# Dynamic Stability of Orthotropic Viscoelastic Rectangular Plate of an Arbitrarily Varying Thickness
## 1 Introduction
Linear differential equations of the third order were obtained for the time function. The prime instability domain position was affected by relaxation time and the correlation between long-term and instantaneous moduli of elasticity. In [30], using the Bubnov-Galerkin method, the equations of motion of a shallow shell in a mixed form under the action of a longitudinal harmonic load were reduced to a discrete system with finite degrees of freedom. The chaotic behavior of systems with different degrees of freedom was analyzed. It was shown that by increasing the parameter of periodical longitudinal exciting, the investigated system starts to lose the regular behavior, forgetting its initial state, and chaotic dynamics occur. The parametric instability of rectangular plates made of functionally graded material, subjected to biaxial periodic loading in the plane, was investigated [31]. The equations of motion were constructed, taking into account shear deformation. Hamilton's Principle was used to transform the basic equations into a linear system of Mathieu-Hill equations. With the help of the Floquet theory, areas of dynamic instability were determined. The influence of various parameters on areas of dynamic instability was investigated. Viscoelastic rectangular plates with constant thickness, shells, and cylindrical panels were studied in a geometrically nonlinear formulation [32; 33; 34; 35]. The models were based on the Kirchhoff-Love hypothesis. The problem was reduced to a system of nonlinear ordinary integrodifferential equations by using the Bubnov-Galerkin method. The resulting system with a weakly singular Koltunov-Rzhanitsyn kernel was solved using a numerical method based on quadrature formulas. The article [36] was devoted to studying the stability of parametric vibrations of an isolated symmetric cross-ply laminated plate system. The plate system's mechanical model was given as an isolated symmetric cross-ply laminated constant thickness rectangular plate subjected to uniformly distributed stochastic in-plane time-dependent loading. The influence of transverse shear on the laminated plate was taken into account. The equations of motion were derived based on the shear deformation theory and the Kirchhoff-Love theory [37]. As models of viscoelastic material, the Maxwell and Kelvin-Voigt constitutive equations and the standard solid model are the simplest and most used by many researchers. These models are reduced to solving differential equations with initial conditions. The use of differential dependencies between stress and strain in dynamic problems of viscoelasticity theory leads to a specific inaccuracy. The inaccuracy is especially noticeable at the initial moment. The reason for this inaccuracy is that the strain rate and stress rate according to these models will always be finite. In contrast, the experiment shows an arbitrarily high rate of deformation. It should be noted that the inaccuracy made at the initial moment significantly affects the final result of the study, i.e., there is an accumulation of errors. There will be no accumulation of errors if the integral relationship between stresses and strains uses a weakly singular Koltunov-Rzhanitsyn kernel with weakly singular features of the Abel type [38; 39; 40; 41]. The use of integral dependence significantly expands the class of problems to be solved in the theory of viscoelasticity. Simultaneously, mathematical models with integral dependence are reduced to solving complex systems of nonlinear integrodifferential equations by numerical methods [42]. The available literature analysis showed a small number of publications on nonlinear vibrations and dynamic stability of thin viscoelastic plates and shells of variable thickness. Reviewed studies show that the calculation of plates and shells of variable thickness faces significant computational difficulties. The calculation of plates and shells of this kind requires the solution of rather cumbersome equations [43; 44; 45].