Non-linear Oscillations of Orthotropic Plates on a Non-linear Elastic Foundation

Abstract

In this paper, the non-linear vibration response of an orthotropic plate rests on a nonlinear foundation that subjected to non-uniform initial stress is investigated. The non-linear partial differential equations of motion for an orthotropic plate are derived by Hamilton's principle. By using these derived governing equations, the large amplitude vibration of an initially stressed plate on a non-linear elastic foundation model was studied. Galerkin's approximate method was applied to the governing partial differential equations to yield ordinary differential equations. The ordinary differential equations were solved by employing a Runge—Kutta method to obtain the ratio of nonlinear to linear frequencies. The initial stress is taken to be a combination of pure bending stresses plus extensional stresses in the plane of the plate. The softening non-linear elastic foundation model is used to describe the plate—foundation interaction. Numerical example of simply supported Mindlin plates subjected to the initial stress and resting on a Winkler non-linear foundation was solved. The frequency responses of non-linear vibration are sensitive of initial stress, amplitude of vibration, material properties, linear foundation stiffness and non-linear foundation stiffness.