# Dynamic Stability of Orthotropic Viscoelastic Rectangular Plate of an Arbitrarily Varying Thickness
## 1 Introduction  
Thin-walled plates and shells of variable thickness are widely used in various technology fields due to the requirements for strength, durability, and thin-walled structural elements' exterior appearance. The study of plates and shells of variable thickness made from homogeneous and inhomogeneous materials is a challenging task, and sometimes it encounters insurmountable difficulties. The difficulties are due to the solution of rather cumbersome equations, which are obtained by the desire to reflect, in mathematical modeling, the real mechanical behavior of the thin-walled structure. Variable thickness plate problems involve computational complexity, i.e., the lack of suitable universal numerical methods for solving the obtained equations, and as a consequence, unified computational algorithms. Studies of parametric vibrations of thin-walled structures have become a separate research area in the mechanics of deformable solids. Thin-walled structures have versatile applications for various mechanical systems, in particular for plates and shells. The solution of dynamic stability problems includes the derivation of the equation of motion, discretization, and determination of areas of dynamic instability of structures. Several publications [1; 2; 3] were devoted to studying the behavior of plates and shells of constant thickness under dynamic loads in an elastic formulation, where detailed reviews of these studies can be found. The research of the dynamic stability of elastic systems was started in 1964 [1]. [PERSON]'s first-order approximation followed by an eigenvalue analysis gave the dynamic behavior including natural frequencies and instability load-frequency margins. The stability of the plane form of the equilibrium position of plates has been studied thoroughly in [2] for the case of compression, typically, in a uniaxial stress field, where it was assumed that the shape of buckling of the plate does not depend on buckling position's coordinate. The article [3] is devoted to reviewing studies carried out from 1987 to 2005 in dynamic stability and parametric vibrations of plate and shell structures. A new approach [4] based on the Amabili-Reddy higher-order shear deformation theory was implemented to modelling the geometrically nonlinear forced vibrations of laminated circular cylindrical shells. The numerical solution of discretized motion equations showed the one-to-one internal resonance for a complete circular cylindrical shell, giving rise to pitchfork bifurcations of the nonlinear response with quasiperiodic vibrations. Theoretical and experimental aspects of nonlinear vibration and stability of shells and plates when the vibration amplitude is larger than the thickness are explained in [5]. A very good agreement between experiments and theoretical predictions was obtained for rectangular plates and circular cylindrical panels in the cases considered. The article [6] provides an overview of publications devoted to studying nonlinear free and forced vibrations of shells made of traditional and modern materials published from 2003 to 2013. In particular, the results of studies of parametric oscillations were considered. Nonlinear vibrations of viscoelastic thin rectangular plates were investigated in [7]. The plates were subjected to harmonic excitation. The plates' material was modeled as a Kelvin-Voigt viscoelastic solid by retaining all the nonlinear terms. It was shown that the nonlinear response of viscoelastic plates is different from the response of the same plate with viscous damping. Experiments presented in [8] have shown that the most significant nonlinear effect of geometrically nonlinear vibrations of rectangular plates is the increase in damping with the vibration's amplitude. At the same time, soft materials present an increase in their stiffness with the vibration frequency. These two phenomena appearing together were explained in the framework of viscoelasticity. The nonlinear damping model for a continuous system was presented in [8].