Active Aeroelastic Tailoring of High Aspect Ratio ... - CiteSeerX

AIAA-2000-1331

Active Aeroelastic Tailoring of High Aspect Ratio Composite Wings Carlos E. S. Cesnik,∗ Miguel Ortega-Morales,† and Mayuresh J. Patil‡ Department of Aeronautics and Astronautics Massachusetts Institute of Technology, Cambridge, Massachusetts

Abstract This paper presents a framework for studying the effects of combined bending and twisting actuation on the aeroelastic performance of highly-flexible active composite wings. The effects of embedded anisotropic piezoelectric strain actuators are consistently accounted for throughout the nonlinear aeroelastic analysis. The nonlinear active aeroelastic analysis consists of an asymptotically correct active crosssectional formulation, geometrically-exact mixed formulation for dynamics of moving beams, and a finitestate unsteady aerodynamics with the ability to model dynamic stall using the ONERA model. Once the nonlinear equilibrium is determined, a linearization about that point yields a set of equations in state space form. From that, different LQG controllers can be designed and the closed-loop system simulated to determine the impact of the anisotropic piezoelectric strain actuators on the stability enhancement and gust alleviation characteristics of flexible wings. A proposed wind tunnel model is presented to illustrate the analysis framework and to show the potential of active aeroelastic tailoring using active fiber composites. Results show that the orientation of the actuators and the electric field profile can be tailored for maximum stability augmentation or for maximum gust alleviation, and that a compromised situation can still have significant improvement in both performance metrics.

aircraft construction brought the possibility of aeroelastic tailoring few decades ago, the recent advancements in smart materials bring a new twist in design freedom: the potential for active aeroelastic tailoring. In fact, embedded anisotropic strain actuators may enable further exploration of structural flexibility to enhance flight performance while controlling aeroelastic instabilities especially in high-aspect ratio wings. Due to the nature of such systems, nonlinear aerostructural interactions are important, and the investigation of these interactions is gathering momentum as the understanding of the dynamics and the availability of the required mathematical tools increase. Since highly flexible wings can reach high angles of attack at which trailing edge control surfaces render ineffective, different control options are required. Distributed sensing and actuation technologies may become necessary if one intends to guarantee maneuverability, extended flight envelope and improve overall performance.

Ref. [1] presents a historical background of aeroelastic tailoring and the theory underlying the technology. The paper provides historical perspective on the development of codes and the activities of various research groups up to that time. It is still true that in many studies, the structural deformation model used is that of a beam-like wing, since tailoring focuses on bend-twist deformation coupling. Restraining the freedom of the chordwise bending mode, which is often done, can result in substantially different natural frequencies and mode shapes for highly coupled Introduction laminates. Also, the prudence of retaining rigid-body modes in flutter analysis during design iterations was While the introduction of composite materials in pointed out. A more recent overview of the different ∗ Assistant Professor of Aeronautics and Astronautics. Senior efforts in aeroelastic tailoring can be found in Ref. [2].

Member, AIAA. Member, AHS. † Graduate Research Assistant. Member AIAA. ‡ Visiting Student. Presently, Post-Doctoral Fellow, School of Aerospace Engineering, Georgia Institute of Technology. Member, AIAA. c Copyright2000 by Carlos E. S. Cesnik, Miguel OrtegaMorales, and Mayuresh J. Patil. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission.

In terms of nonlinear aeroelasticity, the studies conducted by Dugundji and his co-workers are a combination of analysis and experimental validation of the effects of dynamic stall on aeroelastic instabilities for simple cantilevered laminated plate-like wings.3 There,

1 American Institute of Aeronautics and Astronautics

the ONERA stall model was used for aerodynamic loads. Tang and Dowell4 have studied the flutter and forced response of a flexible rotor blade. In this study, geometrical structural nonlinearity and free-play structural nonlinearity are taken into consideration. Again, high angle of attack unsteady aerodynamics was modeled using the ONERA dynamic stall model. Virgin and Dowell have studied the nonlinear behavior of airfoils with control surface free-play and investigated the limit-cycle oscillations and chaotic motion of airfoils.5 Gilliat, Strganac, and Kurdila have investigated airfoil nonlinear aeroelastic behavior both experimentally and analytically.6 A nonlinear support mechanism was constructed and used to represent continuous structural nonlinearities. In order to get a better understanding of the nonlinear aeroelastic phenomena involving the entire vehicle, Refs. [2] and [7] developed a low-order high-fidelity aeroelastic model for preliminary design and control synthesis. It focuses on a very flexible high-aspect ratio (passive) composite wing with discrete flap control, and models the entire aircraft. This is used here as the basis of the new proposed active aeroelastic framework.

the past years for the control of structures. Linear Quadratic Gaussian (LQG) control has been the most common control method adopted so far.18 Lin et al.19 used a sensitivity weighted LQG for better performance by adding sensitivity to parameter variation. This was compared against a compensator designed using a weighted average of LQG cost for a discrete set of plants. Recent approaches to control design take into account real parameter uncertainty optimally by using the Popov stability criterion. These control methodologies have been applied to benchmark problems 26 and the Middeck Active Control experiments27 with great success.

The feasibility of using active piezoelectric control to alleviate vertical tail buffeting was investigated under the ACROBAT program.21 Simpson and Schweiger22 and Manser et al.23 also carried out similar studies. A program to investigate the active suppression of vertical tail buffeting vibrations in the F-18 aircraft is currently under way in a joint US/Australia/Canada effort (see, for example, Refs. [21] and [24]). Also, Larson et al.25 described the use of piezoceramic wafer actuators for active vibrations suppression on skin panels.

or to the orientation of the cross-sectional reference frame. A compact mixed variational formulation can be derived from these equations which is well suited for low-order beam finite element analysis. The exact intrinsic equations are extended to reflect the new constitutive law that includes the effects of the anisotropic actuation. The vector that contains the actuation effects is a function of an external parameter (e.g., applied voltage in the case of piezoelectric actuators) that can be modulated by the controller as function of the sensing parameters (time dependent).

Various approaches have been developed over

The latter work presents a state-space theory for

This paper presents a low-order high-fidelity active nonlinear aeroelastic framework that incorporates anisotropic piezoelectric strain actuators embedded throughout a composite wing construction. It is a significant contribution to the field in the sense that it provides the means for studying the impact of active aeroelastic tailoring, particularly for high-aspect-ratio wings. In what follows, a description of the structural and aerodynamic formulations, the solution of the aeroelastic problem, and description of the control Aeroelastic control can be more difficult than tra- design are presented. Finally, a numerical example is ditional aeroelastic problems, in that the control sys- presented based on a proposed wind tunnel model of a tem introduces a second potential source of instability.8 High-Altitude Long-Endurance (HALE) vehicle wing. There have been considerable efforts in the area of aeroservoelasticity. Refs. [9] and [10] present very Formulation good surveys. The use of smart materials technology is also reviewed in detail by Loewy [11]. Weisshaar An active aeroelastic model that extends the nonand his co-workers have been investigating the use of linear aeroelastic formulation of Ref. [7] is developed active materials in static aeroelastic control.12,13 Alter- here. The basic theory is based on two separate works, native concepts for controlling airplane aerodynamic viz. i) mixed variational formulation based on exsurfaces have also been under investigation, e.g., the act intrinsic equations for dynamics of moving beams,28 Active Flexible Wing technology,14 other variable stiff- and, ii) finite-state airloads for deformable airfoils on ness concepts,15 the Smart Wing effort by Northrop fixed and rotating wings.30,31 Grumman,16 and the NASA/MIT PARTI wing.17,18,19 The former theory is a nonlinear intrinsic formuHeeg reported flutter suppression and gust loads alle- lation for the dynamics of initially curved and twisted viation in a theoretical and experimental investigation beams in a moving frame. There are no approximadescribed in Ref. [20]. tions to the geometry of the deformed reference line

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the lift, drag, and all generalized forces of a deformable airfoil. The theory allows for a thin airfoil performing arbitrary small motions with respect to a reference frame which can perform arbitrary large motions. Let a be the global frame, with its axes labeled as a1 , a2 and a3 . The undeformed reference frame of the wing is named b, with its axes labeled as b1 , b2 and b3 , and the deformed reference frame named B, with its axes labeled as B1 , B2 and B3 . Any arbitrary vector Z represented by its components in one of the basis may be converted to another basis like

 κ

∗

= C

VB∗

ba

θ˜ 2 θT θ 4

∆− 1+

 θ

= C Ba (va + u˙a + ω ˜ a ua )   θ˜ ∆− 2 θ˙ + C Ba ωa = C ba T 1 + θ4θ

Ω∗B

(4)

where ua is the displacement vector measured in the a frame, θ is the rotation vector expressed in terms of Rodrigues parameters, e1 is the unit vector [1, 0, 0]T , ∆ is the 3 × 3 identity matrix, and va and wa are the initial velocity and initial angular velocity of a generic Zb = C ba Za , ZB = C Ba Za (1) point on the a frame. (˜) operator applied to a column vector is defined as: where C ba is the transformation matrix from a to b,   0 −Z3 Z2 and C Ba is that from a to B. There are several ways 0 −Z1  Z˜ =  Z3 to express the transformation matrices. C ba can be −Z Z 0 expressed in terms of direction cosines from the initial 2 1 Ba geometry of the flexible wing, while C contains the unknown rotation variables. To form a mixed formulation, Lagrange’s multipliers are used to enforce VB , ΩB , γ and κ to satisfy the geometric equations in Eq. (4). Structural Analysis

Manipulating the equations accordingly [32], one As described in details in Ref. [28], the variational can obtain the a frame version of the variational formuformulation is derived from Hamilton’s principle which lation based on exact intrinsic equations for dynamics can be written as of moving beams as  t2  l  t2 [δ(K − U ) + δW ] dx1 dt = δA (2) t1 0 δΠa dt = 0 (5) where t1 and t2 are arbitrarily fixed times, K and U are the kinetic and potential energy densities per unit where span, respectively. δA is the virtual action at the ends δΠa = of the beam and at the ends of the time interval, and δW is the virtual work of applied loads per unit span. + Taking the variation of the kinetic and potential + energy terms with respect to VB and ΩB , the linear and angular velocity column vectors, and with respect + to γ and κ, the generalized strain column vectors, + T T   ∂U ∂U − FB = , MB = ∂γ ∂κ T T   − ∂K ∂K PB = , HB = (3) ∂VB ∂ΩB + where FB and MB are internal force and moment column vectors, and PB and HB are linear and angular momentum column vectors. The geometrically exact kinematical relations in the a frame are given by γ∗

= C Ba (C ab e1 + ua ) − e1

t1



l

•

T ab T T ab {δuT a C C FB + δua [(C C PB )

0

ω˜a C T C ab PB ] T

T

δψ a C T C ab MB − δψ a C T C ab (e1 + γ˜ )FB T

•

δψ a [(C T C ab HB ) + ω a C T C ab HB C T C ab V B PB ] T

T

T

T

δF a [C T C ab (e1 + γ) − C ab e1 ] − δF a ua θ˜ θθT ab T  δM a (∆ + + )C κ − δM a θ 2 4

δP a (C T C ab VB − va − ω a ua ) − δP a u˙a T θ˜ θθ T + δH a (∆ − + )(C T C ab ΩB − ωa ) 2 2 T T − δH a θ˙ − δuTa fa − δψ a ma }dx1 (6)

l T T T

− (δuTa Fˆa + δψ a Mˆa − δF a uˆa − δM a θˆ 0

and the rotation matrix C is the product C ab C Ba , and

3 American Institute of Aeronautics and Astronautics

it is expressed in terms of θ as C=

(1 −

θT θ ˜ θθT 4 )∆ − θ + 2 T 1 + θ4θ

δP a

= δP i ,

PB = Pi

δH a

= δH i ,

HB = H i

(7)

where ui , θi , Fi , Mi , Pi and Hi are constant vectors at In Eq. (7), fa and ma are the external force and each node i, and all δ quantities are arbitrary. ξ varies moment vectors respectively, which result from aero- from 0 to 1. dynamic loads. The (ˆ) terms are boundary values of With these shape functions, the spatial integration the corresponding quantities and the (˙) terms repre- in Eq. (9) can be performed explicitly to give sent time derivatives of ( ). The generalized strain and N  force measures, and velocity and momentum measures T T T {δuTi fui + δψ i fψi + δF i fFi + δM i fMi are related through the constitutive relations in the foli=1 lowing form: T T + δP i fPi + δH i fHi      (a)  T FB γ FB + δuTi+1 fui+1 + δψ i+1 fψi+1 = [S] − (a) κ MB MB T T      + δF i+1 fFi+1 + δM i+1 fMi+1 } PB m∆ 0 VB T = (8) ˆ N +1 = δuTN +1 FˆN +1 + δψ N +1 M 0 I HB ΩB T

(a)

(a)

where FB and MB are the piezoelectric (bulk) force and moment vector, respectively. These expressions are then solved for γ, κ, VB , and ΩB as function of the other measures and constants and used in Eq. (7). The stiffness [S] is in general a 6 × 6 matrix, function of material distribution and cross sectional geometry. As described in [29], the 6 × 6 stiffness matrix is related to the 4 × 4 one. The latter is used in this paper, where the stiffness matrix and column vector for the piezoelectric actuation are described in great details in Ref. [33].

T

− δF N +1 u ˆN +1 − δM N +1 θˆN +1 T T ˆ 1 + δF T1 u − δuT1 Fˆ1 − δψ 1 M ˆ1 − δM 1 θˆ1

(11)

where the fui , fψi ,. . ., fMi+1 are the element functions explicitly integrated from the formulation.

Both γ and κ in each element function have to be replaced with a form that is a function of FB and MB using the inverse form of Eq. (8), along with the (a) (a) piezoelectric forcing vector FB and MB . So does VB and ΩB with a form function of PB and HB . This substitution, which accounts for the presence of actuaAdopting a finite element discretization by divid- tors embedded in the structure, leads to the following element functions: ing the wing into N elements, Eq. (5) is written as  t2  δΠi dt = 0 (9) ∆li T ab t1 fψi = −C T C ab Mi − C C · i 2 (a)  (a) where index i indicates the i-th element with length [e1 + {r(Fi + Fi ) + t(Mi + Mi )}]Fi + . . . ∆li , δΠi is the corresponding spatial integration of ∆li T ab the function in Eq. (7) over the i-th element. Due fFi = ui − [C C · 2 to the formulation’s weakest form, the simplest shape (a) (a) (e1 + {r(Fi + Fi ) + t(Mi + Mi )}) − C ab e1 ] functions can be used. Therefore, the following transformation and interpolation are applied within each ∆li θ˜i θi θiT ab fMi = θi − (∆ + + )C · element28 2 2 4 x = xi + ξ∆li , dx = ∆li dξ, ( ) =

1 d () ∆li dξ

δψ a

= δui (1 − ξ) + δui+1 ξ, = δψ i (1 − ξ) + δψ i+1 ξ,

ua = ui θ = θi

δF a

= δF i (1 − ξ) + δF i+1 ξ,

FB = Fi

δM a

= δM i (1 − ξ) + δM i+1 ξ, MB = Mi

δua

(a)

(10) fψi+1

fFi+1

(a)

{tT (Fi + Fi ) + s(Mi + Mi )} ∆li T ab = C T C ab Mi − C C · 2 (a)  (a) [e1 + {r(Fi + Fi ) + t(Mi + Mi )}]Fi + . . . ∆li T ab = −ui − [C C · 2 (a) (a) (e1 + {r(Fi + Fi ) + t(Mi + Mi )}) − C ab e1 ]

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fMi+1

∆li θ˜i θi θiT ab (∆ + + )C · 2 2 4 (a) (a) {tT (Fi + Fi ) + s(Mi + Mi )}

= −θi −

corrections and modifications from experimental data. Thus, corrections such as thickness and Mach number can be incorporated very easily as described in Ref. [31]. It can also be extended to include dynamic stall effects according to the ONERA approach.

Aerodynamic Analysis

LTn = Ln + ρ∞ uT Γn

The aerodynamic loads used are as described in where detail in Ref. [30]. The results are presented here. The theory calculates loads on a deformable airfoil undergoing large deformation in a subsonic flow. Certain aerodynamic parameters for the particular airfoil are required and are assumed to be known empirically or through a CFD analysis. Let the mean chord line deformation of the airfoil cross section be described by h(x2 , t) where x2 is along the mean chord line. The frame motion along x2 , x3 -directions are, respectively, u0 , v(x2 , t). Let λ denote the induced flow due to free vorticity. Let L(x2 , t) denote the distribution of force perpendicular to the mean chord line. Let D be the drag on the airfoil. The integro-differential airload equations can be converted into ordinary differential equations (ODEs) through a Glauert expansion. The ODEs are in terms of the expansion coefficients which are represented by a subscript n.   1 2 ¨n + v˙n {L } = −b [M ] h n 2πρ∞

 uT = Γ¨n +

u0 2 + (v0 + h˙0 − λ0 )2

uT b

η Γ˙n +

2

uT 3 ∆cn b

−ω

(14)

 u 2 T

b

ω 2 Γn =

(15)

d − ω 2 euT dt (uT ∆cn )

The parameters ∆cn , η, ω 2 , and e must be identified for a particular airfoil. The airloads are inserted into the Hamilton’s principle to complete the aeroelastic model. Solution of the Aeroelastic System Eq. (11) yields a set of non-linear equations that can be separated into structural (FS ) and aerodynamic (FL ) terms and written as ˙ E) − FL (X, Y, X) ˙ =0 FS (X, X,

(16)

where X is the column matrix of structural variables, Y is a column matrix of inflow states and E is column (12) matrix of the magnitude of the electrical field distribution shape for each actuation region. Similarly we can separate the inflow equations into an inflow component b[G] {u˙0 hn + u¯0 ζn − u0 vn + u0 λ0 } T  (FI ) and a downwash component (FW ) as   1 ˙ [S] h˙n + vn − λ0 2πρ∞ {Dn } = −b hn + vn − λ0 ˙ + FI (Y, Y˙ ) = 0 −FW (X) (17)  T The solutions of interest for the two coupled sets of +b h¨n + v˙n [G] {hn } equations (Eqs. 16 and 17) can be expressed in the form  T       ¯   X(t) ˇ  X  X u0 h˙n + vn − λ0 [K − H] {hn } = + (18)   ¯ ˇ  Y Y Y (t) T + {u˙0 hn + u¯0 ζn − uo vn + u0 λ0 } [H] {hn } (13) where (¯) denotes steady-state solution and (ˇ) denotes where ρ∞ , b are the air density and semi-chord respec- the small perturbation on it. tively. The matrices denoted by [K], [C], [G], [S], [H], For the steady-state solution one gets Y¯ identically [M ] are constant matrices whose expressions are given equal to zero (from Eq. 17). Thus, one has to solve a in Ref. [30]. set of nonlinear equations given by The required airloads (viz., lift and moment about ¯ 0, E) − FL (X, ¯ 0, 0) = 0 FS (X, (19) mid chord) are obtained as a linear combination of Ln . The inflow is obtained through the finite-state inflow The Jacobian matrix of the above set of nonlinear theory.35 The theory described so far is basically a linear, thin-airfoil theory. But the theory lends itself to equations can be obtained analytically and is found   −bu0 [C] h˙n + vn − λ0 − u0 2 [K] {hn }

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to be very sparse.36 Note that the presence of the ac ∞  ∞ tuation in the wing changes the original terms of the (Js + ρJc )dt = (xT Qx + ρuT Ru)dt (23) J= Jacobian in a similar manner it does in Eq. (11). The 0 0 steady-state solution can be found very efficiently using where the state cost Js is directly related to the toNewton-Raphson method. tal (potential and kinetic) energy in the wing, and the A dynamic response analysis about the steadyactuation cost Jc is related to the applied electrical state with respect to the electric field applied to the power, with ρ being the parameter that modifies the anisotropic actuators embedded in the wing can be perrelative weighting of the two. In addition to being efformed once the solution is obtained. By perturbing fective performance parameters, both control and state Eqs. (16) and (17) about the calculated steady state cost are used to establish the LQG regulator working using Eq. (18), the transient solution is obtained from point. A measure of the control cost necessary to render the wing stable when flying above the flutter speed   is required for flutter enhancement analysis, and is asˇ  ∂FS  X  ∂FL L − ∂F sumed to be37 ∂X − ∂X ∂Y  ∞ + ∂FW ∂FI ¯  W X=X  − ∂F − ˇ ∂X ∂Y ∂Y J= (uT u)dt (24) Y 0 Y =0 ˙  ∂FS  ˇ X L The LQG is more realistic than the LQR, since it allows − ∂F 0 ∂ X˙ ∂ X˙ + (20) ∂FI a reduced number of states to be used for feedback, ¯ X= X  ˇ˙  0 ∂ Y˙ Y thus allowing for the discrete sensing in the wing and Y =0   no feedback of the aerodynamic states.  ∂F  0   S 0 ∂E { E } = The gust model adopted for disturbance rejection 0 0 X=X¯   0 w is a simple harmonic disturbance present in the atY =0 mosphere as8 Now assuming that the axial deformation perturwmax 2πu0 t w= )) (25) (1 − cos ( bation is negligible when compared to the other defor2 xG mations, and that these are small compared to unit, where u0 is the flight speed, and xG and wmax are the the above equation can be simplified to length and maximum amplitude of the gust. x˙ = Ax + Bu + Gw y = Cx + Du

(21)

Numerical Results

where the reduced state vector x contains only the inA wind-tunnel model representative of UAV wings flow variables and the rotation variables and their time is created and different LQG controllers are designed derivatives, that is, and simulated at set flight velocities. The present framework allows to simulate the resulting aeroelasT ˙ λ] x = [θ, θ, (22) tic systems on a digital computer in order to study the impact of actuation orientation in flight performance. and y is the output vector corresponding to any combination of three types of sensors distributed along the Proposed Wind Tunnel Model Wing wing (accelerometers and strain gauge bridges configured for torsion and bending strain measurements), u Typical High Altitude Long Endurance (HALE) is the input vector corresponding to the normalized vehicles fly under M=0.5 and tend to have a tapered magnitude of the electrical field applied to the piezowing with a half aspect ratio between 8 and 15.38,39 The electric actuators (varying from −1 to +1), and w is wing for this study is a proposed wind tunnel model the disturbance (gust). geometrically and dynamically representative of existDifferent controllers can then be designed and ing and potential new HALE UAV wings. The model simulated for set flight velocities in order to study is tapered (taper ratio 1:2) through 75% of the wing the impact of actuation orientation in the aeroelastic (Fig. 1). The most prominent features of this wind performance (i.e., different combinations of bending tunnel model are the following: and twisting actuation). A Linear Quadratic Gaussian (LQG) controller was implemented, where the • Presents non-linear aeroelastic characteristics due to quadratic performance index J adopted is large structural deflections. 6 American Institute of Aeronautics and Astronautics

AFC

50 200

Figure 1: Active wing under consideration in this study–superimpose to its planform representation are the Active Fiber Composite actuators

Nose E-Glass E-Glass E-Glass E-Glass

0/90 0/90 +45/-45 0/90

Leading edge (active) E-Glass 0/90 AFC +45 E-Glass 45/-45 AFC -45 E-Glass 0/90

Trailing edge (active) E-Glass 0/90 AFC +45 E-Glass 45/-45 AFC -45 E-Glass 0/90

0.07 6

0.06

curvature 5

0.05

4

0.04

3

0.03

2

0.02 twist rate

1

0.01

0

0 0

5

10

15 20 25 30 AFC ply angle (deg)

35

40

Peak to peak actuated curvature (1/m)

10

Peak to peak actuated twist rate (deg/m)

20

45

Figure 3: Maximum actuation for twist and bending as function of the AFC actuator ply angle Web E-Glass 0/90 E-Glass 0/90

NACA 0014

Figure 2: Baseline cross section lay-up of the active wing • Presents flutter at subsonic speed.

(AFC) with interdigitated electrodes embedded in the composite.34 The AFC actuator provides a feasible way of integrally actuating the structure with high levels of actuation authority. Detail cross sectional stiffness, mass, and actuation properties can be found in Ref. [40]. Static actuation results for the configurations described above in terms of induced twist rate and bending curvature is presented in Fig. 3. Based on that, five configurations for the embedded strain actuators will be considered in this study:

◦ • Fits in the NASA Langley Transonic Dynamics Tun- • 0 , which provides maximum (flatwise) bending actuation, nel with the appropriate clearances (for potential wind tunnel tests). • ±45◦ , which provides maximum torsion actuation,

The internal structural design, which involves the cross-sectional shape, including number of spars and location, and the composite lay-up, was also studied separately. The decision was to have a single spar at 40% chord, which is the construction used in certain UAVs, as in the case of Aurora’s Perseus. The spar placement and composite lay-up were optimized for low maximum stress and best aeroelastic performance. The airfoil shape used for design was the NACA 0014, approximating the NACA 6514 that is representative of high-altitude flight, presently in use by some of the HALE vehicles. The lay-up of the airfoil is depicted in Fig. 2, where it shows the anisotropic strain actuators placed at ±45◦ (represented by the notation “AFC”). However, the actuator lay-up angle will be varied from 0◦ to the ±45◦ shown in order to investigate the effect of their orientation on the aeroelastic performance of the wing.

• ±22◦ actuated in (flatwise) bending, • ±22◦ actuated in torsion, and • ±45◦ actuated in (flatwise) bending. Finally, the discretization along the span of the wing was done using 10 finite elements based on a convergence study.40 The number of independent active regions in the wing was set to 10, which means that every element can be actuated independently. This provides the maximum control flexibility within the given discretization. Finally, every active station was provided with strain gauge bridges for flatwise bending and torsion measurements, as well as an accelerometer. Note that these last two parameters impact the aeroelastic response as well, and they were studied in Ref. [40].

With the cross section characterized, the present low-order active model allows to analyze different conThe anisotropic strain actuation for this study figurations dynamically. As a preliminary study, conis realized through the use of active fiber composites sider the dynamic stability of the proposed active wing 7 American Institute of Aeronautics and Astronautics

Table 1: Flutter velocities for the different orientation of the AFC lamination angle for a root angle of attack of 2◦ Flutter Speed (m/s) 49.6 46.8 43.4

100

Flutter speed [m/s] / Frequency [Hz]

AFC Orientation 0◦ ±22◦ ±45◦

120

o Flutter Freq. x Flutter Speed

80

60

40

20

in an open loop configuration. Due to the embedded nature of the strain actuators, these become an integral 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 part of the structural wing, that is, by varying the oriRoot angle of attack [degrees] entation of the AFCs, besides changing the actuation characteristics, the (passive) wing stiffness constants change. This results in a variation of the flutter speed Figure 4: Effect of non-linear steady-state on flutter for each case. Table 1 presents the flutter speed for for ±45◦ actuator orientation case each configuration of the actuators. Besides the material anisotropic effect on the flutter speed caused by the rotation of the AFC actuators, the aeroelastic characteristics of the open loop system is function of the “steady-state” loading, reflected here as the root angle of attack. A detailed discussion of the impact of this parameter in the passive aeroelastic stability of a flexible wing is presented in Ref. [41]. The root angle of attack is set to 2◦ for all the cases presented here since it is a relatively demanding condition when one realizes that the flutter speed at this angle of attack is about 60% lower than the same wing set at 0◦ root angle of attack. Fig. 4 exemplifies this variation for the ±45◦ actuation orientation case. The unstable modes about the nonlinear steady condition are the first torsion/first chordwise bending (together) and second flatwise bending modes, and the flutter frequencies are in the 14 to 20-Hz range. In what follows, the effect of actuation orientation on both stability enhancement and gust alleviation will be investigated for the proposed wind tunnel model. The gust study will be conducted both above and below flutter speeds. To account for the changes in the flutter speed for the different active configurations, the velocities are normalized with respect to the corresponding flutter speed. Therefore, in what follow, conditions above and below the flutter speed are set to ±15% of the corresponding flutter speed, respectively. Finally, the control cost Jc has to also be normalized so that the different velocity settings, which correspond to different flow energies, can be factored when comparing costs of different LQG designs. The normalization adopted in this study is given by

Jc (Jc )normalized ←− ¯ kρ∞ u20

(26)

¯ ∞ = 1 kg3 is just an adjustment constant, and where kρ m u0 is the (uniform flow) flying speed. Effect of Actuation Orientation on Stability The current model proves to be useful in establishing the LQG working point by easily determining the relative weighting parameter ρ. Fig. 5 shows the state cost plotted versus the normalized control cost evaluated at 15% over flutter velocity for changing values of the weighting parameter. Notice that the curves are hyperbolic in shape, which indicates that there is a value of ρ for which the equally weighted total cost is a minimum. Taking this as the initial working point for every configuration considered, the normalized stability control costs required by every configuration for speeds between 0% and 15% above corresponding flutter speeds are shown in Fig. 6. Again, the unstable mode is the first torsion/first chordwise bending coalescing with the second flatwise bending mode, at frequencies of 14.3 Hz, 16.7 Hz, and 18.0 Hz for the ±45◦ , ±22◦ , and 0◦ configurations, respectively. Placing the piezoelectric actuators at ±45◦ and actuating them in torsion (“45T” curve in Fig. 6) proves to be the most effective way of stabilizing the plant. At that orientation, maximum twist authority is available from the given set of AFC actuators, and the normalized stability control cost is the lowest. Therefore, by

8 American Institute of Aeronautics and Astronautics

Table 2: Gust amplitudes used in the study of the different active configurations AFC Orientation 0◦ ±22◦ ±45◦

Gust Magnitude (m/s) 5.7 5.4 5.0

Figure 6: Normalized stability control cost vs. percentage over flutter speed for the wing flying at 2◦ root angle of attack (zoomed on the right) Effect of Actuation Orientation on Gust Alleviation

Figure 5: State cost vs. normalized control cost for the wing flying at 15% over their own flutter speed at 2◦ root angle of attack

orienting the actuators at lower angles than ±45◦ and exciting them for twist actuation, the authority will be lower and the control cost higher, requiring more actuation power. This is verified for the ±22◦ being actuated in torsion (labeled “22T” in Fig. 6). Since the 0◦ orientation in torsion does not produce any net twist actuation (Fig. 3), it is not shown in the plot. However, since the unstable mode is also composed of bending deformation, an alternative actuation mode is possible for stabilization of the plant. Even though not as efficient as the “45T,” having maximum bending actuation from the AFC at 0◦ (“0” curve in Fig. 6) presents a reasonable option for stabilizing this plant within the defined speed regime. In fact, this bending actuation is equivalent in terms of control cost to the “22T.” And similarly as explained above, by orienting the strain actuators to angles different than 0◦ will reduce the bending authority. This can be seen on the results of ±22◦ and ±45◦ actuated in bending, labeled “22B” and “45B,” respectively.

From the power spectral density of real gusts,8 it can be inferred that their low frequency content is high. Since the gust model adopted is a single-frequency one, a reasonable low frequency gust of 3 Hz roughly matching the first bending mode of all configurations was chosen as the excitation frequency. The amplitude of the disturbance will be set to 5 m/s for the ±45◦ case, and will be corrected for the other cases accordingly so the normalized gust amplitude by the flying speed is a constant. Table 2 summarizes the gust magnitudes for each configuration. Wing Flying Above Flutter Speed. The previous section detailed how to determine the working point of the regulator by inspection of hyperbolic curves as the ones depicted in Fig. 5. This design point yields an acceptable performance point for any frequency. Since the disturbance frequency is now set (3 Hz), the performance of the regulator for that particular frequency can be improved by refining the weight ρ through iteration. This iteration process implies determining the weight for which the disturbance can be rejected without saturation of the actuators. A very convenient way to do this is to use the sinusoidal-disturbance-tovoltage transfer function Bode plot provided by the proposed framework. Fig. 7 shows the transfer function plot for the ±45◦ orientation at 15% over flut-

9 American Institute of Aeronautics and Astronautics

Figure 7: Sinusoidal disturbance-to-voltage transfer function for ±45◦ actuation orientation for a speed 15% above flutter and a 2◦ angle of attack

ter speed for all 10 actuation regions for the optimum weight (see Fig. 12. for the optimum weight values for every configuration). Fig. 7 provides valuable information on the maximum voltage level required by any disturbance. That maximum corresponds in fact to the spikes located at around 100 Hz, which in turn correspond to the higher chordwise bending modes, of which the system has no direct controllability. This is not an upsetting situation, since their frequency is much higher than that associated with the gust. For the selected 3 Hz, the voltage required (value of the transfer function) is approximately 0.2. That means that 20% of the available voltage limit is required to reject a 1 m/s gust. This confirms that rejection of the 5 m/s gust will exactly require the maximum voltage available (or in this case, tolerated by the AFCs).

Figure 8: Gust alleviation for twist actuation oriented at ±45.◦ (i) chordwise tip deflection (u2 ), (ii) flatwise tip deflection (u3 ), (iii) tip rotation (θ3 ). Gust frequency is 3 Hz at 15% above flutter speed, and root angle of attack is 2◦

Figs. 8 and 9 show the time response of the tip of the wing to the 3 Hz gust disturbance. As one can see, the maximum magnitude of the flatwise tip deflection about the nonlinear equilibrium position is about 13% of the wing semi-span, and the maximum tip rotation is about 4◦ . The chordwise tip deflection remains roughly about its steady value (maximum unstedy deflection of about 2 mm). Figs. 10 and 11 show the voltage history of both configurations to the same disturbance. Note that in all the cases the actuators never saturate, Figure 9: Gust alleviation for bending actuation oriwhich was expected since the gust amplitude was cho- ented at ±45.◦ (i) chordwise tip deflection (u2 ), (ii) sen based on the gust to voltage transfer function. flatwise tip deflection (u3 ), (iii) tip rotation (θ3 ). Gust frequency is 3 Hz at 15% above flutter speed and root Fig. 12 summarizes the time response of the conangle of attack is 2◦ figurations to the gust in a table format. A minimum “maximum state energy times control cost” ratio 10 American Institute of Aeronautics and Astronautics

To: Y8

To: Y7

To: Y6

To: Y5

Voltages [V/Vmax]

To: Y4

To: Y3

To: Y2

To: Y1

Linear Simulation Results 1 0 1 1 1 1 1 1 1

1 0

Figure 12: Effect of actuation orientation on the response settling time due to gust disturbance of a wing flying at 15% above corresponding flutter speed and for a 2◦ root angle of attack

1 0 1 0 1 0 1 0 1 0 1 0

1

To: Y10

To: Y9

1 0 1 0.5 0 0.5

0

0.1

0.2

0.3

0.4

0.5

Time (sec.)

0.6

0.7

0.8

0.9

1

Figure 13: Effect of actuation orientation on the response settling time due to gust disturbance of a wing flying at 15% below corresponding flutter speed and for a 2◦ root angle of attack

To: Y2

To: Y1

Figure 10: Voltage history for twist actuation oriented at ±45◦ under a 3 Hz gust disturbance at 15% above means the regulator is able to dampen down the deflutter speed, and 2◦ root angle of attack flections on the wing with less actuation, thus giving a more effective controller in terms of gust alleviation. As one can see, the bending actuation is the most effective way of achieving such gust alleviation. In fact, even the less direct way of realizing bending actuation through the ±45◦ actuation orientation in bending is more effective than the twist actuation configurations. Linear Simulation Results The settling time, however, is significantly shorter for 1 the twist actuation configurations due to the unstable 0 1 1 condition (15% above flutter speed).

To: Y4 To: Y5 To: Y6 To: Y7

Voltages [V/Vmax]

To: Y3

0 1 0.5 0 0.5 0.5 0 0.5 0.5 0 0.5 0.5 0 0.5 0.5 0 0.5 0.5 0 0.5 0.2 0 0.2 0.05 0 0.05

Wing Flying Below Flutter Speed. Consider now the same configurations discussed before but now flying at 15% below flutter speed. So, stability is not a concern at this point, and all the energy of the actuators can be concentrated in rejecting the gust. Fig. 13 summarizes the results.

To: Y10

To: Y9

To: Y8

The low frequency of the gust is expected to primarily excite the first bending mode, but the results in Fig. 13 indicates that actuating in torsion yields shorter settling times and lower maximum state en0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ergy than actuating in bending. The reason is that the Time (sec.) implementation of the system does not let the actuators saturate, and a closer look to the voltage distribution would show that actuating in bending quickly Figure 11: Voltage history for bending actuation ori- saturates the root actuator while the voltage level in ented at ±45◦ under a 3 Hz gust disturbance at 15% the other regions is still low. Ideally, the root element above flutter speed, and wing at 2◦ root angle of attack would saturate and the rest of the elements would be allowed to draw more voltage, making the bending actuation more effective. Another option is to have less 11 American Institute of Aeronautics and Astronautics

individually controlled actuation regions, that is, from the current 10 regions, to two or one. Ref. [40] discusses the effect of different number of independently actuated regions within this non-saturation condition. Moreover, the much higher control cost required by the actuation in torsion confirms the fact that the bending actuation configurations are not driving as much control energy as available. The metric “maximum state energy times control cost” can be used to account for this limitation. Looking at the results for this metric presented in the last column of Fig. 13, it is obvious that bending-actuated configurations are more effective for gust rejection, providing smaller “maximum state energy times control cost.” Very low results are obtained by all the orientations in bending in the expected order: 0◦ , ±22◦ and ±45◦ . The performance of the two configurations actuated in torsion is well behind those.

Conclusions A framework for the study of the combined bending and twisting actuation and their impact on the aeroelastic performance of highly-flexible active composite wings was presented. It consists of an asymptotically correct active cross-sectional formulation, geometrically-exact mixed formulation for dynamics of moving beams, and a finite-state unsteady aerodynamics with the ONERA dynamic stall model. The resulting state-space equations are then used for LQG control design and simulation. The viability of the present design environment as an effective aeroelastic analysis and tailoring tool has been illustrated with a wing case study representative of HALE vehicles. The results conclude that anisotropic strain actuation tailoring is successful in improving the performance of high aspect ratio wings, in particular in controlling aeroelastic instabilities and performing gust alleviation. By carefully orienting the embedded actuators and selecting the electric field profile, a single configuration may be able to satisfy all the requirements for performance on stability augmentation and gust alleviation. Furthermore, the current framework provides potential ways to understand the mechanisms of active aeroelastic response and instability of highly flexible wings.

Acknowledgments

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