Tingfeng Ma

The fifth-order overtone vibrations of quartz crystal plates with corrected higher-order mindlin plate equations

Higher-order overtone resonators have been widely used in various electronic products for their higher vibration frequencies, which are in the much-needed frequency range beyond the reach of the fundamental mode. However, the existing designs of higher-order overtone resonators and further improvement for meeting more precise requirements are largely based on empirical approaches. As an analytical effort, we have derived the corrected fifth-order Mindlin plate equations with the consideration of electric potential and overtone displacements. The elimination and truncation of the infinite two-dimensional equations has been done to ensure the exact cut-off frequencies of the fundamental, the third-order overtone, and fifth-order overtone thickness-shear modes in comparison with the three-dimensional equations. The frequency spectra are plotted in the vicinity of overtone thickness-shear modes for analysis of couplings and interactions with spurious modes, and the optimal design of quartz crystal blanks for overtone vibrations has been suggested. The equations, solutions, and method will be important in design of the higher-order overtone thickness-shear vibration resonators.

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.

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.

Approximate frequencies of rectangular quartz plates vibrating at thickness-shear modes with free edges

As the core element of a quartz crystal resonator, the thickness-shear vibration frequency of a quartz crystal plate is always of great interest and top priority in the analysis and design. Because of the difficulty in solving plate equations with the consideration of two-dimensional configuration with free edges, the analysis of resonators is traditionally done with one-dimensional solutions based on the straight-crested wave assumption, which has been validated from earlier experiences and lately numerical analysis with the finite element method. In this study, we start with the known Mindlin plate equations for the thickness-shear vibrations of a rectangular quartz crystal plate with the consideration of flexural and thickness-shear modes. Through the separation of variables, we can obtain higher-order ordinary differential equations for the thickness-shear mode and obtain characteristic functions. The special boundary considerations of resonators with free edges are satisfied through the work of stress components of each individual mode. The method starts with the approximation in one direction, then the same procedure is performed in other direction. Eventually, iteration is taken for each direction until the vibration frequency solution is close to approximations from both directions. This is known as the extended Kantorovich method for vibrations of plates and solutions are accurate as compared with known results from the finite element analysis.

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.

Effect of the Ferroelectric Inversion Layer on Resonance Modes of LiNbO3 Thickness-Shear Mode Resonators

The effect of the ferroelectric inversion layer on resonance modes of LiNbO3 thickness-shear mode resonators was investigated. It is found that with the ferroelectric inversion layer, the spurious modes of the LiNbO3 resonators can be suppressed significantly. Moreover, mechanisms of the effect are discussed. This discovery is important to investigate high-sensitivity bulk acoustic wave sensors using LiNbO3 thickness-shear mode resonators.

The Nonlinear Thickness-shear Vibrations of Quartz Crystal Plates under a Strong Electric Field

With higher frequency and miniaturization of piezoelectric resonators, many nonlinear phenomena such as derive level dependency (DLD) and activity dip have emerged and needed to be systematically studied. Our earlier studies shown neither kinematic nor material nonlinearities are the main factors of frequency shifts and performance fluctuation of quartz crystal resonator. With the consideration of material and kinematic nonlinearities, a nonlinear system of two-dimensional equations for the coupled thickness-shear and flexural vibrations of piezoelectric plates is established by expanding the mechanical displacements and the electrical potential into power series in the plate thickness coordinate. The nonlinear equation of thickness-shear vibrations in a strong electric field have been solved by combination of the Galerkin approximation and successive approximation method. Through these effects, we can obtain nth-order electrical current amplitude-frequency relation. We found the higher-order solutions only improve accuracy of solutions slightly while it is extremely complicated to solve directly. The first-order electrical current amplitude-frequency relation is accurate enough to give some nonlinear characteristics of thickness-shear vibration of quartz crystal plate.

The measurement of elastic constants of quartz crystal by resonant ultrasound spectroscopy

The mechanical resonant response of a solid depends on its shape, density, and elastic moduli. We describe here the instrumentation and computational methods for obtaining and analyzing the resonant frequency of very small (0.001 cm3) samples, and provide examples to demonstrate the utilization of the technique. A new material type with six independent elastic constants are formulated and tested with the instrumentation. This study will leads to expansion of the software capability for the testing of novel materials.

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.