Vol. 49, No. 4, December2003 O
Vol. 49. no 4, <lecembre2003
Methodfor FlutterAero-servoelastic
Open
LoopAnalysis
Ruxandra Mihaela Botez * Alexandre Doin * Diallel Eddine Biskri x lulian Cotoi * Dina Hamza * Petrisor Parvu *
NonTEncLATURE
Abstract
Aero-servoelasticity(ASE) is a multi-disciplinarystudy of
interactions among structural dynamics, unsteady
aerodynamics,andcontrol systems. In this paper, the
Aircraft Test Model (ATM) developed by the NASA
Dryden Flight ResearchCenteris used,and the velocitiesat
which flutter occurs are calculatedby use of the Structural
software.
Analysis Routines(STARS) aero-servoelastic
For the validation of our aero-servoelasticstudy, the
STARS aero-servoelasticsoftware, also developed by
tool
NASA, is used.We developeda new aero-servoelastic
in Matlab to consider these interactions,and the results
obtainedthrough our method are comparedwith the ones
obtainedthrough STARS.
words:
aerodynamics.
aeroelasticity,
Key
aero-servoelasticity.
servo-controls,
0)
naturalfiequcncy
n
generalizedcoordinatesas functions of frequency
n(co): [n.(co)n, (ro)l1.1(o)]
l.
generalizedcoordinatesfor elasticmodes
I'
gcneralizedcoordinatesfor rigid rnodes
la
generalizcdcoordinatesfor control modes
C]
non-di mensi onal general i zed coord inat es, wit h
respectto tirle
4c
non-dirncnsionalseneralizcdcoordinatesfbr elastic
rnodes
4r
non-dirnensionalgeneralizedcoordinatesfbr rigid
modes
Qa
non-di rnensi onal
seneralizedcoordinatesfbr control
modes
R6sum6
M
modal incrtia or massmatrix
L'a6roservo6lasticite(ASE) est I'objet d'une etude
multidisciplinaire des interactions entre la dynamique
structurale,l'adrodynamiqueinstable et les systemesde
commande.Dans le prdsentdocument,le matdriel utilizd
est la maquette d'adronef mise au point par le Dryden
Flight Research Center de la NASA. Les vitesses
auxquellesle flottementseproduit sontcalculdesau moyen
STARS.
du logiciel d'adroservoelasticitd
logiciel
Nous
avons egalement utilize le
STARS mis au point par la NASA
d'adroservodlasticite
puis nous
pour validernotre6tudesur I'adroservodlasticitd,
dansle
avonselabordun nouvel outil d'aeroservodlasticite
< Matlab )) pour analyserces interactions.Les r6sultats
obtenuspar le biais de notre mdthodesontcompar6sir ceux
qui ont 6td obtenusir I'aide du logiciel STARS.
.' Aerodynamique, adrodlasticite,
Mots
clds
servocommande,a6roservoelasticite.
D
m o d a ld a r n p i n gm a t r i x
6r
modal dampingcoefficients
K
rnodalelasticstiffnessmatrix
a
modal generalizedaerodynarnicfbrccs
Qr
imaginary part of modal generalizcdaerodynamic
forccs matrix
Qx
real part of rnodal gencralizedaerodynamic forces
rnatrix
o
modal sensormatrix
-.1s
sensorlocations
p
true air density
Po
rcl -crcncc
ai r densi ty
* Ecole de technologicsupcrieurc
Departcmcntdc genie dc la production
automatizeie
1 1 0 0 ,r u c N o t r c D a m e o u c s t ,M o n t r e a l ,Q C
H3C lK3. Canada.
o
V
vo
tr/E
Qa
i!r)2003 CASI
(l)
referencetrue airspeed
equivalentairspeedVr,:.[oV
(2)
ai rspccdrati ov:Lt
(3)
vo
E-rnail: rr"rxandra(z)gpa.etsmtl.ca h
Rcccived7 August 2003
airdensityratioo:o= P
Po
truc airspeed
dynamicprcssureqd
wing chord lcngth
semi-chord,b : cl2
:1,,,
(4)
CanadianAeronauticsand SpaceJournal
k
r edr - r c ed
f r equ e n c yk : Y -' b
2VV
Mach
Mach number
(5)
INrnooucrtoN
In this paper, a rnethod for open-loop fluttcr
acro-servoelasticanalysis is presented. This rnethod is
validatedon the Aircraft TestModel (ATM) with the aid of the
Structural Analysis Routincs (STARS) cornpLlterprogram
developedat NASA Drydcn Flight ResearchCcnterby Gupta
(teet).
Two main bibliographicalresearchthcrncsarc considered
here.Thc first one givesa shortreviewof the aero-servoelastic
analysissoftwarc in the literature.and the second gives a
reviewof unsteadyaerodynamicforcesapprorirnationrnethods
fiom tlie reduced frequencyfr dornain into the Laplacc .r
dornain.
analysissoftwaretools exist
A numberof aero-servoclastic
industry,rnainlyin the U.S.A.Thesetools arc
in the aerospacc
STARSdevelopedby Cupta(1991)at the NASA DrydenFlight
ResearchCentcr;the Analog and Digital Aero-servoelasticity
Me th od( A DA M ) dev c l o p c db y N o l l e t a l . (1 9 8 6 )a t th c Fl i ght
Dynarnics Laboratory Air Force Wright Aeronautical
Intcraction of
Structures.
Laboratories (AFWAL);
Acrodynamics,and Controls(ISAC) dcvelopedby Adarnsand
Ho a d l c y ( 1993) at N A SA L a n g l e y ; a n d F l e x i b l e A i rcrafi
b y P i tt and
Mo d e l ingUs ingS t at eSp a c e(F A M U S S)d e v c l o p e d
Goodman(1992) at the McDonncll DoLrglasCiornpany.
ln this paper,we usc STARS among thc cxisting software
analysis.The STARS prograrnis
tools for aero-servoelastic
cngincering
designedas an efficienttool for analyzingpractical
problerns and (or) supporting relevant research and
activitics,and it hasan interfacewith NASTRAN,
developrncnt
which is still very much used in the aerospaceindustry.
STARS has been applied to various projccts such as the
X-29A, F-18 High Alpha RcsearchVehicle/ThrustVectoring
GenericHypersonics,National
Control System,B-52lPcgasus,
Acro Spac eP lane( NA S P ),S R -7l /H y p e rs o n i cL a u n c hV ehi cl e,
and High SpeedCivil Transport.
analysesis
Another softwareused for aero-servoelasticity
Mcthod (ADAM)
thc Analog and Digital Aero-scrvoclasticity
computer program, which was developed at The Flight
Dynamics Laboratory. ADAM has bcen applied on the
unaugmentedX-29 A and the following two wind-tunnel
m o d e l s :( l ) t h e F D L r n o d e l ( Y F - 1 7 ) t e s t e di n t h e N A S A
Langley l6 ft transonicdynamicstunncl and (2) the Forward
Swept Wing FSW model mountedin the AFIT 5 ft subsonic
w i n d tunnc l ( l f t - 30 .4 8c m).
Thc softwarecalled ISACI(The Interactionof Structures,
Aerodynatnics,and Controls) was developedat the NASA
LangleyRcsearchCcntcr.ISAC hasbeenusedon variousflight
rn o d e l ss uc has DA S T A R W -1 a n dA R W -2 ;D C -1 0w i n d-tunnel
flutter rnodel;genericX-wing feasibilitystudics;analysesof
e l a sti c ,oblique- winga i rc ra ft;Ap ' W (A c ti v e F l c x i b l e W i ng)
180
et spatialdu Canada
Joumal aeronautique
wind tunnel test program; generic hypersonic vehicles;
benchmarkactive controls testing project; high-speedcivil
transport;etc.
Anothcr software fbr developing a statc space model
rcprcscntation of a flexible aircraft for usc in an
aero-servoelasticanalysis has also becn presented.This
techni quei s basedon determi ni ngan equi valcntsyst cmt o
lrequcncyresponse.
The theoryhas
matchthe transf-er-function
in a computercode called FAMUSS at the
been irnplernented
McDonnell Douglas Aircrafl Cornpany.FAMUSS has been
used internallyat McDonnell Douglason its aircraft.
Recently,an acroelasticcode,ZAERO, hasbeendeveloped
atZona Technologyby Chenet al. (2002),which couldbc used
analyscs.The influence of thc
also for aero-servoelastic
acrodynamicstores on the aeroelasticinstability has bcen
studiedusing a number of aerodynamicmodels for the F-16
includingthe isolatcdwing-tip launchcr
aircraftconfigurations.
rnodcl. and the rvhole aircraft with and without stores.The
resultsshou,good agreementbetweenthe prcscntnumcrical
predictionsand the flight-flr-rttcr
tcst data.
A short rcvioi' of eristing aero-servoelastictools is
presentedhere. Nert. a rcvic$, of the aerodynamicforces
approxirnation fl'orn the fi'equency dornain to the Laplace
dornai ni s gi vcn. Thc fol l ol r' i ngrnethods:l eastsquar es( LS) ,
rratrix Pade(MP). and rninin-rurn
state(MS) arc ncccssaryfor
aero-servoelastic
analyses.
The conventional LS method has been used in the
acro-servoelastic
computerprogramcalled ADAM. where its
capability is to determinePad6approximationsof any order
suchthat the sun-rof the numeratorand denominatortermsdocs
n o t e x c c c d1 5 .
includc augmentedstatesthat
Thc state spacc equatior-rs
representthc acrodynamiclagsl their nutnberis dependenton
the nurnberof denominatorrootsin the rationalapproximation.
The aerodynamicentire matrix has been approximatedby a
ratio of matrix polynornials.In tlie MP approximationrnethod
by Roger et al. ( 1975),each tcrm of the aerodynamicrnatrix
rnay be approxirnated
by a polynornialratio in s. Howevcr,it
has also bcen found that comfflondenominatorroots are also
eflective in defining the correspondingpolynomials.Other
rnodificationsof the MP rnethodwcre suggcstedby Karpel
(1982) and D unn (1980).A hi ghcr number of denom inat or
roots is requiredin the MS approxirnationmethodby Karpel
(1990),whcrc the nurnberof augmentedstatesis equalto the
nurnberof denominatorroots.
The capabilities for enforcing or relaxing equality
constraintswere includcdin thc LS, MP, and MS rncthodsby
Ti fl bny and A darns (19U 4, 1987).These cap abilit icswer e
abbreviatcdELS, EMP, and EMS, and they were introducedin
thc aero-servoel aslctl
i c ntputerprograrnIS A C . Tlic r ninir nur n
stateapproach(MIST) was selectedrecentlyin the ASTROS
computerprogramby Chen et al. (1917).This rnethodoffcrs
savingsin the numberof addedstateswith little or no penaltyin
thc accuracyof rnodellingthc acrodyuamicforces.Howcver,its
r) 2003CASI
Vol. 49, No. 4, December2003 I
in the transonicand
applicabilityto the unsteadyaerodynamics
hypersonicregirnesremainsto be established.
As all thesesoftwaresand theorieswcre mainly developed
i n th e U. S . A . .t he nc c d fo r a th e o re ti c aal e ro -s e rv o el asti
c
tool
also existsin the aeronauticalindustryin Canada.The prcscnt
tool is developedin Matlab on the basis o1' thc existing
thcoreticaltools and expertisein the litcrature.
By usc of STARS,the lateraldynarnicsof a half aircrafttest
modcl is stLrdicd.Thc ATM is modelcd by finitc-elernents
rnethods,and the detailsof its modeling arc given by Gupta
(1997). Following the free vibration arralysis of the
flnite-clernentmodcl of thc ATM. tlrrce perf'ectrigid-body
modes,two rigid-controlrnodes,eight elasticrnodes.and thrce
rigid-bodytnodesare gencratcd.
Thc aerodynarnicunsteadyfbrces were getlcratedin the
reducedfrcquencyfr dornainwith the Doublet Latticc Method
(DLM) rnethodin STARS. Then, for flutter calculations.wc
used the p and P/r linear and nonlinearflutter rnethodshere
describedand programmedin Matlab.
Two main comparisonshavc been performcdbctweenthe
resultsobtaincd( l) by introducingthe LS rnethodin the P
flutter methodand comparingthe resultsobtainedby our own
P-LS rnethodversusthe onesobtainedin STARS.(2) rvith the
thrce approxirnationrnethodsLS, MS. and MP. Then. thc
of using thc MS methodarc explained.
advantages
Pk MsrHoD -
LtnsAR SoLUrroN
The formulationfor linearaeroelastic
analvsisin the caseof
the Pk flutter r-nethodis
Mi + Dn + Kn + 4aQ(k,Mac'h)1= Q
(6)
where p, thc n-rodalgeneralizedaerodynamiclorcestnatrix,is
usuallycornplex.The realpart of p denotedby gn, is calledthe
"aerodynamicstiffness",and is in phasc with the vibration
the irnaginarypart of p denotcdby Qt is called
displacement;
the "aerodynarnicdamping"and is in phasewith the vibration
velocity.
This dynamics equation is a sccond-degrccnon-linear
coordinatcsvariable11.
cquationwith rcspcctto thc generalized
The non-linearitycolncs from the fact that the acrodynarnic
generalized
forcesrnatrixQ is a functionof reducedfreqr,rency
/r, dependingof the naturalfiequency0)as shown in E,quation
(s)
If we considerthe problcrnto be quasi-stationary,
wherethe
forccsmatrix Q is indcpendentof the
aerodynamicgeneralized
reduced frequency k, then Equation (6) becomes a linear
by the dynarnicpressureq,1and the
equation,parameterized
Mac'hnumbcr.
The solutionof the problcm becornes
n(1)= v,,e)"'vrtn(o)
(er)
2003CASI
(t)
Vol. 49, no 4, decembre2003
wherc n(0) is the initial value of a generalizedcoordinates
vector;),and Vrare,respectively,
the vectorof eigenvalues
and
the matrix of cigcnvcctors associatedwith the system
represented
by Equation(6); and r represcntsthc timc.
If the general i zcd
coordi nates
vectori s of dim cnsionn, as
thc cquationof acroelasticdynarnicsis of seconddegrec,then
the vector of cigenvalucsis of dirnension2n
tr,lt
(8)
where each cigenr.,alueis written as follows:
) , i- d i + . j ( \
where
1<i<2n
(e)
whcre q is the irlaginary part of thc eigenvalue. represcnting
the liequency, and r/, is thc real part representingthc damping.
The rnatrix ei' is defined as lbllows:
e i r _ d i a g ( e 7 . rsr7 2 r. . . J i t )
(10)
The rnatrix of eigenvectors Vn contains in each of its
colurnns, thc cigenvectors associated with cach eigenvalue.
Thcn. \\re express the systern dcscribcd by Equation (6) in thc
followins r-natrixfonn:
:t '(K:- q,tQ) Ml 'D_lln_l
[n]
ltll-,hl
Ln-l l-M
Ln_l
(,,)
andwe calcr,rlate
thc cigenvalucs
andcigenvectors
of rnatrixl.
The solution (expresscdas the aircraft rnotion) bccomcs
r"rnstablcwhcn the real part of the systcrn eigenvalues
(expressedin tcnrs of darnping)becomespositive.Once the
dynamic prcssLlre(expressedin terms of spced,altitude.or
both) and the Mach numbervary,wc calculatethe parameters
valucs(aircraftfluttervelocities)wherethe flutterphenolnenon
takespl ace.
Pk NlErHoD-
Nox-LrlrnanSolurrox
Now, in the non-stationarycasc, where the aerodynamic
generalizedforcesrnatrixp dependson the reducedfrequency
t, Equation(6) becomesnon-linear.
Many algorithnrslnay give a good approxirnationof the
systerncigcnvalues,
without giving a solutionto the problern.
The first objectiveof the aeroclasticity
is to analyzcthe stability
of the solutionandnot the solutionitself,so thatthe knowledge
of the ei-eenvalues
obtainedis enoughto judge the stabilityof
the sol uti on.
The Pfr methodis one of thc rncthodsthat allows accessto
the systern eigenvalues. This rnethod gives a good
approxirnationof eigenvalues.Its algorithm, presentedin
Fi gure l . consi stsi n fi ri ng a Machnumbcrandcalculat ing
t he
eigenvalucsfbr a sivcn nunrberof speedsthroughan iterativc
Iul
CanadianAeronauticsand SpaceJournal
Fix small V
Approximation
k=abN
onstructQ@ matrix
C a l c u l a t e l v=l d 1 + j a 1
Journalaeronautique
et spatialdu Canada
the form in which the equationsare presentedis different.
Actually, the P method is a non-dimensionalform of the Pft
method,wherethe generalized
coordinates,
speed,and time are
nonnalized.
As Q may be decomposcdinto two parts pp and Qv we
associatepo with 11and Q' with 11.Thercfore,as the Q matrix is
alreadya factorofq, we divide Qrby cqto expressp1 as a factor
of q. Thus, Equation(6) becomes
I
M i + ( D + - qa Q ) n + ( K + q , r p n ) n: 0
B y s u b s t i t u t i nqo
g and cofrom Equations(4) and (5) into
E q u a t i o n( l 2 ) . w e obtai nthe fol l ow i ngequati o n:
Mi+(o.
For each7, calculate
@;bN =k?
(t2)
0)
*,
\/r.)
vcQtln*lK*.pV'QBln=0
(t3)
/\z)
Thc valueof o is givenby Equations( I ) and (2), from where
we obtain the following equation:
p V 2= p o V J
(l4)
Then,both sidesof Equation(1a) are dividedby l and the
definition of ofrom Equation(2) is used,therefore,we obtain
Vr2
V'
Pv:or'i:Prve;:Prvylo
Yes
For V calculate
7"i=di+ jlJ.i
Then -> Next 1",
No l"i -> IncreaseV
F i g u r e l . A l g o r i t h m o f t h e P / <M e t h o d .
processon the reducedfiequencyfr, so that the eigenvaluesof
the system for a given Mach numbcr and speedinterval are
obtained.Of course,as we hypothesizedthatthe Mach number
was constant,the result is invalid only if the speedintervalis
centeredand sufficiently close to the speedcorrespondingto
the Mach number. If it is not close enough,we should restart
the algorithmby choosinga closer Mach number value.
P MnrHoD -
Nox-DrnnpusloNAl
REpnESENTATIoN
The linear and non-linear solutionsof the Pft method have
alreadybeen describedin the previoustwo sections.In this
section,the P method is described.This method comes from
thePk method.The processingof iterationsis the samebut only
182
-
(15)
In the aeroelasticEquation(13), Equation(15) is substitued
into the factor of Qr and Equation( 14) into the factor of pu, and
we obtain
(t\(r^)
M i + lo * ] , p"u , . G -r/r r e , l rnc* l' - p o v l el*n : 0 r l o l
\
4k
\
2
)
A variable change technique is further introduced, this
introduces the reference airspeed V0 through the normal
referencefrequencycr5definedas ob : Vslc.The conversionof
the generalizedcoordinatesq in the codomaininto r1Pin the coP
domain, is realizedwith the following technique:
nP(oP) = n(o)
where
0)P -
0
(17)
('D6
from where
ro : coPob
(18)
and, by deriving Equations (ll)
E quati on(18), w e obtai n
I=ololP
and n=tino
and taking into account
(1e)
O 2 O O 3C A S I
Vo l . 4 9 , N o . 4 , D e c c n rb cr2003 0
To cxpress thc reduced fiequency k as a function of
o)P,v, and o. we substitutecogivenby Equation( l8),
parameters
tr'g i venby E quat ion( 2 ), l t g i v c nb y E q u a ti o n(3 ), a n d ofi orn
Eq u a tions( l) and ( 2) i n to Eq u a ti o n(5 ). a n d w c o b tai n
V ol . 49, no 4, di cembre2003
where Ai are the coetflcients rnatrices and Fi arc lug
coeft-rcients
of the approxinration
rnodel.Further.the objcctivc
functiortto be rninirnizedis deflnedas follows:
1
-e,,{}kp
I w,i,1g,,tjk,t
r :LI
(D(' (' n V6
k=-:-0)'0),-L=-(0'
2V 2
Vr,
.' p I,, Vo
,
,, VF
,
(24)
iil
rvlrere14',,7
i s gi vcn by
(2 0 )
Wilt =
k={o,on
2v
In the end, the aeroelasticnonnalizcdequationis obtained
by the P tnethod, after introducing thc ncw generalized
v c c t orql' b y E q u a ti o n (s1 7 )a n d (1 9 ),a n dby usi ng
co o rd ir lat es
the airspeedratio v from Equation(3)
M P i t ' + ( D P + u . u 6 l u o ) n n+ ( K P + v 2 K ' P ) n P: 0
( 2 1)
I
(2s)
,'0*(t.
lo,tlr ,)
and klis thc /th normalizcdficquency.After minirnization,the
coeftlcientmatricesare lirrthercalculated
|0",,1
:|
wj'1Yr'Yo'
+ nfin,,1
f'l"l i?
where Lf , DP,DgI', KP,and Kul' matricesarc
"
D t ' _! O = D t
cQr
MP =M,
Kt'= Ln x,
oo-
,.P
|
w,r^['0n'ir
1 B[ra''i]
f;
In Equation (26), we havc cxpresscd Bot1,811.Qni1r.and Qtilr
as follows:
Dr' -
|orL'|et(k^
Much\.
lB^u/, - l I o k. lt
t
't-
Ka' = ; P,,c'Qp.(k^Much)
L
In thc following scctions,thc three approximationsfor thc
aerodynamicsconversionfionl the frequencydomain into the
Laplacedomainare described.They are the leastsquares(LS),
the matrix Pade (MP), and thc minimum state (MS)
approxin-rations.
L Elsr-SeulnEs AppnoxrMATToN
The aerodynamic forces approxirnation for Q(k,M
calculatedby the DLM is writtcn in the Laplacedornainas
fbllows:
Q ( r ): A , ,+ A ' . s+ A . . s *l i O , ,
l-l
J
t+F,
A,k 2 +
S o ' - ,- J = k
ik+Fi
7
g,, : [o
" L-
-kt
I
o
^t
- ^;2t
, / l
t i * F i k.i 1+ $ .
-kh.
-k'h'
:
r i * P i ki +lt,
1
l
I
etl
I
I
l
and
Qniir: Reloi(lfrr)l
Qriir:lmlQ,,Q k)l
and
As can be seen in Equations(26) to (28), the coefficient
matricessolutionsdepcndon the lag coefficientsF7 variablc
values.The quasi-Newtonrnethodis further usedto minin-rize
the obj ccti vefuncti on.
(23)
This method is similar to the LS approximation.however
different[3; are calculatedfor each colurnn,so that the next
rnodel is further used.see Tiffany and Adarns (1987)
Q r ( s:) A q r1 A r r s+ A , r r ' * t A , r * ; , r a #
'\ r
|
P,rl
and in the reduccdfiequencyft dornain.it becornes
io 2003 CASI
(2tl)
Mnrntx-Pnne, AppnoxrMATToN
(22)
or, as the Laplace variable is ^s:.7fr, then
Qffr) : A o * A , jk
(26)
(2e)
CanadianAeronauticsand SpaccJournal
Q f i n - A 0 r* A r ri k - A . i k '+) l A 1 , , r ),
- l
,ih
(30)
matriccsof 7th column of the
whcre A,, are the coeft-icicnts
matrix A, and F,,r orc lag cocfficicntsof thc 7th colurnn of
rnodel. In this case. the lbllowing objcctive
approxin-ration
fu n cti o nis r ninir niz c d :
t, =4ZtrilQil(ik)-Q,,t
ir,)l'
( 3 1)
Journaladronautique
et spatialdu Canada
The objectiveis now non-linearwith respectto the E and D
matri ces.The sol uti oni s obtai nedusi ng the i t er at ivelincar
quadrati csol uti on. Matri x D i s now fi xed a nd t he lincar
quadraticproblernis solvcd accordingto thc E matrix. Next,
the E rnatrixis hxed and thc lincarquadraticproblcrnis solved
accordi ngto the D rratri x. Thi s schernei s r epcat edunt il
convergence
is attained.To solvc the linearquadraticproblcrn
accordingto thc E rlatrix with thc D matrix fixcd. thc objcctive
lirnction is rewrittenbv colurnnas fbllows:
(36)
t=\t,
In the MP rlcthod with respectto the LS rnethod,thc
matricesBp7and 8,, arc diffcrcnt for cach column.
t, -4.ltr,lwi e/k)
vtunn-SrnrnAppnoxl MATIoN
M rr,,{r
The MS approximationis written in thc Laplacedornain
I
Q ( s ): A o + A , s + A , 3 ' + D l s l - R l E . r
(3 2 )
In the frcqucncydotnain,this approxirnationbecornes
Rl tr.jt
Qfjtl = Ao * A,.ik+ A,k2 +olikl
(33)
where the D and E matrices are explained in thc next
and arc relatedto the convergencc
of thc solution.R
paragraphs
is thc aerodynarniclags diagonal rnatrix. To rninirnize the
objcctivefunction.wc apply thc next constraints,thus Karpel
( ree0)
ReQ(0)l= ReQ(g)1,
u'herethe colurnnerroris
(37)
where
e i ( f t r=
) Q ; ( l / ' 7 )- Q i ( 7 1 1 )
(38)
Furtherrnore.the objectivefunction (36) is rninimizedfcrr
each col urnr.t.A ccordi ng to E quati on (35). solving t he
r-r-ri ni rni zati probl
on cnr (37; i s equi val cntto solving t he
following cquationfbr eachcolumn of the E rnatrixas shown
b y G o l u ba n d V a n L o a n ( 1 9 8 9 ) :
A"Ei - B"
(3e)
whcrc A,. and 8,, are expressedas follows:
DCy
A" : I clDtw,,, DCor+ cilurwr?7
R c O ( / / ' r ) l= R e Q ( j k / ) i
I
-B,,r)
- Br.,r)
+c/,nfwi(Qr7(frr)
8,,: Icl,DtwT?r(Qni(frr)
I r n Q ltf r * ) l= l m p l r k . ) i
We know that
(:jkr -R) I : (k2r+ R2)r(-/frI-R)
(34)
theseconstraint
applications.
thernodelbecomes:
Following
Qffr r ) = B R1+ DCp T u* j ,B,, + D C ,7 E)
i n wh i c h
(35)
The error is writtcn as a row crror and the linear quadratic
problem,accordingto the D matrix,is equivalentto solvingthe
following equationfbr each row of D:
DiA,t : B,/
(40)
wherc
A,t: ICo,Ewit,EtCL +CyEwit,EtCI
I
t.)
BR/: Qn(o)- * tq*to)-Qn(tr)l
ftl
cnr= nfttir + R2)t tt it+ Rr) ' )r?
= L,
Br1
frQ,tL"t
c , : o [ f t 3 , + R r ) - r- ( k i t + R r ) |
lu4
-lo*,
B,/:
It0,.i
-Br/)wiEtcil
(k)- BB,,)wi?rEtcil,
+(eri(rtr)
is generallybctterthan
In conclusion.the MS approxirnation
the LS and MP approrirnations.When the nr,rmber
of lags is
greaterthan thc nurnberof lrequcncies,the problembecomcs
i l l condi ti oned.
To solve this problern, two adclitional features are
consi dcrcdat cachi terati on:( l ) the l i nearquad r at icpr oblem s
accordingto thc E rnatrix and after that accordtngto thc D
( SVD) as
decornpo sit ion
matri xaresol vcdusi ngsi ngul ar-val ue
show n by Gol ub and V an Loan ( 1989); (2) t he opt im al
r , 2 0 0 3C ' A S I
Vol. 49, No. 4, Dccember2003
0
betweenthe prescntand the lastitcrationis choscn
comprornise
tcl cnsurethc algorithmcouvergcncc.Next, w e w o u l d l i k e t o
cxplain thesctwcl features.
Equat ions( 39) an d (4 0 ) n ra y b e w ri tte n irr the fclllowing
fbrrn:
Ax-B
1al)
i n whic h x is an un k n o w n v c c to r.W h e n th i s s y stcrni s i l l
t hc LS s o l u l i o nc a n b c o b ta i n c cul s i n gSV D of A .
co n d i t ioned.
In l a c t , A c an bc dc c o rn p o s eads l b l l o w s :
A - USVT
(42)
s .n d S i s a di a_qonal
i n w h i c h U and V ar e o rth o g o n anl te ttri c c a
rn a trix .T he diagona lo 1 ' th e S r-n a tri rc o n ta i n s i. tr ordcr.the
si rrg ularv aluc of A s o th a t. i f ra n k (A) - r { n . th c ri I' l ast
e l crn c ntof
s t hc diagon aal rcn u l l . U s i n -e
th i sd c c o n -rp osi ti orr.
thc
n 1 ) c a n b c o b ta i n c da s fo ll ow s:
of E qr , r a ti o(4
L S solLr t ion
Vol. 49, no 4. decernbre2003
Firstly, we have to cxpress tlic aerodynamic forces Q under the
Padi polynomial fbnn
1.*,
Qk)- A,,+ ltiA,+ ( iD A. *'f ,'-ik
7 ik+flt
rvlrereA, arethc coclficicntsof a dirnensioncqr-ral
to the rnatrix
and
fl'orn
obtained
thc
LS
algorithm.
and
O
Fi are the
acrocl ynanrilcags. Thc opti mal val ucs of the se lags. which
mi ni rl i zc the appro.ri rnati on
crror bctw eenthe aer cldynar nic
(.)(/r)nratri x and i ts approxi nrati on
by P adepolynom ials.is
calcLrlatedf urthcr. Equations (5). anc'l the n,cll-knor,vn
crpression.\' : /(Dgivcs ili - hslV trnd by placing this ncw
cxprcssi oni n E quati on(6) w e obtai u
M i + D r l+ K \ + q , r
E - c x E , nr ) + ( l
cr)E1py
(41)
Sirnilarly,fbr the D rnatrix. the ob.jectivefurrctioncan be
rninimizcdaccordinsto cr with
D : c r D , n, , * ( l - c r ) D , n )
(4t)
tJ ,
/ | \l
-,,
+t,rr
.li,)0.*
tL
l.v - ",q.
+ t, In
"1r,,
I
;
F
r
i - t . s+
,
l
]
L
( 4 3)
thc coluntnsof tnatriccsU and
whcrc u; and vr arc,respectively,
V and ^s,is thc ith elernentof diagonaln-ratrixS.
L ct E lprand E , n r ; b c th c o p ti rra ls o l u ti o nto th c p r esentand
last iteration so that the optirnal cornplonrisebetrveenthc
p re se ntand las t it c ra ti o n i s o b ta i n c d b y rn i n i rn i zi ngthe
o b i e c t iv ef unc t ion( 37 ) a c c o rd i n gto u l v i th
(46)
t)l
I
The statcvectorof acrodynamicrnodesis firrthcrintroduccd
\
' -\
:-
-ll
I
(48)
'l
l'
.i+l-0,
and by repl aci ngE quati on(48) i n E quati on(4 7) . r , vcobt ain
:u
ui*on * rn nn,,l'f,'A:*,x,-]
Li-.r
(4el
l
whcre
(45)
Fo r t hes c s c ala r rn i n i rni z a ti o n s " w e u s c d a scal ar
rninimizationalgorithrn.
tI:ivt+
0., D=D. r,,(l)^,
,,,11)'
and
LS MsrHoD lNrnonucEDrNruE P METHoD
Equat ion( 21) dc s c ri b i n gth e d y n a rn i c so f a e ro -s c rvoel asti c
systcmsofl-ersonly serni-lincarreprescntations.
In lbct, all
terms related to the aerodynarnicfbrces present strong
non-lincariticswith respectto the rcdr-rced
lrequcncyk. Then,
th e mu lt it udcof t he a n a l y s i sa n d rn o d c l i n ga l g o ri th r ns
appl i ed
to the linear systemsrequiresthc rnotivatiorrto obtaina linear
system. Various mcthods arc available to
aero-servoelastic
seePitt and Goodrnan(1992),Rogeret
applythis linearrzatron,
a l . ( 1 9 7 5 ) ,K a r p c l( 1 9 8 2 ) ,D u n n ( 1 9 8 0 ) .T i f l a n y a n d A d a m s
, n dC o t o ia n d B o r e z( 2 0 0 1 ) . W e
( 1 9 8 4 .1 9 8 7 ) P
, o i r i o n( 1 9 9 5 ) a
h a vechos ent he s im p l e s o
t n e .th e L S mc th o d b e causei t i s
in STARS- and in this wav we can comparethe rcsults.
r,rsed
i c r2 0 0 3 C A S I
-
/A\
K:K*U,r[
;1,
(s0)
Wc substituteli givenby Equation(3) into Equation(2) and
w e obtai n V as a functi onof V 1,.
v, and o. Then,Equat ion( l8)
gives cq)as a function of crlPand rr) and by equatingthe two
E quati ons(20), gi vi ng the val ue of /t. w hi le t aking int o
consideration
that b : c'12"
we firrtherobtainthc valucof theblV
ratio as function of o, v, and 06, as follows:
h_
L'
^[6
2r,cqy
( sl )
185
CanadianAeronauticsand SpaceJournal
The variablechangetechniqucintroducesZtr: c(erand by
replacingthis expressionforVs into Zp given by Equation(3).
we obtain Vy:.: (:(\1y.Furthcrmorc,wc substitute V, into qr1
g i vcn b y E quat ions( 4) a n d (1 4 ), a n d w e o b ta i n
Journaladronautique
et spatialdu Canada
00
0 ilrP o
0
I
T
0
.p
n'
n'
..P
(s7)
x,o
OI
0
I
, . , = ' Ir r r t ' r t ' - ' i v r
q,t :
;P,,Vt:
(s2)
Thcn,we substituteq6 givenby Equation(52) andblZ gir,'en
by Equation(51) into Equation(49) and into the coefficients
givenbyEquation(50) and we divide the resultingequationby
cofrand we obtain
"l
f l-,,,
0
OI
-fre -fie
OI
0
-rt A!
+l'1
0
OI
Al- T t ,x,t I : 0
t4oi' + Dtnt + RPqP+ vrl F
| /-/
I
Lr-r
l
1,. I/l
Ltl
OI
0
p
0
n'
-v2 A!
n'
xl
1,+ nl,
0
0
0
-01 1
(53)
.p
;,
t1$
REsulrs ANDDrscussroN
where
ilf' : MP + oAl,
DP - DI' + r^[" ,q|
To : KP+v2AP
with the coefficients
^ll
A P c= ,
'^p,,r'2
A r , . A l = L pot'bA1,
Al =
I
^l
A'i : ^prt'-A
L
for
lo,,b''t,
3 <j <2 + n,
The first equationof the systemof Equations(19) and also
the ratio Vlb given by Equation (5 I ) arc substitutedinto
Equation(48), and wc obtain
*,*4roo$iX,
-,on,to
(s4)
!o
Next, salre variablc transfotmationis applied ro X, as the
o n e a p pliedt o 11giv en b y E q u a ti o n(1 9 )
f i' :0fh
*,
(5s)
givenby Equatton
With this lastncw variabletransformatiot-t
(55), and by dividing Equation(5a) by ob, we obtain
X l * P i X i = r l P , w h e r eV' i : ? F ,
r/o
Then, we obtain the final matrix equation
t86
(56)
To validate our rncthod.u'e uscd the STARS Aircraft Test
Modcl devel opedby Gupta (1997) at N A S A Dr yden Flight
elements
ResearchCenter.This rnodclincludesaero-structural
(flexible aircraft)and control surf-aces
(aircraftcommands).
Firstly, a frcc r,'ibrationanalysis was performed in the
to obtainthe free modesof vibration.
absenceof aerodynamics
We obtainedthe samefrequenciesand modesof vibrationby
our method as well as by the one used in STARS.
Secondly, to calculate the aerodynamic forces in the
frequencydomain by the doublct lattice method (DLM), the
simulationparametersare consideredin the STARS computer
program:the referencescmi-chordlength h - 81.50 cm, the
refcrenceair densityat scalcvcl ps : 1.225kg/mt. the altitude
Mach number Mach :
at the sea level Z - 0 km" the ref-erence
0.9,thereferencesoundairspccdat sealevela6: 340.3rn/s.
In Table l, the resultsof an aeroelasticnon-linearanalysis
arc shown"and velocitiesat which flutteroccursare calculated.
In Table l, the resultsgiven by our Pfr methodare cotnparcd
with the results givcn by STARS through the three flutter
open-looprnethods(k method,Pk rnethod,and ASE method).
Actually,we shouldonly comparcthe resultsobtainedby our
P/r method(row 4) with the resultsobtaincdby the P/r method
in the STARSprogram(row 2). We canconcludcthatwe obtain
a goodcomparisonin tennsof fluttervelociticsandfiequencies
for both fl utterpoi nts.
In Table 2, we prescnta comparisonbetweenthe results
obtainedon the linearizedATM with our P rnethodmodifiedto
include the LS fbrmulation shown on row 4 with the ones
obtainedby useof the STARSsoftware,calledthe ASE method
(row 3).
open-loop analysis is further
A lirrear aero-servoelastic
perfbnnedon the ATM. This analysisusesthe P-flutterrnethod
where the aerodynamicunsteadyfbrccs werc approximatcdby
the LS rnethod.Resultsare shownin Figures2 to 4. As shown
in Figure 4, the unstablemodeswhereflutter occursare elastic
mode2 (flutterpoint I ) and mode7 (flutterpoint 2). Mode 2 is
the fuselagefirst bendingand mode 7 is the fin first torsion
mode.
ic,r2003 CASI
Vol. 49,No. 4, December2003 I
Vol. 49, no 4, ddcembre
2003
Table l. Aeroelastic Non-linear Analvsis.
Flutter I
STARS
/c method
Pk method
l,SE method
Our Pk method
Flutter 2
Equiv. airspeed(km/h)
Frequency(radls)
Equiv. airspeed(km/h)
Frequency(rad/s)
821.10
818.74
878.79
817.44
77.4
17.4
77.3
77.4
1596.5
1600.4
l35t.l
1404.7
t47.3
147.l
136.2
145. 6
Table 2. Linear Aeroelastic Analvsis.
Flutter I
STARS
k method
Pft method
,4SE method
Our P-LS method
Flutter 2
Equiv. airspeed(km/h)
Frequency(rad/s)
Equiv. airspeed(km/h)
Frequency(rad/s)
82r.70
8 18 . 7 4
878.79
878.79
77.4
71.4
77.3
77.5
1596.s0
1600.40
t3sl.10
I s 0 8 .01
14 7. 3
147.1
136.2
144.2
35il
i-i Eler:_;tiit-r-rrtr-le
1
.Ir ElAStir_t-il'tde?
3fiil
:5il
I
U]
E
{U
+
+
n
{;}
F
a
Elastirr-nide3
Elastii:tr-rtrje
4
Elastirmo'Je5
Elastictrr'trjEfi
ElasticrxildeT
Elastirrlo,le I
t*
]*t
TJ
C
fi]
?il0
=
{F
il)
!*
LL
150
1ilil
5il
-3ttil
-:[0
- 1ilil
t)amping
{g.20il}
1ilil
Figure 2. P-LS Method - F r e q u e n c y ( r a d / s ) v e r s u s D a m p i n g ( 9 * 2 0 0 ) .
i.) 2003cASI
187
Canadian Aeronautics and Space Journal
0
Journal a6ronautiqueet spatial du Canada
o
x
+
*
tr
+
v
a
;
{n
Elasticmode1
ElaEtic
mode2
Elasticmode3
Elasticmode4
Elasticmode5
Elasticmodefi
E l a s t im
c o d e7
ElasticmodeB
'j(]
s
lF*
L}
C
0)
:5
(r
(I}
: l t t-
5fi0
1il00
15*il
Equivalent
airspeed{krn/h}
2fi*il
Figure 3. P-LS Method - Frequency (rad/s) versus Equivalent Airspeed (km/h).
The lineanzation of aerodynamic forces by the LS method
offers many advantages over the comprehension of the
dynamics of aeroelasticsystems as well as over controller
conception. In our study, we have chosen a linearization
techniquethat is relativelysimple,basedon the optimizationof
aerodynamic lags By and on the LS minimization of Pad6
coefficients4,.
When we observethe equivalentairspeedsfor the flutter
phenomenon,the resultsof the combinedP andLS methodare
closer to the results of the Pft method than those of the ASE
method programmedin STARS. This conclusiondependson
the quality of the hnearrzattonapplied on the ATM, becausethe
results of the non-linear model (Pfr method) and of the
linearizedmodel (LS and P method) are close.Our linearized
rnodel divergeswith respectto the ltneartzedmodel conceived
with STARS. This divergence may be explained by the
precision of each hnearrzation,and on the other hand by the
differenceof the hneartzatronmethodsand more particularly of
the optimrzatron techniques.
188
An optimrzatron algorithm minimizes the quadratic error
denotedby -/ between the exact aerodynamicforce matrix and
the approximation of the aerodynamic force matrix
approximatedby Paddpolynomials.This algorithm varies the
valuesof aerodynamiclags, where the approximationof Pade
polynomials is realized with the LS algorithm.
Resultsof the LS method applied on the ATM for a Mach
numberof referenceM:0.9 are shownin Figure 5. A total of
six aerodynamiclags have been used to obtain a satisfactory
aerodynamicforcesapproximation.The left graph in Figure 5
showsthe evolutionof the norm of the quadraticerror between
the aerodynamic force matrix and the approximated
aerodynamicforce matrix as a function of iterationnumber of
the optimrzatron algorithm.
There is also compromise between the number of
aerodynamiclagsand the quality of approximation.The greater
the number of aerodynamic lags, the better is the
approximation. At the same time, the aeroelastic model
becomesmore cumbersomefrom a computing time point of
view as the number of aerodynamiclags is increased.
( O 2 O O 3C A S I
Vol. 49,No. 4, December
2003 0
Vol.49, no 4,ddcembre2003
o
H
+
#
n
+
v
a
-sfi
{}
{}
E l a s t im
c o d e1
E l a s t im
c o d +2
Elasticmode3
E l a s t im
c o d e4
E l a s t im
c tde5
E l a s t im
c o d eS
E l a s t im
c o d e7
ElasticmodeB
- 1ilfi
f-'l
lc
#J
{}r
r
'a
f; - 15il
ffi
n
-TilS
-?5fi
-300
t^l
LJ
silfi
1ff0fi
150il
Equivalent
airsp*ed
{km/h}
?il*il
Figure 4. P-LS Method - Damping (9*200) versus Equivalent Airspeed (km/h).
o
o
o
c
o
rigid, and control modes,for which the aerodynamicforcesare
defined.
Concerning the existence of the other methods of
optimization,a comparisonis realisedon the useof the LS, MP,
and MS algorithmsfor the aerodynamicforcesapproximation.
Figure 5 showsthe total error for LS, MP, and MS modelsis of
the same order (160). These results shows us clearly the
superiorityof the MS model.
250
200
150
IE
E
x
100
CL
50
o
a,
0
LS wit h 16 0
v ar iables
( 4 lags )
M P w i th 1 6 0 MS w i th 1 6 0
v a ri a b l e s
v a ri a b l e s
(4 l a g s )
(1 6 0 l a g s )
Figure 5. Errors for the Three Methods LS, NIB and NIS.
The reason is that with each ncw aerodynamrclag, we
introduce new aerodynamic l a g s i n th e d e s c ri p ti onof the
aeroelasticmodel dependingon the total number of el asti c.
((,r2003CASI
CoxclusroNs
In this paper, firstly, a comparison was reahzedat the level
of an aero-servoelasticnon-linear analysis. This analysis
concernedour modified Pk methodvalidatedagainstthe flutter
methods programmed by using STARS. The Aircraft Test
Model existing in STARS at NASA Dryden Flight Research
Centerhas been used.A good coherenceat the level of flutter
predictionin open loop, appearedat flutter velocitiesof 834.12
and 1482.90km/h.
CanadianAeronauticsand SpaceJournal
0
Secondly, the influence of the aerodynamic forces
lineanzation LS studies on the flutter prediction, firstly in an
context.
aeroelasticcontext, and later, in an aero-servoclastic
The comparison was good from flutter (aeroelastic) and
points of view.
aero-servoelastic
The aerodynamicforces approximationprecision may be
increasedby augmentingthe number of aerodynamiclags on
one hand,or by useof the most powerful linearizationmethods
that guaranteea good precision,by decreasingthe number of
aerodynamicmodes.
The comparisonof three hneanzationmethods shows the
superiorityof the MS approximations.In the MS method,the
systemis ill-conditionedwhcn the number of approxirnation
lags is larger than the number of reduced frequencics.An
additionalfeatureis presentedto solvethis problem.We should
choose,at each iteration,an optimal compromiscbetweenthe
presentand the last iteration.[n the end, the presentpaper
methodon the
showsa goodvalidationof our aero-servoelastic
ATM model in STARS.
Journal aeronautiqueet spatial du Canada
K a r p e l , M . ( 1 9 9 0 ) . " T i m e - D o m a i n A e r o - s e r v o c l a s t i cM o d e l i n g U s i n g
WeightedUnsteadyAerodynamic Forces".J. Guid. Control, Vol. 13, No. I I,
pp. 30 37.
N o l l , T . , B l a i r , M . , a n d C e r r a ,J . ( 1 9 8 6 ) . ' A D A M , a n A e r o - s e r v o e l a s t i c
A n a l y s i sM e t h o df o r A n a l o go r D i g i t a l S y s t e m s "..1 .A i r c r .V o l . 2 3 , N o . I 1 .
Pitt. D.M.. and Goodman, C.E. (1992). "FAMUSS
A e r o - s e r v o e l a s t iM
c o d e l i n g T o o l " . A I A A P a p .9 2 - 2 3 9 5 .
A
New
Poirion. F. ( 1995). "Modelisation Tcmporclle des Systdmes
A d r o s c r v o i l a s t i q u c sA
. p p l i c a t i o n A I ' E t u d e d e s E f f e t s d e s R e t a r d s " .L a
RechercheAerospaliole.No. 2. pp. 103 I 1,1.
R o g e r ,K . L . , H o d g e s .G . E . . a n d F e l t . L . ( 1 9 7 5 ) ." A c t i v c F l u t t c rS u p p r c s s i o r r
A F l i g h t T e s t D e m o n s t r a t i o n "J. . A i r c r . V o l . 1 2 , p p . 5 5 1 - 5 5 6 .
T i f T a n yS
, . i l . , a n d A d a m s .W M . , J r .( 1 9 8 4 ) ." F i t t i n g A e r o d y n a m i cF o r c e si n
t echniquc
t h e L a p l a c eD o m a i n : A n A p p l i c a t i o no f a N o n l i n e a rN o n - g r a d i e nT
t o M r - r l t i l e v eCl o n s t r a i n c dO p t i r - n i z a t i o n "N. A S A T M - 8 6 3 I 7 .
Tiftany. S.H.. and Adarns,W.M.. Jr. (1987). "Nonlinear Prograrnrling
Methods of UnsteadyAerodynarrics".
E,xtcnsions
to Rational Approxirr-ration
Proc'cetlirtg.so/ thc )8th AIAA/ASME/ASC'l',/AHS Structure.s, Struc'tural
Dt'nomit'.s,ttnd ,)lutet'iul.sConfarenc'c,Montcrey, Cialifornia,6-8 April 1987.
, ew York, New York.
A m e r i c a n I n s t i t u t eo f A e r o n a r - r t i casn d A s t r o n a u t i c sN
pp. 406 .120.
AcTxowLEDGEMENTS
The authorswould like to thank Dr. Kajal Gupta at NASA
Dryden ResearchFlight Centerfor allowing us to usethe ATM
in STARS. Many thanks are due to the other members of
STARS Engineeringgroup for their continuousassistanceand
collaboration:Tim Doyle, Can Bach, and Shun Lung.
RETEnnNCES
Adams, W.M., and Hoadley, S.T. ( 1993). "ISAC: A Tool fbr
A e r o - s e r v o e l a s tM
i co d e l i n ga n d A n a l y s i s " .N A S A T M - 1 0 9 0 3 1 ,p p . I 1 0 .
C h e n . P . C . ,S a r h a d d i .D . , L i u , D . D . . a n d K a r p e l , M ( 1 9 7 7 ) . " A U n i f l e d
Approach
fbr
lnfluencc
Cocfllcient
Aerodynamic
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