# Dynamic Stability of Orthotropic Viscoelastic Rectangular Plate of an Arbitrarily Varying Thickness
## 1 Introduction
The plate's material was assumed to be linearly viscoelastic and modeled by the classical standard linear solid. Then, the equations of motion describing the rectangular plates' nonlinear vibrations were derived by Lagrange equations. Rectangular plates of constant thickness with nonlinear fractional damping were investigated in [9]. It was underlined that damping increases with the vibration amplitude during nonlinear vibrations and, at the same time, soft materials such as silicone rubber increase their stiffness with the vibration frequency. The fractional linear solid model applies to describe the viscoelastic material behavior. The comparison of numerical and experimental results showed a good match. The experimental part of the research in [10] was similar to [9]. A square silicone plate of constant thickness made of silicone with fixed edges was experimentally tested. The plate was excited with harmonic force with a frequency lower than the fundamental natural frequency. Linear and nonlinear plate responses were measured by automatic scanning laser Doppler vibrometer. The increase in damping was around 60% when the vibration amplitude was 1.6 times the plate thickness. Out-of-plane thin membrane actuators have a vital role in micro-electromechanical systems. In [11], the nonlinear vibration control and optimal locations of actuators for a pre-tensioned membrane structure were investigated based on the nonlinear theory of plates. The theoretical and approximate solutions of nonlinear frequencies of the membrane were obtained for a complex parametric study of the behavior of membranes, and the discrepancies between the two solutions under different vibration amplitudes were discussed. The dynamic instability of thin laminated composite rectangular plates with various length-width and length-thickness aspect ratios subjected to harmonic in-plane loading has been theoretically studied [12]. The equations of motion of the plate were developed using the von Karman type of plate equation considering geometric nonlinearity. The equations were solved by using [PERSON]'s technique. Dynamically unstable regions and both stable- and unstable-solution amplitudes of the steady-state vibrations were obtained by applying [PERSON]'s method. Dynamic instability and nonlinear parametric vibrations of multilayer composite plates subjected to the action of periodic compressive loads in the plane applied along two opposite edges were considered in [13]. A system of ordinary differential nonlinear equations of the second-order Mathieu-Hill with periodic coefficients was obtained for determining the regions of dynamic instability and nonlinear parametric oscillations based on the [PERSON] method. The influence of various parameters on the dynamic stability and nonlinear parametric oscillations was investigated: static load, dynamic load, elongation, boundary conditions, orthotropy, and damping coefficient. Functionally graded materials, the material properties of which are varied along the thickness direction, are widely used in aerospace structures, nuclear reactors, and other structures with high functional performance [14]. The nonlinear dynamic instability of a functionally graded rectangular plate of constant thickness was investigated [15]. The plate was made by mixing two different materials, metal on one surface and ceramic on the other surface. The plate was subjected to nonuniform loading. The equations of motion used higher-order shear deformation theory considering von [PERSON] geometric nonlinearity. [PERSON]'s method was used to reduce the nonlinear governing partial differential equations into ordinary differential equations describing the plate's nonlinear dynamic instability. [PERSON]'s method was used to obtain the boundaries of dynamic instability regions. Effects of power index, span-to-thickness ratio, aspect ratio, static load factor, boundary conditions, and various types of linearly varying loadings and parabolically distributed loadings were studied.