FLIGHT DYNAMICS OF HIGHLY FLEXIBLE FLYING WINGS

FLIGHT DYNAMICS OF HIGHLY FLEXIBLE FLYING WINGS Mayuresh J. Pati 1 ad Dewey H. Hodges 1 Departmet of Aerospace ad Ocea Egieerig, Virgiia Poytechic Istitute ad State Uiversity, Backsburg, Virgiia 461-3 USA e-mai: mpati@vt.edu Schoo of Aerospace Egieerig, Georgia Istitute of Techoogy, Atata, Georgia 333-151 USA e-mai: dhodges@aoe.vt.edu Key words: Noiear Aeroeasticity, Fight Dyamics, HALE Aircraft Abstract: The paper presets a theory for fight dyamic aaysis of highy fexibe fyig wig cofiguratios. The aaysis takes ito accout arge aircraft motio couped with geometricay oiear structura deformatio subject oy to a restrictio to sma strai. A arge motio aerodyamic oads mode is itegrated ito the aaysis. The aaysis ca be used for compete aircraft aaysis icudig trim, stabiity aaysis iearized about the trimmed-state, ad oiear simuatio. Resuts are geerated for a typica highaspect-ratio fyig wig cofiguratio. The resuts idicate that the aircraft udergoes arge deformatio durig trim. The fight dyamic characteristics of the deformed aircraft are competey differet as compared to a rigid aircraft. For the exampe aircraft, the phugoid mode is ustabe ad the cassica short-period mode does ot exist. Furthermore, oiear fight simuatio of the aircraft idicates that the phugoid istabiity eads to catastrophic cosequeces. 1 Itroductio The aaysis ad desig of very ight, ad thus highy fexibe, fyig wig cofiguratios is of iterest for the deveopmet of the ext geeratio of high-atitude, og-edurace (HALE), umaed aeria vehices (UAV). The fexibiity of such aircraft eads to arge deformatio, so that iear theories are ot reevat for their aaysis. For exampe, the trim shape of a arge fexibe aircraft is highy depedet o the fight missio (payoad) as we as o the fight coditio; the deformed shape is sigificaty differet from the udeformed shape. Thus, the fight dyamic respose based o the actua trim shape ca be quite differet from that cacuated based o iear, sma deformatio assumptios. I fact, the desiger ca use the deformatio of the aircraft to positivey affect the fight stabiity ad cotro characteristics. The paper presets a theoretica basis for the fight dyamic respose estimatio of a highy fexibe fyig wig. Various reaistic desig space requiremets icudig, cocetrated payoad pods, mutipe egies, mutipe cotro surfaces, vertica surfaces, discrete dihedra, ad cotiuous pretwist, are take ito accout. The code based o the theoretica deveopmet preseted here ca be used i preimiary desig as we as i cotro sythesis. This work is a cotiuatio of work coducted by the authors over the past decade i the area of oiear aeroeasticity. 1, The focus of the preset work is o fight dyamics. 1

Theory The modeig of a fexibe aircraft udergoig arge deformatio requires a geometricayexact structura mode couped with a cosistet arge motio aerodyamic mode. The preset work is based o a fyig wig cocept ad is modeed structuray as a beam udergoig arge dispacemet ad rotatio. The goverig equatios are the geometricayexact equatios of motio from Hodges 3 writte i their itrisic form (i.e. without dispacemet ad rotatioa variabes). However, istead of beig augmeted by the dispacemet- ad rotatio-based kiematica reatios give therei, they are istead augmeted by the itrisic kiematica equatios, derived i Ref. 4 by eimiatig the dispacemet ad rotatioa variabes from the kiematica equatios of Ref. 3. A -D airfoi mode is appropriate for the very high-aspect-ratio wig (without fuseage iterferece) beig aayzed here. The airoads are here based o the fiite-state airoads mode preseted by Peters ad Johso. 5 Before presetig the structura ad aerodyamic theory used for the preset research, there is a eed to preset the supportig preimiaries, icudig the frames ad variabes. The ext sectio presets detais of the omecature, which are essetia to uderstad the theoretica detais preseted ater. y z x Figure 1: Axis system for the aircraft.1 Preimiaries The axis system is as show i Figure 1. The foowig frames of referece are used: i: iertia frame (-z poitig i the directio of gravity) b: udeformed beam frame (x aog the beam axis ad y poitig towards the frot of the airpae; y ad z are the axes i which the cross-sectioa stiffess ad iertia matrices are cacuated) B: deformed/movig beam frame a: aerodyamic frame i which the aerodyamic ift ad momet is defied (y ad z are defied i the airfoi frame with z poitig perpedicuar to the aerodyamic surface; y ad z are the axes i which the aerodyamic coefficiets are cacuated) The foowig rotatio matrices are used to trasform vectors i differet frames:

C a = C ba : trasfers a vector from the aerodyamic frame of referece to the udeformed beam frame of referece. Sice the aerodyamic frame is defied reative to the structura frame, C BA = C ba, where A is the aerodyamic frame of the deformed beam. C r : the rotatio matrix at a ode with a sope discotiuity that trasfers variabes from a referece frame o the right of the ode to oe o the eft of the ode. C = C Bb : determies the deformed beam cross-sectioa frame of referece. For the itrisic formuatio preseted here, C is ot required to sove the equatios but may be used i post-processig. The primary beam variabes are F : itera force measures i the B frame M: itera momet measures i the B frame V : veocity measures i the B frame Ω: aguar veocity measures i the B frame g: the measure umbers of the gravity vector i the B frame The secodary beam variabes are γ: itera strai measures κ: itera curvature measures P : iear mometum measures H: aguar mometum measures. Structura Mode The geometricay-exact, itrisic equatios for the dyamics of a geera, o-uiform, twisted, curved, aisotropic beam, udergoig arge deformatio, are give as F + ( k + κ)f + f = P + ΩP (1) M + ( k + κ)m + (ẽ 1 + γ)f + m = Ḣ + ΩH + Ṽ P () V + ( k + κ)v + (ẽ 1 + γ)ω = γ (3) Ω + ( k + κ)ω = κ (4) where k = k 1 k k 3 is the iitia twist/curvature of the beam, e 1 = 1 T, ad f ad m are the extera forces icudig gravity (f g, m g ), aerodyamic oads (f aero, m aero ), ad thrust (f T, m T ). The first two equatios i the above set are the equatios of motio 3 whie the atter two are the itrisic kiematica equatios 4 derived from the geeraized strai-dispacemet ad geeraized veocity-dispacemet equatios. 3

..1 Cross-sectioa costitutive aws The secodary beam variabes are ieary reated to the primary variabes by the crosssectioa costitutive aws (fexibiity ad iertia matrices), such that { } [ ] { } γ R S F = κ S T (5) T M { } [ ] {V } P µ µ ξ = H µ ξ I Ω where R, S, ad T are 3 3 matrices of cross-sectioa fexibiity coefficiets; ad µ, ξ, I are the mass per uit egth, mass ceter offset, ad mass momet of iertia per uit egth, respectivey. These reatios are derived based o the assumptios of sma strai ad sederess... Fiite-eemet discretizatio To sove the above set of equatios, the beam is discretized ito fiite eemets. The equatios for each eemet are obtaied by discretizig the differetia equatios such that eergy is coserved. 4 For exampe, cosider a variabe X. Let the oda vaues of the variabe after discretizatio be represeted by X ad X r, where the superscript deotes the ode umber, the subscript deotes the eft or right side of the ode, ad the hat deotes that it is oda vaue. The eed to defie two vaues of the variabe at a ode is carified i ater sectio. For the eemet (6) X = X = X = X +1 X r d X +1 + X r I a discretized form the equatios of motio ca be writte as (7) (8) F +1 F r d + ( κ + k )F + f P Ω P = (9) M +1 M r d + ( κ + k )M + (ẽ 1 + γ )F + m H Ω H Ṽ P = (1) V +1 d V r Ω +1 + ( κ + k )V + (ẽ 1 + γ )Ω γ = (11) Ω r d where, as defied above, the barred quatities correspod to the vaues of the variabes i the eemet iterior whie the hatted quatities are oda vaues. The barred ad hatted + ( κ + k )Ω κ = (1) 4

quatities of the primary variabes are reated as F = M = V = F +1 M +1 V +1 + F r + M r + V r Ω +1 Ω + = Ω r (16) The barred secodary variabes are reated to the barred primary variabes as stated above i the cross-sectioa costitutive aw...3 Gravity oads The force term i the equatios of motio icude gravitatioa forces. The gravitatioa force ad momet are (13) (14) (15) f g = µg (17) m g = µ ξg (18) where g is the gravity vector. The measure umbers of g are kow i the i-frame. The measure umbers of the gravity vector g i the B frame at a the odes ca be cacuated usig the foowig equatios: g + ( κ + k)g = which i the discretized form ca be writte as ĝ +1 ġ + Ωg = (19) ĝr + ( κ + d k )g = ĝ + Ωĝ () = The secod equatio above, the time-differetiated oe, is satisfied at oe ode; whie the first equatio, the spatiay-differetiated oe, is used to obtai the g vector at other odes. Both equatios are matrix equatios, i.e. a set of three scaar equatios. The three equatios together ca be show to satisfy a costrait of costat egth for the g vector. Oe ca thus repace ay oe of the equatios by the static form of this egth costrait. This wi remove the artificia root caused by the differetiatio of a costrait. Aso the costrait is satisfied for the steady-state cacuatio whe the dyamic terms are egected. So, the equatio ca be writte as (e 1 e T 1 + e e T ) ĝ g + (e 1 e T 1 + e e T ) Ω gĝ g + (e 3 e T 3 ) ĝ g = e 3 (1) For the case of symmetric fight at the cetra ode, Ω = Ω 3 =. Thus, the above equatios become ĝ 1 = ĝ 1 = () ĝ Ω 1 ĝ 3 = (3) (ĝ ) + (ĝ 3 ) = 1 (4) 5

From the secod equatio above it is cear that for steady-state, symmetric fight, Ω 1 = because ĝ 3 (wig vertica)...4 Egies, oda masses, vertica surfaces ad sope discotiuities Due to oda mass, oda force (thrust) ad sope discotiuities the force o oe side of the ode is differet from the force o the other side of the ode. Thus, F r ĈT r F + f T + µ ĝ r + f aero P r Ω r P r = (5) M r ĈT r M + m T + µ ξĝ r + m aero Ĥ r Ω r Ĥ r V r P r = (6) where f T is the discrete oda thrust force defied i the ( ) r referece frame, m T is the correspodig oda momet, µ is the cocetrated oda mass, ξ is the correspodig mass offset, f aero ad m aero are the aerodyamic forces due to vertica surfaces/pods, ad P r ad Ĥ r are the iear ad aguar mometa of the cocetrated moda mass, give by { } [ P r µ = µ ξ Ĥr µ ξ Î ] } { { V r Ω r Ĥ egie where Î is the mass momet of iertia matrix of the cocetrated mass ad Ĥ egie is the aguar mometum of the egie. A sope discotiuity i the beam wi aso chage a the other variabes, such that V } (7) = Ĉ V r r (8) Ω = Ĉ Ω r r (9) ĝ = Ĉ rĝ r (3) The ( ) r variabes ca be used to repace ( ) variabes for V, Ω ad g, thus reducig the umber of variabes...5 Fia structura equatios The deveopmet preseted above eads to the foowig primary equatios: M +1 M r d F r ĈT r F + f T + µ ĝ r + f aero P r Ω r P r = (31) M r M ĈT r + m T + µ ξĝ r + m aero Ĥ r Ω r Ĥr V r P r = (3) F +1 F r d + ( κ + k )F + f aero + µ g P Ω P = (33) +( κ + k )M +(ẽ 1 + γ )F +m aero+µ ξ g H Ω H Ṽ P = (34) Ĉ +1 r V r +1 V r d Ĉ +1 r Ω +1 r d + ( κ + k )V + (ẽ 1 + γ )Ω γ = (35) Ω r Ĉ +1 r ĝr +1 ĝr d + ( κ + k )Ω κ = (36) + ( κ + k )g = (37) 6

ad, the foowig secodary equatios: { } [ ] { γ R S } F = { } [ P r = Ĥr κ S T T M { } [ ] {V P µ H = µ } ξ µ ξ I Ω F = M = F +1 M +1 + F r + M r V = Ĉ+1 r V r +1 + V r Ω = Ĉ+1 r Ω +1 r + Ω r g = Ĉ+1 r ĝr +1 + ĝr } ] µ µ ξ { V µ ξ Î r Ω r { + Ĥ egie } (38) (39) (4) (41) (4) (43) (44) (45)..6 Boudary coditios The foowig are the boudary coditios for the probem: F 1 = (46) M 1 = (47) F r N+1 = (48) M r N+1 = (49) (e 1 e T 1 + e e T ) ĝ g + (e 1 e T 1 + e e T ) Ω gĝ g + (e 3 e T 3 ) ĝ g = (5) where N deotes the tota umber of eemets ad g gravity. deotes the referece ode for.3 Aerodyamic Mode The airoads are cacuated based o -D aerodyamics usig the kow airfoi parameters. First the veocities i the aerodyamic frame at the mid-chord are writte as V a = Ca T V ỹmcc a T Ω (51) Ω a = C T a Ω (5) where y mc is the vector from the beam referece axis to the mid-chord ad ca be writte i terms of the aerodyamic ceter (at the quarter chord) ocatio as y mc = ȳac b. The ift, drag ad pitchig momet at the quarter-chord are give by: L aero = ρb V T (C + C α α + C β β ) + ρb V T V a C α α rot cos α (53) 7

where D aero = ρb VT Cd + ρb V a C α α rot si α (54) ) M aero = ρb VT (C m + C mα α + C mβ β ρb VT Va C α αrot/ (55) T V VT = Va + Va 3 (56) α si α = V a 3 VT α rot = Ω a 1 b / VT ad, Va ad Va 3 are the measure umbers of V a. β is the fap defectio of the th eemet. The ift, drag ad pitchig momet are aerodyamic forces which ca be writte i the a-frame as: fa = L Va 3 aero D V a aero (59) V T L Va aero VT m a = D aero M aero Fiay, the forces derived above are trasformed to the B frame ad trasferred to the beam referece axis to give the appied aerodyamic forces as.3.1 Usteady Effects V T Va 3 VT (57) (58) (6) f aero = C a f a (61) m aero = C a m a + C a ỹ acf a (6) The above aerodyamic mode is a quasi-steady oe with either wake (ifow) effects or apparet mass effects. To add those effects ito the mode we have to firsty add the ifow λ ad acceeratio terms i the force ad momet equatio. Secody we have to icude a ifow mode that cacuates λ. Here the Peters -D ifow theory of Ref. 6 is used. The force ad momet expressios with the usteady aerodyamics effects are ad fa = ρb (C + C β β )V T V a 3 + C α (V a 3 + λ ) C d V T V a (C + C β β )VT V a C α V a3 b/ C α Va (Va 3 + λ Ω a 1 b /) Cd VT V a 3 8 (Cm + Cm β β )VT >< C m α VT V a 3 b (C α /8 + Cpitch /)V 9 a Ω a 1 b C Ω α a1 /3 + b C α V a3 /8 >= m a = ρb (64) >: >; (63) 8

ad The ifow mode ca be writte as: ( [A ifow ]{ λ V } + T b ) {λ } = ( V a 3 + b Ω a 1 ) {c ifow } (65) λ = 1 {b ifow} T {λ } (66) where λ is a vector of ifow states for the th eemet, ad [A ifow ], {c ifow }, {b ifow } are costat matrices derived i Ref. 6..4 Aeroeastic System A aeroeastic mode is obtaied by coupig the aerodyamic force defiitio give i the previous sectio with the set of equatios preseted i the sectio o the structura mode. The aeroeastic equatios are oiear equatios i terms of the primary variabes (F, Fr, M, M r, Vr, Ω r ad gr ). The set of aeroeastic equatios is soved usig Newto-Raphso method to obtai the steady-state (trim) soutio. The Jacobia cacuated for this soutio is eeded to assess the stabiity of the iearized system at the trim state..5 Trimmig The trim coditios are the same as steady-state coditios, i.e. a the time-derivatives are zero. If a the cotros are give, the the soutio gives the steady-state (trim) correspodig to that fight coditio. O the other had, more ofte it is desired to trim the fight at a specific trim state. To do so, the trim state i terms of the airspeed ad fight age (cimb/descet idicator) are prescribed ad the cotros (thrust ad fap age) are determied. Thus two additioa equatios ad two variabe (cotros) are appeded to the system. The two equatios are give beow. The equatios are sufficiet for a symmetric trimmed state of fight. For asymmetric trimmed fight, the equatios woud be modified ad additioa equatios reatig the radius of tur ad side-sip woud be added. The symmetric trim equatios are ĝ V + ĝ 3 V3 ta φ(ĝ 3 V ĝ V3 ) = (67) V + V 3 V = (68) where φ is the prescribed fight age ad V is the prescribed airspeed. The first equatio is derived from the fact that φ = θ α, where θ is the pitch age (ta θ = bg bg 3 ) ad α is the age of attack (ta α = b V 3 bv )..6 Post-processig ad Graphics The soutio for the itrisic equatios described above ca be used to determie ad pot the deformatio. The foowig equatios reate the strais (ad curvatures) to dispacemets (ad rotatios): r i = C ib e 1 (69) C bi = kc bi (7) (r i + u i ) = C ib (γ + e 1 ) (71) C Bi = ( κ + k)c Bi (7) 9

where r i is the positio vector of the beam axis from the origi of the referece frame i, ad u i is the deformatio i the i frame. The first two equatios determie the geometry of the udeformed wig. The udeformed beam axis coud be potted if r i is kow at a the oda ocatios, whie the deformed beam axis coud be potted if r i + u i is kow. To pot the wig surface oe eeds the vector defiig the positio vector of the poits o the wig surface from the beam axis. For pottig it is assumed that the cross sectio is rigid, the udeformed ad deformed surface ca be geerated by pottig r i + C ib ζ ad r i + u i + C ib ζ, respectivey, where ζ is the cross-sectioa positio vector. The discretized strai-dispacemet equatios are Ĉ bi+1 = ( d + k ) 1 ( d k ) Ĉ bi (73) r +1 i = ri + C ib e 1 d (74) ( Ĉ Bi+1 = d + κ ) 1 ( + k d κ ) + k Ĉ Bi (75) r +1 i + u +1 i = r i + u i + C ib (γ + e 1 )d (76) 8 ft 1 4 ft 6 ft 8 ft 4 ft Figure : Geometry of the aircraft 3 Exampe Cosider the aircraft as iustrated i Figure. The exampe aircraft has a spa of 38.78 ft ad a costat chord of 8 ft. 1/6 th of the spa at each ed has a dihedra of 1. The iertia, eastic ad aerodyamic properties of the wig cross sectio are give i Tabe 1. There are five propusive uits; oe at the mid-spa ad two each at 1/3 rd ad /3 rd semi-spa distace from the mid-spa. There are three vertica surfaces (pods) which act as the adig gear. Two of the pods weigh 5 b each ad are ocated at /3 rd semi-spa distace from the mid-spa. The cetra pod aso acts as a bay for payoad ad weighs betwee 6 b ( empty ) ad 56 b ( fu ). The pod/payoad weight is assumed to a be a poit mass hagig 3 ft uder the wig. The aerodyamic coefficiets for the pods are C α = 5 ad C d =.. 1

Eastic (referece) axis 5% chord Torsioa rigidity.4 1 6 b ft Bedig rigidity.5 1 6 b ft Bedig rigidity (chordwise) 3 1 6 b ft Mass per uit egth 6 bs/ft Ceter of gravity 5% chord Cetroida Mass Mom. Iertia: about x-axis (torsioa) 3 b ft about y-axis 5 b ft about z-axis 5 b ft Aerodyamics Coefficiets (5% chord): C α π C δ 1 C d.1 C m.5 C mδ -.5 Tabe 1: Wig cross-sectioa properties 3.1 Trim Resuts The fight dyamic aaysis of a fexibe aircraft begis with trim aaysis. The trim soutio ivoves the steady-state soutio of the compete oiear equatios. For a symmetric aircraft, oe coud cacuate the trim soutio (the airspeed ad rate of cimb) at a specified thrust ad fap defectio. Here we use a trimmig agorithm to cacuate the thrust (assumed costat for each of the five motors) ad fap defectio (assumed costat throughout the spa) to achieve a specified trim state. For most of the resuts i this paper, the trim soutio is cacuated for a eve fight coditio of 4 ft/s at sea-eve. Figure 3 shows the cotro vaues, the fight age of attack ad the structura deformatio as a fuctio of the payoad weight for the aircraft. The resuts for a rigid aircraft with the same cofiguratio are aso show. Figure 3(d) shows the trim shape of the aircraft. The give aircraft, for the empty cofiguratio, has a amost evey distributed mass (gravitatioa forces) baaced by aerodyamic forces ad thus the equivaet oads ad deformatio of the aircraft are sma. With the additio of the cocetrated payoad at the ceter we get sigificaty higher aerodyamic oads. Such oads ead to arge deformatio i the highy fexibe aircraft. The U shape of the fu cofiguratio has very differet structura as we as fight dyamic characteristics as wi be see throughout the rest of the paper. Figure 3(a) shows the thrust required for the specified trim coditio. The chage i required thrust is isigificat for payoad chages. This is because the primary source of drag for such aircraft is the profie drag ad the ski-frictio drag. This drag does ot chage with the aircraft weight. The iduced drag which is proportioa to the ift (ad thus aircraft gross weight) is mior for very high-aspect-ratio aircraft. The fap defectio required for trim is show i Figure 3(b). The fap defectio is used for pitch cotro of the aircraft. As the aircraft deforms to a U -shape, the ceter of gravity positio moves forward reative the aerodyamic ceter ad thus ower fap defectio is required to provide the pitch-dow momet The (root) age of attack at the trim coditio is show i Figure 3(c). The age of attack icreases with payoad as expected. The differece i the rigid ad fexibe age 11

thrust per motor (b) 1 8 6 4 rigid fexibe 1 3 4 5 payoad (b) fap defectio (deg) 1 8 6 4 rigid fexibe 1 3 4 5 payoad (b) (a) Trim thrust per motor required for specified trim (b) Trim fap defectio required for specified trim 5 age of attack (deg) 4 3 1 rigid fexibe 1 3 4 5 payoad (b) fu eastic rigid empty eastic (c) Trim age of attack at the trim coditio (d) Eastic deformatio of the aircraft at trim Figure 3: Trim resuts for a rigid as we as fexibe aircraft for a rage of payoad at 4 ft/s, sea-eve of attack comes from two sources, the aeroeastic deformatio ad the directio of the ift. The aircraft i its udeformed shape has egigibe aeroeastic coupig sice the aerodyamic ceter ad the shear ceter are coicidet. But as the aircraft deforms the drag o the wigs ead to twist-up momet at the root eadig to aeroeastic deformatio. Thus ower root age of attack is sufficiet to provide the required ift. O the other had, the ift geerated is perpedicuar to the airfoi, ad as the deformatio icreases the directio the ift acts departs more ad more from the vertica directio. Thus, for arge deformatio, because the aerodyamic vertica force is reduced, a arger age of attack is required to geerate the same amout of vertica force. empty payoad fu payoad Rigid mode Phugoid.16 ±.146 i.613 ±.535 i Short Period.84 ± 1.8 i 3.5 ± 1.63 i Fexibe mode Phugoid.18 ±.14 i +.147 ±.586 i Short Period.74 ± 1.76 i Tabe : Fight dyamics roots 1

3. Liear Stabiity Estimatio The aircraft as a fight dyamic/aeroeastic system is oiear. Oce the oiear trim is cacuated, oe coud the simuate the aircraft for various cotro settigs ad/or extera disturbaces usig the compete oiear equatios. It is prudet though to gauge the respose of the system by ivestigatig the iear dyamics of the aircraft about the trim coditio. This is achieved by iearizig the aircraft at the trim coditio. By aayzig the iearized system we ca gauge the stabiity as we as the respose for sma disturbaces. The iear symmetric fight dyamics performace of the aircraft is ormay discussed i terms of the phugoid ad short-period modes of the aircraft. The two modes are the oy symmetric modes of a rigid aircraft. It shoud be oted that the fexibe aircraft has a arge umber of modes which have fexibe as we as aircraft motio. For most covetioa aircraft oe woud sti be abe to separate the modes ito two fight dyamic modes which are domiated by aircraft motio ad the rest as fexibe modes domiated by structura deformatio. For the preset aircraft the ow frequecy fexibe modes of the aircraft are i the same frequecy rage as the fight dyamic modes ad thus there is strog coupig betwee the modes. Tabe presets the short period ad phugoid modes for the rigid ad fexibe aircraft at the empty ad fu payoad cofiguratio. The frequecy of the phugoid mode icreases i frequecy whie the dampig decreases with added payoad mass. For the fexibe aircraft the dampig crosses the imagiary axis ad the mode becomes ustabe for payoad above 6 b. The root ocus of the phugoid mode for the rage of payoad mass is show i Figure 4(a)..5.4 1.5 imagiary.3..1.1. rigid fexibe imagiary 1.5.5 rigid fexibe.3 1.4.5 1.5..1.1. rea (a) Phugoid mode 4 3.5 3.5 rea (b) Short-period mode Figure 4: Root ocus for a rigid as we as fexibe aircraft as a fuctio of payoad; the trim coditio is 4 ft/s at sea-eve (gree star represets empty cofiguratio whie gree circe represets fu ) The short-period mode root ocus is show i Figure 4(b). The short-period mode for the rigid aircraft does ot chage sigificaty with payoad. The fexibe aircraft 13

mode shows a drastic chage i eigevaues of the short-period mode. The root moves rapidy with added mass; ad, for payoad above 95 b, the pair of compex-cojugate short period roots merges to become two rea roots. This ca be expected because for icrease i payoad there is a correspodig icrease i the deformatio. The deformed U -shape eads to a order-of-magitude icrease i the pitch momet of iertia ad so there is correspodig decrease i the frequecy. Thus, this highy fexibe aircraft does ot show a cassica short-period mode i its deformed state. The phugoid mode of the aircraft ca ead to istabiity ad is thus ivestigated i detai. Figure 5 shows the ustabe phugoid mode shape for the fu cofiguratio. The phugoid mode shows the cassica coupig of pitch ad airspeed. The expected exchage of kietic ad potetia eergy is aso see i Figure 5(b). The aircraft oses potetia eergy as it oses atitude but gais kietic eergy (airfois are further apart). I the preset case, the strai eergy due to deformatio is aso ivoved. As see from Figure 5(d), the eastic deformatio is sigificat, but it is ot the domiat factor i the motio of the aircraft. The mode ca be ceary see by observig Figure 5(c). Here, the aircraft motio due to trimmed fight is removed so as to focus o the perturbatios about the trimmed fight coditio. The figure shows the cassica eiptica motio of the aircraft about its expected trim positio. Figure 6 shows the four fexibe modes with the owest eigevaues. Though some of these modes ca be said to be domiat i bedig, the others are couped modes with torsio, bedig (both directios), ad aircraft motio (pitch ad puge). I fact, oe of the modes deveops from the rea eigevaue of the short-period mode. 3.3 Noiear Simuatio Liear stabiity aaysis estimates the respose of the system to sma disturbaces. Liear aaysis does ot provide iformatio regardig respose to arge excitatio or the respose of a ustabe system. To estimate the arge deformatio respose of the system we have to sove the compete dyamic oiear equatios i time. I the preset we use a simpe, secod-order, cetra-differece, time marchig agorithm with high frequecy dampig. Figure 7 shows the oiear respose of fu aircraft at 4 ft/s. The aircraft simuatio is iitiated at the trim state. A extera disturbace is itroduced by addig a fap defectio. A maximum 5 fap defectio is added to trim fap defectio. The shape of the excitatio is a ramp up betwee 1 s ad s, ad a ramp dow betwee s ad 3 s. After 3 s the fap defectio is maitaied at the trim vaue. A time step of. secods is used for the simuatio. The simuatios for a time step of.1 secods as we as.5 secod are practicay the same. As expected the ustabe phugoid mode gets excited ad the ampitude of osciatio icreases. The exchage of potetia ad kietic eergy is see i Figure 7(a). Withi two-three cyces of the osciatios the aircraft starts experiecig very high ages of attack at the highest atitudes. Sice sta is ot modeed i the simuatio, the resuts after sta do ot resembe the rea aircraft motio. The motio of the aircraft is visuaized i Figure 7(b). It is obvious from this figure that oce dyamic sta is modeed oe woud see a sighty differet respose at the highest atitudes. The simuatio is robust eough to ru for a possibe arge aircraft motio as far as the reative structura deformatios are moderate. If the simuatio is ru for oger time the oe sees the aircraft perform pitch oops (Figure 7(c)). Durig the pitch oops, however, the age of attack stays sma; ad sta is ot expected. As expected from a 14

ustabe mode, 7 there is oss of eergy for a costat thrust ad aircraft oses atitude. 4 Cocusio A compete theoretica methodoogy for the aaysis of a highy fexibe fyig wig has bee preseted. The aaysis methodoogy is based o geometricay exact beam theory for eastic deformatio couped with arge motio airfoi aerodyamic theory. The rigid-body degrees of freedom are accouted by icudig the gravity vector i the formuatio. The equatios for the compete aaysis, icudig trim, iear stabiity ad oiear simuatio are itrisic, i.e. the equatios do ot require dispacemet ad rotatio variabes. The aaysis accouts for reaistic desig space requiremets icudig, cocetrated payoad pods, mutipe egies, mutipe cotro surfaces, vertica surfaces, discrete dihedra, ad cotiuous pre-twist. A exampe of a typica, fexibe, fyig-wig aircraft is preseted ad aayzed. The aircraft is very fexibe ad udergoes arge deformatio for fu payoad cofiguratio sice the payoad is cocetrated at the ceter (rather tha beig evey distributed over the wig). The trim shape ad the correspodig trim cotro requiremets are cacuated usig a trim agorithm. The trim shape as expected is a U shape. Due to the chage i the shape, the fight dyamic modes as we as the fexibe modes chage sigificaty. The cassica phugoid mode becomes ustabe with icrease i aircraft deformatio (trim). The cassica short-period mode does ot exist at trim because of the arge pitch momet of iertia of the deformed cofiguratio. Noiear simuatio of the aircraft cofirm that the ustabe phugoid mode ca be catastrophic for such aircraft. Refereces [1] M. J. Pati, D. H. Hodges, ad C. E. S. Cesik. Noiear aeroeastic aaysis of compete aircraft i subsoic fow. Joura of Aircraft, 37(5):753 76, Sep Oct. [] M. J. Pati ad D. H. Hodges. O the importace of aerodyamic ad structura geometrica oiearities i aeroeastic behavior of high-aspect-ratio wigs. Joura of Fuids ad Structures, 19(7):95 915, Aug. 4. [3] D. H. Hodges. A mixed variatioa formuatio based o exact itrisic equatios for dyamics of movig beams. Iteratioa Joura of Soids ad Structures, 6(11):153 173, 199. [4] D. H. Hodges. Geometricay exact, itrisic theory for dyamics of curved ad twisted aisotropic beams. AIAA Joura, 41(6):1131 1137, 3. [5] D. A. Peters ad M. J. Johso. Fiite-state airoads for deformabe airfois o fixed ad rotatig wigs. I Symposium o Aeroeasticity ad Fuid/Structure Iteractio, Proceedigs of the Witer Aua Meetig. ASME, November 6 11, 1994. [6] D. A. Peters, S. Karuamoorthy, ad W.-M. Cao. Fiite state iduced fow modes; part i: two-dimesioa thi airfoi. Joura of Aircraft, 3():313 3, Mar.-Apr. 1995. [7] M. J. Pati. From futterig wigs to fappig fight: The eergy coectio. Joura of Aircraft, 4:7 76, March 3. 15

(a) Iertia 3-D view of the aircraft (aircraft movig to the eft) (b) Iertia -D view of the mid-spa sectio (aircraft movig to the eft) (c) 3-D view of the aircraft for a observer movig at trim airspeed (aircraft motio is cockwise) (d) Eastic deformatio of the aircraft Figure 5: Phugoid mode of the fexibe cofiguratio with 5 b payoad 16

(a) First bedig (domiat):.8583 ±.157 i (b) Couped bedigtorsio-aircraft puge: 3.645 ±.9881 i (c) Secod bedig (domiat):.735 ± 5.186 i (d) Couped bedigtorsio-aircraft pitch: 1.376 ± 6.3915 i Figure 6: Fexibe modes of the aircraft 17

airspeed (ft/s) atitude (ft) age of attack (deg) 1 5 5 1 15 5 5 5 1 5 1 15 5 1 5 5 1 15 5 time (s) (a) Variatio i aircraft parameters (b) Iertia -D view of the mid-spa sectio: time -5s (aircraft movig to the eft) (c) Iertia -D view of the mid-spa sectio: time 5-5s (showig pitch oops) Figure 7: Noiear simuatio of aircraft respose to iitia fap excitatio 18