Do we need to satisfy natural boundary conditions in energy approach to nonlinear vibrations of rectangular plates?

Abstract

Despite a large literature, it seems still unclear if it is necessary to satisfy the natural boundary conditions in an energy approach to the study of nonlinear vibrations of rectangular plates. In the present study, which is based on Lagrange equations, the case of simply supported movable edges is considered. In the past, special nonlinear terms were added to the in-plane displacements to satisfy the natural boundary conditions. However, it was unclear if this was just helping the fast convergence of the series, reducing the number of degrees of freedom in the model, or it was actually necessary. In fact, it is well-known that for linear vibrations only geometric (essential) boundary conditions must be satisfied in variational energy approaches like the Rayleigh-Ritz method. Therefore, this doubt should be clarified. In the present study, accurate expansions of the plate displacements are introduced with additional in-plane trigonometric terms. They allow to satisfy the natural boundary conditions by energy minimization without the introduction of additional nonlinear terms. Nonlinear damping is also introduced in order to obtain results which agree with experiments.