Modal characteristics of in-plane vibration of rectangular plates

Abstract

The equations of in-plane vibration of thin plates are solved for rectangular panels with two parallel edges clamped, while the other two edges are either both clamped, both free, or one clamped and one free. The propagation characteristics of in-plane waves are investigated and the frequency bands of attenuation and propagation of the waves are identified. Simple expressions for the cutoff frequencies of different wave components are derived. The nature of the coupling between in-plane longitudinal and in-plane shear waves is mathematically and physically illustrated. Although all the modes in a plate panel are coupled through Poisson’s effect, it is shown that the coupling is very weak in plate panels with all edges clamped as compared to the cases where one or two parallel edges are free. As a first order approximation, the resonant modes of in-plane waves are classified into uncoupled and coupled mode pairs. By using the coupled mode pairs approximation, it is possible to predict the dynamic response of the plate panel with a reasonable accuracy. Simple expressions are given for approximate estimation of the resonance frequencies of coupled and uncoupled modes. Mode shapes are given, for each case of edge conditions, which satisfy both the displacement and force conditions at the plate edges. A simple procedure is given for the determination of resonance frequencies and mode shapes without excessive computations. The predicted resonance frequencies and mode shapes are compared to the finite element results and good agreement is found. The mode shapes of the in-plane vibration are depicted for the first eight resonant modes for each case of edge conditions.