Modal equations for the nonlinear flexural vibrations of plates

Abstract

HE coupled nonlinear equations of motion based on the von Karman theory of plates can be derived using Hamilton's principle.! A slightly modified form of these equations is the so called dynamic analog of the von Karman equations.M An alternative formulation of the von Karman equations consists of an equilibrium equation for the transverse motion and a compatibility equation for the inplane stresses. The aim of the present Note is to systematically consider higher-order Galerkin approximations to the solutions of the inplane equations of motion as well as the inplane compatibility equation for an assumed single mode expression for w. The computational technique outlined in Ref. 3 is adopted here. The present treatment confines to the assumed space mode approach. The modal equations based on these approximations as well as the approximate solution of the transverse equation of motion are discussed with the help of two typical plate geometries. The significant role of the stretching due to inplane displacements, which has sometimes been overlooked in the literature, is highlighted with the help of the present results. Some of the earlier investigations are considered in the light of the present investigation.