Approximate frequencies of rectangular quartz plates vibrating at thickness-shear modes with free edges

As the core element of a quartz crystal resonator, the thickness-shear vibration frequency of a quartz crystal plate is always of great interest and top priority in the analysis and design. Because of the difficulty in solving plate equations with the consideration of two-dimensional configuration with free edges, the analysis of resonators is traditionally done with one-dimensional solutions based on the straight-crested wave assumption, which has been validated from earlier experiences and lately numerical analysis with the finite element method. In this study, we start with the known Mindlin plate equations for the thickness-shear vibrations of a rectangular quartz crystal plate with the consideration of flexural and thickness-shear modes. Through the separation of variables, we can obtain higher-order ordinary differential equations for the thickness-shear mode and obtain characteristic functions. The special boundary considerations of resonators with free edges are satisfied through the work of stress components of each individual mode. The method starts with the approximation in one direction, then the same procedure is performed in other direction. Eventually, iteration is taken for each direction until the vibration frequency solution is close to approximations from both directions. This is known as the extended Kantorovich method for vibrations of plates and solutions are accurate as compared with known results from the finite element analysis.