Aeroelastic Control Using Multiple Control Surfaces - CiteSeerX

43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Con 22-25 April 2002, Denver, Colorado

AIAA 2002-1632

AIAA 2002-1632

Aeroelastic Control Using Multiple Control Surfaces Mayuresh J. Patil∗ and Robert L. Clark† Duke University, Durham, NC 27708-0300 This paper investigates the use of multiple control surfaces for aeroelastic control. The focus of the present work is to use modal information for intelligent choice of control surface deflection patterns. The theoretical basis for design of optimal flap deflection modeshapes is presented. Optimal flap distribution helps to focus control energy into the mode of interest. This reduces a multi-input multi-output control problem to a single-input single-output problem. In addition, constrained optimization is used to avoid control spillover into other modes. Such simplifications help in the design of simple, robust, physics-based controllers.

Introduction The overall objectives of the work being conducted by the authors is to design a morphing wing that integrates a deformable wing with discrete actuators. The concept chosen for the present study is a highaspect-ratio wing with multiple trailing edge flaps distributed throughout the span. Each flap is actuated by a discrete piezo-actuator. The flaps are configured with slots and tabs so that they can readily interlock from one mechanical assembly to the next leading to a continuously deforming trailing edge. The present work focuses on the possibility of using control surface deformation patterns for specific applications. It is shown that a flap distribution mode can be designed to maximize the modal force affecting the mode of interest. Also, constrained optimization techniques can be used to optimize the effectiveness of the control surface deflection for a given mode while restricting the effect on other modes. The present paper presents analysis and results relevant to aeroelasticity, structural dynamics and adaptive structures as applied to aerospace systems. As such this paper is a good fit for the AIAA-SDM conference.

Theory Comprehensive aeroelastic analysis of a deformable wing is conducted using reduced order analysis methodology similar to Rule, Richard and Clark.1 The wing is modeled as a beam using an assumed modes approach. Aerodynamic analysis is based on an unsteady vortex lattice method (Hall2 ). Balanced-order ∗ Assistant Research Professor, Department of Mechanical Engineering and Materials Science. Presently, Assistant Professor, Department of Mechanical Engineering, Widener University. Member AIAA. † Jeffrey N. Vinik Professor of Mechanical Engineering and Materials Science. Associate Fellow AIAA.

c 2002 by Mayuresh J. Patil and Robert L. Clark. Copyright Published by the American Institute of Aeronautics and Astronautics, Inc. with permission.

reduction (Rule, Cox and Clark3 ) leads to a low order model for analysis, design, and control synthesis. The focus of the present work is on using modal information from the aeroelastic system to design actuator and/or sensor modeshapes for specified goals. Once the actuator mode and sensor is specified, H2 -optimal controller is designed to optimize for the specified performance metric. Given below are details of the formulation. Aeroelastic Model

The equations of motion of the wing are derived using Lagrange’s equations. Lagrange’s equations are written as,   ∂ ∂(T − U ) ∂(T − U ) − = Qi (1) ∂t ∂ q˙i ∂qi where, T and U are the kinetic and potential energies respectively, qi ’s are the generalized co-ordinates, and, Qi ’s are the generalized forces. Generalized forces are derived from applied external (aerodynamic) forces using virtual work principle. The kinetic energy for the beam can be written as,   Z  T  1 l u˙ u˙ m mζ T = dx (2) mζ I θ˙ 2 0 θ˙ where, u and θ are the bending and torsion deformations of the beam, and, m, ζ, I, are the mass, mass offset, and moment of inertia of the beam crosssection. The potential (strain) energy for the beam is given by,   00  Z  T  1 l u00 EI Cbt u U= dx (3) Cbt GJ θ0 2 0 θ0 where, EI, GJ, Cbt , are the bending, torsion, and coupling stiffnesses. Generalized co-ordinates are introduced to make it a finite-dimensional system. The bending and torsion

1 American Institute of Aeronautics and Astronautics Copyright © 2002 by the author(s). Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

deformation are represented in terms of uncoupled modes of prismatic, cantilevered beams. Thus we have the beam deformations given by,   u {w} = = [Φ]{q} (4) θ

The vorticity-downwash relations are static and can be used to write the bound vorticity strengths in terms of the wake vorticity strengths and the downwash as,

where,

Since the number of bound vortices is equal to the number of wing control points, [Kb ] is a square matrix, and is easily inverted. ii) Shed vorticity relations: These relations give the strength of the shed vorticity in terms of the change in bound vorticity. Since vorticity is always conserved, any change in bound vorticity over a time ∆t is accompanied by an equivalent vorticity shed into the wake. Thus one can write the shed vorticity equations for an airfoil section as,

 {φu }1×n [Φ] = 0

0 {φθ }1×m

 (5)

Here, {φu }1×n and {φθ }1×m are the n beam bending and m beam torsion modes. Now, using Lagrange’s equation one can derive the structural component of the equations of motion as, [M ]{¨ q } + [K]{q} = {Q}

(6)

where, [M ] and [K] are the mass and stiffness matrices given by,   Z 1 l T m mζ [Φ] [Φ]dx (7) [M ] = mζ I 2 0 1 [K] = 2

Z

l ? T

[Φ ] 0



EI Cbt

 Cbt [Φ? ]dx GJ

(8)

{Γb } = −[Kb ]−1 [Kw ]{Γw } + [Kb ]−1 {W }

ΣΓt+∆t − ΣΓtb = −Γt+∆t w1 b

 00 {φu }1×n [Φ ] = 0

0 {φ0θ }1×m



∆

(9)

The above equations of motion can be used to solve for the generalized (modal) co-ordinates if the generalized forces are known. The generalized forces are calculated from the aerodynamic forces as described in the following sections. Aerodynamic Model Unsteady vortex-lattice method is used to approximate the unsteady aerodynamic loads on the wing.2 The aerodynamics is represented by discrete horseshoe vortices distributed over the wing and the wake. The strengths of the vortices are the variables for the aerodynamic equations. The aerodynamic loads are determined from the vortex strengths. The aerodynamic equations are derived using three relations presented below. i) Vorticity-downwash relations: These relations are obtained by satisfying the boundary conditions at various wing control points. [Kb ]{Γb } + [Kw ]{Γw } = {W }

(12)

where, the superscript denotes the time, and Γw1 denotes the strength of the first wake vortex. Similar equations can be written at all the airfoil sections. The set of discrete-time equations can then be converted to continuous-time using the following finite-difference equations.

and, ?

(11)

X t+∆t = X t+ 2 +

∆t ˙ t+ ∆ X 2 2

(13)

∆t ˙ t+ ∆ X 2 (14) 2 iii) Convection of wake vorticity relations: It is assumed that the vorticity in the wake is convected downstream with freestream velocity. Now, choosing a time interval (∆t) such that the horse-shoe vortices travel a distance equal to the chordwise vortex spacing, the vortex at a given position is transferred to the next vortex position over the time interval. This can be represented as, ∆

X t = X t+ 2 −

t Γt+∆t wi+1 − Γwi = 0

(15)

Again, the above discrete time equations are converted to continuous time using the finite difference equations (Eqs. 13 and 14). Converting the shed vorticity and vorticity-convection relations to continuous time and replacing the bound vorticity variables using the vorticity-downwash relations, one gets the wake dynamic relations given by,

(10)

where, Γb and Γw are the vortex strengths of the bound vorticity and wake vorticity respectively, [Kb ] and [Kw ] are the corresponding aerodynamic influence coefficients representing the effect of the vortices on the control points, and, W is the downwash at the control points.

˙ } [A1 ]{Γ˙ w } + [A2 ]{Γw } = {W

(16)

This set of dynamic equations can be used to calculate the strengths of the wake vortices if the downwash is known. One can then use Eq. 11 to calculate the strength of the bound vortices. Finally, lift distribution on the wing can be calculated using the bound

2 American Institute of Aeronautics and Astronautics

vortex strengths. The lift distribution (L) at an airfoil section is given by, Z

y

L = ρU∞ Γb + ρb

Γ˙ b dy

(17)

−b

where, ρ. U∞ , and b, are the air density, freestream airspeed, and semi-chord, respectively. Fluid-Structure-Control Interface The formulation up till now gives the uncoupled structural and the aerodynamic equations. The two equations are coupled through the fluid-structure interface. Thus the generalized forces (Q) affecting the structural equations (Eq. 6) are derived from the aerodynamic loads and the downwash (W ) affecting the aerodynamic equations (Eq. 16) are due to the structural and control deformation. The downwash for at an airfoil section can be written in terms of pitch (α) and plunge (h) as, 1 W = U∞ α + h˙ + y α˙ 2

yφθ

   u (19) θ

The downwash at the control points can thus be calculated in terms of the structural coordinates. Similarly, the control deformations lead to down wash distribution over the control points on the control surfaces. This can be represented as, Wcont = [C]{δ}

(20)

where, {δ} is the column of control surface deformations. Finally, the lift equation (Eq. 17) gives the lift distribution as a function of the bound vorticity as L(x, y) = [Υ(x, y)]1×p {Γb }p×1 +[Υ∗ (x, y)]1×p {Γ˙ b }p×1 (21) where, Υ and Υ∗ are the lift distribution functions corresponding to the p bound vortices. Using, principle of virtual work, the generalized forces are given by, Z lZ

b

{Q} = 0

−b

Substituting the interface equations (Eqs. 19, 20, and 22) into the structural and aerodynamic equations (Eqs. 6 and 16) we get the set of coupled aeroelastic system equations as, [M ]{¨ q } + [K]{q} = [Λ1 ]{q} + [Λ2 ]{q} ˙ + [Λ3 ]{¨ q} ˙ + [Λ4 ]{Γw } + [Λ5 ]{Γw } + [Λ6 ]{δ} (23) ˙ [A1 ]{Γw } + [A2 ]{Γw } = [ψ1 ]{q} ˙ + [ψ2 ]{¨ q} (24) where, [Λ]’s and [ψ]’s are the coefficient matrices derived using the interface equations. Now, one can represent the above set of aeroelastic system equations in a state-space form as, {x} ˙ = [A]{x} + [B]{δ}

(25)

  q  q˙ {x} =   Γw

(26)

where,

(18)

The pitch and plunge deformation for the wing can be represented in terms of the beam bending and torsion as given in Eq. 4. Thus, the downwash distribution over the wing due to the structural deformations can be represented as,     u  Wstruc = 0 U∞ φθ + −φu θ

Complete Active Aeroelastic System

[Φ]T L(x, y)dydx

(22)

 I [A] = 0 0

0 M − Λ3 −ψ2

 I [B] = 0 0

−1  0 0 −Λ5  −K + Λ1 A1 0

0 M − Λ3 −ψ2

−1   0 0 −Λ5  Λ6   A1 0

I Λ2 ψ1

 0 Λ4  −A2 (27) (28)

Actuator Mode Selection

Multiple flaps lead to a various possible control architectures. One could directly design a controller which actuates the various flaps independent of one another or one could use all the flaps together as one flap. First possibility leads to a complex controller (which actuates over all modes) while the second one is quite inefficient. Previous aeroelastic control research using piezo patches (Richard, Rule and Clark4 ) has shown that optimization of the patch placement is helpful in focusing the energy into the mode of interest while at the same time avoiding the spillover into other modes. For the present case one can similarly optimize the flap distribution so that it focuses control energy into the mode of interest (flutter mode) while not wasting the energy over other modes. This is accomplished by a gradient based constrained optimization procedure which maximizes the sensor to actuator path for the mode of interest while constraining the control effort in the other modes to be below a chosen value. The sensor to actuator paths are calculated by a modal transformation on the aeroelastic system (Patil5 ).

3 American Institute of Aeronautics and Astronautics

Modal Decoupling The aeroelastic system as described in the Eq. 25 is in the coupled state-space form. The system of equations can be uncoupled by using the aeroelastic modes. Consider the eigenvalue problem based on the homogeneous part of the aeroelastic state-space equations given by, λi {vi } = [A]{vi }

(29)

where, λi ’s are the complex eigenvalues and {vi }’s are the corresponding eigenvectors (modeshapes). Now representing the generalized coordinates in terms of the aeroelastic modes, we have, {x} = [V ]{ξ}

(30)

where, [V ] is the matrix of eigenvectors and {ξ} is the column of modal coordinates. Substituting the modal transformation into Eq. 25 and premultiplying by [V ]−1 , we have,5 ˙ = [Λ]{ξ} + [V ]−1 [B]{δ} {ξ}

(31)

where, [Λ] is a diagonal matrix containing the eigenvectors. Due to decoupling, the equations for any mode can be represented individually as, ξ˙i = λi ξi + {Ξi }{δ}

(32)

where, {Ξi } denotes the ith row of [V ]−1 [B]. Optimal Flap Distribution Now, the problem of focusing energy into a particular mode can be converted to a mathematical problem of maximizing modal force {Ξi }{δ}. The optimization problem can be stated as, max P (δ), δ

where,

{Ξi }{δ} P (δ) = p (33) {δ}T {δ}

The performance metric P (δ) for the above optimization problem is the modal force per unit H2 norm of the control deflection. The solution to the above optimization problem would be trivial if {Ξi } where real. The optimal {δ} for real {Ξi } is constant times {Ξi }. Since {Ξi } consists of complex elements the optimal solution is constant times {Ξi }T . Such a control mode implies phase lag between the various control surfaces and is difficult to implement in practice. Thus, a real control mode {δ} is required which maximizes the magnitude of the expression shown above. The solution to the above optimization problem can obtained by using the following logic. Let us assume that {δ}opt is the optimal solution. Then P (δopt ) will have the largest magnitude possible.

Now, since this number will be complex in general, one can find a unit complex number to multiply the eigenvector to make the solution real. Now a real solution implies that imaginary part of {Ξi eiϑ }{δ}opt is zero while the real part is maximum. The real part can be maximum only if {δ}opt is equal to the real part of {Ξi eiϑ }. Thus the solution is,
{δ}opt = < {Ξi }eiϑ (34) where, tan−1



−2