A Multiple Scale Method Solution for the Nonlinear Vibration of Rectangular Plates

Abstract

In this paper, rst, the equations of motion for a rectangular isotropic plate have been derived. This derivation is based on the Von Karmann theory and the e ects of shear deformation have been considered. Introducing an Airy stress function, the equations of motion have been transformed to a nonlinear coupled equation. Using the Galerkin method, this equation has been separated into position and time functions. By means of the dimensional analysis, it is shown that the orders of magnitude for nonlinear terms are very small, with respect to linear terms. Then, for the rst time, the invariant manifold theory has been applied to the plate problem and it is proved that the nonlinearities are sti ness and inertia types. Finally, the multiple scale method is applied to the equations of motion and closed-form relations for the nonlinear natural frequencies and mode shapes of the plate are derived. The obtained results are in good agreement in comparison with numerical methods. Using the obtained relation, the e ects of initial displacement, thickness and dimensions of the plate on nonlinear natural frequencies and displacements have been investigated. These results are valid for a special range of the ratio of thickness to dimensions of the plate, which is a characteristic of the multiple scale method.