Aerodynamic Load Of An Aircraft With A Highly Elastic Wing

doi:10.14311/AP.2017.57.0272 Ac a Poly echnica 57(4):272–281, 2017 ©Czech Technical Uni e si y in P ague, 2017 a ailable online a h p://ojs.c u .cz/ojs/index.php/ap AERODYNAMIC LOAD OF AN AIRCRAFT WITH A HIGHLY ELASTIC WING Pa el Schoř Ins i u e o ae ospace Enginee ing, Facul y o Mechanical Enginee ing ,B no Uni e si y o Technology , Technicka 2896/2, 616 69 B no Czech Republic co espondence: [email p o ec ed] Abs ac . In his a icle, a me hod o calcula ion o ai loads o an ai c a wi h an elas ic wing is p esen ed. The me hod can p edic a edis ibu ion o ai loads when he elas ic wing de o ms. Unlike he adi ional Eule o Na ie -S okes CFD o FEM coupling, he me hod uses 3D panel me hod as a sou ce o ae odynamic da a. This makes he calcula ion easible on a ypical ecen wo ks a ion. Due o a sho compu a ional ime and low ha dwa e demands his me hod is sui able o bo h he p elimina y design s age and he load e alua ion s age. A case s udy is p esen ed. The s udy compa es a glide wing pe o ming a pull maneu e a bo h igid and and elas ic s a e. The s udy indica es a signi ican edis ibu ion o ai load a he elas ic case. Keywo ds: ae odynamics, panel me hod, ini e elemen me hod, wing load, luid s uc u e in e ac ion. 1. In oduc ion 1.1. P oblem o e iew A ligh o a mode n glide o a High Al i ude Long Endu ance (HALE) ai c a is a ep esen a i e exam- ple o he luid-s uc u e in e ac ion p oblem. These lying ehicles a e a esul o a sea ch o he mos ae odynamically e ec i e shape. This leads o a need o a wing wi h a la ge wingspan and an aspec a io. Mos ecen glide s ha e a wingspan anging om 18 o 30 me e s wi h an aspec a io app oxima ely om 30 o 50. A NASA Helios HALE ai c a had a wing wi h a wingspan o 75 m and aspec a io o 30 [ 1 ]. Du ing a ligh o such ai c a , he wing unde goes la ge de o ma ions, s ill in he elas ic egime, as shown on Figu e 1. The Helios HP-03 expe imen al HALE ai c a c ashed on June 26, 2003 a e encoun e ing a mo- sphe ic u bulence. The ai c a de eloped a wing Figu e 1. The Helios ai c a unde going la ge de o - ma ion [1]. ip displacemen o mo e han 12 m and consecu i e pi ch oscila ions, esul ing in s uc u al ailu e o he leading edge s uc u e, bu lea ing he main spa un- damaged. A e he c ash o Helios HP03 p o o ype, see Figu e 2, a demand ose o new me hods, which a e capable o compu e loads o highly elas ic, “mo - phing” ai c a s [ 1 ]. In his a icle, a me hod o a solu ion o quasi-s a ic load cases o such ai c a is p esen ed. Mo eo e , ecen ai wo hiness egula ions explici ly equi e o calcula e he load edis ibu ion in case “i de lec ions unde load would signi ican ly change he dis ibu ion o ex e nal o in e nal loads” [2, 3]. 1.2. Me hods o e iew Va ious me hods a e adi ionally used in o de o cap u e he change o he ae odynamic load gene a ed in igid and elas ic s a e. The simples op ion is o combine a o ex la ice me hod [ 4 ] wi h a simple beam ini e elemen model [ 5 ] . The mos sophis i- ca ed app oach consis s o Na ie -S okes CFD sol e Figu e 2. The Helios ai c a a e s uc u al ailu e du ing ligh [1] 272 ol. 57 no. 4/2017 Ae odynamic Load O An Ai c a Wi h A Highly Elas ic Wing o ae odynamic inpu combined wi h a de ailed FEM s uc u al model wi h solid elemen s [ 6 ]. The ae ody- namic load is hen di ec ly in e pola ed o he FEM model. Bo h o hese app oaches need o i e a e be- ween he ae odynamic and s uc u al model o ob ain he equilib ium. The simple model wi h he li ing line heo y p o ides he as es solu ion imes, how- e e , e o s may be in oduced because o an imp ope pi ching momen dis ibu ion due o he simpli ied VLM ae odynamic model. The de ailed model wi h he Eule o Na ie -S okes CFD sol e wi h a de ailed FEM s uc u al model can p o ide he mos accu a e p edic ion o he load edis ibu ion. Typically, he CFD olume mesh has o be upda ed acco ding o he esul s om he FEM solu ion. This may cause issues wi h he de o ma ion o he CFD olume mesh when he de o ma ion o he wing is la ge. Recen ly an Imme sed Bounda y Me hod is used o sol e his kind o luid-s uc u e in e ac ion [7] When assuming he cos s o he compu a ional ime o he CFD sol e and also assuming ha up o 10 i e a ions a e usually equi ed o ob ain he con e ged solu ion o one pa icula angle o a ack, he use o he CFD sol e is es ic ed o esend high pe - o mance compu e clus e s, he e o e making i e y expensi e and no easible o a ypical glide manu ac- u e nei he a a design s age no a a load e alua ion s age. In his a icle, a 3D panel me hod is used, combined wi h a geome ically nonlinea ini e elemen me hod wi h beam elemen s o model he s uc u e o he wing. The sa ings o compu a ional ime when compa ed o he CFD N-S sol e and he di e ences in compu ed ai loads jus i y he use o his low ideli y ae odynamic ool, making i possible o compu e la ge numbe o load cases on an a e age wo ks a ion. 1.3. Eligibili y Based on assump ions made in Sec ions 2.1 and 2.2, he me hod p esen ed in his a icle is sui able o analysis o quasi-s a ic maneu e s o an ai c a wi h elas ic wings, ope a ing a a low Mach numbe ( M < 0 . 3) and su icien ly high Reynolds numbe ( Re > 1000000). This includes he HALE ai c a du ing i s low al i ude ligh o a glide pe o ming a pull maneu e . 2. Me hods 2.1. Ae odynamic model A luid mo ion is adi ionally desc ibed by se o equa- ions called Na ie -S okes equa ions [ 8 ]. The numbe o he exac solu ion o Na ie -S okes equa ions is small. Fo p ac ical p oblems, hese equa ions a e sol ed nume ically [9]. Sol ing he Na ie -S okes equa ions p o ides a de- ailed desc ip ion o he low, howe e he compu a- ional cos s a e eno mous. The e o e some assump- ions a e made o educe he complexi y o he p ob- lem. I he low is assumed o be s eady, in iscid, Figu e 3. Po en ial low domain. incomp essible and i o a ional, is called a po en i al low and i can be desc ibed by Laplace’s equa ion [ 8 ] : ∇2Φ=0 (1) Whe e Φis a scala unc ion called eloci y po en ial and eloci y −→ q a each poin can be ob ained by i s de i a ion: −→ q=∇Φ(2) An app oach om [ 10 ] is used o sol e he Laplace’s equa ion. Assume a c oss sec ion o a wing as shown in Figu e 3. The su ace S∞ encloses he p oblem a in ini y, su ace S ep esen s a body (wing) and su ace W ep esen s i s wake. The su ace S + W di ides he domain in o wo egions: he ex e nal egion wi h he low ield o in e es and eloci y po en ial Φand he in e nal egion wi h he ic i ious low and eloci y po en ial Φ i . The su aces a e modeled by a double and sou ce singula i ies. G een’s heo em is applied o bo h he ou e and inne egion and eloci y po en ial Φ P is ob ained by combining bo h exp essions. [ 10 ] The eloci y po en- ial Φ P gi es a eloci y po en ial anywhe e in he wo egions. I is exp essed in e ms o su ace in eg als in e ms o eloci y po en ial and i s no mal de i a i e o e he bounda y su ace as ΦP=1 4πZZS+W+S∞ (Φ −Φi)−→ n· ∇ 1 dS −1 4πZZS+W+S∞ 1 −→ n·(∇Φ− ∇Φi) dS. (3) Whe e is he dis ance om he poin P o he elemen d S on he su ace, and −→ n is a uni no mal ec o o he elemen d S . [ 10 ] The i s in eg al in (3) ep esen s he dis u bance po en ial om a su ace 273 Pa el Schoř Ac a Poly echnica dis ibu ion o double s wi h a densi y o (Φ − Φ ∞ ). The second in eg al ep esen s he con ibu ion o sou ces wi h a densi y o −−→ n·(∇Φ− ∇Φi).[10] Nex , he in e nal Di ichle bounda y condi ion is in oduced. This bounda y condi ions se s he In e nal low equal o he Onse low by Φi= Φ∞,(4) ΦP= Φ∞.(5) Following he p ocedu e om [ 8 , 10 , 11 ], he su ace is disc e ized in o n su ace panels and nw panels in he wake egion wi h a cons an dis ibu ion o singu- la i ies. The (3) is also disc e ized and he Di ichle bounda y condi ion is e alua ed a cen oid o each su ace panel. These poin s a e called Colloca ion poin s. The esul o his p ocedu e is a linea equa- ion [8] n X k=1 Ckµk+ nw X l=1 Clµl+ n X k=1 Bkσk= 0,(6) whe e Ck , Cl a e double in luence coe icien s o body and wake panels, Bk is he sou ce in luence coe icien , µk is he double s eng h o a body panel, µl is he double s eng h o a wake panel, σk is he sou ce s eng h o a body panel. The σkis se as σk=−→ nk·−−→ Q∞,(7) whe e −→ nk is he panel no mal uni ec o and −−→ Q∞ is he ees eam eloci y. The me hod desc ibed abo e is known as he “ i s o de panel me hod” [ 8 ]. I is used as he only sou ce o ae odynamic da a o he luid-s uc u e in e ac- ion. The o mula ion o he panel me hod is iden- ical o well-known and alida ed codes such as he VSAERO [10] and PMARC [11]. Bounda y laye . The po en ial low may gi e use- ul esul s o some applica ions, howe e o applica- ions whe e iscous e ec become mo e impo an , he esul s ob ained unde assump ion o po en ial low may become inaccu a e in e ms o o e p edic ed li cu e slope. A ypical example o a low wi h a signi - ican iscous e ec is a glide wing a a low Reynolds numbe o a low o e a mul i-elemen wing. The iscous e ec s can be in oduced in o he po en- ial low model by displacing he ac ual body su ace by a displacemen hickness o he bounda y laye [ 12 ]. The ac ual displacing o he body su ace is pe - o med by a modi ica ion o panels sou ce s eng h in (7) o σk=−→ nk·−−→ Q∞+d ds(ueδ∗),(8) whe e sis a dis ance on he body, ueis a panel edge eloci y and δ∗ is a displacemen hickness compu ed by he bounda y laye analysis [12]. 2.2. S uc u al model The s uc u e o he wing is modeled using he FEM wi h beam elemen s only. Each node o he elemen has six deg ees o eedom. The s i ness p ope ies a e assumed cons an o each elemen . The s i ness ma ix o such elemen can be ound in many FEM ela ed ex books [ 13 ]. Howe e such elemen s a e usually de i ed o iso opic ma e ials. In ac , wings o mode n glide s a e made o a ca bon ibe ein- o ced composi e o ho opic ma e ial. A wing made o such ma e ial exhibi s coupling be ween bending and o sion. This causes issues, when one ies o educe s i ness p ope ies o a composi e wing in o single s i ness ma ix o a beam elemen . A me hod om [14] was adop ed o sol e his issue. As he wing unde goes la ge de o ma ions i is nec- essa y o cap u e he geome ical non linea i y. To sol e his issue a New on/Rhapson i e a i e me hod is used. The geome ical non-linea i y, howe e , causes di icul ies when calcula ing elemen s s i ness ma i- ces — one mus de e mine he de i a ion o he global s i ness ma ix ( angen s i ness ma ix). A me hod om [ 15 ] was adop ed, which p o ides a nume ically gene a ed angen s i ness ma ix o uss and beam elemen s. 2.3. Ae o-s uc u al in e ac ion Wo k low. A b ie o e iew o he in e ac ion be- ween he ae odynamic and in e nal o ces in he wing s uc u e is shown on Figu e 4. The in e ac ion is done by using a modi ied New on-Rhapson me hod. This me hod and s eps called he “Compu e ae ody- namic o ces” and “Compu e wing displacemen ” om Figu e 4 a e explained in de ail in he pa ag aph Load con ol scheme. Compu ing he igid li cu e as a i s s ep is no necessa y, he main eason o his is o ind he ze o li angle o he a ack α0 , whe e he compu a ion o elas ic li cu e s a s. S a ing om α0 makes he wing almos unloaded and he New on-Rhapson me hod is likely o con e ge. Also, he inc emen o he angle o a ack ∆ α should be small enough o assu e he con e gence, i is usually om 0.5° o 2°. In e ac ing elemen . The in e ac ing elemen is explained in Figu e 5. I consis s o ae odynamic panels, bu he e is only one elemen in he span-wise di ec ion and one FEM beam elemen . Fo ces and momen s a e always in eg a ed om he wing ip o plane o symme y, a a poin which is also a node o he ini e elemen model. The load applied a he FEM node is a sum o o ces momen s om all panels belonging o he in e ac ing elemen as shown on Figu e 5: −→ = n X i=1 −→ i,(9) −→ m= n X i=1 −→ i× i,(10) 274 ol. 57 no. 4/2017 Ae odynamic Load O An Ai c a Wi h A Highly Elas ic Wing Figu e 4. Ae o-s uc u al in e ac ion — wo k low. whe e i is he o ce on i h panel in he global coo - dina e sys em x, y, z , as shown in Figu e 5; i is he ec o be ween he e e ence node and con ol poin o i h panel. The esul o he FEM analysis is a displacemen ec o u a each node. Coo dina es o ae odynamic panels a e upda ed a each load s ep using nodal ansla ional and o a ional displacemen s, acco ding o (11). Also, he nodal coo dina es a e upda ed as explained in [15]. PTSnew =Rx(∆φx)Ry(∆φy)Rz(∆φz)PTSold + ∆U, (11) whe e ∆ φx ,∆ φy ,∆ φz, a e o a ional inc emen s in he global coo dina e sys em calcula ed om he las s ep o he New on-Rhapson i e a ion, see bellow; ∆ U is he displacemen inc emen ; Rx ( φx ) , Ry ( φy ) , Rz ( φz ) a e o a ional ma ices de ined by Rx(∆φx) =   1 0 0 0 cos φx−sin φx 0 sin φxcos φx  ,(12) Ry(∆φy) =   cos φy0 sin φy 0 1 0 −sin φy0 cos φy  ,(13) Rz(∆φz) =   cos φz−sin φz0 sin φzcos φz0 0 0 1  .(14) Figu e 5. Panel me hod — FEM in e ac ion. Load con ol scheme. In he New on/Rhapson me hod, he o al load is di ided in o m load s eps and o each load s ep, a New on i e a ion is done un il an equilib ium be ween he in e nal o ces and ex e nal load is ound. A e ha , he ex e nal load is inc eased by he load s ep. Howe e , in he case o an elas ic wing, he o al ex e nal load is unknown because he load (li ) also depends on how much he wing is de o med. The e o e, he s anda d load con ol in he New on/Rhapson me hod was changed in he ollowing way: The load is con olled by in- c easing he angle o a ack as explained in Figu e 4. Usually he li gene a ed by he wing (sailplane) is he dependen a iable in he u – F cu e. To compu e he load gene a ed by he elas ic wing, a ollowing scheme is used: (1.) The i s i e a ion s a s a an angle o a ack o ze o li o a igid wing. The equilib ium be ween he ex e nal load compu ed om he panel me hod and in e nal o ces is compu ed using he New on i e a ion. (2.) The angle o a ack is inc eased by a small s ep, e.g., 1 ° and he equilib ium be ween he ae ody- namic load and he in e nal o ces is compu ed by he New on i e a ion. (3.) The angle o a ack is u he inc eased un il he o al load (li ) is equal o he desi ed load (li ). The esul is a nonlinea li cu e. Fo each load s ep, he ec o o in e nal o ces is also a nonlinea unc ion o p e ious load s eps. The e o e when com- pu ing a load o an elas ic wing i is con enien o compu e a wing li cu e as desc ibed abo e wi h e y small inc emen o he angle o a ack, e.g., 0 . 5 ° and o sa e he ec o o de o ma ion and in e nal o ces o each con e ged load s ep. When compu - ing la ge numbe o load cases, signi ican sa ings in he compu a ional ime a e achie ed when he p e-compu ed de o ma ion ec o and in e nal o ces ec o a e loaded o he closes lowe angle o a - ack. One hen needs o compu e only he di e ence be ween he p e-compu ed angle o a ack and he desi ed angle o a ack. 275 Pa el Schoř Ac a Poly echnica Figu e 6. Ae o-s uc u al in e ac ion — load con ol in New on-Rhapson me hod. Figu e 7. Coo dina e sys ems. 3. Nume ical s udy 3.1. O e iew A nume ical s udy was conduc ed o demons a e he impo ance o he elas ici y o he wing o i s load. Assuming a ligh en elope om [ 2 , pa . CS22.333], and conside ing he poin A o he en elope o be c i - ical, he ae odynamic load a his poin is e alua ed using h ee di e en me hods: (1.) panel me hod wi h elas ic wing; (2.) panel me hod wi h igid wing; (3.) li ing line heo y. The li ing line heo y is used as a alida ion ool o he panel me hod. 3.2. Coo dina e sys ems A global igh -handed coo dina e sys em OXY Z de- ined in Figu e 7 is used. Addi ional local coo dina e sys ems oixiyizi a e used o e alua ion o he local load. As he wing de o ms, hese coo dina e sys ems P ope y Symbol Value A ea A 11.55 m2 Span L 21.0 m Roo cho d c 0.7 m Tip cho d c 0.4 m Dihed al Γ0 ad Sweep Θ0 ad Table 1. Wing geome y. P ope y Symbol Uni C oss sec ion a ea ASm2 Second momen o a ea Ixm4 Second momen o a ea Izm4 To sional cons an Jym4 O se o neu al axis ∆Xnm O se o neu al axis ∆Znm O se o elas ic axis ∆X m O se o elas ic axis ∆Z m Young modulus EN/m2 Shea modulus GN/m2 Table 2. Wing c oss sec ional p ope ies. ollow he de o med geome y. The ee s eam eloc- i y V wi h posi i e angle o a ack is also shown in Figu e 7. 3.3. Wing specimen A simple ape ed wing is examined in his s udy. The wing is designed as a plana wing in unde o med s a e. The wing has one s aigh main spa loca ed a 0 . 25 c , whe e c is local cho d. The main eason o his geome y is he e alua ion o he load by he li ing line heo y, which can no be used o a non-plana wing geome y and no swep wings. 276 ol. 57 no. 4/2017 Ae odynamic Load O An Ai c a Wi h A Highly Elas ic Wing Figu e 8. Spanwise a ia ion o Iz,Ix,Jy. Figu e 9. Ai oil E603 in iscid ae odynamic da a. Geome y. Geome y o he wing is desc ibed in Table 1. An Epple E603 ai oil is used o all sec ions o he wing. S i ness p ope ies. When analyzing he de o - ma ion o he wing by he FEM analysis desc ibed abo e, an ins umen al s ep is he de e mining o all c oss-sec ional cha ac e is ics de ined in Table 2 o all c oss sec ions o he wing. In his example, he Young’s modulus E and shea modulus G a e assumed as cons an o he whole wing wi h ollowing alues: E= 60 GPa, G = 7 GPa. The wing has only one spa a 0 . 25 c and he nodes a e coinciden wi h he spa . The e o e he o se o he elas ic axis is almos negligible and bo h alues a e assumed o be ze o. The neu al axis is loca ed a he local cen oid wi h non-ze o o se s, bu hese o se s a e no shown in his pape . The spanwise a ia ion o he Second momen s o a ea and o sional cons an is shown on Figu e 8. Fini e elemen model. The wing is di ided in o 162 spanwise sec ions. Each sec ion has one wo- noded beam elemen desc ibed in Sec ion 2. Nodal coo dina es always co espond o a loca ion o a main spa . The node loca ed in he plane o symme y has all deg ees o eedom p esc ibed o ze o. All coupling e ms in he elemen s i ness ma ices ela ed o he bending- o sion coupling a e in en ionally se o ze o o cla i y. Ai oil da a. The ae odynamic cha ac e is ics o he Epple E603 ai oil we e ob ained om he XFOIL in iscid analysis, as shown in Figu e 9. This ae ody- namic da a a e used only o he li ing line analysis. 3.4. Loadcase VA Assuming a maximum li coe icien o he wing CLmax = 1 . 4, om Figu e 10, he ligh condi ions a poin A o he maneu e ing en elope a e shown in Ta- ble 3 The o al mass o he glide is MTOW = 750 kg, bu ine ia o ces ac ing on he wing a e no assumed in he analysis o cla i y. 277 Pa el Schoř Ac a Poly echnica Figu e 10. Compa ison o igid and elas ic wing — li . P ope y Symbol Value Ai speed V64.0 m/s Wing li coe icien CL 1.4 Load ac o n5.3 Table 3. Load case VA. De o ma ion sequence. De o ma ion sequence o a ious wing li coe icien is shown in Figu e 11. I mus be no ed ha his de o ma ion sequence is alid only o ai speed V = 64 . 0m/s. As he de o ma ion o he wing depends on o ces and momen s applied on he wing su ace, o a di e en ai speed, he co - esponding de o med shape will be di e en . In eg al ae odynamic cha ac e is ics. In eg al ae odynamic cha ac e is ics o he igid and elas ic wing a e compa ed on Figu es 10 and 12. A close ag eemen o igid cases can be obse ed. Fo he elas ic case a signi ican dec ease o he slope o he wing li cu e can be seen in Figu e 10. Addi ionally, he inc ease o pi ching momen o he elas ic wing can be seen in Figu e 12. The di e - ences be ween igid and elas ic cases a e discussed in Sec ion 4 Load compa ison. Ac ual compa ison o he load o he igid and elas ic wing pe o ming a s eady ma- neu e wi h he same load ac o n = 5 . 3is shown in Figu es 13–15. Only bending and o sional momen s a e shown. The di e ences be ween he igid and elas ic case a e discussed in Sec ion 4 4. Discussion 4.1. Valida ion o esul s The e a e wo main op ions o alida e he esul compu ed by he p esen ed me hod: Figu e 11. De o ma ion sequence a VA. Nume ical alida ion. A nume ical alida ion means o p e o m he case s udy using a RANS CFD sol e as a sou ce o ae odynamic da a coupled o a wing modeled by shell elemen s o by 3D solid ele- men s. Expe imen al alida ion. Expe imen al alida- ion means o ac ually ly a glide , pe o m he pull maneu e and measu e he displacemen o he wing. Conclusion on alida ion. Un o una ely bo h nume ical and expe imen al alida ion would equi e much mo e e o and unds han wha was spen o his whole wo k. A li e a u e e iew also did no gi e any example, which could be used as a alida ion benchma k. The e o e, his wo k is p esen ed only as a nume ical s udy wi hou any o he alida ion. 4.2. Dec ease o wing li cu e slope As he li gene a ed by he elas ic wing inc eases, he wing ben s mo e, as shown in Figu e 11. As he wing ben s mo e, he slope o he li ing cu e d ops. This is demons a ed in Figu e 10, which shows a educed slope o he li cu e o he elas ic wing, compa ed o he igid wing. This phenomena could possibly be explained by he educ ion o he p ojec ed a ea o 278 ol. 57 no. 4/2017 Ae odynamic Load O An Ai c a Wi h A Highly Elas ic Wing Figu e 12. Compa ison o igid and elas ic wing — pi ching momen . Figu e 13. Compa ison o igid and elas ic wing — No mal bending momen Mx. he wing and he change o di ec ion o he li o ce. This has ollowing consequences: (1.) When he p esc ibed o al li is eached, he wing ope a es a a highe angle o a ack. In he pa icula case p esen ed in Sec ion 3 he di e ence o he c i ical angle o a ack is 6°. (2.) The no mal bending momen is sligh ly educed, see Figu e 13. The angen ial bending momen is, howe e , inc eased due o he inc eased angle o a - ack, see Figu e 15. In he pa icula case p esen ed in Sec ion 3 he angen ial bending momen o he elas ic wing is 1 . 34 imes highe , han in he igid case. The no mal bending momen is, howe e , 0 . 87 imes lowe . 4.3. Inc ease o wing pi ching momen The explana ion o he inc ease o he wing pi ching momen equi es mo e labo han he explana ion o he dec ease o he wing li cu e slope. Fi s , assume a wing c oss sec ion loca ed a he wing ip. The wing is o an unde o med shape and ope a es nea he c i ical angle o a ack. The sec ion gene a es a pi ching momen a ound he main spa MY 1 = 1 2ρ 2S1c1cm1 ( α ). The esul an o he ae odynamic o ce is assumed o ac a he main spa (0 . 25 c ). The di ec ion o he o ce is upwa d and o wa d (nega i e X , posi i e Z ). When he wing has ze o dihed al and ze o sweep, he con ibu ion o his o ce o he pi ching momen is ze o, since he ec o o he sec ion e e ence poin ela ed o he wing e e ence poin ~ 1 = [0 ,− 10 . 37 , 0]. The wing e e ence poin is de ined as a poin (node) on he main spa in he XZ plane. Nex , assume a highly de o med wing shown in Figu e 16. The wings ip sec ion is displaced by a ec o ~ d1= [−0.12,2.5,5.15], he ec o o he sec ion e e ence poin ela ed o he wing e e ence poin is now ~ 1= [−0.3,−8,5.15]. The esul an o he ae odynamic o ce a he wing 279 Pa el Schoř Ac a Poly echnica Figu e 14. Compa ison o igid and elas ic wing — To sional momen My. Figu e 15. Compa ison o igid and elas ic wing — Tangen ial bending momen Mz. e e ence poin is compu ed as ~ 1= [−6.07,49.38,9.31], hen, he esul ing momen a he wing e e ence poin gene a ed by he o ce ~ 1is ~m1=~ 1×~ 1= [−329.05,−28.51,−63.32]. When he pi ching momen o he sec ion is e alu- a ed, he con ibu ion o ∆ MY 1 = − 28 . 51 Nm causes o inc ease he pi ching momen coe icien o he sec- ion by a alue o − 0 . 55 mos ly due o he e ical displacemen o he sec ion and he o ce esul an di ec ing o wa d. 4.4. Wing maximum li In his a icle, he maximum li coe icien o he igid wing was compu ed using he nonlinea li ing line heo y as CLmax = 1 . 4. The ai oil sec ional da a used in he analysis we e compu ed by he XFOIL in iscid analysis, assuming o be close o he expe imen al da a. Since he pu pose o his a icle is o p esen a Figu e 16. Pi ching momen o highly de o med wing. po en ial p oblems o an ai c a wi h a highly elas ic wing, a he han o make a p esc ip ion based on p ac ical esul s, he maximum li coe icien o he elas ic wing is also assumed o be CLmax = 1.4. 280