Forced Vibrations of an Elastic Rectangular Plate Supported by Unilateral Edge Lateral Springs

Abstract

The present study deals with static and dynamic behavior, including forced vibrations, of an elastic rectangular plate supported along its edges by unilateral elastic springs. The plate is assumed to be subjected to a distributed and concentrated load applied eccentrically and the external moments as well. Equations of motion are derived by considering the dynamic response of the plate, assuming a series of the Chebyshev polynomials for the displacement function and applying Galerkin’s method. Effects of the boundary conditions of the plate, i.e., the shear forces, the bending moments and the corner forces, are included in the equation of motion to compensate for the non-satisfied boundary conditions and increase the accuracy of Galerkin’s method. The numerical solution is accomplished by using an iterative process due to the nonlinearity of the unilateral character of the support. Static behavior of the plate under static concentrated load and uniformly distributed load is investigated in detail by taking into consideration a wide range of support stiffnesses. Numerical treatment of the problem in the time domain is carried out by assuming a stepwise change in the external loads, and the linear acceleration procedure is adopted for the solution of the governing differential equation derived from the equation of motion. Various numerical results are presented in the figures focusing on the nonlinearity of the problem due to the plate lift-off from the unilateral springs at the corners of the edge supports.