Fiber-reinforced plastic composites are one of the most widely used composite materials because they balance well between properties and cost. Despite their widespread use in the aviation and automotive industries, there is currently a lack of effective mathematical models for their calculation under various dynamic loads. The research object of this work is an anisotropic viscoelastic fiber-reinforced simply supported rectangular plate. Two dynamic problems are considered: vibrations of the plate under the influence of a uniformly distributed static load; stability of the plate compressed in one direction. Within the Kirchhoff-Love hypothesis framework, a mathematical model was built in a geometrically nonlinear formulation, taking into account the tangential forces of inertia. By the Bubnov-Galerkin method, based on a polynomial approximation of the deflection and displacement, the problem was reduced to solving systems of nonlinear ordinary integro-differential equations. With a weakly singular Koltunov-Rzhanitsyn kernel with variable coefficients, the resulting system was solved by a numerical method based on quadrature formulas. By using experimental studies, considering the directions of the fibers, the values of the rheological parameters of some plastic materials (KAST-V and EDF) were obtained. The plate's dynamical behavior was investigated depending on the plate's geometric parameters, viscoelastic and inhomogeneous material properties. Results show the importance of taking into account the viscoelastic properties of the material when solving dynamic problems of anisotropic reinforced plates made of composite materials. In particular, when studying the problem of dynamic stability of an anisotropic reinforced plate made of KAST-V, the results obtained in elastic and viscoelastic formulations in some cases differ from each other by more than 20 %.