NON-LINEAR AEROELASTIC BEHAVIOR OF HIGHLY FLEXIBLE HALE WINGS

Transcription

1 5 TH ITERATIOAL COGRESS OF THE AEROAUTICAL SCIECES O-LIEAR AEROELASTIC BEHAVIOR OF HIGHLY FLEXIBLE HALE WIGS G.Romeo*, G.Frulla*, E.Ceino*, P.Marzocca** *Poliecnico di Torino, **Claron Univeriy Y Keyword: HALE Fluer Behaviour, on-linear Aeroelaiciy, Limi Cycle Ocillaion. Abrac High apec raio aircraf, uch a High- Aliude Long-Endurance (HALE) UAV, can experience large aic deflecion during normal fligh operaion ha could everely reduce he fluer peed. A uch peed, Limi Cycle Ocillaion (LCO) may occur, reducing heir operaional life. A ime-marching olver of a nonlinear aeroelaic model i propoed a a deign analyi ool for highly flexible aircraf. Wihin hi nonlinear aeroelaic model, moderae o large rucural deflecion and linear/non linear aic and dynamic aeroelaic repone of uch flexible aircraf can be properly conidered. A geomerically conien non-linear rucural model i adoped baed on Hodge and Dowell, including Da Silva econd order geomerical non-linear erm. The rucural model ha been coupled wih an uneady aerodynamic model for an incompreible flow field, baed on he Wagner aerodynamic indicial funcion, in order o obain a non-linear aeroelaic model. Thi procedure enable one o expedie he calculaion proce. The influence of eleced deign aeroelaic parameer, uch a ip deflecion and iffne raio, i udied. In addiion o analyical and compuaional reul, a high apec-raio wing model ha been eleced a a candidae for a feaibiliy experimenal udy and o validae he heoreical model. Preliminary reul on a bala wing model and heoreical comparion are preened. Copyrigh 6 by G. Romeo, G. Frulla, E. Ceino, P. Marzocca. Publihed by ICAS wih permiion. omenclaure a = Elaic axi locaion. b = Semi-chord. e = Secion ma cener from elaic. Axi. E = Modulu of elaiciy. G = Shear modulu. U = Free-ream velociy. m = Ma per uni lengh. g = Acceleraion of graviy. α = Angle of aac. I η, I ζ = Verical and chord-wie area. momen of ineria. ρ = Air deniy. J = Torional iffne conan. ω r = Reference frequency. L = Wing pan. ζ = Dimenionle in-plane generalized deformaion. η = Dimenionle ou of plane generalized deformaion. = Dimenionle orional generalized deformaion., = Wagner funcion conan. vw,, = lag, flap diplacemen and orional roaion. w 3/4c = Downwah velociy a ¾ chord. Fv, Fw, M = Generic in-plane, ou-of plane and aerodynamic momen reulan expreion. EIζ, EIη, GJ = Bending and orion rigidiy. G v, G wg = Added non-linear nd order erm.

2 G.ROMEO*,G.FRULLA*,E.CESTIO*,P.MARZOCCA** General Inroducion Very low weigh combined wih high apec raio give rie o lender and flexible wing rucure. Such high degree of flexibiliy force he deigner o deal wih pecific phenomena no uually conidered in claical aircraf definiion propoing differen deign indicaion [8,9,]. A we rive o reduce weigh and raie performance level uing direcional maerial, hu leading o an increaingly flexible aircraf, here i a need for reliable analyi ool, which model all he imporan characeriic of he fluid-rucure ineracion problem: aeroelaic inabiliie alway conrain he fligh envelope and hu hey have o be conidered fundamenal during he deign proce. One eenial limiaion of linearized analyi i ha i can only provide informaion up o fligh peed a which he aeroelaic inabiliy occur. Furhermore, hee analye are rericed o cae where he ranien aeroelaic repone ampliude are mall. Ofen hi aumpion i violaed prior o he one of inabiliy. To udy he behavior of aeroelaic yem near he poin of inabiliy, nonlineariie hould be included. In recen year, udie of nonlinear fluid-rucure ineracion have been moivaed by evidence ha nonlinear effec in aeroelaic yem may be eiher favorable or unfavorable or a combinaion of boh. For HALE UAV i ha been hown [3-4] ha nonlineariie could induce Limi Cycle Ocillaion (LCO) even below he nominal fluer velociy. Wheher uch nonlinear effec are favorable or no will depend very much on he paricular circumance and parameer involved. onehele i i clear ha nonlinear effec ofen lead o LCO, which in urn, even if no caarophic in naure, can lead o faigue damage. In heir abence he alernaive would be a caarophic fluer and conequenly rucural failure. To achieve an efficien deign of flexible airplane, a beer underanding of all he facor conribuing o he occurrence and increae of he fluer inabiliy boundary, a well a of hoe deermining if he limi cycle ocillaion are able or unable are required. Aeroelaic Model In hi paper a geomerically non-linear moderae o large deflecion rucural model, baed on [] and modified according o he econd order geomerical non-linear erm inroduced in [5], ha been coupled wih an uneady aerodynamic model for an incompreible flow field baed on he Wagner indicial funcion [,3,9]. The rain diplacemen relaion are developed from an exac ranformaion beween he deformed and undeformed coordinae yem. The wing i aumed o be clamped in he plane of ymmery and he equaion of moion are obained from Hamilon principle. Thee equaion are alo valid for beam wih a ma cenroid axi offe from he elaic axi, nonuniform ma and iffne ecion properie, and variable pre-wi. Term up o econd order have been reained in he final expreion. Higher non-linear coefficien have been negleced auming hey would have a negligible effec on he yem dynamic. Roary ineria ha been negleced according o he lender beam hypohei. The effec of hear deformaion ha alo been negleced. The reuling equaion (a-c) are valid o econdorder for long, lender, homogeneou, ioropic beam undergoing moderae o large diplacemen. Such aumpion are conien wih [] and he reader i referred o [,5] and he reference cied herein for furher deail. In Eq. (a-c) G v, G w, G are econd order nonlinear erm derived from Da Silva [5]. ζ ζ η v EI v + EI EI w + G + mv&& = Lv EIηw + EIζ EIη v + G w + mw&& + me&& = Lw GJ + EI EI v w + G + m && + mew&& = M (a-c) ζ η m In order o olve he yem of governing equaion and o udy he ubcriical and upercriical aeroelaic repone a well a he fluer boundarie, he inroducion of a mall dynamic perurbaion abou a non-linear aic

3 O LIEAR AEROELASTIC BEHAVIOUR OF HIGLY FLEXIBLE HALE WIGS equilibrium i applied. In-plane, ou-of-plane and orional deformaion (v, w, ) are conidered o be a ummaion of he aic and dynamic componen in he undeformed reference yem. The deflecion are hown below: v= v + v w= w + w = + (a-c) where v, w, and are he aic in-plane lagging, ou-of-plane bending and orion deformaion due o he aeroelaic rim, correponding o a pecific fligh condiion, repecively. The deformed beam cheme ued in he preen model i repored in Figure. Fig.. Wing reference yem A moderae/large deflecion mall perurbaion approximaion ha been inroduced and he aic variable are conidered only x dependen. The dynamic deformaion v, w, are ime and pace dependen. Uing modal analyi echnique, he oluion of he problem can be expreed a: v (, ) ζ v x v v x = + i i w ηi i (, ) w x w w x = + ( x, ) ( x) = + β (3a-c) i i The form of he oluion i independen of he rial funcion employed o dicreize he yem, bu he proximiy of he dicree yem o he original one i dependen on elecion of he funcion. A well choen e of rial funcion can accuraely repreen he original coninuou yem in erm of a few dicree coordinae. Previou inveigaion on canilever beam [4], uing uncoupled mode hape for flap-lag-orion abiliy analyi, indicaed ha reul would be accurae wih a few a one or wo mode. The example preened in hi paper i obained uing ix mode (wo for each degree-of-freedom) wih mode hape derived from a vibraing uniform canilever beam. The parial differenial equaion governing he dynamic of he flexible beam [,6,9,,5] were reduced o a yem of ordinary differenial equaion uing a erie dicreizaion echnique [], along wih Galerin mehod, o obain he aeroelaic governing equaion. Subiuing Eq. (a-c) ino Eq. (a-c) i i poible o idenify wo e of governing equaion: a aic aeroelaic equilibrium yem and a perurbed dynamic yem. The aic equilibrium non-linear yem can be expreed a: EIζv + EIζ EIη w + G v = Fv EIηw + EIζ EIη v G + w+ mg = Fw GJ + EIζ EIη vw + G + mge= F (4a-c) where G v, G w, G are counerpar of he erm derived from [5] for he aic equilibrium cae. Given a free-ream velociy U and an angle-of-aac α, he aic diplacemen v, w, for he correponding rim condiion can be compued. oice ha, in he perurbed dynamic yem, he aic configuraion will couple wih he dynamic componen hrough he non-linear erm. x = x/ L, w = w x, Inroducing he variable v= v( x ), β β( x ) =, a dimenionle form of he dynamic governing equaion can be ca a: 3

4 G.ROMEO*,G.FRULLA*,E.CESTIO*,P.MARZOCCA** + + ΓΩ + Ω Γ w w ηiwi β i i v b h ζ iv i h w β i i ηiwi v ' v && ζ i i v w v ζivi + β i i + v b h ηiw i h w β i i ζivi v ' && && w ηi i α β i i w w v v w ηiwi ζiv + i + b b α β i i h w v η iwi ζ ivi w ' && α β && i i α η + G + v = L Ω +Ω Γ + + G + w + x = L Ω + Ω Γ + G + r + x w= M + (5a-c) where prime refer o differeniaion wih repec o he dimenionle coordinae along he pan. The erm G,, v G w G couple he equaion hrough orional iffne, even if Γ( EIζ EIη) =. Their full expreion can be found in [9,]. The dimenionle parameer inroduced in he Eq. (5) are expreed a: EI EI e GJ Ωh, Γ=,, Ω, mω L EI b m b L η ζ x 4 α α r η ωr m ω b m µ mb U πρ b m r rα,,, a (6a-h) The uneady aerodynamic force can be obained uing Wagner indicial funcion in Duhamel inegral form [,3] a repored in Eq (7a-b). L= πρb w&& + U & ba&& dw 3/4c πρub w3/4c ( ) Φ + Φ( σ) dσ dσ dw 3/4c M = πρub (.5+ a) w3/4c ( ) Φ + Φ( σ ) dσ + dσ 4 ρπ b + abρπ b ( w&& ba&& ) ρπub (.5 a) b & && 8 where τ (7a-b) Φ i he Wagner funcion and a i he offe of he elaic axi poiion from he mid-chord, poiive af. Approximae expreion of he Wagner funcion have been derived and he R.T. Jone approximaion ha been ued [] in he preen applicaion uch a: bτ b τ Φ τ = Ae Ae, ( τ > ) (8) where he coefficien are: (.65;.335) b i (.455;.3) A i and. Due o he exience of he inegral erm in he aerodynamic inegrodifferenial equaion [3], i i cumberome o inegrae hem numerically. However, applying properie of he inegraion by par, and inroducing he variable [,9,9,]: ( σ) ( σ) W = we dσ, W = we dσ ( σ) ( σ) W3 = e dσ, W4 = e dσ (9) he airload can be rewrien in general form conaining ju differenial operaor leading o a ae-pace form. L = Cw && + C w & && & + C3w+ C4 + C5 + C6 + + CW + CW + CW + C W + F M = Dw&& + Dw& && & + D3w+ D4 + D5 + + D + DW + DW + DW + D W + P (a-b) 4

5 O LIEAR AEROELASTIC BEHAVIOUR OF HIGLY FLEXIBLE HALE WIGS where he coefficien C and i D are funcion i of he elaic axi locaion a, he reduced frequency, he ma parameer µ and he uneady indicial funcion coefficien. F and P are he aerodynamic load due o iniial condiion. Their expreion can be found in [9]. The dimenionle ae vecor T { X} = { x,..., x} become funcion of he diplacemen variable and he addiional aerodynamic lag ae a: x = & ς, x = & ς, x3 = & η, x4 = & η, x5 = & β, x 6 = & β, x7 = ς, x8 = ς, x 9 = η, x = η, x = β, x = β, x 3 = W, x4 = W, x5 = W3, x6 = W4, x = W, x = W, x = W, x = W () Applying he convoluion inegral propery [,9], he remaining conrain equaion can be derived in order o complee he yem. Conrain equaion obained from Wagner indicial heory are expreed a: x& 3 = x9 x3, x& 4 = x9 x4, x& 5 = x x5, x& 6 = x x5, () x& 7 = x x7, x& 8 = x x8, x& 9 = x x9, x& = x x, The aeroelaic yem can finally be wrien in a ae-pace form a: { X} = [ A] { X} + [ G] { X} & (3) LI LI where [ A ] LI i a marix conaining linear erm ha are funcion of he equilibrium oluion and [ G] LI i a marix conaining only nonlinear erm. Fluer peed & frequency are defined by [3] a he lowe airpeed and correponding frequency a which a given rucure flying in a pecific amophere will exhibi uained imple harmonic ocillaion. Fluer condiion i a borderline iuaion or neural abiliy due o he fac ha mall moion mu be able a a peed below linear fluer peed, wherea divergen ocillaion occur in a range of peed above fluer peed. Linear fluer peed calculaion can be performed eing he marix [ G ] LI equal o zero. The abiliy of moion abou he equilibrium operaing condiion i deermined by he eigenvalue of he [ A ] LI marix. The reduced linear perurbaion yem originae a linear approximaion of he behaviour of he yem in he neighborhood of he aic equilibrium poin wih he poibiliy of calculaing a fluer peed for each rim condiion. Linear fluer peed can be compued auming he equilibrium aic configuraion a zero wih no coupling effec compuing eigenvalue of A(..., v =, w =, = ). LI Including equilibrium erm, non-linear fluer peed calculaion can be performed compuing eigenvalue of A(..., v, w, ) LI [9,,]. The yem repone i hen analyzed via ime marching inegraion by he non-linear Mahemaica olver. Linear inegraion, mainaining [ ] LI inegraion, [ ] LI G =, or non-linear G, can be carried ou o inveigae he po-fluer behaviour. 3 Reul and Dicuion 3. Linear and nonlinear Fluer To validae he propoed non-linear aeroelaic model, Pail and Hodge [3,4] and Tang and Dowell [5] model have been conidered. The preen reul are in very cloe agreemen wih [3-5]. High-apec-raio wing undergo large deflecion for relaively low aerodynamic loading a compared o lowapec-raio wing, and he naural frequencie of he high-apec-raio wing are quie low. Under aerodynamic loading he wing exhibi large aic deflecion, ha i, effecively i a curved wing, for which a non-linear coupling beween orion and lagging bending occur []. 5

6 G.ROMEO*,G.FRULLA*,E.CESTIO*,P.MARZOCCA** The claical linear fluer peed, obained by linearizing he non-linear model abou a zero deflecion eady ae, i calculaed fir. Throughou hi paper i will be referred o a VFLEC ha i: fluer velociy in linear equilibrium condiion. Linearizing he yem abou a deformed eady ae, he perurbed moion i influenced by he choen equilibrium poin. The reduced linear perurbaion yem generae a linear approximaion of he behavior of he yem in he neighborhood of he aic equilibrium poin, wih he poibiliy of calculaing a fluer peed for each condiion. In hi cae, he fluer peed i idenified hroughou hi paper a VFLEC ha i: fluer velociy in non-linear equilibrium condiion. To appreciae he imporance of he non-linear effec of aic deformaion, a preliminary analyi wa carried ou inroducing aic deformaion relaive o impoed load diribuion. Srucural and aeroelaic characeriic of he wing were inveigaed by linearizing he problem abou a non-linear aically deformed ae. In Figure he fir four mode obained by ASTRA linear and non-linear analyi are preened for he ame beam. The modal conen of he undefleced wing rucure a zero peed coni of ix mode: fir and hird mode (ω /ωr=., ω 3 /ω r =.386) are ypical flapping-bending mode, econd and fifh mode (ω /ω r =.983, ω 5 /ω r =.5836) are ypical lagging-bending mode, finally fourh and ixh mode (ω 4 /ω r =.35, ω 6 /ω r =.95) are wo orional mode. I i clear from boh, he heoreical and FE analyi, ha he variaion in frequency i moly due o lagging/orion coupling. For even higher deformaion here i furher decreae in he orional/lagging bending frequency, puhing he mode cloer o he fir bending mode. There i alo a furher decreae in he fluer peed, herefore he fluer characeriic of a defleced wing are very differen from hoe of a raigh wing due o he rucural geomeric nonlineariie. A ubequen fluer analyi ha been performed and preliminary fluer calculaion for a ypical lender wing configuraion are repored. The VFLEC ha been deermined fir and correpond o he abiliy characeriic of he unloaded, undeformed wing. Some preliminary numerical reul are preened in he following, referring o he wing daa repored in [3]. A VFLEC of abou.9 ha been obained howing a good correlaion wih reul repored in [3]. The aeroelaic model ha been han linearized abou nonlinear eady ae correponding o differen aic load condiion. Thi analyi give a more realiic predicion of he abiliy of he wing. Fig.. ASTRA and heoreical frequency conen. Fig. 3. Tip deflecion effec I ha been poined ou [3,8], ha he fluer inabiliy i grealy influenced by a aic ip diplacemen. Similar reul ha been obained for he cae under udy. Conidering, for example, he cae of a aic ip diplacemen raio w ip /w ref of.4, he VFLEC reduce o 6

7 O LIEAR AEROELASTIC BEHAVIOUR OF HIGLY FLEXIBLE HALE WIGS.5 and, for w ip /w ref of, VFLEC become lower han.4 howing a reducion of abou 47-54%. The VFLEC behavior a a funcion of he ip diplacemen raio i repored in Figure 3. Thi i an imporan fac o conider when analyzing he non-linear aeroelaic behavior of uch wing. Anoher deign indicaion derived from [8] and confirmed by reul publihed in [9] how a he iffne raio beween in-plane iffne EI ζ and ou-of-plane iffne EI η could play an imporan role in increaing he fluer boundarie; in [8,9] ha been poined ou he poiive effec of increaed fluer peed when an higher value of he iffne raio i inroduced in he airplane rucure deign. The aic wing deflecion, previouly conidered, wa no he real deflecion obained olving he equilibrium yem in Eq. (6a-c). For hi reaon a change in he calculaion procedure ha o be inroduced. A correc procedure o compue he criical fluer peed can be performed according o he cheme in Figure 4. A an example, if we conider ha a every peed a level fligh condiion ha o be aified, hen he fluer peed and frequency could be obained a follow: ) chooe a fligh peed and he correc angle-of-aac o aify he verical equilibrium condiion ) calculae he aic equilibrium deformed hape a he fligh peed, 3) linearize abou he deformed hape, 4) calculae he eigenvalue of he linearized yem, 5) chec for abiliy, if able, increae he fligh peed and repea all of he preceding ep unil inabiliy peed i reached. Fig. 4. Fluer peed calculaion A imilar procedure ha been uggeed by Pail [5] and applying hi approach, a higher VFLEC fluer peed could be obained. 3. Po-Fluer Behaviour The fluer reul obained in he previou ecion give he velociy a he one of fluer. Thee fluer reul imply ha mall diurbance will grow exponenially for velociie higher han he fluer peed. However, a he ampliude of ocillaion grow, addiional non-linear iffne i produced. Thu, he vibraion do no grow o infiniy bu inead converge o a limi cycle ocillaion (LCO). The ampliude of he LCO give an idea of he amoun of re/rain on he rucure herefore i udy could be ueful in analyi and deign. The ampliude of he LCO can only be deermined by ime-marching he non-linear governing equaion of he aeroelaic yem, while qualiaive udie of he characer of he fluer boundary, ha i he idenificaion of a able or unable LCO, can be done, for example, via a Lyapunov baed approach propoed in []. In hi paper, he effec of nonlineariie on he po-fluer behavior ha been inveigaed by conidering a mall iniial diurbance and by analyzing he repone via ime marching inegraion cheme uing he non-linear Mahemaica olver. To provide a beer underanding of how nonlineariie, incorporaed in he aeroelaic model, affec he repone of he yem, Figure 5 how he ime hiorie for he ip deflecion a a peed higher hen he fluer peed. The linear yem how he ypical unable behavior wih growing ampliude a ime unfold. I clearly appear ha he nonlineariie conrain he yem o a limi cycle. For he fully nonlinear yem LCO are preen above he non-linear fluer peed, a preened in Figure 6. Herein he in-plane and ou-of-plane wing ip deflecion reach a able LCO wihin 8 ec. Thi behavior i alo repreenaive of he orional deflecion. Conidering he effec of eady-ae yem configuraion in he fluer and po-fluer behavior, we have aumed a 7

8 G.ROMEO*,G.FRULLA*,E.CESTIO*,P.MARZOCCA** level fligh condiion ha i verified a every peed. Fig. 5. Time hiorie of linear and nonlinear yem A VFLEC of.6 wa compued conidering lagging, flapping and orional aic deformaion and a VFLEC of.86 wa deermined for he cae of an undeformed eady ae equilibrium. Afer he non-linear fluer velociy an LCO wa recorded. Thi clearly demonrae he imporance of incorporaing geomeric nonlineariie in he aeroelaic model becaue i no only affec he fluer peed, bu alo he po-fluer aeroelaic behavior. preliminary experimenal udy ha been conduced on a bala wood wing. Three apecraio wing have been eed boh below and above VFLEC and VFLEC fluer peed. A highlighed in [5], a hyerei phenomenon wa alo found during he e, confirming ha hee highly flexible wing exhibi a ubcriical, unable LCO behavior. Above he non-linear fluer peed, a able LCO occur. Thi LCO remain able and i ampliude increae a he peed increae (Fig ). In he decending peed phae, he LCO remain able even below he VFLEC fluer peed. In all he experimen a magneic enor, Vernier MB- BTA ( ample per econd), wa ued o record he variaion in magneic field produced by a rare-hear magne (low ma, high magneic field) aached o he wing (Fig 7). Magneic field i correlaed wih diplacemen, o i wa poible o record ip diplacemen and LCO po-fluer behavior. The wing were mouned, hrough a variable angle-of-aac uppor, verically ino he wind-unnel o overcome he effec of graviy loading (Fig 7). The main characeriic of he eed wing in erm of dimenionle aeroelaic parameer for he bala wing are repored in he Table. µ Ω h Ω Γ x α variable Table α r α a Fig. 6. Po-fluer behaviour 4. Experimen and umerical Reul In addiion o analyical and compuaional reul, experimen will be conduced on a high-apec-raio wing model wih a lender body a he ip in order o validae he heoreical model previouly decribed. A Experimenal e how a fluer peed raio of U flu /U ref =.54 in he increaing velociy phae and a fluer peed of abou.468 in he decreaing velociy phae. A aic deflecion of approximaely 65 mm a he fluer velociy wa recorded, confirming he heoreical predicion of he deformed equilibrium (Fig.7). Theoreical reul have hown he non-linear behavior of hee highapec-raio wing. A linear fluer peed VFLEC of abou.87 wa compued uing he dynamic perurbaion approach explained in he previou ecion (Fig.8a,b). The linear value i quie far from he fluer peed recorded during he experimen. Inroducing an impoed aic 8

9 O LIEAR AEROELASTIC BEHAVIOUR OF HIGLY FLEXIBLE HALE WIGS deflecion wih a ip diplacemen of 65mm, a non-linear fluer peed raio of approximaely.5 wa compued by he heoreical model (Fig a-b), howing a good correlaion wih he experimenal reul. (damping) veru he flow velociy. When he real par of he eigenvalue become poiive we can recognize he fluer peed for he wo cae of linear and non-linear equilibrium. Fig. 7. Bala wing aic deflecion Fig. 9a. Defleced (65mm) bala wing Fig. 8a. Undefleced bala wing Fig. 9b. Defleced (65mm) bala wing compuaion Fig. 8b. Undefleced bala wing compuaion Figure 8, 9, how a graphical repreenaion of he perurbaion equaion eigenvalue analyi, in he form of real eigenvalue Re(λi) A clear fluer and LCO repone ha alo been oberved in he preen wind unnel e. The LCO reul characerizing he po fluer behaviour how hyereic repone wih increaing and decreaing flow velociy. A hown in [5], hi phenomena i caued by boh rucural and aerodynamic nonlineariy due o eparaion a high angle-of-aac. The evoluion of LCO a peed higher han nonlinear fluer peed wa udied for differen peed and angle-of-aac. One of he meaured ime hiorie for he wing a an angle-of-aac of 8 degree i hown in Figure. A econd experimenal model i alo 9

10 G.ROMEO*,G.FRULLA*,E.CESTIO*,P.MARZOCCA** propoed. The advanced ereo-lihography model ha been conruced and will be eed in he low peed wind unnel a Claron Univeriy [9,]. Reul of hi advanced model will be preened elewhere. The wing i recangular, unwied, and flexible in flap, lag and orion. An exremely fine aerodynamic hape i obained. The wing i compoed by everal piece of ACA 5 rein par uppored by a meallic bar. Rein par are only bonded o he meallic bar leaving microgap beween he rein par in order o reduce orional iffne. ued in HALE wing. The effec of he defleced equilibrium configuraion, obained by nonlinear approximaion, i aen ino conideraion when olving he aeroelaic mall moion yem. The Galerin approach in reduced form i conidered a an analyical ool ued o perform he calculaion. Aerodynamic load are derived according o he Wagner funcion approach and are implified in hee preliminary calculaion. Po-fluer behaviour and LCO evoluion ha been udied uing a imemarching approach by mean of Mahemaica non-linear olver. In addiion o analyical and compuaional reul, a high-apec-raio wing model ha been eleced for a feaibiliy udy cae and for validaion of he heoreical model. Preliminary e have been conduced howing a good correlaion beween he heoreical model and he experimenal reul. More experimen are being conduced on an advanced ereo-lihography model. Fig. a. Bala wing ime hiorie and recorded LCO Fig. b. Bala wing ime hiorie and recorded LCO 5. Concluion High Aliude Long Endurance (HALE) aircraf aero-rucural ineracion are preened and defined. An advanced fluer calculaion i preened and applied o he aeroelaic performance of lender rucure, a Reference [] Hodge DH, Dowell EH. on-linear Equaion of Moion for he Elaic Bending and Torion of Twied on Uniform Roor Blade. ASA T D- 788, 974. [] Sedagha A, Cooper JE, Wrigh JR. Predicion of on Linear Aeroelaic Inabiliie. Proc. ICAS Congre, Harrogae UK, 8 Augu- Sepember,. [3] Pail MJ, Hodge DH. On he Imporance of Aerodynamic and Srucural Geomerical onlineariie in Aeroelaic Behavior of High-Apec- Raio Wing. Journal of Fluid and Srucure, Vol. 9, pp , 4. [4] Pail MJ, Hodge DH. Limi-Cycle Ocillaion in high-apec-raio wing. Journal of Fluid and Srucure, Vol. 5, pp. 7-3,. [5] Tang D, Dowell EH. Experimenal and Theoreical Sudy on Aeroelaic Repone of High-Apec-Raio Wing. AIAA Journal, Vol. 39, o. 8, pp ,. [6] Houbol JC, Broo GW. Differenial Equaion of Moion for Combined Flapwie Bending, Chordwie Bending, and Torion of Twied onuniform Roor Blade. ACA Repor 346. [7] Romeo G, Frulla G, Ceino E, Corino G. HELIPLAT: Deign, Aerodynamic, Srucural Analyi of Long-Endurance Solar-Powered

11 O LIEAR AEROELASTIC BEHAVIOUR OF HIGLY FLEXIBLE HALE WIGS Sraopheric Plaform. Journal of Aircraf, Vol. 4, o. 6, pp. 55-5, 4. [8] Frulla G. Aeroelaic Behavior of a Solar-Powered High-Aliude Long Endurance Unmanned Air Vehicle (HALE-UAV) Slender Wing. Journal of Aeropace Engineering Par G Special Iue, Vol. 8, pp.79-88, 4. [9] Ceino E. Deign of very long-endurance olar powered UAV, PhD Dieraion Poliecnico di Torino, Aeropace Dep., Torino, 6. [] Romeo G, Frulla G, Ceino E, Marzocca P, Tuzcu I. onlinear Aeroelaic Modeling and Experimen of Flexible Wing, Proc. 47 h AIAA/ASME/ASCE/AHS/ASC Srucure, Srucural Dynamic, Maerial Conference, ewpor RI, -4 May, 6. [] Librecu L, Chiocchia G, Marzocca P. Implicaion of Cubic Phyical/Aerodynamic on-lineariie on he Characer of he Fluer Inabiliy Boundary. Inernaional Journal of on-linear Mechanic, Vol. 38, pp , 3. [] Meirovich L. Fundamenal of Vibraion. McGraw Hill, ew Yor. [3] Biplinghoff RL, Ahley H and Halfman RL. Aeroelaiciy, Dover, ew Yor 996. [4] Hodge DH, Ormion RA. on-linear Equaion for Bending of Roaing Beam wih Applicaion o Linear Flap-Lag Sabiliy of Hingele Roor. ASA TM X-77, 973. [5] Crepo Da Silva MRM. on-linear Flexural- Flexural-Torional-Exenional Dynamic of Beam- I. Formulaion. Inernaional Journal of Solid and Srucure, Vol. 4, o., pp. 5-34, 988. [6] Tang DM, Dowell EH. Experimenal and Theoreical Sudy on Aeroelaic Repone of High-Apec-Raio Wing. AIAA Journal, Vol 39, o. 8, Augu. [7] Pail MJ, Hodge DH. Ceni CES. onlinear Aeroelaiciy and Fligh Dynamic of High-Aliude Long-Endurance Aircraf. Journal of Aircraf, Vol. 38, o., pp ,. [8] Timoheno S. Vibraion problem in engineering, Third Ediion, D.Van orand Company Inc, 935. [9] Lee BHK, Price SJ, Wing YS. onlinear aeroelaic analyi of airfoil: bifurcaion and chao. Progre in Aeropace Science, Vol. 3, pp , 999. [] Marzocca P, Librecu L, Chiocchia G. Aeroelaic Repone of a -D Lifing Surface o Gu and Arbirary Exploive Loading Signaure. Inernaional Journal of Impac Engineering, Vol. 5, o., pp. 4-65,. [] Dowell E, Edward J, Srganac T. onlinear Aeroelaiciy. Journal of Aircraf, Vol 4, o. 5, pp , 3. [] Jone RT, The uneady lif of a wing of finie apec raio. ACA Rep 68, 94.