An analysis of thickness-shear vibrations of circular quartz crystal plates

Quartz crystal resonators of circular type are widely used for frequency control and detection functions with the quartz crystal plate vibrating at the thickness-shear mode. Naturally, vibrations of circular quartz crystal plates are of great interests in the design and optimization of resonators. We have derived the Mindlin plate equations for the thickness-shear vibrations of circular plates in a systematic manner. We have further proved that the thickness-shear vibrations of circular plates are consistent with rectangular plates in dispersion relations. By truncating the infinite two-dimensional equations for the thickness-shear vibrations, we obtained a set of equations with only selected vibration modes. Solving the coupled equations for thickness-shear vibration frequencies and mode shapes, we can use these results for the optimal selection of plate parameters to improve performance of resonators. The analytical equations and solution procedure are important in the analysis of quartz crystal vibrations.