Actuator-Work Concepts Applied to Unconventional Aerodynamic Control Devices

Transcription

1 Atutor-Work onepts Applied to Unonventionl Aerodynmi ontrol Devies hristopher O. Johnston *, Willim H. Mson, heolheui Hn, Hrry H. Roertshw nd Dniel J. Inmn ** Virgini Teh, Blksurg, VA, 46 This pper investigtes the resistne to hnge in wing shpe due to the erodynmi fores. In prtiulr, the work required y n irfoil to overome the erodynmi fores nd produe hnge in lift is emined. The reltionship etween this work nd the totl erodynmi energy lne is shown to hve signifint onsequenes for trnsient hnges in irfoil shpe. Speifition of the plement of the tutors nd the tutor energetis is shown to e required for the determintion of the irfoil shpe hnge requiring minimum energy input. A generl, simplified tutor model is dopted in this study whih ssigns different vlues of tutor effiieny for negtive nd positive power output. Unstedy thin irfoil theory is used to nlytilly determine the pressure distriution nd erodynmi oeffiients s funtion of time for rmp input of ontrol defletion. This llows the required power nd work to overome the erodynmi fores to e determined for presried hnge in the irfoil merline. The energy required for pithing flt plte, onventionl flp, onforml flp, nd two vrile mer onfigurtions is investigted. For the pithing flt plte, the minimum energy pithing is is shown to e dependent on the pith rte nd the initil ngle of ttk. The onforml flp is shown to require less tutor energy thn the onventionl flp to overome the erodynmi fores for presried hnge in lift. The energy requirements of vrile mer onfigurtion re shown to e sensitive to the lyout of the vrile mer devie. Nomenlture A n, Fourier oeffiients defined in Eq. (3.6, n,,.., nd is the sme s defined for T, hord length L,n lift oeffiient, n,, nd orrespond to the qusi-stedy, pprent mss, nd wke-effet terms M,n qurter-hord pithing moment oeffiient, n represents the terms defined with L P power oeffiient for the power required to overome the erodynmi fores (P P power oeffiient for the required power input to the tutor (P, (relted to P in Eq. (4.4 W energy oeffiient for the input energy required y n tutor (W D drg (the rred quntity represents the time-verge E energy dissipted to the wke per-unit time (the rred quntity represents the time-verge k rtio of the initil lift to the hnge in lift defined in Eqs. (5.7 nd (6. P power required to overome the erodynmi fores (the rred quntity represents the time-verge P required power input to the tutor (relted to P in Eq. (.7 Q n defined in Eqs. ( , where n,,...,5 q dynmi pressure t time t the time t whih P is zero (s shown in Figure.3 t * time t the end of the unstedy motion T, omponents of the erodynmi lod distriution defined in Eqs. (3. nd (3.7, nd orresponding to the qusi-stedy nd pprent mss terms, nd s nd d orresponding to the * Grdute Reserh Assistnt, Deprtment of Aerospe nd Oen Engineering, Student Memer AIAA Professor, Deprtment of Aerospe nd Oen Engineering, Assoite Fellow AIAA Visiting Sholr, Memer AIAA Professor, Deprtment of Mehnil Engineering ** George R. Goodson Professor, Deprtment of Mehnil Engineering, Fellow AIAA

2 omponents resulting from the stedy nd dmping oundry ondition defined in Eq. (3.3 nd (3.4 K, omponents of the lift oeffiient defined in Eqs. (3.7 nd (3.3, the susripts re defined for T, U free-strem veloity w indued veloity on the irfoil merline W work required to overome the erodynmi fores W required energy input to the tutor (relted to W in Eq. (.8 distne long the irfoil hord ligned with the free-strem veloity pithing is α ngle of ttk defines the time-history of the merline shpe hnge χ lod distriution funtion defined in Eq. (3. δ Dir delt funtion p unstedy pressure loding γ qusi-stedy vortiity distriution η tutor oeffiient defined in Eq. (.7 θ polr oordinte long the irfoil surfe whih is relted to in Eq. (3.6 τ nondimensionl time defined in Eq. (3. τ the vlue of τ t whih P (or P is zero (nondimensionl equivlent to t defined in Figure.3 τ nondimensionl time whih defines the end of the rmp input in Figure 4. ψ shpe funtion of the irfoil merline defined in Eq. (3. I. Introdution REENT interest in morphing irrft, hs initited reserh onerning the hrteristis of unonventionl erodynmi ontrol devies. These unonventionl, or morphing, devies re ment to provide n lterntive to onventionl hinged-flp onfigurtions. For the design of morphing devie, it is desired to determine the hnge in wing shpe tht most effiiently produes the neessry hnge in the erodynmi fores. Thus, understnding the proess of produing hnge in wing shpe is of fundmentl importne for morphing irrft. One of the min design issues relted to understnding this proess is voiding the weight penlty for unneessry tutor pility. For requested hnge in wing shpe, the tutors on the wing must provide the work required to deform the wing while eing ted on y the erodynmi fores. Determining the hnge in wing shpe tht requires the minimum tutor work llows the morphing devie to operte effiiently nd with minimum tutor weight 3,4,5,6,7,8. This pper presents theoretil study of the reltionship etween the hnge in merline shpe of twodimensionl thin irfoil nd the resistne of the erodynmi fores to this hnge. This resistne will e represented y the work required from the tutors on the irfoil to overome the erodynmi fores while produing hnge in merline shpe. The reltionship etween the output work produed y the tutors nd the required input energy will e disussed nd shown to ffet the optiml hnges in wing shpe. A generl tutor model will e presented nd used throughout the nlysis. A new method of unstedy thin irfoil theory for deforming merlines will e presented nd used for the energy lultion. This method llows the erodynmi lod distriution to e represented nlytilly, whih provides insight into the work lultion. The energy required to produe hnge in lift for pithing flt plte will e thoroughly nlyzed. The minimum energy pithing es will e determined for vrious ses. The nlysis of the pithing flt plte is pplile to vrile twist morphing onepts. A omprison nd nlysis of the tutor energy ost for onventionl flp, onforml morphing flp, nd two vrile mer onfigurtions will e presented. The nlyti nture of this study lrifies the fundmentl issues involved with the proess of produing hnge in irfoil shpe. II. The Reltionship etween the Aerodynmi Energy Blne nd Atutor Energy ost For wing moving in n invisid potentil flow, energy trnsfer etween the wing nd the fluid is hieved through the mehnil work required to produe wing motion or deformtion while overoming the fluid fores. This energy lne is stted mthemtilly through the following eqution for onservtion of energy, 9 P + DU E (.

3 where P is the rte of work done y the wing ginst the fluid fores in diretion norml to the onoming flow, D is the drg fore, U is the free-strem veloity of the onoming flow, nd E is the kineti energy dissipted to the flow per unit time. For thin irfoil in inompressile potentil flow, the first two of these omponents re defined s follows z P( t p(, t (, t d t (. z D( t p(, t (, t d S( t (.3 where p is the pressure loding on the irfoil, z defines the merline shpe, nd S is the leding-edge sution fore. Visous effets my e inluded in the energy lne (Eq.. y inluding the skin frition omponent of drg in D nd visous dissiption in E. For the osilltory motion of thin irfoil, Wu shows tht the verge vlue of E over period of osilltion is lwys positive. Wu 3 lter eplins tht this point is redily pprent euse in the frme of referene fied to the undistured fluid, the kineti energy of the si flow is zero. Therefore, ny unstedy motion of ody must inrese the energy of the surrounding flow. It follows from Eq. (. tht for thrust to e generted from osilltory irfoil motion, P must e positive. The se of P < hs meningful interprettion from two different points of view. The first point of view is tht for n irfoil eing propelled through fluid. Although some energy is eing tken from the flow (y definition of P <, more energy is eing supplied to propel the irfoil (euse E <, if P <, then from Eq. (. D >- P >. This se my e interpreted s flutter euse the flow is supplying energy to the struture 4. Ptil 5 points out tht flutter nlyses ssume onstnt flight speed, whih is not prtil euse it implies tht the irrft propulsion system utomtilly ounts for the inrese in drg used y the unstedy wing motion. The seond point of view is tht of fied irfoil osillting in n onoming flow, whih my e interpreted s the power etrtion mode 6,7. The differene etween this se nd the flutter se is tht here there is no energy spent on propulsion euse the onoming flow, suh s nturlly ourring wind, provides the DU omponent of energy. It should e mentioned tht the flutter mode n lso e interpreted s power etrtion mode if the struture is designed for the tsk. The drwk is tht the power spent on propulsion due to the osilltions will lwys e greter thn the hrvested power euse E <. For the trnsient motion or deformtion of thin irfoil, the onsequenes of the erodynmi energy lne re signifintly different from those of the osilltory se disussed in the previous prgrph. The osilltory se onsists of ontinuous motion tht llows for men vlue over period of osilltion to e defined. For the trnsient se, the unstedy motion ends t some presried time (t * while the erodynmi fores ontinue to hnge. This mens tht P is zero fter t *, ut the unstedy drg ontinues to t on the irfoil nd therefore energy ontinues to e trnsferred to the wke. Notie tht in the previous prgrph, no mention ws mde of the men lift ting on the irfoil. This is euse onstnt erodynmi fore does not ffet the men energy lne of n osillting irfoil 8. For the trnsient se, though, onstnt erodynmi fore omponent is signifint. This signifine is understood y reognizing tht the energy required to produe stedy lifting flow from n initilly non-lifting flow is infinite 9. The reson for this infinite energy is shown y Lom to e result of the /t dependene of the unstedy drg s t tends to infinity. With n initil vlue of lift ting on n irfoil during trnsient motion, the flow hs the ility to trnsfer some of the infinite energy present initilly in the flow to the irfoil. If the initil lift on the irfoil is zero, result nlogous to Wu s result tht E > my e stted s follows: if the fluid is undistured t t, then t E ( t dt > (.4 For n irfoil with finite vlue of lift t t, this inequlity does not neessrily hold. Another onsequene of the infinite energy required to produe hnge in lift is tht it mkes ny ttempt to minimize the energy lost in the wke for given hnge in lift invlid. Reognizing tht n infinite mount of time is required for the unstedy drg to trnsfer the infinite energy to the flow, it eomes ler tht the ddition of stedy omponent of drg (e.g. 3

4 visous or 3-D indued drg will lso require n infinite mount of energy to overome. Adding the prtil onsidertion tht these stedy omponents of drg will overshdow the unstedy omponent of drg for most vlues of time, it eomes ler tht the unstedy drg will e n insignifint omponent of the energy required y n irrft propulsion system. On the other hnd, the power required to overome the erodynmi fores nd produe merline deformtions (P, whih is finite, is not ffeted y the ddition of stedy drg omponents. Therefore, the omponent P drives the design of the tution systems on n irrft tht produe merline deformtions. The reminder of this pper will e onerned with the determintion nd minimiztion of the energy required to produe merline deformtions; with it eing epted from the prtil stndpoint mentioned ove tht the infinite energy required to overome the unstedy omponent of drg is eing ignored. Figure. shows one wy of lloting the totl required tutor power (P out for generl irfoil ontrol devie. The struturl fores would e present on ny morphing-type devie tht must deform n outer skin. Fritionl fores my lso e grouped in the struturl fores tegory, whih would lso pply to onventionl hinged flps. The inertil fores re present for ny devie, ut re negligile ompred to the erodynmi nd struturl fores. As previously stted, the urrent study is onerned with the power required to overome the erodynmi fores (P, nd therefore P out is ssumed to equl P in Figure.. For presried hnge in merline shpe long defined pth etween t nd t t *, the totl energy required to overome the erodynmi fores is defined s * t W P( t dt (.5 The power required y the tutor to produe P is defined s P in Figure.. The orresponding energy input to the tutors for presried merline deformtion is defined s * t W P ( t dt (.6 Note tht Eq. (.5 nd (.6 re defined seprtely for eh ontrol surfe or tutor on the irfoil. The vlue of P required for eh ontrol surfe or tutor is distinguished y the dz /dt term in Eq. (.. P s Struturl Fores P Atutor P out P Aerodynmi Fores P i Inertil Fores Figure.. The distriution of the provided tutor power for generl onfigurtion. To otin the quntity P, knowledge of the tutor energetis nd tutor plement is required. For the urrent study, whih is intended to investigte the fundmentls of the tutor energy required to overome the erodynmi fores, generl model of the tutor energetis is proposed. The model is defined s follows for P for P out out, <, P P out P η P out (.7 where η is onstnt rnging from - to depending on the tutor. A seprte effiieny ould e defined for positive vlues of P out (so tht % tutor effiieny is not ssumed, lthough this implies just multiplying W y onstnt (sine η will hnge ordingly. This will not influene omprison etween different ontrol 4

5 surfe onfigurtions nd is therefore not used for this nlysis. Figure. illustrtes Eq. (.7 for three key vlues of η. For η, the tutor requires the sme power input to produe negtive vlues of P out s it does to produe positive vlues. Rell tht positive P out vlues indite tht the tutor motion is resisted y the eternl fores while negtive vlues indite tht the eternl fores t in the diretion of tutor motion. For η, the tutor requires no power input nd llows no power to e etrted while produing negtive vlues of P out. This se is the most onsistent with feedk ontrolled pneumti nd hydruli tutors, whih require only the ontrolled relese of pressurized fluid to produe negtive power. The negleting of negtive work vlues hs lso een onsidered for the energy-ost nlysis of inset flight 3 nd humn musles 4. The η - se llows the tutor to store the inoming energy ssoited with negtive vlues of P out to e used lter to produe positive P out vlues with % onversion effiieny. This vlue of η llows W to e negtive nd zero for ertin ses. P η η P out η - Figure.. The reltionship etween P out, the required rte of tutor work, nd P, the rte of tutor energy, for the proposed generl tutor model. Applying the generl tutor model of Eq. (.7 to Eq. (.6, the eqution for the totl required tutor energy input n e written s W W η W (.8 where W + nd W - re the solute vlues of the positive nd negtive omponents of the integrl in Eq. (.6. An emple of these omponents is shown in Figure.3, where in this se W + is the integrl of P from t to t nd W - is the negtive of the integrl from t to t *. P + + W + t t * t -W - Figure.3. An emple of the seprtion of W into W + nd W - omponents for given trnsient motion. III. Unstedy Thin Airfoil Theory Applied to ontrol Surfe Motions The predition nd understnding of the unstedy erodynmi fores resulting from hnge in merline shpe re neessry to nlyze the energy hrteristis of deforming merline. This setion presents rief desription of unstedy thin irfoil theory in form onvenient for the nlysis of generl deforming merlines. The presenttion of this mteril is neessry euse there re only few pst studies on the unstedy pressure distriution 5,6,7, whih is required for the lultion of P in Eq. (.. Previous reserh on the unstedy pressure distriution hs ll een foused on the osilltory se, nd the etension to the trnsient se hs not een disussed. The method presented here follows diretly from stedy thin irfoil theory, therefore llowing physil 5

6 interprettion of eh of the unstedy omponents. A more detiled disussion of this pproh n e found in Hn, et l. 8 Following the von Krmn nd Sers 9 pproh to unstedy thin irfoil theory, the erodynmi fores ting on n unstedy irfoil n e seprted into three omponents: qusi-stedy, pprent mss, nd wke-effet. The qusi-stedy omponent represents the solution of the stedy thin irfoil prolem to the instntneous oundry onditions. The term instntneous oundry ondition is used insted of instntneous merline shpe euse n etr omponent of indued veloity is used y the rte-of-hnge of the merline shpe. The omponent of the erodynmi fore resulting from this indued veloity usully dmps the unstedy motion, nd will therefore e referred to s the erodynmi dmping. Figures 3. nd 3. illustrte the stedy nd dmping omponents of the qusi-stedy oundry ondition for pithing flt-plte nd defleting flp. Defining the stedy omponent s w s nd the dmping omponent s w d, the totl oundry ondition n e written s w w s + w d (3. From Eq. (3., the qusi-stedy erodynmi fores n e derived seprtely for the stedy nd dmping terms. z z z z U U z z U U Figure 3.. The stedy terms of the oundry ondition for pithing irfoil nd defleting flp. z z z t t Figure 3.. The dmping terms of the oundry ondition for pithing irfoil nd defleting flp. It is onvenient to restrit the present nlysis to time-dependent merlines of the following form z z (, τ ψ ( ( τ (3. where ψ defines the shpe of merline (for emple flpped or proli merline, nd defines the time vrying mgnitude of the merline (for emple the flp defletion ngle or mgnitude of mimum mer. Also, let τ represent nondimensionl time, defined s Ut τ (3. Note tht Eq. (3. nnot represent shpes suh s time vrying flp-to-hord-rtio euse ψ is not funtion of time. From Eq. (3., the stedy nd dmping terms of Eq. (3. n e written s ψ w s (3.3 where is defined s d/dτ. ψ w d (3.4 A. The Qusi-Stedy Terms As mentioned previously, the qusi-stedy omponent of the erodynmi fore is the omponent resulting from stedy thin irfoil theory. While the qusi-stedy fore oeffiients for vrious merline shpes re well known, For ll prtil merline onfigurtions eept for leding edge flps this is true. 6

7 for emple onventionl hinged leding edge 3 nd triling edge 3 flp nd n NAA 4-digit merline 3, there re few nlyti qusi-stedy pressure distriutions presented in the literture (most notle is Allen s 33 solution for triling edge flp. It is onvenient to represent the qusi-stedy irfoil vortiity (γ using Gluert s Fourier series 34, whih is written s ( + osθ + γ θ U A A n sin nθ (3.5 sinθ n where ( osθ A n A w w ( θ, t dθ ( θ, t os( nθ dθ The Fourier oeffiients in Eq. (3.6 my e seprted into stedy nd dmping terms (A,s, A,d, A n,s, nd A n,d y using w s or w d in ple of w. The qusi-stedy lift ( L, nd qurter-hord pithing moment ( M, re written in terms of the Fourier oeffiients s (3.6 L, (A, s + A, s + (A, d + A, d K, s A 4 + K, s, d, d M, (, s, s (, d, d J + J A + A 4 A (3.7 (3.8 where the r over the Fourier oeffiients indites tht they re per-unit or. Similrly, from Eq. (3.5 the qusi-stedy lod distriution is written s (rell tht p /U, γ where p, ( θ + osθ osθ 4A, s 4 An, s sin nθ + + 4A, d + 4 An, d sin nθ sinθ + sinθ n n (3.9 ( A χ( θ + T ( θ + ( A χ( θ + T ( θ, s, s, d, d T T + osθ χ( θ 4 sinθ, s, d ( θ 4 ( θ 4 n n A A n, s n, d sin nθ sin nθ (3. Allen 35 showed tht the infinite series in Eq. (3. my e written in terms of the following integrl 7

8 n An sin nθ ( θ, t w sinθ dθ osθ osθ (3. Thus, the T vlues in Eq. (3. my e rewritten s T T, s, d 4 ( θ 4 ( θ ψ ( θ sinθ dθ osθ osθ ( θ ψ sinθ dθ osθ osθ (3. From Eqs. (3.6, (3.9, nd (3., the qusi-stedy lod distriution my e determined nlytilly for given ψ. It will e shown tht the qusi-stedy omponents re required s inputs for the lultion of the pprent mss nd wke-effet terms. B. The Apprent Mss Terms The pprent mss lift nd pithing moment oeffiients re defined s 6 ( L, γ θ osθ sinθdθ (3.3 τ U ( M, γ θ [os θ / osθ ] sinθdθ (3.4 τ 8U These epressions n e written in terms of stedy nd dmping Fourier oeffiients defined in Eq. (3.6 s follows 3, s 4 L, (A, s + A, s + (A, d + A, d, d K, s + K, d M, (4A, s + A, s + A, s A3, s (4A, d + A, d + A, d A3, d J + J 4 3 (3.5 (3.6 where represents d / dτ. The pprent mss pressure distriution is determined using Neumrk s 36 pproh. After some mnipultion of Neumrk s equtions, whih require the seprtion of γ into irultory nd nonirultory omponents, the following eqution is otined 5 θ θ p, A, s sinθ A, sθ + T, s ( θ + T, d ( θ T, s ( θ sinθ dθ A, d sinθ A, dθ T, d ( θ sinθ dθ + + (3.7 As mentioned previously, the pprent mss terms re dependent upon the qusi-stedy terms. The terms T,s nd T,d represent the most diffiult omponents to otin nlytilly euse they depend upon the integrtion of T,s nd T,d. It is onvenient to reognize tht for merline defined y Eq. (3., the omponent T,d is identil to T,s 5. 8

9 . The Wke-Effet Terms For ritrry time-dependent qusi-stedy lift vritions, the wke-effet lift nd pressure distriution re otined using superposition of step input responses. The lift response to step input in L, is referred to s the Wgner funtion (φ 37. Liner superposition of the Wgner funtion is hieved using the Duhmel integrl 38, whih llows the wke-effet lift ( L, to e written s τ d L,( (3.8 dσ L, ( τ τ φ( τ + ( σ φ( τ σ dσ L, Neumrk 36 shows tht p, hs etly the sme θ-dependene s p, produed y n ngle of ttk. Thus, the eqution for the wke-effet pressure oeffiient ( p, n e written s τ + osθ ( τ L, ( sinθ (3.9 p, τ The Wgner funtion, shown in Figure 3.3, n e pproimted s 39 φ.9τ.6τ ( τ e e (3. Also shown in Figure 3.3 is the result of the Duhmel integrl in Eq. (3.8 for rmp input tht termintes t τ*. This result for rmp input will e used in lter setions of this pper. s indited (nondimensionl Rmp input with τ* (defined s φ in Eq. (4.6 Wgner funtion (φ Figure 3.3. Illustrtion of the Wgner funtion nd the funtion otined from the evlution of the integrl in Eq. (3.8 for rmp input. For sudden step hnge in irultion (due to step hnge in ngle of ttk, flp defletion, et., the totl lift oeffiient n e written s L τ ( τ + φ τ ( + (3. L, ( L, L, τ where the omintion of φ nd L, is L,. The pprent mss lift for step hnge in L, is written using the Dir delt funtion to represent the time-derivtive in Eq. (3.3 s follows δ ( τ L, (A + A 4 (3. Rell tht the Dir delt funtion hs the following properties 9

10 δ ( τ δ ( τ δ ( τ dτ (3.3 The lst property in Eq. (3.3 will e importnt when disussing the work required to overome the erodynmi fores. The lod distriution of this pprent mss pulse is otined y ehnging the Dir delt funtion for the time derivtive of in Eq. (3.7. To onlude this setion, Figure 3.4 illustrtes the omponents of the unstedy lod distriution for onventionl hinged flp with flp-to-hord rtio of.5. These terms re determined nlytilly from Eqs. (3.9 nd (3.7. Note tht the time-dependent mgnitude of these omponents, s indited y Eqs. (3.9 nd (3.7, depend on nd its first nd seond derivtive with respet to τ. For rmp input of, the qusi-stedy term from the stedy oundry ondition, omponent (, vries linerly with while the dmping qusi-stedy term, omponent (, nd the pprent mss term from the rte-of-hnge of omponent (, omponent (, re onstnt with time. The wke-effet omponent, whih hs the shpe of n ngle of ttk lod distriution, is not shown euse its mgnitude depends on the time-history of. From this figure it is seen tht if nd hve the sme sign s, then the hinge moment oeffiient of the flp is inresed ove the stedy vlue..4 s indited (nondimensionl ( ( + T χ 5 ( ( A, s, s ( χ T A, d +, d T,s (d T, d / Figure 3.4. The omponents of the unstedy lod distriution for onventionl hinged flp with flp-to-hord rtio of.5. IV. The Aerodynmi Work for Rmp Input of ontrol Defletion For merline defined y Eq. (3., the time-dependene of the merline deformtion is defined entirely y the funtion. This setion will derive the erodynmi work nd power omponents disussed in Setion II for the funtion defined s terminted rmp, whih will e written s ( τ, τ ( τ +, ( τ +, < τ < τ < τ < (4.

11 where is the initil vlue of, nd is the hnge in etween τ nd τ τ*. These terms re illustrted in Figure 4. long with the orresponding first nd seond derivtives of. Notie tht the seond derivte is defined y two Dir delt funtion impulses. τ d dτ Figure 4.. The speified time-history of the merline deformtion nd the orresponding time-derivtives. d dτ ( τ δ δ ( τ τ* The power required to overome the erodynmi fores ws defined in Eq. (.. It will e onvenient to represent the power y the following nondimensionl power oeffiient ( P P ( τ P qu z p (, τ (, τ d τ (4. olleting the qusi-stedy, pprent mss, nd wke effet omponents of the erodynmi fore from the previous setion, p my e written in terms of s follows " ( A χ( + T ( ( t + ( A χ( + T ( + T ( ( t T ( ( (, t α ( t χ( + + (4.3 p un, s, s, d, d, s, d t For the defined in Eq. (4., the wke effet term, α un, evlutes to the following where φ ( τ [ K, dφ( τ + K, sφ ( τ ] (4.4 τ * αun φ τ ( τ φ( τ.65e ( τ φ( τ σ dσ.9τ e.335e.6τ.6τ +.839e.9τ (4.5 (4.6 Sustituting p from Eqs. ( nd z from Eqs. (3. nd 4. into Eq. (4. llows the power oeffiient for single ontrol surfe to e written s P [ Qφ ( τ + Qφ ( τ + Q3τ + Q4 + ( δ ( τ δ ( τ τ* Q5 ] + Q 3 (4.7 where the Q terms re defined s

12 Q Q K, d Q Q ψ ( d K, s ψ ( d (4.8 (4.9 ( A χ ( + T ( ψ ( d (4. 3, s, s ( A χ ( + T ( ψ ( d (4. 4, d, d Q5 T, d ( ψ ( d (4. Note tht the ½ in the Q 5 eqution is result of the definition of d/dτ t τ nd τ*, whih from Figure 4. n e written s d d ( τ ( τ τ* dτ dτ (4.3 For liner merline shpes, ψ is liner, nd eh term in Eq. (4.7 my e interpreted s omponent of the dynmi hinge-moment oeffiient multiplied y the flp defletion rte (d/dτ. The Q nd Q terms re due to the wke effet fores, Q 3 is due to the qusi-stedy fores, nd Q 4 nd Q 5 re due to the pprent mss fores. The Dir delt funtions in Eq. (4.7 re result of the elertion pulse of the merline s shown in Figure 4.. Hving otined n epression for the output power required y n tutor to overome the erodynmi fores during rmp input of merline deformtion, the input energy required y the tutor (W my e lulted using Eqs. ( The nondimensionl input energy oeffiient is defined s W W q P dτ (4.3 where P is defined through the generl tutor model defined in Eq. (.7, whih n e written in terms of P s for for P P, <, P P P η P (4.4 From Eq. (4.4, the integrtion required y Eq. (4.3 for W n e seprted into positive ( W+ nd negtive ( W- omponents s follows η (4.5 W W + + W whih is equivlent to Eq. (.8 nd is illustrted in Figure.3. Note tht the two Dir delt funtions in Eq. (4.7 result in there lwys eing oth omponent of W+ nd W- present. Assuming Q 5 is greter thn zero, the impulses t τ nd τ τ* provide omponents of W+ nd W-, respetively. These omponents n oth e written s

13 W, δ 5 (4.6 Q whih represent the instntneous trnsfer of energy from the irfoil to the surrounding fluid. Although this is n unrelisti onept, it is epted euse it simplifies the effet of merline elertion y onentrting it t the eginning nd end of the unstedy motion. The diffiulty in pplying Eq. (4.5 is tht the integrtions required for W+ nd W- n only e evluted nlytilly for speil ses. The reson for this is tht τ must e found nd then used s limit of integrtion for the evlution of W+ nd W- (τ is equivlent to t in Figure.3. The nlyti evlution of τ is mde diffiult y the eponentils present in Eqs. (4.5 nd (4.6. For Eq. (4.5 to e evluted nlytilly, τ must e less thn zero or greter thn τ* so tht P remins either positive or negtive throughout the deformtion proess. Detils of these onsidertions re eplined most effetively through n emple, whih is the fous of the net setion. V. Applition to Pithing Flt-Plte Airfoil The pplition of the tutor energy theory developed in the previous setions to pithing flt-plte identifies mny of the interesting spets of the theory. onsider the flt plte shown in Figure 5.. The shpe funtion of Eq. (3. is simply ψ ( (5. nd the time dependent ngle of ttk, ( τ α( τ, is speified to e the rmp input defined in Eq. (4.. z U α(τ Atutor Figure 5.. Definition of the geometry nd tutor plement for pithing flt plte. The Q terms from Eq. ( evlute to the following 3 Q + (5. 8 Q 4 Q3 ( ( Q4 + (5.4 4 Q (5.5 Applying these funtions to Eq. (4.7 for vlue of /.5, the time-history of P ws determined for vrious vlues of τ* nd is plotted in Figure 5.. The es of Figure 5. re normlized with τ* to llow the vrious ses to e shown on the sme figure. This figure shows tht the required positive work ( W+ dereses s τ* inreses, whih is result of redued erodynmi dmping. Beuse the initil ngle of ttk (α is zero for this se, Eq. (4.7 indites tht the vlue of τ t whih P is zero (τ is independent of τ* (this is not ovious in Figure 5. euse of the sling of the es. 3

14 .5 τ*.5 τ* P τ* / α τ* 5 τ* infinity /.5 initil α τ / τ*.8 Figure 5.. The time-history of the power oeffiient for rmp input of α for vrious vlues of τ* It turns out tht this initilly nonlifting se llows for the pproimte nlyti evlution of W+ nd W-. This is possile euse τ my e determined nlytilly y mking use of few vlid ssumptions. The solution proess for τ is initited y setting P from Eq. (4.7 equl to zero, Q φ τ + Q φ ( τ + Q τ + Q (5.6 ( 3 4 where φ nd φ re defined in Eqs. (4.5 nd (4.6. It is oserved in Figure 5. tht τ is less thn one, whih is true for vlues of / >.45. It is lso oserved from Eqs. (4.5 nd (4.6 tht the oeffiients in the eponentils re less thn one. From these oservtions it is onluded tht φ nd φ my e urtely pproimted s follows using the first two terms of Tylor series φ ( τ.5 +.6τ + O( (5.7 τ φ ( τ.5τ + O( (5.8 τ Sustituting these epnsions into Eq. (5.6, τ is found to equl.5q Q4 τ +... (5.9.6Q.5Q + Q From Figure 5., the limits of integrtion for W+ nd W- re identified, whih llows the two terms to e written s 3 W + τ P α ( τ dτ τ QΦ ( τ + QΦ ( τ + Q3 + Q τ + Q 4 5 (5. 4

15 W τ P α ( τ dτ Q [ Φ ( τ* Φ ( τ ] + Q [ Φ ( τ* Φ ( τ ] + Q + Q [ τ ] 3 τ 4 Q 5 (5. where.6τ.9τ Φ( τ φ( τ d τ e e (5..6τ.9τ Φ ( τ φ ( τ d τ τ.9355e e (5.3 Applying the pproimtions of Eqs. ( , the epression for W+ from Eq. (5. simplifies to the following α (.5Q ( Q4 + Q.6Q +.5Q3 W + 5 (5.4 where the Tylor Series pproimtions of Φ (τ nd Φ (τ re used. Similrly, the pproimte eqution for W- is written s W (.5Q Q4 (.6Q +.5 α QΦ ( τ* + QΦ ( τ* + Q3 + Q4τ * Q5 + Q 3 (5.5 where Eq. (5. nd (5.3 re used for Φ (τ* nd Φ (τ*. These equtions re vlid for vlues of / >.45 nd for τ* >.. For vlues of τ* <., τ is greter thn τ* so tht the limits of integrtion in Eqs. (5. nd (5. re no longer vlid. The usefulness of these equtions is tht they urtely predit the vlue of for the minimum W for ny vlue of η nd for vlues of τ* >.. They lso indite tht W- hs more omple funtionl dependene on τ* thn does W+. Figure 5.3 presents the et vlues of W+ nd W-, whih were otined y omputing τ nd speifying the limits of integrtion for eh se. The results of Eq. (5.4 for W+ re shown s dshed line for eh se. It is seen tht the results of Eq. (5.4 re indistinguishle from the et result for / >.45 nd eome invlid s / pprohes.5. The result of Eq. (5.5 is not shown in Figure 5.3, lthough it n e shown to e urte for the sme vlues of s Eq. (5.4. This figure shows tht W+ nd W- onverge to the limit of τ* infinity, whih represents the results of stedy irfoil theory. It lso shows tht, s epeted from stedy irfoil theory, W+ is lrgest for / <.5 nd W- is lrgest for / >.5. The pithing is for minimum W+ is found etly from Eq. (5.4 to equl.57, whih is independent of τ*. Figure 5.3 verifies tht this minimum is loted within the rnge of vlues where Eq. (5.4 is vlid. From the W- plot in Figure 5.3 it is dedued tht s η eomes nonzero nd positive, the minimum W pithing is shifts towrds the leding edge. Similrly, s η eomes negtive, the optiml is shifts to the triling edge. 5

16 W+ / α initil α τ*.5 τ* τ* 5 τ* infinity W- / α τ*.5 τ* τ* 5 τ* infinity initil α / / Figure 5.3. The vrition of W+ nd W- with / nd τ*. The thin dshed lines in the W+ plot represent the result of Eq. (5.4. The ses shown in Figure 5. nd disussed previously speified tht the initil α, nd therefore the initil lift, ws zero. The effet of n initil lift will now e presented. From Eq. (4.7 it is seen tht n initil ngle of ttk (α only influenes P through the lst term, whih ontins Q 3. Dividing this eqution y α llows P to e written s follows P k [ Qφ ( τ + Qφ ( τ + Q3τ + Q4 + ( δ ( τ δ ( τ τ* Q5 ] + Q 3 (5.6 α where α k (5.7 α The vlue k represents the initil lift divided y the hnge in stedy stte lift. Reognizing the term k in Eq. (5.6 is useful euse it indites tht the normlized power oeffiient ( P / α is dependent only on the rtio of α nd α, nd not eh term independently. The presene of k signifintly omplites the prolem of nlytilly determining W+ nd W-, lthough the pproimte method disussed previously n e pplied to ertin vlues of k. The min effet of the initil lift is to vertilly disple the P urves, suh s those shown in Figure 5.. This signifintly hnges τ nd therefore lters the llotion of W into W+ nd W- terms. To gin some insight into the effet of k on W, the limiting ses of τ* pprohing zero nd infinity will e emined. For τ* pprohing zero, the region of integrtion for W+ is τ <, nd the W- omponent omes ompletely from the Dir delt funtion t τ*. From Eq. (5.6, the integrtion for W, with τ* pprohing zero, results in W α Q + Q4 + ( + η Q 5 + O ( (5.8 whih is independent of k. For η > -, the Q 5 term is dominnt. Thus, from Eq. (5.5 the pithing is for minimum W is t the hlf hord. For η -, only the rketed term remins in Eq. (5.8. Sustituting Eqs. (5. nd (5.4 into (5.8 nd setting the derivtive with respet to equl to zero, the pithing is for minimum W is found to e loted t / 3/4. For τ* pprohing infinity, the region of integrtion for W+ nd W- depends upon k nd. This is seen y writing Eq. (5.6 in terms of its lowest order omponents for lrge vlues of τ*. To determine the lowest order omponents, it is neessry to define τ s τ τ (5.9 where <τ <. Sustituting this into Eq. (5.6, the lowest order eqution for P is written s follows 6

17 Q3 lnτ * P ( τ + k + O α (5. Note tht, s pointed out y Lom 7, the symptoti limits of φ nd φ otined from Eqs. (4.5 nd (4.6 re inorret. Therefore, the pproimte Wgner funtion suggested y Grrik 4 ws used insted for otining Eq. (5.. As epeted, Eq. (5. represents the stedy thin irfoil theory result. Eqution (5. shows tht, if k is less thn - or greter thn zero, the lowest order omponent of W is omposed entirely of either W+ or W-. For these vlues of k, W, is written from Eq. (5. nd (5.3 s follows lnτ * F(, k + k ( 4 + O W if F, F(, k (5. α W if F <, η F(, k α For vlues of k etween - nd zero, τ is determined y setting Eq. (5. equl to zero. This vlue of τ is then used s limit of integrtion for W, whih from Eqs. (5. nd (5.3 results in if if < / 4, / 4, W k + k + α W k α 4 ( 4 + ηk k η + k O lnτ * lnτ * ( 4 + O (5. Tle 5. presents the pithing es for minimum W otined from Eqs. (5. nd (5. with the onstrint tht the es remin within the hord. These results re intuitive from the elementry nture of stedy thin irfoil t n ngle of ttk. Tle 5.. The minimum W pithing es s τ* pprohes infinity k < - - < k < -/ -/ <k < k > η / /4 / /4 / /4 / /4 η < / < /4 / /4 / /4 /4 < / < η - / / / / The limiting ses of τ* disussed ove llowed W to e otined nlytilly, whih llowed the optiml pithing es to e determined nlytilly. For the k ses, the pproimte pproh presented in Eqs. ( , ounting for the k term in Eq. (5.6, is vlid for wide rnge of τ* vlues. Where this pproh is not vlid, the integrtion for W is performed numerilly from Eqs. (4.3, (4.4, nd (5.6. Using omintion of nlyti nd numeril pprohes, the minimum W pithing es were otined for η,, nd -. Figure 5.5 shows the vrition of the optiml pithing is with τ* for the η se for vrious k vlues. As determined previously, the es re shown to pproh /.5 s τ* pprohes zero. It is seen in this figure tht s k eomes lrge nd positive, the optiml is is loted t /.5 for most τ* vlues. This is result of W+ eing omposed of only the initil impulse, whih is smllest for the mid-hord is. For negtive k vlues, the optiml is moves towrd the leding edge s τ* inreses. Figure 5.5 shows the vrition of the optiml pithing is with τ* for the η se nd vrious k vlues. It is interesting to note tht for this se, s ws determined previously, the optiml is t oth τ* equl to zero nd infinity is independent of k. This eplins the inresed similrity etween the optiml es urves for vrious k vlues in Figure 5.5 when ompred to Figure 5.4. For irfoils tht must omplete yle, mening they produe hnge in lift (positive k nd then lter produe negtive hnge in lift to return to their initil stte (negtive k, the similrity in the optiml es for negtive nd positive k vlues is dvntgeous. This is euse smller ompromise must e mde, ssuming the pithing is remins fied, 7

18 when hoosing the optiml pithing is for the omplete motion. For the mjority of negtive nd positive omintions of k, the optiml is for the omintion is loted etween the two independent optiml points for given τ*. Thus, Figures 5.4 nd 5.5 re very generl nd pplile to mny prtil ses. Figure 5.6 presents the vrition of the optiml pithing is with τ* for the η - se. It is seen tht the differene etween positive nd negtive k vlues is very lrge ompred to Figures 5.4 nd 5.5. The result of inresing k in Figure 5.6 is seen to e derese in the vlue of τ* t whih the optiml is is the sme s those shown in Tle 5. for the τ* equl to infinity se. The sme onlusion n e stted from Figures 5.5. A similr result ws reported y Ytes 4 for the minimum energy pithing es of n osillting flt plte intended to produe thrust..6 k.5.4 /.3 k 3 k k / k -3/4 k - k -/. k -3. η.5.5 τ* Figure 5.4. The η se for the vrition of the minimum W pithing es with τ* for vrious vlues of k /.3 k k 3 k k / k -/.. k -3 k - k -3/4 η.5.5 τ* Figure 5.5. The η se for the vrition of the minimum W pithing es with τ* for vrious vlues of k. 8

19 k / k η -.8 k k 3.6 /.4 k -/. k -3 k - k -3/4.5.5 τ* Figure 5.6. The η - se for the vrition of the minimum W pithing es with τ* for vrious vlues of k. VI. Applition to Vrious ontrol Surfe onfigurtions This setion desries the ffet of vrious ontrol surfe shpes on the W required for given hnge in lift. The first two ses to e onsidered re shown in Figures 6. nd 6.. A onventionl hinged flp is shown in Figure 6.. The onforml ontrol surfe, shown in Figure 6., is qudrti segment defined to hve zero slope t. The mgnitude of the flp defletion ( is defined in oth ses s the ngle t the triling edge. The rmp input of, defined in Eq. (4., will e used for this nlysis. From the shpe funtions (ψ, whih re shown for eh se in Figures 6. nd 6., the omponents of p in Eq. (4.3 my e determined nlytilly from the equtions of Setion III. The resulting equtions re reltively omple, nd it is therefore onvenient to perform the integrtions required for the Q-terms defined in Eqs. ( numerilly. U <, ψ (, ψ ( + Figure 6.. The merline geometry for onventionl flp U ψ ( ( Figure 6.. The merline geometry for onforml flp < ψ (,, + + ( 9

20 Note tht in the previous se of the pithing flt plte, the L produed y α ws independent of the pithing is. This ment tht the W required for given lift ould e represented y W / α. For ompring vrious ontrol surfe onfigurtions, it is onvenient to insted normlize P nd W y the L. From Eq. (4.7, the normlized eqution for P n then e written s where k [ Qφ ( τ + Qφ ( τ + Q3τ + Q4 + ( δ ( τ δ ( τ τ* Q5 ] + Q3 (6. K P * K * L τ τ, s k (6. L, initil L, s Rell tht the quntity L refers to the hnge in stedy stte lift, whih from Eq. (3.7 is written s K s (6.3 L, onsidering the onventionl nd onforml flp onfigurtions, if k is greter thn zero, then P remins positive throughout the rmp input of. Therefore, W+ is otined y integrting Eq. (6. from τ to τ* nd W- is otined from Eq. (4.6. For smll negtive vlues of k, P hnges from positive to negtive nd therefore τ must e determined. For these ses, the proess desried with Eqs. ( my e used. For lrge negtive k vlues, P remins negtive throughout the rmp input of. Therefore, W- is otined y integrting Eq. (6. from τ to τ* nd W+ is otined from Eq. (4.6. It is desired to ompre the vlues of W resulting from the onventionl nd onforml flp onfigurtions defined in Figures 6. nd 6.. The first se to e onsidered, shown in Figure 6.3, ompres the W required for given L,, nd τ* while vrying k. It is seen tht the W required y the onforml flp is less thn tht required y the onventionl flp for ny k when η. For the η se, there is smll rnge of k vlues where W is slightly less for the onventionl flp. Overll though, the onforml flp requires less W thn the onventionl flp. W / L τ* /.75 dshed lines - η solid lines - η onventionl Flp.6.4. onforml Flp k Figure 6.3. A omprison of the W required for onforml or onventionl flp. The reson for the smller W for the onforml flp is tht it requires less overll merline deformtion for given hnge in lift thn does the onventionl flp. Figure 6.4 illustrtes this result long with the orresponding

21 lod distriution t τ /. It is seen tht the ngle of defletion t the triling edge of the onforml flp is lrger thn tht for the onventionl flp for given hnge in lift, ut the overll z of the merline is less for the onforml flp. The lod distriution for the onventionl flp is entered more towrds the hinge-line thn for the onforml flp, whih is fvorle for the onventionl flp. Nevertheless, the lrger z overshdows the fvorle lod distriution for the onventionl flp. It should e mentioned tht the shpe of the lod distriutions shown in Figure 6.4 pply only t τ /. As shown in Eq. (4.3, the lod distriution does not simply sle linerly with the rmp input of..8 onforml Flp -.5 onforml Flp -. p / L.6.4. onventionl Flp τ / τ* /.75 k / z / / L Figure 6.4. The lod distriution over the flp nd the orresponding shpe of the flp defletions Figure 6.5 shows how W vries with τ* nd for the onforml nd onventionl flp. It is seen tht the onforml flp requires less W for every se. It is lso pprent tht the enefit of the onforml flp eomes lrger s τ* dereses. Hene, the onforml flp is idel in situtions where rpid hnges in lift re required. The vlues of W in the limit s τ* goes to infinity re shown in Figure 6.5. These vlues, whih n e otined from stedy thin irfoil theory, show tht W is 8% less for the onforml flp in the stedy limit. The onsiderle differene etween the stedy nd unstedy vlues in Figure 6.5 indites the importne of inluding the unstedy erodynmi terms in this nlysis. It should e mentioned tht the vlues of W for given hnge in qurter hord pithing moment ( M, produe results similr to those in Figure 6.5. In prtiulr, the vlue of W / M dereses ontinuously s vries from midhord to the triling edge. This is true even though the flp defletion required to produe pithing moment hs minimum t /.75 for the onventionl se onventionl Flp / W / L.5..5 onventionl Flp onforml Flp τ* k η k η /.75 onventionl Flp onforml Flp / dshed lines - limit s τ* goes to infinity Figure 6.5. The effet of flp size nd τ* on the W required for the onforml or onventionl flp. The net ses to e onsidered re the vrile mer onfigurtions shown in Figures 6.6 nd 6.7, whih re defined s NAA 4-digit merlines with time-dependent mgnitudes of mimum mer. onfigurtion A, τ*

22 shown in Figure 6.6, is defined so tht the leding nd triling edges remin on the -is s the mer hnges. onfigurtion B, shown in Figure 6.7, is defined so tht the lotion of mimum mer ( remins on the -is s the mer hnges. Figure 3. shows tht in stedy thin irfoil theory, these two onfigurtions produe the sme erodynmi fores. But, the ddition of the erodynmi dmping omponent, shown in Figure 3., mkes the unstedy thin irfoil results different etween the two ses. In onsidering the tutor energy for eh se, it is ssumed tht eh onfigurtion is tuted with single tutor. This implies tht some type of linkge system is used to produe the desired merline shpe. Also, s hs een done throughout this pper, only the erodynmi fores re onsidered for the tutor energy. It is reognized tht this is ig ssumption for these vrile mer onfigurtions, ut nonetheless, we feel tht the present nlysis provides signifint insight into the tution properties of vrile mer irfoil. U z z ψ ( ( ψ ( <,, + + ( + ( Figure 6.6. The merline geometry for vrile mer irfoil with the leding nd triling edges fied to the -is (onfigurtion A. U z z ψ ( ( ψ ( <,, + ( + ( Figure 6.7. The merline geometry for vrile mer irfoil with fied to the -is (onfigurtion B. The de pendene of W on k nd / is shown in Figure 6.8 for oth onfigurtions nd η. It is seen tht onfigurtion B requires signifint w for positive k ses while onfigurtion A requires very little for these ses. This result is eplined y reognizing tht the merline motion for onfigurtion B is downwrds for positive hnge in lift, whih must therefore move ginst the upwrd ting lift fores. On the other hnd, the merline motion for onfigurtion A is upwrds nd is therefore not resisted y the erodynmi fores. For negtive k vlues, the sitution reverses nd this onfigurtion requires signifint W. Figure 6.8 shows tht onfigurtion B requires less W for given positive k thn onfigurtion A requires for negtive k of the sme mgnitude. This mens tht if the irfoil is intended to produe n equl numer of positive hnges in lift s negtive hnges in lift, then onfigurtion B is fvorle from n energy stndpoint. The seond plot in Figure 6.8 shows tht this onlusion is true for ny lotion of mimum mer (. It is lso seen tht s moves loser to the leding edge, onfigurtion B eomes even more fvorle.

23 W / L onfig. A onfig. B τ* η / onfig. A - solid lines onfig. B - dshed lines k - k - k k τ* η k Figure 6.8. The effet of k nd / on the W required for onfigurtion A nd B. The lod distriution nd orresponding merline shpe t τ / re shown in Figure 6.9. This figure illustrtes the point mde previously tht the merline motion for onfigurtion B is resisted y the erodynmi fores for k greter thn or equl to zero. Note tht the differene etween the lod distriutions shown in this figure omes from the K,d nd A, d terms in Eq. (4.3. This figure mkes ler the resons why onfigurtion B requires less W (when onsidering the entire rnge of k vlues thn onfigurtion A. The first reson is tht onfigurtion B simply requires less overll merline defletion thn onfigurtion A. The seond reson is tht for onfigurtion A, the lrgest merline defletions re towrds the enter of the merline while for onfigurtion B they re t the leding nd triling edges. omining this ft with the shpe of the lod distriution mkes ler the dvntge of onfigurtion B. /.5 p / 5-5 onfig. B onfig. A / τ / τ* k /.4 z / / Figure 6.9. Emple of the unstedy lod distriution nd orresponding merline shpe for onfigurtion A nd B. / onfig. B onfig. A VII onlusions The work required to overome the erodynmi fores to produe hnge in lift through merline deformtion ws shown to depend signifintly on the initil lift of the irfoil. This onlusion rises euse there is infinite energy in lifting two-dimensionl flow. The power required for rmp input of ritrry merline deformtion ws shown to depend on five terms, defined s Q, Q,...,Q 5, whih depend on the results of unstedy thin irfoil theory. The neessity of using unstedy thin irfoil theory for the study ws illustrted. The pithing is required for flt plte to produe hnge in lift with minimum energy input to the tutor ws shown to depend on the energy required y the tutor to produe negtive work. Assuming tht there is no energy ost 3

24 ssoited with negtive work, the minimum energy pithing is for n irfoil with zero initil lift is loted t / equl to.57 for rmp input. For vrious tutor models, the minimum energy pithing es were otined nd shown to depend on the rte of the rmp input (τ*. A onforml flp ws shown to require signifintly less energy thn onventionl flp to produe hnge in lift. This onlusion ws shown to e independent of the initil lift, rte of the flp defletion, nd flp size. A downwrd defleting vrile mer onfigurtion (onfigurtion B ws shown to require less energy thn n upwrd defleting onfigurtion (onfigurtion A if oth positive nd negtive hnges in lift re onsidered. Among the ontrol devies investigted in this pper, the onforml triling edge flp requires the lest energy to overome the erodynmi fores for given hnge in lift. Aknowledgments This work ws performed with the finnil support of the enter for Intelligent Mteril Systems nd Strutures (IMSS. This support is grtefully knowledged. Referenes Stnewsky, E., Aerodynmi Benefits of Adptive Wing Tehnology, Aerospe Sienes nd Tehnology, Vol. 4,, pp Stnewsky, E., Adptive Wing nd Flow ontrol Tehnology, Progress in Aerospe Sienes, Vol. 37,, pp Forster, E., Snders, B., nd Estep, F., Synthesis of Vrile Geometry Triling Edge ontrol Surfe, AIAA Pper 3-77, April 3. 4 Forster, E., Snders, B., nd Estep, F., Modeling nd Sensitivity Anlysis of Vrile Geometry Triling Edge ontrol Surfe, AIAA Pper 3-87, April 3. 5 Gern, F. H., Inmn, D. J., nd Kpni, R. K., omputtion of Atution Power Requirements for Smrt Wings with Morphing Airfoils, AIAA Pper -69, April. 6 Pettit, G. W., Roertshw, H. H., Gern, F. H., nd Inmn, D. J., A Model to Evlute the Aerodynmi Energy Requirements of Ative Mterils in Morphing Wings, ASME Design Engineering Tehnil onferene, Septemer. 7 Prok, B.., Weisshr, T. A., nd rossley, W. A., Morphing Airfoil Shpe hnge Optimiztion with Minimum Atutor Energy s n Ojetive, AIAA Pper -54, Septemer. 8 Snders B., Estep, F. E., nd Forster, E., Aerodynmi nd Aeroelsti hrteristis of Wings with onforml ontrol Surfes for Morphing Airrft, Journl of Airrft, Vol. 4, Jn.-Fe. 3, pp von Krmn, T., nd Burgers, J. M., Generl Aerodynmi Theory Perfet Fluids, Aerodynmi Theory, W. F. Durnd, ed., Vol. II, Julius Springer, 935, pp Grrik, I. E., Propulsion of Flpping nd Osillting Airfoil, NAA Report No. 567, 936. Wu, T. Y., Hydromehnis of Swimming Propulsion, Prt. Swimming of Two-Dimensionl Fleile Plte t Vrile Forwrd Speeds in n Invisid Fluid, Journl of Fluid Mehnis, Vol. 46, prt, 97, pp Wu, T. Y., Swimming of Wving Plte, Journl of Fluid Mehnis, Vol., 96, pp Wu, T. Y., Etrtion of Flow Energy y Wing Osillting in Wves, Journl of Ship Reserh, Mrh, 97, pp Send, W., The Men Power of Fores nd Moments in Unstedy Aerodynmis, ZAMM, Vol. 7, 99, pp Ptil, M. J., From Fluttering Wings to Flpping Flight: The Energy onnetion, Journl of Airrft, Vol. 4, No., Mrh- April, 3. 6 Jones, K. D., nd Pltzer, M. F., Numeril omputtion of Flpping-Wing Propulsion nd Power Etrtion, AIAA pper 97-86, Jn MKinney, W., DeLurier, J., The Wingmill: An Osillting-Wing Windmill, Journl of Energy, Vol. 5, No., Mrh-April, 98, pp Wu, T. Y., Hydromehnis of Swimming Propulsion, Prt. Some Optimum Shpe Prolems, Journl of Fluid Mehnis, Vol. 46, prt 3, 97, pp Jones, R. T., Wing Theory, Prineton University Press, 99, pp Lom, H., Indiil Aerodynmis, in AGARD Mnul on Aeroelstiity, Prt II, hpter 7, June, 96. Fleisher, H., Mnul of Pneumti Systems Optimiztion, MGrw Hill, In., 995. Green, W. L., Airrft Hydruli Systems, John Wiley nd Sons, Weis-Fogh, T., Quik Estimtes of Flight Fitness in Hovering Animls, Inluding Novel Mehnisms of Lift Prodution, Journl of Eperimentl Biology, Vol. 59, 973, pp Aot, B.., Biglnd, B., nd Rithie, J. M., The Physiologil ost of Negtive Work, Journl of Physiology, Vol. 7, 95, pp Postel, E. E., nd Leppert, E. L., Theoretil Pressure Distriution for Thin Airfoil Osillting in Inompressile Flow, Journl of the Aeronutil Sienes, Vol. 5, No. 8, August 948, pp