aiaa-96-3981-ro unsteady adyn & ae models using karhunen-loeve eigenmodes ccae-rom-afoil

8/4/2019 AIAA-96-3981-RO Unsteady Adyn & AE Models Using Karhunen-Loeve Eigenmodes CcAE-ROM-Afoil

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and\"

-70

I

0

0

4

c m = c m . + x - ( i o v a o ) & i&,i = lN a c m+C t q o v a e ) i j

j = 1

The state-space representation of the airfoil motionis obtained by using equations24 - 26 in equations 22and23.Figure I : Wisolakdairfoilaeroelasticmodel

auasi-static correction is not used lo r a large number of [Mil&,= [m210+ [md i+ fomodes are not included. However, the KLmodesgive acomplete reducedordermodel in the sense that no quasi-static correction isrequired toobtain accurate results witha small number ofmodes,thereby simplifying the con-

struction ofthereducedordermodel.

where

AeroeksticModelsConventioMICFDBasedYodel

For a two dimensional airfoil in a compressible flow(Figure 1). free to move in pitch (a)and plunge (h). theequations of motion are:

Equations 22 and 23 can beplaced in amore con-venient form for numerical solution by first defining the

state variable vector

{a}

QI

Q 2

Q 3

Q4The loads on the airfoil at a given time can be ex-

pressed as a function of the interior flow field vari-ables (density, velxities. etc.), { q } , and thebound-arymotion, { a } . Assuming that these quantities rep-resent small, dynamic perturbations about steady state,{ e }= { io)+ ( $ 7 ) ) . and {a) { a o }+ {&(TI}. thenthe aerodynamic lift andmoment coe%cients are

[mz,]

c I O0 1

0 0

0 0

[MI E

Im2l f m2, + [m2.10 0 10 0 0

0 0

-(2koM,)' 0-(2h-*Mw )2

0

Y o 0

(29)

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and

However, equation27 represents a systemof4 equa-tions with N+4 unknowns. The linearized flow fieldequations (equation 12) are used to provide the ad-ditionalN quations to complete the aeroelastic sys-tem. Finally. note that for sufficiently small time steps,a"+' = dr"+ 9 b: + &:+I). Therefore, the completeaeroelastic system ofequarions is given by[ .i. . I. .o. ] { q . i " -r

-M3 : M l - M z

(34)

[;;; . : { . ] + { ...0MI + hi2 F"

where Mz E y m 2 . M JTime marching equation 35directly will give the air-

foil's aeroelastic response. However, this can be costly

for design iterations. since this expensive calculationmustberepeated for each airfoil parameter change (mass,springs, etc.). ?h e aeroelastic stabilityofthe system canalso be directly determined by calculating the eigenval-

ues, fa . ofequation 35 (with F, 3 0) . However, thisis not recommended, since it wouldbe extremely expen-sive, particularly ifdesign changes are tobe examined.

9m3.and F, 5 Arfo.

ReducedOrderModelA better approach is to utilize reduced order aero-dynamic modelling to reduce the size of the aeroelastic

model. This will allow for efficient and inexpensive de-

sign optimization.

The aeroelastic system of equations can be reducedby substituting equation 17 into equation 35 and premul-tiplying the top portion ofequation 35 by the transposeofthe corresponding set ofRcompanion vectors. [WR].to obtain the following reduced order aeroelastic model:

11

wherec~4 [fkf>] VR] .n ofR+4 equations and unknowns, which is much smaller

than the original aeroelastic model. This will result invery efficient time marching aeroelastic analysis, since

for changes in most design parameters. the same reduced

order aerodynamic n d e l can be used. and only the 4structural ( d r ) equations will have to be modified. Theeigenvaluesof the reduccd order aeroelastic system. za ,can also be found much more efficiently than those of theoriginalaeroelastic model, and can be used to determinethe aeroelastic stability directly.

Results

Reduced order models using KL modes were con-structsd using the previously described procedure andthe predicted response of a NACA 0012 Lirfoil at Mach0.5and zero angleof attack, subjected to a step change inangle of attack. lbo hundred discrete flow field "snap-shots" were used to create the ensemble. The eigen-value problem for the correlation matix was solved using

EispackIB . The eigenvalues of the correlation matrixwere examined to determine where to truncate the set ofKLmodes for constructing the reduced ordermodeis.

Figure 2shows that the response of the full 23.560DOF Linearized ST system canbe duplicated by a re-duced order unsteady aerodynamic model formed with

only 50 KL modes! This example is extremely useful inthat this indicates that the 50mode reduced order model

will be valid for unsteady aerodynamic response at any

reduced frequency 4.Once constructed. the reduced order model can be

executed repeatedly for a variety of problems at virtually

nocost. The reduced order unsteady aerodynamic modelexecutes interactively in under 2 cpu seconds and wascreated in approximately the same amount oftime asit takes to run the full linearized ST code for the stepresponse problem to steady state. about 10 cpu minuteson a Cray Y-MF?There is little additional cost for solvingthe 200 x 200 KL eigenvalue problem. Recall, fromReference 3 that it took approximately 4 cpu hours tocreate a reduced order unsteady aerodynamic model with

4 fluid eigenmodes. Also note that this 4 mode reduced

order model was limited to reduced frequencies up to0.3.Figure3 shows convergence of the reduced order un-steady aerodynamic model as the number ofKLmodes isincreased. Note that the scale is blown up to show a small

region of the response relative to Figure 2. The reducedorder model formed with 35KL modes nearly duplicates

Note that equation 36 now represents a sy5


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aiaa-96-3981-ro unsteady adyn & ae models using karhunen-loeve eigenmodes ccae-rom-afoil

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