Guijia Chen

Five-mode frequency spectra of x 3 -dependent modes in AT-cut quartz resonators

We study straight-crested waves and vibration modes with variations along the x3 direction only in an AT-cut quartz plate resonator near the operating frequency of the fundamental thickness-shear mode. Mindlins 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.

The correction factors of mindlin plate theory with and without electrodes for SC-cut quartz crystal plates

Mindlin plate theory was first developed to provide accurate solutions for vibrations of thickness-shear mode, which has a much higher frequency than usual flexure vibrations. It has been widely used in the analysis of high frequency vibrations of quartz crystal plates, which are the core of resonators. The vibration frequency solutions obtained with Mindlin plate theory are proven much closer to the exact solutions. However, due to the truncation and approximation, the plate equations need to be corrected, as compared with the three-dimensional elasticity solutions. This has been done for the high-order Mindlin plate theory with and without electrodes for the AT-cut quartz crystal plates, and correction factors have been obtained though both natural and symmetric procedures. The correction factors could be used in the dispersion relationship and frequency spectrum in the analytical solutions, while the symmetric correction factors can be used in the finite element method implementation. Both correction schemes can provide improved and accurate results in the analysis of quartz crystal resonators. The electrodes are considered though its inertia effect as mass ratio known in resonator analysis.

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.

Correction factors of the mindlin plate equations with the consideration of electrodes [Correspondence]

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.

Mindlin plate equations forthe thickness-shear vibrations of circular elastic plates

The systematic derivation of Mindlin plate equations has been fully presented in the monograph of Mindlin and some of his papers. Most applications followed are focused on the straight-crested wave in rectangular quartz crystal plates due to the necessity in the design of rectangular type resonators and relative simplicity of analysis. Some applications concerning circulate plates vibrating at the thickness-shear mode or the shear effects are analyzed by transforming the relatively simple equations of the thickness-shear and flexural modes to the cylindrical coordinates with a simple procedure, as we can find from a few papers and books. The systematic derivation by following a rigorous procedure of Mindlin plate equations are not presented before, and subsequent applications to coupled vibrations of circular plates at thickness-shear modes have not be studied with the Mindlin plate equations for circular type quartz crystal resonators. Particularly, vibrations of overtone modes with more coupled displacements require a systematic derivation which is more clear with the cylindrical coordinates.

Correction factors of the Mindlin plate equations for vibrations of SC-cut quartz crystal plates

The Mindlin plate equations have been the primary choice for the analysis of high frequency vibrations of AT- and SC-cut quartz crystal plates in the design of resonators for predictions of vibration frequency, mode couplings, and bias effects. However, the plate equations have to be modified to yield exact solutions in certain frequency range and correction factors are chosen in a systematic manner with validation from three-dimensional solutions. We have obtained correction factors up to the fifth-order plate equations in terms of electrode mass ratio as polynomials up to cubic terms through curve fitting for SC-cut quartz crystal. For partially electroded quartz plates, capacitance ratios have also been compared with that from the Lee plate equations in good agreement. We conclude that the procedure have been established to take the advantage of simplicity and elegance of the corrected Mindlin plate equations for analytical and numerical solutions.

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

HIGHLIGHTSHigh frequency vibrations of highly anisotropic SC‐cut quartz crystal plates analyzed.The first‐order Mindlin plate equations are used for the thickness‐shear vibrations.The solution procedure of vibrations of SC‐cut of quartz crystal plates established.Dispersion relations, frequency spectra, mode shapes, and capacitance ratios calculated. ABSTRACT 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.

Correction factors for mindlin plate equations with the consideration of stiffness and mass effects of electodes for SC-cut quartz crystal plates

Mindlin plate theory has been playing a vital role in the analysis of high frequency vibration of quartz crystal resonators, since the coupled equations with infinite number of vibration modes have been truncated to retain essential ones 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. Corrections must be made at the cut-off frequency with stiffness and mass effects of electrodes considered in the plate equations, but correction factors are different between the AT- and SC-cut quartz crystal. The systematic study of correction factors is readily available to the analytical equations with selected modes of strong couplings, and equally convenient for the implementation of the finite element analysis with the Mindlin plate equations.

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.