Vibrations of a viscoelastic isotropic plate under periodic load without considering the tangential forces of inertia

Abstract

A mathematical model of the problem of viscoelastic isotropic plate vibrations based on the Kirchhoff-Love hypothesis in a geometrically nonlinear formulation was presented. The mathematical model was built without considering the tangential forces of inertia. To describe the viscoelastic properties of the plate material, a weakly singular Koltunov-Rzhanitsyn kernel with three different rheological parameters was chosen. To solve the problem of parametric vibrations of a viscoelastic plate with a weakly singular relaxation kernel, a numerical method based on the use of quadrature formulas was applied. A discrete model of this problem was first constructed using the Bubnov-Galerkin method; i.e., a system of integro-differential equations with variable coefficients was obtained, and then, using a numerical method based on the elimination of a singularity of the kernel, the problem of parametric vibrations of viscoelastic rectangular plates was solved. The influence of the viscoelastic properties of the material and the variability of the plate thickness on the oscillatory process was shown.