and the second gives a review of unsteady aerodynamic forces approrirnation rnethods fiom tlie reduced frequency fr dornain into the Laplacc .r dornain.. Two main comparisons havc been p
Trang 1Vol 49, No 4, December 2003 O Vol 49 no 4, <lecembre 2003
Loop Analysis
Ruxandra Mihaela Botez * Alexandre Doin * Diallel Eddine Biskri x lulian Cotoi * Dina Hamza * Petrisor Parvu *
Abstract
Aero-servoelasticity (ASE) is a multi-disciplinary study of
interactions among structural dynamics, unsteady
aerodynamics,and control systems In this paper, the
Aircraft Test Model (ATM) developed by the NASA
Dryden Flight Research Center is used, and the velocities at
which flutter occurs are calculated by use of the Structural
Analysis Routines (STARS) aero-servoelastic software
For the validation of our aero-servoelastic study, the
STARS aero-servoelastic software, also developed by
NASA, is used We developed a new aero-servoelastic tool
in Matlab to consider these interactions, and the results
obtained through our method are compared with the ones
obtained through STARS
Key words: aerodynamics aeroelasticity,
servo-controls, aero-servoelasticity
R6sum6
L'a6roservo6lasticite (ASE) est I'objet d'une etude
multidisciplinaire des interactions entre la dynamique
structurale, l'adrodynamique instable et les systemes de
commande Dans le prdsent document, le matdriel utilizd
est la maquette d'adronef mise au point par le Dryden
Flight Research Center de la NASA Les vitesses
auxquelles le flottement se produit sont calculdes au moyen
du logiciel d'adroservoelasticitd STARS
Nous avons egalement utilize le logiciel
d'adroservodlasticite STARS mis au point par la NASA
pour valider notre 6tude sur I'adroservodlasticitd, puis nous
avons elabord un nouvel outil d'aeroservodlasticite dans le
< Matlab )) pour analyser ces interactions Les r6sultats
obtenus par le biais de notre mdthode sont compar6s ir ceux
qui ont 6td obtenus ir I'aide du logiciel STARS
Mots clds ' Aerodynamique, adrodlasticite,
servocommande, a6roservoelasticite
* Ecole de technologic supcrieurc Departcmcnt dc genie dc la production
a u t o m a t i z e i e
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
H 3 C l K 3 C a n a d a E-rnail : rr"rxandra(z)gpa.etsmtl.ca Rcccived 7 August 2003
0)
n
NonTEncLATURE
natural fiequcncy generalized coordinates as functions of frequency n(co) : [n.(co) n, (ro) l1.1(o)]
generalized coordinates for elastic modes gcneralized coordinates for rigid rnodes generalizcd coordinates for control modes
n o n - d i m e n s i o n a l g e n e r a l i z e d c o o r d i n a t e s , w i t h respect to tirle
non-dirncnsional seneralizcd coordinates fbr elastic rnodes
l
I ' la
C]
4 c
4 r
Q a
M D
6 r K
a
Qr
Qx
o
-.1 s p
P o
o
V vo tr/E
Qa
h
non-dirnensional modes
n o n - d i r n e n s i o n a l modes
generalized coordinates fbr rigid
seneralized coordinates fbr control
modal incrtia or mass matrix
m o d a l d a r n p i n g m a t r i x modal damping coefficients rnodal elastic stiffness matrix modal generalized aerodynarnic fbrccs imaginary part of modal generalizcd aerodynamic forccs matrix
real part of rnodal gencralized rnatrix
modal sensor matrix sensor locations true air density
r c l - c r c n c c a i r d e n s i t y
a i r d e n s i t y r a t i o o : o = P
reference true airspeed equivalent airspeed Vr, :.[oV
aerodynamic forces
a i r s p c c d r a t i o v : L t
vo
dynamic prcssure qd wing chord lcngth semi-chord, b : cl2
( l )
:1,,,
( 2 ) ( 3 ) ( 4 )
Trang 2Canadian Aeronautics and Space Journal Joumal aeronautique et spatial du Canada
k r e d r - r c e d f r e q u e n c y k : Y - ' b ( 5 )
2 V V Mach Mach number
INrnooucrtoN
In this paper, a rnethod for open-loop fluttcr
acro-servoelastic analysis is presented This rnethod is
validated on the Aircraft Test Model (ATM) with the aid of the
Structural Analysis Routincs (STARS) cornpLlter program
developed at NASA Drydcn Flight Research Ccnter by Gupta
( t e e t )
Two main bibliographical research thcrncs arc considered
here Thc first one gives a short review of the aero-servoelastic
analysis softwarc in the literature and the second gives a
review of unsteady aerodynamic forces approrirnation rnethods
fiom tlie reduced frequency fr dornain into the Laplacc r
dornain
A number of aero-servoclastic analysis software tools exist
in the aerospacc industry, rnainly in the U.S.A These tools arc
STARS developed by Cupta (1991) at the NASA Dryden Flight
Research Centcr; the Analog and Digital Aero-servoelasticity
M e t h o d (A D A M ) d e v c l o p c d b y N o l l e t a l ( 1 9 8 6 ) a t t h c F l i g h t
Dynarnics Laboratory Air Force Wright Aeronautical
Laboratories (AFWAL); Intcraction of Structures
Acrodynamics, and Controls (ISAC) dcveloped by Adarns and
H o a d l c y ( 1 9 9 3 ) a t N A S A L a n g l e y ; a n d F l e x i b l e A i r c r a f i
M o d e l i n g U s i n g S t a t e S p a c e ( F A M U S S ) d e v c l o p e d b y P i t t a n d
Goodman (1992) at the McDonncll DoLrglas Ciornpany
ln this paper, we usc STARS among thc cxisting software
tools for aero-servoelastic analysis The STARS prograrn is
designed as an efficient tool for analyzingpractical cngincering
problerns and (or) supporting relevant research and
developrncnt activitics, and it has an interface with NASTRAN,
which is still very much used in the aerospace industry
STARS has been applied to various projccts such as the
X-29A, F-18 High Alpha Rcsearch Vehicle/Thrust Vectoring
Control System, B-52lPcgasus, Generic Hypersonics, National
A c r o S p a c e P l a n e ( N A S P ) , S R - 7 l / H y p e r s o n i c L a u n c h V e h i c l e ,
and High Speed Civil Transport
Another software used for aero-servoelasticity analyses is
thc Analog and Digital Aero-scrvoclasticity Mcthod (ADAM)
computer program, which was developed at The Flight
Dynamics Laboratory ADAM has bcen applied on the
unaugmented X-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 d in t h e N A S A
Langley l6 ft transonic dynamics tunncl and (2) the Forward
Swept Wing FSW model mounted in the AFIT 5 ft subsonic
w i n d t u n n c l ( l f t - 3 0 4 8 c m )
Thc software called ISACI (The Interaction of Structures,
Aerodynatnics, and Controls) was developed at the NASA
Langley Rcsearch Ccntcr ISAC has been used on various flight
r n o d e l s s u c h a s D A S T A R W - 1 a n d A R W - 2 ; D C - 1 0 w i n d - t u n n e l
flutter rnodel; generic X-wing feasibility studics; analyses of
e l a s t i c , o b l i q u e - w i n g a i r c r a f t ; A p ' W ( A c t i v e F l c x i b l e W i n g )
wind tunnel test program; generic hypersonic vehicles; benchmark active controls testing project; high-speed civil transport; etc
Anothcr software fbr developing a statc space model rcprcscntation of a flexible aircraft for usc in an aero-servoelastic analysis has also becn presented This
t e c h n i q u e i s b a s e d o n d e t e r m i n i n g a n e q u i v a l c n t s y s t c m to match the transf-er-function lrequcncy response The theory has been irnplernented in a computer code called FAMUSS at the McDonnell Douglas Aircrafl Cornpany FAMUSS has been used internally at McDonnell Douglas on its aircraft
Recently, an acroelastic code, ZAERO, has been developed atZona Technology by Chen et al (2002), which could bc used also for aero-servoelastic analyscs The influence of thc acrodynamic stores on the aeroelastic instability has bcen studied using a number of aerodynamic models for the F-16 aircraft configurations including the isolatcd wing-tip launchcr rnodcl and the rvhole aircraft with and without stores The results shou, good agreement between the prcscnt numcrical predictions and the flight-flr-rttcr tcst data
A short rcvioi' of eristing aero-servoelastic tools is presented here Nert a rcvic$, of the aerodynamic forces approxirnation fl'orn the fi'equency dornain to the Laplace
d o r n a i n i s g i v c n T h c f o l l o l r ' i n g r n e t h o d s : l e a s t s q u a r e s ( L S ) , rratrix Pade (MP) and rninin-rurn state (MS) arc ncccssary for aero-servoelastic analyses
The conventional LS method has been used in the acro-servoelastic computer program called ADAM where its capability is to determine Pad6 approximations of any order such that the sun-r of the numerator and denominator terms docs
n o t e x c c c d 1 5 Thc state spacc equatior-rs includc augmented states that represent thc acrodynamic lagsl their nutnber is dependent on the nurnber of denominator roots in the rational approximation The aerodynamic entire matrix has been approximated by a ratio of matrix polynornials In tlie MP approximation rnethod
by Roger et al ( 1975), each tcrm of the aerodynamic rnatrix rnay be approxirnated by a polynornial ratio in s Howevcr, it has also bcen found that comfflon denominator roots are also eflective in defining the corresponding polynomials Other rnodifications of the MP rnethod wcre suggcsted by Karpel ( 1 9 8 2 ) a n d D u n n ( 1 9 8 0 ) A h i g h c r n u m b e r o f d e n o m i n a t o r roots is required in the MS approxirnation method by Karpel (1990), whcrc the nurnber of augmented states is equal to the nurnber of denominator roots
The capabilities for enforcing or relaxing equality constraints were includcd in thc LS, MP, and MS rncthods by
T i f l b n y a n d A d a r n s ( 1 9 U 4 , 1 9 8 7 ) T h e s e c a p a b i l i t i c s w e r e abbreviatcd ELS, EMP, and EMS, and they were introduced in
t h c a e r o - s e r v o e l a s l i c c t l n t p u t e r p r o g r a r n I S A C T l i c r n i n i r n u r n state approach (MIST) was selected recently in the ASTROS computer program by Chen et al (1917) This rnethod offcrs savings in the number of added states with little or no penalty in thc accuracy of rnodelling thc acrodyuamic forces Howcver, its
r ) 2 0 0 3 C A S I
Trang 3Vol 49, No 4, December 2003 I Vol 49, no 4, decembre 2003
applicability to the unsteady aerodynamics in the transonic and
hypersonic regirnes remains to be established
As all these softwares and theories wcre mainly developed
i n t h e U S A th e n c c d fo r a t h e o r e t i c a l a e r o - s e r v o e l a s t i c t o o l
also exists in the aeronautical industry in Canada The prcscnt
tool is developed in Matlab on the basis o1' thc existing
thcoretical tools and expertise in the litcrature
By usc of STARS, the lateral dynarnics of a half aircraft test
modcl is stLrdicd Thc ATM is modelcd by finitc-elernents
rnethods, and the details of its modeling arc given by Gupta
(1997) Following the free vibration arralysis of the
flnite-clernent modcl of thc ATM tlrrce perf'ect rigid-body
modes, two rigid-control rnodes, eight elastic rnodes and thrce
rigid-body tnodes are gencratcd
Thc aerodynarnic unsteady fbrces were getlcrated in the
reduced frcquency fr dornain with the Doublet Latticc Method
(DLM) rnethod in STARS Then, for flutter calculations wc
used the p and P/r linear and nonlinear flutter rnethods here
described and programmed in Matlab
Two main comparisons havc been performcd bctween the
results obtaincd ( l) by introducing the LS rnethod in the P
flutter method and comparing the results obtained by our own
P-LS rnethod versus the ones obtained in STARS (2) rvith the
thrce approxirnation rnethods LS, MS and MP Then thc
advantages of using thc MS method arc explained
Pk MsrHoD - LtnsAR SoLUrroN
The formulation for linear aeroelastic analvsis in the case of
the Pk flutter r-nethod is
where p, thc n-rodal generalized aerodynamic lorces tnatrix, is
usually cornplex The real part of p denoted by gn, is called the
"aerodynamic stiffness", and is in phasc with the vibration
displacement; the irnaginary part of p denotcd by Qt is called
the "aerodynarnic damping" and is in phase with the vibration
velocity
This dynamics equation is a sccond-degrcc non-linear
cquation with rcspcct to thc generalized coordinatcs variable 11
The non-linearity colncs from the fact that the acrodynarnic
generalized forces rnatrix Q is a function of reduced freqr,rency
/r, depending of the natural fiequency 0) as shown in E,quation
( s )
If we consider the problcrn to be quasi-stationary, where the
aerodynamic generalized forccs matrix Q is indcpendent of the
reduced frequency k, then Equation (6) becomes a linear
equation, parameterized by the dynarnic pressure q,1 and the
Mac'h numbcr
The solution of the problcm becornes
n(1) = v,,e)"'vrtn(o)
wherc n(0) is the initial value of a generalized coordinates vector; ),and Vrare, respectively, the vector of eigenvalues and the matrix of cigcnvcctors associated with the system represented by Equation (6); and r represcnts thc timc
I f t h e g e n e r a l i z c d c o o r d i n a t e s v e c t o r is o f d i m c n s i o n n , a s thc cquation of acroelastic dynarnics is of second degrec, then the vector of cigenvalucs is of dirnension 2n
where each cigenr.,alue is written as follows:
) , i - d i + j ( \ w h e r e 1 < i < 2 n ( e )
whcre q is the irlaginary part of thc eigenvalue represcnting the liequency, and r/, is thc real part representing thc damping The rnatrix ei' is defined as lbllows:
e i r _ d i a g ( e 7 r r s 7 2 r J i t ) ( 1 0 )
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-natrix fonn:
and we calcr,rlate thc cigenvalucs and cigenvectors of rnatrix l The solution (expresscd as the aircraft rnotion) bccomcs r"rnstablc whcn the real part of the systcrn eigenvalues (expressed in tcnrs of darnping) becomes positive Once the dynamic prcssLlre (expressed in terms of spced, altitude or both) and the Mach number vary, wc calculate the parameters valucs (aircraft flutter velocities) where the flutter phenolnenon
t a k e s p l a c e
Now, in the non-stationary casc, where the aerodynamic generalized forces rnatrix p depends on the reduced frequency
t, Equation (6) becomes non-linear
Many algorithnrs lnay give a good approxirnation of the systern cigcnvalues, without giving a solution to the problern The first objective of the aeroclasticity is to analyzc the stability
of the solution and not the solution itself, so that the knowledge
of the ei-eenvalues obtained is enough to judge the stability of
t h e s o l u t i o n The Pfr method is one of thc rncthods that allows access to the systern eigenvalues This rnethod gives a good approxirnation of eigenvalues Its algorithm, presented in
F i g u r e l c o n s i s t s i n f i r i n g a M a c h n u m b c r a n d c a l c u l a t i n g t h e eigenvalucs fbr a sivcn nunrber of speeds through an iterativc
[n] :t : l ltll -,hl (,,)
Ln-l l-M '(K - q,tQ) M ' D_lln_l Ln_l
( 6 )
(er) 2003 CASI
( t )
I u l
Trang 4Canadian Aeronautics and Space Journal Journal aeronautique et spatial du Canada
the form in which the equations are presented is different Actually, the P method is a non-dimensional form of the Pft method, where the generalized coordinates, speed, and time are nonnalized
As Q may be decomposcd into two parts pp and Qv we associate po with 11 and Q' with 11 Thercfore, as the Q matrix is already a factor ofq, we divide Qrby cq to express p1 as a factor
of q Thus, Equation (6) becomes
qo and co from Equations (4) and
o b t a i n th e f o l l o w i n g e q u a t i o n :
\ / r )
v c Q t l n * l K * p V ' Q B l n = 0
/ \ z )
M i + ( o
* ,
-P v : o r ' i : -P r v e ; : -P r v y l o ( 1 5 )
In the aeroelastic Equation (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 + l o * ] , p u , G r r r e , l n * l r c * ' - p o v l e * l n : 0 r l o l
\ 4 k " - / \ 2 )
I
0)
B y s u b s t i t u t i n g
E q u a t i o n ( l 2 ) w e
nP(oP) = n(o) where 0)P - 0
('D6
from where
ro : coPob and, by deriving Equations (ll) and
E q u a t i o n ( 1 8 ) , w e o b t a i n
( t 2 )
(5) into
( t 3 )
Thc value of o is given by Equations ( I ) and (2), from where
we obtain the following equation:
Then, both sides of Equation (1a) are dividedby l and the definition of ofrom Equation (2) is used, therefore, we obtain
A variable change technique is further introduced, this introduces the reference airspeed V0 through the normal reference frequency cr5 defined as ob : Vslc The conversion of the generalized coordinates q in the codomain into r1P in the coP domain, is realized with the following technique:
process on the reduced fiequency fr, so that the eigenvalues of
the system for a given Mach numbcr and speed interval are
obtained Of course, as we hypothesizedthat the Mach number
was constant, the result is invalid only if the speed interval is
centered and sufficiently close to the speed corresponding to
the Mach number If it is not close enough, we should restart
the algorithm by choosing a closer Mach number value
P MnrHoD - Nox-DrnnpusloNAl
REpnESENTATIoN
The linear and non-linear solutions of the Pft method have
already been described in the previous two sections In this
section, the P method is described This method comes from
the Pk method The processing of iterations is the same but only
1 8 2
( 1 7 )
( 1 8 ) taking into account
( 1 e )
Fix small V
Approximation
k = a b N
onstruct Q@ matrix
For each7, calculate
@ ; b N = k ?
Yes For V calculate
7 " i = d i + j l J i Then -> Next 1",
No l"i -> Increase V
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
I = o l o l P a n d n = t i n o
Trang 5V o l 4 9 , N o 4 , D e c c n r b c r 2 0 0 3
0 V o l 4 9 , n o 4 , d i c e m b r e 2 0 0 3
To cxpress thc reduced fiequency k as a function of
parameters o)P, v, and o we substitute cogiven by Equation ( l8),
t r ' g i v e n b y E q u a t i o n ( 2 ) , l t g i v c n b y E q u a t i o n ( 3 ) , and ofiorn
E q u a t i o n s ( l ) a n d ( 2 ) i n t o E q u a t i o n ( 5 ) and wc obtain
, ( D ( ' ( ' n V6 ' p I,, Vo
k = - : - 0 ) ' 0 ) , - L = - ( 0 '
2V 2 Vr, , ,, VF
( 2 0 )
-k = { o , o n
2v
In the end, the aeroelastic nonnalizcd equation is obtained
by the P tnethod, after introducing thc ncw generalized
c o o r d i r l a t e s v c c t o r q l ' b y E q u a t i o n s ( 1 7 ) and (19), and by using
the airspeed ratio v from Equation (3)
where Ai are the coe tflcients rnatrices and coeft-rcients of the approxinration rnodel Further
functiort to be rninirnized is deflned as follows:
Fi arc lug the objcctivc
M P i t ' + ( D P + u u 6 l u o ) n n + ( K P + v 2 K ' P ) n P : 0
where Lf , DP, DgI', KP, and Kul' matrices arc
M P = M , D t ' _ ! O = D t
cQr
Dr' -
rvlrere 14',,7i s g i v c n b y
I
and klis thc /th normalizcd ficquency After minirnization, the coeftlcient matrices are lirrther calculated
In Equation (26), we havc cxpresscd Bot1, 811 Qni1r and Qtilr
as follows:
f'l"l
: i? wj'1Yr'Yo'
r :LI I w,i,1g,,tjk,t -e,,{}kp 1
i i l
,'0*(t lo ,tl r ,)
B u , - l I o k l ^ / - ^ t
t t t i * F i
g,, : [o -kt o -k'h'
" L- I r i * P i
(24)
(22)
"
f;
w,r^['0n'ir 1 B[ra''i]
Kt' = Ln x,
o o
-, P | ' t
-Ka' = ; P,, c'Qp.(k^ Much)
L
In thc following scctions, thc three approximations for thc
aerodynamics conversion fionl the frequency domain into the
Laplace domain are described They are the least squares (LS),
the matrix Pade (MP), and thc minimum state (MS)
approxin-rations
L Elsr-SeulnEs Appnoxr MATToN
The aerodynamic forces approxirnation for Q(k,M
calculated by the DLM is writtcn in the Laplace dornain as
fbllows:
and
As can be seen in Equations (26) to (28), the coefficient matrices solutions depcnd on the lag coefficients F7 variablc values The quasi-Newton rnethod is further used to minin-rize
t h e o b j c c t i v e f u n c t i o n
Mnrntx-Pnne, Appnoxr MATToN
This method is similar to the LS approximation however different [3; are calculated for each colurnn, so that the next rnodel is further used see Tiffany and Adarns (1987)
^ t
, / l
1 1 e t l
k i + $ l
I
k i + l t , l
Q ( r ) : A , , + A ' s + A s l * i O , ,
l - l
or, as the Laplace variable is ^s
Q f f r ) : A o * A , j k A , k 2 +
J
t + F ,
:.7fr, then
S o , J = k
7 ' - - i k + F i
Q r ( s ) : A q r 1 A r r s + A , r r ' * t A,r*;,ra#
| '\ r P,rl
and in the reduccd fiequency ft dornain it becornes
Trang 6Canadian Aeronautics and Spacc Journal Journal adronautique et spatial du Canada
- l
Q f i n - A0r * A r r i k - A i k ' ) + l A 1 , , r ) ,
, i h ( 3 0 )
whcre A,, are the coeft-icicnts matriccs of 7th column of the
matrix A, and F,,r orc lag cocfficicnts of thc 7th colurnn of
approxin-ration rnodel In this case the lbllowing objcctive
f u n c t i o n is r n i n i r n i z c d :
In the MP rlcthod with respect to the LS rnethod, thc
matrices Bp7 and 8,, arc diffcrcnt for cach column
M rr,,{r vt unn-Srnrn Appnoxl MATIoN
The MS approximation is written in thc Laplace dornain
Q ( s ) : A o + A , s + A , 3 ' + D l s l - R l I E r
The objective is now non-linear with respect to the E and D
m a t r i c e s T h e s o l u t i o n is o b t a i n e d u s i n g t h e i t e r a t i v e l i n c a r
q u a d r a t i c s o l u t i o n M a t r i x D i s n o w f i x e d a n d t h e l i n c a r quadratic problern is solvcd according to thc E matrix Next, the E rnatrix is hxed and thc lincar quadratic problcrn is solved
a c c o r d i n g t o t h e D r r a t r i x T h i s s c h e r n e i s r e p c a t e d u n t i l convergence is attained To solvc the linear quadratic problcrn according to thc E rlatrix with thc D matrix fixcd thc objcctive lirnction is rewritten bv colurnn as fbllows:
t, =4ZtrilQil( ik) -Q,,t ir,)l'
( 3 2 )
t = \ t ,
t, -4.ltr,lwi e /k)
where
e i ( f t r ) = Q ; ( l / ' 7 ) - Q i ( 7 1 1 )
The error is writtcn as a problem, according to the D following equation fbr each DiA,t : B,/
wherc
row crror and the linear quadratic matrix, is equivalent to solving the row of D:
(40)
( 3 6 )
( 3 7 )
( 3 8 )
( 3 e )
In the frcqucncy dotnain, this approxirnation becornes
Q f j t l = A o * A , i k + A , k 2 + o l i k l R l t r j t ( 3 3 )
where the D and E matrices are explained in thc next
paragraphs and arc related to the convergencc of thc solution R
is thc aerodynarnic lags diagonal rnatrix To rninirnize the
objcctive function wc apply thc next constraints, thus Karpel
( r ee0)
I r n Q t l f r * ) l = l m p l r k ) i
We know that
Furtherrnore the objective function (36) is rninimized fcrr
e a c h c o l u r n r t A c c o r d i n g t o E q u a t i o n ( 3 5 ) s o l v i n g t h e
r - r - r i n i r n i z a t i o n p r o b l c n r ( 3 7 ; i s e q u i v a l c n t to s o l v i n g t h e following cquation fbr each column of the E rnatrix as shown
b y G o l u b a n d V a n L o a n ( 1 9 8 9 ) :
A " E i - B "
whcrc A, and 8,, are expressed as follows:
A" : I clDtw,,, DCor + cilurwr?7 DCy
I
8,, : Icl,DtwT?r(Qni(frr) - Br.,r) +c/,nfwi(Qr7(frr) -B,,r)
Q f f r r ) = B R 1 + D C p T u * j , B , , + D C , 7 E )
i n w h i c h
t.)
B R / : Qn(o) - * tq*to) -Qn(tr)l
f t l
cnr = nfttir + R2) t tt it+ Rr)
L ,
B r 1 =
f r Q , t L " t
c , : o [ f t 3 , + R r ) - r - ( k i t + R r ) |
( 3 5 )
' )r?
A , t : ICo,Ewit,EtCL +CyEwit,EtCI
I
B,/ : It0,.i (k) - BB,,)wi?rEtcil, +(eri(rtr)
-Br/)wiEtcil
In conclusion the MS approxirnation is generally bctter than the LS and MP approrirnations When the nr,rmber of lags is greater than thc nurnber of lrequcncies, the problem becomcs
i l l c o n d i t i o n e d
To solve this proble rn, two adclitional features are
c o n s i d c r c d a t c a c h it e r a t i o n : ( l ) t h e l i n e a r q u a d r a t i c p r o b l e m s according to thc E rnatrix and after that accordtng to thc D
m a t r i x a r e s o l v c d u s i n g s i n g u l a r - v a l u e d e c o r n p o s i t i o n ( S V D ) as
s h o w n b y G o l u b a n d V a n L o a n ( 1 9 8 9 ) ; ( 2 ) t h e o p t i m a l
-lo*,
Trang 7Vol 49, No 4, Dccember 2003 0 Vol 49, no 4 decernbre 2003
comprornise between the prescnt and the last
tcl cnsure thc algorithm couvergcncc Next,
cxplain thesc twcl features
E q u a t i o n s ( 3 9 ) a n d ( 4 0 ) n r a y be written
fbrrn:
A x - B
itcration is choscn
w e w o u l d l i k e t o irr the fclllowing
1 a l )
Firstly, we have to cxpress tlic aerodynamic forces Q under the Padi polynomial fbnn
Qk) - A,, + ltiA, + ( iD A *'f ,' -ik 1.*,
7 i k + f l t
-( 4 6 )
i n w h i c h x i s a n u n k n o w n v c c t o r W h e n t h i s s y s t c r n is i l l
c o n d i t i o n e d t h c L S s o l u l i o n c a n b c o b t a i n c c l u s i n g S V D o f A
I n l a c t , A c a n b c d c c o r n p o s e d a s l b l l o w s :
i n w h i c h U a n d V a r e o r t h o g o n a l n t e t t r i c c s a n d S i s a d i a _ q o n a l
r n a t r i x T h e d i a g o n a l o 1 ' t h e S r - n a t r i r c o n t a i n s i t r o r d c r th e
s i r r g u l a r v a l u c o f A s o t h a t if r a n k ( A ) - r { n thc ri I' last
e l c r n c n t s o f t h c d i a g o n a l a r c n u l l U s i n - e t h i s d c c o n - r p o s i t i o r r t h c
L S s o l L r t i o n o f E q r , r a t i o n ( 4 1 ) can bc obtaincd a s f o l l o w s :
rvlrere A , are thc coclficicnts of a dirnension cqr-ral to the rnatrix
O and obtained fl'orn thc LS algorithm and Fi are the
a c r o c l y n a n r i c l a g s T h c o p t i m a l v a l u c s o f t h e s e la g s w h i c h
m i n i r l i z c t h e a p p r o r i r n a t i o n c r r o r b c t w e e n t h e a e r c l d y n a r n i c ( ) ( / r ) n r a t r i x a n d i t s a p p r o x i n r a t i o n b y P a d e p o l y n o m i a l s i s calcLrlated f urthcr Equations (5) anc'l the n,cll-knor,vn crpression \' : /(D givcs ili - hslV trnd by placing this ncw
c x p r c s s i o n i n E q u a t i o n ( 6 ) we obtaiu
M i + D r l + K \ + q , r
The statc vector of acrodynamic rnodes is firrthcr introduccd
( 4 t )
t l ,q , In -,,
L v - " + t
I
i - t s + - ; F r I
t ) l
t
J , / | \ l
" 1 r , , + t , r r l i , ) 0 *
L
whcrc u; and vr arc, respectively, thc coluntns of tnatriccs U and
V and ^s, is thc ith elernent of diagonal n-ratrix S
L c t E l p r a n d E , n r; bc thc optirral solution t o t h c p r e s e n t a n d
last iteration so that the optirnal cornplonrise betrveen thc
p r e s e n t a n d l a s t i t c r a t i o n i s o b t a i n c d b y r n i n i r n i z i n g th e
o b i e c t i v e f u n c t i o n (3 7 ) a c c o r d i n g t o u l v i t h
E - c x E , n r ) + ( l c r ) E 1 p y ( 4 1 )
Sirnilarly, fbr the D rnatrix the ob.jective furrction can be
rninimizcd accordins to cr with
D : c r D , n , , * ( l - c r ) D , n ) ( 4 5 )
F o r t h e s c s c a l a r r n i n i r n i z a t i o n s " w e u s c d a s c a l a r
rn inimization algorithrn
LS MsrHoD lNrnonucED rN ruE P METHoD
E q u a t i o n ( 2 1 ) dcscribing t h e d y n a r n i c s o f a e r o - s c r v o e l a s t i c
systcms ofl-ers only serni-lincar represcntations In lbct, all
terms related to the aerodynarnic fbrces present strong
non-lincaritics with respect to the rcdr-rced lrequcncy k Then,
t h e m u l t i t u d c o f t h e a n a l y s i s a n d r n o d c l i n g a l g o r i t h r n s a p p l i e d
to the linear systems requires thc rnotivatiorr to obtain a linear
aero-servoelastic system Various mcthods arc available to
apply this linearrzatron, see Pitt and Goodrnan (1992), Roger et
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
( 1 9 8 4 1 9 8 7 ) , P o i r i o n ( 1 9 9 5 ) , a n d C o t o i a n d B o r e z ( 2 0 0 1 ) W e
h a v e c h o s e n t h e s i m p l e s t o n e th e L S m c t h o d b e c a u s e i t i s
r,rsed in STARS - and in this wav we can compare the rcsults
i c r 2 0 0 3 C A S I
a n d b y r e p l a c i n g E q u a t i o n ( 4 8 ) i n E q u a t i o n ( 4 7 ) r , v c o b t a i n
( 4 e l
whcre
t I : i v t +
and
( s 0 )
\
\ : - - l l ' -
I l ' ' l
i + l - 0 ,
( 4 8 )
ui*on * rn n n,,l'f ,' A:*,x,-] : u
L i - r l
- / A \
K : K * U , r [
; 1 ,
Wc substitute li given by Equation (3) into Equation (2) and
w e o b t a i n V a s a f u n c t i o n o f V 1 , v , a n d o T h e n , E q u a t i o n ( l 8 ) gives cq) as a function of crlP and rr) and by equating the two
E q u a t i o n s ( 2 0 ) , g i v i n g t h e value of /t while taking into consideration that b : c'12" we firrther obtain thc valuc of the blV ratio as function of o, v, and 06, as follows:
h _ ^ [ 6 L' 2r,cqy
( s l )
Trang 8Canadian Aeronautics and Space Journal Journal adronautique et spatial du Canada
The variable change techniquc introduces Ztr : c(er and by
replacing this expression forVs into Zp given by Equation (3)
we obtain Vy: : (:(\1y Furthcrmorc, wc substitute V, into qr1
g i v c n b y E q u a t i o n s ( 4 ) a n d ( 1 4 ) , a n d w e o b t a i n
( s 2 )
( s 4 )
as the
( 5 s )
0 0 ilrP o
O I
O I
-fre -fie
I T
0 0 0
0
0
O I
p
n '
P
n '
I , , I
q , t :
; P , , V t :
= ' r r r t ' r t ' - ' i v r
with the coefficients
^ l l
A P c , = ' ^ p , , r ' 2 A r , A l
I
Al = lo,,b''t,
1, I-/l Ltl
Thcn, we substitute q6 given by Equation (52) and blZ gir,'en
by Equation (51) into Equation (49) and into the coefficients
given byEquation (50) and we divide the resulting equation by
cofr and we obtain
where
ilf' : MP + oAl, DP
T o : K P + v 2 A P
- DI' + r^[" ,q|
REsulrs AND DrscussroN
To validate our rncthod u'e uscd the STARS Aircraft Test
M o d c l d e v e l o p e d b y G u p t a ( 1 9 9 7 ) a t N A S A D r y d e n F l i g h t Research Center This rnodcl includes aero-structural elements (flexible aircraft) and control surf-aces (aircraft commands) Firstly, a frcc r,'ibration analysis was performed in the absence of aerodynamics to obtain the free modes of vibration
We obtained the same frequencies and modes of vibration by our method as well as by the one used in STARS
Secondly, to calculate the aerodynamic forces in the frequency domain by the doublct lattice method (DLM), the simulation parameters are considered in the STARS computer program: the reference scmi-chord length h - 81.50 cm, the refcrence air density at sca lcvcl ps : 1.225 kg/mt the altitude
at the sea level Z - 0 km" the ref-erence Mach number Mach : 0.9, the reference sound airspccd at sea level a6: 340.3 rn/s
In Table l, the results of an aeroelastic non-linear analysis arc shown" and velocities at which flutter occurs are calculated
In Table l, the results given by our Pfr method are cotnparcd with the results givcn by STARS through the three flutter open-loop rnethods (k method, Pk rnethod, and ASE method) Actually, we should only comparc the results obtained by our P/r method (row 4) with the results obtaincd by the P/r method
in the STARS program (row 2) We can concludc that we obtain
a good comparison in tenns of flutter velocitics and fiequencies
f o r b o t h f l u t t e r p o i n t s
In Table 2, we prescnt a comparison between the results obtained on the linearized ATM with our P rnethod modified to include the LS fbrmulation shown on row 4 with the ones obtained by use of the STARS software, called the ASE method ( r o w 3 )
A lirrear aero-servoelastic open-loop analysis is further perfbnned on the ATM This analysis uses the P-flutter rnethod where the aerodynamic unsteady fbrccs werc approximatcd by the LS rnethod Results are shown in Figures 2 to 4 As shown
in Figure 4, the unstable modes where flutter occurs are elastic mode 2 (flutter point I ) and mode 7 (flutter point 2) Mode 2 is the fuselage first bending and mode 7 is the fin first torsion
m o d e
O I
O I
0 -rt A!
+l'1
0 0
p
n '
p
n ' xl
;,t 1 $ 0
0 -v2 A!1,+ nl,
0 0 -01 1
f l-,,, "l
v r l F A l , x , I
| / - / - T t t I
= L pot'bA1,
^ l
A ' i : ^ p r t ' - A f o r 3 < j < 2 + n ,
L
The first equation of the system of Equations (19) and also
the ratio Vlb given by Equation (5 I ) arc substituted into
Equation (48), and wc obtain
* , * 4 r o o $ i X , - , o n , t o
! o
Next, salre variablc transfotmation is applied ro X,
o n e a p p l i e d t o 1 1 g i v e n b y E q u a t i o n ( 1 9 )
f i : f ' 0 h * ,
X l * P i X i = r l P , w h e r e V i :? F ,'
r/o
With this last ncw variable transformatiot-t given by Equatton
(55), and by dividing Equation (5a) by ob, we obtain
Then, we obtain the final matrix equation
( 5 6 )
ic,r 2003 CASI
Trang 9Vol 49, No 4, December 2003 I Vol 49, no 4, ddcembre 2003
Table l Aeroelastic Non-linear Analvsis.
Equiv airspeed (km/h) Frequency (radls) Equiv airspeed (km/h) Frequency (rad/s) STARS
/c method
Pk method
l,SE method
Our Pk method
821.10
8 1 8 7 4 878.79
8 1 7 4 4
7 7 4
1 7 4
7 7 3
7 7 4
1 5 9 6 5
1 6 0 0 4
l 3 5 t l 1404.7
t 4 7 3
1 4 7 l 136.2
1 4 5 6
Table 2 Linear Aeroelastic Analvsis.
Flutter I Equiv airspeed (km/h) Frequency (rad/s)
Flutter 2 Equiv airspeed (km/h) Frequency (rad/s) STARS
k method
Pft method
,4SE method
Our P-LS method
82r.70
8 1 8 7 4 878.79 878.79
7 7 4
7 1 4
7 7 3
7 7 5
1 5 9 6 s 0
1 6 0 0 4 0
t 3 s l 1 0
I s 0 8 1 0
1 4 7 3
1 4 7 1 136.2 144.2
35il
3fiil
i-i Eler:_;tii- t-r-rrtr-le 1 Ir ElAStir_ t-il'tde ?
+ Elastir r - n i d e 3 + Elastii: tr-rtrje 4
n Elastir m o ' J e 5 {;} Elastic tr r'trjE fi
F E l a s t i c r x i l d e T
a Elastir rlo,le I
U]
E
{U
t*
]*t
TJ
C
fi]
=
{F
il)
!*
LL
?il0
1 5 0
1ilil
5il
t)amping {g.20il}
- Frequency (rad/s) versus Damping (9*200).
Figure 2 P-LS Method
1ilil
i.) 2003 cASI
Trang 10Canadian Aeronautics and Space Journal 0 Journal a6ronautique et spatial du Canada
o Elastic m o d e 1
x ElaEtic m o d e 2 + Elastic m o d e 3
* E l a s t i c m o d e 4
t r Elastic m o d e 5 + E l a s t i c m o d e f i
v E l a s t i c m o d e 7
a Elastic m o d e B
;
{n
'j(]
s
lF*
L}
C
0)
:5
(r
(I}
t-: l t
Equivalent ai rspeed {krn/h}
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 aeroelastic systems as well as over controller
conception In our study, we have chosen a linearization
technique that is relatively simple, based on the optimization of
aerodynamic lags By and on the LS minimization of Pad6
coefficients 4,
When we observe the equivalent airspeeds for the flutter
phenomenon, the results of the combined P and LS method are
closer to the results of the Pft method than those of the ASE
method programmed in STARS This conclusion depends on
the quality of the hnearrzatton applied on the ATM, because the
results of the non-linear model (Pfr method) and of the
linearized model (LS and P method) are close Our linearized
rnodel diverges with respect to the ltneartzed model conceived
with STARS This divergence may be explained by the
precision of each hnearrzation, and on the other hand by the
difference of the hneartzatron methods and more particularly of
the optimrzatron techniques
An optimrzatron algorithm minimizes the quadratic error denoted by -/ between the exact aerodynamic force matrix and the approximation of the aerodynamic force matrix approximated by Padd polynomials This algorithm varies the values of aerodynamic lags, where the approximation of Pade polynomials is realized with the LS algorithm
Results of the LS method applied on the ATM for a Mach number of reference M:0.9 are shown in Figure 5 A total of six aerodynamic lags have been used to obtain a satisfactory aerodynamic forces approximation The left graph in Figure 5 shows the evolution of the norm of the quadratic error between the aerodynamic force matrix and the approximated aerodynamic force matrix as a function of iteration number of the optimrzatron algorithm
There is also compromise between the number of aerodynamic lags and the quality of approximation The greater the number of aerodynamic lags, the better is the approximation At the same time, the aeroelastic model becomes more cumbersome from a computing time point of view as the number of aerodynamic lags is increased
( O 2OO3