Finite element analysis of non-linear vibration of a circular plate with a concentric rigid mass

Abstract

A finite element analysis of non-linear vibration of an elastic circular plate with a concentric rigid mass is examined for different cases of boundary conditions. The non-linear partial differential equations and the associated boundary conditions are derived from an energy method, by using the Hamiltonian principle. The applications of the finite element method to the dynamic problem rely on the use of a variational principle to derive the necessary element properties or equations. The assembled equations for the plate are formed by summing each element equation obtained in consideration of a single element. Then, the boundary conditions are imposed on the vector of nodal field variables, so that the appropriate boundary conditions are satisfied. The assembled equations form an eigenvalue problem and are solved for the eigenvalues as well as the unknown field variables. The relations between the fundamental frequencies and the amplitudes of non-linear vibrations of the circular isotropic elastic plate with a concentric rigid mass are obtained. The results agree well with the solutions obtained by the Kantorovich averaging method. Also, the results yield the upper bounds of the solution to the stated problem.