A semi-analytical model for dynamic analysis of thin plates with plate-type resonators

A semi-analytical method is proposed for free and forced vibrations of a thin square binary material configuration plate containing circular plate-type resonators. The resonator embedded in the host plate consists of a rigid mass and a circular plate made of soft material. Based on the classical plate theory, the host plate and the resonator are modelled separately and then coupled by the condition of displacement compatibility. A set of local admissible functions is incorporated into the global admissible functions in the circular plate domain, to describe the non-smoothness and vibration localization of the displacement caused by the material difference between the host plate and the resonator. The resulting global admissible functions promote the capabilities of the Ritz method in predicting accurately the vibration characteristics of the plate with binary material configuration. The Ritz method with the constructed admissible functions is developed to extract frequencies and analytic mode functions of the plate under different boundary conditions. The plate subjected to a transverse harmonic excitation is discretized into a multi-degree-of-freedom system by the Lagrange approach with the analytic mode functions. The effects of boundary conditions, locations of the mass, and material properties on the vibration are explored. It is demonstrated that the mass location in the circular plate only affects the frequencies of the resonator and has almost no effect on the frequencies and the modes of global vibration, and the resonator with the appropriate material parameters reduces significantly the multi-mode plate vibration.