Coupled thickness–shear and thickness–twist vibrations of unelectroded AT-cut quartz plates

Abstract

The equation of coupled thickness–shear and thickness–twist vibrations for rotated Y-cut quartz plates is applied in the analysis of unelectroded rectangular and circular AT-cut quartz plates with free edges. In the case of the rectangular plate, the appropriate edge conditions are found and the solutions are obtained. In the case of the circular plate, the two coefficients appearing in the planar differential operator are written as a sum of an isotropic part plus a deviation. The eigensolutions for the nearby isotropic case are found and the resonant frequencies for the actual anisotropic case are obtained from the equation for the perturbation of the eigenfrequency from the isotropic solution. For the circular plate, the results have been compared with measurements of the fundamental and the third harmonic overtone, and the agreement is quite good.