The coupled thickness-shear and flexural vibrations of quartz crystal plates under an electric field

A nonlinear system of two-dimensional equations for the coupled thickness-shear and flexural vibrations of electroelastic plates has been established by expanding the mechanical displacements and the electrical potential into power series in the plate thickness coordinate. By Galerkin approximation, these nonlinear partial differential equations have been first converted into two ordinary differential equations which only depend on time. We further employed successive approximation method to obtain the coupled nonlinear amplitude-frequency equations of thickness-shear and flexural vibrations which can be solved for amplitude ratio between these two modes. Then we have obtained electrical current frequency-response relation of thickness-shear vibrations with given amplitude ratio and plotted curves of nonlinear frequency-response behavior with different electrical voltages. With increase of driving electrical voltage, the nonlinear frequency shift will become stronger and unstable, a clear sign of derive level dependency (DLD) phenomenon we have frequently observed in quartz crystal resonators. The basic conclusions we drawn is that the electrical field has a more significant effect on vibration frequency compared with other known factors.