# Dynamic Stability of Orthotropic Viscoelastic Rectangular Plate of an Arbitrarily Varying Thickness
## 1 Introduction
The mathematical setting was carried out within the framework of the classical theory based on the Kirchhoff-Love hypothesis. The proposed method was based on the first-order shear deformation theory and the classical plate theory. A plane stress analysis was carried out using the variational Ritz method and the R-functions theory. The obtained results were applied to investigate buckling and parametric vibrations of laminated plates. In [22], the problem of nonlinear vibrations and analysis of the stability of multilayer symmetric plates of complex shape, loaded with a static or periodic load in the plane, was considered. An approach based on the application of the theory of R-functions and variational methods was proposed. The proposed approach was demonstrated in testing tasks and applied to laminated plates with cutouts. The influence of geometrical parameters, load, boundary conditions on stability regions, and nonlinear oscillations was investigated. The paper [23] proposes a method for studying parametric vibrations of orthotropic plates of complex shape for various types of boundary conditions, based on variational methods in combination with the theory of R-functions. Specific tasks were solved. Regions of dynamic instability and deflection depending on the frequency and excitation time were plotted for a plate of complex shape. R-functions theory was used to study the free vibrations and dynamic instability of a symmetrically layered plate [24]. The plate was subjected to combined static and periodic axial forces. The problem was considered within the framework of the classical theory of plates. The solution of problems was based on the variational Ritz method's joint application, the Galerkin procedure, the theory of R-functions, and [PERSON]'s theory. The influence of the plate's geometric parameters and the mechanical characteristics of the material on the parametric resonance was studied. The combination of the R-function theory and variational methods was proposed in [25; 26; 27]. The parametric vibrations of T-shape, rectangular, and rectangular cutout orthotropic plates of constant thickness were investigated for various boundary conditions [25; 27]. Using the Kirchhoff-Love hypothesis, the motion equations were built. The proposed hybrid approach is based on the combination of the R-functions method and the variation Ritz method. The numerical analysis showed that one mode of approximation cannot be used in all nonlinear dynamics of the orthotropic plates, particularly during their chaotic dynamics. A numerical-analytical method was proposed in [26] for studying parametric vibrations of plates under the action of periodic loads applied in the middle plane. Equations of motion were obtained within the framework of classical theory. The developed approach based on applying the theory of R-functions and variational methods was used to solve the equations, making it possible to study plates of arbitrary geometric shapes with various boundary conditions. The authors used the [PERSON] method to construct the region of dynamic instability. The results obtained using the developed approach were compared with the results of other authors. In [28], the problem of dynamic stability of laminated composite plates under a combination of biaxial periodic loads and bending was considered based on the theory of plates, taking into account shear deformation. A system of second-order ordinary differential equations with periodic coefficients of the Mathieu-Hill type was obtained. The Bolotin method was applied to determine the areas of dynamic instability. The influence of modules' ratio, the number of layers, and static and dynamic parameters of the load on the dynamic instability was investigated. The article [29] considered the dynamic stability of a viscoelastic plate under a periodic load. A differential equation was obtained that describes the dynamic stability of the plate. The solution was sought by the method of separation of variables.