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I
ENeRGY ~MANEUVERAlliLl'l:'£ ( U)
by
John R. Boyu1 B&j 1 USA~'
Tholll.lls I', Chr1st~e
J ames E. Gibson, 1st L~· USI\F
.-·
ti'IQ.U. U JCllUliO IUIQIIJJ.m
1fOT a ~U.UA!LJI T'O
roallQII IIATto:uu
J
/
............. ...
"'
w.... ~ __, • ~ ..~.~ .... ~ ......... .. ,.... MCWM­
~:
,•
Tlaia ..atfti..l c.o"LIJ"' fofo,..,lliV\ dedi~
lho
I doofC>IM o1 lho l!• i .... ! .. tto
efLhi~t U.. ~•"'""' ot toN ltpiov_., Lan
(Tillo U, U..S.C., ll'<\ltM 1U olld '!'H ), lt1
,..,!G..
"''"'Uo"
tr.u.atDf&Uo" er
of • ...,~ fc IA7
... ~Mr &6 • " IMvth.ortwd ,.,... Ll , , .
ltl'b01odb71 ....
In addition to security requirements which must:
be met, t hi s document i s subject to special e x­
port controls and each trunsmittal to foreign
gover nments ur fo•cign nationals may be made
only with prior i!pproViJl of fll'GC (PGTO) Eglin
1\FB, Floridu
~ROUP-3
&I 12 Y£ l• INI(aYA lf:
MO T AUI O.lTitAlll DtCllSSif l£ 0.
OQIX~~AD£0
(This page is Unclassifi ed)
. ...
Jll r~Ri~uRD
/
( U) !his report on Proj ~·ct 0350T4 will be published in three volumes.
Volume I contains the theory, sample application, and the associated mathe­
matical model. This effort commenced on 15 June 1965 and ''as co01pleted on
15 January 1966. Volume~ II ·and Ill will contain a few :>ample comparative
analyses and the Energy- Haneoverability Diagrams of severa l air cr aft for clean
and air-to.-air configurations, This report supersedes 1\PGC-TDR-64-35 (refer- ·
ence 8).
(U) The mngnitude of this progrnm precl udes the listing of all individ­
uals whose efforts huve been invaluable to the progress. of the wor k and ~he.
preparation of this report. H~ever, spedal recognition must be given to
Mr. Carl llavo;, Mathematician, and Hrs, Anthony Bicle1 Programmer, of the .
Mathematical Services Laboratory, and to Miss Betty Jo Salter, Illustrator,
of the Graphics Section,
·
This technical report has been revit>wed and is approved.
~f!&:er
WAlTER P. GLOVER.
USAF
Oirecto'
I •
~~
Colonel ,
for ce &r•a~ent
La bo ~ atory
~~
·
E.
Major Genera l. USAF
J.
R08~R TS,
Coswander, l l r
P19Yfll& Groun<l Ce~te r
L - - -- -- - - -- -- -- --:------ - - - ------··· ·­
,
I
l"
UNCLASSIFitD ABSTRACT
TJiis report sh1>1~s h01~ an eircra ft' s t!ne:rgy state and energy rate capa­
bilities are directly relat~d to operational maneuverability and efficiency
in tenn ~ of energy-ma~euverability theory. It demonstrates also how energy.
maneuverauility theory may be appl.ied to assist the -tactician, ~omander,
piann~r,- and designer in optimizin& aircraft performance. Load factor vernus
velocity (G-V) and altitude varsus ~lach number (H-M) diagrams are employed
to obtain the interacting energy relationships fur.damental to· energy­
maneuverability theory. The G-V diagrams provide a measure of instantaneous
maneuverability while the H-~1 diagrams (the mo~t valuable diagrams) show
sustained maneuverability as a fUnction of energy rate, g, efficiency, and
range throughout an aircraft•s performance envelope. ~he energy diagrams,
as the working toals of energy-meneuverability theory, may be used to deter­
mine operational maneuverability and efficiency of various armament-engine­
airframe combi.natious.
J
I
..
This document is subject to special expert controls and each transmittal
to foreign governments or·farci~n nationals may be made only with prior
approval af APGC (PW:0) 1 Eglin AFB, Florida.
iii
i
.
'
I.
!
CO/n'OOS
• I
·-'
Section
I.
II.
III.
Pege
UW~Dl1CTION
. ............................... ... ..............
.,
INSTANrANEOUS ~~~~UVERABILITY •• •••• ·• · •· ··• · ••••• • • ••••••••••
SUstAINED HAN£1JVERAB1LITY , , , , • • ••••• , , , , • , • , ................ ,
General •..•.. , , . , ... ... , . . . .... .. . . ... , ..•. • , • , . •... ..•. ••••
Energy Rate .. . .•. .• .• • , , , • , • , •. , . , . , • , !
IV.
V.
0
I.
NAS IC/ACAA
DECLASSIFY
( This rnfor matron no
longer
needs to be
classified
III,
, • • , , , • • , , , , , , ••• •• •
l
2
4
4
6
8
G • •• •·• ···· · ······•·· ·· ··· · · ··· ······ ·· ··· ··· ·· ·· ······ · ··•
Efficiency •.. . .... , . . . , •....... , . . , , , . . ; . . , . . , .... , , . , ... , •
10
Range ... . . . .. ....... . . .... .. . . .. . . . .. . . .. .......... , • , , • • , , •
13
TACt'!CAL APPLICATIONS • , •• •• , • , , • •• , , •• , •• , , •• , • ••••• , • , ••• , • • 1 5
R.EQlJI.RDtE'N'l'S •• , .. .... , • • , • •• , • • , • • • , • , , • , • , • • • , • , , , • , , • , • , •••
}l
REFE.RENCES
78
... .......... .......................................
RUt.E-Of-TliUHB PERFORHA~CE CAPA3ILIT'i OF AIM-7£ AND AIM-98/
AA.-.2 MISSii.ES • , , .• . , •.•• . , • .• . • •. . , •... . ••. . .•• , . .... , •• , •• ,
MATHEMATICAL DERIVATIONS ~D MODtLS FOR DEVELOPING Eh~RG'i­
}W\£UVEiW!lLI1'i n£IDR'i AND ASSOCIATED FLIGHT PATHS • , , • • • • • •
A COMPARISON OF THE BRYEON-KELLEY AND THE MODIFIED RIITOWSI<l
T.EQ{}{IQtJE3 •• , • , • • •.•• , •• , ... . , ••. ••. , • , ,·, , . • . • , , , . •• • • , • , • , ,
32
}9
74
ILLUSTRATIONS AND TABLES
Figure
2,
3
5
;.
F-~C G-V Diag:ram at 30,000 Feet .................... . . .. .. ... .
F- 4<: Steady- State Envelope • , • . ••• •• • •. •..•• ••••••••.• •• • • . •• •
F.4C 1-G Energy Rate Diagram with Superimposed Rutowski
Hinirn.u.u'\ Time Path , , , • , .•• , , • . • , ••• • •.••• , •••••..•• • •. •• · • • •
7
4.
F. ijC l-G Energy Rate Diagram with Superimposed Rule-of-
1.
7
Thwnb Path ••. •
r. ' •• • • •.••• • •••••••••• , •
F-IIC }-C Ener~ Rate Diagram • •• ..•• , •• . ••• • •..• • .•. •• •• ••.•• • 9­
F- 4c 5-G Energy Rnte Diagram , •• ••••.••• ••• •.•••• ••.•• ••• • .•• • 9
Conatont (50S) fuel E-M Efficiency Diagram •••• • •••••••• ••••• • lZ
F-4c Varioblc Fuel E-H tfficiency Diagram •••••• • •• · ··•••••··~ 12
l'-lf.c Rance Diagran: . .. ................ .. ·. •.•••.•..•.... . .•.. . •• 14­
I
~
5.
6.
7.
8,
9.
NAS IC/ACAA
DEC LASSIFY
(This tnformatlon no
longer
needs to be class rf ted)
10,
11,
F- 4C G;.V Diagram
•
•
I
•• •••• •
•• •
af 30,000 Feet
, G- 21 G- V ~iagram.at 30,000
v
i
,
I
•• ••
....... . , ... .. ........... ... .. 16
Feet . .. ............. .... .. • ... . . 16
..................-··········
COln'OOS ( Conttnut.d)
'NASIC/ACAA
DEC LASSIFY
(ThiS information
no longer
needs to be
classified)
Figure
12,
Page
F-4C Ha:dmt.om Po~-.-cr l-G Energy Rutc Diagram •••••• •••• • • •••••• • •
HI0-21 Maximwa ?ilwer J...C J:~ergy Rate Diagram • • • , ••••• , • , ••••• •
F-4C Naximum Po~-:er 3- G Er.e.:-gy Rate Dia:;ram •••••. , •••• :.- • •• ••• ,
MrG-2l NllXimwn Power 3-C Energy Rate Diagram •• , , ••• • • • •• • •••••
F-liC Maximum Po:~ei: 5- G EnersJ R3tc Diagram • , •••••••.•••••.•.••
· MIG-21 Maximum l,O't,er 5- G Energy Rnte Diagram ••••••• , •••••.••••
F- 4C Nilitary ·Power l-G Energy Rate Dia~rOim •• , , .. , .... , . . .. , ••
MIG-21 NiUtary. Poiier 1-G Z:nergy Rate Diagram , ..... . ......... ..
F- 4C Military Power ,-G , Energy Rate Di~gram ••••• ••••••••••••••
MIG-21 Miiitaiy · PoWe.~ 3-G' tnergy Rate Diagr&m •••••••••• • •• ••••
F-IIC Hili tar)' Power 5-G Energy Rate Dia:;ram ..... . ............ ,
mc.:.a· Xillt~· 'Power 5;.G Energy Rate Diagr~m , ••••••• • ••••••••
F-4C Hax:Uuum !•ower Constant ( 501>) Fuel E-~ Efficiency Di.a gram •
MIG-21 Maximt1'll Power Constant ( ~) Fuel E-H Efficiency
~7
17
::.9
19
~r
21
21
22
22
23
2}
24
Diagram •.•• , •. ...•.•.• • , ••..•• , •. , .... , .••..•.••..••• . , ..•. ,
24
F-4c Ha:ximum Power Variable Fuel E-.'1 Efficiency Diagram . .. .. ..
MIG-21 Xaximu;n Power Variable Fu~l E-M Effictency Dlagram • • •••
F- 4C Military Power Constant (m) Fu~ E-M Efficiency
25
25
Diagram • •• ••.•• , ••.•.••••.••..•••• ; •• , ............. .. .. ...... ,
MIC-21 Military Power Constant ( ~) Fuel E-H Efficiency
27
. Diagram ••..••.• , .••. , .•..••• , ••..• , , .. , .••.• • • , • , .•• , • , .• .. • 27
Power Variable Fuel E-~ Efficiency Diagram ••••• •
HIG-21 Military Power Variable ruel E-M Efficiency Diagram • • ••
F...4C Range Diagra11 •••• , ••••••• • •• , • ••• ••.• • , " ....... , ••••••••••
HIG-21 Range Diagram, • , .• , • •••.••.• ~ •. .. •..• : , •••••••• , ••••• , • ,
28
29
29
Ruto~ki
F-4C Military
28
Table
Hinimurn Ti~e faths (Hcximum Po~er) •·••· · ···•••· · ···· •
15
II,
~utowski Minimum Time Paths (Hilitary Po~e r) o••·••••·••• • •••••
III.
IV.
RutoloiSki Minimum fuel Paths (MaximlDll Poller) • .. .... .. .. .. .. .. • • •
Rutowsld Mini:.!um Fuel Patba (Military Po~r) • , ••••••• •• , • • • • • •
18
26
26
l,
vi
1
20
­
~
SECfiON I
Im'RODUC7ION
( U) Aircraft maneuverability can be defined as tlle ability to change
direction and/or magnitude of the velocity vector. While this definition
describes 1naneuverability accurately, it provides littLe feel for the fighter
pilot or enginee:: on ho~-1 to acc:>uire. best ( optimwn) maneuverability. However,
fr~ experience, we know that the ~est ·~y t~ maneuver .for positi~n advantage•
or to deny"this sam~ advantage to an opponent depends on the type of ordnance
used and the perfonnance of the aircraft. 'l'he type of ordnance P.mployed
determines the possible delivery conditions needed to effectiv'ely deliver thie
ordnance, whether it be guided missiles, guns, or bOillb&. Quantitatively,
these delivery r.onditions can be depicted by law1ch or firing envelopes. Uncc
the initial delivery conditions a::e kno~olJ 1 the problem beco~r.e'i one of mani!u­
vering into :~~ effective launt"h envelope. Such maneuverability ~.8 depenfent
upon the ability of the pilot to control turn1 al~itude, airspeed, and accel­
eration.
(U) · The purpose of the following discussion is to 6how how enerey­
mancuverability is relate~ to operational 11aneuverability and hQW this rel.a­
~ionship 11ay be exploited by the tactician, co~ander, planner, or designer •
in developing valid maneuverir.g and/or delivery tactice along with better
aerial c~bat we~pons systems.
\
1
/
./
SECl'ION II
rNS!l\li'TJ\NEOUS HAUEIJVER/\BlL1 'I'Y
. ( U) iurn can be described in tenns of radius (r) and/or rate ( w) at
various airspeeds ( V) and radial g ( N, ) b? employin~ the relationships
y?.
r :: -
gNr
and w ;:
gN
~v
in co·.)juncticm ~ith aerodynami'c force system equations. From such equatioos2
numP.roos charts depicting turn r adius and rate can be developed to pxovi de
~vme measur~ of lll!neu•Ter ability. Needless to say, the rt"umerous charl:s and
assoriat ed contoui.c; are difficult to digest. ror simplification and clarity,
load factor versus velocity (G-V) diagrams are employed to depict turn in a
manner con3isten1 with a pilot 1 s background anJ his c~ ckpit instrumentation.
(See figure 1.)
( 11) The intent of this diagram ("Figure 1) is to en<~le a pilot to <let~r-·
mine maxi~um turn in terms of g or load factor by consulting t he ae~odynamic
limit at the left and the structural/stabilator limits &t t he top, bottom, and
to the right. By an overlay comp arison of G-V diagrams, a Rilot can dete:<r~ine
if he, or a possible adversary, has a turn advantage. Any tum capability or
advantage, extracted from such a diagram, provides only a relative measure of
instantaneous maneuverability.
The diagram fails to indicate the eff~ct of
pulling g in terms Qf l osing or gaining altitude and/or air3peed . As a re~ult,
no mea~ure of sustained maneuverability can be acquired from a study of this
diagram, To develop thie i nformation1 a look in a different direction is
nccesaacy,
2
\
CONFIDENTIAL
.
8~~~~--~-- -+-,--~--r-~~.:~:~~
7~-+--~--+---~,/~+---~-+--~
~~:----4----+----~--~---4----+---4----+~
·2~---+-- ·
~;.-: :,
...........
·l t-----t--+-~,c-' .+
. .-... -. -+-.....,-.. ~-.....,.--t-~r-+-1
,
-
0
·f
~
100
200
l o::
•oo
••• ••
' oo
CAL I 811A T(O AI ASP(£0
y
6oo
roo
IIHOTS
Figure 1. F. 4C C-V Diagram at 301 000 Feet.
3
CONfl DfENYIAL
.~
•
aoo
CONFIDENTIAL
:iE:CUON Ill
..
SUSTAINF:O HI\NI:IJVt!WltLI.TY
CENEML
( U) Altitude, <~irspeed, :~nd chances thereto are directly aependent upon
tho fore!! system acting along, nnd no:mal to~ the flight path . Oy mathemati­
cally manipulating the expl"f;~sions describins; this force system, sltitude {h)
aml airspeed (V) can be comllined in the expression for srecific energy (E,)$
. y2
E, ":h .. 2g
~s
..
I
shown in Appendix II.
(U) To J,Jn~uver for a desired change or a combination of chances in
direction, altitude, and airspeed, a pilot must disturb the force system sur­
r ounding his aircraft. Therefore, from 1:he above exprP.s~ion1 ~o-e deduce that
mDneuverability is not only related to directionul change (turn), but is al~o
related to specific energy in terns of altitude and airspeed. From this ex­
pression, y~ can also deduce that all mane~v~ing will be conducted between a
maxin~wn anergy level associated with a best altitude..airspeed combination and
a minintum anergy level a~sociated with 'zero altitude and minimum airspeed,
These maxi~nom ami minimum energy lev'e::ts.•rnay be represented in an altitude
versus Hach number (ll-H) dingram (Figure 2), The maximum energy 'level is
located on Figure 2 at t he,.¢int ~<.here the specific energy ( E, ) contour is
tangent to the steady-state envelope, The minimum ener~;y level is located
on Figure 2 at sea level ~1hcre the appropriate speci'fic energy contour inter~
cepts the stcauy..:state envelope• . (The steady~stilte enve'io(le "is defined as the
level flight operating bJundary determined by angle-of-attack limits; thrust
available, dras, am) st.:uctw-al limits ,)
(U) In an air-te-air battle1 offensive mb~euvering advan~age will b~lonB
to the pilot who can enter an engagement at a higher energy level and maintain
more euergy than his opponent while locked i~o a maneuver and counter-m&neuver
duel. Hancuvering advantage will also belong to the pilot ~oi1o enters an air­
to-air battle at a lower energy level, but can gain more energy than his
opponent durin!; the course of the battle, From a performance standpoint, such
an advnntage is clear beclust' the pilot w).tll t he most encr~ has a better
opportunity to engage or llisengDge at his llWO choosing. On the other hand,
cner.;y-loss maneuvers can be cn•ployed defensively to nullify an attack or to
~;ain a temporar5' offensive !ID.:\euvering position.
Implici.t in the entire
4
CONFIDENTIAL
(This llill'P i" unr.lnsR1 fi Prl)
CONFIDENTIAL
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'"" ~'--".-\
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-=~ 1-j·r--..
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. .-. . . ~. t'-­~ - ~-~-~
=::::~ .. '-~~ - ~t:;: "-~,·
5
1~0
130 140
It
Figure 2.
F-4C Steady-State Envelope,
discussion on energy state and/or energy rate advantages is the fact that a
pilot h~s enough internal energy (fuel~ available to exploit these advantages,
'
..
• ( P.) .. ,\!1. an air- to-surfa::e role, ·a pilot is not as interested in a high
energy :;tate as he is in maintaining energy while maneuvering with a wide
assortment of stores on uoard. tf he cannot maintain maneuvering energy, his
choice of tactics/techniques becomes limited. In addition, if this same pilot
is tappet! by enemy air, his ability to evade or nullify the attack becom~?s
questionable.
( U) Ol>serving the correlation of energy with maneuverability, tt follows
t~at tactical maneuverability is rela:ted to the amount of energy possessed and
how well that energy is m:anaged. From a design standpoint, this means a fighter
pilot must be given a vehicle ~1erein such factors as energy ntate, energy rate,
anu the quantity of internal energy available are pro~erly considered . for uest
maneuverability, 'the fi~hter pilot must know when and how to move to a h,igher or
· lower energy l evel ami nor.{ to !lest conserve hi~; internal energy when locked in
an air-to- air or air-to-surface encounter.
5
CONFIDENTIAL
.,
_ _________:__
__.
~_,.
___
:..:_;~---- -----··· --·
CONFIDENTIAL
!:HERGY MT.E
( U) For an offensive maneuverin~ advantage, a fit;hter pilot must be !It a
higher enargy level or be able to ~ain cner~y more quickly than his adver:;art
before the maneuver and counter-maneuver portion of the battle begins. To gain
energy ~nore . quickly--once GCI, radar, or visual contact is maue-ncce:>sitates a
best path for accu:nplishing this task.
(U) An approximate method for finding a best flisht path was discovered
by t. s. Ruto-..-ski (sec Reference 1) . Using his rnethotlJ as outlined in .•,ppen­
dix II 1 the best ( Rutowski) path for gaining maneuvering energy may be repre­
sentPd on ;Jn altiturle versus Hach number ( 11-N) diagram containing ener~y rate
(specific excess po~1er) contours .,.'ithin the steady.-state envelope, as shown in
figure 3. tnergy gain is maximum at the points ...i 1ere the specific energy (E.).
contours are tangent to the srecific excess power ( P1 ) contour~>, where
(l
'
" (
Tlv,-0) y '
thrust available, D= drag, V ~ velocity, and W; weight. A glance
3 shows a \>est (optimum) path fgr gaining e~ergy most rapidly. Not
noxmally shown is the best path J..-hen the starting point i s located off the
basic Rutows~i path. A solution to this problem becomes easy if the energy
.rate, off the Rutowski path, inside the stead)'-State envelope is assumed to
be zero. Under this assumption, the pilot moves along the specific energy
contou:: ~onsistent with his · energy level until intercept is made with the
Rutows:-i path. As sho1.r1 in Figure 3, the best t-ath consists' nf two segments:
the appropriate specific energy contour and the basic Rutow~ki path. Using
this procedure, pilots can determi~e the best pa•hs from any point in t he enve­
lope, However, these pat hs are approximate for two reasons: ( 1) load factor .
is assumed consta~t (1 g) in developing the ba~ic Rutowski patr and (2) energy
rate is assumed to be zero i.n developing the best path from an point in t_he
envelope,
T.
z
at; Figure
(U) To provide ~ more ·exact solution, A. E. Br yson and H. J , Kelley (Refer­
ence 2 and AppentH-x II) have developed a dizec.t: metltod k'hile H. P. !leeX"'llann (Ref­
erence 3) has developed an indirect metllori for finding best paths, Flight pat11s 1
determined by these metltods, sl1ow that Rutowski is very nearly correct, Rutowski ' s
method, when compared with the Bryson-Kelley and Heermann method's, reveals that
a rule-of- thumb technique can be used by a pile~ or engineer to find best energy
paths. (See Appendix III.) The technique uses the simple rule 6£ 1 : k6H for
fir.cling intercept curves and the subsonic~supersonic transition curve to the
Rutowski path, (See Figure '+.) The value of k c:: 2/5 w11en Mach number must be
decreasecJ and k : -1 ...t en ~!aCh 1 m.lmber mUSt be increased tO intPrCept the basic
Rutowski path.
6
CONfiDENTIAl
(This puge is O'r~C'lassified)
CONFIDENTIAL
10
0
90
100
·r"""r--r- r
~-- t:-.
[',.
t-- ~
110
130 140
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. ." ':\.,. . i"'-
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120
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110
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5
Figure 1, ,
\
\
F-4C 1- G £nergy Rate Diagrum with Superimposed Rutowski
Hi1dmU111 Titae l'ath.
0
1.4
1.2
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IU CH
IHTERCEPT
PATH
_I\
~~ ~
\
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F_t,c 1.- G Ene!'gy Rate Diagram with Superir"~sed Rule-of-!ltwnb Path.
7
CO Nf~DENYIAl
CONFID.ENTIAL
( U) Even though the path developed by thi s yroce<lure :nay be satisfactory
engineer, i t still i s not good enough to enable the pilot to fly the
pattt, because o( tlle constantly changin~ altitude, ~lac·n number, and pitch a11glc.
Observation and analysis of the path just defined, 110\olever, suggest a way to
avoid this predicament. Generally ;;peaking, the su\is(Jnic vortio:l of th~ path
can be represented
a constant Hach numuer climb, while the supersonic portiun
may be approximatpd by an average constant cal!bratcu airopeed. To intercel't
t:hc subsonic or supersonic segments, the pilot vulls up or pushes over, as indi­
cated by the rule-of- thwnb, untii he intercepts the basic -path. At :intercept,
the pilot should leau the Hilch nwuber or calibrated airspeed to prevent a tenL­
porary loss of energy by pulling too much g. 1ntercepts from the SIJbsonic - •
segment to the supersonic segment of the Rut owsk.i. path should be accomp.lishetl
a\ less than 2 g, Mlile intercepts to the subsonic portion of the Rutowsld path
should be al!canplishcd at less than 3 or 4 g at lo11er altitude~> and should
dec~~asc correspondingly as altitude. increases.
to
\
~he
?Y
( U) Al~hough the 11-t-1 Jia:,;ram is useful for appro~irnatin;; be:;t energy r11tc
flight paths, observation reveals that it can also be used fo1• another purpose.
The contours contained within the steau)'-Statc envelope provide a measure of
the ability to &ain energy throughout the envelope. Since gaining energy is
related to maintaining maneuverability~ the l--1{ Energy. Rate diagram provides a
measure of sustained maneuverability as a func~on of energy rate, By overlay
techniques, thP. L.g Energy Ra~e tliagraat can be used by the fighter pilot or .
tactician to tletcrmine i f he can gain energy more quickly than some adversary.
Actual time values, depicting how rapidly the transfer takes place,· can be
provided by the previously'1!1entioned optimization progralns. Such values vill
be provided in "TacticaL Applications," Section IV of this 1·eport. \~'hen this
infonnatioo is correlated with some analysis yet to be presented, the pilot
or t actician can then determine the type of tactics or maneuvers to employ.
G
(U) Energy Rate diagrams of more than 1 g can be helpful in deteDnining
the best tactics to employ in the maneuver and counter- maneuver portions of the
fight. As shown in Figures 5 and 6, these diagrams contain both positive and
negative energy rate (P,) contours within the steady-state envelqpe, As su~h,
these ~iagrams portray the ability to maintain energy while pUlling s; hence,
they provitJe a measure of sustain eo maneuver:1bili ty as a function of g.
( U) Once again, by si:npl~ overlay or comparison techniques, regions of
energy advantage and disadvantage can be easily determined. If a figHter pilot
can gain energy more quickly or lose it less rapidly than some adversary in e
maneuvering fight, he has offensiv~ maneuvering advantage. On the other hand,
8
CO NF iDfl:NTIAl
(This page is Unclassified)
\
CONFIDENTIAl
70
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Figure 5.
...
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CONFIDIENTIAL
- -- -···-..-·
..
- ...
'
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- ·- - - - ----- -- .·--­
1\
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figure 6, F-4C 5-G Energy Rate Diagram,
1\
t\ \
1\
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so
10 .
c
60
F- 4C 3-G £nergy Rate Diogram.
\
2.2
2.4
..
CONFIDENTIAL
if the energy values ure r'!Vel'Sed, the pilot, although forced on the defensive,
may employ energy loss ·•laneuvers to his advantage. In either case, the 3-g am!
5-g Energy Rate dia~rams graphically portray capabilities anu limitations in
the maneuver and counter- maneuver portions of the fiHht . ln the air-to.surface
role, these diagrams may be employed to tletemine maneuvering capabilities aod
limitations with a wide assortment of stores on board. With this infoxmation,
pilots and tacticians can develop pre-a~tack and post-attack tactics and mane~..
vers against a hostile surface complex.
( U) Even thouch the 3- g and 5-g die~grams serve as useful tools to deter­
mine acvantages and disallvantage.s , t!lcy do not specify the exact tactics or
maneuvers neetle~. Tq decide what maneuver:! should be employed, a pilot must
be well versed in the theory, and proficient in the practice, of air-to..air and'
air-to-surface tactics (see References 4 and 5). With this backg=ound, a pilot
can translate relative energy gain or loss relntionships into valuable- t:acti.cal
111aneuvers,
( lT) Recently1 ~he Bryson- Kelley methori has been employed to develop best
three-dimensional maneuvers (see Reference 6). This method appears promising
in f-inding sp.e cific o)>timum mancavers for change of direction, rate of closur~,
and combinations th~reof in minimum time or with minimum fuel until ~eapons
launch, However, as presen~y developed, thiG method fails to consider; (1)
the best relative re~ions · \olithin the flight envelope to maneuver and counter­
maneuver against a kno~ adversary; (2) plausible counter-maneuvers by an
adversary as he obs~ves and/or anticipates the optimum maneuvers; (3) a
sequence of plausible counter-maneuvers or maneuvers by an adversary 'for which
a sequence of optimum mi!neuvers or counter-!llaneuv'e rs will be necessary; and
(~) the possibility that more than one optimum maneuver or counter- maneuver
can be flown agai nst a specific counter-maneuver or maneuller,
(U) Decause of these serious deficienc1es, the Bryson-Kelley method can­
not be used by itself to develop valid tactics for the air- to-uir battle, How­
ever, there may be n possibility of ~eveloping near optimum tactics if the
.. qualitative knowl.Jdge of the tactician concerning plausible maneuvers and
counter-maneuvers· is u·sed in conjunction with the quantitative methods devel­
oped by Rutowski, Bryson-Kelley, and Heermann. The Deputy for Effectiveness
Test, Air Proving Ground Center, and the Air Force Armament Labora'i:ory, Rcscarcl1
and Technology Division (AXO), Eglin Air Force Base, Florida, are investigating
the use or t~ese methods for this purpose.
EFFICIENCY
(U) Until now, the discu~sion ha:; been concerned wjth energy stutc (h,V)
and/or en~rgy . rate (P,) in an effort to describn ~neuvarability and to gain ma­
10
CONFID!ENTIAL
(This page is tfnclassifieu)
.. .
/
·.
(:ONFIDENTIAl
ll~'u\·crin;.; ;ul\·;utta;:c. 1\nuwiu;: Htat ~:ucr)!.,y :;tute nod/or ~rlct·~ rat.c ;ulv:.~n~.. ~~~~ tlt:­
J'':rrcl:' UliUII tJtc illt:Cl'll:ll L'IICI:~j; (fud) >IV<Iil:!IJlv1 WC llill now cor.sidt:t· the :lmOUtlt
~1'. i:rt:t•l:r.t:rl cllct·~...- th;rt c:ur lw cuuvo:rh·cl .into •n<ruuuvoring !!ncr~. To ilcc1uirc thi:;
Jill utwrt rou, :r ntat lli:~u:r t. icill ~:xprc::siun mu::~t be ucvulopull which cun~>iclcrs spcci fi~
\'!tt•rcy ;.;aiuct.J Vl•r:iu~ intl!t11al cut'rJ::Y c.'XltL·udctl. Such an expzession is
pit
c..m: .,-.!.
wr
i-1,
P: :.
'-it~re !:.~It: " cncq:y-utaneuvcrai.Jility efficiency,
average specific excess
I"J~>cr, ~, ."" fuel ,;eight rate 11ow1 and w1 = fuel ~.~eight. (See AppendiX II for
rlotailcd development: of ttlis expression, ) TMl twes of &-.'! Efficiency diagrams,
incorporating these efficiency cuntours witl:In the steady..state envelope, may
be constructed. In the first diagraiT, (see Fi~ure 7} constant fuel weight ( ~
illl:ernal) is assumetl. In the second diagrarn (see fi~ure 8) only the fuel re­
Plainins at a givrm eneruy level i:> consitlered, after reduc.i ng the quantity of
fu.:!L by t he minilllUJII amount of fuel needed to r each t hat energy level. 'Ihe
efficiency contoUt'S depictetl on this diagra111 consitler fuel awilable minus '}1.
iulct'[I'Jl L'ucl illld 20 nriuut;t.!s fuel Cor !lest loitl!r speed ilt l01000 ft. Dotil
of thl.!:;o L:-~1 J.:fficicr:cy diil!,'t'ill'llS cun IJe used to: (l) find the must effi­
cient (ta.in:imunr fuel) pnths !Jy employing the SilmC rule-of- thum!: techniques
u:H:d with t;hc l:ncrgy Ra1:c tliagrilti\S lltltl (2) tlct cnnine the amount of internal
CI\I!I'J;Y t}J<lt cu11 IJc cunvcrtctl into milncuwriug energy .ss well os the cffi­
cicucy of that conversion. Since the uiilgrams cnn be employeu in this fash­
inu1 tht!y provide il measure of sustained manuuvcrilbility as il function of
efficiency. Iu il<lclition, C-H Efficiency tl:Ul!,trilmS con Le used extensively tu
llctcrminc rulathc 01dvant~gc~ Of!d dis.sdv.Jntuges of competint; transport ilntl
llomucr clc:sigus. For thl!sc.typc ilircraf-t, loatl f.:~ctor and energy r~'·e art.:
lcsl:i Jmpurt:mt mcil::>urcs uf opcr.:~tional pcl"l:-orm:J.nce. The second E Efficiency
tJjas.;ram is JQorc meaningful, ~inc~ variavle fuel weight is considcre<l throuslt­
.,·ut tho 11.ight uuvtllop<.!. llowuvcr 1 the con:>tnnt !uel E-H Efficiency tltagl,'am
.i:; importaut in clctcrminin~: regions of best efficiency, .independent of fu el
historius. Tilt.! rt:lativc merit of these t1~o cliagrams lo'ill ·IJe ~iscussed in
"Tilcticill 1\pplications 1 n Sect ion IY of 'i:bis report.
'
.
I'
Oy ~~ploying cumpurativc tcehni~ucs, the tactici.Jn can ~~nerally see'
whcthor , fi~htcr 11ilot or his at!versary will conserve a grenter pere ent.:~gc uf
Cuol in moving frocn unc cnurs_y level to llllothl!r. Higher numerical vnluc:; indi­
cate a gr eater percentage of ·fuel r emaining or a s11aller percentage of fuel
consUIIIell. Thus 1 by correlating the £-N Efficiency diagrams with the !:nergy
Rate dia~rams, the tactician can determine to What clegree a pilot or his ad­
versary can realisticaLly maintain or em~loy ony energy state ~nd/or energy
rate ~dvantages, To assist in this endeavor actual fuel percenta~e vulues
can be provic!cu (by the optuu-.:ati.un prosrams) to show how efficiently the
(U)
11
CONfiDlENYIAL
(This page is Uuclas:>ified)
CONFIDENTIAL
70
Es
60
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3
2
10
IV I If
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2.0
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Fi~ture
60
1\
'
1. 2
7.
Constant {m) Fuel E-M Efficiency Diagr am,
.....,
~~~~
~ '/
L
I~J
·to i Jt,'fl- lLI/
I llN 1f-..l.
0
o .2
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8
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Figure 8.
\ \
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1.8
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F- 4C Variable Fuel C-M •Efficiency Diagram.
12
CO NfiDENTIAL
I
.
\
- - - - - - -·­
CONFIDENTIAL
energy transfer takes place. Such information will be provided in "Tactical
Applications"'" Section I V of ttlis report.
RA."'Ct
(U) Thus far, maneuverability has been descr ibed directly as a fUnction
.,f en2rgy rat:e 1 g1 end efficiency, l!owe\'er1 to completely describe maneuver­
ability, we must consider indirect as well as direct influences. Range in­
directly influ~nc~s ~aneuv=rability as it plays a vital part in d~termining
the area of maneuverability available over the earthts surface. Because of
this relatio1:shiP1 a combat' pilot must have a gl)()d but simple measure of his
available range at any altitude-airspeeJ c~nbina~ion within the steady-state
envelope. To gain ~hi~ information, we must consider: (l) the fU~ consumed
and the distance tr~versed in reaching any altitude-airspeed combination and
(2) the remaining range available as a function of the fuel remaining at any
altitude...a:izspeed conbination.
(U) Ry p:roperly considerlng this information, as outlined in Append.':x
ran~e can be developed (see Figure 9). From
this ~iagram a pilot can determine range at any altitude-airspeed combi­
nation including the distar~e traversed to reach that combination. Th~ range
contours depicted on this diagram a;r(: bosed on ·~e same fuel reserve conaidera­
tiol~. uaed in the variable fuel E-H Efficiency diagram.
The shaded area on the
chart represent£ a transient region in the flight envelope. In this region,
aircraft drag, i.e., thrust r equired, is greater than military thrust available
and lesG than minirnum'>aftcrbu.rner thrust. For this reasor, eteady- t>tate
flight is not possible unless some device, e , g ., ~ecd'orakes, is employed to
i~crease drag.
·
II, an H-M diagram oapicting
( U) By using the Range diagram in conjunction ldth the other energ;.r­
maneuverability diagrams, the tact:ici;m can dete:rmbe to t~at degree a pilot
or his adversary can realistically gain advantage consistent witit distance
from friendly airfields or tanker suppo;-1:. tn .iddition1 planJ>ers and de.
Digners can evaluate the t .rue operational p~rformanc£: of transport an~ b~ber
type aircraft by considering t he Range diagram along with the E-M Efficiency
diagrams.
l3
CO NF~DlENiiAl
(This page
is
Unclassified)
I
. ·--· -··--· - - - - -­
•
···
CO~i FI DENT IAL
'·
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60
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"·L
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40
100
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-
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.6
R• noutic:al miles
.
1.0
I.Z
.
.
1.4
1.6
1.8
2.0
Z.2
2. 4
WA CH
Figure 9; · F- 4C Range Diagram,
T
·,
..
I
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o '
• • ,o•,, , . ,, , , , ' o
t
t
'
I
.,
'
..
..
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-
14
CON FIDENTIAL
····· . ...
..
SECRET
..
SE<..'TION IV
TACTICAL
• ,I
'/
APPLICA!lO~S
( U) Ir t he I)Ocrcfy-ntancuvcrahUi ty concept is incorp oratel.l in the present
l, no~o·lc.dc;l) on 1'\.~llt~r tacticl;;, a pilot can be ptovltle<l rnore.m::oningful infor­
~~ation •Ill lww hi! ~h·Hlltl utaul!uver .to gl!in atlvantll::e. 'rhc resulting infonnation
~Ll L tht.m reveal to t hl' pilot ho1~ he should best exvloit the maneuvering capa­
I
I
or his
aircraft. J\tlJitionally, valid compqt"ative analyses can be
i•<.!l'!'onncll if :>imilar tliagraniS arc constructed fot potential enc111y fighters.
l>i.litic:;
I
'l
'I' he xdot:i V<.! r<?gions of <hlvaotage or tlil>atlvanta:;e are f ound in t ern!! of g,
cncrc;y 1'il\~, efl'icicncy, ami ran~c l>y performdnce comparisons throughout the
flic;ht cnvcl.opc. Fro1!l this compa:ison; the t actician or pilot can easily
llelcnuine wl1 ich of to~o aircraft has the advantage in tc:nns of' ~nstantancnus
lllll llcuver<~bility1 s•1stainel.l 111~n9uverability1 and range. Usin~ this information,
t he tacticiiln can determine ho\~ to best m~neuY.er for advanta::e.
for 'a sample comparisoo, we s hall consitler the f-~C ver sus the Soviet
mc~~?l and tlett!rminc m<:~neuvcring <JC\'unta~cs and disadvantages . The conditions
for <;umpa~1s~'1 will be typical air- to- air configurations, with 5a/. internal
fuel., unle~t\Specified othen.li:>l!. In the G-V dia~rams (figures 10 anl.l U} , t he
a~toclynamic g limit o( the t!IC- 21 lies to t he left of the same limit· Cor t he
f_l,c, At a glance, figur es 10 and ll indicotte the NIC-21 has an eno:nn~ n
c;taat .:meous "'fllo:uveFai>ility advantage over the F- 4C. These llia;;a111S""illso in:.:
dicate that t he HIG- 21 has an advantage when cotllparioi; struc.ttrial limits . 'I'hP.
l-c; Ener~y Rate diagr allS (Hgu:res 12 anl.l U) show that ,tl)e. l•llq~21 l1as .t he ad
va'ntac;e within most of the subsonic portion of the fl:ight envelupe and throt,~sh­
out all vf the s~\,ersonic portion of, the flight envelope. The only r~sion of
advantage for the f -4C lies i11 the sul>:;oni < ·nd transoi11c areas below 13 1 000
feet. The' ma~nitude of the rnaxinJtJJn pr.we~ enerl:ly rat e ~dvant<~ge can be tlctcr
~•inell by .'consul.ting Table I .
N
NASIC/ACAA
DECLASSIFY
(This
information no
longer
needs to be
clas~1fied) ..
..
I
... '
TfiB.Lt .I ~ . RtnOI~SKl HINIHUN TIHE PI\THS ( HJ\XUI~I POI~ER)
'•..
.
' E, "'3, 000 ft to E,
1o·
,ooo
f t t , = 3 1 000 ft to £,
"'
f uel
I
. . Fuel
\
Height._
• h'eight
Fuel
• \
',
,,i.rc1·a ft
Useu
Used
Ti111e
, Used
: Tieo!C ' '
( lb)
' (sec\ '
( sec_}\
[\_(~)
{11>~
32
19?
525
~~) 5
[\ F-4C
·y
. \ .,
.,
.,
...
...
. \. ./
I
\tlC~l
-
167
-
l 58G
...
ft
f uel
U:;e\.1
_(_",.}_
/
l4!>
\ ~6
'
t
..
\
=95, 000
/
.....
SECRE·I
.....:-::::[;::::::;::::::::::~ ~:::::.
8
7
n:::::::·:. :::::.
.. •'"
,:.;.:'·'B
w.
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:;::¥ [8
a:
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3
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l
0
..
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::~:::::;
:;::::::~
~
•3
} JASIC/ACAA
PECLASSIFY
::::::::
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.
. )
v
/
~:::::
I
100
.........
''
201)
::::::::::: ::::::
31)0
400
~00
CALI8RATEO Alf!SPEE O
1
,(This Information no
}fmger
600
TOO
BOO
KNOTS
Figure 10. F-4C G-V Diagram at 30,000 Feet.
needs to be classified)
[!.
ll
0
•. ' 100
Figure ll.
HIG-~
G-V. Diagram ·l\t 30,000. Fee•.
SECRfET
SECRET
..
•
r-t--.._
70
0
90
80
~-."'
I'"r:::
.. ... . . . "~
..
~~-~~-
·· ~
~
-
100
110
IZO
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130
140
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f\
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1.8
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z.z
2.4
MA CH
Figure 12.
F-4C
~axi~um
Power 1- G Energy Rate Diagram,
~
NASIC/ACAA
DECLASSIFY
(ThiS information no
ionger
·needs to be
classified)
•·.
30
z
..
I
0
I
\
Figure 1.3. H!G .21 Haximllll1 Power 1-G .Energy Rilte Diagram.
I
17
SEERfEY
- - ---- - - -- - - --------------------~-------
\
\
.SECRET
In figures 14 through 11, 1--c note that the r -4C retains most of its lol/-:l.ltitullc
suusonic/transonic sustained maneuvering advantage as s is increased from 1 to 5.
In. additi.lln, these fi~uras reveal t llnt tht! mG-21 not only Ita;; a super!>onic atl­
vantage, but alS•l i~ gaining regions of advantage subsonically since it~ can pull
~ in rcgillns ,,here t he r - 4C cannot operate . Essentially 1 such a condition i mli­
cates the NIG- 21 can tun1 mur e quickly t han, or inside, the F'-4C. This samu
c.:onclusion 1-.ras reached uy studying the <l-V uia~ram. Logically, this means the
left-hand uountlaries or t he 1-g, ;i-g, and 5-g Energy Rate diagrams provide a
~toasurc nf instantaneous maneuverability 1 while entire .diagrams p~ovide a
measure of sustained tnancuverability as a function of energy rate and g.
\-&.}. The military po~o;c:r 1- tl Energy Rate diagrams (Figures 18 and 19) sho1
t~e r -4C has the ability to g<~in ener!Jy more rapidly th\ln the HIG- 21
throu~hout most of the envelope. The magnitude of t his advantage i s tlepictP.d
in Table II.
that
TA!l!.E II.
RUTOI~SKI
HlNUliDI
nol~nger
neells to be
clatifted)
I
I
PATHS (HILITMY
POI~!:R)
&, = 3,000 ft to £, :: 45,000 ft
NA~IC/ACAA
DEdLASS IFY
(Tht~ informatton
nm:
Type
Aircr aft
fuel
l~eight
Time
(sec)
(lb)
Used
Fuel
Used
<'-')
12
.309
17 .
~ By cons~t1ng the mllLtary po~er 3-g and 5- g Energy Rate diagrams
(Figures 20 through 23) , we observe that the F- 4C has a su~tained man~uvering
advantage at the lower altitudes and higher Hac-'t numbers, These diagr ams also
reveal that t he HIG- 21 regions of advantage spread to the lower portion of t he
envelope as g is incr eased fro1n 1 tQ 5. In addition, these diagrams (Figures
20 through 23) :reveal that the HlG-2.1 can maneuver in regions unavailable to
the F-~ C. As mentioneo previously, such a condition indicates that the HIG- 21
can outtunt the F-4C,
"(-S.) From Figur~~"24 an~ 25, we observe that the F-4C has the subsonic
advantage, while the HIG-21 has the super s.1nic advantage in terms of t he
maneuvering energy gained versus tne percentage of internal energy expended.
On the other hand , the variable fuel E-H Efficiency diagrams (Figures 26
and 27) XI<!Veal that the P-4C increa'ses its subsonic advantage and acquires
an advantQge through most of the supersonic por t ion of the envelope. A
natural question ari~es at tliis point as to why tl1e F-4C appears to have
a sreater degree of advantage in t he variable fuel diag~<ll11 tnan in the
18
SECRET
..
l
SECRET
I
I
I
.,0
Es
10
·- ·- - r-y....:::
r-r--- ~r::::,...,.r-::.r--::
6 i-IC
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90
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110
i
I
120
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1
VJ~/'~~'~ =- 'X~;;- 1\
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figure 14.
.6
.8
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v/\
IJ, ~
10
1.2
\- ~~ \
i\
14
16
\
r.e
1\
2.0
\\
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2.2
2.4
.WACH
F-4C Maximum Power ) - G Energy Rate Diagram.
NASIC/ACM.
DECLASSIFY
(This i"nformation no
longer
·
needs to 6'e clas~1fi eQ
I
1
l
I
MACH
Figure 15. MIG-21 Maximum Power 3-G En ergy Rate Diagram,
19
S EC~ rtf
- - - - - -·--- - -- -·-----·----- ­
- -·- - ·-·
- - -_j
.•
,. ..,
SECRET
,_r--
1
6
r- :--
4
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30
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2.0
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-~ b
ri p... r- 1
,
It.2
En er~
Z.4
Rate Diagram.
r-- .L
II" .......
I[_ 1'-:v
v
.6
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t\
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10
z
\
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1~0
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\
· Figure 16. F-4C Maximum Power 5-G
NASIC/ACAA
DECLASSI FY
(This information no
longer
\
needs to be
IZO
['.
t'--
./
'"'1 r- :::::,
. •2
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t'---
r--.
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110
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100
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- r- r-
a
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90
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2.4
III'ACH
. Figure 17.. HIG-21 Maximum Poi.'E! r 5-G Energy Rate Di agrsm.
·SECRET
70
'
1
.
r-­
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0
50
--
......, ._..,
;-....
...............
r--r-­
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1.1
1.0
MAC H
F~gurc
NASIC/ACAA
DECLASSifY
(Th is information no
longer
.
needs to be classifi
c,
F-4C
18.
Power 1- G Energy Rate Diagram.
-
•
..
!--..r-..
-
., ·r
-·.
..
'
30
./
20
. .17
. I i/
10
0
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1.0
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1.1
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Ill ACH
Figure 19.
MIC-21 Military Power 1-G Energy Rnte Dlagram.
2l
SEC~fET
.......
• -!- ·,
.
'
I
•
•.
····:t
'
:
SECRET
I
80
10
r--
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60
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-... r-- ,.__
-
·-
IC
r-...._
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1.1
1.0
1.2
MACH
NAS IC/ACAA
DEC LASSIFY
(This information
no longer
needs to be
classified)
Figure 20.
.\
F-4C
~fili tary
Power 3-G Ent!rgy Rate Diagram.
80
10
£~
10
·- --
!10 .
'
H
so
..
10
-
00
'
.s
·.4
.....
.. .........
b
10
.._ :::::: lc:::;
..,.,.. >
_.... :.-"
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;::.- :::;::::
-
r- r-
·'
' I.UCH
7
'
........
........
o-....
r-­
. .......
........
r--..
. ..........
. .Ra~e
.
1.2
1.1
Diagram.
' .
..
/
SECRfET
-··-- ---·-· - -
~
..
..........
1.0
~
22
- -· ··--- ,._ ._.,.,.., _________
..........
..........
1
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t-....
J• 't-....
......_N
'
_
..........
.
HIG-21 Hi.li tary Power 3-G Energy
.
~
.......
..:. :it ~ ::
:- ':-.
h
r-­
r-r ...._
,;....
v
r-
I
I'-­
I'· r­
r­
..,1--c r-c: b-< .-
..,.. v r.!')"-
figure . 21,
-.:
}' v
' / .
J
..
:,?
.i .
-
!10
1--
~
....
/
20,
--
.,. ,,...
-:-
--­ --- -
r- '·
·­
.
..
SECRET
10
'70
E, .
10
~
so
:
....
H
....
30
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----
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2
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.)
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r-­
r-­
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.....
r--.
1"--~
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t- b
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r- ....... 1-
r- ........ .t. ·II 0
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r-::: It!': ~ I
l:;Zl ~ ~ -.......
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t
1.0
........
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........
1'­
I­
........
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......
r--.....
r--...
1.1
1.2
loUCH
figure 22,
F-4C HUitary ?o"oer 5-G Energy R?te Diagram.
I
0
2
.3
.•
.
.s
·'
:1
e
.II
1.0
1. 1
I
1.2
MACH
figure 23.
MIG-21
~lilitary
Power 5-C Energy Rate Diagram.
SEC~ lEY
~---------------------------------------------~-------------
SECRET ·
--
\
-
I•
MA CH
Figure 24. F-4C Maximun Power Constant
NASIC/ACAA
DECLASSIFY
(This information no
longer
\
needs to be
classlfied)
{5~)
Fuel E-H tfficiency
Diagram.
•
70
Es
.•
.·
t-:"t--
..
5
;o •
. , 1:----b.., 1 :--~-..
6011--4-"""=-t-:-4---+--' ~
r-r-:- 1--
eo
- ~ --~k~ •. !':..
c
'" r -
b::;;
~
1-t-- .-;.~~ "~
100
uo -
1":'
110
"
· '\
140
I\
1so
K \
0
\
r- 1\ 1'\
. ~~ 1\ ~1\
J'.... 1-
~~ -~~
o•
120
~
~
~
1\
'0
2
.
10
.4
.G
.8
1.0
1,2
1.4
106
;,e
2.0
2.2
2.4
r.I ACH
Figure 25 • .MIG- _21 Maximum· ?.o~;~er Constant (~) Fuel E-H Effic~ency
Diagra111.
24
SECRET
.
.
SECRET
70
6
'~--....
r--..
t- r- f-..
IAf! ~
~~ ~
.
100
"~-...
130
140
. \
...... ~
" ~ ''\ ,\
::-f>'<
.....
~..,
150
~
\
1\
\
\
.... -+- ......
I'
-~ ~
~
120
'\
~~!% ~ ~;\r-~ ~
t-
110
\
· ~'-
r-....
...... "
t- t--
90
80
-r--t--1--
1\
~\
1\ \
['>..
N ~ ['\ ~
1\
1\
1 }'iJ :>.. f\ \
.,
}J}7.~ -~ ~v~ ''\ ~ ~ .
f-.. WPV:17
~ '\ ~ ~ ~ f(( ' \1
" I\ \j
t W'JI7K: v"" "'- r ~ ~ v
,: If\.
H
3
-Y/1rlf
0
I
h 'fi.Jl.l
II Vf lr-1-.
0
o
.2
r-c­
.4
~
.6
~~
~
Jj
~~~~
f.I V.@
.6
1.0
1\
1,2
\
1\
1.4
f\ l•
~
Nl f\
~ . ' _\ ~
!\
1\
1.6
1\
\1
LA~I 0 l
1\ \ \:
1.8
2.2
2.0
..
2.4
MACH'
figure 26.
NASIC/ACAA
DECLASSIFY
(This informatron no
longer
\
needs to be
classrfied)
~IY I / ~
llftf.V f7
/
~
l
2
\
P- 4<: Maximum Po~-o-er variable Fuel E-H Efficiency Diagram.
·.
10
F:s
60
I\
90
80
1.;;;:: r-- .._
"'­
"r---..
t- ~
~
I
"
.. ::....-
~
40
) %? K ~,
. r-- ~ ·_// V ·V':-\
30
I ii 'It-!. .......
lh f../ lL 1/
,....... lJ
H
.'Ill
2
0
-
00
f l_gure 27.
~
~"'--
7thf-J
II I /. lt'If Ff;. II 1/' ./ ~
III
~v
2
.4
.6
8
120 · tlO
"'
\
"
~
~-
I'-
" !?'
11 K
..... I)
'~
..
~
§;; [? 1\
14 k'L 1'\CJ I,AO
rf.7,['>, IZ
I\
[\. 'r\ .
'1.0 -
1\_
I2
1\
\
1.4
16
1\
· .I 1\
..,...,~
\
f\ \
\
I\
\
\
\
1\ .
I
'\ I
!
~
1\
I6
. ·~
1\
.1' iT J\
1\
1\
"'.I \
.\
.\
~
"' 1\
[\ ·
1\
1'\. 1\ 1\.
•t,
.}
~
~
~· ~ .... r\ !# .·\
P'
/
"""
"
'<::::
150
-~
i.-1"<:
!::.: :::s r-
r>.
.
140
'\
-
!/ ~ ~· !'...
('-.
r- t-- ~ ~ ~
110
100
I',
1\ \ \ I
2.0
2.2
2 ,4
MA CH
. .
MIC-2t Ha.ximuill Power Yariabie Fuel E-l-1 Efficiency Diagram.
SECRET
t.__ _ _ _ _ _ , ________,_,_________
_
- - - - - - - -- · - - - -- - - -- - ­
.
4
.
,.
_
________..
-·· ·- -
- -....
SEERET
.­
0
constant fuel diagram. The greater advantage attributed to the F-4~in the
variable fuel diagram result:s from the fa\!t that the F-4C has e:r.pen·ded a
small~r percental!-:. o{ its .f"uel whE:n it reaches the regions i.iler~the !.UG-2.!.
is more efficient. Certainly such an advantage would not be realistic if tht:
f-4c" entered the engager.1ent with a sr.~aller percentage of fuel on boartl andfor ·
the engagement star ted at a high ~ubsonic energy level. If such a condition
existed, the efficiency advantage attributed to the HIG- 2l in the super~onic
region ·of the <'onstant fuel diagralll ~ould be rt>alir.tic. This point of dif­
ference becomes important to the F- 4C pilot wht'n he encounters a MlG-21 in
hostile territory. · From the milit.l;ry power &.k! Efficiency (liagrams depicted
\in Figures 28 throul!h 311 we see tllat the f- 4C has the arlvantage in both cases,
assuming, of course, that ~ach aircraft has the same assumed or starting fuel
percentages on board. Actual t ime ana percentage values of ~el expended
alon~ mini~um fuel paths between energy levels for these two aircraft Jre dis­
playe~ in Tables Ill and IV.
TABLE III.
r. -::
Type
1\ircraft
NASIC/ACAA
'\f­
DECLASSIFY
(This information
no longer
needs to be
classified)
--
RU'!OWSKI mNIHU:-f rutz. PI\TliS (MAXIMUM POI1'ER)
),000 ft to£.
Fuel·
Weight
'rime
u~ec!
(sec)
(lb)
252
}206
HIG- 21 ·
~
70, 000 ft £, = ,,000 ft to E1 = 95,000 f\:
fuel
Fu~l
Weight
Fuel
\
Used
US<!d
Used
Time
( lb)
(i)
{sec)
~%)
28
370
~70
291
1713
44
""fffflf+
'TABLE IV, . RlffOIYSKI M!NIMUH FI/EL PI\THS (H1LITARY POWER)
r----------~----E ~-3-ooo
-____
ft to r.
'fype
Aircraft
Time
= 45.000 ft
Ft:el
Wclgnt Used
-
Fuel
Used
(sec)
( lb)
' -'>
)25
1292
ll
618
16
1
~ The Range diagr a11s (ficures 32 and 33) r~veal that the F-ht: ha:; a
s•Jbstantial arlva1~tage in the subsonic portion of the envelope and a lesser
advantage in the supersonic portion of the e:welopc.
•
0
..
0
r
\
'\
'
'
·,
SECRET
....--­
·-----..._
I.
•• , ••
, ... i
•
MACH
..
Figure 28, F-4C Military Power Constant (!i~) Fuel E-M Efficie'ncy
&>ia~ram.
/
/.
NASIC/ACAA
DECLASSIFY
(ThiS 1nformat1on
no longer
needs to be.
ctassif!ed)
·•.
LJ
.
"
Figure 29.
·~
I
I.Z
····~
..... ..
.
HIC-2l .Hilita.xi Power Constant (~) FurJ: t-H Ef fi­
ciency Diagram,
I
,.
.. ..
,..,·
\
____
.....
SECRET
80
• TO
- ·-
Es
r- t-
60
ko !<a~
I
40
30
[..<:;
-:;r--.
~ ~!;::
~~
J~ ~ t/
20
~
rz
/
[ti: 1,1
10
zv
v
~
v
,......
,.,
1-"' 1--
/
';?' f.:::
v
::;>
171--t-- y
VI 'J I
.)
v
~ t-- b<
//
2
6
:1
.•
~
T
v
,l_0
­
'
1'--
r-..
r--.
1 ,..
.'
i~~
t-......
w: ~"--t-...
!--...
.......
~~ i'­
17 1--1--J
1/
.6
:­
--r-­
t-- h.
~.n i\\,
!to''
-­
t-- !'-...
i-l•n
•w, ~..-,: ~~
1....-:: ~ E::::: !:"' J< k
1'
r--
:-r--. ,__
f~ ~ 1?:: ~ ~.'?!~
H
,•
t--
r- t- t--
/
'0
--~-- t-
.~
@
r~
t.O
1'-­
I'-..
1,1
1.2
Ill ACH
~igure
NASICIAC AA
DECLASSIFY
(This info rmation
no longer
need s to be
cla ssified)
30. F-4C Mil itary Power Variable Fuel
~M
Efficiency Diagram.
c,
&0
''
50 ·.,
V.
II
,2
Fi_gure 31.
..
I
'II
v·
w
..
r1
r7
·.3
"4
HIG-21 Hi litary Po.wcr :variablr, Fuel t-H tf~iciency .
Diagri.lm.
28
SECRET
'
SECRET
70
eo
r- r-
..
60
....._
~
f- f-
..
. . . D ;r
c
o
60
\
~\
:."'
~
f\ \
A\
~
\
1\
~
.4
a
.6
r.:z
1.0
1.4
z.o
1a
1,6
2.2
2.4
MACH
F.4c. Range Diagram.
90
80
r-- t-- r-
- · - -..
.~
1--
- rt
I
:I
. f!lJ
"'1
·•
~
.,.. ~j,....._ ........
.4
""::-
""':::::.: ~
~< ln.
1--..
L
: ;
P.-.
.... ~- ~ ..;
'\
K
\
'\
>< · 0tO<· ;.:... .
I2
1,0
•.4
\
·I\
\
\
\
•
I.
.
j
l' l
1\
' .·\
.1,6
'\
\
y ~
1\
..7 " .I '1\ '
;...--
v- ;)
\
'!"- ~l ~CA ~0 l ~ .r
il'. ~
140 '1'0
\ .
,..:...
J ..... ~ ~\ ./ \ , \
1 :--... '\ / 1\ 1\ .
"-.!
.6
-
130
]\ · ~ \
· ~
'~
' .i'. .! .
,-......
j- iOb ..!..
1 .. .::::
.,' !.'- '
:..>"•
..
~
--:....::: ...... . '
~ =·
.
"
"
120
110
r .:· ; f r\.
r-..;,
_t,.:,
· ~
r.;..
101.
....... §
H
~
::--0
100
-~
b., .
~ t-- k:
0
0
"\
~
.2
5111
2
~ I:-' 1-- ~
"'
Jll'
70
Es
t\ \
\
t.<r.l-' ~ I>
~ t'"-- ll.
,,0
130 . 14 0
i'\.
-....,
~
1"-
Figu.re 32.
NASIC/ACAA
DECLASSIFY
(This information no
longer
needs to be
classified)
~
--
2
0
"'··"- ;-k
"' -
['.,
120
110
. ":' ::. N t--. · tlv' ,I \
... .
~
.
·:; v ~
?f ~..
~ '\ \
rr ·~ 0 'II'iJ :;:". .. I'\.' )(.. ) ~s... ~. 1\ \
r-- t-..
!,...-' .1( ..... \
"'"t 'L \ !00 '
\
\
v
ff/ \ ......... r-.... rJJ K
\ \ \
.J
\
-c.: ~
. '\. . \ / <-> PL F\~ D WI \
i.tt.J .......,
~
I
..
\
·~.- y \ 00 (!"'
~ \
~
i\
iJ
. / ....., J.;" '::}
\
1\
00
i J / K ~ >ef'- :,t1 /
1\ \ 1\ l \.
--
0
--
1'--
.......
r--...
r-- r- ...._
.
100
90
['-...
\
.
.,
1\'
\
\:
.1!J,
\
II
\
\
1\'. \
.\
1\'
1\
,
'\
r\ \
\
.
:\
\
\ \ \
2.0
2.2
2.4
MilCH ·
Figur~
33 • .HIG-21 Rnnge Diagram.
29
SECttEY
-
- -..
-·- - ··-~
- - - - -- - __ ___
,_,._ ,
...
I '
SECRET .
~ By comparj.ng th... rule-Of- thumb performance of the AL'l-98/AA.-2 and
the AU!..7E ( see Appendix I), ...-e find the F-4<: with four AI H..7.& missiles has
an enormous all-a5pect, first-shot advantage over the HIG-21 equipped with
in~ernal gun(s) and two AA-2 missiles. This advantage prevails against either
a ·maneuvering or nonmaneu\·~rir.:J HIC~21 • . Kowever 1 in spite of tbis advantage,
an F-4C pilot may find it difficult to employ AIH-7E d~ ~ng a re-attack or in
an effo~t to nullif~y an attack1 since the HIG-21 can easily outturn the F-4C
as well as m~rntain more energy While doing so. By exploiting this dual ad­
vantage, a skillful MIG-21 pilot may prevent a successful AIH-7E missile
launch by simply maneuvering away from the front toward the rear hemisphere
of the F-'IC, for close..;in maneuvering, the F-4C can mount a 20 millimeter
centerline. _gun pod in addition tc the four AIH-7 lllisRiles in an effort to get:
inside the missile minimum .firing range restrictions• . Ho"!lever, such a fix
r esults in an ~ven greater tnargin of maneuvering superiority for the MIG-21
by reducin~. the already inferior instantaneous and sustained moneuverability
of the F-4C . 'The w.agnitude of thi:~ maneuverability loss for the F...li<: can be
det~ined by consulti~g the energy diagra~s in Volume III of this repo~t,
to be p~blishod at a later date.
tsJ. From the foregoing analysis, it is clear that the MIC-2::&. enjoys an
enormous instantaneous mane~v~ring advantage and . a substantial sustained
cuneuvering advantage in t~rms of energy :rate and g thl·oughout the supersonic
portion of the flight envelope. Subsonically, at both maximum and milita.ry
~ower, the HIG-21 has a sustained w4neuvering advantage in the upper portions
of the envelope that spread to the lower portions aa g increases. On the
other hand, the F-4C has a sustained maneuvering advan~ge in terw~ of effi­
ciency throughout the entire subsonic portion of the envelope extending
through most of the supersonic envelope . Only in range and first-shot capa­
bilJty does the r~4C enjoy a substantial advantage over the ~G-21.
(If) Naturally, for a complete analysis, additional. information must be
developed . The tactician needs energy-maneuverability diagrams for various
type combat c:onfigurations. In addition, he needs comparative missile firing
envelopes together witn radar and maneuvering constraints that may be impo~ed
on the pilot .a,r radar operator. If this information ia provided, the tacti­
cian can design tactics by using e:1ergy- maneuverability methods. Assuming
that !'oreign Technology Divisions can provide reasonably accurate data con­
cerning enemy perfo~nce, the tactician, for the first time,can develop
effective tactics against any advers~. In addition, tactical commanders
con use the energy-maneuverability co~parative analyses to gain meaningful
perspective for decisions concerning the employment of friendly fighters
against a known enemy.
}0.
SECRET
CONFIDENTIAL
SECTION V
REQ!1IROO:NTS
(U) Presently, in order to meet mission requirements, Atr Force planners
direct that new designs meet certain specifications ·in terms of altitude, air­
speed, acceleration, g1 and ranse. Cont ractors, !n an effort to satisfy the
customer, produce destgns to meP.t these speci:ications. Howeve4 1 no guarantee
can ue [!lade tJ,at the design selected will be the be:;t one aince such specifi­
cations are pnint data (derivatives) and provide no i~dicatton of an airqraftts
integrated l'erformance anu design effic iency throughout the flight envelope.
J
(U) Ho-..ever1 by npplying energy-maneuverability techniques 1 along with
other infoNation dee~:~ed necessary by the tactician, planner~ ~.Uulci have the
advantage of looking at complete performance (including the previously men­
tioned point data) before making decisions concerning aircraft requirements,
As a result, true ope~·ation~l need would be considered by both planners and
designers in determining the best overall combination of armament, engine, and
airframe in future designs .
,31
CONfiiDIENYIAl
L---------------------·----------------------·-·----------~--------------
I
CONFI~ENTIAL
I
APPENDIX
f
RUU:..OF-THUMB f'£RF0Rl1ANCE CA?i\Dli.ITY OF AIH-7!: AND ,qiH-9B/AA..2 Hi:SSIL!S
1GJ The Air force Armament Laboratory (AFATL) 1 Research and Technology
Division, AFSC, Eglin Air Perce Base, Plorida, AIM-9B six-degree-of - free dom,
digital comflllYu!:te!j1r~rn.~:~~~~~~~~~~~llLE!:I::....£lne....lUIJwu:ea..~=~
E.O. 13526, sec1ion 3.3 b)(4)
e
ompany, Bedford, Massachusetts, by means of 3n
analog computer1 produced launch env~lopes ~d a parametric study of AIK-7~
11\is · d agalnst targe-rs atso pUlllng from 1 g to 5 g (see Reterence(J , A
eon
spot che!!k hy AFATL ' s AIJ{...7E fiva- dcgr r.c- o l.'- frccdom, digital compu'ter p:::ograr
sho~<ed excell(lnt agreeme11t witl1 the Raytheon Company r esult s . An analysi s of
t:he A.IX-7£ ond AIM-9!1/AA-2 launr.ll envelopes revealed t he ru.lo..of-'tltl.llllb per­
formance presented in t llis 11ppendi.X f or these m1ssiles . Typical Llur.ch enve­
lopes used to develop tlte r.Ue-of-thu:ob performane:c are shown in rigta"es :-1
ond T,-2.
~ By carefully noting trends or pa~erns, the rule- of-thumb perform­
ance of the AIK-7t and AIH-90/AA-2 missiles can be furt her simplified by
t actical organizations for operational use.
AII'i.-'(1: HI.SSILtS
~
(~B) ,
MHfHl(,'H R.I\NGE: • . Hfnimum :range ( R, 1 a) for nose quarter ( NQ), ob~am
and tail quarter
(T~)
Altitude
( ft )
attacks:
Type of fltta::k
NQ
r.a
( :!"t)
( ft)
'i'Q
( ft)
SL
8,)00
7, Eoo
10,000
9,250
8,100
s,4co
20,000
.30 , 000
10,100
9 , 200
6,300
4o,ooo
o,t.Co
a,4oo
12,eco
9, o€ o
5, 100
5,.300
6,eoo
For t urns into the attack, add 1,000 ft to the above valt!cS.
from the attack, subtract 1 1 000 f1: .
for t urns a>~ay
}2
CO Nf~D(ENYIAl
~-------------------------------------------
CONFIDENTIAL
ue•
It f llUH
n . na
uu•c• u cr e. r
Tlll ll IICW
UNIIIUIIAl
Figure I-1. AIM- 7E Missile Launch Envelope
1. 1
fl
CONFIDENTIAL
·-­
I E.O. 13526, sec tion 3.3(b)(4)
Withheld from public release
under statutory authority
of the Department of Defense
FOIA 5 USC §552(b)(3)
10 usc§ 130
Figure l-2.
AIM-90/AA-2 Hisai le Launch Envelope
3~
L_
CON fiDtEMT!Al
·-- - ---- - -- - -..··-·--
·- - -
.,...,---
···- - - - -
- ·
CONFIDENTIAL
~ HAXIMl/H RANGE. Against a Co-Speed Nonmaneuvering Target. Haximwn
ranse ( \.•) and angle-off ( <l~• ) for a nOS!! quarter, abeam, and tail quarter
::.._.__ __ _ _ _...;:a...;:t"""'ta:..:c:..:..:k--'a=Saiust a co..speed noManeu,vering target:
·
Subsoni.c Target - Mach 0.9:
--
'
Altitude .
NQ/f.
AB
(ft).
( ft/deg)
( ft)
TQ/f.
·( ft/deg)
.SL
&>,ooo/lo
10,000
20,000
30,000
82,000/10
82.,000/J2
15,000
22,000
)0,000
19,000/30
82,000/25
4o,ooo
26,000/30
4o,ooo
82,ooo;r.o
52,000
35,000/30
9, 000/30
1.3,000/30
For a subsonic target Mach o .5 at sea level, NQ/'l:.-·6o,ooo f'/10° 1 A3:: 24,ooo
ft, and TQ/<f: = 181 000 ft/30°. For each additional 10 1000 ft, adc! 5JOOO ft to
NQ, 8,000 ft to AS, and 51 000 ft to TQ.
Supersonic Target:
Altitude
( ft)
SL
10,000
20,000
30,000
4o,ooo
NQ/<
(ft/deg)
82,C<>O/l0
82,000/10
82,000/15
82,000/30
82,000/45
AB
(ft)
TQ/<
( ft/deg)
12,boo
20,000
30JOOO
14,000/30
4o,ooo
28,000/30
7,ooo;-,o
20}000/30
~ i.aainst Nonmaneuverir\g Targets with Attacker Velocity C.reater or
Less than Target Velor.ity, Maximum ranges for nose quar.ter and tail quarter
attacks against nonmaneuvering targets with attacker velocity greater or less
than target vel<Jcity (delta Hach) follow:
Nose Quarter Attacks:
feet
~nen
l, Add 3,000 fe~t ~o R1 1 a for each 0. 1 delta Mach below lO ,COO
target velocity is greater than attacker velocity,
feet
~1en
2. Add 11 500 feet t? R•• • for each 0. 1 delta Hach above 10,000
target velocity is greate~ than attacker velocity.
velocity is
3. Add 1 1 000 feet to R.. a for each 0.1 tlelta Hach when
than target velocity ,
gr~ater
.35
CO NfiD!ENiiAl
&~ttacker
CONFIDENTIAL
Tail•Quarter Attacks;
l.
Aud _21 0UO feet to R••• for each 0.1 delta mach rate of
2.
Subtract 3,000 feet from Raaa for each 0,1 delta much
closu.r~.
separation.
~
1\gainst a }laneuvering Target;
Nose Quarter Attucksr
1.
Reduce R.u ;o1 0oo ff:!et ~\en target maneuvers away from
the attack at 2 g.
2. Reduce Raax 51 000 feet/g for target maneuvers away from
the attack with g greater than 2.
~
Mancuvers .Away frow. Attack, Tail Quarter Attacks•
Altitude
Below 20,000 ft
Below 00 1000 ft
Above 20,000 ft
Above ro 1 000 ft
Target G R( au
3
a)
5
3
2/3 Rau
l/2 R.. 1
l/2 ~ax
5
1/3
RuK
T~e above values do .not include background clutter associated with a target
at low · altitude.
E.O. 13526, section 3.3(b)(4)
Withheld from public release
under statutory authority
of the Department ofDefense
FOIA 5 USC §552(b)(3)
10 usc§ 130
36
CONfiDENTIAl
~~ONFIDENTIAL
,.
I E .O. 13526, section 3.3(b)(4)
With he ld from public release
under statutory authority
of the Department of Defense
FOIA 5 USC §552(b)(3)
10 usc§ 130
I'
I
I
I
. 37
.
CO~f89fEfi\fi8Al
I
I
CONFIDENTIAL
I E.O. 13526, section 3.3(b)(4)
Withheld from public release
under statutory authotity
ofthe Department of Defense
FOIA 5 USC §552(b)(3)
10 usc§ 130
38
CON FID!ENTIAL
'\
'
APPENDJ,XII
HATHOIATICAL DERIVATIONS AND MODELS FOR DEVELOPING
ENEJtGY-MANEUvERABIL!'l'Y THEORY AND ASSOCIATED FLICHT PATHS
· (Appendix Il is unclassified in its entirety)
!n this appendix a discussion 'of t he mathematical methods employed to
develop the Energy-Maneuverability Theory and associated flight paths in the
altitude-Mach number plane will be presented, For convenience, . tP~ derivations
will ~e described in terms of the following computer programs ~~ch have beer.
fonnulated to hanrlle the computational asperts of the t!leory:
Part I
- B;.sic Energy-Maneuverability C:9mputer Hodel
Part II - The Bryson-Kelley Steepest Ascent Optimization Program
Part Ill - Dynamic Profi le Generator Program •
/
39
/
,------·
T•AR!r.
1!:'\S IC EN£RGi'-J.!ANI:lJV£'RABILITY COHPU'I'ER HODE:L
DERIYIIT!ONS
INSTANTANEOUS HANEUVERABU.JTY. For any given aircraft, maximum load
factor (nonnal ac~eleration) ma.v be computed as a function of altitude and
airspeed:
"L.. qSCL
Hll
\.;
where
I
"L =maximum normal acceleration (dimensionless)
q "' ~
p
..
r.1
'F 1 dynamic pre:ssu.;e ( lb/ ft2)
atmospheric density (slugs/ ft~·)
V =true airspeed
I
I
/
·.
(1~/sec)
S "; refer~nce wing area ( f&)
cL, :.x ; maximum coefficient of lift ( dimersionless)
w -= aircraft weicht ( lb)
Since calibratP.cl airspeed (CAS) is more ~eaningful to the pilot than true air­
speed~ the GwV diagrams (see Figure 1, page 3) depict maximum normal accel­
eration versus CAS.
SUSTAINED HANEUVI:RI\fllLITY. En'ergy Rate . The energy (E) possessed by an
aircraft ie the &U.To of.its potential energy (E:,) and its kinetic energy (£x).
Hatheraatically,
E = EP + f...
• loAl +
j rn'f?­
=w(h+~),
..
where
'·
h =altitude (ft)
m :: aircraft mass (slugs)
c = 32.17~
ft/sec2 , the gravitat::.onol acceleration·
The exvression, £
/". + v2 J
'
= w \.h
2g
~ivea
us a measure of the energy stne
.
ot' an aircraft at any altitude-airspeed combination, However, since the main
intcl't!St lies in comparing aircraft witl, diffe1·ent .,•eights at the same altitude­
airspeed combina~ions, it i~ more meaningful to make the above expression inde­
pendent or aircraft weight. Dividing both sides of t he above expression vy w
yields
-
E
y2
:: h + - •
w
2g
The tenn £/w can be· regarded as specific energy (E, ) , with the result that the
energy state of an aircraft can -now be expre&sed as a function of altitude and
airspeed:
y2
E; "'h +2g.
The pr~>blesa of managing energy invol'les controlling the rate of transf"?r
between energy levels . Differentiating the above expression results in
tr·
!Jhere the dot ( ·) inclica'tes the derivative with respect to time,
To
provide more insight into enc:rgy r ate, t~ , we may employ figure II-~ and write
a force balance equation along the flight path.
rr.~
or
= T1
-
D - w sin y
T, - D = w sin y ~
1
w.
gY
1
or
T.- D
-w
=sin
v
I
V
+­ s
'
Hultiplyins both sides of t his expression by V yields
T -~
- W
( '7./Y .. Y sin y+ -;··
\_
Fi~ure
L..__
11: 1• Aircraft Force-Balance Diogram,
_ _ _ _ _ __ __ _ _ _ __ _ _ __ , _ _ _ _ .___ _
_ _ _ _ _ _ _ _ _ __
/
.
•
Since h ::: V sin y, we may write
vV
T.-D'\
•
( w) v"' h
+g-'
The right side of 'the 11bove expression is equ..ai to E1 , Recalling· that
'-'Ork is accomplished in transferrin!: from one enHgy level to another, anrl
tltat power, by definition, is t he time rate of duing l-X:lrk, the left side of
the llbove equation may be equated t'o specific excess power, P, :
• (Ta-D)
w-
P, =£,c
V.
In an attempt to counter an immediiite threat, the energy-oriented fighter
;:: .~... ~t
will strive to increase his maneuvering energy as quickly as possible,
1'his amounts tv maximizing the rate of t~ansfer between energy levels, 'll'!lich
is equivulent to maximizing the int egral
t'
E, =
f
2
P, dt,
J" 1
According to Rutowski (reference l), this is accomplished
[()oP.,V-'£
~ten
-0
, =\; -
.1
= o.
[~~
a h -k
·­
In·the altitude-Hach number plane,
these relationships are 11atisfied nt those
points where the E, contours · are tangent to the .L. g· P, contours, CC'nnect!.ng
these points results in an approximate minimum time path,
Energy-~laneuverabili ty Cfficiency ( E-HE). If the above- mentioned t hreat
is not as imminent, the pilot will attempt to ·increase his maneuvering energy
while conserving internal energy (fuel) for future maneuverability, This is..­
achieved oy maximizing the integral '
E,·=
dJ::
J"Z·dw
-.! dw.
'
..
,
" '1
I
-'-----~----·-· -~-·---·-----'
dw'" -1~ dt ( ~, "' fl!el flow- lb/sec)
ard
we see that
dt,
p.
ct; •• -;;•
•a
and
E1
"
-
J ~ dw.
.
"1
Again, by employing Rutowski's t echnique, we obtain
or
These r elationships are satisfied at thoae points in the altitude-Mach number
pl~ne where t he E, contours lire tangent to the l - g P,/w, contours. Connecting
these points results i n an approximate mini mum fuel path.
The P,/w, contours suggest a measure of efficiency in view of the fact
. thot t hey depict the ~ount of !Pecific energy gai ned per pound of fuel ex­
pended. In order to acquire a more meaningful measure of efficiency, t hese
contours can be modif ied t o portray the amount of.maneuvering energy gained
for the internal energy (fuel) expended. This i e ·d'!:J1le by multiplying the
contours by t he weight of fuel ovaiJ.able, w, ., to obtain the "t'esultin~
expression fox Energy-Maneuverability .£fficiency:
P.Ht
'
1o1here Pt .. the overage
P, aver the fuel wei ght interval "+ 0
( ft/aec), and loT • c w1 0 - fc - IY .,
w, . " i nitial fuel
fc
c
w1 ,
· '"
-
!c
~
w,
!> ~ r
weight ( lb)
fuel consumed in flying fr0111 some reference energy l evel to
any given altitude-Mach number point (lb)
i\:el reseJ.ve ( lb)
,·
44
---·------­
RANG£, for: <!ny altitude-airspeed combination, a vailable range for cruise
condition may be ex~resse~ as
~oherC? ~:
= the average cruise fue!l flow,
'-'to -
and
~.,
over the fuel weig~t interval
fc s wf s "T , ( lb/sec) 1
X = the horizontal distance traversed in flying
energy level to any Hiven
altituue~airspeed
from SOme reference
combination,
ENERGY RATC DIIICIWIS, For any given aircraft, an tnergy Rate diagram may
be constructed by dividing t he altitude- Mach number plane into a rectangular
gl'id 1 co111putin~ energy r ate (l', ) values at all oL' the points of i01tersection of
the grill lines 1 anu t hen connecting points of equal P., The contour defir:ed
~Y P, ; 0 represents the steady- state boundary of the aircraft, An aircraft
car.not uverate out~ide t his contour 11!thout losing energy1 either in the fonn
or altitude, airspeed, or some combination uf both , The steady- state boundary
is furtheX' restricted un the left by the buffet boundal)' (obtoined by connect­
ing pt>int s where CL == CL . ) 1 and on t he ri~:lrt lly placard limits (a combina­
••11
tton uf pressure · (structural} limits. ancl engi ne temperature limits) ,
Con:oiuerable insi ght into tl1e effects of pulling g within t he aircra ft's
Ui.ght l!nvelope can be gained by con:otructins Energy Rnte diagrams of more
than 1 1:!• These cliagrallll' contain both positive and negative P, cor t ours wi t~in
t he 1~1:! stea,!y.stalc envelope, lis ~>uch, they provide ·a measure of sust ained
m<meuvarilllilLty as a rnsult of pulling g within' the envelope,
£..!1 EFflCIWCY DIACfWIS ,
'
Computational aspects of t.his diagr am p'roceetl
in the some manner as for the Energy Rate uiacral!ls 1 except that now we compute
and connect puf.nts of l?CJUa1 E-NI:, T1~o different types of r...m: diagra:ns ure
const:o:uctecl, The first t}'pe is refer: ed to as the path ind~yendent (constant
fuel) !;..~11: lli\\:,'l"ilm. Computatio11s for all point!> in the cnvi!lopo :~re based
on 50~ fue l wd;;ht. Since fuel wci~:ht is held constant, the expression
reduces to
'l'he diagram is called pat h-independent since the amount of fuel at the altitude­
Mach numb~r points where computations are made is independent of the paths re­
quiTed to reach these points.
ln the s ~cond type ~f &-ME diagl·am, called the path--dependent ( variah le
fuel) E-H£ diagram, the amount of fuel req•lired . to r each any given altitude­
Mach nurober point is subtracted from the totlll fuel weight liefore t..H£ cOOiputs­
tions axe made. The assumption is that the pilot has flown a minimlllll fuel path
from some reference energy level (we use E,
~ 3000 feet) to the altitude-
REF
Mach nWll'ller point ut'tler consideration. A more detailed discussion of this as­
SWIIpt:ion will be given later in this appendi x and i n Appendix III. Additionally,
the amount of fuel upon ~ich thP. path- depe.ndent E-Mt comput11tions are based is
reduced by a suitable r eserve (normally ~ of full ~terna1 plus 20 minutes
loiter at 10 1000 ft) ,
RANCE DIAGRAMS . Again, the computational aspects of this diagram are
essent ially the same as for the £nP.rgy Rate and E-H Efficiency diagrams . To
compute range, the program requir es, as an ~dditional input1 a part!al power
setting table, i .e. 1 a table of cruise fuel flow as a function of altitude,
~ch number, and drag (tkrust required) . The subsonic and supersonic'por­
tions of the envelope ill"e computed using partia~ military and partial after­
burner power settings, respectively. A transient region is obse1ved between
the subsonic and ~upersonic portions of t he envelope. In this region1 l evel
unaccclerateu flignt is not pqssible as the thrust required is greater than
military t hrust available, yet less than minimum afterburner thrust available.
For rnnge, only a pi!th-dependent ( \'ariable fuel) Range diagram is con­
structed. For this diagram, the ~mount of fuel available at any given al~itude­
Mach number point is r educed by the amnur.t of . fuel required to fly a minimum
fuel path from some reference energy level ( again1 we use E,
= 3000 feet)
REF
to the point under consi~eration1 and by a suitable fuel reserve {e. g. , 5~ of
full internal plus 20 minutes loiter at 10 1000 feet) . The ·horizonta1 distance
traversed in flying the above-mentioned miniJiwn fuel path 1 x, is considere<l
part of the available rcnge, A discussion of the me thod used to con~ute fuel
cons~~ed and horizor.tal distance traversed i s given later in this appendix and
in Appendix IH,
46
--------~-·--·- ··--------.:;__---:--­
!:XTENSIONS OF THE RtrrOWSKl TECiOOQUE
Earlier i n this appendix, . we saw that Rutowski calculated the lo"ation of
the approximate minimum time a~d mini.nwn f uel paths in the altitude~Hach num­
ber plane. Rut'l\oiSki did not, however, give any measure of the time required,
fuel consumed, or ~ori'zontal distance traversed in flying along these approxi­
mate paths. His two basic S$Sumptions were that (1) the path b' computed for
1 g and ( 2) the we~ght be held constant.
·
Rutowski ' s method has b~en extended to allow an approximation of the time
r equired, fuel consumed, and horizontal distance traversed in flying along the
minimum time or minimum fu el path. A byproduct of this extension has been the
r emoval of his constant weight assumption,
·
Computations proceed a s follows . P:rugra1n inputs include initial , incre..
01ent al1 ' and final values for £, and h:
(liE, ) t ,
£,
1
L
The spec i fic energy equation,
y:!
t , .. h + ­
2g
is rearranged to the form
V=
[2~ ( E,
- h))!_
Then for £, = E, 1_1 E, 1 +
~E.,
E, 1 + 2CX,, • , •
1
E. L , tl1e following array
'
i s constructed:
P, l _
..
h
~
V M T1
L L
L
D
L
(P,/W,h
Actually the array does not nm all t he
~y
from h1 to h •
L
The lower
altitudes result i n Mach nwnbers higher than the aircraft's capability, h11en
this occurs, altitude is incremcmte<.l instead of computing -r&
·, ~, 1 • , • 1 P. IW,.
The higher altitudes result ·in CL's grenter than CL
ts,
This far;:t eliminates
1& 1
many of the lower lines of the array,
l
A•hlitionally, when h > E., the quantity
[2' ( t:, - h)) 2 becomes negative, eliminating line.; of the array, Finally,
other numerical· techniques, beyond the scope of t l1is appen<.lix, are employed
to re<.luce t he si?.c of the above an·ays , therel.ly decr easing the computer time
1:cquiretl til constl'l~ct a patl1.
·
If 8 mini mum time path is being cO:n!>Uted1 thO program sel ects the Bltitude­
nu••licl' point for which I', is maximum in the array and this point becomes a
point. on the minimWII tiJne pat h. Once the line containing the maximum 1• 1 is
sel.~cted, it i s useu, along with a similar line on the previous anay, to ap­
j)rc,xlmate a time increment, At, a fu~l increment, Aw1 and a horizonta t distance
i ncrement , 6X, in thP follo"'ing manner:
~lach
/
1 t he
bar denotes the average, e.g .,
'
P,
l.' ,t ~ P.
1 1
= ----..;.;;;..
2
)
A!:,
Aw: - = "
( I' ,/Ctr)
.
Ah
.l.
AX = [V'! - (At)2J 2 At
The method outlinud auove results in altitu<lc-~!ac!l numuer points through
1nini1~um time path may be cll·a\\n, The 6t 1 S1 Aw• s , and dX'S are summetl
·over tho pat h to ftivc <m ap11rnximation of the time required, fuel consWl.ecl,
and hori~ontal distance traverse!!.
'rohicl1 a
f.ucl1 time ~. is incremented, tl1e weiglat uset.! to computu the quantities in
t he array i s first t.lecrei!Jented by the quantity Aw1 computed in the previous
array. This r esults in a variable weight path,
Computations for il rni nimum fuel path proceed in exactly the sa1r,e mannc:r1
except that, for the minir.nun fuel p<~th, the line is chosen in the ar r ays where
p, / .:; instca<J 0 f pII ill maximum,
To compute a path- t!cpendent £-HC or Range diagram, a Rutc1wski minimlUil fuel
path must be cosap•.sted first und t:1c following ta!Jlc built:
£, 1.
E,
fc~
2
£,
f:::
L
X
L
L
• I
1.nere
i
fc 1 =
L:
.1Wl
I
j=l
and
i
X!
I
OXJ
j::l
· Then, when WIE o.r range is computed for a given altitude-Hath number point,
(h, H), that (h1 ~I) detemines an E, , which; in turn, detennines <~n fc and an x
1
if we asswne that the fuel conswaed and horizontal distance traversed in flying
to any point on a constant specific energy line is the same. This is a r~t~P~
bold asswnption, however, and cannot lie accepted without further discussion,
Appendix III provides ~ detailed treatment of this assumption.
•49
l'MT II,
Till: BRYSON-KI:I.LE'i ST£l.:PEST ASC!:NT OPTIHIZ!'ITION I'ROGRIIH
'l'~t: second computer program cn.ploycd in these analyses is the Bryson- Kelley
Stcc:>est A:.'cent Optimization l'ro(;rarn whi ch provides dynamic flight pr ofiles, in
mini a.u"l time or with n1inimum cxpentlituru of fuel. for transfer between two
ene1·;,y lwels (Ill, Ml) and (lt41 , H,z).
By using the methods of E, S. Rutowski, th~> so-called Rutowski puth, expla i ned
previously under the Encrgy...H aneuverability J•r ogram 1 is obtainec;l and is depi ct ed
in figure ) 1 page 7 of this rt:.t•ort. However1 these methods provide •loth.ing more
tilan very good a!)proximations to the solution of the minimum time or minimum fuel
problems. Tlu~y provide no insight into such pl.lrilmeters as load factor or pi'tch
angle <~long the path, In addition, the methods are predicated on 1- g level njght
p.1rametcr s nr.d do not cons~dcr the forces ,,cting perpendicular to t he flight path.
In essence1 the Rutowski method i:; a static method, in itself1 but a very valuable
too\ leading to tile ll\O~e accuratll dynamic r>ptimum paths .
Even tlle more sophisticated Dynamic Profile Generator Program, discussed in
J'art Ill of this Appendix; p1·ovidcs only appro:dmations to the desired solution
to the minimum time or fuel proul~m. Admittedly, the results of using the
Rutowski p<~ths in conjunction with this program are much more realistic, ilS now
both load factor and pitch angle nre considered throughout the path. H01~evet' 1
the techniques embodied in this program a1.·c still lim:it<:u by the altitutle-Mach
number comuinations input into the program as points describing the approxiwatc
p;ath, and yiel:tl nothing but a better app:·o.x:imntion to the desired optimum p.:~th.
The program is invaluable, though, us a gencriltor. of load factor atl a function
of time for i'1put illto .the Bryson-Kelley Steepest ;.scent l'rog.ram as the nominal
path.
This ~teepest-\lscent method of optimi:.:ation is an iterative scheme· which
beglns with <IllY non- optimal path and prucee<ls to derive :1 slight improvement
eoch iterat ion frcm thi<> ncwninnl puth. This slightly improved path nt each
itero:~tion is used as the new n0111inal path, und the process is ri!pcated until. we
are sufficiently close to the optiMum for our purpose, In this process, ~ach
new p~th is foun.d by taking the trujectory whj,ch yields the largest f:ain in
perfor mance for a given size of perturbation ~n the control variables ,
Tlte \'nlue of il good first guess nt the nominai path is i~M~edi.:~tely evident,
l f this path is close to the optimum, the number of iterations necessary to
arr ive at thi::;. pnfile will be small indeed when co10parcd to those nccessnry if
the n0111inal path i~ far from the optimum. The ubili'ty to input good nominal
p:lths res.ults in tremendous savin.gs O\ vulu.lble computer tiroe,
'
't'he analysis of single !!tage tril;Jectories by the stecpc!!t ascent method
hali l!een thorou~;hly treated in t he litarature, One of the clearest treatments
o:~v<J ilal.ll e i s that of llryson anti Ocnhum i n Rt!fer encr. 2. For convenience, "
!lricf description of the general problem, as formulated by them, i s repeated
here, Some of the detailed derivations which are ornit:ted here are presented
i.n Reference 2.
/\fter rresentation of the geuerJl problem, the specific npplicati\lns ·
mnde in formul<lting tile progr.:tm at the Air l'rov!ug Cround Centar dre given in
dC!tilil.
GCN!Jl'\L PROBLL'l-1
Determine Q(t) in the time interval t 1 :> t s t; . so as to moxim.\ze
:ml!ject to the constraints
()) ~~" n x(t ) , O{t>~ tJ
(4 ) the ~iven initi\11 conditions x(tl )
(5) t 2 dt:tclmincd IJy.
ll
= ll[x(t2 ) 1t2)"' o.
The - over the syml>ols ;1llove indicates a mntrix quantity, . and a more
dutailcd ()e:;cdption or the n!Jove qunntit~cs fellows :
an m x 1 matrix of control variabl e prog):aiiiS,
which we are free to choose,
(6) Q{t) ;;
a. (t)
x,(t)
X.2(t)
(7)
~(t):
an n x l 111atr ix of state variable progr<~ms,
r esulting fr om the choice of O(t) and x(t, ),
51
(8)
•
=
1
a p x 1 matr1x of ter.ninal constraint fJnctions , each
a
(9)
+~the
I<"~ -
fur.-:tiOl\ of X(t ;o ) end t,a 1
pay.off function, . a known funct i on of
ann
X
x(~a)
ann ta,
lmatlo!x of known functi<.mB of x(t), O(t) 1 and t 1
and
(11) n = 0 is the stopping condition that determines final time
and is a known function uf ~(t.a) and t 01 •
t~ ,
The method proceeds as follows:
•
•
Choose a reasonable nominal control variable p~ogram, a (~), and use
it l>'ith the initial conditions (4) and the differ ential e~uetions (:~) to calcu­
late, by numerical methods, the state variable programs i (t) until 0 = o. In
general, this OC111inal path will not satisfy tite terminal conditi ons 1 =0 1 or
yield the· maximum possible value o! t ,
·
1.
2. Cona:ir!cr small pertUrbations 60(t) about ths nominai control variable
progra~, a•(t), where
(l2) 6Q{t)
= ~t) -
a~(t).
As a result of these perturbations, the state variable programs undergo
perturbations 6x(t) 1 where
(13) 6x(t) = x(t) ":' x•'( t).
If the t'elations (l2) and (J.3) are substituted into the differential
equations, given by (}) , the 1inear diff~rential equations
(14)
ft (6x) .. f(t)6x
t
G(t)6a
are obt~ined 1 accurate ~o fir~t order in the perturbations, whe~e
52
(15)
.
(:;)·j
•••
. . . (*)*
. . . (~)*
(~r
(16>
G{t)
=
~~
~
(::)*
ar •
. . ' ((lf")..
ila.
'l'he symbol ( )* indicates tt.at the enclosed partial· derivatives are evalu­
ated along the nominel path.
Using tlie theo1y of adjoint differential equationsJ the f ollowing expres­
sions may be written,
(17) dt
(18)
"' {a~t(t)G(t)6Q(t )dt + At(tl )ox(t1)
dy = f~~r~(t)G(t)6U{t)u~
+ At(t1)6X(t1)
~1
(19)
+ tdta
ll
~ = f."i~(t)C{t)6Q(t)dt + X~(t1 )6X{tl)
t~
~
•
.dta·
+ 6dta 1
~here !he symbol
I
.
•
indicates the transpose of the "atrix and elemen~s o£ the
three i. matrices, appear ing aho,.e1 are obtained through the nl!•nerical integra­
tion of t he diff~£ential ~quati)ns aQjoint to equations (3):
d~
( 20 ) -
dt
_,
­
= - F (t).\(·c)
with the boundary conditions
Note that tt:e ~ 1a a.r e influence functions in that they tell htN much o
certein terminal condition is changed by a small change in some initial state
variable. Note also trAt the adjoint equations (20) must be integrated back­
wards since the ~,undary cor.ditiona (21) a:e given at the terminal point .
54
· - - - - - - - - - - - --··-- - - - - - - ­
For E~eepes~ ascent the 60(t) program that maximizes the dt in expression
(17) must be foun1, given values of dy and dO a 0 in expressions (18) and (19):
respectively. This ~x!mization must also be subject to a given value of the
fn~egral
The valu: of ( dP) 2 is chosen such that the perturbations will be small
enough to insure that the negl ect of second and higher order perturbations
leading to equation (14) is l.·easonablt!. In addition, values of dt are selected
to bring the next solution closer to the desired termu.al constraints, i c o.
The m x m matrix Q(t) is symmetric and contains weighting functions as
'elements . They may be chosen arbitrarily to improve convergence. In the usu~l
case (the APGC 1rogram falls into this. categ?ry), W(t) is taken equal to the
identity man·ix and ( dP)2 becomes the integral. of the squore of the control
variable perturbations, 60{t) • . Observation revealo that all eontrol variables
should have the oame dimensionS for equation (24) to have any meaning. To meet
this requirement the control variables are normalli' required to be nondimensional.
A rather involved series ~f mathematic3l manipulations (p~esented in an
orderly and clear fashion in Reference 2, ~ut cetitted here for the sake of
brevity) leads to the following .Qroper choice of 6~t):
(25)
- 1-,{-'\n - -A.tOI~~It~/
-- ·- '
6l!{t) =±I(" G
) 2 ~ d~~I- 1 dP
..
_, -~ 1 + w-l(;'.\ i-ld~,
' tt - It+Itt~..
tO ;;
i
whcl:'e
(26) d~ = dt-
-
~
l
x;o(tl)6x(tl),
(27) htO ~ A~-
I
i(t:~ ) ....
·-
{(t;~)
.
'j
AG ,
~
r
(28)
- .. >.t~ - fuel­
Ato
•.
t.o
'(t:~)
'
(29}
rtt .. f:l~;ac w-; c'X•otJt,
'l
55
------- _______
,,
____
.....
I
f
(~) r;, =-
[.
3
i
if0 iHr~ a'Xroci-:,
\1
{31)
I~+= fax~ Gw·l G 1A~Idt,
tl
and the ( )- 1 indicates the inverse m3tri~, and the + or - sign before the
radical in (25) is chosen if f is to be tncreased or decreased~ respectively.
If tht! ~elected df is SUCh t hat di is t!JO large1 than tl1e flllmt!rator ir. the
r~dical in (25) might become negative and a limit to the Rlze of d~ ior a given
dP must.be_imposed. Since dP is chosen to insure valid linearization1 the
selected dp must also be limited,
The predict~d change in t for the change in Q{t) given
(32) df
(25) ia
~ ±~(dP)2 - d~tfttdeJ[Iit - 1;trt~ittJ + it~Ytfd~
t
If
then d~
by
r+o<tl )ox(tl).
dt =0 (the terminal constraints having bcen.satisfied) and 6~(t1 ) =0,
=C and
(
equation (32) becomes
df
'' ) dP
... ~ •
I
•-J.._
=·~Itt- •t( Lyy~t+,
whiCh is a g.tadient in function SfBce, since dP'is the length of the stap in the
variabl~ program. As the opt~num program is a~proa!hed and the terminal
constraints are met1 (dl ~ O)J thiS gradient must tend to zero1 and expression
(}2) becomes
control
(34) dt ~ r;tit~df + [rtn(t1)_- r;~~;~1 90ct1)]6x(t1)
· 3·
A new control variable program is now obtained as
·(35)
-Q(t)
'·
-*(t)· + ·6Q(t).
­
= a
This new ·Q(t) is now used in the o::i.g.inal nonlinear differentia: .equation
given by (3) 1 and the .process is repe~t~d until the terminal constraints (2) are
met and the ~~dient (33) becoa-,en nearly zero.
- - - - - -- -
______ ________________________,__,
..
TH£ £CLI.H PRO<::RAH
During the spring of 1964, preparations were begun for the physical test
to validate the Energy-Maneuverability theory, as requested by the Tactical
Air Convnand, The test waa designed to shl11i tha~ the "dipey- dood.lell man~uvers
associated "ith minirr.'!.'!l t!me-tv-.ciirob anc! olinimul'l fuel paths did1 m fal!t , r ep..
r~sent the optimum flight profiles fo-r transfer from one energy level to
ano~l1cr, The above test loi8G co~dueteti under · ~GC Project 0570T1.
To adequately support this test, a pro51'am which could c001pute these optimum
pnths was necesttary. Arrangements had been roade to obtain the Bryson-Kelley
Steepegt A3cent Cornp~tet Program from the Flight Dynamics l~borstory at Wright­
Patterson AFB. The program ~as being :ormulated and developed under Contract
~o. AF 33(657)-8829 by ~cPonnell Aircraft Corporation. When advised that this
program woul:3 11ot be ready in tine for the test, t he decision was made to
develop a pro~am in-house at Egl i~ . Due t o time limitations, a somewhat
s imple Bryson-Kelley Steepest As~ent Program was formulated in ~~o dimensions
and ~ ith one contr~l variable.
The program wFs completed in August 1964 and used extensively during the
conduct of the test. Before giving t:he details of t he Eglin formulation, an
explanation of so11e key features of the program which are not provided in the
general forMulation will be presented.
As mentioned before, the program has but one control variable, n, the
in number of grs . Origina.liy1 t he progrll..ll was formulated
w.ith velocity <md p!tch ar.gle serving as terminal constraints y1 and ta with
alt!tudc as the stopping condition 0, However1 usi ng two terminal constraints
l ed to trouble with the matrix itt· This matrix was found to be nearly singu­
lar, and the existence of its inverse was> thereforeJ quite questionabl~ .
norm<~l accel~ration
Analysis revealea that the constraint on pitch angle was not vital and t hat
two programs should be developed: one with velocit)• serving as the te1111inal con­
straint with altitude in the role of stopping condition> and the other with the
roles of velocity and altitude rever3ed.
ln reducing the application of t t1e Steepest AscE'nt Hethod to a routine
c~putati on,
an aut~atic scheme or c~ntrol ayste~ for determining the step
size, (dP)2 1 must be devised. Ir. t he Eglin program, this control eyatem i e as
ft>llC7.1s:
1.
!Rgin with a desired lmproventent in the quantity to be ()ptimized (time
for mbimum time paths and "total weight ftJr odnimum fu~l vat ha). This desired
ilnprovement, d~ 1 should be reflect:ea in the next iteration if the te:rminal con­
strai~t has been met. Equation (32) io ~olved for (dP )2 - dS'ittd~ as a func­
tion of the given d~ .
?7
~·
~
.f
I
2.
IL' the tl!rmini!l constraint has not been met 7 check to see if
2
( dl') - d& 'fifd6< 'o. If so, scale down dt ;;uch that the quantity is zero. If
nvt, usa th~ (dP? - d~'ff~dat obtained from expression (32), in the expression
for 6~t) giv~n in ·(25).
l
~
f
!
3. The r equested d9 is then c~odified as each iteration comes close.- to
the O?timum. This modification is controllad by the magnitude of the gradient
given l>y ()3). As this magni"tude grO'tls smaller an~ beco111es less tlaan prede­
termined values., the size of dQ i::; :wccessively halved. The given values of
the g~~uient, at which the d~'s are halved, ~re not readily obvious anu appro­
pr!ate volue:; must be learned t!Jrou~h some experience with the program,
4. tven with this sem1autorniltic control devic&, considera;le time mus~
be spent in deternining values of the gradient with the possiLility that con­
. siderable c~mputer time mily still be consumed before a true optimum p~th is
reached. Sever~l nth~r control syst~s arc presently unrter investigation at
Eglin AFB. It i s hopell that a better automatic scheme will be found which will
decrease both computer running time and the manpower requJ~ed to eventually
arrive at the optimum paths.
The formulation of the Bryson- Kelley technique1 presently in use at Eglin
AFB1 is l'resented here such thut one may r eadily follo11 it, havint; been .-.ade
acquainted with the generoL
1.
111
2,
previously.
Control Varic>blz Matrix a'(t),
(m :; l) Matrix
=n(t),
O(t)
wlaere
pro~lem
= l and n =- normal acceleration ln number of g'a (dimensionless).
State
-
x(t)
Varia~le 'Hatrix
=
Xl]
[x.
X~
l<.:J
=
x(t).
[h(t)]
V(t)
Y(t)
1
(n x
l) Matrix
.
w11ere n ~ 41
w(t)
h " altitude above HSL in :feet,
v = true
airspeed in feet per second,
y .. pitch angle {angle between velocity vector and reference hori­
zontal plar1e) in radians,
w = aircraft gross weight in pounds.
.... ....·
~
..
- --·---- - - ---- -- -- - -- - - - - -- ---------- -:-----.
).
Terminal CQnstralnt Hatri~
t
a.
~
$
;
5.
l) !'<J t : t:;t
term.in.:.~
far Progral'l 623 1 ~o~here V 16 t erminal
= V • V<l
stoppin~
oondifjuo,
desir.eo term!nal altitude in feet,
V:;a = C:csirr.d terminal veloci1,.>' in feet per
4.
X
lleccnd~
Pay- Off ruuc·.;i on ~
=-
a.
t
1>.
t = w f)r minimurr. fuel paths.
t for minimum time paths,
Time Derivat.'.vr of 5-cate Varialile Matrix
- [
f
1
f.
(n x l) Hatrix
.
dx
n) tJ ,. •
x,
dt
(n "' 4).
a.
f1
:;
h = V sin
f<l.,
V.. g
-
y• " -g
t3
:o
f.
= ~- .=
v
-
y1
[T·.,- D- sin Y
J,
[ n - co:; YJ,
4- 1
where
g
'l' .
(p "' l )
r.cnstra.int and V is st..,pping condit ~D:l .
constl3int and h is
h~ ~
(p
0.
; = h - h:a for Program 556, wher.e h . i~
-= +
b.
f
= acceler~tion
= thrust
of gravity = ;2.17~ ft/sec2,
c.vailable in pOtiilds,
D .. drag in pollld<- 1
W, .. fJel flow in pounds per seconcl,
59
(p = 1)
6.
Stopping Condition 0 = 0 .
a.
0 : V - V~ 1 for Progt"arn
556.
b. 0 = h - h~ 1 for Program 623 ,
_
of{ t)
F(t) = ox(t ) •
t'(t) = [f1 J (t~ i 1
f\l :: {If, =
. OXJ
j • 11 21
Matri~
n.
2.&
OXJ
ob
o{V sin
a. til ., dFi - --a-h-
y}
c
o,
(n
= 4)
oh
b.
f 14
= 3V = sin y1
c.
f1 3
=-
d.
fl .
oh
=a;; =o,
e . f:n
oh
= v cos
av
=av::
oh
f,
.. . ,
(n x n)
f~a
av=
=av
av...
g. fa, = -
oy
y,
:{&J-!7v)l
an
sin
.!
"'
g o(T.-D}
-; -ah'
o{T.-D)
--w-1
- g cos YJ
h.
f~, ;~
= - !. {~ + ao},
Ow
..... w
Ow
i;
f, l
=
j.
t3~
= av .. -
~~oh = ~~(noh~ eM v)} = o,
av
vrg (n - icos y) ,
··,
-
-- --~--
- k.
0~
f;,3 c -
g
sin y,
.. -
oy - '
l.
f,. "' -awov .. o,
rn.
f.tl .. ..._..-
34
a;,
oh
oh
0~
~
ov
ov
-~
n. f•a = - "' - - ,
ow
= o,
o, fu = oy
P•
8.
ow
t... "' -Ow = o.
al{t).
G(t) "'
(n x m) Matrix
o~t)
_
ai(t)
O(t) = -
on
[MJ
= -
on
1
i
=l 1
• oo 1
4)
4.
n•
{ lllo:l
Let
G'(t) ., (gtJ
i ., 1, ••• , 4,
.
oh
a.
g1
=-en .
b,
~
=-
a(V sin
an
·ov c{~ T\~
()n
c:
vl
..o,
oon
sin
v}
g
a: - lol
an
-
On
~
I
L---------------------------------~'~~-------:
9·
.
Lagrange Multipl i ers ,
a,
.'• "l'f'••··] '
"•·
b.
Xt = "t:~
-
>.ta
["']
A?:~
I
>-t.
lD.
r
-
c,
~
lo= ~ .
Adjoint Differential Equations for Lagrange Multipliers,
a,
d~4
dt
= - F t. 11
b.
~
=-
1
p -=
- ,­
dt
r~ a
fu
0
r'~,,
f:n
0
fa:~
f:~ a
f:n
fa •
fu
0
£.]
fu
0
0
where t he f ,J are given in (7)•
Performing the cnat::r...x multip: icationa indicated in 7a , 7b, and 7c, we hav~:
the following set of differential equations for the individual ele~ents of the
Lagrange Hultiplie~ matriceD:
d.
~.J
.. -
•
E
,,.1
A4, f1 J J
.
(1) ~~ 1
""-~ (>-t:/.il + ~~. fu) 1
At2 =. - (>.~ fu
(2)
+ .1.1:/aa +
>-t,fu
+ f.t,fu)l
•
(') >.§, •- (At. fl.s + ).tafa.t + 1.t,f).s) 1
(~)- ~
••
:0::-
>-~;af~··
j • 11 21 ,, 4•
.
~Va .... <>•tl fu + >-t/u + ).tJ t',a + At,fu)•
(2)
(:~) l.t,
it, ~ -
(4)
f.
~J
a-
:;
(>.t 1 fu +At./»+ At,f.u).
>.•afa••
-2:• A~
fiJI
j • ll 21 3_, 4.
s: l
(1)
~
= -
(~f,;1
(2) .
~
=--
(~ fu +~faa + ~fu
(3)
Xn, .. - (~ fu
•
':n. fu ),
+ An.f•li) • .
+ >.n,;/a.s + ~f,,).
(4) ~~ .. - AO; fat.
ll. -
!Joundary Conditione fer Lagrange Hl.litipliers •
a.
A1(ta)
•
(~1
.
u..!jto:t;~
1
(1) For maximizing t
or
A
~~1 (ta) .. \u;o.tt•ta
f.~ )
-
1
i • 1,2,3,4.
t (minimum time paths).
I
.!
_ _____ _ ________ _ _ _ _
_ __
___:.::,._
--at
0
-~
0
?Jh
?JV
.l. t(ta) •
"'
-~
0
oy
- 2!
~
0
3w
or
J.. t (ta) • 1.1 (ta)
1
;a
= x1 (tz)=
s
.
).. t
.(t;j) :::
4
Q,
.
(2) For maximizing t .. w (minimual fuel paths),
<hi
0
oh
3w
A1(t 01 )
0
av
•
II
Ow
0
Ty"
Ow
. aw
l
t .. ta
or
}.~ (ta)
b.
• l. t 8 (ta) .:. l.t, (t8 ) = 0 1 and At 4 (ta ) • l .
1t(t:~ ) ..[*] t .. ta' or ~ (ta) ·l~!J
t •
:t1/
1 • 1, 2,
~~ 4.
(l) For termjnal constraint on h {Program 556),
oh
l
at;
oh
:\t(t2 )
=
0
av
"
dh
0
oy
oh
0
-~
t . t;
(2) For terminal constraint on V (Program 62,),
ov
''
0
c}h
av
~.(t:~) "
av
l
•
av
oy
av
~
0
0
t •
~~~
or
c.
Ji.t
11 ) •
[~]
~ t .. t 11
,
or
~ (ta)
,.
[ao]
o~
t •
ta
,i
• 1 12,;,4.
(1) For Stopping Condition on V (Program 556)1
av
0
oh
Ao(ta) =
-?Nav
l
"'
~
0
?Jy
av
l)
Ow
t • ta
or
AoaCt.a) ..
1 1 and ~ (t8 )
,.
~ (ta) •
).0. (ta) = o.
(2) For Stopping Condition on h (Program 623),
oh
ah
l
ah
1o{ta) ,.
I
-I
0
3v
iI
•
ch
0
3y
I
ah
-
Ow
t . t8
I.
0
or
~(ta)
1.2.
The Matrices
a
1,
and Ar.g(ta)
Ato and ito•
Ar.a(ta) •
~(ta)
• o.
. .........
a.
or
66
.\
c
t(t:~ ),
'-~o~ ""h~~ - n( t"') "nl'
(l) for maximizing t = -
(2) For maximizing t
e
"'1, 2, ;, 4.
i
t
(minimum time paths),
w (minimum fuel paths) 1
t(t.a) = - ~ (t:~)
(3) For Stopping Condition on V (Program 556)·
O{t2 )
.,
V(t:z)
J!d
- sin yl~
l w
=
t
.,
=t;~
(4) For Stoppi ng Condition on h (Program 62}),
O{t~)
b.
-
>.~ =
c
-
f(t.a)
"t -('(t,a)
~(t:~) = {V sin y]
t
~ ta'
~
"OJ
or
(l) For
te~inal
constraint on h ( Progra m 556),
(2) Fo1: terminal constraint on' V (Program 62})j
t(t:~) =V(t =gf-"~
"
01 )
1},
The :Jatr~x Product
•
(Ato) '~ =!:
...
~
- v}
sin
t ..
(Ato) 'G•
A_;o1g1
"c' rto = AtQag;
+ "toJg3·
t.:~
l~.
Th~ Matrix Pre duct cr~J 'G.
[AtoJ'~ "' ['
~~g' "'G'
r•n ~ A.~g~ + >.~g~ •
l"'l
as ~
c.
1~~
-
=l .
f. ':. l~toJ '<mn-l CGJ 'rtodt
't
l 6•
The
Hat~ix
Pr oduct
d~ • df
.if!/- 1
up
!!
Itt dp•
•
w
>iil(tl )6x(t1)
.. df as -6x(tl) a o.
5561
a.
dt • llh
for Program
b.
di • AY
for P:rog:rB./!1 6231
where Ah and AV are the terminal condition c~anges necessary to bring the next
solution closer to the desired terminal constraints.
c.
6h • h:.a - h(ta)1
where h(ta) and Y(t2 ) arc the alt1tude and true airijpeed at the final t1mc point 1
t ~ t 21 of the previous solution.
I
II·
Combining the results rLom above,
d~'i·ldi" (dt)
2
H
17,
•
IH
The Hatri x Product
i 'j t
it~
iu,
.
"
Since the matrices
I;~ and .Ltt are both one-by-one scalars,
r11 - ft
lB.
l'ft Itt "'
cz,t)z
Itt
J
'
The Matrix ProJuct l't t~ttd~ .
I;fdf
r~,~rnda = - •
ru
19.
The Expression ( dP) 2
From
expr~ssion
-:.
.aD' Ittd~.
(32) is obtained the relationship
a,
which can be reduced to the fo~towing ·expressions, using the results of previous
par agraphs:
b.
If the terminal constraint
c.
J.
ha~
lllhl ~ ~l
l6v l s ~
been satisf!ed, that is,
f~r
Program
556 1 or
for Program 623,
where ~l is some predetermined toleranc~ within which the terminal constraint
on altituqe must fall1 and ~~ the tolerance for a terminal constrain~ on
velocity, then the expression (19.a. ) is usen in the expression for 6Q(t).
·~
'.
If the value of 1~ 1 or IAVj is outsid e the predetermined tolerance, then
- d5' It}·~= 0 1 and use in th expr;!ssio!l for ~~(t).
set (dP) 2
20 .
The Hatrix Product
1~1
w• . .; (.l.~o -
"
21.
-
Itt) •
1
"tl;t\Itt)
,. ~ (')..\ ~~ - ).t~
Itt tt)
"'
+
(
g., )...:q.
The Matrix Product Q-lG'Atnltld~.
-
- ,.. -
-
w-1GAtor'ttd~
22.
w-1a' (X to - ~;o'ifi
=
(A..ri g + ]. f (' g )
, .. ,. ~
IH
..::u.
d•·
The Expression for 6Q(t) .
Combining the results of the preceding pages, the
for 6Q{ t ) is obtained:
follow~Jg
expression
a.
+ (').. ~g:~ + ).. tO.lg~)
df.
I yt
If the value of
t hun
l.llt!
·
or l t~VI is outside the tolerance, ~ t:lr ~~ r espectively,
·
b.
70
'I
1
'
I
PART III.
DYNAHN PROf: C.E GEIIf.RATOR
The Dynamic Profile Generator uses the uerodynamic and performance data
for a gi.,en aircraft to connect points in the altitude-Mach number plane with
an approximate dynamic profile consistent with the capabJJ.itieb and limitations
of the aircraft. The program provides a tJ.me history of the noXlllC:ll acceleration,
~itch an~le, drag, fuel consumed, and horizo~tal rnnge traversed throughout the
flight pilth.
The progrom is deaigned to compute noxmal accelerltion in number of gr s as
a function of time, associated with the dynamic pro file necessary to fly throu!!h
the altitude-Mach nl~ber pQints defining a Rutowski path for a given aircraft.
This g schedule then serves as a good first guess for the nomil)al path i.n the
Bryson-Kelley Steepest Ascent Progra m, leading t o either a m.inimwn ti111e or
minimum fuel path for transfer between different energy levels.
An :initial ail"craft gross weight, w.u no:mal acceleration, nu an'.! pitch
angle, Y11 are assumed at the first altitude-"'ach mtr.~cer i>Oint (h11 H1 ) of the
path . The usual ass~ytion is that the tra11sfer betweeH en~gy levels i.,
initiateu from level flight, i . e. , n1 = 1,0 g and Yl = o, Figure II-2 depicts
a typical pDth co~np,sed of N ( h1 .H) points .
To obtain the values for the quantities at the ith point (h1 1 H1 ) 1 ~
onploy an iterative predictor-co-nector process involVing tlle major steps:
(l)·preu~cting the value of some of the quantities across the i nterval fr~m
(h 1 ,.1. 1 !-: 1 _~) to (h 11 H1 ) based on known vaJu<Js hom the for.net point and (2)
correcting these values durina each of a number of iterations until a desir~d
level of convergence is ~ttained ,
The time interval and fuel consumed nre nstimated for the interval between
the poi.1te1 as discussed previously in this appenuix1 and either cf two methods
employe~ to determine the normal acceleration r~quired by the aircraft to fly
between th~ points,
The first method involves solving simultar.eously tne equations of motion
along and perpendicular to the path to provide on ~ ·r· ;ion for the coefficient
of lift, <1, :
C'
T. - w (sin Y+ V/g) _...:!h.
q SK
X
­
71
•
• • • ••
•
•
•
••
•
•
•
..
•
•
•
II
•
{hj. llj ) ~
(hi-1,
•i-1>
WKU.SSII'Ifll
•
Figure n-2. Typical
Mlere
c0
•
X
Path Used by the Dynamic Profile Gener ator.
is the zcro-J.ift drag coefficientJ
is
is
T. is
!i is
Y is
~ is
g i~
q ie.
S is
~CL
~uta~;ski
the induced dxag parameter,
the value of Cr. for which c0 is minilr.ulll
tne thrust available in polDlds,
the aircraft gross weight in pqunds,
the ai.rcr.Jft pitch angle,
the acceleration along the fl.ight path in ft/ sec?-1
the acceleration due to gravity,
the dynamic pressu.re in lb/ft2,
the aircraft reference wing area in ft:2·,
The bar notation (CL or T.) i ndicates the value of this quantity at tl1e
midpoint of t he interval.
The pitch angle
expression:
sin
Y•
()j at this midpoint is approxi:lated
h! V6t
L__ _ _ _ _ __
_
_ __
.. . ­
by the following
ht~
' .
I
_ _ _ _ _ _ __
_
- --
- --
..•
CON Fl DENTIAL
l<.T!ere .e.t is the tiJie increment_required to fly between (h,_11 Hs-l) ~d
( h11 H1). This value for sin V a1.ong with the other required infozmation are
used to determine CL which provides the value of n, the normal ac~eleration
in number of g's serosa the interval,
An altel"Tlllte method employs an estimate of sin y1 by the f>XPression:
. .
s1n y 1
2th
:: ::--
Yilt
to obtain the angular rate
•
y
nus
y:
- Yl - l
= V1 .e.t
expr ession is then used to compute n:
vv + cos ­Y.
· n :: -
g
A 1110re det<Jiled breakdown of tl:e:;e methods is readlly avallible from the
authorS upon request.
73
CONf~tt)tE~1f~Al
(fhiij page is. Unclassi!led)
L_ _
_ __
_
CONFIDENTIAL
.• •
. _. ,
I~
•
'
A COHPARISON OF 1HE BRYSON..!'XI..ILY .AN:> TilE HODIF.U:D Rlfi'OWSIC[ TE~IQUES
( U) In computing the path-dependent (variable fuel) f-lit and Range dia­
grams, a method is needed for COall>Uting' the fuel consumed ami horizontal dis­
tance traversed in flying from some reference point (h01 M0 ) to any p<l!nt (h, H)
inside the steady..state envelope of an airera ft . This implies sorne path con­
~tecting the points (hg 1 Mg) and (h1 M) in the altitude.Jiach nuxnber plane. Of
course, <my number of "flyable'' paths can be dra1on Which connect (ho , Ho) and
(h1 M) . The foregoitlg £.Ji Ef-ficiency a:~d rangg consideratioJUI suggest a s~mpli­
fication by connecting (h., 1 H..) and ( h1 H) with a minisclll!l fuel path.
(U)
Use ' of a sophisticated technique, such as the one credited to Bryson
ann Keliey 1 bec~ea prohibitive because of the a~ount of computer time involved.
for this reason, an approximate technique becomes exceedingly desiruble.
(U) Heermann (Reference 3) observed that curves of Ct>Mtant minilllum time
are ~pproximate curves of constant specific energy, E., in the altitude-Mach
number plane,. · Hore recent investigations by Heermann indicate that the IKalJe is
true foY curves of constant minizlrum fuel.
(V) Heermann's results, coupled with experience gained with t he Rutowsxi
method, extended in the ~tner described in Appendix !I, suggest that fuel con­
sumed in traversing a Rutowski minimum fuel path to~ given -energy level is
alJ!:ost independent of the altitude-."f~ch ' number combir.ation on that energy l evel •
. Additionally, for range computations, the ·horizontal riistance traversed in
·climbing to a given altitude-l-lach number po!nt is. atnall in comparison Yith the
range r eoaining and, hence, a s~ewh~t leas accurate approximation of horizontal
distance ia acccptnble.
( U) Investigations have revealed that the Rutowski approximations 11rc
extrelliely good ones. That 1s 7 in part, due to the fact that fuel end distance
errors tend tCI COG!pensatc for each other.
(U) Numeroua l~ 7094 computer runs for 1:he f-4C and Hl~ have been
and will be discusst!d he::e to SUfPOrt these .renarka.
s~rized
1'Gj The first er.ample attests to the accuracy of the Rutov.lki approxima­
tion. It is a total·path cooparlson for the F-4C from H :: 0.8 ot 100 feet to
M • 1. 854 at 44,900 feet (E, = 951 00Q feet), with an initial -.,eight of 4o1 ~92
poWlds. The Bryson-Kelley path indicated 1+, 017 poun~ of fuel cnnlll.laled and
68.4 n~uticiiJ. mile& traversed, C~red with ~ 1 993 pounds of f'.Jel COI18UIIIed llTld
64.8 nautical miles ·traveraed ·¥14 the Rutowoki prog~ for the s~e ease. A
74
CONf~t{»ft~¥~l8sl ·
difference of 211 pou :1d~ (0.~) ~nd 3.6 rnlles (5 . ~) exist a betwer. the Bryaon­
Kelley 3nd the Rutoweki paths, The Bryson-Kelley path required 92 J:Unutes of
computer tirne versus 1. ~ minutes for the _Ruto\.oski path.
·
(U) the following ca ses illustl'ate the effects on fuel consumed and dis­
t ance traversed ~~£n the teminal (h, H) lies above or below the Rutowski path
at E. "' 95_.000 feet. Since the sub~onic portions of the paths for the F-4C
liere identical, only the supersonic portions -..ere conaideroo. The nme atate­
~ent applies to th~ MIC-21 paths. Figure II-11 page 46, indicates the general
shape of the paths in the altitu.Je-Jiach niJI!lb~ plan~.
1S) 'rhrP.e Bryson-Killey paths each were l'UJl for t he F-4C and the MIC-21
fron H = 1 . 0 at 391 000 feet and H =1,0 at 44,700 feet, r~spect1vely 1 to
E, -= 951 000 feet. Terminal conditione are given in Table III-1,
TABU: III-1.
NASIC/ACAA
DECLASSIFY
(This information no
longer
needs to be
classified)
case
2
F-4C
F-4C
3
&-4C
5
6
Altitude
(ft)
Aircr:~ft
1
i;
TmfiNAL <DNDITIOM nATA
..
..
Mach No.
.
KIG..gl
.HIG:.2.1
.· . . .
'
KIG-2l. .
I
36, 900
44,900
52,900
39,&:lo
42,ax>
50,a>o
1.997
..
l.B54
1.700
· 1.946
1.1392
1.741
-fiftft-
W Cases 1 , 21 and :5 tem!r.Btoo 81 000 feet below, on, Bnd B1 CXXl feet
above the Rutowski path at E. ~ 95,009 feet, respectively. Cases 41 5, and
6 tera~inated 3,oco feet below, on1 lrlld 81 000 feet !'bove the JWto~oeU p11th at
E, ,. 95 1000 feet, reiipectively. P!Beard conatnirlt~: preve:nttd t he temina­
tion of caae 4 below 391 80o feet.
.
.
tSj Table III-2 presenta a co:rspllr1son of the Bryaon-Xelley and R.utollllld
patha. Th!e table depicta fuel corun~:Ded for the Bryaon-XeUey path, fcBX'
an~ for the Rutow:;ki pat h, fcRi horiz011U!l ciiatance C1Yer the ground for the
Bryson-~elley p11th, :x K, apd for the Rutowlti path, XR• Al.lo th<Ml are the
8
weieflt dtfference41 l.w1 and percer:U!t;e weight differences, ~6'>~1 betl.'een the
Bryoon-l<elle-; and Rutowski paths, a. well u the l!iltlllCe differences, II:1. 1
end percentage dhU:nce differences, '{.ax. All weistht:a are i n pounda, and all
diat~ncea are in nautic4l • i l e5.
-75
----- ......___
'"' ..... ~es.m n
.......
'
t~S£S 3 AIQ 8
"'
H
lltiCU U ifi U
•
figure III-1. Rutowski Path
TABLE Ili-2,
CCHPAAlSOM OF BRYSO!l-K.EUJ:Y A.'ID RITl'Ow'SXI P/a'I'HS
Case Aircraft f csiC
~ASIC/ACAA
!pECLASSIFY
his informatJon no
I nger
eeds to be classified)
.
,
XR
!JW
~6W
ax
'/.l'JX
-
-45.0
28
42
1. 2
1. 8
2. 8
5.9
1. rJ
3.2
-
-
so
3. 1
6. 5
2.1
5. 7
3. 6 JJ . 7 .
6. 8
,., 2
15 , 6
8. 5
6.9
~3.3
f eR
xax
l
F"4c
23~
2
F..4C
2}72
}
F..4C
2420
4
. HIG-21
6}2
5
HI~
6)4
47.13
2}30
46.5
47.7
.. 26.3
591 26.9
6
MI~2l
~
Z"(. 6
~
C~parid on Schc~ .
-22. 7
-
41
4,
55
(II) Observation reveals ! hat the fuel cOl\llllllled via the Rutowelti method
ia conaiatently leaa than the iuel Con5l.lllled via the B:ryaon-ICe.lley method. The
lh'Xme coneilstency holds, ho1:ever, for t he horlzontlll diaumce t1·averaecJ, The
errore introduced by uning thi s apprax~~t ion technique tend to cocpena~te for
each other.
---------- -·------­
-~·· ·
( U) For the entire r ange computation, the percentage errora sholel above
become insignificant since t he climb input is only a part of the totel i Jiput.
However, if more exact computationa for the Range and E-M t!"ficiency diagral!l8
are necessary, the BrysOn-Kelley (or the H~:nnann) poth& can be employed in­
stead of the Rutnwaki paths.
.·
IU:FERDICES
l • . Rutowski, E. s., Energy Appro~~ch to the General Aircraft Performanc!
P.robl~r, Journal of Aerona 11t1cal Sc i ence, Volume 211 No. 3, March l95lt.
2,
Bry&on1 A. £, and Denham, w. F., A Steepeot Ascent Method fo r SolVing·
P!"ogrelllllin$ Problesns, Raytheon CoJIIllany DR-~393, April 1963.
~imum
3 , Heerman.,, H. and !Cretsinger, P., The Mi.nimum Time Probleo~, Journal of
t'ne Astl·ona!lticaf Sciences , Voltm~e XI, No, 4, Winter 1964.
4, Boyd, John R., />erial Attack Study, Nellis Al'B 50-l0- 6c, Revised
ll August 1964.
5.
Tactical Al.r Co:llnilnd Manual 3-l. (teat), Counter Air Interdiction and
£!os~
6.
1.
Support , Harch 1965.
Dennison, J. H., Analyti.:al Approach to F-4 H.Jneuvering in Air-to-Air
Cmbot, McDonnell Aircnft Corpordtlon B-163, 26 March 1965.
AlH-7D/ AlH-7E Maneuvering Target and Hinimum Range Definition StU~
Raytheon
Co~ar.y1
Hissile Systems Division, BR 3361, 26 l.pril 1965.
Boyd, John R. and Chri<.ile, ThQnBs P. , Ent!rgy-ManeuverabUity '!heo:l,
APGC...'l'DR.-64-35, Hay 1964,
9. Boyd, John R. and
Christie,. Th.omos P. 1 l:nP.ril-Moncuvcral>ility Theory
8nd Applications, Paper for 12t~ Annual~ir fo~ce Sc.ence and £ngine~ring
Symposium, 9 October .l965·
\
·.
I
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78
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