Vibration analysis of SC-cut quartz crystal plates with the Mindlin and Lee plate theories

SC-cut quartz crystal resonators have been widely utilized in many applications as core elements of filters and sensors for their higher vibration frequencies. An analysis of SC-cut quartz crystal resonators is required as part of the design process and it has been proven that the Mindlin and Lee plate equations can be used for the vibration analysis of plates at the higher-order overtone modes with accurate prediction of frequency and dispersion relations in the vicinity of cut-off frequencies. We have utilized the first-order Mindlin and Lee plate equations for the analysis of thickness-shear free vibrations of a rectangular SC-cut quartz crystal plate. The procedure has been established for the calculation of dispersion relations, frequency spectra, selected vibration modes, and determination of the optimal length of SC-cut quartz crystal blank. Such detailed studies on the successive overtone vibrations at the thickness-shear mode will provide important design guidelines for SC-cut quartz crystal resonators.