## 2 Problem Formulation
\(E_{c}\), \(v_{c}\), and \(G=E_{c}/2(1+v_{c})\) stand for the material's modulus of elasticity, Poisson's ratio, and shearing modulus, respectively. The outer voltage is applied across the piezoelectric layer thickness. The relationship of strains and stresses for the piezoelectric patch can be expressed as  
\[ \begin{array}{c}\sigma_{11}^{p}=c_{11}\epsilon_{11}+c_{12}\epsilon_{12}-e_{ 31}E_{3},\sigma_{22}^{p}=c_{12}\epsilon_{11}+c_{11}\epsilon_{12}-e_{31}E_{3}\\ \sigma_{12}^{p}=c_{66}\epsilon_{12},D_{3}=e_{31}(\epsilon_{11}+\epsilon_{22}) +\ \epsilon_{33}\ E_{3}\end{array}  \tag{10}\]  
where \(\sigma_{11}^{p}\), \(\sigma_{22}^{p}\) and \(\sigma_{12}^{p}\) represent the stress components. The elastic parameters \(c_{11}\), \(c_{12}\) and \(c_{66}\) can be expressed as \(c_{66}=(c_{11}-c_{22})/2\). \(e_{31}\) and \(e_{33}\) are the piezoelectric and coefficient, respectively. \(V_{0}(t)\) represents the external applied voltage, and \(h_{p}\) is the thickness of the piezoelectric layer. \(D_{3}\) denotes the electric displacement, \(E_{3}=V_{0}(t)/h_{p}\) means the electric intensity along the z-direction [41; 42]. Considering the kinetic energies of the base plate and the piezoelectric layer, and neglecting the influence of rotatory inertia of thin plates considered herein [21], the total kinetic energy for the plate and piezoelectric layer system are obtained as  
\[T=\frac{1}{2}\int_{V}\rho_{c}\dot{w}^{2}dv+\frac{1}{2}\int_{V_{p}}\rho_{p}\dot {w}^{2}dv \tag{11}\]  
where the dot denotes the differentiation with respect to the time. \(\rho_{c}\) and \(\rho_{p}\), respectively, stand for the base plate's and piezoelectric layer's mass densities. \(V\) and \(V_{p}\) represent the plate's and piezoelectric layer's volumes, respectively. The total potential energy of the plate and piezoelectric patch system includes the strain energy of the base plate, The potential energy of the piezoelectric layer concludes the piezoelectric layer's strain energy and electric potential energy. Therefore, the total potential energy of the plate and piezoelectric patch system can be obtained as  
\[U=\frac{1}{2}\int_{V}\sigma_{11}\epsilon_{11}+\sigma_{22}\epsilon_{22}+\sigma _{12}\epsilon_{12}dv+\frac{1}{2}\int_{V_{p}}\sigma_{11}^{p}\epsilon_{11}+ \sigma_{22}^{p}\epsilon_{22}+\sigma_{12}^{p}\epsilon_{12}dv-\frac{1}{2}\int _{V_{p}}D_{3}E_{3}dv \tag{12}\]  
The virtual work includes the effects of the external applied forces and the aerodynamic pressure, which are given as  
\[\delta W=\int_{A}q\delta w+\Delta p\delta wdA \tag{13}\]where \(A\) is the plate's surface area, \(q\) is the distributed load per unit area of the plate surface. On the basis of the supersonic piston theory [1; 2; 23], the aerodynamic load \(\Delta p\) are obtained as  
\[\Delta p=-\zeta\frac{\partial w}{\partial x}-\mu\frac{\partial w}{\partial t} \tag{14}\]  
where \(\mu=\rho_{\infty}U_{\infty}(M_{a}^{2}-2)/\sqrt{M_{a}^{2}-1}(M_{a}^{2}-1)\) is the aerodynamic damping parameters, \(\zeta=\rho_{\infty}U_{\infty}^{2}/\sqrt{M_{a}^{2}-1}\) is the aerodynamic parameters. \(\rho_{\infty}\), \(U_{\infty}\) and \(M_{a}\) represent the free stream air density, velocity, and Mach number, respectively. The transverse displacements variables \(w\) can be assumed in terms of mode shape and generalized coordinates  
\[w(x,y,t)=\sum_{i=1}^{m}\sum_{j=1}^{n}W_{ij}(x,y)r_{ij}(t)=\mathbf{W}^{T}(x,y) \mathbf{r}(t) \tag{15}\]  
where vector \(\mathbf{r}(t)=[r_{11}\ldots\ldots r_{1n},\ r_{21}\ldots\ldots r_{2n},\ldots,\ r_{m1}\ldots\ldots r_{mn}]^{T}\) denotes the generalized coordinate, and vector \(\mathbf{W}(x,y)=[w_{11}\ldots\ldots w_{1n},\ w_{21}\ldots\ldots w_{2n},\ldots \ldots,\ w_{m1}\ldots\ldots w_{mn}]^{T}\) means the mode shape function.