Analysis of Nonstationary Vibrations of a Nonlinear Plate on an Elastic Half-Space via Ray Expansions

The ray method is an effective method for solving problems related to the generation and propagation of wave surfaces of strong and weak discontinuities, including problems of dynamic contact interaction. Nonstationary vibrations could be caused by the action of instantaneous loads on the plate, resulting in the propagation of wave surfaces of strong and weak discontinuity in an elastic half-space. The solution behind the wave fronts up to the contact boundary is constructed using ray expansions. Unknown functions entering in the coefficients of the ray series and in the equation of plate motion are determined from the boundary conditions of the contact interaction between the plate and the half-space. The “manual” procedure (without using any mathematical packages) for calculating the ray series coefficients is rather cumbersome, therefore an algorithm to solve this problem using the Maplesoft has been suggested by the authors for different types of contact conditions first for linear problems. In this paper, the ray method and the developed algorithm are applied to analyze the unsteady response of an infinitely long elastic nonlinear classical von Karman plate of constant thickness lying on an elastic isotropic half-space.