Torsional vibrations of circular elastic plates with thickness steps

This paper presents a theoretical study of torsional vibrations in isotropic elastic plates. The exact solutions for torsional vibrations in circular and annular plates are first reviewed. Then, an approximate method is developed to analyze torsional vibrations of circular plates with thickness steps. The method is based on an approximate plate theory for torsional vibrations derived from the variational principle following Mindlin's series expansion method. Approximate solutions for the zeroth- and first-order torsional modes in the circular plate with one thickness step are presented. It is found that, within a narrow frequency range, the first-order torsional modes can be trapped in the inner region where the thickness exceeds that of the outer region. The mode shapes clearly show that both the displacement and the stress amplitudes decay exponentially away from the thickness step. The existence and the number of the trapped first-order torsional modes in a circular mesa on an infinite plate are determined as functions of the normalized geometric parameters, which may serve as a guide for designing distributed torsional-mode resonators for sensing applications. Comparisons between the theoretical predictions and experimental measurements show close agreements in the resonance frequencies of trapped torsional modes.