Simultaneous resonances involving two mode shapes of parametrically-excited rectangular plates

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 simultaneous 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 directly excited by the excitation. This work deals with the problem of the occurrence of simultaneous resonances in a parametrically-excited and simply-supported rectangular plate. The analysis is based on the dynamic analog of von Karmanʹs 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 resonances and simultaneous resonances. Simultaneous resonances involving two modes of vibration are presented in this paper, and it is shown for the first time that only one possibility out of two cases can really occur. This phenomenon is verified by experimental results and observations.