DYNAMICS AND TRAJECTORY OPTIMIZATION OF MORPHING AIRCRAFT IN PERCHING MANEUVERS

Transcription

1 DYNAMICS AND TRAJECTORY OPTIMIZATION OF MORPHING AIRCRAFT IN PERCHING MANEUVERS A Dissertation Presented to the Facult of the Graduate School of Cornell Universit In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosoph b Adam Michael Wickenheiser Januar 008

2 008 Adam Michael Wickenheiser

3 DYNAMICS AND TRAJECTORY OPTIMIZATION OF MORPHING AIRCRAFT IN PERCHING MANEUVERS Adam Michael Wickenheiser, Ph. D. Cornell Universit 008 Advances in materials, actuators, and control architectures have enabled new design paradigms for unmanned aerial vehicles based on biological inspiration. Indeed, there has been a recent drive to design aircraft features and behaviors based on those of birds, insects, and fling mammals. This dissertation focuses on one such bio-inspired maneuver, perching, which can be described as a low-thrust, aerodnamicall controlled, planted landing. While planted landings are possible in helicopters and jump jets, it has et to be achieved on a low-thrust, high-efficienc platform such as a long loiter reconnaissance aircraft. It is proposed that in-flight shape reconfiguration can enable this class of aircraft to execute this maneuver without sacrificing its cruise performance. This dissertation discusses the perching problem at various levels, from the aerodnamics of the wing to the complete trajector of the maneuver. A lifting-line technique, a variant of Weissinger s classical method, is developed to analze the aerodnamics of morphing wings in the attached flow regime. This tool is used both to stud the effects of morphing on a wing s loading and to populate a database of aerodnamic coefficients for simulating the perching aircraft. Thusl, the aircraft is simulated in the longitudinal plane and compared to a similar fixed-configuration aircraft. Trim and stabilit analses are performed on each aircraft in order to clarif

4 the effects of morphing on the aircraft s longitudinal dnamics. Finall, viable perching trajectories, which bring the aircraft from a cruise configuration to a planted landing, are developed and optimized in terms of their spatial requirements. Three classes of aircraft are simulated in this optimization stud: point-mass, fixedconfiguration (i.e. conventional), and morphing aircraft. Comparisons of these classes show that fixed-configuration aircraft are limited b their pitch maneuverabilit, whereas morphing can alleviate these limitations b increasing the available pitch authorit. It is concluded that morphing increases pitch controllabilit during the maneuver while simultaneousl decreasing the spatial requirements of the perching maneuver.

5 BIOGRAPHICAL SKETCH Adam Wickenheiser received a Bachelor s of Science degree in Mechanical Engineering from Cornell Universit in 00. He also qualified for a minor in Applied Mathematics. As an undergraduate, he was a researcher in the Cornell Universit Fluid Dnamics Laborator and designed experiments for measuring vortex-induced vibrations on oscillating clinders in moving fluids. He also participated in a NASA sponsored joint design project with Sracuse Universit to stud structural and heat transfer properties of thermal tiles for a next-generation reentr vehicle. He attained an overall GPA of 4., the second highest in his graduating class. Upon graduating, Adam continued into the doctoral program at Cornell Universit, receiving a prestigious Cornell Graduate Research Fellowship for his first ear of graduate stud. He joined the Laborator for Intelligent Machine Sstems, directed b Prof. Ephrahim Garcia, in 003. During the same ear, he was admitted into the Graduate Student Research Program at NASA Langle Research Center, receiving a three-ear fellowship for work on morphing aircraft dnamics and control. As part of this program, he worked at NASA Langle during the summers of in the Dnamics & Controls Branch, under the advisement of Mr. Martin Waszak. In Januar 006, he received a Master s of Science degree in Aerospace Engineering for his research entitled Dnamics and Control of Morphing Aircraft in Perching Maneuvers. Subsequentl, Adam continued his research on perching aircraft at Cornell Universit. In 006, he received an ASEE Air Force Summer Graduate Student Award from Wright-Patterson AFB to conduct research on base during the summer. This work iii

6 included the development of a finite-element structural code combined with a panelmethod aerodnamic code for use in modeling morphing wing dnamics. During his doctoral studies, Adam published two papers in the Journal of Aircraft and submitted one paper to the Journal of Guidance, Control, and Dnamics for review. iv

7 ACKNOWLEDGMENTS I want to thank m parents for their ears of support, for smiling and nodding whenever I tried to explain m research to them, and for reminding me that I can t sta a student forever, despite m best efforts. M lab mates Tim Reissman, Justin Manzo, and John Dietl for their great advice, for their terrible jokes, and for putting up with me those times I seemed to lose m mind from overexposure to Matlab. The Big Red Barn for providing a weekl reminder that grad school can be highl amusing at times. Cornell Universit, NASA Langle, Wright-Patterson AFB, and the DSO for funding m research and supporting me during grad school. Dan Sunshine, Carl Whittaker, Jason Harris, Wen Rong Lim, and Joe Andrews for their great help in developing experiments and hardware for m project. Prof. David Caughe and Prof. Jane Wang for their help in preparing for m exams and this dissertation. Finall, I d like to thank m advisor, Prof. Ephrahim Garcia, for his great words of wisdom academic, professional, and otherwise and for inspiring me alwas to earn m take-home pa. v

8 TABLE OF CONTENTS Page Biographical Sketch Acknowledgments List of Figures List of Tables List of Abbreviations List of Smbols iii v viii xii xiii xiv CHAPTER AERODYNAMIC MODELING OF MORPHING WINGS USING AN EXTENDED LIFTING-LINE ANALYSIS. Abstract. Introduction 3. Problem Formulation 5 4. Solution Procedure 9 5. Results 7 6. Conclusion 4 References 5 CHAPTER LONGITUDINAL DYNAMICS OF A PERCHING AIRCRAFT. Abstract 7. Introduction 7 3. Problem Formulation 3 4. Results 38 vi

9 5. Conclusion 5 References 5 CHAPTER 3 OPTIMIZATION OF PERCHING MANEUVERS THROUGH VEHICLE MORPHING. Abstract 54. Introduction Problem Formulation Optimization Results Point-mass Aircraft Conventional Aircraft Morphing Aircraft Conclusion 86 References 88 vii

10 LIST OF FIGURES CHAPTER. Lifting line theor effectivel decouples the 3-D panel problem into a 3 series of -D airfoils. (a) Schematic of the downwash contribution b a segment ds of the 6 lifting vortex (b) Schematic of the downwash contribution b a vortex filament dγ of the trailing vortex sstem.3 Gull wings of constant span with curvature parameter a = 0, 0., Circulation distribution of several gull wings, Λ = 0, α = Downwash angle distribution of several gull wings, Λ = 0, α = Lift per unit length distribution of several gull wings, Λ = 0, α = Drag per unit length distribution of several gull wings, Λ = 0, α = 3.8 Gull wings of constant arc length with curvature parameter a = 0, 0., 0..9 Variation in c.p. and c.g. for twisted and untwisted gull wings on the 3 interval a = [0, 0.], θ max = 5 CHAPTER. The ARES Mars scout 9. The three primar actuations about the pitch axis: A) rotation of the 9 wing incidence angle with respect to the fuselage bod axis, B) rotation of the tail boom, and C) rotation of the horizontal stabilizer.3 Lift curves of ARES-C airfoils 33.4 ARES-C airfoil span locations 33 viii

11 .5 Overview of morphing aircraft simulator 37.6 ARES-C morphing parameters 39.7 Aerodnamic data on the tail calculated b the modified Weissinger s 40 method.8 Aerodnamic data on the blended wing-bod calculated b the 4 modified Weissinger s method.9 Trim results for various angles of attack and smmetric aileron 4 (flaperon) deflections.0 Eigenvalue migration as trim angle of attack varies (δ a = 5 ) 45. Maneuvers for comparison: 46 (a) initiation of perching (b) pitch up. Change in trim conditions throughout the maneuvers 46.3 Eigenvalue migration throughout the maneuvers: the initiation of 48 perching ( ), pitch-up ( ).4 Open-loop position responses to commanded maneuvers 49.5 Open-loop velocit responses to commanded maneuvers 49.6 Open-loop angle of attack responses to commanded maneuvers 50 CHAPTER 3 3. The three shape-change actuations about the pitch axis: A) rotation 57 of the wing incidence angle with respect to the fuselage bod axis, B) rotation of the tail boom, and C) rotation of the horizontal stabilizer 3. Morphing parameters, with directions of increasing value Perching trajectories for a conventional aircraft of varing maximum 6 angle of attack (T/W max = 0.) ix

12 3.4 Division of the perching trajector optimization problem into two 63 phases 3.5 Static mixing parameter p Lift and Moment Coefficients for several elevator deflections Dnamic stall due to rapid angle of attack changes Climb phase trajectories of varing initial velocit Point-mass aircraft climb phase trajectories of varing maximum 7 angle of attack (T/W max = 0.) 3.0 Angle of attack vs. time for point-mass aircraft climb phase 7 trajectories of varing maximum angle of attack (T/W max = 0.) 3. Point-mass aircraft climb phase trajectories of varing maximum 73 thrust-to-weight ratio (α max = 60 ) 3. Thrust-to-weight ratios vs. time for point-mass aircraft climb phase 74 trajectories of varing maximum thrust-to-weight ratio (α max = 60 ) 3.3 Point-mass aircraft full trajectories of varing maximum 75 thrust-to-weight ratio (α max = 60 ) 3.4 Climb phase trajectories of varing maximum thrust-to-weight ratio Full trajectories of varing maximum thrust-to-weight ratio Climb phase trajectories of varing center of gravit location 79 (T/W max = 0.) 3.7 Full trajectories of varing center of gravit location (T/W max = 0.) Climb phase trajectories of varing center of gravit location for the 8 aircraft with morphing (T/W max = 0.) 3.9 Climb phase morphing parameter time histories of varing center of 83 gravit (T/W max = 0.): wing incidence (top), tail boom angle (middle), and tail incidence (bottom) x

13 3.0 Full trajectories of varing maximum thrust-to-weight ratio for the 83 aircraft with morphing 3. Climb phase morphing parameter time histories of varing maximum 84 thrust-to-weight ratio: wing incidence (top), tail boom angle (middle), and tail incidence (bottom) 3. Comparison of fixed-configuration and morphing aircraft perching 84 trajectories (T/W max = 0.) 3.3 Elevator effectiveness for fixed-configuration and morphing aircraft 86 xi

14 LIST OF TABLES CHAPTER. Comparison of several gull wings CHAPTER. Parameter variations in aerodnamic database 38 CHAPTER 3 3. Actuator Constraints 60 xii

15 LIST OF ABBREVIATIONS ARES Aerial Regional-Scale Environmental Surve ARES-C Aerial Regional-Scale Environmental Surve (at Cornell) c.g. c.p. CFD Datcom ISR UAV USAF V/STOL Center of Gravit Center of Pressure Computational Fluid Dnamics Data Compendium Intelligence, Surveillance, and Reconnaissance Unmanned Aerial Vehicle United States Air Force Vertical/Short Take-Off and Landing xiii

16 LIST OF SYMBOLS CHAPTER a b c ĉ c ĉ C l C d C L C D C Y C L C M C N C lα G L l m M Q r S wing curvature parameter wing span local chord length local nondimensional chord length mean aerodnamic chord nondimensional mean aerodnamic chord section lift coefficient section drag coefficient wing lift force coefficient wing drag force coefficient wing side force coefficient wing roll moment coefficient wing pitch moment coefficient wing aw moment coefficient section lift curve slope nondimensional circulation wing lift force section lift force/length number of points used in sine series expansion of circulation function number of points used in trapezoidal approximation dnamic pressure displacement vector wing planform area xiv

17 Sˆ U v w x cp x cg x c/4 0 α α 0L Λ ε Γ Γ η ξ σ L M N nondimensional wing planform area free stream velocit magnitude wind velocit vector downwash velocit position of the wing center of pressure position of the wing center of gravit position of the airfoil quarter-chord point wing semi-span, -coordinate of wingtip wind incidence angle/angle of attack angle of attack for zero lift wing aspect ratio downwash angle at wing ¼-chord line circulation magnitude vorticit vector nondimensional spanwise coordinate nondimensional chordwise coordinate planar densit wing roll moment wing pitch moment wing aw moment CHAPTER D f x, X f z, Z f aircraft drag force x-component of external force z-component of external force external force vector xv

18 g g I I L M m m m q q T T u V v v w x x z α β δ a δ e gravit magnitude gravit vector principal moment of inertia about -axis moment of inertia matrix aircraft lift force aircraft pitch moment aircraft mass -component of external moment external moment vector pitch rate quaternion thrust magnitude transformation matrix from bod- to earth-coordinates x-component of aircraft velocit aircraft velocit magnitude -component of aircraft velocit aircraft velocit vector (bod coordinates) z-component of aircraft velocit forward direction aircraft position vector (inertial coordinates) sidereal direction vertical direction angle of attack sideslip angle aileron deflection angle elevator or smmetric ruddervator deflection angle xvi

19 θ θ b θ t Θ 0 ι ψ ω ω pitch angle boom angle with respect to fuselage tail angle with respect to boom trim pitch angle wing incidence angle roll angle aw angle aircraft angular velocit vector skew-smmetric cross product matrix of ω CHAPTER 3 C D C L C M C M q c c g h I J l m p p 0 q drag coefficient lift coefficient pitch moment coefficient pitch damping coefficient local chord length mean aerodnamic chord acceleration due to gravit vertical position principal moment of inertia about pitch axis cost function characteristic length aircraft mass dnamic mixing parameter static mixing parameter pitch rate xvii

20 S T T/W t V x x cg x cp x np x α γ δ e θ θ b θ t ι κ ρ τ, τ planform area thrust magnitude thrust-to-weight ratio time aircraft velocit magnitude horizontal position aircraft center of gravit airfoil center of pressure aircraft neutral point state vector angle of attack flight path angle elevator deflection angle pitch angle tail boom angle with respect to fuselage tail angle with respect to boom wing incidence angle with respect to fuselage pitch moment scaling factor air densit time constants Subscripts 0 initial f att climb final attached flow regime climb phase xviii

21 dive fuse sep tail wing dive phase fuselage lifting surface separated flow regime tail lifting surface wing lifting surfaces xix

22 CHAPTER AERODYNAMIC MODELING OF MORPHING WINGS USING AN EXTENDED LIFTING-LINE ANALYSIS. Abstract This chapter presents an extension of Weissinger s method and its use in analzing morphing wings. This method is shown to be ideal for preliminar analses of these wings due to its speed and adaptabilit to man disparate wing geometries. It extends Prandtl s lifting-line theor to planform wings of arbitrar curvature and chord distribution and non-ideal airfoil cross sections. The problem formulation described herein leads to an integrodifferential equation for the unknown circulation distribution. It is solved using Gaussian quadrature and a sine-series representation of this distribution. In this chapter, this technique is used to analze the aerodnamics of a morphable gull-like wing. Specificall, this wing s abilit to manipulate lift-to-drag efficienc and center of pressure location is discussed.. Introduction Throughout the histor of aviation, ver little of man s inspiration for flight has manifested itself in aircraft designs. Indeed, manmade flight bears little resemblance to avian morphologies, which are backed b millions of ears of evolution. Birds morph their wings and tail in complex, fluid was, in contrast to the limited range of motion of an aircraft s control surfaces. Most aircraft deplo flaps and slats during takeoff and landing in order to increase lift at slower speeds. This is an example of a From Wickenheiser, A. and Garcia, E., Aerodnamic Modeling of Morphing Wings Using an Extended Lifting-Line Analsis ; reprinted b permission of the American Institute of Aeronautics and Astronautics, Inc.

23 configuration change that occurs continuousl during avian flight. A bird s morpholog allows it to constantl change its wing and tail shapes to suit flight at a wide range of speeds. Recentl, research and development have begun on a new concept that challenges current designs: morphing aircraft []. A morphing aircraft is an aircraft capable of controlled, gross shape changes in-flight, with the purpose of increasing efficienc, versatilit, and/or mission performance. While traditional aircraft are designed as compromises of various performance needs, a single morphing aircraft can excel at numerous tasks [,3]. The same airframe can morph from a highl efficient glider to a fast, high maneuverabilit vehicle. While a traditional wing is designed for high efficienc over a small range of flight conditions, a morphing wing can adapt to grossl different altitudes and flight speeds. Morphing technologies enable new flight capabilities, such as perching, urban navigation, and indoor flight. These capabilities have heretofore been unrealizable due to technological limitations. Modern development of smart structures, adaptive materials, and distributed and adaptive control theor has opened the door to a host of new aircraft designs and flight capabilities [4]. These new capabilities are realized b the careful manipulation of aerodnamic forces and moments. For example, a long endurance aircraft benefits from a high lift-to-drag ratio, while a highl maneuverable aircraft needs high lift and low (or negative) stabilit margins. Highl efficient cruise can be accomplished b morphing the wing cross sections to maintain high lift-to-drag ratios at various flight speeds and altitudes. New capabilities, such as perching, can be achieved b controlling the degree of separated flow over the aircraft s lifting surfaces [5]. Man of these capabilities

24 require levels of actuation far exceeding the bounds of conventional aircraft control surfaces. Unlike most traditional aircraft, morphing aircraft concepts require an aerodnamic analsis for both varing flight conditions and grossl varing geometric configurations. This requirement demands a preliminar analsis methodolog that is fast, accurate, and reconfigurable, without having to rebuild the mesh of the aircraft or flow field, for example. Consequentl, a lifting-line approach is chosen over a computational fluid dnamics (CFD) approach as the aerodnamic modeling method. This method effectivel breaks the 3-D wing into a series of -D airfoils joined b their quarter-chord curve, as depicted in Figure.. Figure.: Lifting line theor effectivel decouples the 3-D panel problem into a series of -D airfoils The analtic nature of this method allows the wing geometries to be programmed as functions into generic software environments such as Matlab or C. Consequentl, changing geometr parameters, such as the wing curvature parameter presented below, ma be placed in a software loop in order to automaticall generate man wing geometr variations. 3

25 Weissinger s method for straight, swept wings is the basis of the present lifting-line theor [6]. His method relates the downwash air velocit at an given span station on the wing to the sum of the downwash contributions of the vortex line attached to the quarter-chord line of the wing and the semi-infinite vortex sheet trailing behind it. This method does not consider the geometr of the wing cross sections or the nonplanarit of the wake. An effort is made, as explained below, to account for the former b introducing real airfoil data for each of the span stations. Although full 3-D analsis tools such as panel methods and CFD software do not require a separate database of airfoil data in this event, computationall it is more efficient to have these data tabulated beforehand. The method presented below onl has to reference these data instead of needing to re-compute them in its algorithm. This method has been extended to curved wings of a specific (polnomial) form b Prössdorf and Tordella [7] for stationar wings and b Chiocchia, et al. [8] for wings in oscillator motion. The problem formulation leads to an integrodifferential equation as shown in the next section. This equation is solved assuming a sine series representation of the circulation, which conforms to the boundar conditions of no circulation at the wingtips. Gaussian quadrature and the trapezoidal rule are then used to compute the integrals. This technique results in a relativel high /M error, where M is the number of function evaluations; however, this number ma be increased independentl from the number of span stations m used in the aerodnamic calculations, as shown below. Prössdorf shows that error in the calculated circulation distribution decreases exponentiall with m, assuming that the quarter-chord curve can be bounded b a polnomial [7]. This analsis is shown to be effective in computing the lift and drag distributions over a variet of wing geometries. Although this method assumes no separation effects, it 4

26 is valid in the Renolds number regime of medium-scale UAVs and larger aircraft at moderate angles of attack, where viscous effects are minimal. This method s speed and reconfigurabilit make it ideal for the preliminar analsis of morphing wings with a large number of varing geometrical parameters. This method is also useful in the construction of an aerodnamic lookup table for use in an aircraft simulation, for example. 3. Problem Formulation A model of the wing geometr and the flow field is developed in order to formulate the circulation distribution along the span. The circulation is found b examining the downwash velocit distribution in the wake of the lifting surface. First, a Cartesian coordinate sstem is established such that the positive x-direction points downstream, parallel to the free-stream velocit U, and the positive -direction points towards the right wingtip. (Thus, b the right-hand rule, the positive z-direction points outward from the page.) The quarter chord of the wing is represented b a continuous, piecewise differentiable function that extends from = 0 to = 0, not necessaril smmetric about the x-axis but contained entirel in the x-plane. The chord and twist distributions are given as piecewise continuous functions of the spanwise coordinate. The model of the flow field consists of a bound or lifting vortex at the quarter chord curve of the wing and a trailing vortex sheet that extends to infinit downstream. The downwash at each point in the flow field is therefore the sum of the velocities induced b the lifting vortex and the distributed vortex sheet. These two contributions are shown in Figure., where the geometr is defined in a similar manner to Prössdorf [7] and DeYoung [9]. Since the flow field is modeled as a superposition of potential flows, this method onl applies when viscous effects are not significant. 5

27 ( x, ) ds r h c/4 curve ( x, ) θ c/4 curve ( x, ) d ( x, ) x (a) x (b) dγ Figure.: Schematic of the downwash contribution b a segment ds of the lifting vortex (a). Schematic of the downwash contribution b a vortex filament dγ of the trailing vortex sstem (b). The contributions of the lifting and trailing vortices to the downwash velocit w can be calculated b the Biot-Savart Law: v 4π = 3 Γ r ds, () r which gives the fluid velocit at an point displaced from a vortex element of strength Γ. Using this law, the downwash caused b segment ds of the lifting vortex is given b dw Γhds =. () 4πr ( x, ) 3 where the geometr is defined in Figure.(a). In terms of the points ( x, ) in the plane of the wing and ( x, ) along the quarter-chord curve, Eq. () becomes dw ( ) ( ) [ x x( ) + x ( )( ) ] x, = Γ 4π ( x x( ) ) + ( ) d 3 [ ], (3) where 6

28 ( ) dx x ( ) =. d = The downwash caused b an infinitesimal vortex filament dγ in the trailing vortex sheet is given b dγ dw ( x, ) = ( cosθ + ) (4) 4πd where the geometr is defined in Figure.(b). In terms of the points ( x, ) and ( x, ), Eq. (4) becomes dw ( x, ) ( ) ( ) x x( ) ( x x( ) ) + ( ) Γ = + d, (5) 4π where ( ) dγ Γ ( ) =. d = Summing Eqs. (3) and (5) and integrating from 0 to 0 gives the total downwash at the point ( x, ) in the flow field: (, ) w x = + + 4π 4π 4π Γ ( ) Γ d Γ ( ) x x( ) ( x x( ) ) + ( ) x x ( ) ( ) + x ( )( ) 3 ( x x( ) ) + ( ) [ ] d d (6) The first two integrals in Eq. (6) have singularities at = ; however, onl the second integral diverges near the singularit. (The singularit in the first integral will be 7

29 8 addressed below.) To remove this discontinuit, the first term is added to and subtracted from Eq. (6), as recommended b DeYoung [9], resulting in ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) [ ] + + Γ + + Γ + Γ = , d x x x x x d x x x x d x w π π π (7) which is referred to as the dimensional form of the modified Weissinger s method. According to the Pistolesi-Weissinger condition [6,0], the overall wind velocit should be tangent to the plane of the wing at the wing s ¾-chord line. In other words, along this line the downwash angle is equal to the local airfoil s angle of attack, which is the sum of the wing s geometrical twist and its overall angle of attack. Thus, the downwash velocit w in Eq. (7) should be evaluated at ( ) ( ) c x x + =, (8) which is half a chord length behind the quarter-chord line. With this substitution, Eq. (7) becomes ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) [ ] Γ Γ + Γ = / / 4 / / 4 d c x x x c x x d c x x c x x d w π π π, (9)

30 where the downwash is now onl a function of the spanwise coordinate. If the geometr for a straight, swept wing is substituted into Eq. (9), then the lifting-line formula derived b Weissinger [6] and DeYoung [9] can be recovered. 4. Solution Procedure Equation (9) gives the downwash caused b the lifting vortex and the trailing vortex sheet at the point along the ¾-chord line; this downwash should be equal to the upwash felt b the wing due to its local incidence to the flow. Therefore, the onl unknown quantit in Eq. (9) is the circulation distribution Γ(). Although Γ() has no explicit solution, it can be approximated to an arbitrar accurac b a sine series, as first shown b Multhopp []. A transformation to trigonometric coordinates will then allow the exact integration of the first term in Eq. (9) and a simplification of the other two terms. The trapezoidal method is then used to integrate the second and third terms. As will be shown, the number of terms used in this integration can be made independent of the number of terms used in the sine series representation of Γ(). It is now convenient to convert Eq. (9) to non-dimensional form b introducing the following dimensionless variables: η =, 0 η =, 0 G = Γ U 0, x ξ =, c w α =. (0) U Here, it is assumed that all downwash angles are small. In dimensionless form, Eq. (9) can now be written as 9

31 0 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] η η η η η ξ η ξ η η η ξ η ξ η ξ η η π η η η η η ξ η ξ η ξ η ξ η η η π η η η η π α η d c G c d c G d G = / / / 4 / / / 4 () In order to simplif the integrals in Eq. () and cast ( ) η G as a sine series, the spanwise coordinates are transformed into angles b the following definitions: ( ) η cos v and ( ) η cos. () For simplicit, let ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] / / /, / / /, η η η η ξ η ξ η η η ξ η ξ η ξ η η η η η η ξ η ξ η ξ η ξ η η η η c R c P (3) After these substitutions, Eq. () becomes ( ) ( ) ( ) ( ) ( ) ( ) ( ) + = π π π π π π α sin, 4, 4 cos cos d G R c d G P d G v v v v v. (4) In order to solve Eq. (4) for the unknown function G(), it is assumed that G() can be represented as a sine series of m terms. (Note that this representation meets the boundar conditions of no circulation at the wingtips, that is G(0) = G(π) = 0.) Let ( ) ( ) = = m k a k k G sin, where ( ) ( ) = π π 0 sin d k G a k. (5)

32 Multhopp s formula [], based on Gaussian quadrature, is used to evaluate the integral in Eq. (5). This method will exactl integrate a sequence of orthogonal functions such as the sine series representation of G() if m points are chosen for the quadrature. These points must be located at the roots of the next function in the sequence, sin [( m +) π ]. Appling this quadrature to Eq. (5) ields a k = m + m n= G ( ) sin( k ) n n nπ, where n =, (6) m + where the n are the roots of the next function in the sine series. Therefore, G m + m ( ) = G( ) sin ( k ) sin ( k ) n= n= n m k = m m G ( ) = G( n ) k sin( kn ) cos( k ). (7) m + k = n With the definitions f h n n m + m ( ) sin( k ) sin( k ) m + k= m ( ) k sin( k ) cos( k ) k= n n G n ( ) G, (8) n Eq. (7) can be written as G m ( ) = ( ) and G ( ) = ( ) n= G n f n m G n h n. (9) n= Substituting Eqs.(8) and (9) into Eq. (4) gives

33 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = = = = + + = π π π π π π α sin, 4, 4 cos cos cos sin d f R G c d h P G d k k k G m n v m n n v n v m n n v m k n m n n v. (0) Although the first integral has a singularit at = v, the integral is finite and given b the formula ( ) ( ) v v v k d k π π sin sin cos cos cos 0 =, () derived b Glauert []. The trapezoidal method is used to evaluate the second and third integrals in Eq. (0). This formula is given b ( ) ( ) ( ) ( ) = M F F F M d F 0 0 µ µ π π π () where + = M µπ µ, for a general function F(). The integer M dictates how man function evaluations are used to compute the integral and is independent of m, the number of terms in the sine series representation of G(). Using Eqs. () and () to evaluate the integrals in Eq. (0) gives ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = = = = = = = m n M n v n v M n v n v n v m n n m k v v n m n n v f R G c M h P h P h P G M k k k G m 0 sin, 4,, 0,0 4 sin sin sin µ µ µ µ µ µ µ π π ϕ α. (3)

34 The sum in the first term of Eq. (3) has an explicit formula, given b m + ( ) ( ), n = v m k sin kn sin kv 4sinv = m + sin k= sin v n, n v ( cosn cosv ) (4) Equation (3) relates the airfoil angle of attack at v to a linear combination of circulation function evaluations G n. Since α() is a known function, it can be evaluated at m distinct points to create a sstem of equations for G n. Constructing the matrix A = m m k= ( M + ) k sin P ( M + ) c( ) ( k ) sin( k ) (,0) h ( 0) + P(, π ) h ( π ) 0 v v n sin v n M µ = R v (, ) f ( ) v µ v n n µ + sin M µ = µ P (, ) h ( ) v µ n µ, (5) where the vn-th component is evaluated at v and n, Eq. (3) becomes α( ) r r =, where α r AG α = M ( ) and α m G r G = M. (6) G m This is a sstem of m equations for the unknowns G n. The vector α r is comprised of the local angle of attack values at,, m, and G r is a vector of unknowns. The G n can be computed b inverting A, and the circulation distribution can be reconstructed using Eq. (9). So far, this analsis has assumed that the airfoil cross sections of the wing are ideal; that is, the have a lift curve slope of π and generate no lift at a zero-degree angle of 3

35 attack. It is desired to incorporate real airfoil data into this lifting-line analsis in order to predict better the lift on real aircraft wings. This incorporation will also help evaluate the effects of airfoil morphing on the entire wing s aerodnamic properties. These data ma be obtained from experiment or computation; however, as stated previousl, the are onl referenced b this algorithm and not recomputed. In order to assimilate non-ideal airfoils, DeYoung suggests the method of distorting the chord length distribution along the wing such that the dimensional circulation about ever span station matches the dimensional circulation of an ideal airfoil with the original chord length [9]. Alternativel, this can be accomplished b offsetting the left-hand side of Eq. (3) b the true angle of attack for zero lift of that section and scaling it b the ratio of its lift curve slope to that of an ideal airfoil. Consequentl, the left-hand side becomes C l α ( v ) [ α( ) α ( )] = L π v 0L v. (7) The downwash angle at each wing station can now be computed. B Munk s analsis [3], the downwash angle is given b half the downwash angle an infinite distance downstream. To calculate this, take half the limit of Eq. (7) as x goes to infinit: 0 lim w ( x, ) = x 0 4π ( ) Γ d. (8) B converting Eq. (8) to non-dimensional form and casting it terms of the sine series coefficients, the downwash angle ε at station v is given b ( k ) sin( k ) m m k sin n v ε ( v ) = ( ) Gn, (9) m + n= k= sin v which is one half the first term in Eq. (3). With the downwash angle given b Eq. (9), the overall wind incidence angle (wing angle of attack + wing twist + downwash 4

36 angle) can be computed at each station. Although this angle is computed using potential theor, it can be used to acquire a good approximation of section lift and drag forces if real airfoil data are available. The overall wind incidence angle is used in lieu of the angle of attack in determining section lift (C l ) and drag (C d ) coefficients. These coefficients are then rotated back into the xz-coordinate sstem b the angle ε as given b xz Cl Cd cosε = sinε sinε cosε wind C C l d, (30) where xz indicates the xz-coordinate sstem and wind indicates the local wind coordinate sstem. Using the standard definitions of lift coefficient (C L ) and section lift coefficient (C l ), L C L = and QS l C l =, (3) Qc a straightforward integration of these coefficients over the entire wing gives the overall lift and pitching moments of the wing. Starting with the relation between lift and section lift b = / L ld, (3) b / the lift coefficient can then be calculated b C = C ˆ cdη, (33) L l Sˆ Similarl, the drag coefficient is given b C = ˆ D Cdcdη. (34) ˆ S 5

37 In this problem formulation, side forces are considered negligible; therefore, C = 0. (35) Y The moment coefficients are also calculated in a straightforward manner. Starting from the definitions of pitch, roll, and aw moment coefficients, C = M M QSc, L CL =, and QSb N CN =, (36) QSb respectivel, the moment coefficients can be calculated using the section lift and drag coefficients as follows: C = C ˆ M lcξdη, C = ˆˆ Sc C ˆ cηdη, L l Sˆ C = C ˆ cηdη. (37) N d Sˆ Two other important parameters in wing design are the centers of pressure and gravit. The displacement between these two points is ver important in determining the stabilit and dnamic response of the wing in an unstead flight condition and in determining the dnamics of the overall aircraft sstem. In the context of the prescribed geometr, the center of pressure relative to the origin is calculated as follows: Lx x cp cp = M = L b / b / lx c / 4. (38) d Similarl, the center of gravit is equal to 6

38 x cg = = b / b / b / b / b / b / b / c b / σxd σd c xd d (39) Here, it is assumed that the densit of the wing is constant, which leads to a square variation of mass with respect to chord length. 5. Results Using the methods described in the previous section, circulation, downwash, and force distributions are calculated for several wing shapes, as well as overall wing parameters such as lift, drag, and center of pressure location. This method s relativel loose requirements for wing geometr enable some non-traditional wings to be analzed. Along the vein of bio-inspiration, a gull wing shape is chosen as the basis of this analsis. This wing features forward- and aft-swept wing sections that can be utilized for c.g. and c.p. adjustments, as discussed below. The wing is described b the quarter-chord curve 4 x ( ) = a (40) k0 k0 and shown in Figure.3 for various values of curvature parameter a. (A value of k = 3 7 is chosen in order to maintain a constant c.g. location for all values of a, assuming a constant densit wing as described previousl.) 7

39 Figure.3: Gull wings of constant span with curvature parameter a = 0, 0., 0. For this analsis, ever wing shape is constructed using an elliptical chord distribution such that a direct comparison of these wings with the canonical straight, elliptical wing can be made. Also, the same root chord length is used throughout in order to maintain a constant aspect ratio of 0. Although this analsis ma be used for an angle of attack within the range of validit of linear theor, an angle of attack of 3 will be used subsequentl as a point for comparison. As an example, the three wings depicted in Figure.3 are analzed using this liftingline theor and compared in Figs. 4-7, using m = M = 0. Figure.4 shows the circulation distributions Γ() for various values of curvature parameter a. As a increases, the circulation increases towards the center, where the wing behaves locall as a forward-swept wing, while towards the wingtips the aft sweep of the wing causes a local reduction in circulation, compared to the elliptical distribution of the straight wing case. The downwash angle ε(), computed from Eq. (39), is plotted in Figure.5. The downwash is nearl constant in the case of the straight elliptical wing (a = 8

40 0), as predicted b Prandtl [4] and Munk [3]. As the wing curvature increases, downwash on the forward-swept sections of the wing increases while decreasing across the aft-swept portions. Similar effects of swept curvature on downwash angle have been shown for parabolic wings b Prössdorf and Tordella. The note that the largest decrease in induced velocit occurs towards the wingtips [7]. Figure.4: Circulation distribution of several gull wings, Λ = 0, α = 3 Figure.5: Downwash angle distribution of several gull wings, Λ = 0, α = 3 9

41 Figure.6 is a plot of the lift force per unit length distribution across the same three gull wings, while Figure.7 displas the drag force. As expected, the lift and drag distributions are both elliptical for the straight wing case. The lift distributions are approximatel equal to the circulation distributions, scaled b a factor of ρ U, since in all cases the downwash angles are small. Similarl, the drag distributions follow the same patterns as the downwash angle distributions. These plots indicate the gull wing s abilit to shift the center of lift forward as the wing morphs from straight to curved, although a drag penalt is incurred. Figure.6: Lift per unit length distribution of several gull wings, Λ = 0, α = 3 0

42 Figure.7: Drag per unit length distribution of several gull wings, Λ = 0, α = 3 The preceding comparison of several gull wings of constant span (and aspect ratio) is desired from a purel theoretical stance since aspect ratio is an important nondimensional parameter when discussing finite wings. For example, aspect ratio is a major factor in comparing the lift-to-drag ratios of several wings of similar shape. However, when developing and analzing morphing wing designs, a constant aspect ratio is often difficult to maintain due to practical limits on planform deformation. Variable-swept wing aircraft such as the F- and the F-4 clearl exemplif the reduction in aspect ratio that morphing wing technologies suffer. To return to the previous example, Figure.8 depicts several gull wings of the same curvature parameters as before but now with constant arc length. These wings more clearl illustrate the deformations caused b bending a straight wing along the quarter-chord line in order to achieve a gull-like geometr. Once again these wings are curved in such a wa as to maintain a constant center of gravit location. These wings are compared with the wings depicted in Figure.3 in Table. below.

43 Figure.8: Gull wings of constant arc length with curvature parameter a = 0, 0., 0. Table.: Comparison of several gull wings Wings of constant span Wings of constant arc length a Lift, N Drag, N Lift/Drag Lift, N Drag, N Lift/Drag e e e e e e As expected, the wings of smaller aspect ratio have reduced lift-to-drag efficienc. This effect is much greater than merel changing the curvature of constant aspect ratio wings. Also note that there is a maximum in drag force within this range of curvature parameter for wings of constant span. Initiall drag increases with curvature due to added downwash over the forward swept sections of the wing; however, at higher curvatures this is mitigated b the reduced downwash over the highl aft-swept wingtip sections. As shown b Table., morphing a straight wing into a gull-like configuration is useful for lift reduction for higher speed flight. For example, a high endurance

44 reconnaissance aircraft ma morph its wings b increasing their curvature parameter a in order to perform a high-speed dive in order to evade an enem. The added feature of forward- and aft-swept wing sections allows the manipulation of the center of pressure if wing twist can be commanded as a function of span. Figure.9: Variation in c.p. and c.g. for twisted and untwisted gull wings on the interval a = [0, 0.], θ max = 5 Figure.9 shows the variation in centers of pressure and gravit with curvature parameter for gull wings of constant wingspan. Untwisted wings, like those discussed above, are compared to twisted wings with twist distributions of the form θ π max, (4) k ( ) = θ sin where once again k = 3 7. Figure.9 indicates that with a maximum twist of onl 5 degrees, the center of pressure can be shifted forward b 8% of the root chord while not moving the center of gravit. Thus, a change in wing configuration from straight to gull can be used to reduce the static margin of the aircraft, for example. 3

45 6. Conclusion An extension of Weissinger s method to curved wings provides a useful analsis tool for the preliminar design of morphing wings. This method can be easil applied to wings whose geometr can be described b piecewise analtical functions. The analtical nature of this technique allows specific geometrical parameters to be varied and their effects on the wing s aerodnamics to be analzed; however, caution must be taken when considering flows where viscous effects dominate. In this chapter, a morphing gull wing is analzed in the cases of both constant span and constant arc length. It is shown that increased curvature of the wing results in reduced lift and liftto-drag efficienc, confirming this morpholog s usefulness in loiter to high-speed dash reconfiguration. Also, this wing s abilit to manipulate its center of pressure location relative to its center of gravit is discussed. Each of these studies demonstrates the usefulness of this analsis technique, as long as the bounds of this method s validit are not overstepped. 4

46 REFERENCES [] Sanders, B., Crowe, R., and Garcia, E., Defense Advanced Research Projects Agenc Smart Materials and Structures Demonstration Program Overview, Journal of Intelligent Material Sstems and Structures, Vol. 5, 004, pp [] Bowman, J., Sanders, B., and Weisshaar, T., Evaluating the Impact of Morphing Technologies on Aircraft Performance, AIAA Paper 00-63, 00. [3] Wickenheiser, A., Garcia, E., and Waszak, M., Evaluation of Bio-Inspired Morphing Concepts with Regard to Aircraft Dnamics and Performance, Proceedings of SPIE The International Societ for Optical Engineering, Vol. 5390, 004, pp. 0. [4] McGowan, A. R., AVST Morphing Project Research Summaries in FY 00, NASA TM , 00. [5] Wickenheiser, A., Garcia, E., and Waszak, M., Longitudinal Dnamics of a Perching Aircraft Concept, Proceedings of SPIE The International Societ for Optical Engineering, Vol. 5764, 005, pp [6] Weissinger, J., The Lift Distribution of Swept-Back Wings, NACA TM-0, 947. [7] Prössdorf, S., and Tordella, D., On an Extension of Prandtl s Lifting Line Theor to Curved Wings, Impact of Computing in Science and Engineering, Vol. 3, No. 3, 99, pp. 9. [8] Chiocchia, G., Tordella, D., and Prössdorf, S. The Lifting Line Equation for a Curved Wing in Oscillator Motion, Zeitschrift fuer Angewandte Mathematik und Mechanik, Vol. 77, No. 4, 997, pp

47 [9] DeYoung, J., and Harper, C. W., Theoretical Smmetric Span Loading at Subsonic Speeds for Wings Having Arbitrar Plan Form, NACA Report No. 9, 948. [0] Multhopp, H.. Die Berechnung der Auftriebsverteilung von Tragflgeln (The Calculation of the Lift Distribution of Wings), Luftfahrtforschung, Vol. 5, 938, pp (translated as British RTP translation No. 39). [] Glauert, H., The Elements of Aerofoil and Airscrew Theor, The Universit Press, Cambridge, 943. [] Munk, M. M., The Minimum Induced Drag of Airfoils, NACA Report, 9. [3] Prandtl, L., Essentials of Fluid Dnamics, Hafner Publishing Compan, New York, 95, pp This chapter originall appeared as: Wickenheiser, A., and Garcia, E., Aerodnamic Modeling of Morphing Wings Using an Extended Lifting-Line Analsis, Journal of Aircraft, Vol. 44, No., 007, pp

48 CHAPTER LONGITUDINAL DYNAMICS OF A PERCHING AIRCRAFT. Abstract This chapter introduces a morphing aircraft concept whose purpose is to demonstrate a new bio-inspired flight capabilit: perching. Perching is a maneuver that utilizes primaril aerodnamics as opposed to thrust generation to achieve a vertical or short landing. The flight vehicle that will accomplish this is described herein with particular emphasis on its addition levels of actuation beond the traditional aircraft control surfaces. The dnamics of this aircraft are examined with respect to changing vehicle configuration and flight condition. The analsis methodologies include an analtical and empirical aerodnamic analsis, trim and stabilit analses, and flight simulation. For this stud, the aircraft s motions are limited to the longitudinal plane onl. Specificall, cruise and the perching maneuver are examined, and comparisons are drawn between maneuvers involving vehicle reconfiguration and those that do not.. Introduction One of the major goals of the morphing aircraft program is the enabling of new flight capabilities and missions [-3]. With additional levels of sensing and actuation, morphing aircraft are able to mimic more closel the capabilities of man s inspiration for flight: birds. The gross extents to which birds morph their bodies allow them to perform maneuvers irreproducible b conventional aircraft. One such avian maneuver is perching. Perching can be described as a high angle-of-attack approach, with the From Wickenheiser, A. and Garcia, E., Longitudinal Dnamics of a Perching Aircraft ; reprinted b permission of the American Institute of Aeronautics and Astronautics, Inc. 7

49 purpose of using high-drag, separated flow for braking, followed b a vertical or ver short landing [4]. This maneuver is based off of several avian landing techniques, including maximizing drag b flaring the wings and tail, and diving under the intended landing site and then pulling up into a climb to reduce speed. While vertical landings have been accomplished b rotar and V/STOL aircraft, it is desired to perch using primaril aerodnamics, with little input from thrust-generating devices. This will alleviate the need for the heav thrust generators required to land verticall, which are not compatible with long endurance aircraft sstems. Thus, perching will be especiall useful for small, efficient reconnaissance aircraft, for example, whose thrust-to-weight ratios might be on the order of /0. This chapter presents a concept for a perching aircraft and an analsis of its longitudinal dnamics. This concept is based on the Aerial Regional-Scale Environmental Surve (ARES) Mars scout craft, an aircraft designed to unfold from a Viking derivative aeroshell and fl for approximatel 70 minutes over a Martian landscape, collecting data on atmospheric chemistr, geolog, and crustal magnetism [5]. The idea to tr to perch a similar airframe grew from the challenge to save the ARES scout from a high-speed crash landing at the end of its mission b using drag to slow it down enough to land with its instruments intact. It is desired to perform the perching maneuver without complicating the aircraft sstem unnecessaril and b adding the fewest number of additional actuators. The original ARES craft features a blended-wing bod with folding tail boom, tail surfaces, and wings, shown in Figure.. The inverted V-tail features two ruddervators that combine the functionalit of a rudder and elevator. In order to add perching capabilities, actuators are incorporated into the tail degrees of freedom, and variable incidence is added to the folding wing 8

50 sections. These additional degrees of actuation in the perching flight vehicle, dubbed the ARES-C, are shown in Figure.. Figure.: The ARES Mars scout [3] Figure.: The three primar actuations about the pitch axis: A) rotation of the wing incidence angle with respect to the fuselage bod axis, B) rotation of the tail boom, and C) rotation of the horizontal stabilizer 9

51 The level of geometric reconfigurabilit required to recreate the perching maneuver in a manmade aircraft falls far outside the bounds of conventional aircraft designs. In order to maintain stabilit and controllabilit at the high angles of attack required for aerodnamic braking, the aircraft s wings are rotated to a negative incidence angle in order to moderate their angle of attack with respect to the oncoming air as the bod s angle of attack increases past stall. Additionall, the tail is rotated down and out of the resulting unstead wake of the bod, and the horizontal stabilizer is also actuated in order to remain in the linear angle of attack range as the tail boom rotates. These actuations keep the standard aileron and ruddervator surfaces effective at trimming and control. The also allow a larger degree of control over the aircraft s dnamics through a wider range of flight conditions. For the purposes of this initial stud, the aircraft is modeled in the longitudinal plane onl; that is, roll, aw, and sideslip dnamics are not considered. B eliminating lateral dnamics, the number of parameters in the aerodnamic model is greatl reduced, and asmmetric geometric configurations need not be considered. Also, it is assumed a priori that the optimal perching trajector, in terms of curvilinear distance, time to land, or overall energ consumption, will be confined to the longitudinal plane. Even restricted to this plane, lateral dnamics enter into the model when considering post-stall aerodnamics, which feature asmmetric flows, and disturbance rejection from sideslip wind gusts; however, both of these phenomena are beond the scope of this stud. In the present aerodnamic model, the fuselage and wings are modeled as a blendedwing bod, and the tail is considered as a separate lifting surface. The aerodnamic forces on the aircraft components are calculated using a modified version of 30