3C-5 The Analysis of Thickness-Shear Vibrations of the Third-Order Overtone Mode of Quartz Crystal Plates with Mindlin Plate Theory

Extensive efforts have been made on the analysis of quartz crystal resonators vibrating at the fundamental thickness-shear mode, but demands for overtone products with higher frequencies are calling for the study of vibrations of higher-order thickness-shear modes. The challenges of such research are multifold, including demands of appropriate equations to describe the vibrations of overtone modes of piezoelectric plates as the foremost. Our study starts from exact solutions of overtone modes of thickness-shear vibrations of a quartz crystal plate with symmetric electrodes, which is the ideal model of quartz crystal resonators, will provide exact resonant frequency for reference. We extended the Mindlin plate theory suitable for quartz crystal resonator analysis to include higher- order deformations so overtone modes can be included systematically. Since the flexural mode and the thickness-shear mode are strongly coupled, the same procedure in the derivation of higher-order equations also reveals that the flexural mode and its overtones are strongly coupled with the fundamental thickness-shear mode and its third-order overtone mode. Further studies of higher-order overtones can be done in a similar fashion except that the number of equations will increase due to expanded couplings. For insights about couplings of the fundamental and overtone modes, we concentrate on the third- order equations which are required for a detailed examination of analytical solutions. We calculated and plotted the dispersion relations of AT-cut quartz crystal plates for frequencies up to the third-order cut-off frequency of the thickness-shear mode. In a manner similar to our analytical procedure for the fundamental modes, we calculated the frequency spectrum based on analytical solutions from coupled equations of the flexural and thickness- shear modes and their overtones.