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

In this paper, first, the equations of motion for a rectangular isotropic plate have been derived.this derivation is based on the von karmann theory and the effects of shear deformationhave been considered. introducing an airy stress function, the equations of motion have beentransformed to a nonlinear coupled equation. using the galerkin method, this equation has beenseparated into position and time functions. by means of the dimensional analysis, it is shownthat the orders of magnitude for nonlinear terms are very small, with respect to linear terms.then, for the first time, the invariant manifold theory has been applied to the plate problemand it is proved that the nonlinearities are stiffness and inertia types. finally, the multiplescale method is applied to the equations of motion and closed-form relations for the nonlinearnatural frequencies and mode shapes of the plate are derived. the obtained results are in goodagreement in comparison with numerical methods. using the obtained relation, the effects ofinitial displacement, thickness and dimensions of the plate on nonlinear natural frequencies anddisplacements have been investigated. these results are valid for a special range of the ratio ofthickness to dimensions of the plate, which is a characteristic of the multiple scale method