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

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 eects 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 stiness 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 eects 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.