Three-dimensional analysis for rectangular AT-cut quartz plates with bi-mesa structure

This paper describes a method to analyze 3-D vibrations of rectangular AT-cut quartz plates with bi-mesa structure. The method is based on a classical approach. As in the 2-D analysis of the previous paper, the displacement field for each of thick and thin portions of a plate 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. We also incorporate a new expression for orthogonality relation of guided waves into the continuity condition of displacement and stress over part of a junction cross section. A computational advantage of this method is that it can greatly reduce matrix size compared with the 3-D FEM. As an example of calculation, the frequency spectrum for an X elongated plate of bi-mesa shape is presented along with one for a flat plate.