Analysis of 3-D vibrations of rectangular AT-cut quartz plates with a bi-mesa structure

A method to analyze 3-D vibrations of rectangular AT-cut quartz, bi-mesa-shaped plates is developed. The method is based on a classical approach. As in 2-D analysis, the half structure of a plate is separated into a thick bi-mesa and a thin-side portion, and the displacement field of each region is represented by a linear combination of guided waves. In the 3-D analysis, we apply the 2-D finite element method (FEM) to obtain the waves guided by two pairs of parallel surfaces. The orthogonal property of guided modes is incorporated to approximately fulfil the continuity conditions at the interface between the thick and thin portions. The stress-free conditions on the plate edges are satisfied by employing the method of weighted residuals (MWR). The computational advantage of this method is that it can greatly reduce the matrix size compared with the 3-D FEM. As a numerical example, the frequency spectra are calculated for X-elongated plates of bi-mesa shape, and the strong energy-trapping effect on the fundamental thickness-shear (TS) resonance is verified.