
fo u n d t o v a ry
in
a
mo
re
.
an d
the
i
eng ne
to rqu e
as
é
!
t d ar d a t m
osp h ere p
)
t p u tti n g ¢ (h )
Eq
u a ti ng d
a nd g
In
a
s an
(22)
z
is
a
P
X
Di fferen ti a ti ng p a rti ally with f
I,
V
n
a
s
Qp nl —Y J"
V
é
fu ncti on
i
v
es
g
,
5
2
" 1
k no w n
"
V V
an d
a
1
3
V
t
co n s a n
l a
fi
z
_
3
z:
t l ea d s t o
’ nl
(
W
(
, P
of
t he
h eight
h
.
APP L IED AER ODYN AMICS
4 72
in
Th e seco n d
t h e e x t re e
m
It
my th
o t
c ns an
ter mi n
is
to
seen
be
o n e-
qu ar t er of
t h e fi rs t
ca se
.
l
ta ken
be
a
en
t
for t h e
r i v a t i ve
b racket
t he
a s a sat i sfact o ry ap p roxi
mt i
a
on ,
th a t
o d iti on s of si m
ilar m
otio ns an d t h e res i st ance de
n
r
w
n
d
e
s
with w eight (m
d
ity
a
o
d
i
g
to
a
c
c
n
t
e
a
h
l
(p )
g)
c n
X v a ri es
,
,
,
zv
z
p
m
mom t d
Th e
t he
sa
e
en
m
xp ression follow s for
er i v a ti v es
e
t he
1
8
( )
oth er fo rce d eri v a ti v es
Fo r
.
,
wh ere k i s t h e rad iu s of gyr a tion Th e n ecessary th eo remfor t h e
b etw een st a bi lity at a gi v en h eight an d a gi v en l oa di ng an d t h e
ula t ed
at any o th er h eight a n d lo a d i n g can n ow b e fo r m
l tion
re a
.
.
L et p o,
“
V0 an d Y be On e
')
s
co n dition s
se t
of v a lu es of d en sity v elocity
,
an d
loa d
o f s t ea d y m
whi ch t h e
otion h av e been sati sfi e d an d t h e
in e d
resi st an ce d er i v a ti v es d et erm
otion i n whi ch t h e density v elocity an d loa d in g
For an oth er sta t e of m
i ng for
.
,
a re
pl
,
t he
V 1 and
o d iti on s for s tea di ness will
c n
?!Y
fi
be
sa
ti sfi ed
z
v
o
P o
W1
32
( )
m
m b fo
mt h v lu
“
Py
f
i
a P?
dv an ce p er r ev olution of t h e ai rscrew be ad e t h e sa
en t o f t h e en gi n e th ro ttl e
by a n a dju st m
Th e d er i v a ti v es i n t h e n ew st ea d y m
ot i on a re obt ai n ed f ro
an d
t he
a
.
i n t he
forces
o ri
gi n al
an d
by
as
I f the
mti
o
on
p 1V ,
PoVo
by
my b
a
u
cou
e seen
d er i v a ti v es of ( 18 )
i
ffi ci en t s
8 n ew se r es o f co e
in
mlti p lyi
s o f th e mb u t
t erm
,
n
g th emby
the
ra
ti o
e as
e
re
e
a
es
'
1
for
fo r
p l es
by
be
Th e fi rs t
.
u se
ra
ti o
is
u
l
a
q
e
to
ot 0
o
Y
_
VI
or
or
to
of
i d en ti fied wi th
for t h e
if
d ensity p a
t bility eq u a ti on
s a
d ensity p , a nd
W
:
l oa d i n g
-
b
ca n
an d
be
Th ey
2
d
loa d in g
w ri tt en
a re
7°
own
4 78
ST AB IL ITY
Coeffi cien t
‘
of Al
‘
o ffi ci en t
A,
1
,
c e
of
A13 ,
i
o
c effici en
t of
A12
W1
P0
+Z
.
—
wo
X“
+B
¢
o ffi ci ent of
01
c e
1
P
_1
W1 P0
”i f
.
B
kT
g
W
M,
M,
'
W
Al l ,
WI PO
xi x”
1
00
Zl
wi p e
Z0
i t0
+
WoP1
Zn
M“ M,
g
+ XG +
(
W
”
0P1 ”Co
M,
o ffi ci en t s of A1
0
c e
9
B
,
W
0
_
W1
p_1
Po
k‘
—
f X, X“
k?
,
Z"
Z
w Si n
M,
.
005 00
90
33
( )
0
It will be seen fro m(83 ) th at sev eral m
od i fi ca tion s are in t ro d u ced in to
t h e s ta bility e qu a ti on by t h e ch an ges of loa d i n g a nd d en sit y
For ch an ges of d ens ity o nly k ,
kc If t h e w eight of an aero p la ne
be ch a nged i t wi ll u su a lly follow th a t t h e rad i u s of gy ra tion
If t h e m
a ss es a r e
a s t h e a dd ed w eight will be n ea r t h e cen t re of g ra v ity
so d i sp o sed d u r in g a ch an g e of l oa d i ng th a t k,
kg a nd t h e h eight i s so
.
.
,
.
,
vVQ
— L}
h osen th a t
c
W1
.
0
Po
= 1 (83 ) l ea ds to
31
3
A
A1 1
W1
the
is
my i
n
not
ea s
i
.
ily
sa
B
ti sfied
2
1
1
,
‘
3
4
( )
‘
i
s n ce
t he
Th e
.
h ea v y l oa d in g
o d ition
c n
in
on e
case
v ol v e t h e u se of too grea t a h eight i n t h e corr esp on d in g li ghtly l oa d ed
Th e
}?
is
Po
a
I
t bi lity
s a
im
p l e formo f equa tion
s
h i CI A] D 1 0
e xa ctly th a t of t h e o ri gin a l m
otion
“
and
t he
,
a.
fact or
the
h
in
g
c an e of
m
1
1
2;
13
du
.
.
-
p
t
re rese n s
t he
q u an tity
et o ch ange of fli ght Sp eed a t con stan t altit u d e
t h e a i rscr ew thi s
A p ar t f r o
For
t h en zero for all sp eed s
t h a t t h e t ail p la ne d
a ll
w i ll be v ery s m
8
8
( )
o es n ot
.
q u an tity woul d always be z er o sin ce M i s
an a ero p lan e with twi n en gi n es so far a p ar t
p roj ect i n to
the
tail
ra ces
th
e
v a lu e
oi
Mg
B
”
APP L IE D AERODYN AMIC S
4 74
As
an e a
x
l 20,
mp l
;
fli h
I
an d
"
:
e of
t he
0 74 , i
.
e.
of (3 8) i t will be ass um
ed th a t
1
t h e lo a di ng h as been increa sed by 20 p er
1
2
u se
1
-
0
t
cen
.
0
ta ki n g p lace at 10 000 ft inst ead of nea r t h e groun d The
l eas t s ta bl e con d ition of t h e aerep lane h as b een ch osen T a ble 1 s hows
th a t it occu rs for V0 60 ft s Th e co nd itions l ea d t o
an d
t he
g t
is
.
,
.
.
-
.
.
an d
mp l
I n t h e origi na l e x a
s ta bi lity e q u a tio n w er e
A,
With
e,
p age
V1 = I 27VO
467, t h e
v al u es
se e, B , = 1o o, cl = 1 se
-
m
ft
°
of
an d
~
the
D,
coe
olution of it
an d a s
2
'
3
A
2I
7
°
3
00A
O 9GA
2 82
°
.
ffi ci en t s
of
t he
4 39
m
“
A
s
-
60, o
12 a n d t h e v a lu es of t h e d eri v a ti v es
23 7, t h e n ew e qu a ti on for st a bi lity b eco es
28 5—
no
.
in
gi v en
0
is
A
0
5
1
1
+
(
Th e secon d fac t or sh ows
Th e n ew an d o ri gi n a l
0
00
1
2 3 110
+
0
0 655
1
)
0
tha t t h e m
oti on i s o nly ju s t st a bl e
moti on s are co mp ared i n t h e T a bl e b el ow
.
.
TAB L E 2
.
mt i
o
m
o n ne ar
New ot i o n at
feet wi t h an i ncrease
of
20
Pe ri od of p h ugoi d os ci lla t i on
D a p g fa ct or
Ti e t o h alf di s t u r ba n ce
mm
m
general effect of t h e in creased loa di ng an d h eight i s seen to be
p in g
an i ncr ea se i n t h e p er i od of t h e o scill a ti on s a n d a r ed u c ti on m
t h e d am
Th e t en d ency i s cl ea rly t ow ar d s i nst a bility of t h e p h u goi d oscill a ti on
u at i on for Longi tu dinal
uadratic Eq
Approxi ate Solut i ons of th e B iq
Stabi li ty —I f t h e p eri od a n d d am
p in g of t h e r ap i d oscill a ti on be v ery
mu ch gr eat er th an those of t h e p h u goi d osci ll a ti on t h e biq u ad ra ti c can be
e r a p i d ity
di vi d ed in t o two a pp roxi m
a t e q u a d ra ti c f act o rs with ex t rem
Th e o rigi nal equ a tio n b ein g
Th e
.
m
.
.
,
.
A
‘
t he
a
pp roxi
mt
a e
A1”
2
A
B1
Ci d
D1
factors a re
?
A
“
B,
A11
B,
h
i s?
B,
O
APP L IED AER ODY N AMICS
4 76
Ga a vxr a rrox a n ATTR AC TI O N
'
Th e
wh ere
co
mp
is
on en
t of t h e w eight of t h e aero p l an e along
mll
a s
a
m
g
cos
Th e
gl e
an
.
a
00
pp ro x i
mF
A
Gen erally , t h e
lat era l force
d ep en d on v p and r With a
s
Y L and N t a ke t h e fo rm
,
8 in
¢
mtio
a
i
ax s o f
Y
is
88
( )
.
n si n
s
5
e
l
wi
l
b
9
a
u sed
.
oncn s
ent
en t
olling m
om
om
an d y a wi n g m
r eserv at io n as to light e r th an a i r cra ft
r
-
.
,
.
t he
-
,
,
Y = j 7 (v P T)
L =h M p n )
N =fl ( v P»7)
r
:
.
r
m
Th ere are no uns tea dy otions ex clus i v ely la teral su ch as tha t o f
oti ons as tu rn in g an d s p inn i n g
otion Su ch m
1
00 p i ng for longit u di na l m
t h e l on gitu d i n a l
a lth ough st eady cann ot th eo re ti ca lly be t r ea t ed ap a r t fro
motion For th ese reason s Y L an d M d o not con t ai n t erms of zero
or d er i n v p an d r and exp an sion of (39 ) l ea d s i m
medi at ely t o t h e d eri v a
ti v sa Exp an d in g by T aylor s th eo rem
,
m
.
,
.
.
,
,
,
’
.
e t c. ,
,
with
or
a
t tion
i
no a
s
il
m
mp l y d
to th at
ar
e
e
o
for
longit u d i n a l
an d
4
( 1)
d eri v a ti v es
wi th si m
ilar exp ress i on s for L an d N
F o rm
i ng t h e eq u a ti on s for sm
a ll oscill a ti on s
.
i 1+ uor = g
E
T
A
P
—1>E
i
mm(8 7)
l ea d s
to
+ rY§
cos
”Le
l PL,
v N, +
N
p ,
’
-
“
71
1,
r
m
N,
B e fo re e qu a tion s (42) can b e use d as si ult an eo u s equ a ti on s in v, p
> in t e r
s of
a
n
r
d
an d r , i t i s n ece ssa ry t o e xp re ss <I
p
e th o d i s
To obt ai n t h e p o si tion d eno t ed by 00, ct 1
b t h e st an d ar d
1 th en a b out ( i Y th r o ugh
t o r ot a t e t h e a erOplan e a bout GZ th rou gh 11
00, an d fin a lly a b out GX th rough 95 Th e i n iti al rot a tion a bo ut GZ h as a
an d con se qu en tly
i s not e qu a l to p
p onen t a bo u t GX (Fig
co
Th e t wo od es of exp r e s si n g a n gul ar v elociti es l ead to t h e rel atio n s
m
.
,
m
,
m
.
m
.
.
p
z
mbi
Co
i g
n n
t h e t wo
e
—
l v sin 9
i
0
q l
cos
qu a ti on s we h av e
)
t
a
t
n
r
g
2
}
00
,
:
00
ST AB IL ITY
4 77
m
Eq u a ti on s ( 43 ) ight be us e d t o co nv er t e qu a ti on s (42) t o t h e v ari a bl es
Th e alt ern a ti v e an d e qui v al en t
etho d i s to use t h e kn ow
6 an d III
0, 5
l ed ge th a t d Ad in o r der t o exp res s 4»i n t er s of p an d r Eq u a ti on s
m
m
.
1
3
u or
= g cos
g
.
sin
Go
0L.
i C — pE :
h
e ,
'
Th e
so
luti on of (45) i s ob t a i n ed by
1
3 = Av p
,
wh ere 01
,
r
r
a
e
a
n
d
l
,
p
Eq u a ti ons ( 4 5)
t he
initi a l
ubstituti ons
Ap , i
'
z
va
s
lu es of
Yr
)? (
9
3
AA
t he
d ist u r bance
.
of a ny
u
tio
n
f
o
w
i
h
a
r
m
h
c
q
Ai s
i !) 80
i
AE
30
_ N9 0
e
= Ar
becom
e
(A
— Lr v
t he
NJ?
two of t he qu an titi es
d e t erm
in ed i s to
,
.
0
0
v,
p
an d
r
l ead s
(4 7)
to the
.
If t h e first row b e m
in a t o rs t h e equa ti on
u lti p li ed by A to cl ear t h e d en om
will be seen to b e a biqu a d rati c in A t h e co effi ci ent of t h e fi rs t t ermb ein g
,
AC
E2
m
.
For t h e p u r p oses of co p ar iso n of res ults it i s con v eni en t to di v i d e
all co effi ci en t s of p ow ers of A by AC by d i v i di ng t h e secon d row by A an d
t h e thi r d by C
Th e co effi ci en t s obtai ned , af ter th es e ch anges, by ex
p ens io n of (48) i n p owers of Aare
.
4 78
APP L IED AERODYNAMICS
Co effi ci en t
of
A2 E : coeffi ci en t
A4 , 1
A3
of
,
1
1
A
C
m
E
i
B2
coe ffi c en
1
5
t
Yp
L,
L,
of
N'
—
h
r
n
e
s .)
A2,
of
Yo
02 E :co effi ci en t
L,
+
1
0
Ye
_ a
0
f yr
'
Y.
Y,
Y.
N.
N,
L,
“
i
L
‘
L!
A,
l
I
g
L,
N
‘
,
N:
cos
D2
o i i
c eff c en
00
t of AO
,
0
co s
sin
00
00
th a t (4 9) i s g rea tly s im
p li fi ed in fo rmi f t h e axes of X a nd Z a r e
ch osen so a s t o coi n ci d e with p rin c i p a l a x es o f i n er ti a sin ce E i s th en
z e ro
I t app ea rs from
a gni tu d es of t h e v a ri o u s
p a ri son of t h e m
a co m
s th a t th ose con t a i ni n g E a s a fa cto r a r e n ev er i m
te rm
p o r t an t for a ny
118 a
ch oi ce of a x es
Th e t erm
s of (49) whi ch do n ot co n t a i n E s h ow a s t ron g g en e r a l
si m
i la rity of fo rm
t o th ose fo r l on git u di n a l s t a bility
Th e con d itio ns for sta bi lity a re th a t A2 B 2 02 D 2 a n d An Cz
2
z
02
Az Dg sh a ll a ll b e p ositi v e
It is
l
c ea r
,
.
.
.
,
,
,
.
”0
Y,
1
E
Y,
90 ft S ,
0 105,
o
’
o
—0051
0
15
l
.
l
’
&
Su bs t i t u t i ng t h e
N'
v a u es of
A,
D, i s
n ega t i ve and
l
00 142,
l
N,
E
( 50) i n ( 40)
B,
003 2,
.
N'
é
leads t o
515
2, 0,
i nd i ca t es i ns ta bili t y
1
D,
r
0 960
480
APP L IE D
for t h e
la t e ra l
mti
AE RODYN AM IC S
m
h a s t wo rea l roo ts an d one p ai r of co p l ex ro ot s
Wh en t h e ae rop lane i s st a lled or ov ers t all ed t h e osci lla ti on b eco es
v ery uns t a ble, an d stallin g i s a co
on p r eli i na ry t o an i nv ol u n t ar y
sp i n
Fo r sp ee d s b e tw ee n 70 ft - s an d 100 ft -s t h e oscilla ti on i s v ery
st a bl e, an d n eith er t h e p eri od nor t h e da
p i ng s hows uch ch ange
The da p in g of t h e rollin g sub si d ence 18 co p ared b elow with t h e
o
on
.
.
m
v alue
a
bo v e
of
i
t he
L,
o
on acc u n
mi imump
n
mt
oss
t
t he
of
mm
m
m
m
m
m k bl
.
.
re
ar
a
m
.
.
.
at
e
p d s w ell
S ee
i ble
.
ugges t s t h a t (A
monly a facto r of t h e
i s co m
otio n in d i
biqu a dra ti c for s ta bility excep t near st a lli n g sp eed Th e m
oti on of a win g du e to t h e in crea se
ca t e d i s t h e s topp i ng of t h e d ow n w ar d m
of angle of i n ci d en ce This i s t h e n earest a p p ro ach to sim
p l e ot ion
It i s
in an y of t h e di stur b ances to w h i ch an aero p lan e i s subj ect ed
s ent ered un d er Sp ir a l sub si d ence r ea lly
po ss ibl e t ha t t h e fi rs t two t erm
belong t o t h e ro llin g subsi dence as t h e ana lys is u p t o thi s p oi n t d oes n ot
a gn i
p erm
i t of di scr i m
in a t ion wh en t h e roots ar e roughly of t h e sa e m
g
Th e
s
en
a ree
.
m
.
.
m
.
tu d e
In
.
ei
th er
t he
case
di screp an cy be tw een
an d
m
im
p l mti
the
da p in g
o on
e
gr ea t and in itself in di ca t es a m
u ch l ess s
fo r a n a ero p lan e whi ch i s ov ers ta ll ed an d th en d i stur be d
O v er a consi d era ble ran ge of sp ee ds (70 ft s to 180 ft s ) inst a bility i s
in di cat ed in wha t h as b een ca lle d t h e sp ir a l subs i d en ce
This i s no t a
d an g erous ty p e of in st a bili ty a n d h as b een accep t ed for t h e reason tha t
an oeu v ri g
an y a dv an t a g es for ra p i d
cons i d era bl e r u dd er con t rol h as
n
a s i n ae r i a l fi gh t ing an d t h e con d itions for large co n t r ols a r e n ot easi ly
reco nci l e d with th ose for st a bili ty
e
For nav iga ti o n su ch in st a bility i s un d esi ra bl e sin ce as t h e na m
im
p li es t h e a ero p lane t en d s to t rav el i n sp i ra ls unl ess cons tant ly cor
Thi s m
oti on can be an aly s ed som
ewha t ea sily so a s to j u st i fy
root ed
Sp i r al
t h e d escri p ti on
Sp i ra l in st a bili ty i s a sso ci at ed with
As wa s i ndi ca t ed in equ a tion
a ch ang e i n s i gn of D2 fr o mp ositi v e t o n ega ti v e w h ils t Cg i s th en
facto r at
lt
.
s. i s
-
,
.
.
-
.
.
-
.
"
.
,
m
m
,
.
,
,
,
,
.
”
.
,
,
A PP L I E D
484
mod
AERODYN AMIC S
a ll
t ly large If D2 is v ery sm
r esp on d i n g wi t h t h e Sp i ra l sub si d en ce i s
era e
A+
00 i s
o between
00 i s zero
90 ft
zer
wh en
the
.
.
-s .
an d
r
oot of
t he
biqu adra ti c
cor
g§
100 ft
.
-
s
.
,
an d e
qua tion (4 9) sh ow s tha t
en t s an d y a wi ng m
D , d ep en ds on t h e rollin g m
om
om
en ts d u e to
si d esli pp in g an d turni ng an d ch ang es Si gn wh en N, L, i s n um
er i cally
an d
,
Cons i d er t h e m
oti on of t h e aero p lane wh en b anked but not t u rn ing
t h e aero p lane be gin s to si d esli p d own w ar d s an d t h e si d esli pp ing act ing
th ro u gh t h e d ih edr al an gl e p rodu ces a rolling cou p l e L. t en di n g to red uce
e t h e si d esli pp i ng ac ti ng on t h e fi n an d
e ti m
t h e ban k
At t h e sam
ru dd er p ro d u ces a cou p l e N
tur n in g t h e aero p lan e t owar d s t h e low er
win g Th e u pp er wi ng t rav els th rou gh t h e ai r fas t er than t he l ow er
as a res ult of t h is turni n g an d p r o d u ces a co u p l e L t end ing to in c rease
p ed by t h e cou p le N
Th e tu rni ng i s d am
t h e b ank
Th ere are th en two cou p l es ten di n g t o afi ect t h e b ank i n o p posi te
d i rections an d t h eraeroplan e i s st a bl e if t h e rightin g cou p l e p repo n d era tes
If on t h e oth er h an d t h e aero p lan e i s un st a bl e i t ov erb anks s i d es li ps
o re rap i dly an d so on t h e resu lt b ein g a Sp i ral Th ere i s a li m
i t to
in m
ore f orm
a l t r eat m
en t of d i stu r b ed m
otion
t h e ra t e of tu rnin g but t h e m
mu st be deferred to a la t er p art cf t he ch ap t er Enough has b een sai d
t o j ustify t h e t er s u sed
,
.
,
.
,
,
p
.
'
,
.
,
,
,
,
,
.
,
m
.
.
Owin g t o t h e twi st in t h e ai rscrew race t h e effec t of v ari a ti on of
ay be v ery co nsi d era bl e
th rust on t h e p o sition of t h e ru dd er m
Th e
d eri v a ti v es also ch ang e b eca u se of t h e ch an ge of Sp eed of t h e ai r ov er
t h e fi n an d r u dd er
An ai rscrew whi ch h as a v elocity not a lo n g i t s a xi s
exp eri en ces a fo rce e qui v a l en t t o th a t on a fi n i n t h e p o sition of t h e
en t s as w ell as f orces
Y awin g an d si d esli pp i ng p ro d u ce m
om
a irscrew
us t i n gen era l be app roach ed by t h e
an d t h e ca l cula tio n of sta bility m
otion an d n ew d eri v ati v es
es ti m
a tio n of new con dition s o f st ea d y m
.
.
.
,
.
V ARI ATI O N
or
L ATERAL STAB IL I TY
wrr n H aronr
AN D
LO AD I N G
Th e d eri v a ti v es ch ange with d ensity an d lo ading acco rd in g t o t h e
law alrea d y d ed u ced for l on gitu din al s t a bili ty wh ere i t w as sh own th at
en t d er i v a ti v es d i vi d ed by t h e
o
t h e f orce d er i v ati v es an d t h e
ass
2
V
V S
V
p
p
— w e re k ep t
i f t h e qu an t i t i es
o f t h e ae ro p la n e v a ri ed a s
a nd
7?
W
7
715
mm
m
,
“
"
4 85
ST AB IL ITY
t
cons a n
t
d en sity
an d
d en sity
a re o
for
one
o
o
en
mt i
t dy
oth er then
,
If
on s.
o
s ea
btaine d f romth ose
mm t
t he
s ea
for
an
mt i
t dy
in t h e
an
and
po
p
co r r es o n
E
g
1
-
a nd
fo rce d eri v a ti v es
t he
t he
S
on
by
multi p lyi
multi p lyi
g facto r
i n t h e fi rs t
de ri v a ti v es
WO
n
n
d with loa di n g
w
ith
pl
in t h e
loa ding
secon
d
an d
mti
o
on
Pl
g by
is
or
mo
re
I n w riti ng d ow n t h e coeffi ci en t s o f t h e biq u a d ra ti c for st a bility it
will b e a ssu ed tha t t h e ax es of X an d Z h a v e b een ch osen t o b e p ri nci p al
a x es of i n erti a , so th a t E i s z ero
Th e coefli ci ent s a re :
m
.
Coeffi ci en t
o ffi ci en t
4
A1 ,
of
1
AI 3 ,
c e
of
coeffi ci en t
2
A
of
1 ,
Y'
‘3
L,
“
t
s
o /r
e
L,
la
Y
i
‘
y"
“
Po
w n
”
W
k,
.
o
N,
L”
N,
1k
(3( k
W1
)
P
o
l
i
co e ffi c en
t
A,
of
(
2
P
Pl
N
,
co s
00
L,
0
a
a
0
1
1, t h e
l,
Re
“
We
)o wl
( kl :
e
B
g
If
W1 Po
N,
Np
t bili ty i s a gain t h e sam
e
s a
origin a l st a bili ty
It h as been p oi n t ed out th a t Sp i ra l i n st a bi li ty o ccu rs wh en D 2 ch a nges
Sign a n d f ro m
5
4
r
i
n
it
l
e
th
f
t
will
c
h
ge t h e
s
c
a
n
e
w
c
o
n
a
rs
a
t
t
e
a
o
t
h
)
(
as
t he
,
.
APP L IED AER ODYN AMI CS
48 6
ay a ffe c t
d iti on a lth ough th ey m
Sp i ra l i n st a bi lity can no t b e e li m
i na t ed
o r l oa d i n g
mg
tu d e
p rod u c ed by
t he
co n
or
ni
a
I t follow s th a t
ch an ge s of h eight
.
.
Exa
m
pl
e
.
—In crea se
0 740
-
.
of
loa i g 20 pe r ce n t
’
ko
— 1.
Speed OO ft s q
gp
(i i )
-
is
a nd
Fo r t h e
ra ti o whi c h
fee t , w h ere
,
V,
1 27V.
of
.
.
s.
ffi ci en t s of t h e
t he
bi
coe
qd
ua
12 D 2 = 0 104
— 3
73:
,
764 ft
'
l oad in g we a n d po t h e v a l u es
corr esp on d with T a bl e 8 a r e
=3 48 B 2 = 2 3 3 , 02
A2 =
,
from(54) t h e v a lu es fo r t h e i ncrea se d l oa d i ng a nd h eight a re fou n d as
,
A2
,
biq u a d ra ti c
fact o rs b ei n g
A
(
Th e
e
02
m
m
2 4s) (i
-
ca s
d
igin a l
or
D2
q u ati on with th ese
(A
n ew a n
,
B2
Th e
the
h eigh t
t he
)
°
a nd
a nd
.
m
0 25
z
coefli ci en t s
0 728 ) (A
olv ed
h a s been
s
,
0 03 62)
-
0 127 t 0 s525
i
)(
o
,
0 03 62)
-
ti on s a re co m
p a red
i n t he
T a bl e b elow
TAB LE 5
.
O r i g i nal
ml
ot on near
load i ng
60 ft
Pe ri od
mp i
mt
Da
Ti
e
s
.
.
of
76 4 i t
.
o
s.
o f la t e ra l osc illa t i o n
fa c t or of s p i ra l s u bs i de
h a lf d i s t u r b a n ce
ng
o
23
se es .
19
se cs
.
l li n g su b si d ence i s so m
ew h a t le s s h e a v ily
dam
p ed fo r t h e
i ncreased lo ad i ng a n d h eigh t w h il st t h e Sp i ra l su b si d en ce i s m
ore h e a v ily
d am
p ed Th e p eri od of t h e l a t e ral oscill a tion i s i n crea se d an d i t s
d am
p i ng m
u ch r ed u ced
I n b oth l on git u di n a l an d l a t er a l m
oti o ns t h e m
o st m
a r k e d effect o f
r e d u ce d d ens ity a n d i n cr ea s e d l oa d i ng h as b e en t h e d ecrea se of d am
p i ng
of t h e slow er oscilla ti on s
Th e
ro
,
.
.
.
STAB I LI TY
mCm m F
cn
c
Lron r
l ongit u d i n a l a n d l a t era l st a biliti es of a n aere p lane can o nly be
co n s i d er e d s ep a r a t ely wh en t h e s t ea d y m
oti on i s rec tilin ea r a n d i n t h e
met ry an d it i s n ow p rop osed t o deal wi th those cases i n
p l an e of sy m
w hi ch t h e se p a ra ti on ca nnot be assu m
ed t o h ol d wi th s u ffi ci en t accura cy
Th e
,
.
APP L IED AE RODYN AMICS
48 8
en t ally
v alu es a re d e t erm
in ed ex p er i m
X
Th e shor t er n ot a tion X
in t ro d u ced by B ry an is al so r et ain ed I f u o + u be w ritten for a 00 4 1:
N u p t o fi rst
for c et c in e qu a tio ns ( 56) an d t h e exp ans i ons of X
di fi erent i al coe ffi ci en t s u se d i ns t ea d of t h e gen eral f u nctions t h e eq u a ti o ns
Th e t er s of zero o r de r
ca n b e di v i d ed i n t o p ar ts o f zer o an d firs t o r d er
v an i sh i n vi rt u e of t he co n diti ons of s t ea d y otio n as gi v en by
and
th ere re ai n t h e first or d er t er s as below
,”
.
.
.
,
,
-
,
,
'
m
,
m
.
m
i
m
g
t
1
3
d?
wog
o
ur
m
-
n r
+
o
0
q
m+p
==
gd
wop
wp o
cop
vp o
cor
cro
11K .
d
n
g z
=
=
n3
d
qg
u o
uo
u Y,
+p
uZ
3
1
X
-
(
)
q
q
r 0
o
g
(
- f
Pc l
’
wY,
’
v
,
+ rYr
u
q
'
J
L
P p
A
q
Y
1X ,
'
Z,
'
cL ,
’
q
'
rZ
Lc
)
'
'
r
’
wL,
'
’
'
w
'
uL
ro
"
'
p
C
vX,
’
f Lr
'
"
Po
PM;
(q
po
u
Nu
+ PNP +
ml
11,
e
v,
e
q
l f Nr
l
'
'
'
m
p resen t t h e s all di s
q
with t h e suffi x zer o app ly t o t h e s t ea dy
ot ion an d ar e th erefore con
st ant d ur in g t h e fu r th er cal
cu lat i ons
Th e d as h es us ed to
t h e l ett ers X
N in dica t e
th at t h e p arts du e to ai r on ly
ar e in v ol v e d ; t h e d er i v a ti v es
an d
w, p ,
tt ers
e
'
'
'
In th ese equ ati ons
t ur bances , whils t t h e sa
q
'
re
r
m
~
,
.
ar e
o t
t
c ns an s
m
.
Evalu ati on of d , ri n g and
713 i n ter
(1
s of p ,
and r
B efore p rogress can be
a de
wi t h eq ua tions (61) it i s necessa ry
t o r ed u ce all t h e qu an titi es t o
d ep en d ence on p
an d r
In
dev elop i n g t h e rela ti on thr ee
a u x ili a ry s
all angl es or, B an d
y ar e u sed whi ch rep resen t di e
p lace ents fro
t h e or igi nal
p osition an d exp ressi ons for
a
n
d
r an d du b ( 1
112 a n d ( 1
11
p
3
y , an d t h e r ot a tions i n t h e st ea d y
m
q
,
q
m
m
Fro 244
.
mt
he
o i on
writt en d own
.
.
in
t er
mf
s o
,
a,
B
,
q
,
m
.
.
,
m
ST AB I L ITY
If
p rese nt t he d own war d ly di r ect ed v erti ca l d efin e d
by t h e di recti on cos in es 111 712 a n d 113 befo re di sp l acem
e n t a n d by
et c a ft erw ar ds i t i s r ea d ily d e d u ce d from
t h e figu re th a t
GP
of
Fig
4 89
244
.
re
,
.
,
,
(In )
n,
with si m
i l ar exp ress io ns
for
113
11,
712 a n d a s.
I
du g
d na
Th e
3
1
112y
” 17
" 3“
713
( n,
7
127
+
8
h
g
c a n es of
d ir ecti on
cosi n es
-
7120:
a d e u p o f $ 2 a bo ut t h e v er ti ca l an d
ulta n t velocity bei n g m
o
i fl an d 9 a bo ut t h e a xes of X Y a nd Z t h e ch a nges fr omp g go an d ro
ca n be ob t a in ed by r esolutio n alon g t h e n ew a x es an d h ence
Th e
res
.
,
,
,
,
,
=
5
P
§—
a
f ofl HIoY
i
'
'
In t h e
so
of
case
s
mll
+ 1303 + 7
“
90
“
oscill a tions it
a
fr o mt h e genera l ty p e
i s known
lution th a t
=
5
t =
M
usin g th ese v a lu es in (64 )
f ormfor whi ch t h e soluti on i s
a nd
A
l
fo
“
—
l 90
d uces t h e e qua tions to si
90 P
P0 9
A
7
“
90
“
p
711
q
90
”3
9
2
4
2
)
r
1
0
p
2
ar ex r ess
is
ous lin ear
an e
m
re
It
u
t in t h e d en o m
i na to r of t h e l ast exp ressi on i s easily evalua t ed
d
a
e
ce
6
a
n
d
6
6
n
b
e
to be A ?
it
d
d
u
th
t
ca
an d fro
8
( )
( )
n an
A
m
il
mlt
70
l
m
m
i
Th e d et er
an d fo u n d
re
( 65)
—B
m
Si
of
ll
" zd f
A
g
ions
o v i
c n en en
t
112
for ( 1
to
an d t h i s
mk t mp o
a e
e
foll ow fr omsym
metry by
rar
y
u se of a
t he
6
3
( )
or di n ar y
q u an tity p d efin ed by
.
9
2
11
With
t h e ai d
qu ati o ns
e
of
4
-
?
1
l tion s d evelo p ed i t is
ore co nv eni en t f orm
as
( 61) i n m
t he
re a
n ow
p oss i bl e to
w rite
re
4 90
APP L IED AERODYN AMI CS
Mu
(X
‘
( X.
)
r. v
.
'
2o
0
( Yrs
,
"
A) ”
Pc lw
_
fl ap o dn allp +
'
e
l ( Yo
,
“
+wo
.
(Z n l
l lZ ,
" "
‘
l (Z
‘
,
P0) ” l ( Z
‘
n
‘
'
e
n
‘
,
0
Alw
n
'
“
‘
0
L.
'
L. w
+
A
0
B
M
’
.
-
7?
C
L;
3
-
0
v
B
’
N.
A
—
E
q)
—'
1>
o
’
B
N,
u
c
_
)
Aq
’
N, w
’
u
c
N,
'
-
~
A—B
’
-
.
"
C
'
C
An
q}
O
,
(
m
c
3
1 —A
I
y
po
0
"
.
C
r
’
x in ati on of t he equa ti on s will sh ow th a t
be gr ou p ed t ogeth er a nd t rea t ed as new d er i va ti v es
be co n v en i en t for r e feren ce t o t h e e q u i val en t s use d
e a
t i
t
t
cer a n con s a n s
Th e
.
ta bl e
b el o w
my
a
w ill
.
A
—B
0
p t)
Ta bl e ( 71) need s littl e exp l anati on ; it in d i cat es th a t i n t h e fu rth er w o rk
an e
ress n s c a s
xp
io u h X i s used ins t ea d of t h e l ong er one X
re a n d 80 on
I f n ew t h e v a ri a bl es
r u v an d w be eli m
in a t ed fro m
p q
eq u a ti ons (7
t he s a i
e
t bil ty qua ti on i n A i s o bt a in ed an d i n d et erm
in an t al fo r i s gi v en
’
,
,
,
,
,
,
m
,
,
ri
g
p
-
A x,
Y.
Yv
“
Z
Z
-
IL .
.
d
Yu
Z .
.
x, +p c ( 1
Yv fl ‘i n zp o
Z,
'
L,
11.
1
x.
x,
11,
N,
11,
M,
N,
N,
1
x,
”
"a
“(
m
o
Yv i fi nd1_ n ‘r)
Z'
'
“
L,
11,
iv,
.
—n y \)
x.
Y,
—n
'
a
l
L.
1
,
11,
1
N,
-
A
Th e
fu rth er p rocedu re cons is ts i n a n a li cat i o n of
a nd t he
oi n t
p
p
p
a t whi ch a na lyti cal m
eth o ds a r e u se
d before i n t r od uci n g nu m
eri ca l v a lu es
i s a t t h e ch oi ce o f a worker
Th e a na lysi s h a s el sewh er e b een car ri ed t o
t h e s t a g e a t whi ch t h e coef
fi ci ei 1t s of Ah a v e all been f u d
g
l fo
.
o n
in
en era
m
r
,
APP L IE D AERODYN AMI CS
4 92
m
Th e e qu a ti on p ro ves t o be of t h e eighth d egr ee t h e t er whi ch a p p ears
1
f
r
r
A
to be o o d e
h a vin g a z er o coefli ci en t Th e exp ress i ons whi ch oc c ur
wh en t h e l on git u di n a l an d l atera l oti ons ar e sep ara bl e ar e un d er li n e d in
whi ch th erefore con ta ins t h e oc ti c
t h e firs t d et er i n an t of equ ati on
,
“
m
m
An” +
4
A
(
2
131A
W
01A
DO
-
.
”
3
+
W 2
023 + 132)
PA
s a re negl ec t e d it
w
r itt en for C, wh en t h e 9 t er m
0 0
th a t t h e seco n d d et erm
in an t cont ai ns a t erm
be
I f C1
,
:
9
Fr o
'
2
0
3
mt h
+ A1
th ir d
e
A
B
+ 1
2
A
fifth
an d
3
A
)(
C,
0
”
d et er
2
A
A2
-
m
i
na n
t
be
ca n
m
0 S
i
lb
.
Th e
fou r th d et erm
i n an t furn i sh es
LP
.
,
9
11
o
b vi ous
t er m
"3
'
I
.
,
i l ar t erm
im
‘
A
"
9
3
A,”
”1
Mu Mw
m
m
m
t it m
t he
_ ’
9 \ iL o
N ,N
N,
a s
bt a i n ed
713
is
5
7
( )
B 2A
o
4
7
( )
Th e r e a ini ng t er s of (78) ar e
er i cal
on e or two n u
w ay , b u t fro
m
too co m
p li cat ed to analyse i n a gen era l
ex am
p l es it woul d a pp ear tha t t h e m
o re
im
p ort an e s are shown i n
o tio n s
Th e facto rs of ( 74) ar e e xactly those whi ch w oul d be used if t h e m
wer e sep ar abl e but with t h e d er i v ati v es ha vin g t h e v al u es for t h e c ur v i
,
m
Exa p le
ac t ua l fli ght
f
o
the Calculat i on
f
o
the Sta bili ty
mt i
.
I ni t i al
con di ti ons of
th e
s tead y
n, = 0
c.
t h e a xi s of
o
n3
°
X i s h oriz on ta l
an d
l l3 5ft
.
t he
erop la ne
when tu r ni ng d u ri ng Lor i
= o 7o7
°
aerop lan e
v°
S.
-
i
on
= 0 707
n,
f mA
o
b a nk ed
at
wo = 0
=0
m
t h e fl i gh t sp eed i s 113 5 ft -a , and t h ere i s no s id esli p ping or nor a l
v eloc i t y
Th e las t condi ti on const i t u t es s s peci al cas e in whi ch t h e t e
s alt an t
ot i on h as b een ch ose n as ly i ng a long o n e of t h e p ri n ci pal axes of
m
r
.
.
°
t . e. o n e co
mp l t
e e
t u rn i n
Q
ab o u t
22 secs
°
=
4
5
o
¢
for i ts
v alu e i s
r ed s
02 84
t h at wh i ch gi ves 11
.
sec.
-
.
u
The
d a i u ced
,
q ti
second e
woPo
from" q" g and fl a
ua
on of
( 59 ) i s
Yo
( 79 )
for t h e condi ti on of n o sid es li pp in g Yo d epen ds only on gravi t a t i onal a t t rac t i on
and i s e u al t o 11g
sin ce r o=n ,0 , whi lst w0 and p , a re zero , e u a t i on ( 79) b eco
es
a nd
'
q
q
m
.
ST AB IL ITY
re lat i on
4 98
m
q
bet ween a and
uan t i t i es defined i n ( 78 ) wh i ch
ust be sat i sfi ed
Th e
ot h er e ua t i on s of ( 69)
us t be sa ti sfi ed, and t h e su b ject is d ea lt wi t h i n Ch a te r V
p
Si nce t h ere are only four con t rols a t t h e disp os al of t h e pi lot . so e ot her au to a t i c
adjus t
en t besi d es ( 80 i s re u i red , an d i s i nv olv ed ab ove i n t h e s t a te
en t t h at a o= 113 5
)
ft -s when war 0 The st at e of s tead y oti on i s fi xed by e ua ti ons
an d t h e s
all
v ari a ti ons of u
r a b ou t t hi s st ea dy st a t e lead to t h e resi st an ce der i vat i v es
In
t h e presen t st at e of kn owledge i t i s a pp a ren t ly su ffi ci en t to assu e t h at d eri vat i ves are
funct i ons of angle of in ci den ce ch i efl y and li t t le d epen den t on t h e agn i t u de of co, p o,
Progress i n a p pli ca t i on of t h e la ws of
ot i on dep en ds on an i n crease i n
9 0 and r e
kn o wledge of t h e aerod yna i cs
Wi t h t hese re arks in t er posed as a cau ti on , t h e deri v at i ves for an aer oplan e of ab ou t
°
2000 lbs wei gh t fl yi ng at an angle of i nci den ce of 6
a y be t
y pi cal y rep re sente d by t he
a
m
q
m
m
q
-
.
m
.
m
m
m
m
q
.
'
m
m
m
m
.
.
.
.
m
.
'
Res i sta nce Den w ti
m(
see
l
Tab le
0
107
—
0
I
0
0
94
015
The v alu es
of
A, B
an d
0 occu r
only
mt h
e
0
der i va t i ves ,
an d
t he
M
L
u se of
AB
’
q
a nd
N
6
Th e wh ole of t h e u a nt i fi es i n ( 8 1)
i n ( 73 ) d oes not affect t h e con di t i on for st a b ili t y
b
are essen t i a ll
r
e
b
e
o
t
a
i
n
e
d
f
r
o
t
h
e
x
e
a
s
h
o
st u dy of
r
a
n
d
e
n
t
l
u
t
t
e
r
e
f
e
i
p
y
da t a
Wh en t h e efi ects of air screw sli p st rea are i nclu ded t h e dedu ct i on fro gen eral
da ta is lab ori ous and needs consi derab le exp eri en ce if seri ous error is t o be avoi ded
Th e n u er i cal values of t h e d eri va t i ves as gi v en i n ( 8 1) can b e su b st i t u t ed i n ( 73 )
an d t h e d eter i na nt s red u ced su ccess i v ely 1 t il t h e oct i c h as been d ete r i ne d
I t is
a so ewh a t hi gh degree of accu racy i n t he p rocess i n order to avoi d
Th e fi n a l
cer t ain errors of opera t i on whi ch afi ect t h e so lu t i on to a large ext ent
resu lt ob t ain ed in t h e p resent exa
p le is
m
m
m
.
m
A
20
m
.
m
‘
m
.
m
m
m
m
m
1513
-
a
q
»
»
490
.
.
»
687
719
»
15
0A
'
+109 A
0
( 82)
has t wo real root s only , wh i ch can b e ext r act ed i f desi red b y Horner s
A general et h od for afl root s h as been gi ven b y Graefi a and as t his d oes not
process
a pp ear i n t h e En glis h te xt
ap pen di x t o t his ch ap te r
B y use of t h e et h od i t was found t hat e ua t i on ( 82) has
t he facte r s
Thi s
e ua t i on
’
m
.
m
.
q
( 83 )
an d
t he di st urb ed
A
mti
o
on consi sts of
t hree
oscilla ti ons , one of
whi ch is
m
i
u nst a ble , and
t wo
a ble cases of lon i t u di nal and
s
a
i
h
t
o
f
t
h
e
l
e
r
83
i
n
t
h
e
)
(
g
p
g
la t er al di st u rb an ces sh ows t ha t t h e facto rs i n t h e order gi v en corres p ond wi t h ( a ) Ra pid
longi t u di nal oscilla t i on ; ( b) Ph u goi d osci llat i on ( u nst ab le ) ; ( c) Rolli ng su bsi den ce
( d ) Spir al su bsi den ce and ( c) Lateral osci llat i on I t app ea rs fro fur t h er calcu lat i ons
°
t ha t at an angle of i n ci den ce of 6 t h e efi ect of t u rni n g s h ows chi efly i n t h e p h u go i d
osci llati on and i n t h e sp i r a l su bsi den ce , t h e fo r er b e co i ng less st a b le and t h e la tt e r
car efu l exa
na ti cn cf
m
.
m
m
AER ODYN AMI CS
A PP L IE D
4 94
m
Co parison 0! St raig h t Flyi ng and Ci rcli ng Fli gh t —For r easons gi ven
earli er as to t h e i na d eq u acy of t h e d a t a for ca l cu l a tin g d eri v a ti v es too
u ch w eight shoul d not b e a tt ach ed to t h e followin g t a bl es as rep re
flight
Th ey d o how ev er illus tra t e p oin ts of
sen t at i v e of ac t u a l
i p ort ance i n t h e eflect of t u rni n g on s ta bility Four co n d iti ons are
m
m
,
.
,
,
'
.
o i d er ed :
r
a
a
o
i
o
n
t
l
s
t
ight fl ight
1
H
r
z
( )
i
s
r
a
li
d
g
t
ight
fl
ght
i
n
2
G
( )
r
n
c
r
c
n
o
i
o
t
a
l
i
li
g
8
z
H
( )
4
i
ra
l
gli
di
n
g
S
( ) p
p tion t h a t t h e ai rscrew gi v es a thr ust
Th e d at a 18 b a sed on t h e assu m
o nly an d th er ef or e 1gn 0r es t h e effect s of sli p st r ea m
on t h e t a il whi ch m
odi fy
t he
om
en t coefli ci ent s 111 both t h e lon git u di n a l an d l a t er al m
oti o ns
A
recen t p a p er by Mi ss B M Ca v e B r own e C a v e shows th a t ou r kn owl e d e
g
i s r eachi n g t h e st age at whi ch t h e full eflect s can be d ea lt with on
so m
ewh a t wi d e g enera l gr ou n d s
Th e t a bl es ar e b ase d on flight in all
an d t h e sp ee d h as been v ar i ed to m
cases a t an angl e of
a i n t ai n th a t
con di ti on
Th e angl e of b ank i n t u rn in g h as b een t a ken as
c ns
.
.
.
.
m
,
.
-
.
.
-
'
.
.
mpi
fac t o r -Z- veloci t y
Mod u lus —veloci t y
Da
ng
.
m
m
f actors for cu r vi lin ea r flight s ar e both a pp reci ably gr ea ter
th an those for r ectilin ear fli ght an d it will be seen fr omt h e t h ir d row
of t h e t a bl e th a t t h e in cr ease i s enti r ely accoun t ed for by t h e ch a nge of
s p ee d
The da
ng
,
.
H ori zont a l
st rai g h t
fa ch
r -é-veloci t y .
m
t
.
00405
0 28
00004 88
00029
l
00555
0 28
00005
80
00030
m
uch l ess t han
Th e d a p i n g factors for cur v ili near flight are v ery
th ose for r ectili near otion whi l st t h e od uli ar e gr ea t er Th e o sc ill a tion
or e rap i d but l ess h ea v ily d a
p ed , whi ls t t h e e ffect
i s th erefore r ath er
e ch aract er for both s t ra ight a n d cu r v ed fl ight
of d escen d i n g 13 of t h e sa
p a th s an d d escen t gi v es i ncreased st a bility i n all cases
,
,
,
m
m
m
,
m
,
,
m
.
.
APP L I ED AER ODYN AMIC S
496
M
m
m
m
on th o flt a
ti li n gu al th c l p orh nt Du i vat i
i ty o!
—
Th e d eri vati v es consi dered w ere
Straight and ( fircii ng Horizontal Fli ght
M” , L , and N , with conse u en ti al cha nges of M, an d N " an d are i wrt ant
M“, can be v ar i ed by ch an gin g t h e p osi tion of t h e
i n d i ffer en t respect s
en t of t h e la t eral
cen tr e of gra v ity a n d t h e t a il p la n e area, L, by a djus t
di h edral an gl e an d N , by ch ang e of fin and r u dder ar ea All are ap p reci
e
a bly at t h e choi ce of a d es igner , an d t h e foll o wi ng ca l cula tio ns gi v e so
i dea of t h e p ossible efiects whi ch ay be p r o d u ced At a gi v en angle of
i nci d ence r es ist an ce d er i vati v es ar e p r o p or tional to v el oci ty an d si p li city
of co p ari so n h as been assi st ed by a r ecogni tion of t his fact
const ant
Vari ati ons c
L and
ul
m
q
m
.
-
m
lO x darnp i ng
facto r
veloci t y
‘
i
md
Hori z on t al
o u lns
,
.
.
mg
m
Horu ont ai ci
-
lO‘ x
Hori zont al st rl i gh t
h g
m
n
m
.
m
m
’
m
.
,
u
se
-
we
s as
as s
6 17
6 15
l
i
fg?
e ct
h
543
-
95
-
mm
a)
o l(
( 3 90)
-
m
c
3 08 )
g gi v en to M i s p ar ti cular ly l arge an d t h e m
ost n oti cea bl e
a ll effec t of tu rni n g on t h e rap i d lon gitu din a l osci l
featur e of (89) i s t h e sm
lation Th e figur es in brackets corresp on d with a p air of r eal roo t s vi z
n r ep resen t ed i s
otio
an d it will b e seen th a t t h e m
5
0
4
x
(
For a v ery unst a bl e aero
always sta bl e b u t not alw ays an osci lla tion
e in d i ca t ion of
p l an e as r ep resen t ed oy t h e low est v alu e of M th ere i s som
p l ex int erch an ge bet ween t h e longitu dinal and la teral otions wh i ch
a co m
ean in g coul d be cl early
wou l d need f u r th er i n v esti gation before i t s m
Th e
ran e
,
,
,
.
.
.
m
,
x velod t y
100
.
H or izont al st r ai ght
4 5
-
4 3
-
,
see
sea
m
,
3
3
m
h ere very arked Th e for er shows st a bility at all p os iti v e va lu es of
st a bility of t h e osci lla tion to i ns ta bility i n a n ose
M an d t h e ch ange fr o m
edi a t e s t age of an uns ta bl e oscilla tio n
di ve occurs wi thout t h e int erm
In
howev er t h e gen eral res ult of a red u cti on of M is t o
circli n g fl ight
p ro d u ce i n i ncreasin g o scil la tion I n all cases t h e dam
p ing is very
ot ion at an angl e of bank of 45 as com
p ared with th at
s a ll i n ci rclin g
In
in st raight flyin g a nd a grea t er v alu e of M is n eed ed for st a bili ty
.
,"
.
m
.
,
,
m
,
.
°
,
.
ST AB IL ITY
t ight flyin g th ere is in d ica t e d a li m
it to
p h u goi d oscill a ti on whi ch can be a tt a in ed
4 97
t he
s ra
d egree
m
d a p i ng
of
of
t he
.
100
x
v eloci t y .
In r ectili nea r fl ight t h e sp i ra l
a n d t h e n eg ati v e v a lu e in di ca t es
all i ns t a bility i n to a
conv ert a s
m
o d ary or d er of v ar i ation on
a sec n
moti i u ff t d by h g f M
i t bility Th ff t of tu i g i to
m k d t bility wh i h i d p d t f
t h mg itu d of M
on
s
ns a
ar
ec e
c a n es o
e e
.
e
e
na
ec
rn n
s
c
s a
a n
e
,
e en
en
”,
s
or
.
i
r
i
L
l
ll
I t a pp ea rs th a t neith er
O
s
ate a
c at on
Rolli ng Subsi dence and
of th es e q u a n titi es i s a pp reci a bly a fl ect ed by eith er t h e v ar i atio n of M,
or of ci rclin g , beyon d t h e chan ges whi ch ar e p ro p or tiona l t o t h e v el ocity
o f fl ight
Th e ex p ress ions cor reSp on di ng with tho se used i n (90) ar e th en
For t h e r olli n g s ubsi d en ce
con s t a n t s for t h e co n diti ons now i n v es tiga t ed
da p i ng fact or/veloci t y ha s t h e v alue 00686, w hils t for t h e l a t era l
d a p in g fact or/veloci t y is eq u al to
whi l st
oscill a ti on
x
h as t h e v alu e
odu lu s/v eloci t y
x
M“, Constant Th e chan ges of ra p i d l ongi
Vari ati ons of L, and N,
t u d inal oscill a tion du e t o ch ang e of l a t eral d eri va ti v es ar e in app r eci a bl e,
an d t h e d i fferen ces be tw een s tra ight fl yi n g and ci r cli n g a re p r o d u ced only
by t h e ch anges i n t h e v elocity of fli ght Si il ar re ar ks app ly t o t h e rollin g
'
.
m
m
.
m
—
.
m m
t d f mt h v
ly i m
p t tv
ubsi dence as m
ight h a v e been exp ec e
otion an d t h e f act th a t t h e o n
of t h e m
i c L h as not b een su bj ect e d t o ch a n ge
.
s
,
.
ro
m
y sime charact er
ari a bl e of t h e m
otion
e
or an
er
,
.
.
N IC x l O‘ I ve loclt y
'
Da
m
pi
ng
Mod u lu s
‘
fac t o r x lo
1 54
lveloci t y
o se
-
-
1
-
.
’
lO /veloc i t y
0 25
-
I
l2 0
°
S t ra ight
fli ght
.
Da
mp i
ng
Mod u lus
40
fa c t o r x lO‘lveloc i t y
‘
20 2 fo r a ll
x l O lvcloci t y
va lu es o f
3 6i
°
l
L . an d N,
.
um
er i ca lly s m
allest va lu es of L a n d N t h e cen t rifuga l t er m
s
i n t rod u ced by tur n i ng con v ert a st a bl e p hugoi d t o a n uns t a bl e one
I ncrease in t h e d ih edr al an gl e has a coun t er bal ancin g effect an d t h e p hugoi d
b ecom
Th e l on gi
es st a bl e ov er t h e ran g e of N co v ered by t h e t a bl e
oti on 18 of co ur se un change d by a dih ed r a l
t u di nal st a bili ty of r ectili n ear m
a n gl e or by t h e si z e of t h e fin an d r u dd er whi ch a re t he p ar t s of t h e
i n e L an d N
arily d e t erm
aero p l an e whi ch p ri m
For t h e
n
,
,
,
.
,
.
,
,
APP L IED AERODYN AMI CS
4 98
Da
mpi g f
n
act or x
I O‘ Ivcloci t y
.
N-IC x l O‘ lv eloclt y
l +o 5
0
5
4 10002045
”
2s
-
°
o
07
69
4 01
see
4
03
705
601
P
—0 70
-
i
+
-
2 00
1o 1s
-
—0 72
-
504
9 05
M
2005
3 03
0001
.
4 09
Th e v alu e of N , ch anges sign wh en t h e aer o p la n e, r egar d ed as a w ea th er
In t h e a b
coc k r ot a t i n g a bou t t h e a xi s of Z , j us t t en ds t o tur n t ai l first
sen ce of a di h edral an gl e t h e st ea dy s t at e i s n eut ra l i n s t ra ight fl ight , b u t
b eco es s ta bl e on t u rni n g For both st raight fl yin g an d turni n g st a bi lity
ay be p r o d u ce d i n an aerop l an e sho win g w ea t h ercoc k i ns ta b i li ty by t h e
It i s n ot kn own h ow far t hi s
use of a suffi ci en tly l arge d ih e dra l an gl e
ay be a pp li ed a t oth er an gl es of i n ci d en ce
co nclus io n
.
m
m
.
,
m
.
.
00
l
l 28 2 00
0
°
'
,
l
Horizont al
'
80
‘
86 56 l
'
'
Th e fi gures i n (94 ) show th a t t h e la t er a l osc ill a ti on i s v ery d e p en d ent
o n t h e si z e of t h e d ih e d r a l a ngl e an d li ttl e d ep en d en t on t h e ra t e of tur n in g
e xcep t wh en t h e aerOp la ne i s d e v oi d of w ea th ercock s t a bili ty , i s N , 3> 0
—
u
i
r
N
h
n
e
c
a
l
Alth ou gh a ll t h e calcu
R
su
l
t
e
k
s
o
e
ts
Gener al Re ar
an d to cir cli n g at an an gl e of
la t i ons r ef er t o on e angl e of in ci d en ce
°
b a n k of 45 wh en turni ng i s p resent , th ey h av e n ev erth el ess shown th at t he
m
m
.
.
APP L I ED AERODYN AMI CS
500
Thi s de t erm
i nan t
(
eas
BM
An
s
4
A
is
ily red u ced
to
w
ne
ea
—A x
,
z,
,
Z
,,
As
Y,
x
—A
B a 4» ca + 112)
s
g
—A
Y, +
—
L, A
L,
o
wh ere t h e quantiti es A,
D 1 A2
D 2 are those for longit u di n a l
an d l a t eral st a bili ty wh en gy r oscop i c cou p l es are ign ore d
ina tion of (99) i n a par ti cular case show ed th a t t h e coeffi ci en t s
An exa m
s w er e all p ositi v e an d sm
of p ow ers of A i n t h e gyrosco p i c term
d
ar
all co m
e
p
wit h t h e co effi ci ent s obta in ed fr omt h e p rod u ct O f t h e biqua dr atic f act o rs
Th e rap i d m
oti ons lo ngitu d in a l an d l a t er al will th er efore be littl e a fl ec t ed
It a pp ears fur th er th at t h e ch an ge i n t h e p h u goi d oscill ation i s a sm
al l
i ncrease i n sta bility Sin ce t h e gy roscop i c t erm
s d o n ot con t ai n on e i n
ar k as t o signs of t h e coefi ci en t s s ho w s tha t
d ep en d en t of A t h e a bo v e r em
a Sp ir ally s t a bl e or uns t a bl e a er o p l an e w1t h ou t r ot a tin g air scr ew will re ai n
s t a bl e o r u n s t a bl e wh en gy rosco p i c effec t s ar e a dd ed
I n any case o f
im
p ort an ce how ev er e q u ation (99) i s easy to a pp ly an d t h e conclus i on
n eed n ot b e reli ed u pon as m
ore th an an in di ca ti v e exa p l e
,
.
.
'
,
,
.
,
,
.
m
,
.
,
,
,
m
TH E Sr a s x
rv
OF
Arnsn rr s
AND
m
.
K rr s B ALL O O NS
m
m
th a t
Th e t rea t ent O f t h e st a bi lity of light er- th an ai r cra ft di flers fr o
fo r t h e a erop l an e i n se v era l p ar ti culars all of whi ch ar e conn ected with th e
atio n of t h e f orces acting
es ti
Th e effect of t h e buoy ancy of t h e gas i s
e qui v a l en t to a red u cti on of w eight so far as f or ces alo ng t h e co-o r di n a t e
b u t t h e co bin ed effect of w eight an d buoyancy
a xes ar e con cern e d
otio n whi ch w ere n o t
i nt rod u ces t er s i n to t h e equ ati on of a ng u l ar
p rev i ous ly p resen t Th e oorin g of an airsh i p t o a ca bl e or t h e effect o f
o en t equa ti ons
a kit e wi r e in t r o d u ces t er s i n both t h e force a n d
Th e ath e a ti cal th eory i s d ev el op ed i n t er s of r esi s t ance d er i va ti v es
with out s eri ou s d i ffi cu lty, b u t t h e n u b er of d et er i nati ons of t h e l a tt er
r
s
a ll th a t t h e a pp li ca ti ons ca nn o t
a
i
s
l
t
h
ar
t
o f a su fli ci en t l co
e
e
c
c
e
s
o
p
y
b e sa i d t o b e a d e qu a t e Thi s i s i n p ar t d u e t o t h e l ack o f f u ll scal e t est s o n
whi ch to ch eck calcul a ti ons an d i n p ar t t o t h e fact th a t t h e ai r f orces a n d
o
en t s on t h e l ar g e bul k O f t h e e n v el o p es O f light er th a n a i r cra ft d ep en d
n o t o nl y o n t h e lin ear a n d an gula r v el ociti e s t h rou gh t h e a i r , but als o on
t h e li nea r a nd a n g ular accel e ra ti ons
I n a s i p l e exa p l e it w ou l d a pp ear
tha t t he l a t e ra l a ccel era ti on of a n ai rsh i p i s littl e ore th an half th a t
w hi ch w oul d b e ca l cula t e d on t h e a ss u p ti on th a t t h e l at eral r esi s tan ce i s
d et er in e d only by t h e v el ociti es of t h e envel op e
Th e n ew t er s a ri sin g fro b u oyancy will b e d ev el op ed generally an d
a ca bl e l eft t o a sep a ra t e s ecti on sin ce th ey d o n ot
t h e t er s a ri si ng fr o
otio n of an a i rs hi p
a ffect t h e fr ee
Th e sep ar ation in to lon gitu dinal an d
l a t era l st abi liti es will b e a d op t ed an d t h e g en eral case l eft un til su ch ti e
en t al d a t a a r e s u fli ci en t ly a dv an ced as t o
as it a pp ea rs th a t t h e exp er i
p er it of t hei r u se
'
-
m
,
.
m
m
,
m
m
.
m m
m
m
m
m
mm
m
m
.
-
.
mm
m
m
m
m
m
,
.
m
.
.
,
.
m
m
m
m
.
m
m
-
-
m
501
ST AB IL ITY
Gravi tati onal and Bu oy ancy Forces —I f t h e u p war d f orce d u e t o
bu oyancy be d eno t ed by F , t h e v alu es of t h e co p onent forces along t h e
m
a
x es
are
m
m
X = n l ( g —F)
m
m
—
fl ight m
F
i
g
1
0
0
m
( )
mp t t m v i h
m
—
=
n 2( g
F)
Y
Z =n3(
—
F
)
g
o
For an a irs hi p i n fr ee
on en
es
an s
s z ero a n d t h e co
I n t h e kit e b allo on r eserv e b u oy ancy i s p resent an d i s b al a nced by t h e
v erti cal co pon en t of t h e p u ll i n t h e kit e wi re
Gr avi tati onal and B u oy ancy cou ples —Th e cen t r e of gra v ity O f light er
th an ai r cra ft i s u su ally w ell b el ow t h e cent re O f buoyancy ,
b el ow t h e
cen t re O f v ol u
e of t h e d i s p l ace d a ir
Th e l a tt er p oi n t will v ary with t h e
co n d iti o n of t h e b a lloon et s an d
us t b e sep arat ely ev al u at ed i n each cas e
as p ar t of t h e s t at e
en t of t h e con ditio ns O f st ea d y
oti on
B oth t h e
cen t r es of gr a v ity an d bu oy a ncy will be t a k en t o li e i n t h e p l an e of sy
et ry ,
an d t h e cO-or di n a t es O f t h e l a tt er a r e d en ot ed by a: an d 2 r el ati v e to t h e
body a xes th rough t h e cen t re of gravi ty Th e b u oyancy f orce F act s
v erti cally u p war ds an d t h e co p o nen t s o f force at (x o z) are th er efor e
m
-
.
m
m
m
.
m
m
.
om
en t s a b ou t
T a ki ng m
co m
p on en ts ar e
L = n gF
3
M
,
,
n
on
u
d i na
(
71| Z
1
en s
o
F
)
on
e
ee
u an
e
on
thi s
n gF
,
ons as
acce era
o
0
1
( 1)
body a x es sh ows th a t
713 5
1
3
,
—
n3 F
a nd
m t —T m t t h
l ti
w ll
l mti t h q
titi X
Ai r Forces and Mo
an d
o
en t s d ep en d on
a
t he
mm
.
.
—n 2F
mm
t h t i l git
f m
.
e
es
acc
oun t
t he
1
0
2
(
)
x
fe a t u re
th a t t h e forces
ed
as on v el oc ity it i s a s su m
Z a n d M h a v e t h e ty p i ca l
new
,
.
or
X =j x( u
o ti on th r o u gh
ult o f m
e x p a n d ed a s
a s a r es
X ri
Th e
(
nu
u o. 100.
go)
mb
er Of
n X,
t he
w, (1
,
.
u
Of an
w
,
)
q
0
8
1
(
)
fo ll o wi n g t h e
air
q
i nt ro d u ce d
a e ro p l a n e
twi ce
as
e
0X,
°
is
mth
p r e v i ou s
X, + 1LX § +113
10K ,
d eri v a ti v es
longi t u d i nal st a bi lity
,
great
as
o
d X
is
0
1
4
(
)
th a t
for t h e
.
m
an d 713
h g d ep en d on t h e v ari a ti ons of t h e di r ec ti on cosi nes
en t s o f t h e a x es a n d m
a y b e d e t e rm
di spl ac e m
i ned d i rectly or
a ri s i n g from
e q u al
gi v en i n ( 68) by p utti ng 42 710 (10 To an d
fro m
t h e genera l form
to zer o Th e ch ang es O f th e d i recti on cos i nes ar e
c a n es
,
,
,
.
.
'
.
_
f q ( 1712
’s
A
A
713
(1
of whi ch t he first and l as t refer only t o l on git u d i n al st ability an d t h e secon d
t o l ate ral s ta bility
.
APP L IE D AERODYN AMIC S
5
02
q
Divi si on oi (56) i nt o E u atlonr oi Steady Hot i on au d Distu rbed l ot i on
—Usi n g t h e sep arat e exp ressi ons for forces du e to gr a vity , bu oyancy an d
o
e
b
a i r e q a ti ons ( 5
6
ec
)
u
.
m
,
In
t dy
s ea
=
q
i
w
w
u
i
mti
o
q
m
(
+i x (u
g
(
n,
g
W
lq
it , iv,
é
al l (
,
.
n,
m1
g
+ f x( “o
{n
l)
)
q
2 T
)
.
‘ ' I" 1
o
a re z e ro , a n d
b
w,
u,
,
f
f
”
h ence ( 106)
beco
m
es
m 1)
‘
(
0
“
q
3
1
3» 13 »f
.
q
w,
" 12
on ,
to»
,
o
v
,
(
0
I f i n ( l 06) u 0 +u i s w ritt en for 11, et c , a r t du h for n ,, et c
o f d i s t ur be d
o ti on are o bt a i ne d
t h e t er s o f z e ro or d er bei ng those
t h e fi rs t o r d er te r s are
an d th er e fore i n d ep en d en tly s a ti s fie d
m
-
m
.
0
,
m
-
2
1
+ woq
W
e
s t
z
wz.
in
e
:
ax,
“
.
t
2
F(n l a:
M
u
,
en ;
m
x
i ;
2
1,
ioZ e
(
w
M
q,
ivMs
We
m
a d e in
Coll ecti ng th ese t er s i n accor d an ce with t h e n ote
carri ed o u t for t h e a er o p l a n e i n e qu a ti ons ( 11) an d ( 12) l ea ds t o
—
(Z
of
{
A) w
,
Z
AZ
9
M
an d
0
1
( )
fill
F
+
11
0
)(
=
q0
0
9
1
)
(
mp i g ( 109) with ( 11) h w th t t h h g o i t of t h w iti g
f g X
AX, f
X t
x p t th t i t h
of M t h
fy
F/m
A
p
i M AM,
w
itt
i
t
d
of
F
i
M
) /
f d i tu b d m
i ti g
w
d q
f mt h th
q tio
oti
Eli m
l d t
q ti i Awhi h i of t h fou th d g
of t h
i th
E
pl
p t f t h t mi d p d t f A t h
fli i t i t h
Co
or
O
ex
res s on
na
ea
s
ar n
s
a e ro
o an e
an e
.
.
or
,
“,
a
e c
u,
ua
xce
on
an
ro
n
c
or
e
er
.
e c
,
e
s
u 3c
,
n
s
o
e
s
e
es c ns s
a
ce
n
en
n s ea
r ee e u a
ns o
e
n
an
en
r
r
en
e r ee a s
o
e
e case
,
s
n
e coe
r
,
n
e
.
r
on
e
e cas e
c en s
e
n
e
APP L IED AERODY N AMI CS
504
wh ere 11112 on ly app ears beca use 112 i s zer o as
co ns i d era tion O f t h e l a t era l an d lo n gitu di n a l
o di tion of t h e sep arate
motions Si mi lar ly 110 p,
a c n
.
,
m
by t h e forces an d cou p l es du e to t h e ai r b ein g a lso zero fr o mt h e sym et ry
of t h e m
otion Th e v alu e Of du g h as
oti on i n t erm
s
d r ar e
e qu ati ons O f d1st ur be d m
ti
+
n or
lz
7
=
y
)
i (
+ p Y, + rY,
+ r L,
11
1
31
w
(
=
+ vNo + PNp +r N,
6N1»
Arran gi ng t h e
t erm
s as f act ors o f v
Y, +AY +
A
Y
Y
o
+
a
(
~
(
X
-
(
,
p
F
g
1A +
i
5N
’N »
'
i
l ds to
an d 1 ea
°
F
Y, +AY, —
uo
’
N,
31
1
2 5
s )
p+
AN,
N,
AN;
li m
i na tio n O f v p an d r frome qua ti ons ( 113 ) gi v es
eq
u a ti on with t h e followi ng coeffi ci ent s
4
oe
e
C ffi ci n t Of A
—
Yg l Yb
Y
L,
15— A L ;
N;
N
N; — ( i
Th e
e
,
,
,
1
°
Coeffi ci en t
Of
3
A
,
3 }
11 + 1 Fan
°
a
biq u a d r a ti c
505
STAB IL ITY
Coeffici en t
Y» Y,
L, L,
N
of
A,
11.—u , +71, Y . Ye
L,
L, Lb —A
N,
,
+
Coeffi ci en t
o
—Fz
.
—
m
Y
F
;
g
/
Fz
L
| ,
L;
Yv
L,
Yp
N
N,
,
L,
of
—
—
E
l
Yc
g
/
—Fz
Fit
m
/ ) +
—F
F2
—F:1:
L5
N5
A,
—Fz
mY —
—(g— E/m
1
1
+
3
)
Y,—1 Y,
L,
L,
N;
Np
i nd ep en dent
ml
'
11,
N
F2:
m
—F
/
g
Ye
713
F2
—Fa:
,
no
,
'
'
L,
N,
_
Yr u 0
11,
N
4
1
1
( )
,
whi ch m
ost u se h a s hith er to b een m
a d e i n a i rs hi p
d ed uced fr om( 114 ) by con si d eri n g hor i zon t al flight with t h e
en v el op e h or i z on t al ; n , i s th en zero
Th e r eserv e b u oy an cy
—
m
O
en t r e of b u oy ancy i s v er ti cally a bo v e t h e
m
a
n
d
t
h
e
E
c
/
g
grav ity so th at a: i s zero Th e coe ffi ci en t in d ep en d ent of A i s
Th e f or
s t a bi lity i s
a xi s of t h e
z er
-
Fa:
11,
is
mw a r
— e F/ )
u a
of
.
,
,
t
cen re O f
.
th en
713
12
3
1
'
Y
Y'
,
N
N.
"
uo
5
1
1
( )
,
s t a bility to i ns t a
if thi s qu antity ch anges si gn th ere i s a ch ang e from
bi li t y t h e l a tt er corresp on d i ng with a p ositi v e sign u n d er u su al co n ditio n s
es
For a n a i rshi p to be l a t er ally st a bl e t h e con d itio n b ecom
an d
.
,
m
N
) ,
Y, N, > ( Y,
110
q
Exa p les of th e use of th e E uati ons 01 Di sturbed Ai rsh i p Moti on
Th e fu r t h e r re a r ks will b e con fined t o h ori zont al flight i n whi ch ca se
us t
Th e nu eri ca l da t a a re n o t a ll th a t coul d be d esi red an d u se
111=r0
a de o f g e n e r a l i d eas
as y et b e
—
For a n ai rs hi p o f any ty p e
Re arks on th e Values of th e Deri vati ves
et ry n o t o n ly a bo ut a v e r ti ca l
i n p resen t u se th e re i s a pp roxi a t e sy
p la ne b u t a l so a b o u t a h or i zon t al p l an e th rou gh t h e cen t re of b u oy ancy
Th e re ar e th en so e Si p l e rel ati ons b etween t h e forces a n d cou pl es du e t o
a y be e xp ect ed th a t
It
r i si n g a nd falli ng a n d th o s e d u e t o si d esli pp i n g
t h e for ce s o n an ai rs hi p will not be a ffect ed a pp reci a bly by a slow r o t ati on
p ti on b e a d e it i s easily
a b ou t t h e a xi s o f t h e en v elO p e a n d i f thi s ass u
seen th a t t h e r el a ti o ns h i p b e tw ee n d eri v a ti v es d u e t o r olli n g a n d d er i v ati v es
d u e t o s i d es li pp i n g i s s i p l e
Th e r el a ti on s whi ch ay b e si p ly d ed u ce d
a s a r es ult o f t h e a b o v e h y p oth eses are
.
m
.
m
m
m
,
,
m
.
m
.
mm
m m
-
.
.
m
.
m
,
m
.
=
a
th ere i s n o ch an ge
t h e ai r shi p t o t h e wi n d
of
'
t e
.
.
X
i t
res s ance
m
m
m
=o
for
li ght i nclin ations of
s
t he
i of
ax s
.
Thi s rel a ti on exp resses t h e fact th a t
t h e li ft
an d
l a t eral
force
on
t he
i hi p
a rs
APP L IED AERODY N AMI CS
506
e i n clin a ti on s
ha v e t h e sam
e v alu e for t h e sa m
i n p it ch a n d y a w resp ecti v ely
of
-
the
a
xis
of
X
wind
t o t he
.
X, = X
1
8
1
(
)
,
wh er e X, t he v ari ation of r esi st an ce du e t o p it chin g di fi ers f romX t he
va ri ati on of resi st an ce du e t o ya win g beca u se t h e a xi s of X li es at a di s t ance
z b el ow t h e cen t re of b u oya n cy whil s t t h e a xi s of Z p asses thr o u gh th at
p oin t Sym
met ry a bou t a v ertical p l a ne i s su ffi ci ent t o ensur e th a t X i s
'
,
,
,
,
.
z er o
.
i
As t h e
ca r an d a rs cr e
w
t
t he
are n ear
cen re o f
gra v ity
X
'
will
be
lm
ost w h olly d u e t o t h e r esi st ance o f t h e env el op e in i t s fo re a n d a ft
moti on du e t o p it chi ng a bout t h e a xis of Y Th e ch ange of r es i st a nce
of t h e wh ol e a i rs hi p d u e t o a ch an g e of forw ar d s p ee d 11 will b e grea ter
th an K mp art ly on accou n t of t h e a dd ition al resi st ance of t h e ca r bu t
t h e ai rscrew
a l so b eca u se o f r ed u ced th r u s t f rom
,
,
a
.
.
'
o
,
.
.
_ Yr
Z0
a l f o r ce d u e t o p it chi n g a n d l a t er a l fo r ce du e t o
v ari ati ons of n orm
y awi ng will b e rou ghly rel a t ed as sh ow n in
B oth ar e ass oci a t ed
d i rectly with u , i n t h e cond iti ons o f s t a bility an d th eir val u e i s n ot kn own
with any d egr ee of a ccu racy Th ere i s a p ossibility th a t Y m
ay b e h a l f as
gr ea t a s no
( 120)
Th e
,
.
,
.
lift d u e to win d i s zero t h e rat eof ch ange of Z with ch ange
of fo rwar d Sp eed will also b e zero
en t d u e t o ch an g e of fo rw ar d sp ee d a
Th e p it chi n g m
om
ay
le M m
not b e z ero
If t h e a i rs crew s ar e a t t h e l ev el of t h e car an d th er efore
n ea r t h e C G
it w ou l d a pp ea r th at t h e ch ange of a i rscrew thr u s t wi th
It can th en b e s t at ed
ch a nge of for w ar d sp ee d will not g rea tly a ffec t M
as p r o b a bl e th a t
Sin ce t h e
,
.
.
,
.
.
u
,
,
.
,
.
M,
M,
s
’
,
N,
mp i
Eq u ati on ( 121) gi v es a r el a ti on b etw een t h e d a
p it ch and y aw, assu i ng eq u al fi n a reas hori zon ta lly
'
2
e
x , occu rs b ecau se t h e a x i s of X i s b el ow t h e
t r
m
mm
1
2
1
(
)
g deri v ati v es i n
t he
an d v e rti ca lly
cen t r e of buoy a n cy
n
.
( 122)
m
l ti on whi ch assu es equ al fi n area s B oth M an d N are
gr ea tly d ep en d en t on t h e ar ea an d d i sp osition of t h e fin s and ar e t wo of
t he m
ore i m
p ort an t d er i v ati ves
Th e a pp ro xi m
a t e r el a tio n
L
m
zY
( 123 )
is
a
fu rth er
re a
,
.
,
,
.
-
,
,
d ed u ce d fro mt h e consi d erati on th at t h e l a t eral force on t h e car is
u ni m
p ort a n t com
p are d with th at on t h e env el op e an d th at t h e r o t at ion
can
be
,
508
APP L IED AERODYN AMICS
Routh s d i scri m
i na nt whi ch m
ight l ea d to a n ew criti cal Sp eed bu t t he
fu r th er ana ly si s will b e co nfi n ed t o an exa i na tion of t h e app ro xi m
at e
es
u
a
r
a
i
c
e
equ a tion s ( 125
a
n
d
h
fi
rs
t
o
f
th
e
h
as
t
e
s
t
a
bili
ty
biq
d
t
h
T
)
—
—
—
A
A
I
z
Z
“
+
a
a
{
s
(
0
e
’
m
Mm
—{ z
an
m
,
,
M
( g
z,)
B A)
FZ /A
8
1
2
(
)
0
It i s not st ri ctly l e giti at e to say th a t resi s t ance deri vat iv es d u e t o
=O s i n ce sli ght r es i d u al t er m
s o f hi gh er
ch a n g es of v el ocity v a ni s h wh en V =
or d er a re p r es en t but i n acco r d anc e with t h e th eo ry o f sm
all osc il la ti ons
a s d e v el op e d thi s will b e t h e case an d with t h e a irs hi p s t e pp ed e q u a ti on
( 128) red uces t o
,
.
,
2
AB
Si nce
wh il s t B an d F
un d a m
p ed oscilla ti on of p eri od
2
is
n ega t i ve,
,
Fz
9
1
2
(
)
o
p ositi v e this
a re
,
i s t he
e
q u a ti on
B
30
1
(
)
F2
If
,
Xu ltl,
m
pp ears p ro b a bl e we ay n egl ect
e qu a ti o n ( 1
28 ) h as on e root gi v en by
as a
,
o f an
3
in
om
p ar iso n with
c
X“
whi ch i n di cat es th a t a vari a ti on o f forwar d sp ee d i s d am
p ed ou t ap eri odi
s a re th os e a r i si ng f ro m
ca lly
Th e n egl ect ed t erm
ch anges o f d r a g o f t h e
en v e l op e d u e t o p it chi n g a bo ut a n a xi s b el ow t h e cent r e o f fi g u r e
—
i
r
x
m
r
i
r
i
t
io
Ap p o
ate C e
n ! “ Longi tu di nal Sta
na Eq u ati o n ( 128 ) n ow
t akes t h e f orm
.
m
1
(
“0
“
.
l Zq
"
M
( 9
B ") i F2/
‘
‘
s u s i n g t h e th eory of eq
u a ti o ns an
by a consi d erati on o f t h e t er m
im
p o rt an t a pp ro xim
i na n t for l ongitu d i nal s t a bility i s o b ta i ned
a t e d i scr i m
u s t th er ef or e h a v e a t l eas t o ne real
Th e eq u a ti o n i s a cu bi c i n A an d m
an d
,
,
.
,
i
r
oot
.
Th e
p ro d u ct
of
t he
t
is
r oo s
f
t he
valu e of
whi ch i s
ess ent i
lly n ega ti v e a nd i m
p ort a nt Thi s foll ows fr om
gen eral knowl ed g e for z is
n eg ati v e F p os iti v e Z
a nd Z ; n eg a ti v e a n d B p o siti v e i n a ll a i rcraft
co n t em
p l a t ed I f only one real roo t occu rs it m
us t th erefore b e nega ti ve
whil st if all t h e root s ar e rea l th ey m
u s t eith er all b e n ega ti v e or t wo
p os iti v e a nd one n eg ati v e A ch ange of sig n of a r eal r oot ca n only occur
by a p a ssage th rough z er o an d i n t h e p resen t i nst ance thi s d oes not occur
ay r ep resent
s i n ce t h e p r o d u ct o f t h e r oo t s ca n no t b e z ero
Th e cu bi c m
a s ubs i d en ce an d a n oscill a ti on a n d t h e o nly p oss ibility o f i ns t a bility ar ises
fro m
a n i n cre ase i n t h e a m
p litu d e of t h e l att e r
a
,
.
,
,
,
,
,
.
.
,
.
,
.
ST AB IL ITY
509
m
Th e co n d iti on for ch a n ge of sig n of t h e d a p i n g co effi ci en t
o sci ll a tion can eas ily be d ed u ced for t h e su of t h e root s i s
m
,
of
t he
( 13 3 )
dam
p i ng o f t h e oscillation will be z ero if t h e real root i s e qual t o
t h is v al u e Ma kin g t h e sub stit u tion for Amequation ( 13 2) l eads to t h e
cr it eri on for st a bility
M,
an d
the
.
Fz ( l
M”
m
Z
Z
hi ? “
p erio d i c ti e of t h e osci llati on at
of t h e r oot s a n d i s
t h e p ro d uct a n d su m
Th e
QHB
Z
B
/
w
a
‘
e
t he
cr
iti cal ch ange is foun d f rom
,
m
m
i n t h e b rack et i s alw ays p ositi v e co p a ri son of
Sin ce t h e seco n d t er
t
h
e
o
sc
a
n
i
n
h
a
ith
h
w
t
t
ill
tio
c
r
iti
a
l
o
tio
s
l
w
1
3
0
s
o
s
c
n
i
s
o
er
1
3
5
w
( )
( )
th an th at at rest Th e criti cal v eloc ity a bo v e w h i ch t h e oti on i s u ns t a bl e
an d a kn owl ed ge of t h e
a nn er of v a ri a ti on
i s ea sily d et er i n ed fro
of t h e d eri v ati ves w ith chan ge of sp eed I f u , be t h e criti cal v el ocity a n d
n o t h e v el o city for w h i ch t h e d er i v a ti v es w er e ca l cul a t e d , t h e e xp r essi on for
is
m
m
m
m
,
m
.
.
0
_
M0
‘
2
Z
( )
l
(
Z
2
) /B
w
uc
F rome q u a ti on ( 13 6) ca n be s een t h e con d iti on gi v e n by Crocco (see
p age 4 1 T ech ni cal R ep o r t of t h e Ad v i sory Com
mitt ee for Aeronauti cs
—
9
0
9
40
fo r t h e n o n e xi s t en ce of a cr iti ca l v el ocity i l e
1
Con
v or t e d i nt o p r ese n t n ot a ti on Grocco s con di ti on i s
“
.
,
,
f
-
,
.
’
,
1
3
( 7)
a ri s o n wit h 11
w
ed th a t Z
a s n e gligi bl e i n co m
xcep t th a t Crocco a ssu m
p
0
,
Hi s ex p re ss i on for l a t era l s t a bility h as an e xactly a nal ogou s for m
a in i ng
110
Z , i s p osi t i v e whil s t M, Z a n d z a r e nega ti v e a n d t h e re m
If M be nega ti v e 1 e i f a
t erm
s ar e p os iti v e with t h e exce p ti on o f
r es t o ri ng m
om
en t d u e t o t h e wi n d i s i n t rod u ce d by a n gu l a r d i sp l ace
men t e xp ressi on ( 13 6) shows th a t t h e a irshi p 11 oti on i s s ta bl e at all
ay be o bt a i n e d with M
It w ill be seen h owe v e r t h a t s t a bility m
sp ee d s
p ositi ve an d thi s i s t h e u s u a l s t a t e o wi ng t o cons t ru cti onal d i ffi cu lti es i n
a tt ach in g l ar g e fi ns
—
t
a
ili
Th e biqu a dr a ti c equ ation
o
f
r
L
a
t
ra
l
b
S
ty
e
Ap proxi at e Cri teri n o
for sta bility whi ch 18 o bt ai n ed fr o e qu ati on ( 126) i s
e
.
t”
,
,,
,
,
’
.
.
m
A“
Y,
-
mY
z
N
,
,
Ye)
Y,
— zL,
—zN,
"
.
.
,,
,
m
,
m
,
,
.
.
.
Z
AA
Fz/A
0
APP L IED AER ODYN AMI CS
510
t h c d eri v a ti v es
If t h e m
o ti on t hr ough t h e a ir i s v ery sl ow
e z ero an d ( 13 8) r e d u ces t o
ch a ng es of v el ocity b eco m
,
d u e to
-
,
z
—
A
Fz/A z a O
1
9
3
(
)
g fr omt h e exp r essi on conta inin g p cl ear ly refers t o a n osc illa tion
It a pp ea rs th a t n o a i rs hi p i s p rov i d e d with con t rols whi ch a fl ect
i n roll
ay b e exp ect e d a t a ll fli ght S ee ds
t h e rolli ng a n d an oscilla ti on in roll m
p
T hi s s ugges ts th a t t h e t erm— zY i s us ually uni m
p ort an t i n whi c h case
t h e o sci ll atio n i n fli ght i s gi v en by
a n d a ri si n
'
m
,
.
,
,
Re
1
{2
A
a nd
is
fact or
m
( 140)
A
m
d am
p i ng t er du e t o t h e otion
of t h e s t a bility equ at i on i s th en
—uo + Yy = 0
Y —Ml —Ya)
—
N
AO
N
seen
t o h a ve
a
Th e
.
r
em
ai n i n g
o
,
,
m f t h oot of
mf ll i h i p
l
qu ation ( 14 1) is
a n d i s n e ga ti v e
é
Y
a rs
s;
or a
If e qua tion ( 141) h as co m
p l ex r oot s th ere
i n each t er
fore t h e rea l p art ust be n egati v e an d t h e corresp o n din g oscilla ti on
s t a bl e
L at er al ins ta bi li ty can th en on ly occur by a ch ange i n t h e si gn
in dep en d en t of A and t h e criterio n for st a bili ty i s
of t h e t erm
Th e
su
s
e r
o
e
!
-
'
9
m
,
,
“
.
,
Yo
Ne
N are b oth n egati v e it i s an i m
medi ate ded u cti on from
om
a r es t ori n g m
en t a bout a v er ti cal a xi s th r ough t h e
du e
a p ositi v e v alu e for N i s n ot an essen ti al for st a bili ty
t o wi n d forces
bin ed with t h e con d itio n th at equally efl ect i v e fin s be u sed
Moreo v er com
both v erti cally an d hori z ont ally ( 142) i s su ci ent t o ens ur e t h e co m
p let e
st a bi lity of an ai rs hi p a t all sp ee d s
In t h e crit erion of st a bili ty Y an d Y ar e i n v ersely p rop o rtional
ass of t h e ai rs hi p an d it i s in t er es ti n g t o e x a m
to mt h e m
in e t he
p ossibiliti es of v ari ation of Y and Y at v ar ious h eight s 1 e when m
e d i n t h e p r ecedi n g an aly si s th a t t h e m
It h as b een ass u m
v a r i es
ass of
oved
t h e a i rshi p in clu d e d th a t of t h e hy dr ogen zl e th a t t h e hy d rogen m
Thi s i s ob v io us ly o nly an a pp ro xi m
ation
as a soli d with t h e env el op e
en t s of t h e gas a r e cl ear ly p o ss i bl e but
ov em
t o th e t r uth as i nt ernal m
ass co n cern ed i n t h e m
otion i s th a t of t h e ai r
so far as it hol d s goo d t h e m
ass i s i n d ep en d en t of t h e co n d iti on of
di sp l aced by t h e ai rshi p Thi s m
ou n t of a i r i n t h e b alloon et s
on t h e o th e r hand
t h e hy dr ogen or t h e am
it i s p rop orti on al to t h e d ens ity of t h e ai r an d th erefore v ari es wit h h ei ght
e v elocity a l so v ar y d i r ec tly as the
Th e f orc es on t h e a i rshi p a t t h e sam
ai r d en sity a n d h ence Y an d Y ar e i n dep en d en t o f h eight
Th e s t a bilty
of a n a i rs hi p i s n ot a fl e c t e d by h eight a t l ea st t o a fi rs t a pp r o xi
a tion
As Y,
a
th
t
4
1
2
)
(
an d
,
,
,
,,
m
,
,
.
'
.
,
,
,
,
,
°
,
,
,
.
.
.
,
.
.
,
,
,
,
.
,
.
,
,
,
.
'
,
m
.
APP L IED AERODYN AMIC S
512
H o ri z ont al fins
d enot ed i n Fig 245by a an d b
Ver ti ca l fin s a re d en ot e d i n Fig 24 5by c d c f g a nd h
Of t h e v er ti cal fi n s f g an d h w er e a rr an ged as bi p lanes Th e p resence
of t h e hori zon t al fins was fou n d not t o a ffect app reci a bly t h e for ces o n t he
v ert i ca l fins a n d vi ce cared
ar e
.
.
.
,
,
,
,
.
,
,
.
.
.
,
Monop la no fi ns
.
No fi ns
.
1
—4
‘
1
7x 0
40
‘
2 7 x 10
—
1
+ 35
—a 3 x 10 —3 8
‘
—59 x 10'
‘
1
x 0
—54 x lO‘
f
—l
‘
1
0
7x
—7 8 x 10‘
i
'
-
z ex 1
73 x 1
—
Owi ng to t h e shi el d i ng of t h e fins by t h e bo dy of t h e en v el op e t h e
met ri cal flight than wh en y a wed
is l ess for sym
eri cal v a lu e o f Y
num
It i s p rob a bl e th er efore th a t turn in g t en ds
to p r o d uce gr eat er sta bility as i t in t r o d u ces
s ca n be
Th e a dd iti onal t erm
s i d es li pp in g
e di sc u s si on
i n t rod u ce d as requir ed a nd som
of t h e s u bj ect h as a lr ea d y b ee n gi v en by
a ti ca l th eory
a th em
J on es a n d Nayl or Th e m
i s w e ll a h ea d of i t s a pp li ca ti on s a n d n o d i lfi
c u lty i n e x t en d i n g it as re quir ed h a v e as y et
a pp e a re d
Forces i n a Moori ng Cable or Ki t e B alloon
Wi re du e to i ts Weig h t and th e Efi ect of the
—
E
n
Th e a x es OS
h
r
d
t
Movem
en of t e Up pe
01) a n d 01
; ( Fig 24 6) ar e ch osen as fi xed
r el a ti v e t o t h e ear th t h e ca bl e or wire being
en t of t he
fi xe d a t O
Th e p oin t of a ttachm
r
a
f
s
n
e
a
r
d
a
bl
a
i
t
P
h
v
a
t
o
t
h
e
c
i
a
c
e
y
Fro 24 6
mov ement in v ar ious directions
—
o
i
r
I f t h e s ti ffn ess of t h e wi re a n d t h e w ind
u
t
o
th
e
e
W
F r ces at P d e
of t h e wi r e will b e a ca t en a ry a n d it is
forces on it be n egl ect e d t h e f orm
.
,
,
,
.
,
.
,
.
,
.
,
m
.
.
,
.
.
,
,
ST AB IL ITY
513
l
th a t t h e f orces i n it will n ot be a fi ect ed by rotations a bout t h e a x i s
of r Th e p ro bl emso far as it affec ts t h e forces at P du e to t h e kite wi re
p l et ely sol v ed by cons i d erin g d eflections of P i n a p l ane
can th en b e co m
i tt ed al ong t h e
I n any actu al cas e it i s cert ai n th a t w a v es will be t rans m
wire but t h e a bo v e assu m
p tions wo u l d a pp ea r t o rep r esen t those of
'
c ear
,
.
,
.
,
horiz on tal com
p onen t of t h e t ens i on i n t h e wi re (cons t an t
a t all p oin ts wh en wi n d forces a re n egl ect e d ) t h e e q u a ti on t o t h e ca t ena ry
ca n be s h o wn to b e
If
k be t h e
,
o h
c s
Z
eu s
weight of t h e wir e p er un it l en gth a n d 5
s
o
t
t
i
a
s
c
n
a
n
of
0
i nt egra tio n so chosen th at
wh en E
Fr o mt h e geo m
et ry o f
t h e ca t enar y it will rea dily be seen th a t So i s t h e d i st ance f ro m
t h e p oi n t
en t of t h e wi re t o t h e v ert ex of t h e ca t ena ry t h e d is t an ce b ei n g
o f a tt ach m
measu red along t h e nega ti ve di recti on of E This foll ows fr omt h e fact
wh en 5 —20
w is t he
,
,
.
173
It
.
i s con v eni en t to u s e as a sep ara t e e xp ress i on t h e l ength
If s be u se d t o deno t e thi s
t h e p oi n t P t o t h e gro u n d
fro m
th en
,
,
.
s
=
wi r e
l en gth
of
,
£ g
as
si nh
m
Equ ati ons ( 148) an d ( 144) d efin e k t h e hori z o n ta l co ponen t of t h e
t ensi on in t h e wi re and t h e l ength 3 i n t er s of t he p osition of t h e p oi n t
P an d t h e weight of u n it l engt h of wi re In t he case of an aircraft t h e
ay be c ha n ged by a gu s t of wi n d , a n d it i s n ow
co o r d i na t es of P
p rop ose d t o fin d t h e v ari a ti ons of k whi ch r es u lt f ro any ar bit ra ry
A fu rth er app rox i a tion wi ll
o ti on o f P i n t h e p l an e of t h e wi re
a d e h ere i n tha t t h e ex t ensibi lity o f t h e wi re wi ll be negl ect ed
be
a th e a ti cally wi ll be cons i d er ed as one o f s all
As t h e p roble
osci ll ations , this ass u p tio n falls wit h in t h e li ita ti ons us u ally i p osed
m
,
,
,
m
m
m
.
m
.
mm m
m
m
m
m
.
m
by such analys is
oti o ns of P u n d er
Si nce t h e l ength of t h e wi re i s cons t an t i n t h e m
co ns i d era ti on it follows by di ffer en ti a tio n o f ( 144 ) th at
.
,
o
z
g
dk
s
2
1
0 0
Ii i
’
;
-
l,
7
g
c sh
o
w
.
g
k
It will be o b v iou s fr o mt h e d efin iti on of So gi v en p re viously th at any
v ari a ti on i n P wi ll p ro d uce a correspon di ng ch ange in So and a lth ou gh
us t be i nclu d ed
a co ns t an t of in t egra tio n wh en P i s fixed i t s vari atio ns
i n t h e p res en t cal cul a tio ns
m
,
,
.
APP L IED AERODYN AMI CS
514
Di fferenti atin g eq u ati on ( 14 3 ) gi v es an exp res si on corresp on d i ng to ( 14 5)
dc
dk
5
k
g
5
1
0 0
2
k
i
m
na
S lnh
8
k
n
5+ So) + 115
0
+ d€ s in h
Eli
5(
” r
“
"
I
"
l
r
k
ti ng (150 be tween e qua tions ( 145) a n d ( 146) t he r el ati on
m
is obt ai ned , w h i ch gi v es t h e v ari a ti ons o f h ori zo nt al f orce die i n t er s o f
en t s o f t h e u pp er end o f t h e wi re
t he
ov e
To fi nd t h e v ari a ti on of t h e v e rti ca l co p onen t of t h e t en s io n of t h e
wi re a s a co n se q u en ce of ch a nges d f a n d dZ i n t h e p ositio n of t h e p oin t P
it i s us ef u l t o e p loy eq u a ti ons ( 14 7) an d
Th e sl op e of t h e wi r e
e q u a ti on
4
er en ti a ti o n ,
by
d
i
1
3
f
f
a t t h e p o i n t P ca n be o bt a i n ed fr o
(
)
m m
m
.
’
m
.
.
m
i
v
i
n
g
g
mh Ea
d
?
w
a
and
£0)
s
s
i
4
)
(
th er efore t h e v erti cal com
p onen t of t h e t ensio n T , i s
T1
di sp l ace d p osition of
t ensi on T2 will be gi v en by
In t h e
,
i h
k
s n
-
( E £0)
p oin t P
t he
t he
( 14 9)
v er ti cal com
p onent
of
t he
,
+ w oosh
Usi ng t h e va lu e
es
5
b
o
0
1
e
c
( )
m
dk
T,
T,
i;
t
my
5 whi ch
of 11 0,
cos
h
[ g
a
e
.
5
1
£0)
t
si nh
6 + 50)
+ dEJD OOSh
be
e—
lt
o
bt ai ned
w
oh
c s
fro
m
e
qua tio n
g
et
}
5
c
008 1
]
mqu
Subs tit u ti n g i n e q u ati on ( 151) t h e va lu e of die obt ai ned fro
a n exp ressi on fo r T2
T , i n t er s of £15an d d); is obt ai ned
m
e
.
a
tion ( 14 7)
516
APP L IED AERODYN AMI CS
en t a t t h e a i rcraft i t will be p ossi bl e t o h a ve
th an one p oi n t of att achm
metry i n t h e v er tica l p l a ne
with out h avi ng t h e p l ane of sym
e q u ili b r i um
If h owev er sym
metry i s assumed
cont a ini n g t h e win d d i recti on
en t a bo u t any a x is p ara ll el t o 0;
om
be n ecessary t o arr ange t h a t t h e m
,
.
,
,
,
Wit h t h e a id of t h e a bove eq u a tions it is p o ss ibl e to d et erm
i n e b oth t h e
for a ca p ti ve a i rcraft an d t h e deri v a ti ves d u e t o
co n d iti ons of eq u ilib r iu m
t h e swayi n g of t h e r ap e
.
THE
su
mm
r y or
a
11—Ta n Da r a rns
PART
CHAPT ER X
.
TH E
or r u n
MOTI ONS or A I RCRAF T
Mo r ro n
Di s r u a a a n
m m
m
or AN
Aa a oe na xn
a th e
a ti ca l th eory of st a bility it was sh ow n th a t
I N d evel op in g t h e
t h e p eri ods an d d a p in g fact ors of oscillati ons coul d be obt ai ned togeth er
with t h e r at es of su bs i d ence or d i vergen ce of n on p eri odi c otions It
e th od s d ev elop ed to show h ow t h e
was n ot , how ev er, p ossibl e by t h e
-
m
m
-
.
lt an t m
otion was d i v i ded between forw ar d m
oti on v erti cal m
oti on
a n d p it chin g for longit udi na l di s t u r ba nces or b etw een s i d esli pp i n g r olli n g
an d y a wi ng for l a t eral di s tur b an ces
a th em
a ti ca l a n a ly si s i n
It i s now p rep osed to t ake u p t h e fu r th er m
t h e case of sep ar a bl e m
otions an d t o illus t rat e t he th eo ry by a nu i n b er of
p l es inclu di ng fli ght in a natural win d Th e su bj ect in clu d es t h e
ex a m
co nsi d era ti on of t h e effect of co n t r ols an d t h e ch an g es whi ch o ccur a s an
on e st ea d y st a t e t o an oth er
It 1s p oss ibl e th at
a er Op lan e 1s b ro ught from
t he m
eth o d of a tt a ck wi ll b e f oun d s uit a bl e for i n v est iga tions r el a tin g t o
a ti c s t a bility
t h e light n ess of con t r ols an d t h e d ev elo p m
ent of a u tom
d evi ces
R eference t o t h e eq u ati on s of di s tur bed m
otion (8) an d
will show
th a t thr ee e q u a ti ons ar e d efined for l ongitu din al an d th ree for l a t er a l
moti on an d th at in each case a combin ation of themh as led to a single
Th ere a re l eft t wo oth er r elations whi ch can
fin al equa ti on for st a b i li ty
b e u se d to fin d t h e rel ati v e p r op o rtio ns i n t h e d i stur b ance of t h e v ar io u s
p on en t v elo citi es and an gu l ar v elo citi es
com
—
i
r
u
n
Th e co n di tion for st abili ty was obt a i ne d
Longi tudinal D st ba ce
by eli m
i natin g a w an d g fr o mt h e eq ua ti ons of m
oti on a n d d e t erm
i ne d
va l u es of A f romwhi ch t h e p eri ods an d d am
p i ng f act ors w ere cal cul a t e d
Th e m
eth o d of sol u ti on of t h e di ffer en ti a l e q u ati on d e p en ds on t h e kn ow
l e d ge of t h e fact th at
r esu
,
,
,
.
.
,
.
.
,
,
.
.
,
.
==ae“
u
“
=
be
w
(g
r -
“
( 158)
cc
p r essions whi ch when in t ro d u ce d in t o t h e di fferenti al eq u a ti ons of
( 1 b an d c are t h e
d i stur b e d m
t o alg eb r a i c equ atio ns
otio n r e d u ce th em
i niti al v al u es of t h e di stu r b an ces i n u w and q
whi ch corresp on d with
t h e ch osen v a lu e o f A An exa in ati on of t h e s t a b i lity equ ation shows
that th er e ar e f ou r v alu es of A i n t h e cas e of an a erop l an e so m
e of whi ch
p l ex an d oth ers r eal Us in g ( 158) t h e e q u a tions of di s tur b ed
ar e com
motion become
a r e ex
.
m
.
,
,
,
.
AERODYN AMI CS
A PP L IE D
518
— wo
(X
.
159
Z . a + (z
Mwh
up
(M,
m
m
B A) c
0
on e co
bi n a ti on of th ese thr ee equa tio n s , o n ly
Si n ce A is kn ow n fr o
two of th e can be consi d ered as in d ep en d ent rel ations between a , b an d c,
9) is
an d ch oo sin g t h e firs t two , a soluti on o f ( 15
m
b
x
m j
—
—
x,
o g
—
x, w,
,
"
—
A
x
3
Xu
,
—A x
,
l
zw —A
Z
m Thi i
i
uit
f Ath f m
Ab
om
p l x it i
ti ally t h e so l uti o n
r eq u i r ed a n d for r ea l v a l u es o
e or
s s
e r i ca l
a bl e fo r di r ect n um
a pp li ca ti on
e c
e
s n ecessary to con si d er a p a i r
I f h owev er
o f corr esp on di n g r o ot s an d t o s ep ara t e t h e r ea l a n d i m
agi n ary p ar t s of ( 160)
b efore co m
p u t atio n i s p ossibl e
If t h e root s be A1=h +i k and A2 =h —i k t h e two v al u es of su ch a t erm
a s 14 gr ou p t og eth er as
an d
t he
ra
tios b/a
.
/
,
.
d et er in ed
an d c c a re
,
,
s
.
s essen
,
.
,
( 123
u
or
in
t er m
s of s in es a n d cosi n es i ns t ea d
-
of ex
i“
)
p onenti als
,
kt + 1(a 1 — 0 2) si n kt
)
}
e q u a ti ons 1
es i r e d t o fin d t h e v al u es of 0 , + 0 3 a nd of
it
d
s
an d f r o m
0
6
i
( )
i (a l
( 12) i n or d er t o
i
v
rea l f or
of a d a m
p
ed o scill ati on
u
t
e
e
h
g
On s ub stitutin g h
i h for A i n equ a ti ons ( 160) t h e exp r essio n s b eco e
co
p l ex a n d of t h e f or m
+
u
'
cos
02
1
2
6
(
)
m
.
m
Ml
0
co rresp on d i ng OXPreSSi OIl
0 1 11
with
a
“
M)
wh ere
t he
v alu es
1 ,
of p l , 1
2
1
bi ll‘ s l i vzl
“
13
,
zw — h
z, + u 0 +
v2 a n
i k,
roo
6112)
beta s
‘
th
for t h e
w
u
'
d
whi ch
3 a re
is
i V3 )
0
f ou n d f rom
02 13
v
{In ch
1
1
i
t 3)
'
z
’s’
1
rz + h 1
‘
fif f
ns i - k:
7 :
£312
k
(
2
c
”0
i
13 h
g
-
la n
a
i
m
APP L IE D AER ODYN AMIC S
520
mil
xp ressi ons follow for w an d g In t h e case of rec ti li n ear
= 0 an d h ence in t egr a ti on
otion i n t h e p l an e of sy
et ry a nd i n s till ai r q
0
x is of X t o t h e hor iz o n t a l
i
v
s
t
v
a
lu
f
i
t
e
i
li
n
a
tio
o
f
h
a
o
t
e
h
e
e
e
h
n
c
n
e
g
I t app ears tha t it
ay be n ecessary t o d ea l with
ual Rea
Eq
etho d out lin ed a bo v e t h en b r ea ks
eq u al or n ear ly equ a l r o ot s an d t h e
d own Followi ng t h e us u al ath e ati cal etho d it i s assu ed th at
Si
m
ar
mm
e
m
.
,
,
'
.
,
.
-
,
.
m
m
m m m
Fr om( 160)
b
q
an d
w=
c
«s
a
S(A)
the
s
m
,
2
1
7
)
(
“
c
(
u
.
D08
m
ol ution
for w is
e
It i s th erefore n ecessary i n t h e case of equal rea l r oots to find t h e v alue
of
as w ell as th a t of
Th e di fferen ti a tion p res en t s no seri ous
di ffi culti es an d does not occur su fii ci ent ly oft en for t h e com
p l ete f orm
ul as
to be r ep ro d uce d
( 178 )
Do
-
.
m
Exa p le —T M d eri vat i ves
X.
ass u
o a pp ly
e
= —O l 4
Z
n
Z
'
L
md t
in
pa r t i cu lar
a
=
.
M =
l
3
1
B
n,
m
M, =
wo = o
1
.
0
.
=
,
ca se are
u,
—s 4e
= 30
( 175)
Fro ( 175) i t will be seen th at fl i gh t is h ori zontal wi t h t h e axi s of X i n t h e d irec ti on
Proceedi ng t o t h e b i u ad r at i c for s ta b i li t y an d i ts so lu t i ons, sh o ws t h a t
of fli gh t
q
.
—502
A,
Applyi ng t h e for
—502
A,
-
ml
u ae of
A,
-
( 165)
v
,,
,
0003 96
0 350
,
o,
N A)
s o)
a nd
by
su bs t i t u t i on
‘
u
c
wh e re A, B . C an d D
ot i on
of t h e
m
to
:
in
q
°”
v
M
177
6 92
e u a t i ons
WA
008
{
—
Wfl f
r
3
0 214A
-
a
o
o
o
zu
u
-
m
1
55
12
008
( i 7s)
to
=
,
—0003 13
00143
1 12
o,
-
1
1)
204
( 177)
553
( 158 )
and
( 169 )
B
0 28 3!
si n
are a rb i t ra ry cons t an t s
00 14 73
D l)
0 2830}
t o be fi xed pr esently by t h e ini ti al co nd i t i ons
y
-
)
cos
o 283t
m
'
-
00002773
)
cos
( 0000277A
553 1)
0020
O 257 and
°
m
( 00 14 7A 0 214 3 ) si n 0 28 30}
177 )
e
a77o 2041)
00002773 ) cos 0 2831 ( 0000277A
0002743 ) si n 0 28 34
—
—e °W ( s 920 5031)
{
Wh ere
0075t 0 28 3
.
—
°W
o
7
,
-
-
-
9
d 71,
( 167) leads
1
1
an
s in
y
( 0 28 3t
7)
°
000274 3
(
)
002D 1+
si n
( 0283!
-
r)
i
1
( 179 )
—
T
L
ITY
DI ST URB ED MOT ION
B
I
S A
I n it ial
—I
co nd i t i ona
t a”
fl
to”
A
u;
q
q
of u ,
8 , be t he values
” and
w.
521
q
an d
C
5B3 B
00002773
69 20
0 00274A
- 00
033 3 A - 0 00883 B + 123 C + 1 204D
°
'
,
0,
t h en
0wh en
‘
'
'
‘
q
m
four li n ea r e u a t i on s are p rod u ced t o gi v e A, B , C and D in te r s of t h e i ni ti a l
Illust r a t i ons of t h e ot i on are gi v en in Fi g 24 7
valu es of co p onen t s o f t h e di s t u r b an ce
for t he fo u r si p le i ni ti al di s t u r ba n ces i n u , w, 9 and 0 For t h e fi rs t of t h ese , wh ere
=
=
W
h
0
u 1=u
wl = 0
M
d
e
n
l
0
l
a nd
m
m
m
.
.
.
q
t h e values of A
D
are
l ool a ,
A
'
B =3
c
D
h
e di s t u r bed
n
s
f
o
r
t
o
re
s
s
i
e
h
x
1
7
v
es
e
a
n
a
l
c
a
l
t
i
i
t
9
p
(
)g
y
and
ee t i n g a h ead on gu s t
Th e co ple ted for u lae are shown i n
ot i on d u e t o
th e
t h e curves of Fi g 24 7 ( 0 ) were ob t ai ned fro
s u b s t i t u t i on of
Th e
m
th ese i n
m
‘
.
.
—m?
it
w
q
1
1 16
”5
7
4
si n
008
“
i t ‘s
008
0 28 3t
0 28 38)
0042 s in
‘
00028 1 008 o 28 3t
000104
cos
0 28 3:
a le
000070 si n 0 28 3t )
°
-
a le
m
u
si n
'
a
m
m
m m
.
“
W
( 182)
Move ent of a ControL—If an aere plan e be flyin g stea dily
u n d er gi v en con ditio ns an d t h e el e v a t or b e
ov e d or t h e engin e thr ottl e
a dj us t ed i t will b egin to
o ve to so e new con dition of e q u ili briu if
t h e aerop l ane is s t a bl e
Th e di stu r b ances of otio n ay th en be r ega r d ed
as t he differences b etw een t h e ori gina l s t ea d y
o ti on a n d t h e fi na l s t ea dy
otio n an d if s all can be co v er ed by t h e th eo ry of s all osci llations A
o ve ent of t h e el ev ator wi ll be d enot ed by p and a change of thr us t by
ed to
v ; t h e ch a ng es i n t h e f or ces an d
en t s whi ch r es u lt will b e as s u
o
be p ro p ortion al to p an d v Eq u a tio n (5) th en beco es
"X
Sin
1
8
3
we
003 00 9 + u X l
( )
wh ere no wo an d 00 still a pp ly to t h e o rigin al otion and t he firs t t wo
t er s are th erefo re z ero whi ls t a w an d 0 ar e t h e chan g es in t h e s t ea dy
t h e el eva to r
o v e ent p an d t h e thr ust chan g e v
otion whi ch aris e fro
T hr ee equ ations are obt ain ed whi ch d efin e t h e di stur b ances u w and 0 i n
Efb ct
of
the
'
,
.
m
m
mm
,
m
m
m
m m
m
m
mm
m
-
:
t er
m
m
,
,
s o
.
v
m
u
,
m fp
m m
.
.
,
an d v, an d ar e
x
+m
—g cos
—g sin
+w
u
Th e
.
.
.
m
m
m
M 4
»
u
w
,,
=
O
X
v
X
,
+y “ +
=
v
Z
0
Z
+ p ,, + ,
=
V
M
M
0
+y ” +
y
ol u tion of th ese eq u ations p res en t s
s
di fli cu lt y
no
m
0
X,,
Z.
M
'
,,
Xy
X
p XV i
Z
2»
l
—
v
M,
M
u
l
y
-
,
w
M
,
“
y s in 00
0
Z
,
M
a
J
X
x
,
l
p p
°
cos
v
v
,
g
s
i
9 n
0
00
0o
Z
aXp
p
.
all ,“
+v
Z
v
,
M
v
,
w
M“
,
l
w
— g cos 00 X
l ea ds to
u
-
'
an d
Z
a s
l
l
i
p “
-
-
M
y
M
v
,
522
AERODYN AMI CS
A PP L IE D
mti f t h
um
p tio th
w
m d
fo u n d for ch anges of el ev ator an d th rust
a t t h e old st ea d y con d itio ns p ersi st e d w hi ls t X ” X
on
ass
n
er e
easu r e
t h i s being t h e us u al assu m
p tion u n der lyi n g t he
et c
ca l cula ti on of d er i v a ti v es
Th e
t he
o
e aer Oplan e
on o
is
”
,
.
m
in
m
Exa p le
Use of Ele lor onl y
In ad di t i on to t h e d eri v a ti ves p re vi ous ly gi ven
t h i s s ec t i on i t rs necessary t o ha v e t h e valu es
.
'
X
in
to
-
.
ord er
a
to
calcu late
m mt
ove
en
of
the
“
l
I
=0
mt
e e
B
m
.
“
q
=
0
q
‘
and
t h ese va lu es t o get he r wi th
and D , w h i c h a re
A =9 5
an d s u ffi ce
m
i
qi
q
d is t u r ban ces
u , w,
an d 0 wh i c h
E u a t i ons ( 185) t h en lea d t o
en ary
t h e e le vato r
( 186)
q
e u a t i on s
( 180)
°
a
.
mi
ne
t he
o
v alu es o f
D = - 0 23 1o
°
mt i
( 187)
°
t o d ete r
C =0221p
B = ss ss
t h e wh ole
se rve
—0 58 1y
0
m q ti
mt i
th y
t mk t h i
u va len t
ar e e
A, B , C
( 188 )
As ca lcu la t ed , t h e
e
v a lu es o f u , e t c , refer t o t h e fi nal s tead y
o on
can be u sed re la t i ve t o t h e
ot i on b y add in g co ns t a n ts
a e
ori gi n a l s te ad y
o
e ni t i al d i s t u r b a n ces
u,
g
Th is was t h e p roced u re follow ed i n p rod u cin g Fi g 24 8 fro
t he ana ly t i cal
an d 8 ze ro
exp ress i ons
Cha nge of Ai rscr ew Th at onl y —I n o rd er t o gi ve X a v a lu e i t wo u ld b e n eces sa ry
to d e fin e v as so e u a n t i t y d epe nd i ng o n t h e pos i t i on of t h e t h ro t t le , v i z t he re v o
lu t i ons of t h e ai r scre w If h owe v er, a s i p le e xa p le be t ak en i t rs p er is si ble to wr i te
t o d e te r
ne
o
on
m
.
fr o
ons
e ua
.
m
.
.
mq
m
m
m
,
.
m
.
+v
.
m
m
= 8T
m
.
’
.
M =O
( 189 )
,
Si nce
wh e re 8T re p rese n ts t h e in cre ent of t hru s t w hi ch co ns t i t u t e s t h e d i s t u r ba nce
nen t di s t u r b a n ces a re
o
a nd t h e co
ot i o n was h ori zo n t al,
t h e ori gi na l s tea d y
p
m
m
.
ST
m
a nd a red u c t i on o f
a ri ly to a d esce n t
t h ru s t lea ds p ri
co r re s po nd i ng w i t h ( 1
90) i s t h e t yp i ca l
m
Th e di agra
Fi g 24 7 ( d)
.
a n d not
si
mp l
to
a ch an ge o f s ee d .
p
d i s t ur b a n ce
e
s
h ow n i n
.
Ds s c arr r ros
or
F108 247
.
AND
CAL C U LATI O NS
or
m
24 8 rr w s r nxr rxc r n n R e s u
L o xc rr v n x n Drs r v a s xs cs s
m
e
m
e
or
—
f
F
i
i
n
th
e
D
i
re
c
t
i
o
n
o
l gh t
Gust
Th e r es ult i s sh ow n i n Fig 24 7 (a ) t h e
agn itu d e of t h e i n cr ea se of
or d i n a t e s of w h i ch ar e p rop or ti on a l t o t h e
wi n d Sp eed , 111 an d t h e a b sci ssae t h e ti es i n secon ds a ft er en t er i n g t h e g u st
r
e
S
r i a ti on s o f g us t with ti
r
h
ee
h
r
d
lt
w
th
l
a
t
p
d
t
gh
e
i
ou
e
e
a
e
t
V
a
a
e
T
h
)
(
t o z er o i n l ess th a n fi ve seco n ds
a i r i s s ee n t o f a ll r a p i d ly fr o
u =u , a t
—o 5u 1 in n ea rly 10secs Th e recor d i s that
a n d t o con t i n u e i t s f a ll t o u
p ed oscill a ti on of i nsigni ficant a p litu d e a t t h e en d of on e i nu t e
of a d a
Th e v a lu e of w a t firs t f all s ra p i d ly sh owi n g a r ap i d a dj u st en t of a ngl e of
i nci d ence t o t h e new con ditions a n d i s acco p ani ed by a v ery s i ila r bu t
Th e incli n a ti on of
Opp os it ely d i sp ose d cu rv e for t h e a n gul a r v el ocity
t h e a ere p lan e a xi s t o t h e grou n d i s seen t o v a ry consi d er a bly , a nd t o h a v e
i ni u nearly a qu ar t er of a p eri o d l a t er th a n t h e
a xi
u
and
its
v el ocity , whil s t th at of w i s a h alf p erio d l a t er and 9 a l os t in p h as e Thi s
r el ati on constit u t es a ch ar a ct er i s ti c of t h e p h u g oi d os cill a ti on a n d a pp li es
t o t h e l a t er p a rt s o f a ll t h e d i ag ra s
Th e f a ct ca n b e d ed u ce d fr o
,
m
m
m
.
,
.
m
.
.
'
m
m
.
m
,
,
q
m
m
m
.
.
m mm
m mm
m
-
m
.
.
,
m
A PP L IE D AE RODY NAM IC S
524
os t e q u a lly
ge t s a n an gul ar v elocity v ery r ap i dly and l oses it al m
ra p i dly so th a t t h e an gl e of in ci d en ce h as a dj us t e d it self a t an ear ly s t age
t o t h e v alu e s u it a bl e for t h e res i d u a l p h u goi d osci llati on
—
A h or iz on t al whi r lwi n d i s t h e o nly
Di sturbance of Ang ular Veloci ty
mea ns of p ro d u cin g su ch an effect and coul d not con tin u e without p ro d u ci ng
p erm
Th e only v a li d d ed uctio n to be dra wn fr o ma
an en t inclin a tion
e ra p i di ty a n d will
d istur bance i n q
i s t h a t it wi ll be t a ken u p with ex tr em
a ll am
l eav e a p hugoi d of sm
p litu de
a er e p la n e
,
.
.
,
.
.
SECONDS
20
Fro
.
248
.
—D i s t u r ba nce du e
m mt
t o t he
ov e
en
30
of an e le v a to r .
—
Ch an ge of t hr ust i s t h e
oi Path
( Ae ro p la ne
s t a b le .
)
t e qui v al en t t o
515
”
is
Si nce t h e v a l u e of F
t hi s dis tur b ance whi ch is m
a inly p h u goi d
a
t
h
e
s
s
ss
o
n
re
e
ca
a
n
a
e
t
h
ll
h
lf
o
n
d
lyti
l
xp
i
how
th
t
s
ec
a
f
a
n
d
o
at t h e e
a
sm
a
n
a
o
n
sc
o
e
o
f
t
h
b
q
t
tio
i
s
t
s
p
hug
i
d
o
ill
ti
with
n
s
co
s
a
s
e
ll
c
a
n
i
n
o
su se u en m
a
se
n
c
d
s
r
f
t
h
e
o
e
n
p
lit
d
w
i
h
d
p
d
oth
g
itu
d
i
tu
bi
g
u
m
a
t
h
e
o
n
b
n
s
e
e
hc
u e
am
a n d on i t s ty p e
s iti v e
—
o
a
n
u
o
f
res
s
t
e
8
h
s
Fig
4
how
lt
gi
v
i
p
2
g
Move ent of the N ew ton
o
um
n
ro
c
o
e
c
n
t
h
w
n
d
s
rr
eS
o
c
o
v
t
l
v
to
t
i
ith
t
l
l
s
h
r
a
e
mo emen t o t h e e
p
r
e
c
a
a
n
r
a
a
e
s
i
s
r
e
w
d
l
v
t
d
w
T
h
ult
p
i
d
gul
v
lo
ity
n
o
r
e
a
o
for a r a n d t h e e
whi ch ra iScs t h e t a il and red uces t h e a ngl e of inci d ence ; t h e aero p l a ne
Disturbance
.
,
‘
,
m
n ea r es
.
.
.
—
ST AB I L ITY DI STUR B ED M O T IO N
525
m
m
di v es an d ga i ns sp eed
Aft er t h e da p in g of t h e oscill ati on i s co p l et e
t h e aerOp lane h as n o a n gular v elocity , a red u ce d a ngl e of i n ci d ence an
i n creased sp eed and a d own war d p ath Th e final otio n was i n di cat ed
by t h e s i p l er etho ds of Ch ap t er I I , but t h e p resen t r esu lt shows exactly
h ow t h e n ew st at e i s r each ed
.
m m
m
.
,
,
.
.
Di s w
ns a x cs s o r
L ATE RAL Mo r ro w
en ts follow ed ar e those a lrea d y d ea lt with
um
d e t ail wi ll th erefore be o m
itte d If t h e dis tur bances be
“‘
“
“
=
=
=
=
r
cc
v
ac
b
e
p
Th e
arg
,
an d
mu h of t h
c
e
.
the
valu es of a
,
b
—
A
A
.
p
gi v en by t h e rel ations
an d c are
L.
.
N, —AO
N,,
-
.
wh ere p rin ci p al a xes of i ner tia h av e been chosen so th a t E i s zero If A
whi ls t if com
p l ex
be rea l t h e v al u es of b/a a n d c/a a re obt a in e d from
t h e p r oced ure i s th a t f ollow ed in co nn ec tion with lo ngi t u di n al s t a bility
p l ex roo ts a n d th er e fore for
Th e exp r ess i on s for v p an d r for co m
ewh a t d i fferen t form but are es se n ti ally
will be gi v en so m
oscill a ti ons
il ar to those gi v en i n
si m
.
,
,
.
,
,
,
cos
( H ow
i
e
m)
—
a
P3
2
+
Valu es of 4) a n d a];
Th e
valu es of p g
,
as r e
}
qui red ar e d et er m
i ne d from
t h e rel a ti ons
t
e
c
pg
,
m
—
k
t
(
Si "
008
z
Vs
si n
.
,
are
e
as
e
n
i
v
b
low
g
hA
L
,
MAC
526
APPL IED
Exa
mpl —Th
e
e
d e ri va t i ves
AE RODYN AM IC S
md t
ass u
e
o a
l
p
p y in
a
a rt i cu la r ca se a re
p
00 154
l
L,
A
1
—100
L,
A
n,
Th e
solu t i on o f
=0
n,
t he bi
00 157
A,
qd
ua
=0
ra t i c
A,
l
n,
.
q
b/a
whi ls t fo r A,
Fo r t h e
co
mpl
p
,
l
,,
e xp ress i ons co rres
v
—
0
8
l ( O OI I OA
—0 00338 A
‘
cond i t i on s
t h en
mm
0
De
m
‘"
‘
—
1
95
000298 B )
000
s in
'
Illust r ati ons of t h e four si
t h e fi rs t of t h ese
m
pl
v,
.
0014 906
.
D
e
t yp es
( 0 984!
y)
y
t he
'
r,
A
si n
p l , r ,, an d e, be
C
1)
A
0008 328
0 00376A
000324 A
00 1103
0004 39 8
000103A
p,
va lu es of
)
CO B
=
y
—Let
v,
-
00 157:
0 246 De
‘
'
5fi '
y)
v alu es of v,
0000690
500 14 90
004400
p,
r a nd
=
h
w
e
n
o,
t
a
2 24D
1)
0 24 5
0271D
'
200
'
of in i t i al d i st u r ba nce are
4
9
ve
n
i
n
F
i
2
i
g
g
.
.
For
=
n o
=v
are
A
B
C
D
ua
06
000338 B ) s in
cos
an alyt i ca l e xp ress i o ns
q ti
000213
-
v3
B si n 00844)
0003 24 3
'
‘
( 0 00298 A
e
t :
O OI I OB )
‘
in
—0000144
'
c
Th e
u ns t a b le.
000332B ) cos 0084 1
0003 76A
( 00033 2A 0003703 ) si n 00840}
‘
t he
is t h e re fo re
-
f
and
valu es
wi t h t h ose for longi t u d i n al d i s t u rb a nces ,
n
n
d
i
o
g
p
O 984l
cos
p
I n it i al
A,
'
80
c a
00000170
00 164
00 125
I"
Usi ng
an d
ae ro p la n e
n-
no
/
000069
A,
e x roo t s
0
for sta b i li t y gi ves to A t he
A, a nd A.
Th e fi rst v a lu e o f A is posi t i v e a nd t h e
Fo r A. 0 0l57 e u a t i o ns ( 19 2) gi ve
'
w“
0l
—00020501
for t h e di st u rb a nce
can
be
ob t ai n ed
by
u si ng
t h ese
v alues
ons
m
Ii 3 an d 7) be u sed t o d eno t e t h e
Effect of th e Move ent of a Control
o v ed and th ese angl es
a n gl es t h r ou gh whi ch t h e a il e ro n s a nd r u dd er a re m
en t s a n d for ces a re
om
all q u a n titi es for w h i ch t h e m
b e res t ri ct ed t o be sm
o tion
p ro p or ti onal to t h e a ngl e of ail eron an d r u dd er t h e effect on t h e m
-
.
,
528
A PP L IE D AE RODYNAM IC S
—
T
A
B
I
L
ITY
DI ST UR B ED MOT ION
S
529
— Th e
rol lin g i s
i gust striki ng th e Left Wi ng
Rolli ng du e to Un
a n d l ea v es t h e ae re p la ne with
with grea t ra p i di ty ( Fig 249
b a nk ; s i d esli pp i n g th en occurs a nd t h e a erop l an e fin i sh es with
-
.
,
t u rn
for
l a t era l gus t
t pp e d
a sm
all
a sp i ra l
s o
—
Xawi ng due to a Gust whi ch strik es th e Tail from
h e e ffec t
the Righ t
o f tu rni n g i s t o i ncreas e t h e v el ocity of t h e l eft wi ng and i ncrease i t s lift
um
a xi m
c a us i n g a b an k for a right h a n d t u r n
Th e ba n k r each es i t s m
i n a b out 2 secs
U n d er t h e acti on of cen t ri f u gal forces t h e a ere plane
b egi ns t o si desli p t o t h e l eft a nd 1m
til t h e b an k i s su ffi ci en t t o r ev erse thi s
e ffec t t h e ch a n g es ar e r a p i d
Th e aer op l a ne r ea ch es an u ns t a bl e turn
a g ni tu d e
o f a pp r eci a bl e m
ent s a re
in ary ra p i d m
o v em
Efi ect o! a Su dden Bank — Th e p reli m
t o war ds t h e fina l s p i ra l t ur n of l arge m
a gn it u d e ; t h e a er op la ne with i t s
l eft wi n g u p b egi ns to si des li p in war ds a n d to t u rn to t h e r ight
as
a
.
"
,
-
.
.
,
.
.
,
.
20
F
m250 —Di
In
all ca ses
.
.
s t u rb a n ce
du e t o t he
m mt
ov e
en
30
S EC O N O S
o f a ru dd er
o nly i m
p or ta n t d i st u rb ances
a r e th ose of t h e u ns t a bl e Sp i ra l t u r n
i ti ng
t he
m
( Aerop lan e
.
ex s
at
u ns ta b le
t he
.
)
en d
of
10 secs
Fig 250 to be
Th e Efl eot of 8. Move ent of th e Rudder i s seen fr o
t h e i niti a tion o f a sp ir al tur n , an d t h e con t rol of su ch an aerop lan e in v ol v e s
Th e or din at e of t h e cur v es i s ar bit rary,
t h e p ilot as an es s en ti a l f eatur e
but i s p r op ortiona l to t h e ov e ent of t h e ru dder
.
m
.
m m
.
.
.
Ta n MATH EMA TI C AL Ta noa v o r D i s r u n s s n Mo r t o n A ND Co u r a o r WHEN
e Ca u s e s a ns Va a ra nnn WI TH Tu rn
r a n Di s r v a n
m
.
genera l theory of t h e sol u tion of di derent i al equati ons of t he typ e
a ll oscill a ti ons show s th a t t h e
s of t h e st a b i li ty of sm
met with in p robl em
p l e di stur ban ces coe xi st as though in d ep en d en t of each
e flect s of two si m
oth er It i s th er efore p er m
i ssibl e to r egar d t h e new p r obl ema s a sear ch
en t ar y d i s t u r b an ces du e t o gus t s
b er of el em
eth od of a ddi ng a n um
for a m
u
r
a
t
a
n
o
m
cc
e
m
a
c
o
v
t
o
t
l
s
whi
h
ti
n
r
o
f
t
h
e
c
s
o
m
e
n
e
or t o m
y
y
For any aerop l a ne su bj ect t o di stur ban ce it h as al r ea dy been sh own
s of t h e ty p e
th at t h e m
ber of t er m
otio n ca n b e exp ressed i n a n u m
Th e
'
.
.
1
4
‘ z‘
Aa
B
t z‘
e
Ceh
‘
‘
De
t
t
‘
AERODYN AMI CS
A PP L I E D
53 0
wh ere
m
ean i n g of ch an g e of v elocity al ong t h e a x is of X
h as i t s u s ua l
Th e coef fici ent s A, B C a n d D are, i n g en era l li n ear fun ctio ns o f t he
I f, for ins t a nce, a h ori zon t al g u s t of v el oci ty u , is
di stu r bi ng ca u ses
It i s con v eni en t t o
enco u n t er ed A, B , et c , are all p rop orti ona l to u l
ha v e a shor t ene d not atio n for (206) an d it i s p re p osed to d enot e it by
u
.
,
,
.
.
,
mi l
Si
ar
ly
an e
.
u
2
0
7
)
(
w
(208 )
xp ressi on
m
i s writt en for t h e effect of a h oriz on ta l g us t o f a gn itu de u , a t t =0 on
Si il ar e xp r es si ons foll ow for and 0
t h e nor a l v elocity w
r
W
a
fi
u
a
n
d
ll
a
e
d
e
n i t e functio ns o f t for a gi ven
( ),
( ),
p l es h a ve b een gi v en i n Fig 247 (a )
aerOp la n e, a n d exa
It i s now n eces sary to define a secon d ti e r an d to exp l ain i t s r elati on
et c , gi v e t h e
agnitu d e of a di st u r b an ee t
Th e exp ress i ons
to t
It will be e vi d en t tha t i n a
s ecs a ft er t h e di s tur bin g ca use o per a t e d
e t will be t h e su
of t he
su ccess i on o f gus t s t h e fi n a l d is tur ban ce a t ti
es , a n d r i s use d to d i s tin gu is h
e fi ect s o f d i s tur bin g caus es a t a ll p r ev ious ti
t h e ti e at whi ch t h e di stur bance occ urre d
Th e exp ressi on ( u u ),
e t fro
a d is tur ban ce a t r , t he
rep resen t s t h e d is tur b ance fact o r a t ti
By
agn it u d e of whi ch ca n b e r ep resen t e d by t h e el e en t
ea n s of t h i s d e fin iti on i t will b e s een th a t j (r ) r ep rese n t s a co n ti n u ous
di stur bing ca use o v er any ran ge of ti e wha t ev er , an d th a t t h e r esi d ual
di stur b ance in u is
m
q
.
m
m
m
m
m
m
m m
q
.
.
.
.
.
m
.
.
'
m
‘
.
m
m
m
m
o d of ho ri zont al v elocity as obt ai ne d fr om
om
an an em
et er
es l a bori o us
an d i n th a t ca s e t h e differ en ti a tio n t o fi n d f (t) beco m
The
ay be av oi d ed by a pa r ti al in t egra ti o n whi ch
n ecess ity fo r t h e o p era ti on m
l ea ds t o
f(t)
my b
a
e a rec r
,
’
.
(
u
Th e differen ti ation of t h e kn own alge b rai c
seri o us d iffi culty , a n d t h i s l att er for
is in
form
er
m
“u
h Jt
-
r
d
l r
'
xp r ession
p resen t s no
any w ays p ref era bl e to t he
m
e
m
.
of w qan d 0 t h e in t egr als at t h e li its are zer o for a
horiz on ta l gus t wh il st for u t h e qu an tity (u u ) o=l and f(o) i s zero if con
t i n u i t y i n u i s ass um
ed as an i ni ti al co n d iti on
I n t h e l a t t er case as
e
t
e
i
n
c
u
a
i
s
t
h
c
n
e
t
t
h
g
v
lo
ity
of
wi
d
o
v
g
r
ou
d
ti
h
e
n
er
h
e
a
t
d
n
m
e
t
a
n
j( )
i s t h e ch an ge of v elocity of t h e aer o p l an e rel ati v e to t he ai r i n t h e sa e
in t erv al of ti m
e it ca n be seen tha t
In t he
cas e
,
,
.
,
m
,
z
is t he
h
g of
c an e
t he
v elocity
of
t he
aero
(211)
p l an e rel ati v e t o t h e groun d
.
582
e b as e
ti m
d ott ed cur ve rep r esen ts
velocity
Th e
re
AE RODYN AMI CS
A PP L IE D
as o r
v ers ed
for
d in ate
con
and
on a
cov ers
exce
v en i en ce Wh en t = 4 9
secs
p t th a t t h e
i
—
f
t
i
whil
t
h
v
l
i
M
Q
f( )
g m t Th p d u t of MQ d MP th
t h i t g l of
d i p lott d
QM i
ti m i m
il ly bt i d o m
pl t t h
t th
of
s
r
.
,
s
l
s ca e of 1
M Fi g
at
.
,
.
40
secs
.
been
h as
val u e
251, t h e
MP owin g to t h e
s
a ue o
e
s
,
p er i o d of a bo u t
a
i l
Sp ec a
p resents an el em
en t o f
e n e ra
an
s
e as
Po in t s
l
l n t h e fi gur e b elow
o
e e
e l ow er cur v e t h e ar ea of
ar
a ne
c
es s
a
o
er
whi ch r ep resen t s t h e t ot al di stur bance a t 60 secs d u e to p ass a ge
thr ou gh gus ty air Fur th er cur v es are sh ow n for oth er ti es to e p h asi ze
ar kedly on t h e tim
t h e fac t th a t t h e effect of a gu s t d ep en ds m
e whi ch h as
el ap se d si n ce it was enco u n t er ed
A l arge effecti v e in crease sho wn by F
6 7 secs a t H an d h as a co ns i d era bl e n ega ti ve
a t 60 secs beco m
es z er o at 5
v alu e a t K at
secs
Th e tot al areas show so m
ewh a t s i
i lar
ch ar act eri sti cs
A r ep etition of t h e cal cul ations us in g
( ag)g an d (a d) . woul d l ea d
an d 0as h
ct i ons of ti
to t h e det erm
in ation of w q
e
I n efl ect ( see foo t
thi s h as b een d one an d t h e ore i m
p or tan t i t e s ar e s hown i n
n ot e)
arra n e
en
e
.
ro
an
c
en re
.
,
”
.
.
m
m
.
m
,
.
.
.
.
m
m
,
,
,
ti
o
m
e sca le , an d
md t K
t i mb
wu
a e a
m
0beern t ory ,
ew
mm
u e of
m
.
'
m
Dines Record
u si ng a
m
u p ied fl O sec
fiO soos
n t he
Si nce n o b et te r infor a t i on is a vaila ble E
wi n d sp ee d bei ng 20 ft s
t h rou gh t he ai r are t he
e t by an aero lane t r a ve llin g SO ft
t ha t t h e gu st s
en t
reg is tered d u r ing t h e passage 0 80 ft o f ai r ove r t h e o bser va tor y i nst ru
p art repr od u ced on a
T
m
.
o
c
m
.
m
.
.
Th e d ot te d
cur v e of
a sa vi ng of la b ou r
Fi g 251 was
.
was t h ere b y
q
not e u al t o
effec t ed .
It
re p resent s
cos
e
m
By a ch ange of t h e t i e epoc h t o t he po i nt
i t is clear t hat t h e u lt i plyi n g fa ct or
m
g
ui lds
t he
b u t to
-n
3
mmt
g Ane
, t he
e u
a
“
mg
m
.
o
u
sa
t h ose
e as
.
p ort i on
only Of
i t» si nce
valu e of
03283 !
( 212)
wh i ch t h i s d ot te d li ne
at
o
cu t s
t he
axi s at
ti
m
e,
—
Ice W 7“ si n 02 838
is
si
mlt
u
a n eo u sly app lied ,
e xp ir a t i on of a
q
u ar t er of a
1
: bei ng
per i od
.
kn own
Tw o cur ves
q
00t
a
—
w
yen
[
o
e
u al
e
t o t he
valu e of e
-
W 'I Q
at
t he
rep resen t i ng
" coo
oessu
)
'
r ar
0
f
l ew
"
( 213 )
0
fun ct i ons of t in t hi s w
ad e for t h e ra i
I n ad d i t i on ca lcu la t i ons were
p
su fli ci en t d eg ree o f accur acy t h e i n t egra ls
we re
o b t a i n ed as co nt i nu ou s
m
a
y
r
.
oscilla t i ons , a nd
“
i t was fou n d t h a t t o
a
)e
I“
7
)d7
t h e valu es b ein g 0 17?
we re p re po rt i onal t o
an d 00308 1
17 ) resp ect i vel
ot i on o f t h e ae rop lan e re rese n t e d b
eans t h a t t h e
p
y ( 2 14 ) i s so rap
t u r ban ce i s a co pa ra ti vely e xac t co u n te rpa rt of t h e net
Th e c u rv es of d i s t ur b a nce i n u , w, 9 a nd 0 ca n
o b ta i ned fro
t h e abo ve cu rves by
ad di ti on i n v ar i ou s p ro o rt i ons d ete r
i n ed b y t h e ar bi tr ary co nst a n t s A, B , C an d D
p
m
m
m
.
m
m
.
ST AB I L IT Y—DI ST UR B ED
MOTI ON
Fig 252 One of t h e q u an titi es esti m
a t e d h as b een
a b ov e t h e groun d a n d i n v ol v es t h e r el a ti on s
.
t he
.
53 8
vari ati on
h eight
of
,
u 09
li ==
mh vi
a
n
g
m
p i l ch aract er si nce i n t he st ea dy oti on t h e a xi s of
X i s alo ng t h e d i r ection of fli ght an d i s hori zon t al
Descri pti on of Fi g 262 —Th e u pp er c ur ve sh ows t h e wi n d r ecor d fr om
O t o 60 seco n ds
Th e aerop l ane was su pp os ed to h a v e a fl yi n g Sp ee d of
80 1t s in i t s st ea dy st a t e an d to h av e thi s v elocity r el ati v e t o t h e a i r
at t =0
I t s v el ocity o v er t h e gr ou n d was th en 60 i t s an d t h e win d
Sp ee d of 20 i t s
oti on Th e f u ll cur v e
a gai ns t
t h e a erop l a ne s m
a r ke d
v ari ati on of a i r sp ee d was cal cu l a t ed at t h e p oi n ts i n d i cat ed
wh il s t t h e d ot t e d cu r ve sh ows t h e v ari a ti on of grou n d sp ee d Th e di fler
ence b etw een t h e cu r v es sh own by t h e sh a d e d a rea i s e qu a l t o t h e v ari a ti on
o f t h e win d fr o m20 ft s
i s t h e o r di n at es ar e e q u al to th ose of t h e
u pp er di agra m It wi ll b e n oti ce d th a t t h e in er ti a of t h e a erep lan e i s
gr eat enough t o av erage out all t h e m
ore r ap i d ch an g es of wi n d sp eed a n d
sh ows t h e a d v an t ag e of Sp ee d of fl ight as a
eans of p r o d u cin g av er ag e
s t ead i ness
It will be seen th at v ari ations of sp eed of 1 12 ft s are
i n di cat ed and th ese ay be consi d ere d t oo lar ge t o com
e withi n t h e defini
tion of sm
all oscill a tio ns
Cert a i n o th er a pp roxi ati ons will h av e b een
n oti ce d by a car efu l s tu d en t whi ch co u l d b e m
e t by m
ore ri gorou s t rea t
men t if d esi red Th e ad v ant ages of t h e p resen t metho ds ar e how ever
th ou ght to be su ffi ci ently gr eat t o w arr an t th ei r u se
Th e cur v e
b du e to i nclinati on
ar k ed u OO r ep res en t s t h e r a t e of cli m
of t h e a xi s of X whil s t w s h ow s t h e ch a n g e of n orm
Th e
a l v eloc ity
or d i na t es of t h e Sh a d ed ar ea th en r ep resen t t h e t ot a l r at e of cli
b h
I n t egr a ti on with r esp ect t o t th en l ea d s to t h e l ast cu r v e for ri se an d fall
On t h e wh ol e t h e aer op l an e gai ns h eight t h e m
umgai n b ein g 40f eet
a xi m
It is
i n one p l ace a f all b elow t h e origi nal l ev el of a bo u t 15fee t i s sh own
p ossibl e th a t t h e aerop l ane shown w ould j u st be a bl e t o l an d itself as t h e
v er ti ca l d own w a r d v elocity d u e t o t h e g u s t s i s n ot m
or e th an 5 i t s
t h e for
a S ec a
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TH E EFF EC TS
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or
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MO V E ME NTS
u ou s
USE
OF
Co n r a o ns
,
n e ce s s an v r o C O U N TE R TH E
A ND
mC
s
EF F EC TS
AL C U L A T I O N
or A
Gu s r
p robl emt o be att acked will be t h e fi n d i n g of a n el ev at or
o v em
i na t e t h e e ffect s of
en t of a co nti n u ous ch ara ct er w h i ch will eli m
an i s ol at ed gu st
etho d an alogou s t o th a t f ollow i n g i n a dd i ng
By a
ot i on of a n
t h e e ffect s o f gu st s it i s cl ear ly p ossibl e t o ca l cu l a t e t h e m
Th e
oti on of t h e el ev a t or
aer o p l an e w hi ch resu lt s f roma p res cri b ed m
p r obl e n ow consi der ed i s t h e conv ers e of t hi s si nce it i s p rop osed to fi nd
t h e el ev at or
o v em
en t whi ch corr esp on d s with a p r escrib ed aer Op la n e
motion
o ti on th at ch an ges
It h as b een seen i n t h e d i scu ssion of di st u r be d m
i sol at ed d i stu r bi n g ca u ses i n any of t h e
a nd 0 whi ch ari se fr om
in u w q
Th e firs t
m
.
m
m
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A PP L IE D AE RODY N AM IC S
534
50
Fl o 252
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-
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s ccou os
a na t u ra
40
l wi n d
( AerOpla ne s t abl e
586
A PP L I E D AE R ODY N AM IC S
t
g
t h e p ar ti cu l a r di s t u r b an ce d u e t o p utti n g o ver
u s t a l s o he
t h e el ev a t or i s z ero wh en t
0 th en t h e ra t e of ch ang e of 2 m
a ll an d it i s t h en
ay b e sm
I n any p racti ca l case t h e r a t e of ch ange m
zero
ov em
en t w h i c h i
in e so m
it to t h e el ev ator m
e li m
n ecessary to d et erm
w hi ch
p erm
itt e d Thi s i m
p o ses so e slight lim
it a tion t o t h e v alu es of
ca n be p r o d u ced
If du e to a h ori zont al gu st a d i st u r bance E h as b een gi v en t h e fi nal
moti on is
If t he
h
ra e of c an e of
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h
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ll
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)
(
—A sol u tion of thi s equ a tion can a l w ays
Solu ti on of Eq
uati on
b e f ou n d by a p r ocess of t r i a l a n d err or ev en wh en li m
it a ti ons are gi v en
t o the m
oti ons o f t h e con t r ol s
I n so e cases an a nalyti ca l e xp r ess ion
in a ti on of (220) wh en E O a n d E a n d 3 ,
can h e f oun d for an exa m
o f exp on e n tia l
h av e t h e formof (216) s h ow s th a t P(i ) m
u s t b e a su m
s of t h e ty p e
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Assu i n g t h e r esu lt i t wi ll b e sh ow n th a t t h e exp ressi on i s corr ect
a n d th at A ,
D 1 a n d K 1 K 2 a n d K 3 can b e d et er in e d
Writi n g d own t h e exp r essions for E“ an d E “ i n accor d an ce wi th
es
21
6
gi
v
( )
m
‘
,
3
;
for t h e
mm t
o
en
t
is:
h
A26
g
tim
e 0
m
ag
m
( 222)
e
ti
at
.
w
!
s
e
v er ti cal v el ocity
ra e of c a n e of
at
w
e
B
+ z
Al 1
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e
t du e
to
l v at or
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(223 )
e
v erti cal v elocity at ti e t du e t o a h ori zont al gu s t at ti m
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Th e t er s of e q u ation (220) a re n ow Sp eci fi ed a n d t h e in t egra ti on
p r esen t s li ttl e di cu lt y Th e result i s to obtai n
m
for t h e
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K1 A
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( 224
6
as a n
e
i d en ti cal
l tio n
re a
.
Th e
ffi ci en t
coe
x n‘
of c
in th e
s
i d en tity gi v es
t he
q u ati on
K1
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+
K i —X
2
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10 5
l
ST AB IL ITY— DI ST UR B ED
537
M O T ION
“
im
i l ar e xp ressi ons foll ow fo r K 2 a nd K 3
I t will th en b e seen th a t
K 2 a n d K 3 are d et erm
i ne d by t h e s ol u ti ons o f t h e cubi c eq u atio n i n K
an d t wo
K 1,
s
.
B2
A2
I<
Eq u a ti ng t h e
A2
-
coefli ci en t s of e
“
A“
B3
D1_
J
r
(E
D1
A1
K 7 A2
‘
B1
Al
Al
K2
Al
K2
B1
C1
”
A2
K,
A2
K2
m
pp aren tly
,
K — A4
(224) gi ves
N
whi ch t o
fi v e e q u a ti ons fro
A 1, B 1 C1 an d D 1
h i d in g t h e e qu a ti ons
s een t h a t th ere i s ob t a i n e d t h e exp r essi on
or a
L
’
-
(
A1
C2
.
K2
d et er m
i ne t h e four qu antiti es
togeth er h ow ev er it will b e
,
thi s i s oft en in t r i n si cally sati sfi ed In th a t case A,
min ed I f E3 be zero th en D , i s zero and t h e additi on
s ti ll r equi r es th a t
an d
.
.
A2
,
,
(228)
be d et er
D 1 can
of t h e e quati on s
olution 18 t o b e p ossi bl e
u s t all b e n ega t 1v e 1f t h e aer Op lan e 18 to
Th e v al u es of K 1 K 2 and K 3 m
b e p erm
an ently co n t roll a bl e oth erwi se t h e el ev a t or a ngl e will i ncr eas e to
t h e p erm
i ssibl e li m
it and fail u r e wi ll th en occur
e p hys i cal i d eas ill u st rat e t h e v alu e of t h e r est ri ctions (228) a n d
Som
It i s for exam
p l e not p ossibl e to eli m
i n at e all t h e efi ect s of a h ea d o n
u st of i ni ti al am
p
litu
d
e u l for E 18 th en zer o an d t h e v a l u e of A+B +C+D
g
i s un ity
It i s cl ea r th a t n othin g sh ort of an infini t e force could n eut rali se
e o bj ecti on d oes n ot
Th e sa m
t h e assu m
ed i n st a n t a n eous i ncr ea se i n c l
er of th es e it i s r ea dily seen th a t t h e
app ly t o W q
a n d 0 b u t for t h e f orm
am
en t w oul d be p r ohi biti v e i n t h e i niti al
p li t u d e of t h e el ev ator m
o v em
st ag es
p t ed woul d b e th at of p ro d ucin g a change
Th e Op erati on a tt em
of d own lo a d on t h e t ail e qual to t h e ch an ge of u p loa d on t h e wi n gs q
an d
m
ay b e gi v en a l ar ge v al u e
a
re
e
r o a s a res u lt of t h e gu st whil s t
b
th
o
z
q
q
by u se of t h e el ev at or Eith er qor 0 i s th erefor e a s uita bl e qu an tity
for com
p l et e eli m
i nati on by t h e u se of t h e el ev ator In rel a tion t o t h e
figu r es p rev iou sly gi v en for an el ev ato r it a pp ears th a t a soluti on whi ch
lea d s t o t h e eli m
i n ation of 0i s
if a
s
.
,
,
,
.
'
,
,
,
.
.
,
,
.
.
.
,
.
.
K1
A1
th at
wh en t
so
l v t
gr ea t
the
is
K2
0
”
B1
01
t t
fr o
e e a or s a r s
.
Th e
con
—O 192
'
=0
mi t
d iti on
s zer
K3
D1 :
o p osition at ti m
e t an d r et u rn s th er e
e li ttl e d i fficu lty i n
i n t ro d u ces som
APP L IE D AERODYN AMI CS
58 8
i nt erp r et ati on a n d a ppe a rs t o i n vol v e
di ti
mental co nsequence so th a t in t he case for wh ich
i s p ro b a bl e th a t a fu rth er i d en ti ca l r el ati on h ol ds
t he
,
Approxi
mt
Solu ti on
a e
0!
q
E
uati on
on
con
—
m
(
l
b tit ti
TlM E
FI G 253
.
se of
t he
n
da
it
n ec ess
S ECOND S
S ECO N DS
elev a t or
mo v m t i
fu
to
eli
mi
na t e a
di s t u rban ce
( Aerop la n e
.
s ta
ble )
.
of t ri a l an d error bu t p r es en t s no seri ou s
I t h as al rea d y b een p oi nt e d ou t th a t equa ti o n (217) p ro v i d es
di ti i cu lt i es
its
ea ns o f ca l cul a t i ng t h e d i s t u r ba nce d u e t o an el ev a to r wh en
the m
movemen t is kn own t he p rocess of t ri al an d error assu mes a cu rve
e a r bit r ar y li m
it ( 1 5i n Fi g 253 ( b) ) an d
e b eginn in g a t so m
for a s h or t t i m
otio n
thi s su fli ces fo r a ca l cul ati on of t h e m
Th e r esu lt is co m
p are d w ith
a ll s t ep s
oti on it i s d esi re d t o r ep ea t a nd corr ec te d accord in gly If sm
t he m
e ar e t a ken a t t h e b e gi n ni n g t h e ca l cu l a ti o n will b e w ell es ta blis h ed
o f ti m
en t
o v em
Th e r esu ltin g m
or e r a p i d ly i n t h e l a t er s t a g es
a n d ca n p r oce ed m
t he
l v tor
—U
IO
a
e e a
e
en
»
ity fo r d ea l i ng
e thod
of a gr a p hi ca l m
Th e p r ocess of fi n d i n g
Th e
.
IO
—B 1 as
.
wi th p arti al eli m
i na ti on l ea d s t o a s u s u on
for t he exa ct a nalyti cal exp ress i ons j us t gi ven
TIME
A1
s one
,
.
'
.
,
.
.
,
.
APP L I ED AERODYN AMI CS
d i stu r b an ce cont em
p l at ed (at p resen t all di st ur ban ces ar e co n si d ered
a s i sol at ed a n d s ust ain ed i s a c h a ng e i n t h e wi n d
th
e
n
£1 £2 £3 a n d g,
)
are all z ero an d t h e a er o p lan e u lti m
a t ely settl es d own to t h e sa
e st ea dv
otion r el ati v e to t h e new wi n d If t h e di stur b an ce 18 du e to an in t ernal
cau s e su ch as a
en t of t h e el e v a tor w e h a v e
ov em
0 q
w==
wh en t =o Th e v alu es of a B 7 an d 8 ar e foun d as in di cat e d
ua
eq
an d fi nally to sati sfy t h e i niti al con d itio ns u st gi v en
ti ons
j
I f t he
,
m
,
,
,
m
,
m
.
,
.
,
m
,
,
,
,
si n
3
5
2
(
)
215
3 + 171?
ot ations in t h e wi n d are assu m
e d to o ccu r { 3 wi ll be z er o
It will b e seen from
e q u atio n s
tha t t h e p h ase di fl er ences
an d 0 ar e i n d ep en d en t of t h e v al u es of
b etw een t h e oscill ations mu w q
att er wh at t h e n a ture of t h e d i st u r b an ce t h e m
oz a n d 3 a n d n o m
o ti ons
an d 0 wi ll f oll ow each o th er 1n t h e sam
e o r d er with t h e sa m
in u w q
e
p h ase d i ffer ences an d t h e sam
p litu des T h i s rela ti o n betw een
e rel ati v e a m
t h e o scill a tio ns can b e cl early seen fr o mt h e cur v es of Fig 24 7 s in ce t h e
oth er t erm
s ar e v ani shi n gly sm
all a t t h e en d of 2 secs
‘ ‘
s in c
can b e d i vi d e d i n t o two p art s t h e r el atio ns b etw een
Th e t erm
t h e d i stur b an ces i n u w qan d 0 for t h e p ar t s b ein g in d e pen d ent
of y an d 8
Th e rel atio ns b etw een t h e os cillatio n s j u s t ref erred to will be found
l at er t o sim
p lify t h e an alysi s of m
otions du e to an el evat or It i s cl ear
a ti on all si m
th at to an exceedi n gly high d egr ee of a pp roxi m
l
d
i
t
b
i
e
s
u
r
a n ces
l
e si m
il ar d am
p ed osci ll ations of p h u goi d ty p e
r ap i dly t en d t o b eco m
il ar p or tion s li m
it ed to a ti e of
a n d it a pp ears th a t only t h e d i ssim
p assin g
i n t h e wo r st cas e n ee d Sp eci al t r eat m
en t i n
a bo ut 5 s ee s
f r o mv ari ation of v erti cal v el ocity say to th a t of v ari atio n of fli gh t
sp ee d et c
ove ent
The elevat or
) having been ch osen so that there i s no
Un l ess
r
.
'
.
,
,
,
,
,
,
,
.
.
,
.
x
,
,
,
,
.
.
,
m
,
,
.
mm
.
,
resu lt ant ver fical veloci ty ,
due to g ust
,
m
it i s
and elevator .
now
Th e
,
desired t o flnd th e ch
m
g e of fli 8ht speed
v alu e i s
m
j
'
u.
+
es se s
but for t h e general p robl em11 will be r ep l aced by
li near com
bin a tion of t h e v ari ations u w q
an d 6
F r ome qu ati ons (23 1)
(284 ) t h e v al u es of E!
b e e xp resse d r esp ecti v ely as
2
,
,
al
3
52
and
u
e
m
’
M
«
a
'
cos
S
fi)
b
,
c ,c
si n
si n
cos
wh ere
’
( kt
’
'
di ffers fr omE
cos
in
3,
b
a,
(
'
( Id
m
wh ere
?
.
“
an d
3
,
any
clear lv
( 23 7)
c.
f ,”
.
a
'
i
E is
my
B,”
an
sin
2 3,
"
( l sg
( 23 8 )
0
e di ffer en t li n ear fun cti on
bein g som
of u
,
w,
q
ST AB IL ITY—DI ST UR B ED
MOTIO N
54 1
0 ; t h e v a lues o f (1 1 fll y , a n d 8 , a r e th e r e fore t h e s a
'
'
a n d E , but a , b,
e t c , h a v e be e n ch ang e d t o a , b ,
et c
a nd
,
,
.
l
~a
l
a
v al u e of
t he
b
E
with
a so
9”
mwh
e
“
a
cos
t si m
il ar
a
_S in
b eco m
es
s
9
;
:
'
e
or
'
.
2
l
mf
7
( kt
s,
n
f“
( 23 80 )
m
'
)
p res si on
n
,
”
M
1
2
3
+
y“ , )
( 23 9 )
.
for
ex
1
Th e
v al u e
in t egra l
whi ch i s t h e d i stur bance
o f v erti ca l v el ocity d u e to t h e el ev a to r
in e d
h as alrea dy b een d et er m
n
i
s
w
Fi
o
r
It
p
op osed to m
ake t h e in t egra l
( g
0
d ep en d on t h e kn own in t egr al whi ch for brev ity will so m
eti m
es b e
r ef err e d to as j (t)
I t s v alu e 1n t he co m
p l et e not a tion a bov e i s
t he
Of
,
[
.
,
.
—
0,
'
a
J
b'
1
6
2 0 v
My
be
810
v
)
3, + 11}
+1
w
e“ )
m
)
h
c
a
t
g
l
en
g ti v e i n thi s
It h as alr ea dy been p oin te d ou t tha t a ft er t h e l ap se of a sh ort i n t er v a l
e a ll cu r v es for i sol a t e d s u st a in e d d i s t ur b a n ces a r e si m
of ti m
il a r i n gen eral
p litu d e of t h e osci lla tio n Ch an ges
char ac t er b u t d iffer in t h e p ha se an d a m
p litu d e wi ll th erefore b e i n trod u ced su ccessi v ely i n to
in p h ase a nd a m
t h e ex ressi n
Th e p h ase difference b etw een t h e oscill a ti ons
_
p
o
in
a nd
i s seen t o b e 11
e p h as e of
n or i s e q u i v al en t t o a ti m
an d
i
cann
ot
,
l
H
{
ne a
.
,
'
'
”
-
,
(_
n
)
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E
secs
It
.
th en
n
o
tion wi ll h av e b een m
a de
thi s subs tituti on (240) beco m
es
c rr ec
is
s
ub stit u t ed
for t in
for t h e
p hase di fference
—t
oos
a s
.
As
ui ta bl e
a res
ult of
,
H
F ( r xs
n
'
l
m
'
h
df
7
F1
s
1?
v -r
cos
n
e
w
l
e
—
t
{ H r H pr f u i
'
-
-
( 242 )
om
p ar i son with ( 24 1) it will be seen th at t h e p h ases of t h e
oscill a ti ons of z an d
wh i ch refer t o flight Speed and verti cal v el ocity
resp ecti v ely
d u e to u s e of t h e el eva t or onl y h ave been b rou ght i nt o
an d on c
’
,
,
APP L I ED AER ODY N AMI CS
a
greem
but
en t
t he
,
tu d e
an d
mp litu d
d by mlti p lyi
b m
u
;
till d i fferen t
n
mt
Agr ee
.
g bo th si d es
a
mp li
m
by
24
1
( )
of
of
en
e
es
eco
h"
"
are s
es
p ro d u ce
e q u atio n th en
be
ca n
t he
a
m
2
s
m
r
)
1
1k< t
cos
0
+
1 (
F
[
”
0
_M
1
1
Jr
u
2
( 243 1
I n thi s exp ressi on t h e secon d i n tegral h as b een o bt a in e d by a s u bs t i t u
’
ti on of 7 +( n —n) k for r i n th at ar i sin g fr o t h e sep ara ti on of t h e in t egra l
of (241) in t o t w o p ar t s.
Fr o (242) an d (248 ) an exp ression can be wr itt en d o wn for
m
m
in
t erm
s of f (t)
f
m
{
_
k
'
F
+
J
1
.
T
“
k
“
,
2
i d ua l in t egra ls
l)
cos
'
r
—,
l
'
a n d r es
—
k
i u T li fl d
'
n
A
3
e
a
-
+ (m
t
‘
N
“
12
m
B y a few si p l e
t h e fir s t in t egr al
,
flight
(
v el ocity)
or
i
-
S
m
-
e
k
p eed )
is
.
6
.
an d
-
w
r
mfg
—h ( ”
n)
’
h(n
-
n
)
z
1
:
m
"
—
M a)
n
k
df
i
‘
,
i nt erp ret ati on of
h an d si d e
w
as,
hi
n
)
2m
-«
1
1
1
1
.
v
t
-
'
t ransform
a tio ns s u ch as t h e s ubs tit u tio n
es
of t h e r ight h an d si d e (244 ) becom
-
Th e
.
r'
"
of
t —r for
1
in
,
—
n)
is
n ot
”
s
di fli cu lt
.
Th e firs t
t er
m
on
the
r
ight
hows th at p ar t of t h e di stur bance
p ro p ortional to t h e d i st u r b an ce in
di ffers in p hase by
a
ti
m
c
(
p ar t
E:
or
v er ti cal
of
t he
54 4
is
A PP L IE D AE RODYN AM ICS
g ti v e
ne a
an d
If
.
we
ca
ll thi s d i ffer ence
8
4
2
( )
l
Eg
a
—
m
x
-
k
c
i
f
k l ‘i
g
I )
ly .
l
t
Al - h Xit
'
fro
mO t
m
as
Th e o ti on
t h e r esu lt of
E
t he
both
ti on
of
9
2
4
( )
0
b
fr o m
0to
g t ve
n 18 n e a i
k
of
na
n
n
o
mbi
+
4
c
k
n
co
—
wh ere X r ep resen ts t he v al ue of
an d
m
” —
fl
th en by
X»
if i t —n is p o s i ti ve
’
.
op l a ne i n E
t he var i a ti on o f fl ight S p eed
win d and el eva tor m
o vem
en t s i s
'
a er
t he
.
:
1
r
Ee
) (3
v
( cont i nuous
u se of ele va to r a nd
Iso late d h o ri zo nt a l gus t )
m
m
m
-
,
2
1
5
)
(
dr
f
‘
x tly si ilar exp r ess i on t o t ha t for 5:fr o mwhi ch P(i ) was o ri g in ally
5
fou n d B y an ex a in ation of t h e t er s i n (245) a n d (2 0) it will be seen
The
tha t t h e t er d ep en din g on f(t) v anish es fro t h e exp ressi on for E
an e ac
.
re
mi
a n
m
d er of t h e t erm
s in i t
'
“
m
ar e easi
m
ly evalua t ed
'
—
h( —)
m
‘
n
n
(
Af t +
e
.
.
n
r
n
k
m
ai ni n g t er s
p lott ed fr o t h e k nown v alu e of Af(t) wh ilst t h e r em
e
i n v ol ve o nly calcul ations for ti m
es u p to 2 secs
wh en t h ey becom
can
be
,
,
Add in g (24 5)
together
(250)
an d
t he
—h( n —
n)
v alu e of E
'
is
seen
to
be
'
m
t
'
'
E E Eu
+
1
x
e
lm
0
A r
f
L
7
)
cos
0
+
‘
b
0
F
(
(
r
Wh en n
'
-
m=l
”
i
t
1+
“
n)
r
7
[
et c
.
,
2
2
5
( )
c
.
,
-
O
—n
re
)l
a ny case
‘
m
u
)
2
5
2
(
)
to
Af(t )
e
i n whi ch n
excee d s 71 it woul d be
’
—
n
n
a
s
t
h
e
e
x
cess o v er 1: i n st ea d of o v er z er o
(
)
c h a n gi n g t h e s ign of
an d p r ocee d i n g a s b efore
In
'
-
)
ml m
d u ces
dr
.
it
‘
t
t
(
,
];
?
2 2
1
fast
MW
-
m
s
n
b
'
,
[g
81
l
8
[vi ly a + 3(
te
n,
n
PX
mu
'
8
,
,“
7
n
ni h
( In
—h(
fl
-
k
W
“
i
y
“fi
k
’
.
.
a
(253 )
dv an t ag eous t o ta ke
This i s e q ui v al en t to
—
ST AB IL ITY DI ST UR B ED MOTION
m
m
q
m
54 5
Illustra ti on “ th e Mathe ati cal Proceases o
u ati ons ( 24 1) to ( 25
2) by
R eference to a Parti cular Cu — Th e ass u ed el ev ator
o v e en t wi ll be
t ha t in di ca t ed i n Fig 258, an d i s one w h i ch p racti cally eli in a t es v ari a
t io ns of v er ti ca l v elocity of t h e aerep lane wh en
o ving thr ough an
i s ol a te d ho ri zo nt al gust In this case E = u 00—w, and t h e v ari a
t i on of v er ti cal v elocity du e to t h e gus t whils t t he el eva t or re ains
fi xed i s (nod
an d i s s hown i n one of t h e c ur v es of Fig
258
Th e al os t si i lar
oti on p ro d uced by t h e o ve en t s of t h e el ev a t or
a lo n e with n o wi n d i s als o shown i n t h e sa
e figu re an d i s t h e v alu e of
m m
m
m
.
~
m
.
m m m
m m
.
.
m
betw een
Th e di fference
two
t he
cu r
v es h as b een
e fi gur e an d corr esp on d s with t h e m
i n di ca t ed i n t h e sam
a ti ca l
a th em
Th e v ar i a ti on of v er ti cal v elocity du e
e xp r ession Af(t) of e qua tion
—
t
d
u
s
n
i
s
e
n
i
w
n
a
e
ca
i
d
ti
l with j (t) of t h e sa e equ ati on
t o a ho ri zo t l g t
o
)
(
It i s now d esi red to fin d t h e vari ation of forw ar d Sp ee d of t h e aerop l an e
a th em
ati ca l e qu a ti o
n s i s th en u
r el a ti v e to t h e ai r
5 of t h e m
Th e
v a ri a tio n of t h e v elocity of t h e aero p l an e du e t o t h e win d alon e i s
a in ly
p ed p hugoi d a n d i s sh own in Fi g 24 7 (a ) Th e v ari ation of v elocity
a d am
o d i fi ed by t h e el ev
a to r m
will be m
o v e en t and t h e v alu e of
i a t he
v ar i a tion of v elocity du e to a sin gl e s u dd en o v e ent of t h e el eva tor i s
ar k ed a i n Fig 25
4 (c) si nce t h e el ev at or
r ep resen t ed by t h e cu r v e m
en t is kn ow n
movem
.
o
.
,
m
,
.
'
m
.
.
.
m
.
mm
,
.
.
.
,
3
'
E
u
'
c
s
-
2
( 54 )
dr
‘
t
i n t egr al for a whi ch gi ves t h e cur ve of Fig 254 could h a v e b een
d e t er m
i ne d with ou t an y r eferen ce to t h e fact th a t F (r ) was ch osen to
i na ti on of
It is howev er alr ea dy clear th a t
ens u r e t h e eli m
e s uch r el a ti onsh i p e x i st s an d in fin di n g t h e v al u e of u fr om
so m
( 254 ) t h e
r esult s w ere so arra n g e d as to in d i ca t e thi s relations h i p
I n t h e actual worki n g it was f oun d to be co nv eni en t to t rans fo rm
by
p
ar ti a l in t egr a tio n obt ai n in g
a
4
2
)
(
and
t he
.
,
,
,
.
5
,
5
1
: 5
1
F0
5u +
,
E bb d ?
(
;
-
m
ro
4
a
a
n
d
i
s c ur v e a n d t h e
d
w
f
th
n i n Fig 25
s
ra
i
c
i
(
)
( y)
o v e en t i n Fig 258 t h e v alu e of t h e in t egral of (255) has been
el ev a t or
fo un d Th e or d in at e SQ of t h e cur v e B QMNP, Fig 254 (a) , is p l ot t e d
ilar to tha t p rev iously a d o p t ed for d eter i n in g th e
a nn er s i
in a
o v e en t F(t) i e t h e el ev a to r p osition a t 12 secs obtai n ed
el ev a t o r
fr o Fig 258 i s ulti p li ed by t h e or di na t e of t h e cur v e u ” of Fig 254 (a)
—
5
8
12) secs t o obt ain t h e el e en t SQ of t h e in t egr al repre
for a ti e ( 1
Th e ar ea of t h e fi gu re B QMNP i s
8 secs
sen tin g t h e di s tur ban ce a t 15
th en t h e di stur b ance a t 158 secs whi ch r es ults fr o t h e ov e en t
of t h e el ev ator alone, an d th e co p l et e di stur ban ce at any ti e du e
to el evator alone i s rep r esent ed by t h e cur v e AB CD of Fig 254
Th e cur ve u, of t h e sa e fi gur e i s t h e di s tur bance du e t o win d alone, a n d
t h e d i stur b ance of t h e achi n e as a co nsequ ence of both wi n d and el evator
Th e
u
m m
m
m
m m
m
m
m
c rve
,
.
.
m
.
.
,
.
.
.
m
.
.
.
m
.
m
m
.
m
m m
m
.
2
s
APP L I ED AERODYN AMI CS
546
mo v m t
m
i s obt a i n ed by si p l e a d di t io n of t h e sep ara te effec t s a n d i s
gi ven by t h e c ur ve EB F Th e cur v e h as t he general characteri s ti c s of
t h e c ur v e i n Fig 258 , si c t h e vari a ti ons in u ar e rough ly p rop or ti o n a l t o
t h e el ev a tor
ov e
en t i n thi s cas e of si
p le ini ti al d istur bance
e
en
.
m m
.
Tt
m( S
e
ec on d s )
m
.
.
Ti
.
m (S
eco n d s
e
.
)
.
'
Res u lt ant Di st u rbance of a n Ae r o p lane a ft e r c hoo s i ng
Ele va t or Move e n t t o Eli i na t e Va r ia t i ons o f Ver t ic a'Ve loc i t y
W h t C h w o u ld o t h e r-w:
b e ca u sed by a s u dd e n Hor i zon tal Gu s t
m
m
,
Fl u
.
m—
il
m
20
Tn c
Res id u a l d i st ur b a nces
is
md
a
.
50
( 5e cond 3 )
o f a con t ro lle d aerop la n e
e
in
.
w h en fu ll u se
ai n t a i ni n g lev el fli g h t
m
of
t he
elev a t o r
.
R eturnin g t o a cons i d era ti on of Fig 254 (a) it will be s een tha t a cur v e
AB CED has been draw n whi ch coi nci d es wi th in for all p o in t s aft er B Th e
p art AB CED i s a rep rodu ct i on t o anoth er scal e of t h e cu r v e for ( M
but
o v ed along t h e axi s of tim
e so tha t t h e p oin t K of Fig 254 (b)
has b een m
4 (a) Th e ch ange of ti e i s e qui v al ent
coi n ci d es with t h e p o i n t E of Fig 25
n
7
t o s u bs titutin g t + 1 — for t as was d o ne to obt ai n equ ation
whilst
,
.
,
.
.
.
m
.
APP L IED AER ODY N AMI CS
54 8
—
i
n
W
d Th e p r ocess to be foll ow e d
a
fr omthi s po in t on war ds i s i d en ti ca l with tha t d escri b ed mt h e p r evi ou s
s ec tio n on t h e d is tur b e d long t u di nal m
oti on of an aer op lane fl yin g i n
a n at ur al wi n d ( p ag es 5
29
p tions i s
Th e di fi eren ce i n t h e in iti al ass um
p l et ely if t h e cur v es of Fi g 254 (c) ar e used in t h e n ew cal cu l a
co ver ed com
tions in ev ery case in whi ch t h e curv e of Fig 247 (s ) wer e used i n t h e
ear li er ca l cu l a tio ns
par is on with th ose p rev i
A b ri ef re f erence t o t h e res ult s and a co m
onely obt a i n e d for an u nco n t rolled a er o p lan e wi ll show how di fi ere n t
t h e d is tur b an ces m
p ort an ce of t h e r esul t s app ea rs t o li e
ay b e
Th e im
onst r ation th at such r ed uctio n of distur ban ce i s p ossi bl e
n ot i n t h e d em
b u t i n in di cating a m
a ti c in v es tiga tion an d d es ign o f
etho d for t h e sy st em
a uto m
a ti c d e v i ces for a ere plan es
Th e r esult s also s how th at th er e is a
p oss ibili ty of g ettin g m
o re an d m
or e a dvan t age fromt h e use of in h eren t
s t a bility wi thout t h e a tt en d an t di sa dv an t ages of v iol en t m
otion in wi n d s
i f i n a ddi tion som
e m
ech ani ca l d ev i ce can h e in v en t ed whi ch will Op er a t e
t h e con t rol s so as to r ed u ce t h e d is tur b an ces whi ch t h e i nh er en t s ta bili ty
i na t e
h as t o eli m
In co ns i d eri ng t h e res ult s of t h e cal cu l a tio ns r ef erri n g t o t h e longi tu d i nal
motion of an aereplane i n a natur al wi n d it shou l d be r ememb ered t ha t t he
p tion th at a p erf ect p ilot h as
cal cu l ati on h as b een carr i ed out on t h e assum
i i i s t an t aneou s kn owl ed ge of t h e v ar i a ti ons in t h e win d an d i s a bl e t o m
a ke
t h e necessary co rr ec t m
o v em
en t
An actua l p ilot wo ul d p r o d u ce a l ess
p t t o gi ve
e x ac t a pp r o x i m
ation an d in p ar ti cu l ar woul d p r ob a bly n ot a tt em
p li cat ed m
o v em
Thi s i n t r o d u ces a fu r th er
s u ch co m
en t s to hi s elev ato r
mo dification and it wi ll b e in t eres tin g l at er t o fin d t h e efi ect of a slow
o v em
it i s how e v er
en t whi ch a v er ag es out rap i d fl u ct u a tio n s
el ev a tor m
a tio n an d not of p r i n ci p l e
a qu es ti on of o r d er of a pp r o x im
o v em
en t re quis it e to cu t ou t t h e v ar i a tions of v er ti c al
Th e el ev a tor m
v el ocity du e to t h e win d d escri b ed i n t h e p revious section i s gi v en in Fig
255 tog eth er wi th t h e anem
ogram
ewha t close ly
I t s g en eral ch ar act eris ti cs follow th ose of t h e win d som
Th ere i s h ow ev er a su p er p osed var i a ti on whi ch d o es n ot bear an y
si m
p l e r el ation to t h e wi n d at t h e inst an t and i s in fact d ep en den t to a
l ar ge ex t ent on p r ev ious hi s tor y d urin g t h e l ast m
in u t e
Th e r esi d u al v ar i ation in v er ti ca l v elocity h as b een p lot t ed t o t en ti m
es
for t h e u ncon tr oll ed m
ac hin e as
t h e scal e of t h e co rr esp on di n g d i a gr am
all to see cl ear ly
oth er wi s e it woul d h av e b een t oo s m
Th e r esi d ual v er ti ca l
v el ocity i s sho wn in Fi g 255 t ogether wi th t h e v er ti cal v eloci ty of t h e
unco n tr oll ed aero p l an e Th e m
a xi m
umv er ti cal v elocity wh en t h e a er o
p l an e i s con t roll ed as assum
ed i s o nly a fr action of a fee t p er secon d ins te a d
of t h e 10 i t s p r evi ously f oun d at t h e en d of a m
in u t e T h is i n di ca t es
in ation of t h e v erti cal v elocity an d i s t h e bes t whi ch can
t h e p racti ca l eli m
st a n ces
be d on e un der a ny ci rcum
en t m
i ght h av e been chosen so as
ilar way t h e el eva tor m
o vem
I n a sim
p r acti cally to eli m
inat e t h e v ari ation of sp eed o v er t h e gr o un d or th e
in cli na tion of t h e ax i s of t h e aer o p l an e Wi t h t h e el ev a to r m
o v em
en t
as s u m
ed whi ch was n ot p ri m
ar ily arra ng ed to red u ce anythin g but t h e
Extensi on to
Motion i n
Nat ur al
,
.
.
.
'
.
,
.
,
.
,
.
,
.
'
,
‘
,
,
.
.
.
,
.
,
,
,
,
.
,
.
.
,
.
.
.
.
,
.
,
STAB IL ITY —D ISTUR B ED
ve
mt i
en
MOT IO N
n
t o ( he Gr o u
'
en u nc o n t r o lled
so
TlM E
Fl a
.
I
o n l ro
llc-d
fl ig h t
in
a na t u ra l
u nco n t ro lled
(li g h t
( Se c ond s )
wi nd
o f sa
40
( ae rop la n e
m
e a e ro
s ta
p la ne
.
b le )
co
mp
a red
wi t h
550
A PP L IE D
AERODYN AMIC S
v er ti cal v elocity t h e v ari ations of hori zont al Speed
r e d u ce d
5
as wi ll b e seen by r ef erence to Fi g 25
One of t h e cu rv es of thi s fi gu r e sh ow s t h e v ari ati on of h o r i z o n ta l
v elocity of t h e a er o p l an e i n t h e n atur al wi n d for t h e elev ator o v e e n t s
shown a bo v e 1
16 wh en t h e a erOplan e i s con t r oll ed
Th e v ar i a t i o n s o f
Sp eed are n ot grea t a n d v ar y be tw een a n i n cr ea se of 2 ft s
a n d a
d ecr ease of 4 ft s Th e co p arati v e c ur v e of v elocity i s rep ro d u c e d f r o
t h e p r evi o u s secti on an d sh ow s a sp ee d of fli ght o v er t h e grou n d v a r y i n g
froman i ncr ease of 10 ft s t o a d ecreas e of 12 ft 8
,
,
.
.
m m
.
,
.
m
,
-
.
.
.
,
.
-
.
.
.
-
.
m
552
APP ENDIX
m
q
The p roces s followed i st h at of u l t i p lyi ng t wo e ua ti on s t og et h er t h e
t he first i n t h a t t h e s igu o f A h as b ee n ch a ng e d
h d ifi ers fr o
m
,
s e c o nd
T he
‘
+
+
09.
2
03
2
02
+ 2020 4
01
+ 00
2
01
+
.
2
00
20103
(3 )
e r i cal
p rod u ct requ ired I n a nu m
e s ec o n d l i n e
t
t
y
ll
il
a
u
n
h
e
s
m
i
s
v
a
n
d
r
,
o f t h e roo t s h as t he n b ee n e fi ec t e d
ddi t i on of t he res u l t s l eads t o
p l e t h e p rocess is rep ea te d u nt il 2u
e xa m
s
is d evoi d of cons ecu ti ve t er m
a nd
t he
m
a
t he
.
.
I n t he
nu
mi
e r ca
l
e xa
mp l
b
e a ov e
.
p rocess
t he
i
carr ed o u t as
is
bel ow
+ 21 6
6
6
4
+
1
000
2nd
power o f r oot s
3 55
3
+
1
+
2
1 72
117
2
+
1
+33 0
Th e
power
,
p a ra t i on is p roved by t h e l as t line t o have been com
p l ete a t
se
a nd
t he
md li f t h
g mtf w
i t arran
conven en
o u
e
en
Lo s
m
m
A
as
Di ll
.
.
o f logs .
3 55
0
06 68
3 1 18
i t
coe ffi c en s o f
a nd
,
a
lit t l e
Di
0 l 58 ,
°
fr o
m
/2
.
V ii
d
secon
46 6
5a nd
375
Ant lloa
.
59 6
2559
00 362
qu adrati c fac tors th ey can be obt a in ed
a nd
a nd
mt h
as
75
Ai n t h e
=0
14 6 9 1
Th ese fact ors di ffer
will b e s h own l a te r
fou nd
g is
3 55
0
Th e v al u es a re p l
'
a re
orki n
or
3 3 53
I f p , a nd p 2 b e t h e
fro t h e for u la
e root s
o
—
t he
os e
t h e fa cto rs
2
A
+ 04 58 )
i
v en
g
a n a cc u ra t e a ns we r ca n
of
( 1) a re
=O
q u a t i on
p
a l ways b e o b t a i n ed
in
e
( 5)
.
4 6 1, b u t ,
fr o
a ny
m
A PP E ND IX
a ti on
pp roxi m
n ot requ i red
Ill ust rati on
i
d
q
wh en ev er i t i s
a
r e u re
Greece s
p
.
493
,
a nd
for
um
l
.
’
,
t
q ti
558
my p
an
u ati on of th e Ei gh th
q
t h e Solu i on of a N
eri ca l E
Meth od —Th e e u a on t o be s o v ed
of
.
a nd
is
pos es h igh a ccu r acy
ur
will
tak en
be
as
e
is
Degree b y
qu a t i on
( 6)
m
Th e root s a re known t o b e p a rt ly rea l a nd p artly co p l ex, b u t t h is kn owl edge
i s n ot o f assist a nce i n t h e a pp li ca tion of t h e et h od
Gra efi e for s t h e eq u at i on
w h ose roo t s a re t h e s q u ar es of t h e r oot s o f
a nd t rea ti ng t h e n ew e q u a t i on
i n t h e sa e way , for s t h e e u at i on wh os e roots a re t h e fou r th p ower of t h os e
of
Aft er cont i nu i ng t h e p rocess for a nu ber o f ti es ( 71) t h e roo ts w ill
h av e b een ra i sed t o t h e ( 2n) t h p ower a nd it i s a l ost ob v i ou s wi t hou t for a l
p roof t h a t th is wi ll l ea d t o a s ep arat i on o f t h e r oot s , a t a ny ra t e wh en th ey are
rea l a nd u n eq u al
One p o i n t i s , h owev er, wort hy o f n oti ce h ere, a nd t h at i s
t h e s u pp ress i on of s ig n whi ch ta k es pl ace o n s q u a ri ng
Th is l ea ds t o no grea t
di ffi cu l t y wh en ta ki ng t h e (2n ) t h root of a r ea l q u a n tity , b u t int rod u ces t h e
n ecess i t y for s p eci a l co ns i d era t i on of co
p l ex root s I n t h e cas e of r ea l ro ots
’
t h e s ig ns
u s t b e fou n d by t ri a l i f neces s ary , b u t t h e u s e o f D es ca rt es ru l e of
m
m
m
q
m
.
m
,
m
m
m
,
.
.
m
m
i gns
my
.
i al u nnecessa ry
To fi nd t h e eq
u a t i o n wh os e r oo ts a re t h e s q u a res o f ( 6) it is on ly necessa ry t o
ch a ng e t h e s ig n of At o form
ua
a n ew e q u a t i on a nd th e n t o m
u l t i p ly t h is n ew e q
e
t i on by e q ua t i on
Th e m
eth od o f a rra ngi ng t h e m
u l t i p li ca t i o n i s o f s om
im
p ort ance a nd t h e formadOp t ed by Graefi e is as follo ws
Su pp os e t h e or igi na l eq u a ti on i s
s
a
d er
ren
tr
.
,
au
.
n
ix
—l
-
( l am
a‘
l
(lg
‘
Wri t e d own only t h e coeffici ents a nd benea t h th emt h e s igns of
e ffi ci ent s for m
ed by ch a ngi ng t h e s ig n o f z
u l t i p li ca t i on p rocess
The m
rea dily seen t o foll ow as b el ow
t h e n ew co
,
,
(a
a
(
n
(a
(a
1) (a —
a) l fl a
6
1
( 4)
n
rr -
)
2
fl
an-
n
o
i s t h en
ir
‘
‘
n-
9
) (a
n-
4
)
.-
-
l
c
m
m
Th e p rod u cts a re con ti nu ed i n s u ccess i ve r ows as fa r a s p oss ibl e, and t h e su s
ns giv e t h e new e q u a ti on wh ose roo t s a re t h e s q ua r es of t h e ro o t s
of t h e col u
After rep eti t i on it will b e noti ced i n a nu er i ca l exa p l e
of e q ua ti o n ( 6)
t hat t h e t er s I n t h e l owes t rows soon beco e v er y s a ll , a nd if all t h e r oot s
a re r ea l a nd u neq u al, t h e p rocess r a i dly l ea ds t o a ll t h e t er s i n t h e s econd
p
i ng neghgible, and t h e sep arat ion of t h e root s i s
a nd su cceed i ng ro ws b eco
th en co p l et e If a co p l ex p a ir occurs, one of t h e p rod u ct s i n t h e s econd
s
e u ni
rt a nh a nd t h e ca lcu la t i on i s s topp e d wh en t h e t e r
row will not beco
i edi at ely t o t h e r igh t a nd l eft of i t b eco e n egli gibl e More t h an one
pl ex p ai r l ea ds t o or e t h an one i p ort a nt t er i n t h e s econd row, b u t i n
co
t he a bs ence o f r ep ea ted r oo t s t h es e t er s ca n n ever b e co ntigu ous
o f ca l cu l a t i on
Th e p rocess p r es u es t h e exi s t ence o f a li it o f a cc u racy z
m
m
m
mm
m
.
m
m
m
m m
m
m
m
m
m
m
m
m
m
m
m
m
.
.
'
554
APPE ND IX
m
t h e w ork wh i ch foll ows , t h is li it wi ll b e tak en t o b e t h at whi ch ca n be
o bta i ned by a 20 in ch slid e ru le
Th e exact li i t t a ken affect s t h e accu r acv
o f d et er
i na ti on of t h e roots , b u t i t wou l d p rob ably not b e a dva n ta geou s t o
get h igh accur a cy di rectly , b u t t o do t h is as a s econ d a nd entir ely s ep a ra te
a nd I n
m
l i
m
-
.
One
.
o
ther
oi n
t
mi
d ily b e seen
t hat t h e rais ing of 11011316 18 t o high powers will l ea d t o t h e in trodu ct i o n of
extr em
bers i n t h e fina l eq ua ti on I t i s a
ely l arg e a nd extr em
ely sm
all nu m
conv eni en ce t h eref or e t o hav e s om
e m
eans of rea dily i n di ca ti ng pow ers o f 10
b er
Th e not a ti on used by Von Sand en will be a d op ted and this exp ress es a n u m
°
u
s ch as l 323 x 1
0 by 1
Th e not a t i on
b etter al t ernat i ve su ggest s i t s elf Proceedi ng now t o solve equ a ti on ( 6) t he
fir s t s tep co rres p ond ing wit h ( 8 ) i s
calcu a t on
i
of conv en ence re
a ns
.
it
will
rea
.
.
,
.
z
’
'
z
l
204 '
z
l
+3 026
“
4 9 01’
—2 000
+4 7 19
—l
—5 l 68 '
s
'
+2 080
'
188
‘
‘
l9
4
+ 7
’
+2060
—l 068
+04 54
+013 7
'
+0003
+ 0 02l
'
+0001
2n d p ower
’
l
l 135
—
—6 l 2
—l 925 —4 l 55
‘
°
'
‘
°
+4 7 19
‘
'
"
ti
on ( 9 ) i s t h e eq u a ti on w h ose roots are t h e s q u a res of t h e roo t s of
q
e q ua t i o n
t h e p owers of 9:
n
bers i n t h e four th
a pp ly all d own t h e col um
Th e nu m
a ll a n d I n cont i n u i ng t o t h e fou r th p owers of t h e r oot s it will b e fo u n d t h at
sm
t h ey beco m
e n egli gibl e
u ence of sign s du e t o changi ng a: t o —z i n
Th e seq
o
w
i
s
gi
v
e
n
as
t
h
e
n
e
x
t
ro
w
a
n
lti
p
l
i
t
i
o
n
i
s
t
n
i
n
u
e
d
or
t
9
co
t
d
h
m
u
h
e
n
f
ca
t
e
( )
E
ua
.
,
.
,
+3 7 05
—5083
“
—16 44
—O 094
'
"
l 726
+04 80
+0 0l l
-
°
“
55
1
7
+
—l l 77
'
‘
4t h power
l
—l 23 5
“
'
—18 83
7
+ 1568
"
—3 008 "
m
—
2
2
0
5
l 525
+
“
07 92
a
8 t h po wer
’
109 1
I
—
4
“
195
4
c
"
020
4
'
—0022
o
1 35
"
”
"
6
5
4
1
+
—29 9 1
a
=+24 70"
—I s47=
-
—l 168
°
“
+ 4 9w
( l l)
-
.
m
m
I n t h e p r ocess of fin di ng t h e eq u at i ons wi th root s of t h e 8 t h p ower fro
t h a t wi t h r oots o f t h e 4 t h , i t will b e not i ced t h at only on e t er , and t h a t a v an
I n t h e s econ d row t h e 3 rd , 5t h a nd 7t h t e r s
a ll on e, occu rs i n t h e t hi r d r ow
s
a re ve ry s
a ll a nd will di sa p ea r i n t h e p rocess of fin di ng t h e l 6t h p ow ers
This
i nd i ca t es t h at a c o ns i de ra b e d egree of s ep a ra t i on o f t he ro ots has a lread v b een
m
m
m
.
.
556
A PPE NDIX
mt h 3 2 d t b t k
Th
w h w
32 m
big i t i
p d i g wi t h t h 3 2
t
f
ity d
ll t h
mp l
t
im
pl m
fd t m
i i g th fi l
i im
i t
mdi t ly pp t Th mt h d p mp d by G fi i h
l ( 16)
q d ti q ti
bt i g
mbig i ty ly L wi t h
f
p it i
g ti
ig f t h
l p t f th
t B y t i l i ( 13 )
th
ti g th
t
f th
will b f d t b i d m
q
i ibl
E t
mbig it y
d
wh i h
b
m d by t i l i
t th
l b t it i
l
Th mt h d will b
t
b p f tly g
mt h d by wh i h t h m
t i d th
pl f t
b
t d I t will b
t
d i t ly f m( 16) t h v l
f t h md l
f th
t f th
igi l q t i
an d
mld
by
a
a re no
ng u se of
ev e r ,
o
,
De
Moi vr e s t h eore
’
u
a
e
s
a e
so v e
ua
as a
v e or n e a
os
e
es e
a
e
a re n
a
ou n
u
occ u rs
e
e
c e
n
o
ua
or
e
ro
e
.
e
e ex ra c e
u
,
e
s o
a
.
one o
3
ur
er
.
no
e no
o
an
a nsv a
n
s
.
e r oo
bta ined as t h e 3 2nd root of
necessa rily t h e p os i t i v e on e
Th ere is
s o on
,
a ca s e
u are roo
e s
an
na
,
r a
n
en era
o u us o
on
en
.
un
e
n su c
.
n
e r ec
e
e
u
e r oo
r a
a c o rs ca n
a ue o
o ne a
x ra c
e
ov e
o
r ae
o
ar
ss
e re
ex
e co
a ni n
on , o
n n
e er
ose
e rea
e s ee n
c
o
e
na
ca n
c
o
o
r ec
ra
ve s
e
.
en
a
roo s o
e
eans o
e
ex , no s
e roo s a re co
as a
e
ca n
roo
n
on
es c o rres
n
e
na
e or
e on y
at
ce
e ua
on
ign t o be t a k en i s
u a ti o n o f
t h en a fa ct o r of t h e origin a l eq
l
t h e for m
m
Wh er e r is kn own b u t p i s t o b e fou n d
I f t h e or igi na l
( i
e ua t i on i s d i vi d ed b y t h i s fa ct o r a r em
i
d
w
li
i
n a: will b e l eft
n
er
hi
c
h
i
s
n
ea
r
a
q
a n d s in ce r i s k nown t h e coe f
fi ci ent s o f a: a nd t h e coe fii ci ent i nd ep en d en t of 1
v e t wo e ua t i ons fro m
t
h
e
c
t
e
d
i
w
i
d
i
T
i
by
h
c
h
t
o
t
e
r
m
n
e
h
s
m
a
b
e
e
fi
e
e
g
q
y
u c h o f t he
rocess
o
f
n
t
e com
or
a
t
m
fi
d
i
n
g
h
m
c
ill
l
t
h
n
t
w
w
a
a
t
o
I
b
e
s
h
t
e
r
f
n
o
p
d i vis i on ca n b e ca rr i ed ou t genera lly a nd in a ny p art i cu lar cas e t h e hi ghes t
er
w
r
t
t
a
o
e
o
f
t
o
b
e
d
ea
l
wit
n
d
e
il
e
d
i
h
m
e
t
i
ca
l
d
i
i
i
o
n
i
s
o
g
t
a
n
re
h
i
t
h
e
t
a
r
t
v
s
p
p
n
l wh ere n is t h e h i gh es t p ower of a: i n t h e o r igi nal eq u at i on For
t han
e ua t i ons u
d
e
d
t
o
w
e
as
e
e
n
rr
u
n
i
c
l
d
i
e
l
i
i
i
b
i
n
t
c
a
t
o
a
n
n
h
e
6
h
r
d
r
t
h
e
h
o
d
v
s
o
h
q
p
g
o u t ge ner ally gi v i ng t h e foll owi n
f
u l ae for
o
r
m
g
p
Fo r a c u bi c t h e val u e ca n be o b t a i ned fro mt h e su mo f t he r oo t s a nd t he
v a l u e o f t h e rea l r oot
I f t he c oe ffi ci ent s h ave t h e s ig ni fi ca nce gi v en t o t h e mi n e q
t he
u at io n
fo rm
u l ae for
a re
p
ca n
be
a nd
o
t he
s
.
.
,
,
.
,
.
,
.
.
B iq u ad ra t i c
p
Qu i nt i c
(
“fi
Sexti c
-
wri te
f
o
fl r
"5
as
do
a1
13
f
or
y
0fi ?)
14
Go
r
8 for
05
04
m
m
For eq u a t i ons of h ig h er d egrees t h e for u l a get s app reci a bly l onger a nd av
not b e a d va n t a eou s
Th e for u l a e gi v en ab ov e cov er t h e us u al cas es occ u r ri ng
g
fo r t h e t a b i li t y of a n aerop l ane, as i n g eneral t wo of t h e root s of t h e oct i c e u a t i on
.
a re r ea
m
q
p
pp ly
m
t h e foreg oi ng a nalys is t o e qu a t i on ( 6) t h e
od u li ar e r eq u i re d , a nd
t h es e c a n b e o b t a i ne d a t t he sa e t i e a s t h e nu e ri ca l v a l u e o f t h e r ea l r o ots
Th e fu r t h e r ca l cu l a t i o ns a r e gi ven b el o w i n a for s ugges t ed by If v o n Sa n den
To
a
m m
m
m
.
.
.
APP E NDIX
t he
nu
mb
ers
mb i
l
i n t h e fi rs t
e ng o
n
co u
557
b t a in ed directly
mq
fr o
e
i
u a t ons
( 14)
a nd
Anti lo g
32
3 0 20"
28 516
28 516
08 9 2
779 20
49 4 04
1545
2 709
0-3 385
106 28
12 3 48
2 828 5
.
l nega ti ve root
modu l us of comp lex root
rea
350 7
r;
.
.
19 4 8 0
22 189
"
1168
16 067
-
72
0172
r,
00 674
rea
l n ega tive root
( 20)
p rocess d oes not n eed d et a iled d escrip tion as it is t h e sam
e as t ha t
foll owed in ext ract i ng t h e nt h roo t by m
s
ea ns o f l ogar i t hm
Th e origi nal eq u a t ion will now b e red u ced t o on e of t h e 6t h d egree by
t h rou gh by t h e fa ct o rs A
t h e v a l u es of t h e
an d A
00674 obta in ed fr om
rea l root s
Th e origi na l eq
ua ti on i s r ep res en te d by i t s coe ffi ci ent s in t h e fi rs t
li ne a nd t h e s econd and t h ir d giv e t h e figures obta in ed for t h e su ccess i v e qu oti ents
a nd rem
ai nd ers wh en di v i d ing by A
2
3
2
2
1
2
1 20 4
15
0
1 09
e
1513
4 9 01 saw
( 21)
Th e
,
.
.
,
m
-
I
u se 053 1
”
0765
-
-
0138 6
09 00 01746
00 088 1
3
108 1
term
s u n d er lined gi ve t he se v en t h d egr ee eq ua ti on re q u i r ed a n d t h e
fi rs t t er m
i n t h e t hi r d ro w v i z
s h ows t ha t t h e r oot is a pp roxi m
a tely
I t i s essentia l for su ccess t ha t t h e di v is i on by l a rge roo t s s h ou l d begin fr om
t he
t ermi nd ep end en t of A a nd for sm
all roots s h ould b egin at t h e t erm
conta ini ng
t he h igh es t p ower o f A
a l not a ti on
Di vi di ng by ( A
a nd worki ng i n t h e or d i na ry d ec im
Th e
,
.
,
,
.
1
= 0 ( 22)
900
08 8 1
“
A
12 52A5
°
52 3 414
Th e l as t l i ne i s a s ext i c
v
i
g en i n ( 19) with f l
117 22 + 13 07
wi t h t hr ee p a i rs of com
p l ex roots Us ing t he form
ul a
t h e v al u e of p is ob t ain ed as bel ow
.
q ua d ra t i c fa cto r i s t herefore A 11 22A
Di vi de ou t by t h is facto r rem
b eri ng t h at i t
em
2
On e
°
,
1
I
13 22
0 (23 )
5
23 5
2 215
0
is
a
la rge roo t
0 3 70
3
7 54
°
qu oti ent is i ndi ca ted by t h e figu r es whi ch are d ou bly u nd erlined a nd
t he a pp roxim
at e c orrect ness of t h e fa ct or i s in di ca ted by t h e agreem
en t of t h e
firs t nu m
bers in t h e t hi r d and fifth rows with those i n t h e q uadra ti c fact o r
The
,
.
558
A PPE ND IX
Th e
biq uadr a ti c ( 26)
ca n now
be
l ved
so
,
ml
i g
for
us n
u a
rs
an
d
a nd
02 15
73
t h e t wo
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re
a n ng
qua dratic factors are
?
(I
u
r
(
a nd
m+
1 3 25A
e oea
°
( 29 )
( 30)
0172)
wh ol e ca l cu la ti on t o this st age ca n be ca rr i ed ou t t o slide ru le a ccuracy
by t wo com
p u ters i n a bou t 3 h ou rs It i s necessar y t o work in d e p e n dently
for s u ccess i ve st eps a nd t o m
pa riso ns a t t he en d of each s t e p When
a k e com
t h e p owers of 10 in t h e l a t er stages becom
e grea t co ns i d era bl e d is cr ep a n ci es in
t he si g nifi ca nt figur es occur and seem
Thi s i s not
t o in dica t e wan t o f accuracy
u s u a lly t he case a nd t h e root s u l t im
p u te rs w ill be
bot h com
a t ely d ed u ce d b
fou n d t o agree even wh en t h e di s crep anci es m
a bove app ear t o b e v ery
en ti on
grea t Th e reason for th is is ob vi ous wh en col u m
i ned
n 2 i n t a bl e ( 20) i s ex a m
—
Meth od of obtaini ng Any Resu lt m
l
I n exam
r
t
e
ini ng t h e s ta bili ty
ore accu a
y
of a p ar ti cu lar aerop la ne it i s p ro ba bl e t hat t he roots t h us obta i ned a re su ffi c i en tly
a ccur a te for all p rac ti ca l p ur os es
I n a n inv es tiga t i on concer n ing t h e effect
p
of cer t a i n m
eth od s wil l be
odi fi ca ti ons of d et a i l h igh er accu r acy i s d es ir abl e a nd m
descr ibed for i ncreas ing t h e accu racy o f a ny com
p lex root p rogress i vely wi t h out
t h e necess it y for a kn owl edg e of t h e rem
Th e p roce d u r e i s as
ai n ing r oot s
foll ows : Di v i d e t h e origina l eq
a t e q u a dra ti c fa ctor
ua ti on by t h e app roxi m
o bta i ning a rem
a in d er of t h e formB 13:
R0 a nd a qu oti en t
Ag a in d ivide
t h is q u oti ent by t h e a pp roxim
a i n d er B ar i R1
a te qua dr at i c fact o r l ea vi ng a re m
2
I f t h e app roxim
at e qua dra ti c fa ctor b e 3 +p x +r t h en t h e correct ed q u adrat ic
fact or is
wh ere
The
~
.
.
,
.
,
°
.
.
.
.
,
,
,
,
,
.
-
,
.
,
RI
3
Bo
Rs 2
Ra
R
B
a
S
P
f Ra
RI
R
P s
Rs
a nd
3
R2
Sr
p rocess ca n be r ep ea t ed t o gi ve a ny desi red degree of a ccur acy I t is
p roba bl e t ha t t he val u es of R2 an d R3 e nce o bt ai ned will b e s uffi ci en tly accurate
for us e i n several s u ccess i v e d i visi ons
Nu m
eri cal Illus trati on of t h e Use of t h e Above Meth od of Su ccessi ve Approxi a
”
—
A
t i on
1 3 25A
i s kn own t o b e an
Th e factor gi v en by
It i s d es ired t o fin d a factor whi ch is !
app ro xim
at e s ol u ti on of e qu at i on
more accu rat e solu tion Th e sli de rule is h ere ra laced by a ca lcu lating
u l t i p li ca ti ons and
u i red are
mac hi ne on which bo th t he m
tract i ons req
ang
F or real a s Ne wt o n s a nd H om
m
et h od
er
descri bed i n Engli sh t ex t books N?
Th e
.
m
.
°
.
.
’
,
’
s
s
as
A PPE NDIX
560
R0 is
7199
a nd
n ow
RI
R3
Re
R2
4
164
R0
d 8p
81 z
'
q ua d ra ti c factor is
2
A
wi t h
as
of
an
Th e
It , is
1133 550621
5
219246 1
y whi ch p ro ba bly ex tends t o t h e las t di gi t
ati on will be s een t o corresp on d wi t h
et h od of a pp ro xi m
Th e m
a t i on t o t h e v a l u e of r ea l r oo ts a nd it i s g rea t ly ass is t e d b
method of app roxi m
y
t he u s e of a ca l cu l a t i ng m
ac h i ne
No cou n t erp ar t o f H om
er s p rocess fo r
ach t o it bein g o ne d es cr ib ed by Jeli n ek
r oo t s is
All
s u ch m
et h ods req u ir e
cas e of rep ea te d root s
but
s u ch c as es a re n ot of s
mon occu rrence t o m
y com
a k e a d e ta i l ed di sc us
s i on n ecessary
a n acc u rac
.
,
,
’
.
.
,
.
562
IN D EX
Ca b les , st ru ts
an d
.
wi res, I
251
w
i
o
a
i
rs
c
re
f
t
c
h
,
p
Ca
Ch an u te
,
Coe ffi ci e n t ,
cen t re o f
fac t o rs
p ressure ,
of
of
m mt
ov e
en
of,
521
Fronde
’
s
la w ,