# Dynamic Stability of Orthotropic Viscoelastic Rectangular Plate of an Arbitrarily Varying Thickness
## 1 Introduction
Analytical studies on the dynamic instability analysis of a functionally graded constant thickness skew plate (in the form of the parallelogram) made by mixing ceramic and metal materials were presented in [16]. The plate was subjected to uniform and linearly varying in-plane periodic loadings with four different types of boundary conditions. The total energy functional of the plate was formulated based on [PERSON]'s third-order shear deformation theory. The energy functional was converted into a set of ordinary differential equations (Mathieu-Hill equations) using the Rayleigh-Ritz method in conjunction with BCOPs. The solution of the Mathieu-Hill equations described the dynamic instability behavior of the skew plate. The instability regions were traced using the Bolotin method. The parametric instability of a functionally graded polymer-based graphene-reinforced nanocomposite plate of constant thickness was investigated in [17]. The plate comprised multiple graphene platelet reinforced composite layers in which graphene platelets are uniformly distributed in each layer with graphene platelet concentration varying layerwise across the plate thickness. The plate was subjected to periodic uniaxial in-plane force and a uniform temperature rise. The basic equations were derived from the theory of first-order shear deformation. The problems were solved using the differential quadrature approach in combination with the Bolotin method. The instability of rectangular plates due to parametric resonance was considered in [18]. The partial differential equation was derived using the extended Hamilton principle. Using the [PERSON] method, the resulting equation was reduced to a system of ordinary differential equations. Semianalytical results were obtained for various plate boundary conditions, confirmed by numerical simulations and other published results. The boundaries of instability of axially accelerating plates with internal resonance were studied in [19]. A connection was introduced between acceleration and tensions changing in the longitudinal direction. The main equation and the corresponding boundary conditions were derived from the generalized Hamilton's principle. The effects of internal resonances and inhomogeneous boundary conditions at the boundaries of instability were distinguished. Multiple scales established modified conditions of solvability in the main parametric and internal resonances. The Routh-Hurwitz criterion was introduced to determine the boundaries of instability. The influence of the coefficient of viscoelasticity and the coefficient of viscous damping on instability boundaries was investigated. Abnormal instability boundaries were detected upon the application of the internal resonance. The phenomenon of local zigzag and V-shaped boundaries was explained in terms of modal interactions. Numerical calculations of differential quadrature circuits for the first four complex frequencies, the first four complex modes, and stability boundaries were used to confirm the analytical method's results. The R-functions theory was used for solving dynamic problems of the theory of plates and shells, connected with the construction of the basic function in the case of an arbitrary domain [20; 21; 22; 23; 24; 25; 26]. In [20], dynamic instability and nonlinear parametric vibrations of multi-layer plates of complex shapes were investigated. A numerical-analytical method based on a combination of the theory of R-functions and the variational method was proposed. Instability regions and response curves were plotted for various types of materials and load parameters. In [21], using the theory of R-functions, a method was proposed for studying dynamical instability and nonlinear parametric vibrations of laminated plates. The plates had complex shapes and different cutouts. The structure of the plate was symmetrical concerning its middle surface, consisting of layers of constant thickness. The loads consisted of a periodic in-plane load and static load.