Nonlinear Flutter Response of Heated Curved Composite Panels with Embedded Macrofiber Composite Actuators

0e nonlinear flutter response of heated curved composite panels with embedded macrofiber composite (MFC) actuators in supersonic airflow is investigated. Prescribed voltages are statically applied to the piezoelectric actuators, inducing a prestress field which results in an additional stiffness effect on the curved panel, and it will change the aeroelastic behavior of curved composite panels. 0e aeroelastic equations of curved composite panels with embedded MFC actuators are formulated by the finite element approach. 0e von Karman large deflection panel theory and the first-order piston theory aerodynamics are adopted in the formulation.0emotion equations are solved by a fourth-order Runge–Kutta numerical scheme, and time history, phase portrait, Poincaré map, bifurcation diagram, and Lyapunov exponent are used for better understanding of the pre/postflutter responses. 0e results demonstrate that the nonlinear flutter response characteristics of the curved panel differs from those of the flat panels significantly, and the transverse displacement of the curved composite panels with embedded MFC actuators in the preflutter region shows a gradual static displacement; the chaotic motions occur directly after static motion because of the effect of the temperature elevation. 0e applied voltages can increase the critical dynamic pressure and change the bifurcation diagram of the curved composite panels with embedded MFC actuators, and the response amplitudes can be reduced evidently.


Introduction
Since aircraft reached supersonic speeds in the early fifties, a supersonic panel flutter is a well-known phenomenon and many researchers have done intensive theoretical and experimental investigations [1]. e panel flutter is a kind of dynamic aeroelastic instability resulting from the interaction of aerodynamic force, inertial force, and elastic force. For supersonic or hypersonic aircraft, thermal stress induced by aerodynamic heating plays an important role and leads more complex dynamic behaviors.
In the published literature on the panel flutter, a large number of efforts were dedicated to investigate the flat panel flutter in the supersonic or hypersonic flow regime [2][3][4][5][6][7], while the curvature of the aircraft skin panel exists in the engineering practice of supersonic aircraft structure design. Owing to the inherent curvature, static aerodynamic loading is experienced by a curved panel, and it will result in the static aerodynamic deformation of the curved panel, so the aeroelastic problem of the curved panel will be more complex than that of the flat panel. Dowell [8] had discussed qualitative and quantitative features of the flutters of 2-D and 3-D curved plates in detail without considering the temperature elevation effects, and he showed that the stream-wise curvature reduces the flutter critical dynamic pressure and increases the flutter amplitude, simultaneity. Azzouz et al. [9][10][11] had done investigations in static deflection, flutter boundary, and response of curved panels by using the finite element method, and they showed that the flutter response of curved panels is quite similar to the flat panels when the panel's height rise is very small, and as the panel's height rises increase the curved panel flutter response reveals a new variety of dynamic behaviors. Ghoman and Azzouz [12,13] studied the nonlinear flutter of curved panels under yawed supersonic flow at elevated temperature by using the frequency-domain method and the time-domain method, and they showed no critical buckling temperatures are found out for curved panels. Yang et al. [14] proposed a flow field modified local piston theory, which is applied to the integrated analysis on static/dynamic aeroelastic behavior of the curved panel. ey showed that the existing curvature modified method is nonconservative compared to the proposed flow field modified method.
How can the panel flutter be suppressed when panel flutter occurs? at is an important subject. Most of the studies on panel flutter suppression mainly rely on smart materials, and piezoelectric materials are the most representative smart materials because piezoelectric materials are capable of altering the structure's response through sensing, actuation, and control. By now on, the main piezoelectric materials such as piezoceramic [15][16][17] and MFC [18,19] are used in panel flutter suppression. Lai et al. [20] studied to control the nonlinear flutter of a simply supported isotropic plate by using piezoelectric actuators. ey concluded that the bending moment was effective in flutter suppression. Zhou et al. [21] used the finite element method and LQR full-state feedback to control isotropic and composite panels with surface bonded or embedded piezoelectric patches. e norms of the feedback control gain were used to provide the optimal shape and location of the piezoelectric actuators. Numerical simulations showed that the critical flutter dynamic pressure can be increased up to four times and two times for simply supported and clamped isotropic panels, respectively. Moon [22] employed an optimal control scheme of LQR with output feedback to study the nonlinear flutter suppression of a composite panel with piezoelectric actuators and sensors. e shape and location of the actuators and sensors were determined using the genetic algorithms to obtain maximum control effects. Li et al. [23][24][25] studied the active aeroelastic flutter properties and vibration control of the supersonic composite-laminated flat panel in recent years. However, the typical piezoceramic actuator has some disadvantages. ey have the brittle nature of piezoceramics which makes them vulnerable to damage. In addition, it is hard to make them conform to the curved surface. To overcome these drawbacks of the typical piezoceramic actuator, the MFC actuator has been developed by Hagood and Bent [26]. e polymer matrix can protect piezoceramic fibers against impact and make MFC flexible; therefore, they can be made to conform to the curved surface. Park and Kim [18] applied MFC actuators to passively suppress the nonlinear panel flutter under uniform temperature distribution. ey showed that in-plane actuation of the MFC actuators can increase the critical temperature and the critical dynamic pressure and decrease the large thermal deflection. Li et al. [19] showed that the piezoelectric MFC is more efficient than PZT5A. Guo et al. [27] studied the nonlinear dynamical behavior of a multilayer piezoelectric MFC laminated shell without temperature elevation, and the results indicate that piezoelectric parameters have strong effects on the vibration control of the MFC laminated shell.
In the broad published literature on panel flutter suppression, there is no research on the nonlinear flutter response of curved composite panels with embedded MFC actuators under combined aerodynamic, thermal, and piezoelectric loads. In this paper, the nonlinear flutter response of curved composite panels with embedded MFC actuators in supersonic airflow is investigated, and time history, phase portrait, Poincaré map, bifurcation diagram, and Lyapunov exponent are used for better understanding of the pre/postflutter responses. Figure 1 depicts a 3-D curved panel with its geometry defined by the length a, the width b, the maximum height H, and the thickness h. A supersonic flow with airspeed V a along x is imposed on the panel. With the assumption of small in-plane strains and moderately large transverse displacement, the total strains including inplane, bending, and von Karman nonlinear strains due to moderately large deflection and the Marguerre strain due to the panel curvature are as given in the following equation:

Constitutive Equation.
where ε 0 m is the linear membrane strain vector, ε 0 b is the von Karman nonlinear membrane stretching strain vector, ε 0 w 0 is the curvature strain vector, and κ { } is the bending curvature: (2) Figure 2 shows the layers of the composite panel with embedded MFC actuators. Considering the temperature elevation and piezoelectric loads, the constitutive equations for the composite panel with embedded MFC actuators can be represented as where ε E is the total stain vector, ΔT is the temperature elevation, α { } is the transformed thermal expansion coefficient vector, d is the transformed piezoelectric constant vector, and e is the electric field. e electric field with the applied voltage V to the actuator is defined as [24] where h n p indicates the electrode spacing of the interdigitated electrode for the MFC actuator and the subscript n p means the number of the actuator. e in-plane forces, bending moments, and the shear forces for the curved composite panels with embedded MFC actuators can be expressed as

e First-Order Piston eory.
e first-order piston theory with the static aerodynamic loading is employed for the aerodynamic pressure load: where w denotes the transverse displacement of the curved panel, w 0 is the curved panel geometry, and the subscripts, x and, y denote the partial differentiation. In addition, the nondimensional aerodynamic pressure and aerodynamic damping are given as D 11 is the first entry of the laminate stiffness matrix [D], and and then,

Advances in Materials Science and Engineering
2.3. Governing Equations. By using the principle of virtual work, the governing equations of the curved composite panels with embedded MFC actuators subject to the combined aerodynamic, thermal, and piezoelectric load can be derived as follows: e virtual work of the internal forces over the curved panel element is given by where α s is the shear correction factor for the laminated composite element. e virtual work of the external forces over the curved panel element is given by where ρ is the density of the panel.
In addition, Application of the principle of virtual work as shown in equation (11) leads to the equations of motion for nonlinear flutter of curved composite panels with embedded MFC actuators at elevated temperatures and applied voltages, which is expressed as 1 the equations of motion for the nonlinear flutter of curved composite panels with embedded MFC actuators can be simplified as 1

Solution Procedures
e system equations of motion presented in equation (17) are not suitable for numerical integration because of two shortcomings: (1) the number of nodal freedom is too large and (2) the nonlinear stiffness matrices are functions of the system displacement vector. erefore, to investigate the nonlinear postflutter time response, a straightforward and efficient approach is used to solve directly the system by transferring it into modal coordinates.
Assuming that the panel nodal displacements W { } can be expressed as a linear combination of selected natural modes and associated modal coordinates, where n is the number of the truncation modal. By substituting equation (18) into equation (17), modal coordinate transformation relations can be introduced:  (17) are transformed into the following reduced nonlinear system in the modal coordinate as where e system of equation (20) can be written in the form of the state-space differential equations as _ a A fourth-order multimode Runge-Kutta scheme can be used to solve equation (22) for the nonlinear flutter response. Time history, phase portrait, Poincaré map, bifurcation diagram, and Lyapunov exponent are used for better understanding of the pre/postflutter responses of the curved composite panels with embedded MFC actuators.

Results and Discussion
In this paper, a clamped curved composite panel with embedded MFC actuators is taken as an example to study the nonlinear flutter response. e lay-up of the curved composite panel is [0/θ MFC /−45/90] s . θ MFC indicates the lamination angle of the MFC actuator. e planar dimension of the panel is a � 0.381 m and b � 0.305 m. e material properties for graphite/epoxy and MFC used in this example are shown in Table 1. Figure 3 depicts the bifurcation diagram versus dynamic pressure for H/h � 0.2 and ΔT/ΔT cr � 1 without applied voltages. Parameters H/h and ΔT/ΔT cr are used as the measure of the curvature and temperature elevation, respectively. Unlike the flat panel system, the transverse displacement in the preflutter region shows a gradual static displacement between 0 ≤ λ < 410, which is because of the existence of curvature, and the static aerodynamic loading will be experienced by the curved panel and will result in the static aeroelastic deformation. Figure 4(a) shows the typical time history response and phase maps of the curved panel at λ � 400 and the panel converges at a stable equilibrium point (−0.08, 0) finally. With the increasing of dynamic pressure, a Hopf bifurcation happens indicating the flutter onset by a direct jump into limit cycle oscillation at λ � 410. e nonlinear flutter response is mainly limit cycle oscillation between 410 ≤ λ < 1000. e typical time history response, phase portrait, and Poincaré map at λ � 900 are shown in Figure 4(b), and it is a one periodic limit cycle motion.
ere are only one maximum value (peak value) and one minimum value (valley value) in one period shown in the time history diagram; a 2-D close loop is shown in the phase diagram and one point is shown in the Poincaré diagram. Between 1000 ≤ λ < 1140, the curved panel will experience quasiperiodic motion; however, the LCO exists in this range. e typical time history response, phase portrait, and Poincaré maps at λ � 1010, λ � 1040, and λ � 1100 are shown in Figures 4(c)-4(e). e quasiperiodic motion is similar as chaotic motion from time history response and phase maps; however, a closed loop cycle is shown in the Poincaré map which indicates a quasiperiodic motion. When λ ≥ 1140, the curved panel starts to fall in the chaotic motion; Figure 4(f ) shows the typical time history response, phase portrait, and Poincaré map of the curved panel at λ � 1150 and the largest Lyapunov exponent is 0.1495 which indicates a chaotic motion. Otherwise, in this region, the quasiperiodic motion also exists as shown in Figure 4(g). Figure 5 shows the bifurcation diagram versus dynamic pressure for H/h � 0.2 and ΔT/ΔT cr � 2 without   Advances in Materials Science and Engineering applied voltages. Compared to Figure 3, this bifurcation has a big difference which illustrates the effect of the temperature elevation is very strong. It also shows that an aerostatic deflection occurs between the dynamic pressures 0 ≤ λ < 250 as shown in Figure 6(a) at λ � 240. Beyond λ � 250, it is seen that the chaotic motion occurs and there is no gradual motion transition from static phase to LCO and then to chaotic oscillations; the shift is sudden, and the flutter is onset directly into chaotic motion. Figure 6(b) shows the typical time history response, phase portrait, and Poincaré map of the curved panel at λ � 250 and the largest Lyapunov exponent is 0.1271. Compared to the stability boundary in the case ΔT/ΔT cr � 1, the critical dynamic pressure decreases with the increasing of the temperature elevation. When 250 ≤ λ < 990, LCOs and quasiperiodic motions all exist in this range as shown in Figure 6(c) at λ � 780 and Figure 6(d) at λ � 930. When λ ≥ 990, the curved panel starts to fall in the chaotic motion; Figure 6(e) shows the typical time history response, phase portrait, and Poincaré map of the curved panel at λ � 990 and the largest Lyapunov exponent is 0.1573. Figure 7 shows the bifurcation diagram versus dynamic pressure for H/h � 0.2 and ΔT/ΔT cr � 1 with the applied voltage V U � V L � 1000 V. Compared to Figure 3, the bifurcation diagram changes greatly and the response  Advances in Materials Science and Engineering amplitudes of the curved composite panel with embedded MFC actuators are reduced evidently under the applied voltage. An aerostatic deflection is occurred between the dynamic pressures 0 ≤ λ < 500 as shown in Figure 8(a) at λ � 490. With the increasing of dynamic pressure, a Hopf bifurcation happens at λ � 500, which indicates that the applied voltage will increase the critical aerodynamic pressure. And the nonlinear flutter response is mainly limit cycle oscillation in the range of 500 ≤ λ < 1190. Figure 8( Figure 8(e) at λ � 1600.

Conclusion
In this paper, the nonlinear flutter responses of curved composite panels with embedded MFC actuators are studied subject to thermal loading and the applied voltage; some concluding remarks can be drawn:    (1) Unlike the flat panel system, the transverse displacement of the curved composite panel with embedded MFC actuators in the preflutter region shows a gradual static displacement. is is because of the existence of curvature, and the static aerodynamic loading will be experienced by curved panels and will result in the static aeroelastic deformation. (2) e dynamic behavior of the curved composite panels becomes more complex because of the temperature elevation. e critical dynamic pressure will decrease with the increasing of the temperature elevation, and the system does not evolve chaotic motions through period-doubling bifurcation while the system parameters vary and the chaotic motions occur directly after static motion.   Under the in-plane actuation with the applied voltage, the response amplitudes can be reduced evidently. So, the MFC actuator can be used to suppress the panel flutter.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.