Aerothermoelastic flutter analysis and active vibration suppression of nonlinear composite laminated panels with time-dependent boundary conditions in supersonic airflow

Abstract

Aerothermoelastic flutter properties of nonlinear composite laminated panels in supersonic airflow are studied, and investigations on active flutter and aerothermal postbuckling suppression for the panels with time-dependent boundary conditions are also carried out using macro fiber composite actuator and sensor. The von Karman strain–displacement relation in conjunction with the supersonic piston theory is applied in structural modeling. Nonlinear dynamic equations of motion for the structural system are established using Hamilton’s principle and the assumed mode method. Frequency- and time-domain methods are used to investigate the aerothermoelastic characteristics and active flutter and aerothermal postbuckling suppression of the panels. Effects of aspect ratio and ply angle on the nonlinear aerothermoelastic behaviors are studied. The displacement feedback control is used to conduct the active flutter and postbuckling suppression. The nonlinear output controller consisting of a linear quadratic regulator and a nonlinear state estimator of extended Kalman filter is also used in designing the controller. Controlled vibration responses of the structural system under the two different controllers are compared. The results show that the developed linear quadratic regulator/extended Kalman filter controller is more effective than the displacement feedback control controller in flutter and postbuckling control of the panels with time-dependent boundary conditions in supersonic airflow.

Keywords

Nonlinear composite laminated panel time-dependent boundary conditions aerothermoelastic analysis macro fiber composite aerothermal postbuckling active flutter suppression

Introduction

Flutter is a kind of aeroelastic instability phenomenon caused by the coupling effect of the elastic force, aerodynamic force, and inertial force of the structure. The flutter of the aircraft structures will occur when the airflow velocity reaches a critical value. Moreover, the thermal buckling may happen on the aeroelastic structure because of the high velocity of the airflow. This phenomenon may cause fatigue failure of the flight vehicles in supersonic airflow. Consequently, the nonlinear flutter properties of the flight vehicles are of great interest to researchers (Donadon and Faria, 2016; Koo and Hwang, 2004; Li and Song, 2014; Librescu et al., 2004; Yang and Han, 1983; Zhou et al., 2012).

The composite laminated panels have been widely applied in the flight vehicles because of their super properties of low specific weight and high strength. There are a great many of literatures have investigated the aeroelastic flutter problem of the composite laminated panels. Zhou et al. (1994) presented a finite element time-domain modal method to study the nonlinear aeroelastic flutter behaviors of laminated panels under elevated temperature. Based on a developed hyperbolic shear deformation theory, Grover et al. (2015) analyzed the flutter characteristic of multilayered composite plates in yawed supersonic airflow. Kouchakzadeh et al. (2010) carried out the nonlinear aeroelastic analysis of a general two-dimensional laminated panel subjected to supersonic airflow. The influences of different parameters on the limit cycle amplitudes of the composite laminated panel were also studied. Gray and Mei (1993) investigated the limit cycle oscillation of nonlinear thin composite laminated panels subjected to hypersonic flow using the finite element method (FEM). The third-order piston aerodynamic and von Karman large deflection theories were applied in the structural modeling. Using the FEM, Prakash and Ganapathi (2006) calculated the critical flutter aerodynamic pressure of a functionally graded panel in the thermal environment. The effects of the panel thickness, aspect ratio, aerodynamic damping, and boundary condition on the flutter bound were studied in details. Ibrahim et al. (2011) presented a nonlinear finite element model to calculate the amplitude of limit cycle oscillation for shape memory alloy hybrid composite panels in supersonic airflow under the effect of the aerodynamic heating.

The active aerothermoelastic flutter control of panels subjected to supersonic airflow using the piezoelectric materials has drawn a large number of researchers’ attention. Li et al. (2007) designed a linear quadratic regulator (LQR) combining with Kalman filter controller to suppress the nonlinear response and thermal postbuckling of composite laminated panels with thermal effect. Frampton et al. (1996) researched the aeroelastic properties and active flutter control for linear panels in supersonic airflow using the piezoelectric actuators. Kim et al. (2008) presented an effective controller based on aeroelastic modes to improve the aeroelastic flutter characteristics of composite laminated panels in supersonic airflow using the FEM. Nam et al. (2000) studied the aeroelastic stability and active flutter suppression of smart composite wing with the effect of delamination. Based on displacement feedback control (DFC) and LQR, Zhang et al. (2016) carried out the active flutter suppress analysis of carbon nanotube–reinforced functionally graded composite panels subject to supersonic airflow. Moon et al. (2005) developed a feedback linearization control approach to study the active flutter control for composite panels with piezoelectric ceramic actuators. Lai et al. (1995) used the piezoelectric material to suppress the limit cycle motions of panels based on the linear optimal control theory. Moon and Hwang (2005) investigated the flutter suppression of composite panels with LQR controller based on the linear and nonlinear models. Using the proportional integral derivative controller, Korbahti (2010) researched the aeroelastic flutter control of a rectangular panel in supersonic airflow. The detailed analyses on the influences of feedback control gains on the flutter bounds of the panel were carried out. Based on the FEM, Guo et al. (2007) carried out the nonlinear response suppression for shape memory alloy composite laminated panels under the condition of elevated temperature. Song and Li (2017) used the displacement feedback controller to improve the flutter and thermal buckling properties of sandwich panels with triangular lattice core subject to supersonic airflow.

Although the piezoceramics are widely used in structural active flutter control field, the brittle nature of the piezoceramics makes them difficult to operate in bonding procedure. In order to overcome these drawbacks of the typical piezoceramic actuator, macro fiber composite (MFC) has been developed by the NASA Langley Research Center (Wilkie et al., 2000; Williams et al., 2002). Recently, the MFC transducers were applied to investigate the active vibration suppression of structures because of the flexibility and durability of the MFC which make the bonding procedure easy. Sodano et al. (2004) studied the vibration control and structural health monitoring with MFC sensor and actuator employing experimental method. Kim et al. (2011) and Sohn et al. (2011, 2014) used the MFC actuators to suppress vibration responses of the end-capped cylindrical shell, the smart hull, and the ring-stiffened cylindrical shell structures. Park and Kim (2004) carried out the active vibration control of nonlinear limit cycle oscillation for composite panels embedded with the MFC actuators based on the nonlinear FEM. Song et al. (2010) designed an energy harvesting device with a cantilever beam utilizing the MFC materials. The results calculated by the theoretical model were consistent with the energy harvesting device quite well. Vadiraja and Sahasrabudhe (2009) investigated the vibration properties and optimal control for thin-walled composite beams with the MFC material based on linear quadratic Gaussian control method.

Most of the previous literatures about the flutter suppression of panels subjected to supersonic airflow are on the base of the piezoelectric ceramics, and few researchers have carried out the investigations on the active flutter control of composite laminated panels using the MFC materials. Because the excitation often transmits to the structure through supports, the boundary conditions of the structures in many common engineering fields are time dependent (Chai et al., 2017; Ramachandran, 1972). However, few investigations on the active flutter and thermal buckling control of nonlinear composite laminated panels with the time-dependent boundary conditions have been reported. Inspired from these, in this article, the aerothermoelastic flutter properties of the composite laminated panels in supersonic airflow are analyzed, and the active flutter and thermal postbuckling control for nonlinear composite laminated panels with time-dependent boundary conditions are investigated using the MFC actuator and sensor. The influences caused by the boundary displacement are transformed into the equivalent excitations. In the structural modeling, the von Karman large deflection and supersonic piston theories are used. Hamilton’s principle and the assumed mode method are applied to formulate the equation of motion. The flutter bounds of the composite laminated panels are computed under different ply angles and aspect ratios. Time-domain responses of the nonlinear structural system with time-dependent boundary conditions are calculated using the Runge–Kutta method. The controlled nonlinear responses of the structural system are calculated using the DFC and LQR/extended Kalman filter (EKF) methods. The performances of the different control methods are discussed in detail.

Formulation for the equation of motion

Figure 1 shows a composite laminated panel subject to time-dependent boundary conditions with the MFC actuator and sensor in supersonic airflow. The MFC patches are well bonded on the top and bottom surfaces of the base composite laminated panel to act as the actuator and sensor. The length, width, and total thickness of the composite laminated panel are donated by a, b, and h_c. The thickness of each layer for the panel is h_l, and the thicknesses of the MFC actuator and sensor patches are both h_p. So the total thickness of the structural system is h = 2h_p + h_c. In this study, the composite laminated panel is fully covered with the MFC actuator and sensor patches. The transverse displacement is w_r on the boundaries y = 0 and b.

Figure 1.

Composite laminated panel with time-dependent boundary conditions in supersonic airflow and MFC actuator/sensor patches structural system.

It is well known that the deformation of the aerothermoelastic structural system is relatively small before the flutter occurs. In order to investigate the post-flutter performances of the structural system, the geometric nonlinearity should be considered. So in the structural modeling, the von Karman large deflection theory is applied. On the basis of the Kirchhoff plate theory, the displacement fields of the structural system are given as

u = u_{0} - z \frac{\partial w}{\partial x}, v = v_{0} - z \frac{\partial w}{\partial y}, w = w

(1)

where z is the transverse coordinate of the z direction, u and v are the in-plane displacements of the panel in the x and y directions, u₀ and v₀ are the displacements of the neutral plane, and w is the transverse displacement which is composed of two parts

w (x, y, t) = w_{s} (x, y, t) + w_{r} (x, y, t)

(2)

where w_s satisfies the regular boundary condition without time-dependent conditions and w_r is the displacement caused by the movement of the boundaries.

The von Karman nonlinear strain–displacement relationship of the composite laminated panel can be written as

ε = {[ε_{x}, ε_{y}, γ_{xy}]}^{T} = ε_{0} + z κ

(3)

where ε ₀ and κ are the membrane strain and bending curvature vectors which can be given as follows

ε_{0} = {[\frac{\partial u_{0}}{\partial x} + \frac{1}{2} {(\frac{\partial w}{\partial x})}^{2}, \frac{\partial v_{0}}{\partial y} + \frac{1}{2} {(\frac{\partial w}{\partial y})}^{2}, \frac{\partial u_{0}}{\partial y} + \frac{\partial v_{0}}{\partial x} + \frac{\partial w}{\partial x} \frac{\partial w}{\partial y}]}^{T}

(4a)

κ = {[- \frac{\partial^{2} w}{\partial x^{2}}, - \frac{\partial^{2} w}{\partial y^{2}}, - 2 \frac{\partial^{2} w}{\partial x \partial y}]}^{T}

(4b)

In the general coordinates, the stress–strain relation of the kth lamina for the composite laminated panel is expressed as

σ^{(k)} = {\bar{Q}}_{ij}^{(k)} (ε - α Δ T), i, j = 1, 2, and 6

(5)

in which α is the thermal expansion coefficient vector which depends on the lamina thermal expansion coefficients α₁ and α₂, ΔT is the temperature change, and ${\bar{Q}}^{(k)}$ is the transformed reduced stiffness matrix that depends on Young’s and shear moduli E₁, E₂, and G₁₂, Poisson’s ratios υ₁₂ and υ₂₁, and the ply angle θ of the kth lamina of the composite laminated panel (Jones, 1975).

The bending moment resultant and membrane force resultant vectors M ₀ and N ₀ of the composite laminated panel can be obtained by

M_{0} = B_{0} ε_{0} + D_{0} κ - M_{Δ T}, N_{0} = A_{0} ε_{0} + B_{0} κ - N_{Δ T}

(6)

where

(A_{0}, B_{0}, D_{0}) = \sum_{k = 1}^{n_{l}} \int_{z_{k - 1}}^{z_{k}} {\bar{Q}}_{i j}^{(k)} (1, z, z^{2}) d z, i, j = 1, 2, and 6

(7a)

N_{Δ T} = \sum_{k = 1}^{n_{l}} {\bar{Q}}_{ij}^{(k)} (z_{k} - z_{k - 1}) α Δ T, M_{Δ T} = \frac{1}{2} \sum_{k = 1}^{n_{l}} {\bar{Q}}_{ij}^{(k)} (z_{k}^{2} - z_{k - 1}^{2}) α Δ T, i, j = 1, 2, and 6

(7b)

where n_l is the layer number of the composite laminated panel; z_k and $z_{k - 1}$ are the coordinates of the upper and lower surfaces of the kth lamina of the composite laminated panel in the z-axis.

The piezoelectric MFC is mainly composed of epoxy matrix, isotropic piezoceramic fiber material, and electrodes, and it can be homogenized to an orthotropic material with arbitrary piezoelectric fiber angles like composite laminated structures. The fiber reinforcement direction is parallel to the mid-plane, and the direction of polarization is along the thickness of the MFC. The constitutive equations of the piezoelectric MFC material in the general coordinates (x, y) system can be expressed as follows (Deraemaeker et al., 2009; Xue et al., 2014; Zhang et al., 2015)

σ^{p} = c (ε^{p} - α^{p} Δ T) - e^{T} E

(8a)

D^{p} = e ε^{p} + χ E

(8b)

in which superscript p denotes the piezoelectric MFC material; σ ^p and ε ^p are the stress and strain vectors of the MFC material; α ^p = [α_x, α_y, 0]^T is the thermal expansion coefficient vector of the MFC material, in which α_x and α_y are the thermal expansion coefficients in the x and y directions; D ^p and E are the electric displacement and the electric field intensity vectors, respectively; χ is the dielectric constant matrix; and $c = T_{as}^{- 1} {\bar{c} T}_{as}^{- T}$ and $e = {\bar{e} T}_{as}^{- 1}$ are the stiffness coefficient matrix and piezoelectric constant vector, respectively. These vectors and matrices are expressed as

\begin{matrix} T_{a s} = [\begin{matrix} \cos^{2} θ_{a s} & \sin^{2} θ_{a s} & 2 \sin θ_{a s} \cos θ_{a s} \\ \sin^{2} θ_{a s} & \cos^{2} θ_{a s} & - 2 \sin θ_{a s} \cos θ_{a s} \\ - \sin θ_{a s} \cos θ_{a s} & \sin θ_{a s} \cos θ_{a s} & \cos^{2} θ_{a s} - \sin^{2} θ_{a s} \end{matrix}], \\ \bar{c} = [\begin{matrix} \frac{{\bar{E}}_{1}}{1 - {\bar{υ}}_{12} {\bar{υ}}_{21}} & \frac{{\bar{υ}}_{12} {\bar{E}}_{1}}{1 - {\bar{υ}}_{12} {\bar{υ}}_{21}} & 0 \\ \frac{{\bar{υ}}_{12} {\bar{E}}_{1}}{1 - {\bar{υ}}_{12} {\bar{υ}}_{21}} & \frac{{\bar{E}}_{2}}{1 - {\bar{υ}}_{12} {\bar{υ}}_{21}} & 0 \\ 0 & 0 & {\bar{G}}_{12} \end{matrix}] \bar{e} = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ {\bar{e}}_{31} & {\bar{e}}_{32} & 0 \end{matrix}], \\ E = {\begin{matrix} 0 \\ 0 \\ E_{3} \end{matrix}}, D^{p} = {\begin{matrix} 0 \\ 0 \\ D_{z} \end{matrix}}, χ = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & χ_{33} \end{matrix}] \end{matrix}

(9)

where θ_as in matrix T _as represents the MFC fiber angle; ${\bar{E}}_{1}$ , ${\bar{E}}_{2}$ , and ${\bar{G}}_{12}$ are the elastic and shear moduli; ${\bar{υ}}_{12}$ and ${\bar{υ}}_{21}$ are Poisson’s ratios of the MFC material; ${\bar{e}}_{31}$ and ${\bar{e}}_{32}$ are the piezoelectric constants; E₃ = V₀(t)/h_p is the electric field intensity in which V₀(t) is the external voltage applied in the z direction; and D_z and χ₃₃ are the electric displacement and dielectric constant.

To establish the governing equation of motion, Hamilton’s principle is applied and written as

\int_{t_{1}}^{t_{2}} [δ (T_{k} - U_{p}) + δ W_{Δ p}] d t = 0

(10)

where T_k and U_p are the total kinetic energy and potential energy of the structural system, δW_Δp is the virtual work done by external loads, and t₁ and t₂ are the integration time limits.

The total kinetic energy and potential energy of the composite laminated panel and MFC actuator and sensor are calculated by

T_{k} = \frac{1}{2} \int_{V} ρ ({\overset{\cdot}{u}}^{2} + {\overset{\cdot}{v}}^{2} + {\overset{\cdot}{w}}^{2}) d V + \frac{1}{2} \int_{V_{a} + V_{s}} ρ_{p} ({\overset{\cdot}{u}}^{2} + {\overset{\cdot}{v}}^{2} + {\overset{\cdot}{w}}^{2}) d V

(11)

U_{p} = \frac{1}{2} \int_{A} (ε_{0}^{T} N_{0} + κ^{T} M) d A + \frac{1}{2} \int_{V_{a} + V_{s}} ε^{T} σ^{p} d V - \frac{1}{2} \int_{V_{a}} E^{T} D^{p} d V

(12)

where A is the surface area of the base composite laminated panel; ρ and ρ_p are the mass densities of the base composite laminated panel and MFC material, respectively; and V, V_a, and V_s are the volumes of the composite laminated panel and the MFC actuator and sensor patches, respectively.

The virtual work δW_Δp done by the aerodynamic pressure can be expressed as

δ W_{Δ p} = \int_{A} Δ p δ w d A

(13)

where Δp is the aerodynamic pressure which can be computed by the supersonic piston theory as (Prakash and Ganapathi, 2006; Shin et al., 2006)

Δ p = - \frac{2 q_{\infty}}{β} (\frac{\partial w}{\partial x} + \frac{M_{\infty}^{2} - 2}{M_{\infty}^{2} - 1} \frac{1}{U_{\infty}} \frac{\partial w}{\partial t})

(14)

where

q_{\infty} = \frac{1}{2} ρ_{\infty} U_{\infty}^{2}, β = (M_{\infty}^{2} - 1)^{1 / 2}

(15)

where ρ_∞, U_∞, and M_∞ are the air density, velocity, and Mach number of the free stream, respectively. For convenience, a non-dimensional aerodynamic pressure λ = 2q_∞a³/(βD) is introduced in which D = E₁h³/[12(1 − υ₁₂υ₁₂)]. In the case of $M_{\infty} >> 1$ , [(M_∞² − 2)/(M_∞²−1)]μ/β → μ/M_∞, where μ = ρ_∞a/(ρh).

In order to discretize the continuous structural system, the assumed mode method is used. The mid-plane displacements u₀ and v₀ and transverse displacement w_s of the structural system are expressed as follows (for convenience, subscript “0” is omitted)

u (x, y, t) = \sum_{i = 1}^{m} \sum_{j = 1}^{n} ψ_{ij} (x, y) p_{ij} (t) = ψ^{T} p (t)

(16a)

v (x, y, t) = \sum_{i = 1}^{m} \sum_{j = 1}^{n} ζ_{ij} (x, y) s_{ij} (t) = ζ^{T} s (t)

(16b)

w_{s} (x, y, t) = \sum_{i = 1}^{m} \sum_{j = 1}^{n} W_{ij} (x, y) g_{ij} (t) = W^{T} g (t)

(16c)

in which m and n are the number of modes taken in the x and y directions; ψ , ζ , and W are the vectors of the assumed modes; and p , s , and g are the vectors of the generalized coordinates. For a simply supported composite laminated panel without time-dependent boundary conditions, the vibration modes of the structural system can be assumed to be

ψ_{ij} (x, y) = \cos \frac{i π x}{a} \sin \frac{j π y}{b}, ζ_{ij} (x, y) = \sin \frac{i π x}{a} \cos \frac{j π y}{b}, W_{ij} (x, y) = \sin \frac{i π x}{a} \sin \frac{j π y}{b}

(17)

The displacement w_r caused by the movement of the boundaries can be expressed as (Ramachandran, 1972)

w_{r} (x, y, t) = q (x, y) η (t)

(18)

where the function q(x, y) must be found fitting in with the given time-dependent boundary conditions of the composite laminated panel, and η(t) = cos ω_rt is the driving function, in which ω_r is the frequency of the boundary excitation. The boundaries y = 0 and y = b of the composite laminated panel are subjected to a time-dependent displacement, and the time-dependent boundary conditions are written as

q (x, 0) = d_{0} \sin (\frac{π x}{a}), q (x, b) = d_{0} \sin (\frac{π x}{a})

(19a)

M_{y} = - D_{12} \frac{\partial^{2} q}{\partial x^{2}} - D_{22} \frac{\partial^{2} q}{\partial y^{2}} = 0 at y = 0 and y = b

(19b)

in which d₀ is the amplitude of the boundary motion. According to the Levy-type solution, the function q(x, y) can be obtained. Then the displacement function w_r that satisfies the time-dependent boundary conditions can be given as (Ramachandran, 1972)

\begin{matrix} w_{r} (x, y, t) = d_{0} {\cosh β_{r} y + \frac{ν - 1}{2} β_{r} y \sinh β_{r} y + \frac{1 - \cosh β_{r} b}{2 \sinh β_{r} b} [1 - \frac{β_{r} b (1 - ν)}{2 \sinh β_{r} b}] \sinh β_{r} y + \frac{(1 - ν) (\cosh β_{r} b - 1)}{2 \sinh β_{r} b} β_{r} y \cosh β_{r} y} \sin β_{r} x \cos ω_{r} t \end{matrix}

(20)

where

β_{r} = \frac{π}{a}, ν = \frac{D_{22}}{D_{12}}

(21)

Substituting equations (1) to (4), (6), (8), (14), and (16) to (18) into equations (11) to (13), the expressions of the total kinetic energy T_k, potential energy U_p, and virtual work δW_Δp done by the aerodynamic pressure can be obtained. Then, substituting the expressions of T_k, U_p, and δW_Δp into equation (10) and conducting the variation operation, the equation of motion of the electromechanically coupled structural system can be obtained as follows

M \overset{\cdot\cdot}{X} + C_{Δ p} \overset{\cdot}{X} + [K_{l} + K_{n} (X) + K_{Δ p} + K_{Δ T}] X + K_{a} V_{0} = F_{f}

(22)

where X = [ p ^T, s ^T, g ^T]^T is the generalized coordinate vector, M is the structural mass matrix, C _Δ _p is the aerodynamic damping matrix, K _l and K _n( X ) are the linear and nonlinear structural stiffness matrices, K _Δ_p is the aerodynamic stiffness matrix, K _Δ_T is the thermal stiffness matrix, K _a is the electromechanical coupling vector, and F _f is the forcing vector caused by the movement of boundaries.

Equation (22) is an electromechanically coupled equation of motion which relates the applied voltage to the structural displacement and will be used to study the active aerothermoelastic flutter and postbuckling control of the nonlinear composite laminated panel with the time-dependent boundary conditions in supersonic airflow by different control methods.

The design of the active controllers

In this section, the active aerothermoelastic flutter and postbuckling control of the nonlinear composite laminated panel with the time-dependent conditions are studied using the DFC and the LQR/EKF methods. It is generally known that charges will be generated on the piezoelectric sensor when the sensor is deformed. The charge quantity of the piezoelectric sensor can be obtained by (Reddy, 1999)

Q_{s} (t) = \int_{A_{s} | z = z_{s}} D_{z} d A = [K_{sl} + K_{sn} (X)] X (t)

(23)

where A_s is the surface area of the MFC sensor, z_s is the transverse coordinate of the mid-plane of the MFC sensor from the neutral plane of the composite laminated panel, and K _sl and K _sn are the linear and nonlinear coefficient vectors, respectively.

The MFC sensor voltage V_s can be calculated by

V_{s} = \frac{Q_{s} h_{p}}{χ_{33} A_{s}} = K_{s} X (t)

(24)

where

K_{s} = \frac{h_{p}}{χ_{33} A_{s}} (K_{sl} + K_{sn})

(25)

DFC method

In this section, the DFC method is used on the aerothermoelastic structural system, and the control voltage V₀ applied on the MFC actuator can be expressed as

V_{0} (t) = G_{p} V_{s} = G_{p} K_{s} X (t)

(26)

where G_p is the feedback control gain in the DFC method for the MFC actuator. Then substituting equation (26) into equation (22), the controlled equation of motion based on the DFC method can be obtained as

M \overset{\cdot\cdot}{X} (t) + C_{Δ p} \overset{\cdot}{X} (t) + [K_{l} + K_{n} (X) + K_{Δ p} + K_{Δ T} + G_{p} K_{a} K_{s}] X (t) = F_{f}

(27)

It can be seen from equation (27) that the DFC method can produce an active stiffness for the aerothermoelastic structural system.

LQR/EKF method

According to equation (22), the standard state-space equation for control design can be expressed as follows

\overset{\cdot}{Z} = \bar{AZ} + BU + D_{f} F_{ff}, Y = \bar{CZ}

(28)

where

\begin{matrix} Z = [\begin{matrix} X \\ \overset{\cdot}{X} \end{matrix}], B = [\begin{matrix} 0 \\ - M^{- 1} K_{a} \end{matrix}] \\ \bar{A} = A + A_{n} = [\begin{matrix} 0 & I \\ - M^{- 1} (K_{l} + K_{Δ p} + K_{Δ T}) & - M^{- 1} C_{Δ p} \end{matrix}] \\ + [\begin{matrix} 0 & 0 \\ - M^{- 1} K_{n} (X) & 0 \end{matrix}] \\ D_{f} = [\begin{matrix} 0 & 0 \\ 0 & M^{- 1} \end{matrix}], F_{ff} = [\begin{matrix} 0 \\ F_{f} \end{matrix}], U = V_{0}, \\ \bar{C} = C + C_{n} = [\begin{matrix} K_{sl} & 0 \end{matrix}] + [\begin{matrix} K_{sn} (X) & 0 \end{matrix}] \end{matrix}

(29)

where I is the unit matrix.

In this section, the LQR/EKF method is used to suppress the flutter and thermal postbuckling of the nonlinear composite laminated panel with time-dependent boundary conditions. First, the LQR method is applied to obtain the control voltage for the linear full-state feedback problem, which is given as

U = - G_{L} Z

(30)

where G _L is the feedback control gain. U minimizes a cost function that is proportional to the required measure of the system’s response. The cost function is written as

J = \int_{0}^{\infty} (Z^{T} QZ + U^{T} RU) dt

(31)

where Q and R are the semi-positive-definite and positive-definite weighting matrices on the outputs and control inputs, respectively. By solving the functional extremum problem, the feedback control gain G _L can be obtained as G _L = R ⁻¹ B ^T P , where P satisfies the following Riccati equation

\overset{\cdot}{P} + PA + A^{T} P - PB R^{- 1} B^{T} P + Q = 0

(32)

In order to apply the LQR, all the states of the structural system should be known. However, it is difficult for the MFC sensor to give the accurate information on every state. Thus, the EKF is introduced in the calculation to carry out the state estimate, and the nonlinear effects are considered during the process. The nonlinear state estimation and measurement are given as follows (Kim et al., 2008; Li et al., 2007)

\overset{\cdot}{\hat{Z}} = \bar{A} (\hat{Z}, t) \hat{Z} + BU + K_{e} (t) (Y - \bar{C} \hat{Z}), \hat{Y} = \bar{C} \hat{Z}

(33)

In the EKF, the nonlinearity of the system dynamics is included, which is linearized on the basis of the traditional Kalman filter. Actually, the linear approximation is carried out for each step of the iteration for the nonlinear structural system. According to Taylor’s series, the first-order linear approximation of the state matrix can be expressed as

F (t) \approx A + {[A_{n} (Z, t) + \frac{\partial A_{n} (Z, t)}{\partial Z} Z] |}_{Z = \hat{Z} (t)}

(34)

Subsequently, the gain K_e of the EKF can be obtained at every time interval by solving the differential Riccati equation as

{\overset{\cdot}{P}}_{e} (t) + F (t) P_{e} (t) + P_{e} (t) F (t) - P_{e} (t) C^{T} R_{e}^{- 1} C P_{e} (t) + Q_{e} = 0

(35)

from which the gain of the EKF can be obtained as

K_{e} (t) = P_{e} (t) C^{T} R_{e}^{- 1}

(36)

The optimal state estimation of the nonlinear structural system $\hat{Z}$ can be obtained after solving the gain $K_{e}$ of the EKF, and the control voltage can be expressed as

U = - G_{L} \hat{Z}

(37)

Substituting equation (37) into equation (28), the state of the structural system at the next time interval can be obtained. The iterative process is repeated until the state of the structural system is controlled to convergence.

Solution of the equation of motion

It can be seen from the equation of motion of the structural system, equation (22), that the movement of boundaries contributes only to producing the forcing vector, and it has no relation to the stiffness matrix of the structural system. That is to say, the movement of boundaries does not influence the natural frequencies of the structural system. So, the movement of boundaries is not considered in the process of analyzing the aerothermoelastic characteristics of the composite laminated panel by the frequency-domain method.

Solving the homogeneous differential equation of equation (22), the complex eigenvalue Ω = Ω_R + iω can be obtained, where Ω_R and ω =2 πf are the real part of the eigenvalue and the circular frequency of the structural system, in which f is the natural frequency (Hz), and they are both related to the aerodynamic pressure λ. As it is well known that certain two adjacent frequencies of the aerothermoelastic structural system will close to each other and then coalesce at certain λ_cr with the increase of the aerodynamic pressure λ. When the non-dimensional aerodynamic pressure reaches λ_cr, the flutter of the structural system will occur. λ_cr is called the critical flutter aerodynamic pressure.

In order to obtain the critical buckling temperature change of the structural system, all the terms relating to the derivatives with respect to the time, the aerodynamic stiffness, the nonlinear stiffness, and the forcing term at the right-hand side of equation (22) are dropped, and the steady-state equation of motion under the thermal load can be written as

(K_{l} + K_{Δ T}) X (t) = 0

(38)

in which the thermal stiffness matrix K _ΔT is related to the temperature change ΔT. The critical thermal buckling temperature change ΔT_cr can be obtained by letting the determinant of coefficient of equation (38) equal to zero.

It can also be found from the equation of motion, that is, equation (22), that the movement of boundaries has a significant effect on the time-domain responses of the structural system. In this study, the time-domain responses of the nonlinear structural system with time-dependent boundary conditions are calculated using Runge–Kutta method. The active flutter and thermal postbuckling control effects are also investigated by comparing the time-domain responses under different control methods.

Numerical simulations and discussions

Verification

Prior to the study of the main topic, the mode convergence analysis and verification of formulations correctness in the article are carried out by comparing the limit cycle oscillation amplitudes calculated from the present study with those obtained by Zhou et al. (2012). In the verification process, the movement of boundaries is not considered by setting the amplitude of the boundary displacement d₀ = 0. The geometrical sizes and material properties of the composite laminated panel, which are the same as those applied by Zhou et al. (2012), are as follows: a = b = 0.305 m, h = 0.00127 m, ρ = 1550 kg m⁻³, E₁ = 155 × 10⁹ N m⁻², E₂ = 8.07 × 10⁹ N m⁻², G₁₂ = 4.55 × 10⁹ N m⁻², υ₁₂ = 0.22. The composite laminated panel is symmetrical and consists of eight layers of lamina whose lay-up angles are [0°/45°/−45°/90°]_s. The flutter bounds of the simply supported composite laminated panel and the limit cycle oscillation amplitudes of the nonlinear structural system are calculated under different numbers of assumed modes as shown in Figure 2. It can be observed from the figure that the critical flutter aerodynamic pressure and the limit cycle oscillation amplitude of the structural system when the mode numbers are chosen to be m = 4 and n = 1 are almost the same as those computed using m = 6 and n = 1. It can be concluded that the mode and limit cycle oscillation amplitude are convergent at m = 4 and n = 1 for both the linear and nonlinear composite laminated panel and four orders of the assumed mode can be taken to study the aerothermoelastic problem of the composite laminated panel in supersonic airflow. It can also be seen from Figure 2(b) that the limit cycle oscillation amplitudes of the composite laminated panel with ply angles of [0°/45°/−45°/90°]_s are the same as the results calculated by Zhou et al. (2012) using the FEM, which verifies that the present formulations and the MATLAB codes applied in the numerical calculations are correct.

Figure 2.

Convergences of (a) natural frequencies and (b) limit cycle oscillation amplitudes for a simply supported square composite laminated panel.

To verify the accuracy of the investigation considering the thermal effect, the limit cycle oscillation amplitudes of an isotropic simply supported panel as studied by Dowell (1966) under different temperature rise ratios are investigated. By setting E₁ = E₂ = E, υ₁₂ = υ₂₁ = υ, ply angle θ = 0, and the amplitude of the boundary displacement d₀ = 0, the composite laminated panel with time-dependent boundary conditions degenerates into the same isotropic simple supported panel as that considered by Dowell (1966). The comparison results of the limit cycle oscillation amplitudes with different temperature changes are shown in Figure 3. It can be observed that the present results coincide very well with those by Dowell (1966), which indicates that the analysis on the aerothermoelastic problem of the composite laminated panel in this study is correct and feasible.

Figure 3.

Limit cycle oscillation amplitudes of an isotropic simply supported panel under different temperature changes.

Linear aerothermoelastic analysis

In this subsection, the movement of boundaries is not considered because the natural frequency of the structural system is independent of the movement of boundaries. Based on the frequency-domain method, the analyses on the flutter and aerothermal buckling of the composite laminated panel in supersonic airflow are carried out by omitting the nonlinear terms of the equation of motion. The effects of the ply angle and aspect ratio on the flutter and buckling properties of the composite laminated panel are studied in detail. The structural and material parameters of the orthotropic composite laminated panel are as follows: h_l = 0.00025 m, ρ = 1600 kg m⁻³, E₁ = 150 × 10⁹ N m⁻², E₂ = 9.0 × 10⁹ N m⁻², G₁₂ = 7.1 × 10⁹ N m⁻², υ₁₂ = 0.3, α₁ = −0.07 × 10⁻⁶/°C, and α₂ = 30.1 × 10⁻⁶/°C. The ply angles of eight layers of lamina for the symmetric composite laminated panel are [θ°/−θ°/θ°/−θ°]_s. Without special statement, a = 0.2 m and b = 0.2 m.

First, the flutter behaviors of the symmetric composite laminated panel under different ply angles are studied. Figure 4 displays the variations of natural frequencies of composite laminated panels with aerodynamic pressure under different ply angles when temperature change ΔT = 0. It is seen from Figure 4 that the flutter bounds of the composite laminated panel are clearly different for different ply angles. For a square composite laminated panel with a = 0.2 m and b = 0.2 m as shown in Figure 4(a), the non-dimensional critical flutter aerodynamic pressure decreases with the increase of the ply angle. The composite laminated panel with ply angle of [0°/0°/0°/0°]_s has the highest flutter bound, and if the ply angle is [60°/−60°/60°/−60°]_s, the critical flutter aerodynamic pressure is relatively low. It is also observed that for the composite laminated panel with a = 0.3 m and b = 0.2 m, with the increase of the ply angle, the critical flutter aerodynamic pressure increases first and then decreases. The aeroelastic stability of the rectangular composite laminated panel (a = 0.3 m, b = 0.2 m) with the ply angle of [30°/−30°/30°/−30°]_s is better than those of the panels with other ply angles as shown in Figure 4(b). It can be concluded that the results obtained in such condition are quite different from those of the square composite laminated panels as shown in Figure 4(a). Therefore, it is necessary to study the relationship between the aspect ratio and optimal ply angle of the composite laminated panel.

Figure 4.

Variations of natural frequencies of eight-layer symmetric composite laminated panels with the non-dimensional aerodynamic pressure under different ply angles. (a) a = 0.2m, b = 0.2m and (b) a = 0.3m, b = 0.2m.

Variations in critical flutter aerodynamic pressure of the composite laminated panel with the ply angles under different aspect ratios at ΔT = 0 are shown in Figure 5. It can be observed from the figure that for the composite laminated panels with aspect ratios a/b = 0.5 and 1, the critical flutter aerodynamic pressures decrease with the increase of the ply angle for the composite laminated panel, and the maximum flutter bound can be obtained when the ply angle is θ = 0°. It is also seen that for the composite laminated panels with aspect ratios a/b = 1.5 and 2, the critical flutter aerodynamic pressures first increase and then decrease with the increase of the ply angles. The increasing trend of the critical flutter aerodynamic pressure for θ changing from 0° to 45° is not as obvious as the decreasing trend of the critical flutter aerodynamic pressure for θ changing from 45° to 90°. It should be observed that the optimal ply angle and the aspect ratio of the composite laminated panel are closely related, and the critical flutter aerodynamic pressure reaches the minimum value when the ply angle is θ = 90°. The variations of the critical flutter aerodynamic pressure with the aspect ratio is relatively small when the ply angles are in the vicinity of θ = 0° and θ = 90°, and the flutter bound of the composite laminated panel with θ = 45° is more sensitive to the change of the aspect ratio. It can also be noted from the figure that the flutter bound increases with the increase of aspect ratio regardless of the value of the ply angle. The cause of this phenomenon may be that the airflow is along the x direction. Therefore, increasing the dimension in the x direction can more effectively improve the stiffness of the aerothermoelastic structure system.

Figure 5.

Flutter bounds varying with ply angles under different aspect ratios.

The aerothermoelastic properties of the composite laminated panel are investigated by considering the effects of aerodynamic heating. The varying curves of the natural frequencies of the composite laminated panel versus non-dimensional aerodynamic pressure at ΔT = 0, 0.5ΔT_cr, ΔT_cr, and 2ΔT_cr are presented in Figure 6. Here, the ply angle of the composite laminated panel is [0°/0°/0°/0°]_s. From this figure, it can be seen that the critical flutter aerodynamic pressure decreases with the increase of the temperature change ΔT. It is also noted that when the temperature change ΔT increases to ΔT_cr, the natural frequency of the composite laminated panel at λ = 0 will be equal to 0, which means that the panel is buckling. Moreover, when the temperature change is ΔT = 2ΔT_cr, the composite laminated panel will be always buckling until the non-dimensional aerodynamic pressure λ reaches 151. However, as soon as the non-dimensional aerodynamic pressure λ is beyond 151, the thermal buckling of the panel disappears, which means that the process for flutter is contrary to that for thermal buckling.

Figure 6.

The natural frequencies of the composite laminated panel varying with non-dimensional aerodynamic pressure under different temperature changes.

The influences of the ply angle of the eight-layer symmetric composite laminated panel on the critical thermal buckling temperature change under different aspect ratios are also researched in detail. Figure 7 shows the variations of the critical thermal buckling temperature change with ply angles of the composite laminated panel. It can be observed that for the composite laminated panels with different aspect ratios, the critical thermal buckling temperature change decreases with the increase of the ply angle. The peak value of the critical thermal buckling temperature change can be obtained when θ = 0°. It can also be seen that when the ply angle is equal to 90°, the corresponding critical thermal buckling temperature change is much smaller than that for θ = 0°. That is to say, the composite laminated panel with smaller ply angles exhibits better thermal buckling characteristics.

Figure 7.

Critical buckling temperature changes of composite laminated panels with different ply angles under different aspect ratios.

Figure 8 displays the stable regions of the composite laminated panels with different ply angles. It can be seen from the figure that both the critical buckling temperature change and critical flutter aerodynamic pressure decrease with the increase of ply angle for the composite laminated panel. By comparing the flat and stable regions of the composite laminated panels at different ply angles, it is observed that the aerothermoelastic stability of the composite laminated panel with θ = 0° is the best.

Figure 8.

Stable regions of composite laminated panels with different ply angles. (a) θ = 0° and 15° and (b) θ = 30°, 45° and 60°.

The stable regions of the composite laminated panels with ply angle θ = 45° under different aspect ratios are also calculated and shown in Figure 9. It is noted that with the increase of the aspect ratio, the critical flutter aerodynamic pressure increases and the critical buckling temperature change decreases. That is to say, the increase of the aspect ratio can improve the flutter characteristic of the structure, and the decrease of the aspect ratio can enhance the thermal buckling behavior of the structural system. The reason for this phenomenon may be that the x-axis is the direction of the airflow and the thermal expansion coefficient vector α mainly depends on the lamina thermal expansion coefficients α₂. Based on the above analysis and calculation, the aspect ratio of the composite laminated panel with ply angle of [45°/−45°/45°/−45°]_s should be neither too small nor too large in the engineering structure design.

Figure 9.

Stable regions of composite laminated panels with different aspect ratios.

Nonlinear response and postbuckling control

In this section, the nonlinear vibration and thermal postbuckling control of the nonlinear composite laminated panel with time-dependent boundary conditions in supersonic airflow are studied using the MFC material. The ply angles of the symmetric four-layer composite laminated panel studied here are [30°/−30°/30°/−30°]. The geometrical sizes of the composite laminated panel are as follows: a = 0.2 m, b = 0.2 m, and h_l = 5 × 10⁻⁴ m. The material parameters of the composite laminated panel are the same as those given in section “Linear aerothermoelastic analysis.” The geometrical sizes and the material properties of the MFC material are as follows: h_p = 2 × 10⁴ m, ρ_p = 5540 kg m⁻³, ${\bar{υ}}_{12} = 0.31$ , ${\bar{υ}}_{21} = 0.162$ , ${\bar{E}}_{1} = 30.34 \times 10^{9} N m^{- 2}$ , ${\bar{E}}_{2} = 15.86 \times 10^{9} N m^{- 2}$ , ${\bar{G}}_{12} = 5.52 \times 10^{9} N m^{- 2}$ , d₃₁ = 530 × 10⁻¹² m V⁻², d₃₂ = 210 × 10⁻¹² m V⁻², χ₃₃ = 1.5 × 10⁻⁸ F m⁻¹, and α = [4 × 10⁻⁶, 4 × 10⁻⁶, 0]^T/°C. The amplitude of the boundary displacement is d₀ = 2 × 10⁻⁵ m, and the frequency of the boundary excitation is ω_r = 1000 Hz. The nonlinear controller is designed by the DFC and the LQR/EKF methods. The designed parameters used in LQR/EKF controller are chosen as follows: Q = 10⁴ × I , R = 1000, Q _e = 1000 × I , and R _e = 0.001. The feedback control gain in the DFC method for the MFC actuator is G_p = 10.

First of all, the time-response histories at the point (0.75a, 0.5b) of the nonlinear composite laminated panel with time-dependent boundary conditions are computed under different temperature changes and aerodynamic pressures. The composite laminated panels with and without time-dependent boundary conditions in supersonic airflow with a given initial displacement are considered, and the differences of the time-domain responses between the composite laminated panels with and without time-dependent boundary conditions are also investigated in detail. Figure 10 shows the time-response histories of the structural system with and without time-dependent boundary conditions when aerodynamic pressure λ = 200 which is much smaller than the critical flutter aerodynamic pressure λ_cr = 365. It is seen from Figure 10(a) that the time-response history of the structural system is relatively small and converges to zero quickly when the amplitude of the boundary displacement is d₀ = 0. However, as shown in Figure 10(b), the time-response history of the structural system does not converge to zero, and the vibration response is a constant amplitude vibration due to the time-dependent boundary condition.

Figure 10.

Time-response histories of the nonlinear composite laminated panel with and without time-dependent boundary conditions under λ = 200 and ΔT = 0. (a) d₀ = 0 and (b) d₀ = 2 × 10⁻⁵m.

The time-response histories of the structural system with and without time-dependent boundary conditions at λ = 400 and ΔT = 0 are shown in Figure 11. It can be observed that the time-response history of the structural system is in a flutter region and the deflection in a flutter region exhibits the limit cycle oscillation when the amplitude of the boundary displacement d₀ = 0. The time-response history of the structural system with time-dependent boundary condition is similar to a double period motion when the amplitude of the boundary displacement is equal to d₀ = 2 × 10⁻⁵ m. It can also be seen from the figure that the vibration amplitude of the structural system is larger than that shown in Figure 10 for the smaller aerodynamic pressure (λ = 200).

Figure 11.

Flutter time-response histories of the nonlinear composite laminated panel with and without time-dependent boundary conditions under λ = 400 and ΔT = 0. (a) d₀ = 0 and (b) d₀ = 2 × 10⁻⁵m.

Considering the influences of the thermal effect, the time-response histories of the structural system with and without time-dependent boundary conditions at λ = 100 and ΔT = 1.5ΔT_cr are calculated and shown in Figure 12. It is noted from Figure 12(a) that the vibration response of the structural system is convergent when the amplitude of the boundary displacement is d₀ = 0. However, the vibration response does not converge to zero, which means that the thermal postbuckling occurs. It can also be seen in Figure 12(b) that when the time-dependent boundary conditions are considered, the vibration response of the structural system will be a constant amplitude vibration around the thermal postbuckling equilibrium position. Moreover, when the amplitude of the boundary displacement increases to d₀ = 2 × 10⁴ m, the thermal postbuckling phenomenon disappears, and the vibration response returns to limit cycle oscillation, which can be noted from Figure 13. It can be concluded that the time-dependent boundary conditions can eliminate the thermal postbuckling phenomenon of the composite laminated panel in supersonic airflow in a certain degree.

Figure 12.

Thermal postbuckling time-response histories of the nonlinear composite laminated panel with and without time-dependent boundary conditions under λ = 100 and ΔT = 1.5ΔT_cr. (a) d₀ = 0 and (b) d₀ = 2 × 10⁻⁵m.

Figure 13.

Time-response history of the nonlinear composite laminated panel with time-dependent boundary conditions under λ = 100 and ΔT = 1.5ΔT_cr when d₀ = 2 × 10⁴ m.

Next, the DFC and LQR/EKF control methods are used to suppress the flutter of the nonlinear composite laminated panel with time-dependent boundary conditions, and the control effects of the two control methods are compared. Figure 14 displays the controlled fluttering time-response histories of the nonlinear composite laminated panel with time-dependent boundary conditions using the two different control methods. It can be observed from the figure that the vibration response is the limit cycle oscillation for the uncontrolled structural system. When the two controllers are applied, the limit cycle oscillations become convergent responses. It is concluded that the DFC and LQR/EKF methods can both effectively suppress the flutter of the nonlinear composite laminated panel with time-dependent boundary conditions. It is also noted from the figure that the convergent speed of the nonlinear structural system using the LQR/EKF method is faster than that using the DFC method. Furthermore, the control voltage used in the MFC actuator of the LQR/EKF controller is much lower than that applied in the DFC controller, which can be clearly seen from Figure 14(b).

Figure 14.

(a) Controlled flutter time-response histories of the nonlinear composite laminated panel with time-dependent boundary conditions and (b) actuator voltages under different control methods.

Figure 15 shows the time-response histories of the nonlinear composite laminated panel with time-dependent boundary conditions using different controllers when the aerodynamic pressure is λ = 500, which is much greater than the critical flutter aerodynamic pressure λ_cr = 365 of the uncontrolled structural system. It can be observed in Figure 15(a) that when the DFC method is used, the amplitude of the limit cycle oscillation of the structural system is reduced, but the flutter will continue to occur. However, when the LQR/EKF is applied, the limit cycle oscillation is convergent quickly. On the basis of the above analysis, one can conclude that the LQR/EKF control method developed in this study is more efficient than the DFC method in the flutter control of the nonlinear composite laminated panel with time-dependent boundary conditions.

Figure 15.

Controlled flutter time-response histories of the nonlinear composite laminated panel with time-dependent boundary conditions under different control methods. (a) DFC method and (b) LQR/EKF method.

The two control methods are also applied to suppress the thermal postbuckling of the nonlinear composite laminated panel with time-dependent boundary conditions in supersonic airflow. The controlled time-response histories and the control voltages are shown in Figure 16. It is noted that the thermal postbuckling of the structural system disappears after the two controllers are opened, and the convergent speed of the thermal postbuckling time-response history of the structural system with the LQR/EKF controller is faster than that with the DFC controller. That is to say, the DFC and LQR/EKF methods can both effectively suppress the thermal postbuckling of the structural system. Moreover, the control effect of the LQR/EKF method is better than the DFC method in the thermal postbuckling control of the nonlinear composite laminated panel with time-dependent boundary conditions.

Figure 16.

(a) Controlled thermal postbuckling time-response histories of the nonlinear composite laminated panel with time-dependent boundary conditions and (b) actuator voltages under different control methods.

Figure 17 displays the controlled thermal postbuckling time-response histories of the nonlinear composite laminated panel with and without time-dependent boundary conditions using the LQR/EKF method when the aerodynamic pressure and temperature change are λ = 100 and ΔT = 2ΔT_cr. It can be noted from the figure that the thermal postbuckling time-response history of the nonlinear composite laminated panel without time-dependent boundary conditions is convergent to zero quickly when the LQR/EKF is used. For the nonlinear composite laminated panel with time-dependent boundary conditions, the thermal postbuckling time-response history is also convergent as shown in Figure 17. However, in this case, the thermal postbuckling time-response history of the structural system does not converge to zero because of the time-dependent boundary conditions. This is the primary difference between the cases with and without time-dependent boundary conditions.

Figure 17.

Controlled thermal postbuckling time–response histories of the nonlinear composite laminated panel with and without time-dependent boundary conditions using LQR/EKF method.

Conclusion

In this article, the aerothermoelastic behaviors of the nonlinear composite laminated panels with time-dependent boundary conditions in supersonic airflow are analyzed. The active thermal postbuckling and flutter control for the nonlinear structural system using the MFC actuator and sensor are also studied. In the structural modeling, the time-dependent boundary conditions are considered. Supersonic piston theory is used to evaluate the aerodynamic pressure. Hamilton’s principle with the assumed mode method is applied to formulate the equation of motion. The DFC and the LQR/EKF methods are used to design the controllers. The aerothermoelastic properties and the active flutter and thermal postbuckling control effects of the nonlinear composite laminated panel with and without time-dependent boundary conditions are analyzed. From the numerical simulation results, the conclusions can be drawn as follows:

With the increase of the dimension in the x direction for a composite laminated panel, the critical flutter aerodynamic pressure increases at arbitrary ply angle. The critical flutter aerodynamic pressure is minimum when the ply angle of the composite laminated panel is θ = 90°.

The stable region is associated with the ply angle and aspect ratio of the composite laminated panel. The stable region of the panel with θ = 0° is larger than that with other ply angles when aspect ratio a/b = 1, and in order to obtain a reasonable stable region, the aspect ratio of the panel with θ = 45° should be neither too small nor too large in the engineering structure design.

The time-dependent boundary conditions can eliminate thermal postbuckling phenomenon of the composite laminated panel in supersonic airflow in a certain degree.

The limit cycle oscillation and thermal postbuckling of the nonlinear composite laminated panels with time-dependent boundary conditions can be effectively suppressed using the DFC method. However, the control voltage of the actuator is relatively high.

The LQR/EKF method applied in this article is more efficient than the DFC method in controlling the flutter and thermal postbuckling of the nonlinear composite laminated panels with time-dependent boundary conditions, and the voltage applied on the controller is much lower than that used in the DFC method.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research is supported by the National Natural Science Foundation of China (no. 11572007, 11172084). Fengming Li is grateful to the financial support by the Alexander von Humboldt Foundation for his scientific visit at the Chair of Structural Mechanics, University of Siegen, Germany. Zhiguang Song is grateful to the financial support by the Alexander von Humboldt Foundation for his scientific visit at the Dynamics and Vibrations Group, Technische Universität Darmstadt, Germany

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