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

Zhou, Jian; Xu, Minglong; Yang, Zhichun

doi:https://doi.org/10.1155/2018/3103250

Advances in Materials Science and Engineering

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Abstract Introduction Results and Discussion Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2018 | Article ID 3103250 | https://doi.org/10.1155/2018/3103250

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

Jian Zhou,¹Minglong Xu,¹and Zhichun Yang²

Academic Editor: Paweł Kłosowski

Received26 Sept 2018

Accepted21 Nov 2018

Published26 Dec 2018

Abstract

The 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. The aeroelastic equations of curved composite panels with embedded MFC actuators are formulated by the finite element approach. The von Karman large deflection panel theory and the first-order piston theory aerodynamics are adopted in the formulation. The motion 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. The 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. The 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.

1. 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]. The 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 [2–7], 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. [9–11] 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. They 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? That 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 [15–17] 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. They 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. The 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. The shape and location of the actuators and sensors were determined using the genetic algorithms to obtain maximum control effects. Li et al. [23–25] 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. They 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]. The 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. They 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.

2. Nonlinear Finite Element Formulation

2.1. Constitutive Equation

Figure 1 depicts a 3-D curved panel with its geometry defined by the length , the width , the maximum height , and the thickness . A supersonic flow with airspeed along is imposed on the panel. With the assumption of small in-plane strains and moderately large transverse displacement, the total strains including in-plane, 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:where is the linear membrane strain vector, is the von Karman nonlinear membrane stretching strain vector, is the curvature strain vector, and is the bending curvature:

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 aswhere is the total stain vector, is the temperature elevation, is the transformed thermal expansion coefficient vector, is the transformed piezoelectric constant vector, and is the electric field.

The electric field with the applied voltage to the actuator is defined as [24]where indicates the electrode spacing of the interdigitated electrode for the MFC actuator and the subscript means the number of the actuator.

The in-plane forces, bending moments, and the shear forces for the curved composite panels with embedded MFC actuators can be expressed aswhere

2.2. The First-Order Piston Theory

The first-order piston theory with the static aerodynamic loading is employed for the aerodynamic pressure load:where denotes the transverse displacement of the curved panel, 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

is the first entry of the laminate stiffness matrix , andand then,

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:

The virtual work of the internal forces over the curved panel element is given bywhere is the shear correction factor for the laminated composite element.

The virtual work of the external forces over the curved panel element is given bywhere 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 aswhere the matrix denotes the system mass matrix; is the system aerodynamic damping; is the system linear stiffness matrix; is the system aerodynamic stiffness matrix with respect to the direction; is the system linear shear stiffness matrix; is the system linear stiffness matrix due to shallow shell geometry; , , , , and are the system nonlinear first-order stiffness matrixes; is the system nonlinear second-order stiffness matrix; is the thermal load; is the piezoelectric load; and is the static aerodynamic loads causing the static deflection of the panel.

Assuming,the equations of motion for the nonlinear flutter of curved composite panels with embedded MFC actuators can be simplified as

3. Solution Procedures

The 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. Therefore, 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 can be expressed as a linear combination of selected natural modes and associated modal coordinates,where is the number of the truncation modal.

By substituting equation (18) into equation (17), modal coordinate transformation relations can be introduced:

The nonlinear first-order modal stiffness matrices are evaluated with the correspondent linear natural mode . The nonlinear second-order modal matrix is evaluated with a combination of the correspondent linear natural modes and simultaneously.

The equations of motion (17) are transformed into the following reduced nonlinear system in the modal coordinate aswhere

The system of equation (20) can be written in the form of the state-space differential equations as

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.

4. 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. The lay-up of the curved composite panel is . indicates the lamination angle of the MFC actuator. The planar dimension of the panel is and . The 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 without applied voltages. Parameters and 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 , 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 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 . The nonlinear flutter response is mainly limit cycle oscillation between . The typical time history response, phase portrait, and Poincaré map at are shown in Figure 4(b), and it is a one periodic limit cycle motion. There 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 , the curved panel will experience quasiperiodic motion; however, the LCO exists in this range. The typical time history response, phase portrait, and Poincaré maps at , , and are shown in Figures 4(c)–4(e). The 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 , 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 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).

(a)

(b)

(c)

(d)

(e)

(f)

(g)

Figure 5 shows the bifurcation diagram versus dynamic pressure for without 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 as shown in Figure 6(a) at . Beyond , 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 and the largest Lyapunov exponent is 0.1271. Compared to the stability boundary in the case , the critical dynamic pressure decreases with the increasing of the temperature elevation. When , LCOs and quasiperiodic motions all exist in this range as shown in Figure 6(c) at and Figure 6(d) at . When , 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 and the largest Lyapunov exponent is 0.1573.

(a)

(b)

(c)

(d)

(e)

Figure 7 shows the bifurcation diagram versus dynamic pressure for with the applied voltage . Compared to Figure 3, the bifurcation diagram changes greatly and the response 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 as shown in Figure 8(a) at . With the increasing of dynamic pressure, a Hopf bifurcation happens at , 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 . Figure 8(b) shows the typical time history response, phase portrait, and Poincaré map of the curved panel at . The curved composite panel with embedded MFC actuators will experience quasiperiodic motion in the range of and chaotic motion in the range of ; Figures 8(c) and 8(d) show the typical time history response, phase portrait, and Poincaré maps of the curved panel, respectively. When , the motion of the curved composite panel with embedded MFC actuators will experience LCO motion as shown in Figure 8(e) at .

(a)

(b)

(c)

(d)

(e)

5. 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. This 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)The dynamic behavior of the curved composite panels becomes more complex because of the temperature elevation. The 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.(3)The applied voltage has a great influence to the nonlinear flutter response of the curved composite panel with embedded MFC actuators. It can increase the critical dynamic pressure and change the bifurcation diagram of the curved panel response. 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

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

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by National Natural Science Foundation of China (Grant Nos. 11072198 and 11702204).

References

C. Mei, K. Abdel-Motagaly, and R. Chen, “Review of nonlinear panel flutter at supersonic and hypersonic speeds,” Applied Mechanics Reviews, vol. 52, no. 10, pp. 321–332, 1999.
View at: Publisher Site | Google Scholar
K. Zhou, X. Huang, Z. Zhang, and H. Hua, “Aero-thermo-elastic flutter analysis of coupled plate structures in supersonic flow with general boundary conditions,” Journal of Sound and Vibration, vol. 430, pp. 36–58, 2018.
View at: Publisher Site | Google Scholar
E. Carrera and E. Zappino, “Aeroelastic analysis of pinched panels in supersonic flow changing with altitude,” Journal of Spacecraft and Rockets, vol. 51, no. 1, pp. 187–199, 2014.
View at: Publisher Site | Google Scholar
X.-D. Yang, T.-J. Yu, W. Zhang, Y.-J. Qian, and M.-H. Yao, “Damping effect on supersonic panel flutter of composite plate with viscoelastic mid-layer,” Composite Structures, vol. 137, pp. 105–113, 2016.
View at: Publisher Site | Google Scholar
V. Vasily, “Interaction of panel flutter with inviscid boundary layer instability in supersonic flow,” Journal of Fluid Mechanics, vol. 736, pp. 216–249, 2013.
View at: Publisher Site | Google Scholar
C. Shao, D. Cao, Y. Xu, and H. Zhao, “Flutter and thermal buckling analysis for composite laminated panel embedded with shape memory alloy wires in supersonic flow,” International Journal of Aerospace Engineering, vol. 2016, Article ID 8562716, 12 pages, 2016.
View at: Publisher Site | Google Scholar
H. Zhao and D. Cao, “Supersonic flutter of laminated composite panel in coupled multi-fields,” Aerospace Science and Technology, vol. 47, pp. 75–85, 2015.
View at: Publisher Site | Google Scholar
E. H. Dowell, “Nonlinear flutter of curved plates,” AIAA Journal, vol. 7, no. 3, pp. 424–431, 1969.
View at: Publisher Site | Google Scholar
M. S. Azzouz and W. Mokthar, “Fluid structure interaction-nonlinear-flutter of cylindrical panels,” in Proceedings of 44th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, USA, January 2006.
View at: Google Scholar
M. S. Azzouz, X. Y. Guo, A. Przekop, and C. Mei, “Nonlinear flutter of cylindrical shell panels under yawed supersonic flow using FE,” in Proceedings of 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference, Palm Springs, CA, USA, April 2004.
View at: Google Scholar
M. S. Azzouz and C. Mei, “Finite element time domain-modal formulation for nonlinear flutter of cylindrical panels,” in Proceedings of 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Newport, RI, USA, May 2006.
View at: Google Scholar
S. S. Ghoman and M. S. Azzouz, “Supersonic aerothermoelastic nonlinear flutter study of curved panels: frequency domain,” Journal of Aircraft, vol. 49, no. 4, pp. 1075–1090, 2012.
View at: Publisher Site | Google Scholar
S. S. Ghoman and M. S. Azzouz, “Supersonic aerothermoelastic nonlinear flutter study of curved panels: time domain,” Journal of Aircraft, vol. 49, no. 4, pp. 1179–1183, 2012.
View at: Publisher Site | Google Scholar
Z. Yang, J. Zhou, and Y. Gu, “Integrated analysis on static/dynamic aeroelasticity of curved panels based on a modified local piston theory,” Journal of Sound and Vibration, vol. 333, no. 22, pp. 5885–5897, 2014.
View at: Publisher Site | Google Scholar
K. A. Motagaly, B. Duan, and C. Mei, “Active control of nonlinear panel flutter under yawed supersonic flow,” in Proceedings of 44th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Norfolk, VA, USA, April 2003.
View at: Google Scholar
K. Yuan and Z. Qiu, “Flutter control of composite panels with embedded piezoelectric materials,” Journal of Beijing University of Aeronautics and Astronautics, vol. 35, no. 12, pp. 1429–1433, 2009, in Chinese.
View at: Google Scholar
A. Almeida, M. V. Donadon, A. R. de Faria, and S. F. M. de Almeida, “The effect of piezoelectrically induced stress stiffening on the aeroelastic stability of curved composite panels,” Composite Structures, vol. 94, no. 12, pp. 3601–3611, 2012.
View at: Publisher Site | Google Scholar
J.-S. Park and J.-H. Kim, “Suppression of aero-thermal large deflections and snap-through behaviors of composite panels using Macro Fiber Composite actuators,” Smart Materials and Structures, vol. 13, no. 6, pp. 1448–1459, 2004.
View at: Publisher Site | Google Scholar
Q. Li, C. Mei, and J.-K. Huang, “Suppression of thermal postbuckling and nonlinear panel flutter motions using piezoelectric actuators,” AIAA Journal, vol. 45, no. 8, pp. 1861–1873, 2007.
View at: Publisher Site | Google Scholar
Z. Lai, D. Y. Xue, J.-K. Huang, and C. Mei, “Panel flutter limit-cycle suppression with piezoelectric actuation,” Journal of Intelligent Material Systems and Structures, vol. 6, no. 2, pp. 274–282, 2016.
View at: Publisher Site | Google Scholar
R. C. Zhou, C. Mei, and J.-K. Huang, “Suppression of nonlinear panel flutter at supersonic speeds and elevated temperatures,” AIAA Journal, vol. 34, no. 2, pp. 347–354, 1996.
View at: Publisher Site | Google Scholar
S. H. Moon, “Finite element analysis and design of control system with feedback output using piezoelectric sensor/actuator for panel flutter suppression,” Finite Elements in Analysis and Design, vol. 42, no. 12, pp. 1071–1078, 2006.
View at: Publisher Site | Google Scholar
F.-M. Li, “Active aeroelastic flutter suppression of a supersonic plate with piezoelectric material,” International Journal of Engineering Science, vol. 51, pp. 190–203, 2012.
View at: Publisher Site | Google Scholar
Z.-G. Song and F.-M. Li, “Active aeroelastic flutter analysis and vibration control of supersonic composite laminated plate,” Composite Structures, vol. 94, no. 2, pp. 702–713, 2012.
View at: Publisher Site | Google Scholar
Z.-G. Song and F.-M. Li, “Flutter and buckling characteristics and active control of sandwich panels with triangular lattice core in supersonic airflow,” Composites Part B: Engineering, vol. 108, pp. 334–344, 2017.
View at: Publisher Site | Google Scholar
N. W. Hagood and A. A. Bent, “Development of piezoelectric fiber composites for structural actuation,” in Proceedings of 34th Structures, Structural Dynamics and Materials Conference, La Jolla, CA, USA, April 1993.
View at: Google Scholar
X. Guo, D. Liu, W. Zhang, L. Sun, and S. Chen, “Nonlinear dynamic analysis of macrofiber composites laminated shells,” Advances in Materials Science and Engineering, vol. 2017, Article ID 4073591, 17 pages, 2017.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2018 Jian Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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