Vibrations Of Quartz Crystal Plates Research Articles

The Integrated Analysis of Vibrations of Quartz Crystal Plates through Artificial Coupling Factors

We introduce smaller artificial factors into the elastic constants matrix and manage to make Mindlin first-order plate theoryequations of motions coupled for a uniform and integrated analysis. The energy distributions of the five coupled modes are obtained and all the five vibration modes are identified through the energy calculation. This analytical approach based on artificial couplings of vibration modes suggests that all vibration modes of structural components can be analyzed through the same procedure and computer code if the right elastic constants are modified and the mode identification can be done with the energy method. This is a new technique to study multimode vibrations of structures in a broad frequency range with just one procedure and calculation tool for simplification.

Examinations of vibration frequency and mode shape variations of quartz crystal plates in a thermal field with strain and kinetic energies

The frequency-temperature behavior has long been analyzed to ensure the smoothness of functioning frequency of a quartz crystal resonator during the temperature variation and maintain its operations in a stable state. It has been proven that vibrations of quartz crystal plates in the thickness-shear mode can be well described by the theory of incremental thermal field in conjunction with the Mindlin plate equations, demonstrating the possibility that vibration frequencies and mode shapes can also be well predicted with the temperature as a targeted variable. By studying the vibrations with temperature changes in a given range, the frequencies and vibration modes can be calculated as a function of temperature, then they can be used in the calculations of both strain and kinetic energies for vibration mode identification or characterization based on energy proportion and changes. The trends of changes of vibration frequency, amplitudes, and energy with temperature variation make it possible for characterization the variations of vibrations and particularly the weakening or enhancement of the dominance of specific vibration modes. With such relations between vibration properties and temperature known, possible analytical tools can be developed to aid the optimal design by avoiding significant mode changes or occurrences of mode conversion.

Effects of Elliptical Ring Electrodes on Shear Vibrations of Quartz Crystal Plates

A theoretical analysis is performed on the thickness-shear vibrations of an AT-cut quartz piezoelectric crystal plate with elliptical ring electrodes. The scalar differential equation by Tiersten and Smythe is used. An analytical solution is obtained. Numerical results from the solution show that the thickness-shear mode of interest may be trapped by the ring electrodes and can have a convex, concave, or nearly flat vibration distribution near the plate center, which is fundamentally important when the plate is used as an acoustic wave mass sensor. The vibration distribution is found to be sensitive to both the geometric and physical parameters of the electrodes. Therefore, a careful design is needed to realize the desired trapped mode with suitable center convexity for sensor application

An analysis of nonlinear thickness-shear vibrations of quartz crystal plates using the two-dimensional finite element method

ABSTRACTA nonlinear analysis of high-frequency thickness-shear vibrations of AT-cut quartz crystal plates is presented with the two-dimensional finite element method. The Mindlin plate equations are truncated to the first-order ones as an approximation, and then they are used for the formulation of nonlinear finite element analysis with all zero- and first-order displacements. The matrix equation of motion is established with the first-order harmonic approximation, and the generalized nonlinear eigensystem is solved by a direct iterative procedure. A displacement amplitude versus frequency curve and corresponding mode shapes are obtained and examined. The nonlinear finite element program is developed based on the earlier linear edition and can be utilized to predict nonlinear characteristics of miniaturized quartz crystal resonators in the design process.

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.

The non-linear thickness-shear vibrations of quartz crystal plates under an electric field

With the consideration of material and kinematic non-linearities, a non-linear system of two-dimensional equations for the strongly coupled thickness-shear and flexural vibrations of electroelastic plates is established by expanding mechanical displacements and electric potential into power series in the plate thickness coordinate and integrating over the thickness. Since the non-linear equations are too complicated to be solved directly by known methods, we utilized the Galerkin approximation to convert the non-linear equation of thickness-shear vibrations into an ordinary differential equation depending only on time by assuming the mode shape of linear vibrations. This non-linear forced vibration equation has been solved by the successive approximation method and we plotted frequency–response curves with different amplitude ratios and electrical voltages. Numerical results showed that the electric field has a more significant effect on vibration frequency compared with other known factors.

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.

An analysis of nonlinear vibrations of coupled thickness-shear and flexural modes of quartz crystal plates with the homotopy analysis method

We investigated the nonlinear vibrations of the coupled thickness-shear and flexural modes of quartz crystal plates with the nonlinear Mindlin plate equations, taking into consideration the kinematic and material nonlinearities. The nonlinear Mindlin plate equations for strongly coupled thickness- shear and flexural modes have been established by following Mindlin with the nonlinear constitutive relations and approximation procedures. Based on the long thickness-shear wave approximation and aided by corresponding linear solutions, the nonlinear equation of thickness-shear vibrations of quartz crystal plate has been solved by the combination of the Galerkin and homotopy analysis methods. The amplitude frequency relation we obtained showed that the nonlinear frequency of thickness-shear vibrations depends on the vibration amplitude, thickness, and length of plate, which is significantly different from the linear case. Numerical results from this study also indicated that neither kinematic nor material nonlinearities are the main factors in frequency shifts and performance fluctuation of the quartz crystal resonators we have observed. These efforts will result in applicable solution techniques for further studies of nonlinear effects of quartz plates under bias fields for the precise analysis and design of quartz crystal resonators.

Resonant frequency function of thickness-shear vibrations of rectangular crystal plates

The resonant frequencies of thickness-shear vibrations of quartz crystal plates in rectangular and circular shapes are always required in the design and manufacturing of quartz crystal resonators. As the size of quartz crystal resonators shrinks, for rectangular plates we must consider effects of both length and width for the precise calculation of resonant frequency. Starting from the three-dimensional equations of wave propagation in finite crystal plates and the general expression of vibration modes, we obtained the relations between frequency and wavenumbers. By satisfying the major boundary conditions of the dominant thickness-shear mode, three wavenumber solutions are obtained and the frequency equation is constructed. It is shown the resonant frequency of thickness-shear mode is a second-order polynomial of aspect ratios. This conforms to known results in the simplest form and is applicable to further analytical and experimental studies of the frequency equation of quartz crystal resonators.

Mechanical effects of electrodes on the vibrations of quartz crystal plates.

A system of approximate first-order equations is extracted from an infinite system of 2-D equations for piezoelectric crystal plates with thickness-graded material properties, which is deduced from the 3-D equations of linear piezoelectricity. These equations are used to study mechanical effects on the thickness-shear (TS), flexural (F), and face-shear (FS) vibrations of an AT-cut quartz plated with two identical electrodes. Dispersion curves are calculated from the present 2-D equations as well as the 3-D equations. The comparison of these curves shows that the agreement is very close for all three frequency branches of TS, F, and FS modes in a range up to the 1.5 times the fundamental TS frequency and for gold and aluminum electrodes with R, the ratio of the mass of the electrodes to that of the plate, equal to 0.05. without introducing any correction factors. In order to assess electrode effects, spectra of omega vs. a/bq (length-to-thickness ratio of the quartz) are computed for plates with gold and aluminum electrodes and different R ratios. And the spectrum of omega vs. R is computed for plates with aluminum electrodes and a given a/bq ratio. For a plate with gold electrodes, the frequencies of predominant TS, F, and FS modes are decreasing as R increases, but the amount of frequency changes for the TS mode is much greater than those for the other two modes. However, for a plate with aluminum electrodes, the frequencies of the TS and FS modes are decreasing, but those of the F modes are increasing as R increases.