Rongxing Wu

The analysis of the third-order thickness-shear overtone vibrations of quartz crystal plates with Mindlin plate theory

The design and analysis of quartz crystal resonators in the fundamental thickness-shear mode have been extensively studied with many methods including the simple model based on finite plates for the vibration frequency and Mindlin plate theory for couplings of the fundamental thickness-shear and spurious modes. These methods are widely used in the design process for the optimal determination of crystal blanks and electrode configuration. In order to study the overtone vibrations of quartz crystal resonators, Mindlin plate theory is used in the form of the third-order equations with selected modes to obtain the dispersion and frequency spectra in the vicinity of the third-order thickness-shear mode. In our earlier studies, a set of correction factors have been suggested for the Mindlin plate equations to be accurate at the cut-off frequency at the third-order thickness-shear mode. By checking the accuracy at the cutoff frequencies and of the dispersion relations, the third-order plate equations of selected modes are chosen for the calculation. The coupling of modes and effect of electrodes for the third-order overtone vibrations at the thickness-shear mode will be used for the design of quartz crystal resonators of overtone types.

Solutions of nonlinear thickness-shear vibrations of an infinite isotropic plate with the homotopy analysis method

As a preliminary attempt for the study on nonlinear vibrations of a finite crystal plate, the thickness-shear mode of an infinite and isotropic plate is investigated. By including nonlinear constitutive relations and strain components, we have established nonlinear equations of thickness-shear vibrations. Through further assuming the mode shape of linear vibrations, we utilized the standard Galerkin approximation to obtain a nonlinear ordinary differential equation depending only on time. We solved this nonlinear equation and obtained its amplitude–frequency relation by the homotopy analysis method (HAM). The accuracy of the present results is shown by comparison between our results and the perturbation method. Numerical results show that the homotopy analysis solutions can be adjusted to improve the accuracy. These equations and results are useful in verifying the available methods and improving our further solution strategy for the coupled nonlinear vibrations of finite piezoelectric plates.

An analysis of vibrations of quartz crystal plates with nonlinear Mindlin plate equations

The nonlinear effects of material constants and initial stresses and strains in quartz crystal resonators is well known f on the frequency-temperature curves, drive-level dependency, acceleration sensitivity, and stress compensation. Consequently, accurate predictions on resonator behavior and their electrical circuit parameters require the use of nonlinear vibration equations and their solutions. The effectiveness of nonlinear analyses has been shown by a few researchers with the finite element and perturbation methods. The Mindlin plate theory, which has been used extensively for understanding plate modes and their coupling effects in plate vibrations analysis, is not enough in the study of the nonlinear behavior of quartz resonators. We have followed the Mindlin plate theory to derive the nonlinear equations with the inclusion of large displacements and higher order elastic constants. The coupling of vibration modes due to nonlinearity is clearly observed and it is quite different from linear cases that we are familiar with. We start from the equations of vibration for the thickness-shear mode to validate the solution techniques, which could be the perturbation method and the latest Homotopy Analytical Method (HAM). Then the methods are applied to the coupled equations of thickness-shear and flexural vibrations which are the two dominant modes of quartz crystal resonators of the thickness-shear type. These solutions, in the absence of the strong electrical field, can be used to study the frequency, deformation, and mode conversion in nonlinear vibrations. We hope the frequency spectra and spatial variations of the thickness-shear and flexural displacements from the accurate solutions of nonlinear equations will provide insights on the changes in each mode when compared with their linear vibrations. The further extension of nonlinear plate equations with the inclusion of piezoelectric effects will also provide useful examination of nonlinear behavior of quartz crystal resonators.

Nonlinear Mindlin plate equations for the thickness-shear vibrations of crystal plates

Mindlin plate equations have wide applications in engineering fields with piezoelectric crystal resonators in particular due to its accurate prediction of high frequency vibrations of plates in the vicinity of thickness-shear mode. As a linear theory based on the assumption of infinitesimal deformation, its applications have been limited mainly to vibration frequency analysis. Many important properties of quartz crystal resonators such as the electrical circuit parameters, and nonlinear phenomena such as the drive-level dependence (DLD), have to be studied with the consideration of higher-order material constants and subsequent inclusion of nonlinear strain components. This, in turn, implies that the consideration of large deformation related to the driving electrical field. The need of nonlinear theory for the analysis of high frequency vibrations of piezoelectric crystal plates have been noticed before, and research work concerning large electrical fields and deformation have been initiated for the calculation of electrical properties and DLD of quartz crystal resonators by many investigators recently. In this study, we start with higher-order constitutive relation which includes higher-order elastic constants. Then, nonlinear terms of strain components are considered in a compatible and systematic manner. By following Mindlins procedure in deriving the two-dimensional equations, nonlinear Mindlin plate equations for large deformation are obtained. The equations take the familiar form of Mindlin plate theory except the inclusion of nonlinear terms in the strain tensor.

Thickness-shear modes of an elliptical, contoured at-cut quartz resonator

We study free vibrations of an elliptical crystal resonator of AT-cut quartz with an optimal ratio between the semi-major and semi-minor axes as defined by Mindlin. The resonator is contoured with a quadratic thickness variation. The scalar equation for thickness-shear modes in an AT-cut quartz plate by Tiersten and Smythe is used. Analytical solutions for the frequencies and modes to the scalar equation are obtained using a power series expansion that converges rapidly. The frequencies and modes are exact in the sense that they can satisfy the scalar differential equation and the free edge condition to any desired accuracy. They are simple and can be used conveniently for further studies on other effects on frequencies and modes of contoured resonators.

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 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.

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.