Adaptive aeroelastic composite wings—Control and optimization issues

Composites Printed

Engineering,

Vol. 2, Nor 5-7, pp. 457-476.

1992.

0961.9526/92 $S.tM+ .W QJ 1992 Pergamon Press Ltd

in Great Britain.

ADAPTIVE AEROELASTIC COMPOSITE WINGSCONTROL AND OPTIMIZATION ISSUES TERRENCE A. WEISSHAAR School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907-1282, U.S.A.

and STEVEN M. EHLERS Advanced Concepts, McDonnell-Douglas Technologies, San Diego, CA 92127-1713, U.S.A. (Received

17 October

1991; final

version

accepted

21 January

1992)

Abstract-High-performance aircraft are adaptive machines composed of internal structural skeletons to which are attached control surfaces operated by hydraulic muscles to allow them to maneuver. The flight crew, avionic sensors and systems function as the brain and nervous system to adapt the machine to changing flight conditions, such as take-off, cruise and landing. The development of new materials that can expand or contract on command or change stiffness on demand will blur the now distinct boundaries between the structure, actuators and the control system. This paper discusses the use of imbedded active piezoelectric materials to change the aeroelastic stiffness of a lifting surface to allow this surface to control the aircraft. Expressions are developed for the piezoelectric material effectiveness when these active materials are combined with advanced composite structural materials for a swept, high-aspect-ratio wing. The interaction between advanced composite material properties and piezoelectric electromechanical properties is examined. The importance of choosing the proper active control laws is also illustrated. INTRODUCTION

Aircraft are multi-function devices with the ability to adapt to changing flight conditions. As a result, the aircraft geometry must change on command to take-off, land and cruise efficiently. In general, an adaptive structure is an integrated engineering system containing sensing elements (the nervous system), actuators (the muscles), and a signalprocessing device (the brain) all wrapped around a skeleton (the main structure). The control of the aircraft motion and attitude, including trimming, climbing, turning and descending, requires movable aerodynamic surfaces operated by hydraulic pumps. These pumps and their fluid are reliable, powerful and heavy. The control surfaces are attached at only a few points on the structure, creating the need for heavy joints. This arrangement differs markedly from the natural approach of integrating distributed muscular control into animal structures. The purpose of this paper is to demonstrate that active materials can be combined with advanced composites to design lifting surfaces to control the motion of the aircraft. Until recently, analogies between animal structures and machines were tenuous at best. However, man-made materials that can increase their size (self-strain) and change mechanical properties in response to a controlled stimulus such as an electric field are now emerging as candidates for the muscles and sensors for man-made structures. These materials are sometimes called “smart materials”, “active materials” or “servoelastic materials”. The use of these new actively controlled materials to augment or replace the aerodynamic surface control of aircraft is the subject of this paper. Aeroelasticity High-performance flight vehicles are aeroelastic. The word “aeroelastic” is used because the lifting surface deformation shape depends on the structural stiffness, structural inertia and aerodynamic loads. These aerodynamic loads also depend on the surface shape. As a result, we create a system where shape and loads interact. 457

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T. A. WEISSHAAR

and S. M.

EHLERS

Aeroelastic phenomena can be divided into two parts: static, where the inertia loads are either slowly changing or not time dependent at all; and dynamic, where there is motion of the vehicle or its components sufficient to create unsteady, time-dependent aerodynamic forces that cannot be ignored. In both cases there is load/deflection interaction. Dynamic aeroelastic phenomena include the serious condition of flutter. Flutter is a dynamic structural instability caused by interaction among aerodynamics, structural deflection and inertia. Static aeroelastic effects control the aerodynamic load distribution on the aircraft lifting surfaces. In some cases they determine the static stability of the flight vehicle itself. These effects are important design criteria for flight-vehicle structures (Wilson, 1986; Schneider and Zimmerman, 1986). The ability to increase or decrease the steady lift on a wing by actively changing the strain or the modulus of structural material could be an important design tool. The active structure might be used to decrease drag, improve performance of conventional ailerons, increase the stability of the vehicle or even maneuver the vehicle. To understand this potential design improvement, consider Fig. 1. Figure I shows a flexible swept-back wing planform that bends and twists when airloads are applied in the upward direction. With the airstream coming from the left, as indicated in Fig. 1, the wing is called a “wash-out” wing because bending and twisting produce a local decreased stream-wise incidence or angle of attack. This, in turn, reduces the size of the original airload on the wing. A wash-out wing is “lift ineffective” because it requires a larger initial wing angle of attack to create the same lift as a similar, but rigid wing. If the sectional incidence and airloads increase as the wing bends and twists, then the wing is called a “wash-in” wing. This would be the case if the airstream in Fig. 1 were reversed so that the wing was swept forward. This wing is “lift effective” since it produces more lift per unit angle of attack than a similar rigid wing. On the other hand, this type of wing is subject to a static instability called wing divergence at high dynamic pressures (Bisplinghoff et al., 1955). Aeroelastic

tailoring

Aeroelastic tailoring concerns itself with the intentional use of advanced composite directional stiffness to control the structural deformation and resultant aerodynamic forces that act upon a wing or a rotor blade (Shirk et al., 1986; Weisshaar, 1987). When the wing cover skin is composed of laminated composites, the structural stiffness distribution and deflection shape can be controlled by laminate plies in the proper directions. The result is that the distance between the airload resultant forces and moments and

Flexible

Wing

Rigid Wing

/ -
A-A

Wash-Out Fig. 1. Swept-wing airfoil incidence (wash-out) caused by bend/twist deflections.

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the so-called wing flexural axis can be controlled. In many cases, the designer can create wash-in or wash-out features as required. Aeroelastic tailoring and optimization were used to design the X-29 wing structure cover skins to delay the appearance of a static aeroelastic instability, called wing divergence. Uncontrolled, the tendency of the X-29 wings to diverge at low flight dynamic pressures would lead to a flight vehicle dynamic instability called body freedom flutter inside the operational flight envelope. Aeroelastic tailoring has more recently been used to design the vertical stabilizer of the highly maneuverable X-3 1. While laminate design can control aerodynamic/structural deformation load interaction, it is a passive, single-point design technique that cannot be changed in flight. This is unfortunate, because different flight conditions demand different, sometimes opposite, laminate orientations to create different stiffness orientations. Tailoring can only address the most important design condition, often to the detriment of other design requirements. For instance, the best laminate design to increase the aircraft roll rate might degrade the stability of the structure. To aid the reader who is unfamiliar with some of the areas that must be integrated into this technology, we will review, briefly, aeroelasticity, aeroelastic tailoring and active materials. The discussion of active materials will be confined primarily to piezoelectric materials that develop strain when a voltage is applied to their upper and lower surfaces. Active adaptive structures-smart

structures?

Wada ef al. (1989) define an adaptive structure as a structure that has the ability, through the presence of actuators, to alter its behavior or characteristics in a controlled manner. A structure capable of sensing its state is called a sensory structure. The combination of adaptive and sensory structures is called a controlled structure. One subset of controlled structures is the so-called active structure. These structural systems have sensors and actuators that are integral elements of the structure. These elements carry loads and function as actuators or sensors. The most advanced subset of active structures is the “intelligent” structure. Intelligent structures include built-in control logic as well as integral sensors and actuators. Using these definitions, the active structures proposed in this paper are “controlled structures” but are not “intelligent”. We will refer to these structures as active adaptive wings. The effectiveness of structures constructed with advanced composite materials can be increased by adding active materials to the laminate design. As a result, structural stiffness can be matched to changing flight conditions. Active materials have mechanical properties, for example Young’s modulus or strain, that change in response to external stimuli such as an electric field (Cross, 1989). The addition of a pre-programmed control law to control the stimulus and sense the results of this control means that these active material properties can be “slaved” to the external stimulus. Active materials show promise for aeroelastic flight-vehicle structural design. The presence of the airstream is a key consideration in the study of adaptive structures for flight vehicles, since it represents an energy source to be exploited. As a result, an active aeroelastic structure uses the airstream as an energy source to drive the structure toward its intended shape. Scope The focus of this paper is limited to simplified approximate, mathematical models of active adaptive flexible swept wings controlled by linear piezoelectric materials. The adaptive-structure wing model will be based on classical laminated plate theory using a linear piezoelectric material model. The laminate will be assumed to be a sandwich type with embedded piezoelectric material layers. In the next section we will discuss this model and the tailoring issues that drive the design. COE 2:5/7-K

T. A. WEISSHAARand S. M. EHLERS

460

AEROELASTIC

TAILORING

STRUCTURAL

MODEL-REVIEW

Laminated wing structures are complex built-up configurations. Within these configurations, there are numerous advanced composite plies, oriented so that material stiffness and strength is used in an optimum way. Analysis of such structures can range from sophisticated to elementary, depending upon the objective of the analysis. At the most elementary level, beam models are used to represent the deflection behavior of the wing. This model’s effectiveness is restricted to wings where the ratio of wing length to mean chord is in the range of 5 or above. Because these models are useful for trade studies and for identifying the interaction of design parameters, this type of model is used in this paper. A laminated-beam model developed by Weisshaar (1978, 1980) uses wing laminate geometry and ply lay-up details to develop the relationship between beam(wing) bending curvature and twist rate and the internal resultant bending moments and torques applied to the beam cross-sections. These relationships are developed with respect to a wing reference axis, shown as the y-axis in Fig. 2. This figure also shows the location of the so-called flexural axis and its dependence on the cross-sectional stiffnesses. The most important beam deformations are the bending slope C#Iand the wing twist angle, 8, measured with respect to the reference axis in Fig. 2. The relationship between the bending and twisting deformations and the cross-sectional stiffnesses is given by the following:

The notation ( )’ refers to differentiation with respect to the wing reference axis coordinate y. The terms EI, GJ and K are the equivalent beam cross-sectional stiffnesses. There are several different ways of developing expressions for these coefficients, depending upon whether or not the beam is assumed to be plate-like or whether or not it is of thin-wall closed-cell construction with one or more cells (Hodges, 1988). The stiffness term K is referred to as a stiffness cross-coupling term. Its magnitude and sign are a function of laminate geometry, in particular the ply orientation (Weisshaar, 1980). The presence of the K term is essential to controlling the wing deformation using laminate ply orientation. Figure 2 shows the flexural axis and provides an equation for the location of this line with respect to the reference axis (Kobyashi, 1984). The flexural axis location is defined by removing the aerodynamic loads and then placing a point load upward on the wing/beam. Next, the stream-wise rotation at the cross-section where the load is applied is computed. There will be a unique fore or aft position of the point load, qEA, such that combined beam bending and twist along the reference axis will create zero stream-wise %A = (1) /

GJsinA

2 EI COS*A+ GJ sia*h

for Swept Wings Fig. 2. Beam reference axis and flexural axis position, qEa.

Adaptive aeroelastic composite wings

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incidence at the cross-section where the load is applied. The locus of all such points along the wing is the flexural axis. The expression for the flexural axis location depends on EZ, GJ and K, as well as on the wing sweep angle. The position of the flexural axis with respect to the wing centers of pressure, or more accurately, the aerodynamic centers, is important. In subsonic flight conditions, these aerodynamic centers are located approximately one-quarter of the wing chord behind the wing leading edge. This relative position between the wing quarter-chord and the flexural axis determines whether the wing will tend to rotate downward under load (the wash-out wing) or whether it will rotate nose up (the wash-in wing). The best values of K, EZ and GJ are found by optimizing the laminate ply geometry. On the other hand, these ply angles are fixed. Because the laminate may be design-critical for some load conditions that it seldom sees, the performance of the structure may be less than optimal for other conditions. Incorporating active materials will increase the effectiveness of the structure by using both laminate design and active, controllable, inflight stiffness. ACTIVE

MATERIALS

Active materials respond to external stimuli such as electric fields, electric current, magnetic fields, and temperature. When such a stimulus is applied they will change shape, develop internal strain or change some of their mechanical properties. Active materials include polymers, metals, ceramics and composite forms of these materials. There are several different materials that might be considered for adaptive aeroelastic design. These include: l l l

shape-memory effect materials have two shapes-they are formed at one temperature, but return to their original shape when heated; magnetostrictive materials that change size when a magnetic field is applied; and piezoelectric materials that change shape when an electric field is applied.

Let us briefly consider these possibilities. Shape-memory

effect @ME) alloys

The ability of a material to recover (change) its shape upon exposure to an external stimulus (such as heat) is known as the Shape-Memory Effect (SME). It was first discovered at the Naval Ordnance Laboratory in near equiatomic Nickel Titanium compounds, in what became known as Nitinol (Buehler and Wiley, 1965). Nitinol can recover up to about 8% plastic strain and generate stresss on the order of 100 ksi. Nitinol also increases its Young’s modulus by almost a factor of three during the transition caused by heating. SME materials cannot change shape quickly, and as a result they are not effective as actuators that must respond to dynamic conditions. On the other hand, they can generate large strains and have been used to actuate “tuned” panels to create changes in the vibration frequencies (Rogers, 1990). Scott and Weisshaar (1991) have proposed using Nitinol patches to tune frequencies to increase panel flutter airspeeds, using heat generated at high supersonic speeds. Magnetostrictive

materials

The creation of an elastic strain by an external magnetic field is called magnetostriction, an effect present in ferromagnetic materials. The converse effect in which an applied stress changes the magnetic induction of a material also occurs. Unfortunately, the strains induced by magnetic fields in common magnetostrictive materials, such as iron, nickel and cobalt, are usually only on the order of tens of microstrain. These levels are too small to create an effective structural actuator for aeroelastic control. On the other hand, recent research has produced a new generation of magnetostrictive materials capable of inducing strains up to 0.2%. While these materials

T. A. WEISSHAAR

462

and S. M. EHLERS

can provide large enough strains to be used as an actuator, they are not effective because it is difficult to apply the necessary magnetic field to the embedded actuator. Piezoelectrics Piezoelectric ceramics operate on the principle of piezoelectricity (cf. Jaffe et al., 1971). This term was coined from the Greek words meaning “pressure electricity”. The piezoelectric effect involves the generation of an electric charge by application of stress. When an electric current is applied to a piezoelectric crystal, there is an internal lattice deformation created by anion/cation attraction within the crystal (Uchino, 1986). The converse piezoelectric effect, the change in dimension induced by an applied voltage, was first demonstrated by Jacques and Pierre Curie (Mason, 1981). Piezoelectric material deformation can be represented as the sum of two effects. The first effect produces deformation proportional to the electric field. The second effect is called electrostriction and produces a deformation proportional to the square of the electric field. The principal candidate piezoelectric materials for engineering design are actually classified as ferroelectric materials (IEEE, 1986). These materials exhibit spontaneous polarization along an axis that can be reoriented if a large enough electric field is applied. Piezoelectric materials have some undesirable characteristics, such as nonlinear response at high field levels, hysteresis and loss of properties due to aging. This behavior is most severe in the piezoceramics. Despite these potential difficulties, our focus in this paper will be concerned exclusively with the use of piezoelectric actuators for aeroelastic control. ACTIVE

AEROELASTIC

STRUCTURES

The use of integrated active materials in a structure has been considered to control orbiting space structures. The critical design requirements of these structures are to damp vibrations or control static shape. Control of such structures, using materials integrated into the structure, was first introduced by Swigert and Forward (1981) who used piezoelectric transducers to damp bending vibration of a flexible cylindrical mast. Their work represents one of the first uses of piezoelectric materials as both a sensor and an actuator. At nearly the same time, Chiarappa and Claysmith (1981) developed a system for actively controlling the surface contour of a mirror. This control was achieved using piezoelectric materials in a stacked actuator configuration. The actuators were separate from the mirror surface and attached to the back surface at discrete locations. This system represented an early example of adaptive optics using active structure design. Another example of a deformable mirror element using multi-layered electrostrictors was designed and built in Japan (Uchino and Nomura, 1981; Sato et al., 1982). A multilayered laminate, divided into separate modal actuators, was used. Unlike Chiarappa and Claysmith this design embedded active material in the mirror substrate. This Japanese design is considered to be the first use of electrostrictive materials as the active material element in an adaptive structure. Active control of flight-vehicle

structures

Credit for the first actively controlled flight vehicle with a tailored stiffness structure must certainly go to the Wright Brothers. Their aircraft was unstable to the extent that it had to be actively controlled by the pilot. For roll control, a system of cables was pulled when the pilot moved his hips from side to side. These cables warped the biplane wing structure so that the aircraft would roll (Wolko, 1987). In more recent times, the Mission Adaptive Wing (MAW) Program successfully used static camber-shape control on an F-l 11B to reduce drag (DeCamp and Hardy, 1984). This control used a hydraulic actuator arrangement, housed in the wing, to actively deform the wing structure. Aeroelastic effects were incidental and not part of the design objective.

Adaptive aeroelastic composite wings

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Active control with hydraulically powered ailerons and wing flaps also has been used successfully for gust-load alleviation and aircraft-stability augmentation (No11 and Perry, 1989). The primary objectives of these projects has been the design of the control system and the control law relating the surface motion and the sensor output. Combined optimization of the structural configuration and control system also has been studied. The Active Flexible Wing (AFW) Program (Perry et al., 1988) is an example of successful integrated structural and control-system development. The objective of this program was improvement of aircraft performance and stability using active control technology and the inherent flexibility of the wing structure. Leading-edge and trailingedge surfaces controlled the deformed shape of the wing to improve roll effectiveness at high speeds. Active adaptive aeroelastic structures There are few published active material/adaptive flight vehicle structures studies. One such study (Crawley et al., 1988) has examined the use of laminated piezoelectric configurations to control hypervelocity missiles. Piezolectric actuation was used to create chord-wise bending and span-wise twisting to control aerodynamic loads. In another study, Bohlmann and Lazarus (1990) examined the response of piezoelectrically actuated flat plates in an airstream. Chord-wise curvature of the plate to generate lift loads was created by piezoelectric material actuation. In a different study, Spangler and Hall (1989) investigated the use of a piezoelectric servotab actuator to control high-frequency vibration of helicopter rotor blades. Both of these latter studies used piezoelectric materials as actuators. However, neither used them as an integral part of the lifting surface structure. In addition, no feedback of the state of the structure was used to drive the actuators. More importantly, aeroelastic interaction was not examined in either case. This exclusion limits the effectiveness of the actuators. Because external aerodynamic loads and internal structural forces both depend on wing structural distortion, the effective stiffness of a wing is changed by aerodynamic forces. In addition, the total lift that a wing would produce if it did not deform may be very much different than the lift created considering structural deformation. These changes are so important that the Federal Aviation Regulations (FAR Part 25) governing structural design of transport aircraft require that these changes be considered in analysis of loads. The ratio of total lift produced by a flexible wing to the lift produced by a similar, but rigid wing is called lift effectiveness. For a swept wing, this ratio is usually less than unity. Changing the lift effectiveness and divergence airspeed with an active adaptive wing structure has been examined by Ehlers and Weisshaar (1990). They considered the use of piezoelectric materials embedded in a laminated wing skin. The strain in the active materials was controlled by a feedback control law. This law specified the size and direction of voltages applied to the thin piezoelectric layers. More recently, a study by Weisshaar and Ehlers (1991) examined the control of wing-lift effectiveness and wing-static stability by changing the feedback gain between the wing root load and the applied voltage. These results indicate that there are limits to the amount of lift change that can be obtained. These limits are due to an effect called actuator saturation and will be discussed later in this paper. Song et al. (1991) have also examined active material control of the static aeroelastic response of a swept composite wing. They used a thin-walled tubular laminate model for their wing structural stiffness development. Actuator limitations were not addressed in their study. Scott (1990) has investigated the use of active materials to increase the supersonic panel flutter speed and supersonic flutter speed of a low-aspect-ratio wing. His results indicate that controlled piezoelectric materials may not be effective for panel flutter control because of their limited strain capabilities. In addition, piezoelectric materials may add unwanted weight to the panel that far outweighs the benefits of active stiffness.

464

T. A. WEISSHAAR

and S. M.

EHLERS

The use of both the shape-memory effect and piezoelectric materials to change the supersonic flutter speed of a rectangular panel has also been investigated by Scott and Weisshaar (1991). These results show that the flutter speed of a panel can be increased by using SME materials to create in-plane tensile forces to increase panel stiffness. Recent work by Hajela and Glowasky (1991) reinforces the conclusions of earlier panel flutter control studies. Lazarus et al. (1991) have investigated the effectiveness of active material actuators to control flutter. In their study, they reduce the three-dimensional wing to a twodimensional typical section idealization. Several in-depth analyses link the effectiveness of the controller to the task of controlling flutter. Their work indicates that active materials may be an effective means to control flutter of certain types of wings. As mentioned previously, this study will confine its scope to piezoelectric actuators to control static aeroelastic deformation and the loads that arise from this deformation. We now turn to the development of an analytical model to study the effectiveness of piezoelectric layers to control static aeroelastic deformation and the lift effectiveness and stability of swept composite wings. We begin with a discussion of the wing-active structure model. WING-ADAPTIVE

STRUCTURE

MODEL

The wing planform for this study is shown in Fig. 3. This wing has a uniform structure with beam-like load/deflection characteristics. The wing cover structure is laminated. It includes two thin sheets of active piezoelectric material covering the entire wing box. The thin laminated cover skin structure has a chord-wise width, w, and depth, h, as shown in Fig. 3. The wing planform has a chord-wise dimension, c, and a length, L. The wing root is placed at an angle of attack O,,, measured with respect to the swept reference axis, y. The size and direction of the active piezoelectric material strains are controlled by a feedback control law. This control law relates the voltages applied to the piezoelectric layers to the bending and twisting strain at the wing root caused by aerodynamic loads distributed along the wing. Figure 4 shows the thin piezoelectric layer and the position of this layer within the structural laminate. The piezoelectric material has conducting upper and lower surface coatings and is assumed to be in a state of plane stress. A voltage, V, is applied across the thickness to create an electric field, E, = V/t, where t is the thickness of the layer. The relationship between the stress and strain in directions 1 and 2, shown in Fig. 4, and the transversely applied electric field, E, , in direction 3 reads:

(2)

Fig.

3. Swept-wing

planform

model

(wing

box shaded).

Adaptive aeroelastic composite wings

465

3

Fig. 4. Piezoelectric layer and position within the laminate.

The coefficients d, measure the material strain developed per unit of electric field applied across the actuator thickness. This relationship is similar to the relationship between free strain and temperature change. When a piezoelectric layer of material is oriented at an angle with respect to the structural axis of the wing (the y-axis in Fig. 2), the relationship between stress, mechanical strain and piezoelectric strain is:

(3) The 1, 2, 6 directions in eqn (3) and Fig. 4 transform to the x,y directions in Fig. 2 (cf. Tsai and Hahn, 1980). The notation s refers to shear in the xy plane. The notation dss represents the term d,, in this notation. Note that, because the piezoceramic material is isotropic, this term will be zero unless the material is applied in a special manner (cf. Barrett, 1990). The wing structure is idealized to be a laminated sandwich beam with a piezoelectric material integrated into the wing as shown in Fig. 5. The beam idealization is developed by first considering a plate model and then reducing this plate model to a beam model. The kth ply of composite material is located a distance zk away from the plate mid-plane. It is assumed to follow the Kirchoff-Love relationship between stress, in-plane strain and curvature. In addition, the influence of the electric field is as in eqn (3). An inactive advanced composite layer has an applied field E, of zero. The relationship between the in-plane moment resultants and plate curvatures and applied voltage is: [$]

=

Dijl[$]

+

tcMlv*

(4)

Equation (4) is the result of multiplying the ply stress by its distance, z, from the laminate mid-surface, and then integrating with respect to z. The plate-bending stiffness coefficients Dij depend upon the lamina properties, including the stacking sequence, stiffness er skin

referencdaxis

ae

lower

+

skin

Fig. 5. Idealized laminated wing-structure cross-section.

466

T. A.

and S. M.

WEISS-MAR

EHLERS

and orientation with respect to the y-axis. The matrix {CM) is a column matrix composed of elements given by: mxx e3x c mYY

Slc

mxy

e3y

= k

il

e3s

(5)

zk. k

Equation (4) is restricted to have only one voltage input, V, applied to two layers arranged symmetrically top and bottom on the laminate. The voltage is applied antisymmetrically (or the layers may have opposite polarization) to create a bending or twisting moment. A similar expanded relationship can be derived for multiple active layers. The plate constitutive relations in eqn (4) can be presented in a slightly different form as:

Equation (6) is called the “actuator the following relationship:

equation”.

The actuator matrix (r) is computed from

trJ = [Dijl-ltcM~~

(7)

Assuming that the wing is unrestrained in the chord-wise direction so that the chord-wise bending moment MIX is zero, the relations between spanwise bending and twisting moments are as follows:

(8) where the reduced plate bending stiffnesses are given as: (9)

The beam constitutive

equations are written as:

where the beam cross-sectional properties are defined as: EI = w&, GJ = 4wDs6

(11)

K = 2wD,,.

The beam actuation coefficients are related to the plate-bending

stiffness coefficients by:

r, = -r, r, = +r,. These coefficients may be approximated

by the following

(12)

relationships: (13) (14)

These actuator coefficients are functions of the laminate cross-sectional stiffness and the electromechanical coefficients of the piezoelectric material.

Adaptive

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This active-beam stiffness model can now be used to represent a wing with airloads applied to it. We can then determine how we can control the magnitude of these airloads and examine how much control we have available. SWEPT-WING

AEROELASTIC

DEFLECTION

A closed-form solution for the deflection and resultant lift generated by an idealized, conventional flexible swept wing with a uniform beam-like structure was first developed by Diederich and Foss (1952). Since their development occurred two decades before active interest in advanced composite materials, their solution did not include the stiffness cross-coupling, K. A closed-form solution to the same problem, but including the stiffness crosscoupling, was developed by Weisshaar (1979). This model was modified by Ehlers (1991) to include layered, active, piezoelectric actuators and sensors. The bending and torsion static equilibrium equations for the wing model with two active layers of piezoelectric material, spread uniformly over the upper and lower wing box surfaces, are written as: EZ $

- K $

= l(J)

GJ$

- K$

= t(j)

(15)

The distributed lift, I(y), and torque, t(y), are approximated by strip-theory aerodynamic expressions. These expressions assume that the fluid dynamic interaction among airfoil sections along the wing is weak (cf. Bisplinghoff et al., 1955). In this case I(y) and t(y) are given as: I(y) = qca, co? A(& + 0 - 4 tan A)

(17)

t(y)

(18)

= qcea, cos2 A(& + 0 - 4 tan A).

Equations (17) and (18) contain aerodynamic information such as the dynamic pressure, q, a function of aircraft speed and altitude, and the wing section lift-curve slope a,,. This lift-curve slope is assumed to be constant along the wing by this strip-theory approximation. These coupled bending and twisting equilibrium equations are combined into a single equation in terms of an effective angle of attack, a,, defined as: (Y, = 8 - $tanA. The resulting differential coordinate q = y/L is:

equation a;

Two nondimensional

(1%

with respect to the nondimensional

+ A~: + he

=

span-wise

-Dee.

(20)

constants, A and D, appear in eqn (20). They are defined as: A=q

$os’A(’

; It;iA)

(21)

where k = K/EZ and g = K/GJ; D = q$cos2A(ta;t;gg).

(22)

When the stiffness cross-coupling, K, is small then the size of A reflects the importance of aeroelastic torsional flexibility to the solution for the deformation of the wing. The magnitude of D indicates the importance of bending deformation and sweep to the problem. Notice that the electromechanical features of active material inclusion in the laminate do not appear in eqn (20).

468

T. A. WEISSHAAR

and

S. M.

EHLERS

The ratio A/D does not depend on 4. Small values of A/D mean that the wing static aeroelastic deflection and load distribution are determined primarily by bending deflection and the size of the wing sweep, A. Large A/D values indicate that torsional deformation will be more important than bending. Another meaningful nondimensional dynamic pressure parameter for swept wings is defined as: q=di?X (23) This parameter is useful when comparing aeroelastic deformation of wings with an identical A/D ratio, but with different values of A and D. Three boundary conditions are required to compute the solution for CY,. Then the lift on the wing can be calculated. First of all, since the incidence angle of the wing root is fixed, the following boundary condition is required: q(O) = 0.

(24)

As yet, the active material properties do not appear in any of our equations. The presence of active material appears because both the resultant cross-sectional torque and bending moment at the wing tip must be zero. This is written, in dimensional form, as: (25)

These two requirements

can be written in terms of a single boundary condition

involving

E, and the aeroelastic angle of attack, CY,:

a:(l)

= (I, - I, tanA)E,L.

(26)

Note that E3 is an independent input in eqn (26). The final boundary condition reequires that there be no transverse shear resultant at the wing tip. This condition is: (r;(l) + Aa, = -A&. (27) The active control law formulation

If we were to specify the values of both wing incidence 0, and electric field E3, then the aeroelastic deflection CY,along the wing would appear as a function of two independent inputs. Since the lift on the wing is computed by integrating the lift per unit length, eqn (17), the total lift would be a linear function of &, and E3 and a transcendental function of A and D. The lift generated by the flexible aeroelastic wing can be compared to the amount of lift generated when the wing is rigid. If we call this “rigid” lift L, and the lift on the flexible wing L,, then the ratio L,/L, is called the lift effectiveness. The lift effectiveness is a function of A, D, 0, and the ratio E3 /8,, . If E3 is independent of other input quantities, the wing will be active, but not adaptive. To make the wing structure active and adaptive, we will require that the electric field be proportional to an internal load at the wing root. This suggests that one or more derivatives of the wing displacement function should be sensed and used to control the voltage applied to the piezoelectric layers. A control law is defined so that: E3 = K&(O).

(28)

The term KP is a free feedback gain parameter under the control of the designer, but subject to material limitations. The boundary condition in eqn (26) then becomes: o:(l) For simplicity,

= K,(I,

- I, tan A)a:(O).

(2%

define a new feedback gain parameter KP as follows: KP = K,(I,

so that the boundary condition

-

r,

tan A)

(30)

in eqn (29) becomes: C$(l) = K&(O).

(31)

Adaptive aeroelastic composite wings

469

Equation (31) removes the electric field as a visible, independent input to the system and modifies the stiffness of the wing. The piezoelectric layers appear as an effective wing-tip load that responds to deformation at the wing root. Equation (31) also represents a design “promise” because we have assumed that the electric field required by this boundary condition can be supplied. In reality, all piezoelectric materials have limitations on the size of the electric field that may be applied to them without saturating them and causing damage. The maximum value of electric field, called the “depoling value”, is a material parameter denoted as ET. As a result, the range of Kp is limited. We will examine this limitation later. For now, we will assume that we can supply the electric field required by the gain parameter K,, and simply use it as a design parameter in place of E3. Control of static stability

boundaries

The solution to eqn (20) combines a particular solution and the homogeneous solution to the differential equation. The boundary conditions determine values for the arbitrary constants contained in the homogeneous solution. The homogeneous solution and the boundary conditions are also used to solve for unique combinations of A, D and Kp that indicate the static stability, or divergence, boundaries for the wing. When Kp is zero, these wing divergence boundaries are functions only of A and D (Flax, 1961). When K,, is not zero, the wing divergence dynamic pressure can be changed by the active piezoelectric material. Figure 6 shows wing divergence boundaries for four values of Kp, including zero. At values of q above the lines shown in the figure, the wing will be statically unstable. When Kp is 1 the wing stability boundary passes through the A, D origin. This indicates that the active wing is statically unstable even when the airspeed is zero. This instability is caused solely by the feedback control system interacting with the beam deflection to produce large deflections.

Fig. 6. Wing static stability boundaries for values of feedback gain, Kp, equal to 1, 0, -1, -2.

Any negative value of Kp will improve wing stability. On the other hand, positive values of Kp degrade stability. Notice that the stability boundary pivots about a point in the upper right quadrant of Fig. 6 and Kp changes. This pivot point is located at the A/D ratio of 0.710. This special value represents a wing configuration operating at a dynamic pressure for which the adaptive structure has no effect on wing stability. This special value occurs because the wing deflection combination chosen by the control law in eqn (31) is unobservable to the sensor at the wing root. As a result, no matter what value of K,, is chosen, no voltage will be applied to the piezoelectric layers. This situation can be changed by choosing a different feedback control law. When the active material layers reposition the wing divergence boundary, the lift effectiveness of the wing also changes. Changes in lift effectiveness at a fixed angle of attack are important because these changes can be used to control the motion of the aircraft.

470

T. A. WEISSHAARand S. M. EHLERS

Lift effectiveness-examples Lift effectiveness was defined as the ratio of the lift generated by the flexible wing to the lift generated by a similar but rigid wing. Since the initial angles of attack for each of these conditions are identical, the wing root incidence 0, does not appear in lifteffectiveness expressions. Also, E3 is controlled by KP so that it does not appear explicitly in the expression for wing lift. Lift effectiveness is a function of KP, A and D. Let us consider an example to illustrate how lift effectiveness, and the total amount of lift acting on the wing, can be changed by an activation of the piezoelectric material layers. For this illustration, let us examine a forward-swept wing with a stiffness parameter ratio A/D equal to - 1. This negative ratio could also represent a swept-back wing to which stiffness cross-coupling has been added to induce wash-in. Contours of constant lift effectiveness for this wing are plotted as a function of 4 and KP in Fig. 7. The divergence condition for this wing when there is no feedback (KP = 0) is at point A in Fig. 7. There is a control-system-induced instability at point B, at zero airspeed, when KP is 1. 6

KP

Fig. 7. Contours of constant wing-lift effectiveness for a swept-forward wing with A/D = -1, plotted as a function of 4 and K,. The contour labelled 1.O in an aeroisoclinic wing.

At a constant flight speed and altitude, 4 is a constant. A horizontal line drawn across Fig. 7 at any value of 4 will intersect lines of constant lift effectiveness. Assuming that we operate below the 4 value that intersects point A, it can be seen that changing the size of KP will change the lift effectiveness of the wing, and thus the total lift acting on the wing when the angle of attack at the root remains fixed. As a result, the lift on the wing can change when KP is changed. Assume that the wing is initially trimmed (the aircraft is in straight and level flight) at a fixed wing angle of attack 19~when KP is 0. Now, let us increase the gain (KP > 0). As a result, the wing-lift effectiveness will increase and the total lift on the wing will also increase. If nothing else is changed, the aircraft will increase its altitude. Conversely, if the gain parameter decreased (KP c 0) then the lift effectiveness will decrease and the aircraft will descend. Notice the nearly straight line shown in Fig. 7 as the line L,/L, equal to 1. This line defines a special relationship between KP and Q so that the lift effectiveness does not change with dynamic pressure. This condition is known in aeroelasticity as the “aeroisoclinic” condition. The wing deforms so that the effects of the airloads created by bending and twisting deformation will cancel. Now consider a different wing configuration for which A/D is 0.5. This special value was chosen so that the wing will not diverge for any value of Q when KP is zero. This ratio is typical of a conventional aft-swept wing or a laminated forward-swept wing with negative stiffness cross-coupling. Contours of constant lift effectiveness are plotted as a function of q and KP in Fig. 8. Two regions of instability appear in Fig. 8. One region occurs at low dynamic pressures when the feedback gain is positive. The other region occurs at higher dynamic pressures with negative feedback gains. This wing is unstable at zero airspeed when KP = 1.

Adaptive aeroelastic composite wings

471

Fig. 8. Contours of constant wing-left effectiveness for a swept-back wing with A/D plotted as a function of 4 and K,, .

Limits on the feedback-gain parameter &--strength

= 0.5,

parameters

The size of the gain parameter is limited because of limits on E,, To examine the nature and effect of these limits, two nondimensional parameters, called strength parameters, are developed to relate material properties to design objectives. One of these parameters can be used when the angle of attack at the wing root is constant, such as when a wing is mounted in a wind tunnel. The other will be used when the resultant wing lift must be constant, such as when a wing operates at a constant speed and altitude. The development of the “fixed-root strength parameter” begins by setting eqn (26) equal to eqn (31). The result is: (r, - r, tanA)E,

= K,ol:(O).

(32)

The derivative of CY,at the wing root can be written symbolically 4(O) = e,foU,

as:

D, K,).

(33)

Substituting eqn (33) into eqn (32) and dividing both sides of the resulting expression by the wing-root incidence angle yields:

(re - r$;A)E3L

=f,(A,J)K,).

0

Equation (34) is nondimensional. The left-hand side represents the ratio of the wing-tip angle of attack (created by the active material when the flight speed is zero) to the wing-root incidence angle, measured with respect to the swept reference axis. The term on the left-hand side of eqn (34) is denoted by So. It is called the fixed-root strength parameter. The right-hand side of eqn (34) depends on the parameters A, D and KP. It gives the required magnitude of So for a given flight condition. Substituting the approximate expressions for the actuation coefficients into eqn (34) gives an expression for So that contains wing geometry, adaptive structure and active material properties: s

=

0

KK

-

GJtan

W3,

- (EZ - K tan A)e3,]E3L Bo(EZGJ - K2)

[tadowl.

(35)

To change the value of So we can change the stiffness of the wing, change the actuator thicknesses or width or use actuator materials with different electromechanical constants. In trimmed free flight the wing root incidence angle 0, in eqn (35) depends on the flight dynamic pressure. As a result, relations between vehicle size, performance and active material properties need to be established. First of all, the wing-root incidence angle is computed using the expression for lift effectiveness. Next, the adaptive winglift effectiveness is calculated by integrating the lift distribution. This is a function of

412

and

T. A. WEI~SHAAR

A, D and Kp and is represented symbolically

S. M.

EHLERS

by:

LF - = fr(A, D, K,).

(36)

LR

Note that eqn (36) is independent of &,. Using eqn (36) and setting the flexible lift, L,, equal to the vehicle weight, W, the following expression for 0, in terms of A, D and Kp is obtained: (37) In eqn (37) the aircraft weight is W, the symbol S represents the wing surface area, while a, is the lift-curve slope of the wing sections. Substituting eqn (37) into eqn (33) and placing f, on the right-hand side, we obtain: qa, cos2 A

[(K - GJtan A)e,, - W - K tan W3,1E3L t d w = .hM9 Q KJ a (I (W/S)(EZGJ - K2) f,(A~,K,)’

t38j

Equation (38) can be manipulated to develop the constant lift-strength parameter. We first replace the term qa, with the following expression obtained from the definitions of A and D: dc-5 (39) 4a. = cL2 cos2 A [e2(EZ - K tan A)2 + L2(GJtan A - K)2]-“2. We then move terms involving

A and D to the left-hand side. The result is:

KK - GJtan *)e,, + (EZ + K tan A)e3,1~3Va4~cl = f cA D K ) ( W/S)Lde2(EZ

- K tan A)2 + L’(GJtan

L

A - K)2

’

The left-hand side of eqn (40) is denoted as S, , the constant lift-strength nondimensional and defined as:

sL = Kwfi

- R tan *)e3, + (1 - wfi

(W/S)(L/c)J7e/c)2(1

- yfi

’ ’ ’ parameter.

tan A)e~,lE3[(t,/h)(d,/h)(w/c)l(h/c)2.

(40) It is

t41I

tan A)2 + (L/cm

In eqn (41) the stiffness ratio EZ/GJ is represented by R, while the ratio K/(EZGJ)“2 is represented by w. The size of SL is limited by the piezoelectric material properties, laminate geometry and vehicle design parameters such as W/S (the wing loading). The piezoelectric stress coefficients, e3X and e3$, are fixed material properties. The magnitude of the applied electric field, E,, is limited by depoling or saturation considerations. A plot of lines of constant S,, plotted against Kp and 4, is shown in Fig. 9 when A/D = -1. Note that relatively small values of SL (of the order of l/2) will allow the 6 5 4 zj3 2 1

0

y/l

0

Fig. 9. Contours of constant lift-strength parameter S, as a function of gain parameter and a for A/D = -1.

Adaptive aeroelastic composite wings

473

Fig. 10. Contours of the constant-strength parameter S, as a function of 4 and K,, for A/D

= 0.5.

active wing to change its divergence speed by a large amount. Considering Figs 7 and 9 together shows that smaller values of S, are required at larger values of dynamic pressure if we wish to make the wing behave aeroisoclinically. A contour plot of the constant lift strength parameter for the second example is presented in Fig. 10. Comparing Figs 10 and 8, we see that the ability to reduce lift effectiveness at low speeds is limited by the maximum value of the strength parameter. At higher dynamic pressures this requirement is reduced by the contribution of the displacement-dependent aerodynamic forces. The magnitude of Sr required to maintain rigid-wing lift effectiveness increases only slightly with increasing dynamic pressure. The size of the strength parameter necessary to increase lift effectiveness above the value of a nonactive wing (Kp = 0) also increases with dynamic pressure. Limits

on effectiveness and prospects for tailoring

Previous expressions for the strength parameters include the vehicle wing loading W/S as a parameter. This indicates that the physical size of the vehicle affects the size of the required strength parameter. Equation (41) can be rearranged by multiplying by W/S and dividing by SL to obtain the following expression for wing loading as a function of wing structural and geometric parameters: ;

= 4(g9[

w’p’]

[(u/a - R tan A)e,, + (1 - I,&? tan A)e,,]E, S,~/C)~(I+VI& - R tan A)2 . (42)

Equation (42) must be satisfied if the active piezoelectric material is to control the wing. The first group of terms on the right-hand side of eqn (42) indicates the relative thickness and location of the piezoelectric layers in the laminate (see Fig. 5). The second group includes geometric terms describing the wing geometry. These include the thicknessto-chord ratio, structural aspect ratio, L/c, and wing box size, w/c. The final group includes terms such as the structural stiffnesses, sweep angle, aspect ratio and reference axis/aerodynamic center offset, etc. The piezoelectric stress coefficients, eJwand eXs, and the applied electric field also appear. Equation (42) can be written in the following form: (43) The symbol P in eqn (43) is defined as: p = [(I,&? - R tan A) + (1 - wfi AA)‘(e/L)’

tan A)e,,/e,,] *

(44)

474

T. A. WEI~SHAARand S. M. EHLERS

There are two limiting cases for P. First of all, when K is zero and GJ tends to infinity, then P = 1. This is the situation known as “pure bending” in which torsional deflection has no effect. Equation (42) reduces to a relation between wing loading and normalized pieoelectric strength for a swept wing with only the aeroelastic effects of wing bending. When K and A are zero and EI + OQthen the factor P represents the value for the case where only torsional deformation (pure torsion) is included in the aeroelastic analysis. In this case: p = (e3Je3JL/c) (45) (e/c) Substituting eqn (45) into eqn (42) gives a relationship between the wing loading W/S and normalized piezoelectric strength for a wing with pure torsion:

WWW2 [(L/c)(e/c)]

e3sE3

I* Even without torsional actuation capability (e 3s = 0) laminate the size of P. When e3s is zero, the expression for P reduces to:

(46)

I S,

tailoring

can control

(47)

The maximum value of P in eqn (47) is unity. Laminate design tailoring will find the best combination of P and SL to maximize control. Equation (43) relates wing loading to required normalized piezoelectric strength. This linear relationship is graphed in Fig. 11. The slope of the line in Fig. 11 depends on the position of the active material layers in the structural laminate and other parameters, as indicated. Let us define the normalized piezoelectric strength as the piezoelectric stress divided by the required value of SL and then multiplied by a factor P. The size of this parameter is limited by the maximum applied field, E3, and the piezoelectric stress constants for the material, e3x and e3s. The structural parameters and geometry also have an effect, as indicated by their inclusion in P. Material limitations are indicated by the vertical, hatched-line boundary in Fig. 11. The intersection of this boundary and the inclined straight line provides the maximum value of wing loading that is controllable by the active wing. To increase the wing loading requires either a piezoelectric material with better performance, a smaller required strength parameter or a larger value of P. The design space available is enclosed by the triangle shown in Fig. 11. P can be increased by changing the wing cross-sectional stiffness ratio R = EI/GJ and the stiffness cross-coupling parameter. When the stiffnesses are changing, the values of the parameters A and D may change. If this is so, then the required magnitude of SL also changes. This indicates that there is an interaction problem that requires trades between design parameters.

-P I%A I SL

Fig. 11. Wing loading versus normalized piezoelectric strength for an aeroelastically tailored adaptive swept wing.

Adaptive aeroelastic composite wings

475

In general, this interaction becomes a design optimization problem in which conflicting objectives must be resolved. In the end, the structural and active material properties will be configured to achieve desired performance levels within the limits of the active-material properties. CONCLUSION

Forming adaptive structures from active materials embedded in composite laminates extends the versatility of passive tailoring to an active capability. This capability provides “tailored stiffness on demand”. On the other hand, many obstacles must be overcome before these ideas can be translated into reality. These include manufacturing and reliability considerations, as well as cost. Other practical problems remain to be solved. Current piezoelectric materials are isotropic. Currently, shear actuation (e,,) such as that described in this study is not possible unless laminate extension-shear coupling is used or special construction techniques are used. The development of other material forms such as fibers and anisotropic, directional actuators would provide more design options. In addition, the capability of embedding reliable actuators within a laminate is in an exploratory stage of development at best. This study suggests that theoretical size or scale effects may limit the size of aircraft wings that can be controlled with active structures. This is the direct result of the actuator cross-sectional area growing as the square of the scale dimension while the size of the vehicle to be controlled grows as the cube of this scale dimension. This “square-cube” law conflict, together with practical manufacturing and maintenance issues, must be overcome by developing better actuator materials. On the other hand, the development of more effective materials, the integration of these materials with structural configurations such as thin-walled, multi-cell configurations and the development of effective fibrous piezoelectric materials promises to move this technology forward. The expressions developed in this paper for the effectiveness of materials for aeroelastic use show that there is an optimization problem relating the best combination of materials, not just the best material. Finally, there are ways to control wing deflection and motion other than deforming the entire structure. For instance, panel flutter stabilization, cited previously, uses active materials to control small local phenomena and achieve large results. A similar application is to use small actuators to control wing drag at transonic speeds. Shock-wave control in inlets may be within the capability of active-material devices. Challenges such as these will provide opportunities to exercise creativity by combining advanced material forms to create better, more efficient flight vehicles. Acknowledgement-The authors acknowledge the support of NASA Langley Research Center under Grant NSG l-157 to Purdue University. The technical monitor was Jessica Woods-Vendeler. REFERENCES Barrett, R. (1990). Intelligent rotor blade actuation through directionally attached piezoelectric crystals, American Helicopter Society National Forum, Washington, DC. Bisplinghoff, R. L., Ashley, H. and Halfman, R. L. (1965). Aeroelusticity. Addison-Wesley, Reading, MA. Bohlmann, J. and Lazarus, K. B. (1990). Active structures technology deveiopment. GD-ERR-FW-3064, General Dynamics Corp., Fort Worth, Texas. Buehler, W. J. and Wiley, R. C. (1956). Nickel based alloys. U.S. Patent 3,174,851. Chiarappa, D. J. and Claysmith, C. R. (1981). Deformable mirror surface control techniques. J. Guid. Control 4(l), 27-34. Crawley, E. F., Warkentin, D. J. and Lazarus, K. B. (1988). Feasibility analysis of piezoelectric devices. MIT-SSL 5-88, Space Systems Laboratory, Massachusetts Institute of Technology, Cambridge, MA. Cross, L. E. (1989). Piezoelectric and electrostrictive sensors and actuators for adaptive structures and smart materials. Adaptive Structures (Edited by B. K. Wada), ASME AD-15, pp. 9-17. ASME, New York. DeCamp, R. W. and Hardy, R. (1984). Mission adaptive wing advanced research concepts. AIAA Paper No. 84-2088. Diederich, F. W. and Foss, K. A. (1952). Charts and approximate formulas for the estimation of aeroelastic effects on the loading of swept and unswept wings. NACA TN 2608. Ehlers, S. M. (1991). Aeroelastic behavior of an adaptive lifting surface. Ph. D. Thesis, School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN. COE 2:5/7-L

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T. A. WE~S~HAARand S. M. EHLERS

Ehlers, S. M. and Weisshaar, T. A. (1990). Static aeroelastic behavior of an adaptive laminated piezoelectric composite wing. In Proc. 31st AIAA/ASME/ASCE/AHS Structures, Structural Dynamics and Materials Conf., Long Beach, CA, pp. 1611-1623. Flax, A. H. (1961). Aeroeleasticity and flutter. In High Speed Problems of Aircraft and ExperimentalMethods (Edited by A. F. Donovan and H. R. Lawrence). Princeton University Press, Princeton, NJ. Hajela, P. and Glowasky, R. (1991). Application of piezoelectric elements in supersonic panel flutter suppression. AIAA Paper 91-3191. AIAA/AHS/ASEE Aircraft Design Systems and Operations Meeting, Baltimore, MD. Hodges, D. H. (1988). Review of composite rotor blade modelling. AIAA Paper 88-2249. In Proc. 29th AIAA/ASME/ASCE/AHS Structures, Structural Dynamics and Materials Conf., Williamsburg, VA, pp. 305-312. IEEE Standard Definitions of Primary Ferroelectric Terms (1986). ANSI/IEEE Std 180-1986, Institute of Electrical and Electronics Engineers. Jaffe, B., Cook, W. R. and Jaffe, H. (1971). Piezoelectric Ceramics. Academic Press, New York. Kobyashi, S. (1984). Overview-Divergence of forward swept wings strengthened by composite material. Committee on Aircraft Structures (CAS), Research Report (A), Aeronautical Engineering Dept, Tokyo University. Lazarus, K. B., Crawley, E. F. and Lin, C. Y. (1991). Fundamental mechanisms of aeroelastic control with control surface and strain actuation. In Proc. 32nd AIAA/ASME/ASCE/AHS Structures, Structural Dynamics and Materials Conf., Baltimore, MD. Mason, W. P. (1981). Piezoelectricity, its history and applications. .I. Acoust. Sot. Am. 70(6), 1561-1566. Noll, T. E. and Perry, B., III, (1989). Aeroservoelestic wind-tunnel investigations using the active flexible wing model-status and recent accomplishments. AIAA Paper No. 89-l 168~CP. Perry, B., III, Dunn, H. J. and Sandford, M. C. (1988). Control law parameterization for an aeroelastic wind-tunnel model equipped with an active roll control system and comparison with experiment. In Proc. 29th AIAA/ASME/ASCE/AHS Structures, Structural Dynamics and Materials Conf., Williamsburg, VA., pp. 41-56. Rogers, C. A. (1990). Active vibration and structural control of shape memory alloy hybrid composites: Experimental results. .I. Acoust. Sot. Am., pp. 2803-2811. Sato. T.. Ishikawa. H.. Ikeda. 0.. Nomura. S. and Uchino. K. (1982). Deformable 2-D mirror usine multilayered electrostrictors. Appl.’ Optics 21(20), 3669-3672. . ’ Schneider, G. and Zimmerman, H. (1986). Static aeroelastic effects on high-performance aircraft. AGARD Report No. 725, Static Aeroelasticity in Combat Aircraft. Scott, R. C. (1990). Control of flutter using adaptive materials. M.S. Thesis, Purdue University, West Lafayette, IN. Scott, R. C. and Weisshaar, T. A. (1991). Controlling panel flutter using adaptive materials. In Proc. 32nd AIAA/ASME/ASCE/AHS Structures, Structural Dynamics and Materials Conf., Baltimore, MD. Shirk, M. H., Hertz, T. J. and Weisshaar, T. A. (1986). Aeroelastic tailoring-theory, practice and promise. I. Aircraft 23(l), 6-18. Song, O., Librescu, L. and Rogers, C. A. (1991). Static aeroelasticity behavior of adaptive aircraft wing structures modelled as composite thin-walled beams. International Forum on Aeroelasticity and Structural Dynamics, Aachen, Germany. Spangler, R. L., Jr and Hall, S. R. (1989). Piezoelectric actuators for helicopter rotor control. MIT-SSL l-89, Aeronautics and Astronautics Dept, Massachusetts Institute of Technology, Cambridge, MA. Swigert, C. J. and Forward, R. L. (1981). Electronic damping of orthogonal bending modes in a cylindrical mast-theory. I. Spacecraft Rockets 18(l), 5-10. Tsai, S. W. and Hahn, H. T. (1980). Introduction to Composite Materials. Technomic, Westport, CT. Uchino, K. (1986). Electrostrictive actuators: materials and applications. Ceram. Bull. 65(4), 647-652. Uchino, K. and Nomura, S. (1981). New electromechanical materials and their applications. Jap. I. Appl. Phys. 21 (Supplement 20-4), 225-228. Wada, B. K., Fanson, J. L. and Crawley, E. F. (1989). Adaptive Structures. ASME Winter Annual Meeting, ASME Vol. AD-15. ASME, New York. Weisshaar, T. A. (1978). Aeroeleastic stability and performance characteristics of aircraft with advanced composite sweptforward wing structures. AFFDL-TR-78-116, Air Force Flight Dynamics Laboratory, Wright-Patterson AFB, OH. Weisshaar, T. A. (1979). Forward swept wing static aeroelasticity. AFFDL-TR-79-3087, Air Force Flight Dynamics Laboratory, Wright-Patterson AFB, OH. Weisshaar, T. A. (1980). Divergence of forward swept composite wings. I. Aircraft 17(6), 442-448. Weisshaar, T. A. (1987). Aeroelastic tailoring: creative uses of unusual materials. AIAA Paper No. 87-0976~CP, 28th AIAA/ASME/ASCE/AHS Structures, Structural Dynamics and Materials Conf., Monterey, CA. Weisshaar, T. A. and Ehlers, S. M. (1991). Adaptive static and dynamic aeroelastic design. Workshop on Smart Material Systems and Structures, International Forum on Aeroelasticity and Structural Dynamics, Aachen, Germany. Wilson, E. G., Jr (1986). Static aeroelasticity in the design of modern fighters. AGARD Report No. 725, Static Aeroelasticity in Combat Aircraft. Wolko, H. S. (1987). Structural design of the Wright flyers. In The Wright Flyer-An Engineering Perspective, pp. 97-106. Smithsonian Institution Press, Washington, DC.