MORPHING AIRFOIL SHAPE CHANGE OPTIMIZATION WITH MINIMUM ACTUATOR ENERGY AS AN OBJECTIVE

Transcription

1 9th AIAA/ISSMO Sympsium n Multidisciplinary Analysis and Optimizatin 4-6 September, Atlanta, Gergia AIAA -54 MORPHING AIRFOIL SHAPE CHANGE OPTIMIZATION WITH MINIMUM ACTUATOR ENERGY AS AN OBJECTIVE Brian C. Prck * Terrence A. Weisshaar William A. Crssley Purdue University Schl f Aernautics and Astrnautics West Lafayette, Indiana Abstract Mrphing aircraft are multi-rle aircraft that use innvative actuatrs, effectrs, and mechanisms t change their state t perfrm select missins with substantially imprved system perfrmance. State change in this study means a change in the crss-sectinal shape f the wing itself, nt chrd extensin r span extensin. Integrating actuatrs and mechanisms int an effective, light weight structural tplgy that generates lift and sustains the air lads generated by the wing is central t the success f mrphing, shape changing wings and airfils. The bjective f this study is t explre a prcess t link analytical mdels and ptimizatin tls with design methds t create energy efficient, lightweight wing/structure/actuatr cmbinatins fr mrphing aircraft wings. In this case, the energy reuired t change frm ne wing r airfil shape t anther is used as the perfrmance index fr ptimizatin while the aerdynamic perfrmance such as lift r drag is cnstrained. Three different, but related, tpics are cnsidered: energy reuired t perate articulated trailing edge flaps and slats attached t flexible D airfils; ptimal, minimum energy, articulated cntrl deflectins n wings t generate lift; and, defrmable airfils with crss-sectinal shape changes reuiring strain energy changes t mve frm ne lift cefficient t anther. Results indicate that a frmal ptimizatin scheme using minimum actuatr energy as an bjective and internal structural tplgy features as design variables can identify the best actuatrs and their mst effective lcatins s that minimal energy is reuired t perate a mrphing wing. Backgrund A mrphing aircraft is a multi-rle aircraft that, thrugh the use f mrphing technlgies such as innvative actuatrs, effectrs r mechanisms, changes its state substantially t cmplete all rles - with superir system perfrmance. Fr instance, fr a hunter-killer aircraft, rle A is lng-duratin liter while rle B is high-speed dash and rle C is sme frm f energy depsitin t neutralize the target. Mrphing aircraft are a majr part f a system that reuires technlgy integratin t manipulate gemetric, mechanical electrmagnetic r ther missin critical features - n the grund r in-flight - t match vehicle perfrmance t a well-defined envirnment and missin bjective. Mdern aircraft already cntain cmplex mrphing devices t allw them t balance ne missin demand against anther but still perfrm a missin well. The simplest example is the use f landing and take-ff flaps t allw transprt aircraft t perate frm shrter length airprts and still cruise at high speed. The trade-ff that favrs mrphing is the fuel saved when a smaller wing is used fr efficient high-speed flight and deplyable flaps generate increased lift at lw landing and take-ff speeds. The cst f this system perfrmance balancing is expressed in metrics such as weight, cmplexity and cst. * Graduate Student, Schl f Aernautics and Astrnautics, 8 Grissm Hall, West Lafayette, IN Prfessr, Schl f Aernautics and Astrnautics, 8 Grissm Hall, West Lafayette, IN , currently n leave at DARPA, Arlingtn, Virginia. Assciate Prfessr, Schl f Aernautics and Astrnautics, 8 Grissm Hall, West Lafayette, IN Cpyright by the authr(s). Published by the American Institute f Aernautics and Astrnautics, Inc., with permissin.

2 Mrphing Aircraft Ptential cmbining many systems int ne ISR missin: high altitude, lng range / endurance Hunt: sar, bserve cmbine functins / reduce system cmplexity Attack missin: high speed dash, high speed turns Attack: dive, maneuver Figure Cmbining system functins Mrphing shape/state changes are als necessary when lng liter time is cupled with a high-speed dash reuirement r when stealth features are nt reuired during the entire missin. In the latter case, mrphing int a faceted brick shape like the F-7 fr the stealthy prtin f the missin wuld then change the aircraft radar crss-sectin state while preserving aerdynamic perfrmance during lng cruise segments. Mdern mrphing aircraft cncepts address system prblems such as that depicted in Figure. The upper part f Figure shws the different sensr aircraft reuired t identify targets fr cmbat aircraft t engage and attack. In additin t the variety f aircraft used, this bserve, identify, target and attack system reuires cmmand and cntrl as well as human peratrs. The result is a cmplex system with large time lags that impede its effectiveness. The lwer part f Figure shws nature s answer t a sensing and attack system. All capability, including system peratin, is cncentrated nbard the vehicle. Wing mrphing cannt be applied t all systems and all missins. In fact, it reuires a majr effrt t identify the new missins that might result frm mdern mrphing. Hwever, nce these missins and system architectures are identified and the aircraft that use them are develped, the result will be t create game-changers. Minimizing actuatr energy reuired t generate lift Mdern mrphing cncepts include wings with a variety f mving surfaces, such as: articulated flaps and slats; surface flw cntrl devices; and cntinuusly defrming surfaces. In the latter case, the defrmatin f the surface is generated either by internal elements that exert frces and mments n the aer structure t defrm it r external devices such as cntinuusly defrming trailing edge surfaces. The latter case includes Active Aerelastic Wing cncepts (AAW). Because wings are lightweight, structural flexibility and aerelasticity are essential features f mrphing wing design. The energy reuired t change wing crss-sectinal shape t generate lift has tw parts, the energy reuired t strain the structure and the energy reuired t mve against the air pressures n the wing surface. The strain energy stred during the lift generatin is wasted energy since it cannt be recvered by the pwer system nbard the aircraft. In fact, in sme cases, such as ailern induced distrtin that leads t ailern reversal, all f the energy ges t strain the structure and change the pressure distributin, but the result is n net lift and thus there is n useful utput despite a large energy expenditure. We will cnsider three related aer-structure interactin studies t illustrate hw actuatin energy plays a rle in sme mrphing wing design activities. The first f these studies is determining the energy reuired t perate leading and trailing edge devices with different flap-t-chrd ratis at different airspeeds. This study prvides a useful illustratin in hw external aerdynamic frces can be used t assist the nbard cntrl energy t generate lift. The secnd study illustrates the interactin between aerelastic effects and energy reuired t generate lift n a flexible wing. This study has tw parts, ne with a simple strip thery mdel and the ther with a mre accurate aerdynamic cnfiguratin. The third study lks at a cntinuusly defrming surface and fcuses n the strain energy reuired t change frm ne airfil shape t anther, but withut including aerelastic effects. These airfil shapes are limited t NACA 4-series airfils t ensure that all f the shapes cnsidered are smth. A secnd part f the third study examines a set f shell-like finite element mdels and illustrates hw the cmbinatin f actuatr placement,

3 deplyment and structural design can make a large difference in the energy reuired t perate mrphing systems. Minimizing flap deflectin energy fr an idealized -DOF System The first f ur three studies determines the energy reuired t perate leading and trailing edge devices with different flap-t-chrd ratis at different airspeeds t generate lift. A -DOF mdel with pitch (trsin) freedm, θ, abut a hypthetical elastic axis, and a cntrl deflectin, δ, is illustrated in Figure. The airfil chrdwise pressure distributin at three different dynamic pressures is shwn in Figure fr the same ailern input. The integral f the pressure distributin ver the entire surface the lift decreases as airspeed increases and becmes negative when the ailern reverses. This is due t aerelasticity. The hinge mment created by the aerdynamic frces n the ailern aft f the hinge line als changes as dynamic pressure increases. Figure indicates that the mment abut the hinge line, due t integrated pressure distributin abut the hinge line, declines as airspeed increases. Pressure distributin pre-reversal reversal pst-reversal The pressure distributin (psitive up) n the airfil is the linear superpsitin f the pressure distributin (psitive upward) due t the ailern deflectin (psitive dwn, as in the figure) written as p δ due t δ This is written as p δ Figure Aerdynamic pressure distributins n a flexible airfil mdel at three different airspeeds. = and a twist angle θ (psitive nse-up) that ccurs when the ailern is deflected. p due t θ given as θ = f δ where = θ. Static euilibrium cnsideratins lead t the relatin between θ and δ ο p θ divergence C Lδ f = reversal C Lα and =. In general, f is a negative number. divergence The hinge mment due t the aerdynamic pressure n the flap is written as Mδ = Hδδ where xtrailing δ Hδ = ( p fp )( x xh ) dx x δ + θ. The wrk dne by the ailern is Wailern = H δ. This wrk is zer if the hinge hinge mment is zer. This special case ccurs at airspeeds where the fllwing is true. This ccurs when CL c C δ macδ f = + CL e C α L α xtrailing ( ) = ( ) () xtrailing p x x x H dx f p x xh dx. () δ hinge x θ hinge = xtrailing xhinge θ ( ) p x δ x xh dx hinge a5 =. The dynamic pressure at xtrailing p x x dx a6 ( ) which this ccurs is defined as and is fund t be a5 = = divergence CL c C δ macδ a + a CL e C α Lα 6 5 H (3) 3

4 Plts f and the nndimensinalized reversal dynamic pressure reversal reversal = are divergence presented in Figure 3, pltted as functins f the ailern hinge line lcatin (r flap-t-chrd-rati) and nndimensinal dynamic pressure =. Reversal ccurs befre the zer divergence mment dynamic pressure ccurs. Near this cmbinatin f dynamic pressure and hinge mment lcatin, this airfil/cntrl cmbinatin reuires nly a small amunt f wrk t generate lift. Theretically, at the ailern wrk reuired is zer because the airstream des all f the wrk. Flap t Chrd Rati / Figure 3 Nndimensinal reversal D dynamic pressure (blue) and nndimensinal minimum energy dynamic pressure (red) as a functin f flap-t-chrd rati divergence Reversal Lw Hinge Mments = div Figure 4 Lift per unit actuatr energy vs. = and flap-t-chrd rati. Dark blue shading represents lw lift inefficient regins; red/yellw regins represent high efficiency regins. Slid lines represent lines f cnstant lift. panel V panel ailern ailern Figure 5 Tw-degree-f-freedm segmented airfil mdel with intercnnecting trsinal springs and ailerns. The efficiency f the ailern in generating lift is measured as the lift generated per unit f wrk input int the ailern cntrl surface. A cntur plt f lift generated by the ailern, divided by the wrk dne by the ailern, is prvided in Figure 4. In this plt, the lgarithm f the rati is pltted. Figure 4 als shws lines f eual lift. Thse nearest reversal are smallest because f lift ineffectiveness. Nte that the mst efficient flap-t-chrd and dynamic pressure cmbinatins are thse near. The mst inefficient peratin ccurs near reversal, the dark blue line in Figures 3 and 4. Several cnclusins may be drawn frm this simple study. First f all, aerelasticity matters and affects peratin f the cntrl device. Secnd, the stiffness f the structure and the size f the actuatr can be tailred fr lw energy perfrmance. A similar study with a leading edge device, nt shwn here, shws that there is a similar leveraging f aerelastic effects fr airfil/structure design, althugh reversal des nt ccur. Wing actuatr efficiency mdeling The secnd prblem t illustrate energy ptimizatin fr mrphing wings uses the mdel shwn in Figure 5. This wing mdel cnsists f tw spanwise segments, each with a trailing edge r leading edge articulated surface. We want t find hw t create a fixed amunt f lift by deflecting cntrl surfaces, but distribute the lift ver the wing segments s that the ttal energy reuired t deflect the surfaces is minimized. Fr mdels f this type, the articulated surface hinge mment, M i, is related t the ensemble f cntrl 4

5 deflectins δ j by the euatin M = H δ (4) i j j= where the cefficients H are hinge mment cefficients that prvide the influence f each surface n the M = H δ s that hinge mment in uestin. The expressin fr the ensemble f hinge mments is { i} { j} the wrk dne by the cntrllers is J = δi { Mi} = δ i H { δ j} (5) We nrmalize this expressin by dividing by a reference cefficient s that J H = δ = δ δ H H { M } { } i i i j (6) If we simplify ur aerdynamic mdel t remve the crss-cupling terms H and H and assume that H and H are eual then the perfrmance index J simply becmes J = δ + δ (7) The bjective is t generate a fixed amunt f lift L with the cntrl surfaces that deflect amunts δ and δ. Figure 6 illustrates the fact that the energy minimizatin prblem in which we minimize En. 7 while generating a specified lift has a clsed-frm slutin. When we use simple aerdynamic strip thery, the euatins f static euilibrium prvide tw simultaneus euatins in terms f the twist angles f each sectin and trsinal stiffnesses f the springs cnnecting the panels with each ther and the wall. The reuirement fr cnstant lift L plts as a straight line L = A δ + A δ, as shwn in Figure 6. ( ) ( ) δ = LA pt A + A J =cnstant δ L = A ( + δ ) δ A ( ) L =cnstant δ Figure 6 Graphical depictin f energy ptimizatin with cnstant lift. The cefficients A and A are functins f dynamic pressure,, because f aerelastic effects. The least energy slutin is fund by lcating the tangent between the circle defined in En. 7 and the cnstant lift line. When this tangency pint is input int En. 7 the result is J pt = L A + A (8) Since A and A depend n aerelasticity, J is a functin f the aerelastic features f the design. If, at a certain dynamic pressure, ne f the cntrl surfaces reverses, then the cefficient A i fr that surface will be zer and the cnstant lift line in Figure 6 will be parallel t ne f the axes. The minimum energy slutin will reuire that nly ne f the surfaces generate lift while the ther is idle. 5

6 Figure 7 shws hw the ailern deflectin n each panel changes with nn-dimensinal dynamic pressure =. The vertical divergence axis measures the amunt f ailern deflectin (per radian) per unit f reuired wing lift. Nte that the sum f the lift generated by each sectin is cnstant. As airspeed increases, if the ailern n ne f the panels reverses, it will then deflect upward (a negative number in Figure 7) t create psitive lift. Figure 8 plts the nn-dimensinal energy functin J is pltted as a functin f nndimensinal dynamic pressure =. divergence Figure 8 shws that the energy reuired is large between the reversal pints f individual surfaces shwn in Figure 7. The maximum amunt f energy reuired des nt ccur exactly at the reversal pint fr either ailern. Instead, there is a pint where the cmbinatin f structural stiffness and ailern arrangement creates the need fr large actuatr energy inputs t achieve the reuired lift. The dynamic pressure at which this ccurs can be lcated exactly and this slutin will be discussed in the next sectin. delta pt /L bar J pt /L Panel reversal Variatin f Optimal Flap Energy delta delta Panel reversal Figure 7 Ailern deflectin reuired fr each panel segment t generate a fixed amunt f lift n a DOF wing vs. =. divergence Multi-degree-f-freedm wing actuatr mdel A mre accurate, but still simplistic, mdel f a flexible wing with leading and trailing edge actuatrs prvides further insight t the minimum energy actuatin lift generatin prblem. The mdel shwn in Figure 9 uses a distributin f hrseshe vrtices n a flexible, bar Figure 8 Objective functin J per unit wing lift fr the DOF wing mdel vs. =. divergence beam-like wing t prvide matrix euatins fr spanwise lift as a functin f wing spanwise angle f attack distributin and cntrl surface deflectins. The static euilibrium f frces and mments alng the wing reference axis is written in matrix frm as 3 A C l = + Z + E { } { α } { δ } j r j r B { lj} = { αr} + Z + E { δ j} (9) () s that the lift distributin is {} l { } i = B αr + B Z + E { δ j} () Figure 9 Idealized beam-like wing mdel shwing hrseshe vrtex ensemble. 6

7 The matrices Z and E are functins f cntrller aerdynamic derivatives, lcatin and wing bending and trsinal stiffness. The matrix [ B ] = [ A ] [ C ] is the aerelastic flexibility matrix fr the symmetrical, cantilevered wing. This matrix is a functin f the distributin f aerdynamic derivatives alng the wing, either estimated r measured, as well as planfrm gemetry and structural stiffness. The distributin f knwn initial angles f attack alng the wing is given by { α r } while the distributin f δ. Fr this study, we set { α } = s that cntrl surface deflectins is given by { } i r {} l i = B Z + E { δ j} () If the design bjective is t use active cntrl surfaces t increase lift (t any distributin favred by the designer), then we can use cmbinatins f full span ailerns r full span leading edge devices t d this. If there are as many cntrl elements as there are panel segments then there is a clsed-frm slutin t this prblem. Fr instance, if we wish t cmpute the reuired ailern deflectins t create a lift distributin with knwn resultant and an elliptical distributin fr minimum drag, then we can cmpute the distributin t be { δi} = Z + E B { lj} In this case, the slutin is clsed-frm and des nt reuire minimizatin because there are n free design variables. Euatin 3 reuires the inversin f the matrix Z E +. This matrix is a functin f the wing trsinal and bending stiffness, as well as the size and lcatin f the leading edge and trailing edge devices. This suggests the fllwing eigenvalue prblem elliptical (3) r { } {} Z E + δ = j { δ } { δ } (4) E Z j = j (5) Euatin 5 shws that there is a self-euilibrating deflectin shape { δ i } at the dynamic pressure,, fr which the deflected surfaces and wing distrtin cmbinatins are in static euilibrium n matter what the magnitude f the deflectin. Hwever, at this airspeed the ailerns cannt create lift. This is a multi-degree f freedm reversal speed, but it differs frm the classical reversal speed, since the classical divergence speed is fr a single ailern instead f an ensemble f ailerns. Euatin 5 is an eigenvalue prblem and the cntrl deflectins are an eigenvectr. Near r at the special eigenvalue, the ailerns will generate lift if they are nt deflected in the eigenvectr shape, but there is a high actuatr energy cst assciated with this cntrl-induced lift. T study this phenmenn, we created a cnstant chrd, swept wing mdel representative f a high-altitude flying wing. Flap deflectin [rad] Tip Rt Figure Ailern deflectins fr a cntinuus set f trailing edge cntrls pltted against Mach number. 7

8 Energy Panel Lift Rt Tip Figure Spanwise lift distributin created fr the actuatr minimum energy prblem Figure - Minimum actuatin energy functin J as a functin f Mach number Flap deflectin [rad] Tip Rt Rt Tip Figure 3 - Ailern deflectins fr a cntinuus set f trailing edge cntrls vs. Mach number. The wing sweep angle is 5.5 while the span is nearly feet. The euatins fr static euilibrium given in En. 9 were prgrammed int a MATLAB surce cde. The ptimizatin prblem is t minimize J subject t the cnstraint that L = 4 lb. This prblem was slved using the MATLAB ptimizatin tl kit. This cnfiguratin was fund t have a reversal eigenvalue at M=.38. Figure shws the slutin fr minimum energy trailing edge flap deflectins (in radians) as a functin f Mach number. Depending n their lcatin alng the span, the utbard cntrl surfaces begin t reverse at relatively lw Mach numbers (indicated by the change in deflectins frm psitive t negative in Figure ). The inbard panels d nt reverse in the Mach number range cnsidered. Figure shws hw the lift distributin adjusts as Mach number increases. As Mach number increases, the inbard panels (n the left side in this figure) must generate mre f the lad as the utbard ailerns begin t reverse. As Mach number increases, the reversed trailing edge surfaces becme mre effective in their reversed state than the inbard panels, causing the majrity f the lad t shift utbard. The spanwise lift distributin is distrted and induced drag changes. Figure indicates that there is a dramatic increase in the inbard lift at abut M=.38, fllwed by an increase in the utbard lift as the reversed panels begin t generate mre lad. Figure shws the minimum actuatin energy functin J as a functin f Mach number. In cntrast t the -degree-ffreedm mdel, there is n peak in actuatin energy, but instead there is a steady decline in the energy due t aerelastic effects, in this case the peratin f reversed ailern segments. As presently psed, there is n cnstraint n spanwise lift distributin, nly a reuirement that the lift have a specified net value. As a result, the ailerns are free t avid actuatr shapes that resemble the eigenvectr at ur reversal eigenvalue f.38. If we add an additinal cnstraint t the prblem and frce the spanwise lift distributin t be elliptical s 8

9 that we include minimum induced drag int the prblem, the results are very different. Because the number f design variables (deflectins) euals the number f cnstraints (elliptical panel lift distributin with a specified net value), there is nly ne cmbinatin f trailing edge deflectins that will prduce an elliptical lift distributin. Therefre, energy is nt minimized, it is simply calculated and the frmal ptimizatin prblem disappears. Figure 3 shws the trailing edge surface deflectins reuired t create an elliptical lift distributin. The deflectins becme large at the critical reversal Mach number defined in En. 5. Figure 4 shws the cntrl energy reuired t generate an elliptical lift distributin. As the determinant f the matrix Z E + becmes small and the trailing edge deflectins becme large and the actuatin energy functin als becmes large, but nt infinite since the cntrl surface deflectin pattern is nt exactly the same shape as a system eigenvalue. When bth leading edge and trailing edge surfaces are included in the cntrl prblem with elliptical lift distributin reuirements, an ptimizatin prblem nce again appears since there are mre design variables than there are cnstraints. Figure 5 shws the leading and trailing edge deflectins that prduce an elliptical lift distributin with minimum cntrl energy. At the critical reversal Mach number, the trailing edge surfaces are unable t prduce the reuired distributin withut substantial energy expenditure s the mre effective leading edge surfaces begin t be deplyed. Figure 6 shws the minimum actuatin energy functin J reuired t create an elliptical lift distributin with a cmbinatin f leading and trailing edge surfaces. The energy des have a peak assciated with the system eigenvalue, but it n lnger is extremely large. The additin f leading edge flaps allws the lift distributin t be maintained with reasnable cntrl surface deflectins. Summary These reversal pints and the assciated reversal prblems are revealed by an eigenvalue prblem such as that discussed in this sectin. Fr trailing edge surfaces, this eigenvalue prblem will mst likely define regins f ineffectiveness that are within the flight envelpe. These regins are functins f the usual ensemble f aerelastic parameters, including trsinal and bending stiffness and flap-t-chrd rati. Flap deflectin [rad] Energy System determinant = (Energy peak) Figure 4 - Cntrl energy J reuired t generate elliptical lift distributin vs. Mach number Leading Edge Trailing Edge Figure 5 Optimal ailern deflectins fr a cntinuus leading edge and trailing edge cntrls vs. Mach number. Energy Figure 6 - Cntrl energy J reuired t generate elliptical lift distributin with leading edge and trailing edge cntrls vs. Mach number. 9

10 Leading edge surfaces have eigenvalues that are usually negative and thus cannt experience reversal. When a cmbinatin f leading edge and trailing edge surfaces are used and a frmal ptimizatin rutine is emplyed, effective cmbinatins f lift generating surfaces are revealed. Minimizing strain energy assciated with airfil shape change Mdern mrphing wing cncepts cntain segments with internal cntrllers that cntinuusly change the shape f the wing in flight withut the hinge-line discntinuity assciated with articulated surfaces (cf. References 4-9). In such cases, the efficient design f mrphing structures t create lift and mments in flight must cnsider the structural strain energy reuired t change the lift n the surface as it transitins frm ne airfil shape t anther. The final part f ur study cnsidered hw t defrm tw dimensinal airfil sectins using the least amunt f energy t achieve the gal f creating lift. The stated gal is t determine hw t pse a useful, frmal ptimizatin prblem t help us in the design prcess and actuatr selectin prcess. When an airfil changes its shape, the strain energy change reuired depends n the type f sub-structure enclsed within the airfil shape. This internal tplgy must be designed, tgether with the actuatrs and their lcatins. T examine hw much energy is reuired t change frm ne airfil shape t anther, we assumed that the airfil structures are slid, cellular structures, idealized as a cntinuus cllectin f internal springs. Aerdynamic and aerelastic effects were nt cnsidered fr this study. The airfil shapes used in the study were limited t the NACA 4-series t ensure that all f the shapes were smth and realizable. These airfils have numbers such as NACA 4. This airfil series is described by three design parameters: the first number is the maximum camber-t-chrd rati, being %, the chrdwise lcatin f maximum camber, 4 being the 4% psitin, and the thickness-t-chrd rati, being % thick. All airfils cnsidered fr this study have the maximum camber lcatin fixed at the 4% chrd psitin. Any candidate airfil shape is described by nly the camber-t-chrd and thickness-tchrd design variables. Because n aerdynamic pressures are included in ur mdel, the nly expenditure f energy is the strain energy change reuired t transitin frm ne airfil shape t anther. Chsing an initial airfil shape allws cnstructin f an energy respnse surface t describe the strain energy assciated with shape changes. Fr this study, the maximum camber varied between % and 9% f the chrd and the thicknesst-chrd rati varied frm % t 5%. Strain energy is develped in the springs when frce actuatrs displace the initial airfil cntur t generate a new shape specified by new maximum camber and thickness values. There are tw springs at each chrdwise lcatin. One spring cnnects the camber line t the upper surface while the ther spring cnnects the camber line t the lwer surface. The strain energy assciated with the change in spring lengths is given as: U = k L n i i (6) i= EA where ki = is the stiffness f spring i and L i is the change in length f spring i. Euatin 6 becmes L i EA U = n Li (7) i= Li Euatin 7 can be nn-dimensinalized with respect t the chrd length, c, t give U ND.. n = U L/ c EAc = / (8) i= ( ) ( ) i L c i

11 During airfil defrmatin, the camber line f the riginal airfil remains grunded, s that the change in spring length is gverned slely by the displacement f the airfil s uter surfaces with respect t the initial camber line. This mdeling apprach is illustrated in Figures 7a and 7b. This mdeling scheme reflects a shape change created by actuatrs applying vertical frces. Original airfil Initial upper spring length Initial lwer spring length Original camber line New airfil Original camber line Defrmed upper spring length Defrmed lwer spring length Figure 7a: (left): Representative initial spring setup fr the starting airfil. Figure 7b (right): Spring defrmatin after changing t the new airfil shape. By chsing an initial airfil shape with a given lift cefficient and displacing it int all ther allwable shapes, it is pssible t determine the strain energy respnse surface. Figure 8 shws this respnse surface using the NACA 4 airfil as the initial, reference airfil. The reference strain energy is zer. The cnstant strain cnturs are shwn as bld lines. Figure 8 als shws lines f cnstant c l (thin slid lines) and cnstant c.8 d (thin dashed lines). Lines with larger c l values are lcated nearer the tp f Figure 8 while larger drag values are.6 lcated in the upper right part f the figure. Figure 8 shws that there are many airfil shapes capable f satisfying a cnstraint n c l, but there exists nly ne that can be mdified fr the least amunt f energy expended t defrm the surface. If cnstraints exist n bth c l and c d there is n ptimizatin prblem because nly ne airfil shape satisfies bth f the cnstraints. Hwever, since the aerdynamic perfrmance cnstraints are predetermined, results such as thse in Figure 8 can help t determine what these cnstraints shuld be by shwing the tradeffs between aerdynamic perfrmance and energy. Fr instance, it might be seen that fr a small increase in allwable drag, large reductins in actuatin strain energy can be btained. Adaptive shell mdels The internal spring mdel strain energy is based n a cellular wing structure with cntrlled displacements frm the riginal camber line. Hwever, the linear spring mdel des nt explicitly describe the strain energy assciated with a shell structure like that usually used in wing design. In particular, the shell skin may underg substantial perimeter changes as the airfil shape changes. Figure 9 shws a Max Camber/Chrd f Final Airfil Thickness/Chrd f Final Airfil Figure 8: Strain energy assciated with changing a 4 airfil. The new airfil shape can be determined frm the thickness/chrd and max camber/chrd ratis; the lcatin f maximum camber is 4% f the chrd. Figure 9: Representative airfil shell finite element mdel. Figure : Typical strain energy distributin resulting frm the shape change f a 4 airfil via vertical ndal displacements.

12 representative finite element mdel created using ASTROS System). (Autmated STRuctural Optimizatin There are tw ways t change frm ne airfil shape t anther. We can simply mve pints straight upward frm the initial psitin r we can mve pints at an angle. Althugh the final shape is the same, the amunt f bending and stretching f the shell is different. In bth cases the changes in the airfil shape were created by defining the ndal displacements f the final airfil shape. When we mve the initial pints vertically, the strain energy distributin is shwn in Figure. Nte that strain energy is cncentrated at the leading edge because leading edge elements have large bending and in-plane strains, while mid-chrd elements underg nearly rigid-bdy translatins. When ndes are displaced alng a vectr that yields the shrtest distance between the initial and final airfil shapes the strain energy distributin is shwn in Figure. These results indicate that this alternative displacement methd distributes the strain energy ver a larger area f the airfil and the leading edge stress cncentratin is nt as large. Figure : Typical strain energy distributin resulting frm the shape change f a 4 airfil via minimum distance ndal displacements Max Camber/Chrd f Final Airfil Thickness/Chrd f Final Airfil Max Camber/Chrd f Final Airfil Thickness/Chrd f Final Airfil Max Camber/Chrd f Final Airfil Thickness/Chrd f Final Airfil Figure - Strain energy respnse surfaces fr a 4 airfil: (a) (left) mdeled with internal springs; (b) (center) mdeled with finite elements, vertical ndal displacement scheme; (c) mdeled with finite elements, minimum distance ndal displacement scheme. When the shell mdel is cmpared t the riginal cellular structure mdel the shapes f the strain energy respnse surfaces are different. There are als differences between the tw ndal displacement methds. These differences are shwn in Figures a-c. These results imply that the strain energy develped during a shape change depends nt nly n the initial and final airfil shapes, but als n the manner in which the shape change is carried ut. They als imply that ne type f actuatr may be better than anther s that the shape change has minimum actuatin energy. Optimizatin t minimize the strain energy assciated with an airfil shape change is dependent upn the actuatin apprach used t create the shape change. Cnclusin - actuatr/structure integratin and aerelasticity matter The develpment f smart materials and new actuatrs has created the ability t design and use cntinuusly defrming surfaces t generate aerdynamic frces and mments. Aerelastic effects allw the airstream t d wrk n the airfil t generate lift with reduced actuatr effrt. As a result, there are flight regins where certain types f actuatrs perate mre efficiently. Hwever, cntinuusly defrming surfaces encunter different prblems than cnventinal surfaces because they may be deplyed in an infinite number f different shapes and sme deplyment schemes are better r mre effective. Cntinuusly defrming cntrl surfaces are subject t a special reversal phenmena that limit their effectiveness. These reversal prblems are revealed by an eigenvalue prblem that has a critical dynamic pressure and a mde shape. The result is a range f Mach numbers r dynamic pressures where the surfaces are ineffective. Fr trailing edge surfaces, these ineffectiveness regins are functins f an ensemble f aerelastic parameters, including trsinal and bending stiffness and flap-t-chrd rati.

13 Cntinuus leading edge surfaces have critical reversal eigenvalues r dynamic pressures that are negative; these surfaces d nt experience cntrl reversal. When a cmbinatin f leading edge and trailing edge surfaces are used and a frmal ptimizatin rutine is emplyed, effective cmbinatins f lift generating surfaces are revealed. When the surface itself is defrmed by internal actuatrs that strain the structure, sme shapes that generate the reuired lift are easier t mrph int. The results shwn here indicate that strain energy is a useful metric t use t judge hw t g frm ne lift reuirement t anther. What is needed is a mdel with an ptimizatin prcedure that, unlike tday s weight minimizatin, is multi-disciplinary in that it cnsiders metrics such as strain energy as the perfrmance index and takes int accunt changes in the structure and aerdynamic measures such as lift and drag. ACKNOWLEDGEMENTS This research was spnsred by the Air Vehicles Directrate, Air Frce Research Labratry, Wright- Pattersn AFB, Ohi. The authrs thank Dr. Brian Sanders f the Air Vehicles Directrate fr technical cmments and valuable suggestins during the curse f this wrk. REFERENCES. McGwan, A.R., et al., Recent Results frm NASA s Mrphing Prject, SPIE Paper N , 9 th Internatinal Sympsium n Smart Structures and Materials, March 7-,, San Dieg, Califrnia.. Perry, B., Cle, S. and Miller, G., Summary f an Active Flexible Wing Prgram, Jurnal f Aircraft, Vl. 3, N., Jan.-Feb. 995, pp Weisshaar, T.A., Frward Swept Wing Static Aerelasticity, AFFDL-TR , Air Frce Flight Dynamics Labratry, Wright-Pattersn AFB, Ohi. 4. Saggere, L., Kta, S., Static Shape Cntrl f Smart Structures Using Cmpliant Mechanisms. AIAA Jurnal, Vl. 37, N. 5, May 999, pp Spillman, J. J. The Use f Variable Camber t Reduce Drag, Weight and Csts f Transprt Aircraft. Aernautical Jurnal, January 99, pp Austin, F., Siclari, M. J.; Van Nstrand, W., Weisensel, G. N., Kttamasu, V., Vlpe, G., Cmparisn f Smart-Wing Cncepts fr Transnic Cruise Drag Reductin. SPIE, Vl. 344, pp Austin, F., Van Nstrand, W., Shape Cntrl f an Adaptive Wing fr Transnic Drag Reductin. SPIE, Vl. 447, pp Mnner, H.P., Breitbach, I., Bein, T., Hanselka, H. Design Aspects f the Adaptive Wing the Elastic Trailing Edge and the Lcal Spiler Bump. Aernautical Jurnal, February, pp Campanile, L. F., Seack, O., Sachau, D. The Belt-Rib Cncept fr Variable-Camber Airfils; Recent Develpments. Prceedings f SPIE Vl Smart Structures and Materials : Smart Structures and Integrated Systems. Pp. -.. Abbtt, I. H.; Vn Denhff, A. E. Thery f Wing Sectins. Dver Publicatins, Inc., New Yrk, Jhnsn, E.H., and V.B. Venkayya, Autmated Structural Optimizatin System, Vl. I, Theretical Manual, Technical Reprt 88-38, Air Frce Wright Aernautical Labratries, March