An analysis of thickness-shear vibrations of doubly-rotated quartz crystal plates with the corrected first-order mindlin plate equations

The Mindlin plate equations have been widely used in the analysis of high-frequency vibrations of quartz crystal resonators with accurate solutions, as demonstrated by the design procedure based on analytical results in terms of frequency, mode shapes, and optimal parameters for the ATcut quartz crystal plate, which is the core element in a resonator structure. Earlier studies have been focused on the AT-cut (which is one type of rotated Y-cut) quartz crystal plates because it is widely produced and has relatively simple couplings of vibration modes at thickness-shear frequencies of the fundamental and overtone modes. The simplified equations through the truncation, correction, and modification of the Mindlin plate equations have been widely accepted for practical applications, and further efforts to expand their applications to similar problems of other material types, such as doubly-rotated quartz crystals, with the SC-cut being a typical and popular one, are also naturally expected. We have found out that the Mindlin plate theory can be truncated and corrected for the SC-cut quartz crystal plates in a manner similar to the AT-cut plates. The analytical results show that the corrected Mindlin plate equations are equally accurate and convenient for obtaining essential design parameters of resonators for the thickness-shear vibrations of SC-cut quartz crystal plates.