Free and forced vibrations of SC-cut quartz crystal rectangular plates with the first-order Mindlin plate equations.

Mindlin plate theory was used to provide accurate solutions to thickness-shear vibrations of plates, which have a much higher frequency than usual flexural vibrations and are the functioning modes of quartz crystal resonators. The vibration frequency solutions obtained with the Mindlin plate theory are proven being accurate along with mode shapes. In this paper, straight-crested wave solutions of free and forced vibrations of doubly rotated SC-cut of quartz crystal plates of rectangular shapes with four free edges are obtained with validated Mindlin plate equations. A procedure has been established for the calculation of dispersion relations, frequency spectra, mode shapes, and capacitance ratios of forced vibrations needed in resonator design.

Forced vibrations of SC-cut quartz crystal rectangular plates with partial electrodes by the Lee plate equations.

Lee plate equations for high frequency vibrations of piezoelectric plates have been established and perfected over decades with the sole objective of obtaining accurate predictions of frequency and mode shapes to aid the analysis and design of quartz crystal resonators. The latest improvement includes extra terms related to derivatives of the flexural displacement to provide much accurate solutions for vibrations of the thickness-shear mode, which is the functioning mode of resonators and has much higher frequency than the flexural mode. The improved Lee plate equations have been used in the analysis of high frequency vibrations of quartz crystal plates as an essential step for analysis of AT- and SC-cut quartz crystal resonators after validations with fully electrode quartz crystal piezoelectric plates. In this study, closed-form solutions of free and forced vibrations of SC-cut quartz plates with partial electrodes are obtained. A procedure has been established for the calculation of dispersion relations, frequency spectra, selected vibration modes, and capacitance ratios of forced vibrations. The vibration solutions obtained with the first-order Lee plate equations are proven to be close to solutions from the Mindlin plate equations. It is now clear that both the Mindlin and Lee plate equations can be used in the analysis and design of quartz crystal resonators.

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 AT-cut 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.

Correction factors of the Mindlin plate equations with the consideration of electrodes.

The Mindlin plate theory plays a vital role in the analysis of high-frequency vibrations of quartz crystal resonators in the thickness-shear mode. The coupled equations with an infinite number of vibration modes have been truncated to retain essential modes with strong couplings for simplified analysis, and correction factors have been introduced to ensure exact solutions at cut-off frequencies as part of the validation procedure. Starting from plates without complications, correction factors have also been modified to include electrodes for their mass effect. Such results have been important for higher-order correction factors to ensure accurate consideration of electrodes in AT-cut quartz crystal plates. The findings in this systematic study of correction factors are readily available for the analytical equations with selected modes with strong couplings, and equally convenient for the implementation of the finite element analysis with the Mindlin plate equations.

Five-mode frequency spectra of x3-dependent modes in AT-cut quartz resonators.

We study straight-crested waves and vibration modes with variations along the x(3) direction only in an AT-cut quartz plate resonator near the operating frequency of the fundamental thickness-shear mode. Mindlin's two-dimensional equations for anisotropic crystal plates are used. Dispersion relations and frequency spectra of the five relevant waves are obtained. It is found that, to avoid unwanted couplings between the resonator operating mode and other undesirable modes, in addition to certain known values of the plate length/thickness ratio that need to be avoided, an additional series of discrete values of the plate length/thickness ratio also must be excluded.