Simultaneous Resonances Involving Three Mode Shapes of Parametrically-Excited Rectangular Plates

Abstract

It is known that, for multi-degree-of-freedom systems under time-dependent excitation, the combination of an internal resonance with an external resonance will give rise to simulataneous resonances, and these resonances are characterized by the fact that the system in question can resonate simultaneously in more than one normal mode while only one resonant mode is diectly excited by the excited by the excitation. The present work deals with the problem of the occurrence of simultaneous resonances in parametrically-excited and simply-supported rectangular plates. The analysis is based on the dynamic analog of von Kármán's large-deflection theory and the governing equations are satisfied using the orthogonality properties of the assumed functions. The nonlinear temporal response of the damped system is determined by the first-order generalized asymptotic method. The solution for simply supported plates indicates the possibility of principal parametric resonance and simultaneous resonances. Simulataneous resonances involving two modes of vibration have been presented in a number of papers by the present author. Numerical results of simultaneous resonances involving three mode shaoes are presented in this paper for the first time.