DEVELOPMENT OF A DISCRETE-TIME AERODYNAMIC MODEL FOR CFD- BASED AEROELASTIC ANALYSIS

Transcription

1 AIAA DEVELOPENT OF A DISCRETE-TIE AERODYNAIC ODEL FOR CFD- BASED AEROELASTIC ANALYSIS Timohy J. Cown * nd Andrew S. Aren, Jr. echnicl nd Aeropce Engineering Deprmen Oklhom Se Univeriy Sillwer, OK 7478 Kjl K. Gup NASA Dryden Fligh Reerch Cener Edwrd, CA Abrc Syem idenificion i ued o develop n ccure nd compuionlly efficien dicree-ime erodynmic model of hree-dimenionl, unedy CFD oluion. Thi erodynmic model i hen ued in plce of he unedy CFD oluion in coupled eroelic nlyi reuling in ubnil ving in compuionl ime. The mehodology h he dvnge of producing n explici mhemicl relionhip for he erodynmic force cing on rucure while ill reining he ccurcy of he complee unedy CFD oluion. The explici erodynmic model cn hen be coupled wih he known rucurl model nd rec in e-pce form. Uing he combined e-pce form, biliy nd conrol nlyi for he eroelic yem cn be compleed bed on n eigenvlue oluion for he e mrice. Reul ddre he exen o which hi mehodology i pplicble o eropce pplicion. Nomenclure ρ = free rem deniy = free rem peed of ound CFD = Compuionl Fluid Dynmic [C] = generlized dmping mrix f = frequency f = generlized erodynmic force vecor [I] = ideniy mrix [K] = generlized iffne mrix [] = generlized m mrix = ch number nr = number of roo or mode hpe q = free rem dynmic preure q = generlized diplcemen vecor * Grdue Reerch Ain, Suden ember AIAA. Ain Profeor, Senior ember AIAA. Aeropce Engineer, ember AIAA. Copyrigh 999 by Timohy J. Cown nd Andrew S. Aren, Jr.. Publihed by he Americn Iniue of Aeronuic nd Aronuic, Inc., wih permiion. Inroducion Predicing inbiliie in he eroelic behvior of eropce rucure i imporn in he deign of modern ircrf which opere over wide envelope. However, complee eroelic nlyi i ofen difficul o complee due o he complex rucurl, erodynmic, nd conrol inercion ocied wih even he imple fligh vehicle. In order o obin he mo ccure predicion for ircrf fligh chrceriic, conemporry reerch h urned owrd he developmen of n inegred compuionl model cpble of cpuring hee complex inercion. One of he mo powerful compuionl ool vilble for erodynmic nlyi i he CFD model. Such model i ofen deirble for dvnced eropce pplicion ince i mke he fewe umpion bou he chrceriic of he flow field nd i cpble of ccurely predicing complex hock inercion for rnonic nd uperonic flow round compliced hree-dimenionl geomery. For eroelic nlyi, one cn ke dvnge of hee benefi by coupling n unedy Euler or Nvier-Soke CFD lgorihm o n ccure rucurl dynmic olver nd predic he complee eroelic repone of rucure. However, he compuionl ime required for CFD-bed eroelic imulion h ypiclly limied he ue of uch model in n operionl environmen even wih recen dvnce in CPU peed. Furhermore, hi ype of formulion i no menble o conrol nlyi ince rnfer funcion or e-pce repreenion for he CFD model i no explicily defined. When running coupled eroelic imulion, i i he unedy CFD oluion ech ime ep which require he overwhelming proporion of CPU ime. Hence, recen reerch h rgeed he ccelerion of hi oluion hrough vriou modeling echnique. In priculr, yem idenificion h been hown o yield ignificn ving in compuionl ime for CFD-bed eroelic nlyi. Americn Iniue of Aeronuic nd Aronuic

2 The yem idenificion mehodology llow one o develop n efficien mhemicl model for he inpu-oupu relionhip of n unedy CFD oluion. The yem model hen replce he unedy CFD oluion in he coupled eroelic nlyi reuling in compuionlly efficien eroelic imulion while ill reining he ccurcy of he originl CFD model. Thi mehodology h he dvnge h i i pplicble o wide rnge of flow regime, including rnonic, nd ny rbirry geomery. I will lo be demonred h hi mehodology h he furher dvnge of llowing he problem o be rec in epce form. The e mrice for he eroelic yem cn hen be ued for biliy nlyi by compuing eigenvlue or for eroervoelic pplicion where conrol lw re implemened. The emphi of he preen work i o furher demonre he efficcy of uing yem idenificion echnique o ccelere CFD-bed eroelic nlyi for hree-dimenionl rucure. The previouly developed yem idenificion procedure i ummrize nd n nlyi i preened on elecing he opimum inpu ignl for ucceful idenificion of he dicree-ime model prmeer. A derivion of he e-pce form for he coupled eroelic yem i preened long wih he procedure for idenifying eroelic inbiliie from he reuling e mrice. Reul re preened which ddre he exen o which hi mehodology i pplicble o geomerie nd flow of prcicl inere in eropce pplicion. Compuionl Tool Compuionl nlyi for hi udy w performed uing he eroelic cpbiliie of he STARS code developed NASA Dryden Fligh Reerch Cener. STARS 2 i n highly inegred, finie elemen bed code for mulidiciplinry nlyi of fligh vehicle including ic nd dynmic rucurl nlyi, compuionl fluid dynmic, he rnfer, nd eroervoelic cpbiliie. Srucurl nlyi in STARS i ccomplihed uing he finie elemen mehod o compue he eigenvecor nd eigenvlue which decribe he elic mode for rucure. Arbirry moion of he rucure cn hen be repreened by muliplying ech eigenvecor by generlized diplcemen nd pplying modl uperpoiion. STARS unedy CFD nlyi i ccomplihed uing ime-mrched, finie elemen pproch o olving he unedy Euler equion. The CFD oluion i performed on meh coniing of unrucured erhedr uing he rnpirion mehod o imule rucurl deformion. A complee eroelic nlyi i ccomplihed by coupling dynmic olver, uing he modl vecor, wih he unedy CFD olver which compue he generlized erodynmic force cing on he rucure. The coupled oluion i hen ime mrched mehodology for olving Equion (), he mrix equion of moion for n rbirry rucure uing generlized coordine. [ ] q + [ C] q& + [ K] q = ( ) && () Conidering he decribed oluion cheme, he STARS eroelic oluion mehod cn be grphiclly repreened by he imple block digrm hown in Figure. FE Solid Anlyi odl Prmeer Aero. Force Globl Time Sep Dynmic Solver Unedy CFD Soluion f Sedy Se CFD Soluion I.C.' CFD B.C.' Figure : Block Digrm Repreenion of STARS Aeroelic Soluion Noice in Figure h eroelic oluion cheme require wo iniil piece of informion bou he problem; modl prmeer for he rucure nd edy e CFD oluion. The modl prmeer define he rucurl dynmic of he problem, nd he edy e CFD oluion ure he ime ccurcy of he unedy CFD oluion by providing i wih iniil ring condiion. Thee wo iem re required for ny problem before beginning n eroelic nlyi. Syem Idenificion Procedure A hown in Figure 2, i i he unedy CFD oluion ech ime ep which require he overwhelming proporion of CPU ime when implemening he decribed eroelic oluion cheme. In n effor o mke CFD-bed eroelic nlyi more prcicl in n operionl environmen, yem idenificion procedure h been developed 2 Americn Iniue of Aeronuic nd Aronuic

3 which llow one o develop n efficien mhemicl model which cloely pproxime he erodynmic prediced by he unedy CFD oluion. Thi erodynmic model i hen independen of he rucurl prmeer nd cn be ued o erch for eroelic inbiliie by vrying he dynmic preure, generlized m, ec. Figure 2: Typicl CPU Frcion During STARS Aeroelic Anlyi The yem idenificion procedure mke he umpion h mo eroelic yem cn be reed dynmiclly liner. Th i, he erodynmic repond linerly o mll perurbion bou poenilly nonliner edy-e men flow. Thi umpion i uppored by modern eroelic reerch which indice h i i he ic nonlineriie which re imporn in n eroelic nlyi. 3 Furhermore, hi umpion require no exr compuionl effor ince he nonliner edy e oluion i lredy required for ll problem. By borrowing from Figure, one cn repreen he STARS unedy CFD olver imple dynmic yem uing he block digrm of Figure 3. Noice h he unedy CFD oluion i muli-inpu mulioupu, IO, yem wih n inpu vecor coniing of generlized diplcemen for he rucure nd n oupu vecor coniing of generlized erodynmic force. q() Inpu() CFD Soluion Unedy CFD Solver Dynmic Syem Dynmic Soluion f () Oupu() Figure 3: Block Digrm Repreenion of STARS Unedy CFD Solver In order o model he inpu-oupu relionhip for hi IO yem, he yem idenificion procedure uilize vecor form of he dicree-ime ARA model rucure defined by Equion (2). f n nb ( k) = [ Ai ] f( k i) + [ Bi ] q( k i) i= i= (2) Noice h he ARA model rucure of Equion (2) eque he curren oupu, or generlized erodynmic force, o liner combinion of n p oupu nd nb p inpu of he unedy CFD olver. Hence, model order, or ize, cn be defined uing hee wo ineger, n-nb. Wih he model rucure defined, yem idenificion i hen proce for compuing he mrice of conn coefficien which will reul in model h ccurely mche he rel dynmic of he unedy CFD oluion for priculr flow field. The mehodology implemened in STARS uilize lequre numericl echnique o compue he model coefficien which minimize he error beween he model nd e of ime hiory d, or rining d, ken from he unedy CFD oluion. Thi rining d i he erodynmic repone ime hiory compued for precribed moion of he rucure. Once n opimum fi o he rining d i idenified, he model cn hen be implemened o predic he erodynmic for ny rbirry moion of he rucure. By implemening i in plce of he unedy CFD oluion in coupled eroelic nlyi, one cn predic complee eroelic ime hiorie in frcion of he ime while ill reining he ccurcy of he originl CFD oluion. Furhermore, he erodynmic model my be ued repeedly wih vriou combinion of rucurl prmeer o predic eroelic inbiliie ince i w developed independen of he rucurl dynmic for he problem. The model mu only be rerined when he flow phyic of he problem re lered by chnging he ch number. Inpu Opimizion The ucce of ny yem idenificion procedure i highly dependen on he moun nd quliy of ime hiory d vilble when idenifying he model prmeer. There mu be much informion bou he yem dynmic poible pcked ino he rining e of d in order for he idenificion procedure o ucceed. Hence, he precribed inpu ignl ued in he iniil e of rining d mu be choen crefully. When chooing he rining inpu, convenionl widom indice h i hould excie brod pecrum of frequencie in he dynmic yem. Hence he hrmonic conen of he inpu hould be exmined o enure i i uible. 8 For yem uch n unedy CFD olver, one h very creful conrol over he inpu, o n lmo unlimied moun of ignl re 3 Americn Iniue of Aeronuic nd Aronuic

4 vilble. The only limiion i h he inpu mu be mhemiclly decribble in erm of he boundry condiion for he flow olver o h he flow phyic re ccurely repreened. Alhough he erodynmic model of Equion (2) only conider generlized diplcemen, noher vecor of generlized velociie i lo needed o complee he boundry condiion required for CFD oluion. Furhermore, i i required h here be mhemicl coniency beween he wo e of boundry condiion or he flow phyic will no be ccurely repreened. Th i, inegring he velociy inpu mu yield he diplcemen inpu, nd differeniing he diplcemen inpu mu yield he velociy inpu. If hi condiion i no me, one will derive n erodynmic model which h improper phyic nd cnno be ued o model n eroelic problem. Hioriclly, he 32 muliep inpu h been widely uilized in fligh eing pplicion of yem idenificion. In ddiion o being ey o implemen, he 32 muliep h brod frequency conen pcked ino hor ignl. Afer evluing vriey of inpu ignl wih he unedy CFD oluion, he 32 muliep w iniilly choen he opimum inpu for yem idenificion. However, he muliep inpu w implemened in n unorhodox wy in order o ify he condiion required for mhemicl coniency beween he CFD boundry condiion. A een in Figure 4, he 32 muliep w pplied o he velociy boundry condiion nd hen inegred o compue he diplcemen boundry condiion. Implemening he muliep in hi wy reuling in wo inereing boundry condiion ime hiorie nd voided he mhemicl diconinuiie h would hve ppered in he velociy boundry condiion if he muliep hd been pplied o he diplcemen nd differenied. Furhermore, noice h he muliep inpu for ech mode re pplied ou of phe o llow he idenificion procedure o idenify he dynmic effec of ech unique inpu in he ime hiory d. To de, hi muliep inpu h been uilized on vriey of eroelic problem nd yield model which re ypiclly in excellen greemen wih he unedy CFD oluion., However, ligh inbiliie which do no exi in he unedy CFD oluion were omeime oberved high frequencie for hee model. Hence, more recen work h relied on reworked muliep inpu which h beer frequency conen o id in he idenificion of he higher frequency dynmic. A wih he 32 muliep, he vrible mpliude muliep, Figure 5, i pplied o he velociy boundry condiion, nd i inegrl, Figure 6, i pplied o he diplcemen boundry condiion x v x v -2-4 v ( ) Time, Figure 5: Vrible Ampliude uliep Inpu..8.6 x x2 v Figure 4: 32 uliep Inpu Signl for Unedy CFD Flow Solver v x ( ) Time, Figure 6: Inegrl of Vrible Ampliude uliep Inpu 4 Americn Iniue of Aeronuic nd Aronuic

5 The mpliude for ech ep of he new inpu re choen uch h he re under ech ep remin conn. Addiionlly, he lengh of ech ep i choen uch h he ol lengh of he vrible mpliude muliep i he me he previouly implemened 32 muliep. For comprion beween he frequency conen of he wo inpu, he power pecrl deniy, PSD, for ech diplcemen ime hiory i ploed veru he rio beween cul frequency nd criicl frequency, f / f c, in Figure 7. Noice h here i n order of mgniude improvemen in frequency conen for he vrible mpliude muliep cro he enire frequency rnge. In priculr, he increed power high frequencie help o elimine he model inbiliie previouly oberved when uing he 32 muliep for yem idenificion. ix mode ece. The ime hiory hown in Figure 8 i obviouly no divergen, however i i no obviou how o eime he dmping rio or qunify how cloe he poin of inbiliy i. Hence, he erch for inbiliie in he eroelic yem become he k of mking quliive comprion beween obviouly ble nd divergen ime hiorie in order o eime where he croover poin would be x PSD. Vr. uliep 32 uliep Figure 8: Individul Repone Time Hiory From Six ode Tece f / fc Figure 7: Comprion Beween Power Specrl Deniy for Diplcemen Inpu Se-Spce Repreenion Afer compleing he yem idenificion procedure, one hen h n explici mhemicl model for he erodynmic repone of he rucure. Thi dicree-ime erodynmic model ke he plce of he unedy CFD oluion in he coupled eroelic oluion. The reuling eroelic model cn be ued o predic complee repone ime hiorie in econd rher hn he dy one would wi when employing he unedy CFD oluion. Hence he erch for eroelic inbiliie, or he croover from ble o divergen ime hiorie, my be compleed by vrying he dynmic preure nd re-execuing he eroelic imulion lmo no compuionl co. However, i i ofen difficul o pin down n eroelic inbiliy by exmining repone ime hiorie, epecilly for rucure wih muliple mode hpe. The inercion beween differen rucurl mode ofen reul in erric repone ime hiorie uch h hown in Figure 8 which w ken from Rher hn udying ime hiory d in he erch for inbiliie, one could ined rnform he problem ino e-pce form nd compue i eigenvlue by king dvnge of now hving n explici mhemicl model for he erodynmic of he yem. Noice in Figure h he STARS eroelic oluion mehod eenilly coni of wo dicree-ime yem coupled ogeher in liner feedbck loop. Now h n explici repreenion for he unedy CFD oluion exi, one cn collpe hi yem ino coupled e-pce form by connecing he inpu nd oupu of he wo yem nd implifying. The fir ep in he eroelic e-pce formulion i o derive he e mrice for ech individul yem; he rucurl dynmic nd erodynmic. One migh be emped o derive he rnfer funcion form for he wo yem nd ue block digrm lgebr o implify hem ino ingle cloed-loop rnfer funcion, however hi i no convenien for compliced IO yem i i difficul o uome genericlly in compuer lgorihm. Fir, conider he rucurl dynmic model implemened by STARS. A dicued previouly, STARS olve Equion (), he mrix equion of moion for n rbirry rucure in generlized coordine. Noe h he force vecor on he righhnd ide of Equion () cn be ny forcing funcion for he rucurl yem, no ju n erodynmic 5 Americn Iniue of Aeronuic nd Aronuic

6 forcing funcion. Hence, he erodynmic ubcrip will be dropped in he derivion of he rucurl yem in fvor of llowing ny generic forcing funcion, f(), o c on he yem. The e pce form for he rucurl equion of moion cn hen be hown o be follow: where x ( ) ( ) = [ A ] x ( ) + [ B ] f( ) x& (3) q ( ) [ C ] x ( ) + [ D ] f( ) q& q = (4) ( ) ( ) [ ] A = [ ] [ C] [ ] [ K] [] [ ] I [ ] [ ] [ ] B = [ C ] = [ ] [] I ] [ D ] = [ ] Thee mrix equion re olved numericlly in STARS uing e rniion mrix oluion. Thi oluion mehodology i he dicree-ime equivlen of pplying zero-order hold o he inpu nd mpling he oupu of he originl coninuou-ime yem hown in Figure 9. f(k) ZOH f() Srucurl Dynmic q() Figure 9: Block Digrm Repreenion of Equivlen Dicree-Time Srucure q(k) Wih hi in mind, one cn conver he bove coninuou-ime yem, Equion (3) nd (4), o i dicree-ime equivlen follow: where x ( k ) = [ G ] x ( k) + [ H ] f( k) q [ ] [ A e ] G = + (5) ( k) [ C ] x ( k) + [ D ] f( k) = (6) [ A] λ [ A ] [ H ] e dλ [ B ] = ( e [] I )[ A ] [ B ] = Nex, conider he dicree-ime erodynmic model obined uing he yem idenificion procedure. A een in Figure 3, hi model mp he relionhip beween generlized diplcemen nd generlized erodynmic force. The unedy CFD olver in STARS ue he free rem dynmic preure liner cling fcor for convering from he CFD unknown ino he urfce preure needed o compue generlized force. Hence, he dicree-ime erodynmic model h h been developed for generlized force cn be implemen independen of he dynmic preure by cling i o mch whichever curren dynmic preure i deired. Bed on he definiion of dynmic preure, Equion (7), one ypiclly ue he free rem deniy he model cling prmeer ince he ch number mu be held conn for given model. ( ) 2 q = ρ (7) 2 For convenience, we hen define cled erodynmic force, fˆ ( k), which h he iniil rining deniy, ˆρ, divided ou. Thi cled erodynmic force i reled o he originl erodynmic force he model w rined o predic by Equion (8). f ( k) ˆρ fˆ ( k) = (8) By ubiuing hi relionhip ino he erodynmic model of Equion (2) nd rerrnging, he new equion for he properly cled erodynmic model i given by Equion (9). When implemening hi model in n eroelic problem, one mu imply muliply by he deired free rem deniy o ge he correc erodynmic force. fˆ n ( k) = [ A ] ˆ i f( k i) + [ Bi ] q( k i) nb i= ρˆ i= (9) In order o derive he e-pce form for hi model, we define e-vecor, x (k), coniing of n + nb vecor e follow: ( ) x ( k) ( k ) fˆ fˆ ( ) k n = q( k ) ( ) q k nb + () The e pce form for he dicree-ime erodynmic model cn hen be hown o be follow: x ( k ) = [ G ] x ( k) + [ H ] q( k) + () ( ) = [ C] x( k) + [ D] q( k) fˆ o fˆ k + (2) 6 Americn Iniue of Aeronuic nd Aronuic

7 where [ ] G [ A] [ A 2] L [ An ] [ An] ˆ [ B] ˆ [ B2] L ˆ [ B 2] ˆ [ B ] ρ ρ ρ nb ρ nb [] I [ ] L [ ] [ ] [ ] [ ] L [ ] [ ] [ ] [] I L [ ] [ ] [ ] [ ] L [ ] [ ] = [ ] [ ] L [] I [ ] [ ] [ ] L [ ] [ ] [ ] [ ] L [ ] [ ] [ ] [ ] L [ ] [ ] [ ] [ ] L [ ] [ ] [] I [ ] L [ ] [ ] [ ] [ ] [ ] [ ] [ ] [] I L [ ] [ ] O O [ ] [ ] L [ ] [ ] [ ] [ ] L [] I [ ] [ ] O O [ ] [ ] [ A ] [ A ] L [ A ] [ A ] [ B ] [ B ] L [ B ] [ B ] [ ] = [ ] C = 2 n n ρ ˆ ρ ˆ 2 ρ ˆ nb 2 ρˆ nb H D = ρˆ [ B] [ ] [ ] [ ] [] I [ ] [ ] [ ] ρˆ B Noice in Equion (2) h he oupu for he dicree-ime erodynmic model include vecor of ic offe, fˆ. Thee re he cled ic offe o ubrced from he erodynmic ime hiory d in he de-rending proce of he yem idenificion procedure. To keep he e-pce derivion conien wih he STARS eroelic oluion cheme, hi vecor of ic offe mu be dded o he erodynmic prediced by he yem model. However, i will be een ler h he ic offe vecor i no imporn o he overll biliy of he eroelic yem. Wih boh he rucurl nd erodynmic yem now defined, he block digrm model from Figure 9 nd Figure 3 re conneced in liner feedbck loop hown in Figure below. To be conien wih he STARS numericl oluion cheme, he erodynmic force re conneced negive feedbck while n impulive diurbnce force, f I, i inroduced o excie he iniil moion of he rucure. ubiuing he oupu of he rucurl dynmic yem, Equion (6), ino he erodynmic yem, Equion () nd (2). Nex, he modified oupu equion for he erodynmic yem i ubiued ino he rucurl dynmic yem, Equion (5) nd (6). Afer ome mnipulion nd implificion, he reuling coupled eroelic yem in e-pce form i defined in erm of he erodynmic nd rucurl dynmic e vecor follow: x x ( k + ) ( k + ) x [ ] ( ) k x ( k) [ H] [ ] [ H ] f ρ ˆ ( ) = F + fi k (3) [] o where [ ] x ( ) [ ] [ ] ( ) k q k = C x ( k) (4) [ G] ρ [ H][ D][ C] ρ [ H][ C] [ H ][ C ] [ G ] F (5) f I(k) + ρ f(k) fˆ ( k) Srucurl Dynmic Aerodynmic q(k) Figure : Block Digrm Repreenion of Coupled Aeroelic Syem q(k) Bed on he lyou of he yem hown in Figure, coupled eroelic yem cn be derived by fir Sbiliy Anlyi Wih he coupled e-pce model for he eroelic yem now defined, i i poible o direcly evlue he biliy of he yem wihou exmining muliple e of ime hiory d. Ined, one cn urn o dicree-ime biliy crieri nd exmine he roo, or eigenvlue, of he eroelic yem in order o evlue i biliy. Hence, he eroelic biliy crieri for hi dicree-ime yem i hen h ll eigenvlue of he mrix defined by Equion (5) lie wihin he uni circle when ploed in he complex plne. 7 Americn Iniue of Aeronuic nd Aronuic

8 The k hnd i hen o produce roo locu by compuing he eigenvlue for he eroelic emrix vriou deniie. To ccomplih hi k, mrix eigenvlue olver i employed which compue he nr (n + nb +) eigenvlue for he previouly defined e mrix. To de, hi mehodology h been ued uccefully o predic eroelic inbiliie for everl geomerie over wide rnge of ch number. One uch geomery i he AGARD wing configurion which i ndrd eroelic e ce h h been inveiged experimenlly in he Lngley Trnonic Dynmic unnel. A plnform view of he configurion i hown in Figure. Thi wing geomery i ofen ued in he lierure vlidion ce for compuionl eroelic code in he rnonic flow regime. Recen work h hown h he STARS eroelic nlyi module i cpble of predicing he experimenl d for hi wing geomery including he rnonic dip in he fluer boundry round ch.. In STARS, he AGARD i modeled rucurlly uing he wo dominn eigenvecor which repreen he fir wo nurl vibrion mode of he rucure. Thee mode hpe phyiclly repreen wing fir bending nd orion. The CFD meh for he AGARD coni of 7,36 node nd 376,25 erhedrl elemen. To begin he eroelic nlyi of he AGARD wing configurion, he yem idenificion procedure previouly dicued w ued o develop dicreeime erodynmic model for he unedy CFD oluion ch.96. During he idenificion procedure, 4- model order yielded he be greemen wih he CFD rining d nd w choen he opimum model. Wih n explici erodynmic model vilble for he AGARD ch.96, he coupled eroelic e mrix define by Equion (5) w embled for hi yem by following he previouly derived procedure. Thi reuled in 3 3 mrix which i funcion of he free rem deniy. A roo locu for he AGARD eroelic yem cn hen be developed by compuing he 3 eigenvlue for he e mrix over rnge of deniie. Figure 2 preen he AGARD roo locu plo produced uing en evenly pced deniy vlue rnging from.4-9 o linche/in 3. Uing hi plo, eroelic inbiliie cn be idenified by erching for eigenvlue which cro he uni circle, or whoe mgniude i greer hn one. A een in he plo below, he fir inbiliy occur ner deniy of linche/in 3. Figure : AGARD Te Wing Geomery nd Surfce Dicreizion 8 Americn Iniue of Aeronuic nd Aronuic

9 .25.4E-9 ρ = linch/in 3 5.E Inbiliy Occur Ner ρ = Figure 2: Complee Roo Locu for AGARD Dicree-ime Aeroelic Syem ch.96 Including Cloe-up of he Inbiliy Croover Poin To vlide he oberved inbiliy i correc, one cn reurn o he ime hiory nlyi ypicl ued in he STARS eroelic nlyi. Boh he dicree-ime model nd he unedy CFD oluion were ued o predic he eroelic ime hiorie for he AGARD ch.96 nd he inbiliy croover deniy of linche/in 3. A een in Figure 3, boh he model nd CFD oluion re in excellen greemen nd boh oluion eem o hve prediced neurlly ble or undmped ime hiorie. Thi prove h he erodynmic model i n ccure repreenion of he unedy CFD oluion nd h he e-pce biliy nlyi i cpble of ccurely cpuring eroelic inbiliie. x.3 Euler.2 d.. model x 2.4 Euler.3 d.. model Figure 3: Comprion Beween odel nd Euler Aeroelic Time Hiorie for he AGARD ch.96 nd ρ = Americn Iniue of Aeronuic nd Aronuic

10 Aide from he benefi of being ble o ccurely qunify eroelic inbiliie, he e-pce biliy nlyi provide furher inigh ino he phyic of n eroelic yem no redily vilble uing ime hiory nlyi. For he wo mode AGARD yem, we ee h here re wo dominn eigenvlue long he righ-hnd edge of he uni circle. Noice h he deniy incree, one of hee roo move inwrd nd become more ble while he oher roo move ouwrd unil i finlly debilize. Furhermore, we recognize h he unble roo i cully he lowe frequency mode for he yem. A een in Figure 4, he generlized diplcemen for ech mode of he AGARD follow divergen ph beyond he inbiliy deniy. However, we now know from he e-pce biliy nlyi h i i mode one which drive hee divergen ime hiorie. x 2 x Figure 4: Unble Time Hiorie For AGARD Aeroelic Syem ch.96 Anoher inereing geomery o udy i h of he Generic Hyperonic Vehicle (GHV). The GHV i ece developed by NASA o e he eroelic effec h migh be oberved on hyperonic vehicle. Figure 5 how he CFD urfce meh ued o model he GHV. The CFD meh for he GHV coni of 58,786 node nd 323,47 erhedrl elemen. Figure 5: GHV Geomery nd Surfce Dicreizion Srucurlly, he GHV i much more compliced hn he AGARD i i modeled uing nine eigenvecor which repreen vriou bending nd orion mode for boh he wing nd body. A wih he AGARD yem, he yem idenificion procedure w ued o develop dicree-ime model for he erodynmic of he GHV ch 2.2. Thi ime, 2- model order yielded he be greemen beween he erodynmic model nd he unedy CFD rining d. The reuling eroelic e w hen defined by qure mrix. Figure 6 preen he GHV roo locu plo produced uing en evenly pced deniy vlue rnging from.4-7 o linche/in 3. A before, we idenify inbiliie by erching for eigenvlue which cro he uni circle. A een in he plo below, he fir inbiliy occur ner deniy of linche/in 3 nd econd croover occur ler deniy ner linche/in 3. Alhough he econd inbiliy i of no prcicl inere ince he fir inbiliy i criicl nd would mo likely reul in he derucion of he vehicle, i i inereing o noe h he econd inbiliy doe exi nd cn be qunified uing he e-pce nlyi. For even more compliced yem, one migh find hi informion more ueful if eigenvlue hppen o become unble nd hen loop bck inide he uni circle, mking he higher order inbiliie more criicl. Americn Iniue of Aeronuic nd Aronuic

11 .25.4E-7 ρ = linch/in 3 5.E Inbiliy Occur Ner ρ = Figure 6: Complee Roo Locu for GHV Dicree-ime Aeroelic Syem ch 2.2 Including Cloe-up of Unble Croover Poin.5 Euler d.. model.3.2 Euler d.. model x x Figure 7: Comprion Beween odel nd Euler Aeroelic Time Hiorie for ode 2 & 4 of he GHV ch 2.2 nd ρ = linche/in 3 A wih he AGARD, we vlide he oberved inbiliy by compring he eroelic ime hiorie prediced by he dicree-ime model nd he unedy CFD oluion ch 2.2 nd he inbiliy croover deniy of linche/in 3. A een in Figure 7, boh he model nd CFD oluion re in excellen greemen for mode wo nd four nd boh oluion predic ime hiorie ner, bu lighly below he cul neurl poin. Noe h only mode wo nd four re hown her o conerve pce, bu reul re imilr for ll nine mode. Agin, hi prove h he erodynmic model i n ccure repreenion of he unedy CFD oluion for he GHV nd h he epce biliy nlyi i cpble of ccurely cpuring i eroelic inbiliie. Concluion The objecive of hi udy w o develop n ccure nd efficien mehod dicree-ime erodynmic model for ue in CFD-bed eroelic nlyi. The yem idenificion mehodology preened here h he benefi h only one unedy CFD oluion i required o produce model for ech ch number, reuling in ubnil ving in compuionl ime for eroelic nlyi. The echnique i pplicble o differen rucurl Americn Iniue of Aeronuic nd Aronuic

12 geomerie over wide rnge of ch number including he rnonic regime. The yem idenificion mehodology h he furher benefi of providing n explici mhemicl model for he unedy CFD oluion which hen llow one o develop e-pce form he eroelic yem. Wih e-pce form defined, eroelic inbiliie re eier o qunify uing dicree-ime biliy nlyi bed on he eigenvlue of he yem. Alhough no demonred here, he epce form i lo beer uied for eroervoelic pplicion well. For uch n pplicion, conrol lw could be coupled round he eroelic yem mking conrol nlyi more efficien. Bed on he benefi demonred here, hi pproch my mke he ue of CFD imulion rouine in he eroelic nlyi of eropce vehicle. Acknowledgemen Fund for he uppor of hi udy hve been lloced hrough he NASA-Ame Univeriy Conorium Office, under Inerchnge Number NCC2-55, nd Oklhom Se Univeriy. 7. Hollkmp, J. J., nd Bill, S.., A Recurive Algorihm For Dicree Time Domin Prmeer Idenificion, AIAA Pper Hmel, P. G., nd Jegonkr, R. V., Evoluion of Fligh Vehicle Syem Idenificion, Journl Of Aircrf, Vol. 33, No., 996, pp Pre, W.H., Teukolky, S.A., Veerling, W.T., nd Flnnery, B.P., Numericl Recipe in Forrn 77: The Ar of Scienific Compuing, 2 nd Ediion, Cmbridge Univeriy Pre, Gup, K.K., Developmen of Finie Elemen Aeroelic Anlyi Cpbiliy, Journl of Aircrf, Vol. 33, No. 5, Sepember-Ocober 996, pp Cown, T.J., Efficien Aeroelic CFD Predicion Uing Syem Idenificion, er Thei, Oklhom Se Univeriy, y 998. Reference. Cown, T.J., Aren, A.S., nd Gup, K.K., Accelering CFD-Bed Aeroelic Predicion Uing Syem Idenificion, AIAA , Augu, Gup, K.K., STARS An Inegred Generl-Purpoe Finie Elemen Srucurl, Aeroelic, nd Aeroervoelic Anlyi Compuer Progrm, NASA T-4795, y Dowell, E.H., e l, A odern Coure in Aeroeliciy, 3 rd Revied nd Enlrged Ediion, Klewer Acdemic Publiher, Ljung, L., Syem Idenificion: Theory For The Uer, Prenice Hll, Inc., New Jerey, Hollkmp, J. J. nd Bill, S.., Auomed Prmeer Idenificion nd Order Reducion for Dicree Time Serie odel, AIAA Journl, Vol. 29, No., 99, pp Pinkelmn, J. K., Bill, S.., nd Kehoe,. W. Tol Le Squre Crieri in Prmeer Idenificion for Fligh Fluer Teing, Journl of Aircrf, Vol. 33, No. 4, 996, pp Americn Iniue of Aeronuic nd Aronuic