Trapped torsional vibrations in elastic plates

We have observed that torsional vibrations can be trapped in elastic plates with circular regions of slightly thicker steps or with smooth convex contoured surfaces. An electromagnetic acoustic transducer (EMAT) was used to generate oscillatory surface traction. The resonant frequencies and Q-values were measured. It was found that these trapped torsional modes have Q-values exceeding 100,000 with pure in-plane motion, which is of practical importance for acoustic sensor applications. In this paper, a set of approximate two-dimensional equations is developed to study vibrations in axisymmetrically contoured or stepped elastic plates. By assuming circumferentially independent motion, the first-order equations are decoupled into four groups, with torsional modes uncoupled from flexural and extensional modes. Analytical solutions for torsional modes are obtained for stepped and linearly contoured circular plates. It is found that the first-order torsional modes can be trapped in an infinite plate with a stepped or contoured region if critical conditions for the geometrical parameters are met. The analytical results are compared to experiments and finite element analyses with good agreements.