Curve fitting approach for transonic flutter prediction

Transcription

1 THE AERONATICAL ORNAL SEPTEMBER 6 Cuve fitting appoa fo tansoni flutte pedition A. Sedagat Sool of Meanial Engineeing nivesity of Leeds, K. E. Coope,. R. Wigt and A. Y. T. Leung Maneste Sool of Engineeing nivesity of Maneste, K ABSTRACT Tis pape outlines an initial investigation fo detemining nonlinea aeodynamis fo unsteady tansoni flows toug te use of uve fitting unsteady omputational fluid dynamis (CFD) data. Te full aeodynamis inluding linea and non-linea aeodynamis an be identified as a polynomial funtion. Toug te uve fitting metod, te impotant non-linea tems an be identified and te smalle tems an be negleted. Having modelled te non-linea aeodynamis and inluded into te aeoelasti model, te aateistis and stability of non-linea aeoelasti system an ten be investigated using nomal fom teoy. Te metodology is demonstated upon a simple two-degees-of-feedom aeoelasti wing model wit stutual and aeodynamis nonlineaity. A good ageement is obtained fo all ases studied between analytial and simulation esults.. INTRODCTION It is usual fo aeoelasti alulations and flutte leaane to be made assuming linea aeodynamis and linea aiaft stutue. Howeve, te influene of non-lineaities on moden aiaft is beoming ineasingly impotant () and te equiement fo moe auate peditive tools gows stonge. Tese non-lineaities an be due to stutual (fee-play, baklas, ubi stiffness), aeodynami (moving soks and tansoni effets) o ontol (time delays, ontol laws) penomena. Non-linea flutte beaviou su as limit yle osillations (LCO) an only ou in non-linea systems (,). Consequently, it is not possible to pedit LCO using a puely linea analysis and su an analysis is beoming less auate as aiaft stutues ae designed to be moe effiient. Te study of LCO as beome of ineasing impotane ove te last few yeas, altoug su poblems ave been noted sine te 97s. Tee is an ugent need fo a peditive apability of non-linea aeoelasti penomena. Su ability would enable fligt flutte tests to be ompleted faste and wit a geate degee of safety. It is of patiula inteest to detemine: wete an aiaft will epeiene bounded limit yle osillations and/o atastopi flutte weeabouts in te fligt envelope LCO penomena will ou te peise natue of te LCO. Altoug not desiable, LCO is essentially a fatigue poblem, weeas flutte is usually atastopi and must be avoided at all osts. An auate LCO/flutte pedition apability would edue signifiantly te amount of fligts equied in any fligt leaane test pogamme wit uent osts being estimated at aound $7k pe test fligt. Tee as been mu wok in eent yeas (-) devoted towads te aateisation of non-linea aeoelasti beaviou. Tis wok as pimaily onsisted of simulating te esponse of te aeoelasti system toug numeial integation, altoug tee ae a few known instanes of epeimental veifiation (). Nealy all of te effot as been devoted towads te effets of stutual non-lineaities, toug some attention as been made towads investigating te effet of non-lineaities in aeosevoelasti systems (6). Tee as also been a signifiant effot devoted (7,8) to impoving unsteady modelling toug te oupling of te aeodynami and stutual models. Signifiant eadway as been made towads solving te poblem, patiulaly in te tansoni egion. Howeve, tee ae still majo poblems ineent due to te enomous omputational esoues equied fo even te simplest ases. Tis wok is investigating te use of nomal fom teoy (9) to pedit te beaviou of non-linea aeoelasti systems. Te idea of Pape No. 68. Manusipt eeived 9 May, evised vesion eeived Febuay, aepted Apil.

2 66 THE AERONATICAL ORNAL SEPTEMBER Fung (), and used ee in non-dimensional fom. Te equations epessed fo invisid inompessible flow wit no stutual damping at speed ae witten as Figue. Impulse esponse of CFD ode at diffeent Ma numbes. te appoa is to do away wit te need fo etensive omputational simulations at evey point in te fligt envelope. Howeve, te metodology is not seen as eplaement fo etensive CFD modelling, but as a guide to detemine te itial pats of te fligt envelop tat sould be investigated using te sopistiated CFD metods. So fa te metodology as been applied suessfully to systems ontaining ontinuous stutual non-lineaities. Tis wok is also desibes te uent pogess in using te appoa to model aeoelasti beaviou in te tansoni egion, wit patiula empasis on te use of uve fitting metods to model te non-linea aeodynami aateistis. A two-degees of feedom wing model was used to evaluate te peditability of nomal fom teoy fo aeoelasti systems. Te metodology takes te following steps:. Define te govening aeoelasti equations. Tansfom te equations to inlude diffeential opeatos instead of te unsteady Wagne funtion tems (fo te linea aeodynamis ase). se te uve-fitting appoa to model aeodynami foes obtained fom unsteady CFD solutions o epeimental data (fo te non-linea aeodynamis ase). Tansfom te equations into modal anonial fom. Redue te system ode using te ente-manifold tenique 6. se te aveaging metod to find nomal foms and investigate te system beaviou in te tansfom domain 7. Tansfom bak to pysial spae to detemine limit yle osillations amplitude and fequeny Eamples of te use of te appoa ae given using a simple two degees of feedom aeoelasti model wit stutual and aeodynamis non-lineaities.. GOVERNING AEROELASTIC EQATIONS Conside a two-degee-of-feedom aeofoil osillating in pit and plunge as sown in Fig.. Te eave defletion is denoted by, positive in te downwad dietion, α is te pit angle about te elasti ais, positive wit te nose up. Te elasti ais is loated at a distane a b fom te mid-od wile te mass ente is loated at a distane ab fom te elasti ais. Bot distanes ae positive wen measued towads te tailing edge of te aeofoil. Te integodiffeential aeoelasti equations of motion ave been deived by ù G M () () C L ðì () ô () ô C M ðì wee α is te aeofoil inidene and ξ /b is te non-dimensional vetial displaement wee b is te semi-od. Te pime supesipt denotes diffeentiation wit espet to te non-dimensional time τ wi is defined as τ.t/b. _ is a non-dimensional veloity defined as _ /bω α, and ω _ is given by ω _ ω ξ /ω α, wee ω ξ and ω α ae te unoupled eaving and piting modes natual fequenies, espetively. m µ/(πb π) is te atio of te aeofoil to fluid mass. α is te adius of gyation, epesenting te effet of te moment of inetia about te elasti ais. G(ξ) and M(α) ae stutual non-lineaities. C L (τ) and C M (τ) ae te lift and piting moment oeffiients, espetively. Fo te linea fomulation of te inompessible invisid flow, te epessions fo C L (τ) and C M (τ) ae given as Ref. C L + ð () ô ð( a α + ) + ðö C M τ ö () ô () + () + a α () ( ô ó) () ó + () ó + a () ó ð () τ a ( a α ) + ð + a ö + ð + a ð a () ô () + () + a () ô ö ( ô ó) () ó + () ó + a () ó dó wee φ(τ) is te Wagne funtion. Te integal Wagne opeatos in Equation () ad to be eplaed by some diffeential opeatos fo nomal fom analysis. Tis was aieved by intoduing fou additional vaiables, wi ineased te size of govening equation fom Ref.. A emedy to avoid te above poedue is to use te uve fitting metod intodued in te setion. Te uve fitting appoa also simplifies epessions fo non-linea aeodynamis. Fo a twodimensional aeoelasti system at tansoni egime, a simple ypotesis was assumed fo aeodynamis foes as FL F FM F L M wee M is te fee steam Ma numbe and F L and F M ae te lift and te moment foes, espetively. Tese foes inlude linea and non-linea aeodynamis and an be obtained using te uve fitting of unsteady CFD solutions.. CRVE FITTING APPROACH (,,,,, M, ù ) (,,,,, M, ù ) ð 6.. ().. ().. () A matematial model of an aeoelasti system an be obtained by uve-fitting te aeodynamis foes by epessing tem as some funtion of te displaement and veloity data. Tese data an be dó

3 SEDAGHAT ET AL CRVE FITTING APPROACH FOR TRANSONIC FLTTER PREDICTION 67 obtained eite fom epeiment o CFD solutions. A powe seies up to tid ode was used fo appoimating aeodynamis oeffiients fo a two-dimensional wing model (Fig. ) as follows C L i a i i + i j i b ij i j + g + g + g + g C d + e + f M i i ij i j ij i j i i j i i j wee te veto is epessed as {,,, }{,,, } Te system of unknown oeffiients,,,, a b g i i ij ij d, e, i ij f ij, i i,,,, i,,,, + ij i j i j j,,, j,,, wi ae in geneal funtions of te fee steam Ma numbe, te edued veloity and te edued fequeny, an be solved using te following linea system [] T.{B } {C } [] L L T.{B } {C } M M wee {C L } and {C M } ae te time vetos of lift and moment oeffiients. Te unknown oeffiients in veto fom ae pesented as {B L } fo te lift and {B M } fo te moment oeffiients. Te mati of all powe seies tems is epesented by [T]. Fom CFD solutions, te time veto {} Equation () and te aeodynamis time vetos {C L } and {C M } ae known. Equation (7) ten an be solved to obtain te unknown oeffiients {B L } and {B M }. Te oot mean squae (RMS) of all linea and non-linea tems in Equation () an be used to edue te size of model. Tose tems tat ave lage RMS ontibution an be kept and te emaining smalle tems an be omitted. Eamples of uve fitted models ae given in setions 8 and 9.. AVERAGING METHOD.. ().. ().. (6).. (7) One of te basi tools, in te study of qualitative beaviou of nonlinea systems, at te viinity of a bifuation point is nomal fom teoy (9). Te basi idea of te metod of nomal foms is employing suessive o-odinate tansfomations to systematially onstut te simplest possible fom of te oiginal diffeential equations. Te simple fom an eibit all possible dynamial popeties of te oiginal system in te neigbouood of te bifuation point. Finding nomal foms of a non-linea system is not an easy task. Te aveaging metod is a useful matematial tool fo solving weakly non-linea dynamial systems. As wit nomal fom tansfomations, te aveaging metod () uses nea identity o-odinate tansfomations to simplify a given system of odinay diffeential equations. Te lassial nomal fom tansfomation applies to autonomous systems wile te aveaging metod applies to nonautonomous systems. Te autonomous diffeential equations an also be tansfomed into a non-autonomous system using te metod developed by Leung and ten analysed using te aveaging metod outlined ee and disussed in detail in Ref.. Conside te non-autonomous diffeential equations (, ô,å), å X å X << wee te funtion (X, τ, ε) is a T-peiodi veto field. sing te non-autonomous T-peiodi tansfomation of te fom k ( ç, ô) + å ( ç, ô) + å ( ç, ô) X ç + å + ten te esulting aveaged equation as tems autonomous up to O(ε k ) and an be epessed as () ~ k ~ k + ç + å () ç + + å () ç + å R( ç, ô,å) ç å ~ Te non-autonomous pat, te emainde R, is tunated and te emaining system of equations is analysed. By using an asymptoti teoy, it an be sown tat te solution of te final system appoimates te solution fo te oiginal system Equation (8). Fo a two-dimensional system, te Equation () an be e-witten in pola o-odinate system η (, θ) as f è f () () wee and θ epesent te amplitude and te fequeny of limit yle osillations in te tansfomed domain, espetively. Te amplitude of LCO an be easily obtained as a funtion of non-dimensional time by integating te fist epession in Equation (). By substituting tis esult of te LCO amplitude into te seond epession in Equation (), te fequeny of LCO is also obtained. Ten, tese esults sould be tansfomed bak into te pysial domain using te evese tansfomation outlined above. Tese opeations wee failitated toug using te symboli pakage Matematia.. NON-LINEAR ANALYSIS Conside a geneal non-linea system as ~ X AX + G( X, ô) k.. (8).. (9).. ().. ().. () wee A ~ is a onstant squae mati of wi all eigenvalues ave non-zeo eal pats. Te tem A ~ X and G(X, τ) epesent te linea and non-linea pats of te system. Tis system is fute etended to analyse bifuation beaviou of te aeoelasti system using te system paamete δ defined as L ä.. () wee L is onstant and equal to te linea flutte speed. By substituting te Equation () into Equation (), te non-linea system of Equation () an be ewitten as () ( )( ) X AX + B ä X + ä F X, ô.. () δ Te mati A is te aobian mati evaluated at te equilibium point and bifuation value (i.e. δ ). Te mati A as at least one pai of puely imaginay eigenvalue λ, ± iω. Te seond and tid tems in () ae non-linea funtions of δ, X and τ. Epessions fo te maties A, B, and te non-linea funtion F(X,τ) ae given in Ref. fo an aeoelasti system wit linea aeodynamis and nonlineaity in stutue. To apply nomal fom teoy using te aveaging metod, te system Equation () is fist tansfomed into its modal anonial k

4 68 THE AERONATICAL ORNAL SEPTEMBER fom. As an eample, a tansfomation mati Q is obtained fom te eigenspae of A in Ref., su tat Q.A.Q b b ë ë, ë ë 6 ù ù,.. () Intoduing a new vaiable, Y, su tat T Y Q. X ( y, y,, y 8 ).. (6) te system Equation () beomes ( Q. B() ä. Q) Y + ( ä ) Q. F( Q., ô) Y. Y + Y ä.. (7) Figue. C l and C m time istoies and fequeny ontent of C m at M 8. Te dynami esponse of te above system an ten be investigated toug a two-dimensional system obtained fom te following edution tenique. 6. CENTRE MANIFOLD REDCTION Te idea of te ente manifold teoy is also suessive non-linea tansfomations. It edues te oiginal system to a ente manifold assoiated wit te pat of te oiginal system aateised by te eigenvalues wit zeo eal pats at a bifuation point. Te appoa based upon te ente manifold edution an be used to edue a multi-degee of feedom dynamial system into a two-dimensional mati system oesponding to its itial mode. Tus, ige ode nomal foms an be obtained fo te edued system. Intoduing v, u [ y y ä], [ y y y y y y ] Te system Equation (7) an be split into two systems su tat u. u + f v. v + f ( u, v) ( u, wee te fist equation in Equation (9) oesponds to te itial mode. f, f ae non-linea funtions of u and v stating fom te seond ode tems. Te fist ode tems ave aleady been inluded in te fist pat assoiated wit u and v. Sine a solution nea te oigin is sougt, if an appoimate funtion v (u) an be found nea te oigin su tat v D u ( u) u D u ( u) + f ( u, ( u)) ( u)[ u + f ( u, ( u))].. (8).. (9).. () ten only te fist equation in Equation (9) will be adequate fo te non-linea analysis. In te above elation, te mati D u (u) is te aobian mati of (u). Te funtion (u) an be appoimated by any ode funtion as defined in Refs and. 7. NSTEADY CFD MODEL An unsteady tansoni CFD ode () developed at te nivesity of Glasgow as been used in tis wok. Two types of motion wee used to podue data fo tis identifiation. Tese motions ae speified as. Fee esponse to a step input in pit and plunge,. Pesibed sinusoidal motion in pit and plunge, Te fee esponse ase was used to detemine wete te motion at a patiula ondition was damped o wete flutte o LCOs esulted. Te CFD omputations wee aied out ove a ange of Ma numbes and speeds using te step esponse mode to detemine te type of esponse. It was obseved tat at Ma numbe.8, te aeoelasti system eibit a simple lassial LCO. Howeve fo te sligtly ige Ma numbe of 8, te inteesting featue of a non-linea divegene ombined wit a LCO is obseved (6). Fo te ige Ma numbe of 7, te osillation damped away, as te tansoni non-linea effet is diminised (see Fig. ). Fo tee ases, Ma numbes of 8, 8, and 7, te wing model was eited wit a pesibed sinusoidal motion in te pit mode using te detemined LCO fequeny obtained fom te step esponse solutions (see Figs -). At te lowest speed, te lassial LCO fequeny ontent of te lift oeffiient ontains a dominant fequeny along wit signifiant tid and fift amonis. Tee is a mu geate fequeny ontent at Ma numbe 8, suggesting a stonge non-linea flow-stutue inteation at tis patiula flow ondition. Fo te ige Ma numbe of 7, te fist mode is only obseved in te fequeny ontent of te moment oeffiient implying tat te aeodynami foes ae linea. Fom te estoing foe plot fo te moment oeffiient and te oesponding eo plot fo diffeent ode of polynomial fits (see Ref. 6), it was obseved tat a ubi non-linea funtion an adequately epesent te non-linea beaviou of te aeodynami foes fo tansoni invisid flows.

5 SEDAGHAT ET AL CRVE FITTING APPROACH FOR TRANSONIC FLTTER PREDICTION 69 Figue. Aeofoil geomety fo two degees of feedom motion. 8. CRVE FITTING VERIFICATION In ode to assess te appliability and auay of te uve fitting appoa intodued in te setion fo te wing model in Fig., te desibed aeoelasti model Equation () wit linea aeodynamis Equation () and te stutual non-lineaity given by Ref. 7 G M () () â â fo te two ases: () Figue. C l and C m time istoies and fequeny ontent of C m at M 8. Figue. C l and C m time istoies and fequeny ontent of C m at M 7. â, â () â, â.. ().. () wee used. Having applied tis appoa, te following model was obtained fo te ase in Equation () as + + l α + + m α ù α l m + m + m Te oeffiients in Equation () wee obtained at diffeent speeds. Ten, tese oeffiients wee elated to te edued veloity using a simple polynomial fit. p to te auay of te data fit softwae te following esults obtained l l l l l m m m m Te limit yle osillations solutions fo te oiginal model (te aeoelasti Equation () wit te linea aeodynamis Equation () and te stutual nonlineaity Equations (-)) (simulation), te uve-fitted model Equation () (model) and te nomal fom metod wee ompaed in te pitfok diagam of Fig. 6 fo speeds above te linea flutte speed. Te geneal epession fo te nomal fom solution is obtained as follows a + a + a è b + b + b wee and θ. ae te amplitude and te fequeny of LCO in te tansfomed domain, espetively. Fo eample, at d 9 te following nomal fom solution wee obtained. Te sape of limit yle osillations in pit and plunge wee ompaed fo tis speed in Fig. 7 using diffeent metods outlined above. l è l l.. ().. ().. ().. (6)

6 7 THE AERONATICAL ORNAL SEPTEMBER ì 8 ù.. (8) Te CFD solutions fo te above stutue wee obtained at diffeent Ma numbes. Having applied te uve fitting appoa on te CFD data, te following aeoelasti model was deived Figue 6. Amplitude of limit yle osillations in pit (degee) and plunge (non-dimensional) fo te test ase wit ubi non-lineaity in bot pit and eave motion. + ù + l l l l l6 l7 + l + + m m m m wee again te unknown oeffiients above wee elated to te edued veloity using a data fit softwae up to its auay as follow.. (9) l ; l ; l ; l ; l l l ; ; ;.. () and fo te moment oeffiient as follows Figue 7. Te sape of limit yle osillations fo te test ase wit ubi nonlineaity in bot pit and eave motion and at te two speeds of δ 9 and δ. m ; m 99 ; m ; m ;.. () 9. NON-LINEAR AERODYNAMICS MODELLING In ode to develop aeoelasti models at tansoni egion, a speified stutue was osen su tat te following elation between te edued veloity and te fee steam Ma numbe is maintained at fligt eigts above km (see Sliting et al (8) fo moe details). 87 M.. (7) Te wing model Equation () wit te following aateistis wee onsideed: Te linea flutte speed of _ LF 6 was obtained fo tis system as it sown in te eigenvalue plot of Fig. 8. Note tat te linea pat of te system Equation (9) may sow diffeent tends depending on te numbe of points and te ange of speeds was taken fo te uve fitting of CFD solutions. Tus, it is eommended tat te uve fitting appoa to be applied at viinity of te Hopf bifuation point (te linea flutte ondition) wit at least tee sets of data oesponding to tee speed levels. It sould also be noted tat uve-fitting appoa would not yield to a unique solution and pysially unaeptable esults may also be obtained fo te linea pat. Te amplitude of limit yle osillation obtained fom simulation of aeoelasti model Equation (9) was ompaed wit CFD solutions in Fig. 9 fo pit and plunge motions. A vey lose ageement is obtained fo te amplitude of motion in pit; oweve, te amplitude of osillations in plunge was sligtly unde pedited. Hee te fous is on pediting te pit motion auately wit simple models. Howeve, by keeping moe non-linea tems of te uve

7 SEDAGHAT ET AL CRVE FITTING APPROACH FOR TRANSONIC FLTTER PREDICTION 7 Figue 8. Eigenvalue plot fo damping atio and natual fequeny. Figue. Compaison of te sape of limit yle osillations obtained at vaious Ma numbes (sifted by a degees inteval) using CFD solutions (solid line) and simulation of te developed aeoelasti model (symbols). Figue 9. Te amplitude of limit yle osillations in pit and plunge motions at vaious Ma numbes using CFD solutions (+) and te developed aeoelasti model (o). fitted model tat ontibute to te plunge motion, te plunge solutions may also be impoved. Te sapes of limit yle osillation ae ompaed qualitatively fo seveal Ma numbes in Fig. between CFD solutions and simulations, wi ae in oveall ageement. To inease laity and to avoid ovelapping diffeent plots oesponding to diffeent Ma numbes in Fig., ea suessive plot was sifted degees fom te pevious one. Note tat tese models ae not seen as a geneal solution fo a geneal wing model. But, it gives a guideline ow to simplify a diffiult engineeing poblem to investigate te pysial beaviou of te omplete system. Te ypotesis given in Equation () an be fute etended to ove moe ange of system paametes. It would also advisable to ompae diffeent CFD solutions wit ea ote and epeimental obsevation in ode to use te most eliable data fo su effots.. CONCLSIONS Te metodology disussed in tis epot fo alulating limit yle osillations was suessfully applied fo seveal aeoelasti systems inluding te wing model wit te stutual nonlineaity. Te uve fitting appoa was intodued fo development of aeoelasti models at tansoni egions. Te unsteady CFD ode of te Glasgow nivesity was used to podue data fo a ange of Ma numbes at tansoni speeds. Te non-linea aeodynamis was suessfully modelled fo a speified stutue. Te esults of te developed aeoelasti model ave sown tat te model an epodue LCO amplitude in pit faily lose to CFD data. Te omputational ost of te model is substantially less tan te atual CFD alulation; oweve, tese types of models intended to seve as a guide as to wi pats of te fligt envelope tat sould be investigated using sopistiated CFD metods. Resea is ontinuing wit te appliation of te metodology to ige ode systems and systems ontaining non-linea aeodynamis and disontinuous non-lineaities. ACKNOWLEDGEMENT Tis wok was funded by te EPSRC toug gant GR/L97. Te autos ae also gateful fo te suppot fom te nivesity of Glasgow fo poviding tei CFD odes. REFERENCES. AGARD CPs66, Advaned aeosevoelasti testing and data analysis, 99.. PRICE, S.., ALIGHANBARI, H. and LEE, B.H.K. Te aeoelasti esponse of a two dimensional aeofoil wit bilinea and ubi stutual nonlineaities,. Fluid and Stutues, 99, 9, pp DIMITRIADIS, G. and COOPER,.E. Limit yle osillation ontol and suppession, Aeo, 999,, pp YANG, Z.C. and ZHAO, L.C. Analysis of limit yle flutte of an aifoil in inompessible flow,. Sound and Vibation, 988,, (), pp -.. HOLDEN, M., BRAZIER R. and CAL, A. Effets of stutual non-lineaities on a tailplane flutte mode, 99, Int Foum on Aeoelastiity and Stutual Dynamis pape 6.

8 7 THE AERONATICAL ORNAL SEPTEMBER 6. COOPER,.E. and DIMITRIADIS, G. Caateisation of non-linea aeosevoelasti beaviou, 999, RTO Speialists Meeting on Stutual Aspets of Fleible Aiaft Contol Pape AGARD CP 7, Tansoni unsteady aeodynamis and aeoelastiity, AGARD Repot 8, Numeial unsteady aeodynamis and aeoelasti simulation, LENG, A.Y.T., ZHANG, Q.C. and CHEN, Y.S. Nomal fom analysis of Hopf bifuation eemplified by Duffing s equation,. Sok and Vibation, 99,, pp -.. FNG, Y.C. An Intodution to te Teoy of Aeoelastiity, 99, Wiley, New Yok.. SEDAGHAT, A., COOPER,.E., WRIGHT,.R. and LENG, A.Y.T. Pedition of Non-linea Aeoelasti Instabilities,, ICAS.. LENG, A.Y.T. and QICHANG, Z. Hige-ode nomal fom and peiod aveaging,. Sound and Vibation, SEDAGHAT, A., COOPER,.E., WRIGHT,.R. and LENG, A.Y.T. Limit yle osillation pedition fo non-linea aeoelasti systems wit osillatoy aeodynamis,, Poeedings RAeS Aeodynamis Confeene, Apil.. SEDAGHAT, A., COOPER,.E., WRIGHT,.R. and LENG, A.Y.T. Limit yle osillation pedition fo aeoelasti systems wit ontinuous non-lineaities,, AIAA--97, st SDM Confeene, Apil.. BADCOCK, K.., SIM, G. and RICHARDS, B.E. Aeoelasti studies using tansoni flow CFD modelling, 99, Int Foum on Aeoelastiity and Stutual Dynamis Pape SEDAGHAT, A., VIO, G.A., COOPER,.E. and WRIGHT,.R. Modelling of non-linea aeodynamis duing limit yle osillations,, ISMA, Leuven. 7. LI, L., WONG, Y.S. and LEE, B.H.K. Appliation of te ente manifold teoy in non linea aeoelastiity, 99, Int Foum on Aeoelastiity and Stutual Dynamis, pp SCHLICHTING, H., TRCKENBRODT, E. and RAMM, H.. Aeodynamis of te Aiplane, 979, MGaw-Hill, pp -.