Y.-K. Yong

On the accuracy of Mindlin plate predictions for the frequency-temperature behavior of resonant modes in AT- and SC-cut quartz plates

The frequency spectra of resonant modes in AT- and SC-cut quartz plates and their frequency-temperature behavior were studied using Mindlin first- and third-order plate equations. Both straight-crested wave solutions and two-dimensional plate solutions were studied. The first-order Mindlin plate theory with shear correction factors was previously found to yield inaccurate frequency spectra of the modes in the vicinity of the fundamental thickness-shear frequency. The third-order Mindlin plate equations without correction factors, on the other hand, predict well the frequency spectrum in the same vicinity. In general, the frequency-temperature curves of the fundamental thickness-shear obtained from the first-order Mindlin plate theory are sufficiently different from those of the third-order Mindlin plate theory that they raise concerns. The least accurately predicted mode of vibration is the flexure mode, which results in discrepancies in its frequency-temperature behavior. The accuracy of other modes of vibrations depends on the degree of couplings with the flexure mode. Mindlin first-order plate theory with only the shear correction factors is not sufficiently accurate for high frequency crystal vibrations at the fundamental thickness-shear frequency. Comparison with measured resonant frequencies and frequency-temperature results on an AT-cut quartz plate shows that the third-order plate theory is more accurate than the first-order plate theory; this is especially true for the technically important fundamental thickness shear mode in the AT-cut quartz plate.

A perturbation method for finite element modeling of piezoelectric vibrations in quartz plate resonators

When the piezoelectric stiffening matrix is added to the mechanical stiffness matrix of a finite element model, its sparse matrix structure is destroyed. A direct consequence of this loss in sparseness is a significant rise in memory and computational time requirements for the model. For weakly coupled piezoelectric materials, the matrix sparseness can be retained by a perturbation method which separates the mechanical eigenvalue solution from its piezoelectric effects. Using this approach, a perturbation and finite element scheme for weakly coupled piezoelectric vibrations in quartz plate resonators has been developed. Finite-element matrix equations were derived specifically for third-overtone thickness-shear, SC-cut quartz plate resonators with electrode platings. High-frequency piezoelectric plate equations were employed in the formulation of the finite element matrix equation. Results from the perturbation method compared well with the direct solution of the piezoelectric finite element equations. This method will result in significant savings in the computer memory and computational time. Resonance and antiresonance frequencies of a certain mode could be calculated easily by using the same eigenpair from the purely mechanical stiffness matrix. Numerical results for straight crested waves in a third overtone SC-cut quartz strip with and without electrodes are presented. The steady-state response to an electrical excitation is calculated.<<ETX>>

Mass-frequency influence surface, mode shapes, and frequency spectrum of a rectangular AT-cut quartz plate

The mass-frequency influence surface and frequency spectrum of a rectangular AT-cut quartz plate are studied. The mass-frequency influence surface is defined as a surface giving the frequency change due to a small localized mass applied on the plate surface. Finite-element solutions of R.D. Mindlins (1963) two-dimensional plate equations for thickness-shear, thickness-twist, and flexural vibrations are given. Spectrum splicing, and an efficient eigenvalue solver using the C. Lanczos (1950) algorithm are incorporated into the finite-element program. A convergence study of the fundamental thickness-shear mode and its first symmetric, anharmonic overtone is performed for finite-element meshes of increasing fineness. As a general rule, more than two elements must span any half-wave in the plate or spurious mode shapes will be obtained. Two-dimensional (2D) mode shapes and frequency spectrum of a rectangular AT-cut plate in the region of the fundamental thickness-shear frequency are presented. The mass-frequency influence surface for a 5-MHz rectangular, AT-cut plate with patch electrodes is obtained by calculating the frequency change due to a small mass layer moving over the plate surface. The frequency change is proportional to the ratio of mass loading to mass of plate per unit area and is confined mostly within the electrode area, where the magnitude is on the order 10/sup 8/ Hz/g.<<ETX>>

A laminated plate theory for high frequency, piezoelectric thin‐film resonators

A high frequency, piezoelectric, laminated plate theory is developed and presented for the purpose of modeling and analyzing piezoelectric thin‐film resonators and filters. The laminated plate equations are extensions of anisotropic composite plate theories to include piezoelectric effects and capabilities for modeling harmonic overtones of thickness‐shear vibrations. Two‐dimensional equations of motion for piezoelectric laminates were deduced from the three‐dimensional equations of linear piezoelectricity by expanding the mechanical displacements and electric potential in a series of trigonometric function, and obtaining stress resultants by integrating through the plate thickness. Relations for handling the mechanical and electrical effects of platings on the top and bottom surfaces of the laminate are derived. A new matrix method of correcting the cutoff frequencies is presented. This matrix method could also be used to efficiently correct the cutoff frequencies of any nth order plate laminate theories...

Modeling resonator frequency fluctuations induced by adsorbing and desorbing surface molecules

Resonator frequency fluctuations due to adsorption and desorption of molecules on plate electrodes are studied using the principle of mass-loading effects of adsorbed molecules. The study is based on a 525 MHz, AT-cut quartz resonator enclosed in a small crystal holder. Equations relating the surface adsorption rates of the crystal holder to pressure were derived and found to be quadratic polynomial functions of the adsorption rates. Calculations based on these equations show that a contaminant gas with a higher desorption energy creates larger changes in pressure when the temperature is varied. The function describing the frequency fluctuations due to any one contaminant site is a continuous-time Markov chain. Kolmogorov equations and an autocorrelation function for the Markov chain are derived. The autocorrelation and spectral density function of resonator frequency fluctuations are derived. The spectral density of frequency fluctuations at 1 Hz is studied as a function of pressure, temperature, and desorption energy of molecules. The noise levels for a contaminant gas with one type of molecules are found to be lower for lower desorption energies, and higher at lower pressures.<<ETX>>

Resonator surface contamination-a cause of frequency fluctuations?

The mass loading effects of adsorbing and desorbing contaminant molecules on the magnitude and characteristics of frequency fluctuations in a thickness-shear resonator are studied. The study is motivated by the observation that the frequency of a thickness-shear resonator is determined predominantly by such mechanical parameters as the thickness of the resonator, elastic stiffnesses, mass loading of the electrodes, and energy trapping. An equation was derived relating the spectral density of frequency fluctuations to: (1) rates of adsorption and desorption of one species of contaminant molecules; (2) mass per unit area of a monolayer of molecules: (3) frequency constant; (4) thickness of resonator; and (5) number of molecular sites on one resonator surface. The induced phase noises were found to be significant in very-high-frequency resonators and are not simple functions of the percentage of area contaminated. The spectral density of frequency fluctuations was inversely proportional to the fourth power of the thickness if other parameters were held constant.<<ETX>>The mass loading effects of adsorbing and desorbing contaminant molecules on the magnitude and characteristics of frequency fluctuations are studied in a thickness shear resonator. An equation was derived relating the spectral density of frequency fluctuations to: rates of adsorption and desorption of one species of contaminant molecules; mass per unit area of a monolayer of molecules; frequency constant; thickness of resonator; and number of molecular sites on one resonator surface. The induced phase noises were found to be significant in very high frequency resonators and are not simple functions of the percent area contaminated. The spectral density of frequency fluctuations was inversely proportional to the fourth power of the thickness if other parameters were held constant. Since the resonator frequency is inversely proportional to the thickness, the spectral density is, in effect, proportional to the fourth power of resonator frequency.<<ETX>>

Numerical algorithms and results for SC-cut quartz plates vibrating at the third harmonic overtone of thickness shear

Finite element matrix equations, derived from two-dimensional piezoelectric high frequency plate theory are solved to study the vibrational behavior of the third overtone of thickness shear in square and circular SC-cut quartz resonators. The mass-loading and electric effects of electrodes are included. A perturbation method which reduces the memory requirements and computational time significantly is employed to calculate the piezoelectric resonant frequencies. A new storage scheme is introduced which reduces memory requirements for mass matrix by about 90% over that of the envelope storage scheme. Substructure techniques are used in eigenvalue calculation to save storage. Resonant frequency and the mode shapes of the harmonic third overtone thickness shear vibrations for square and circular plates are calculated. A predominant third overtone thickness shear displacement, coupled with the third overtone of thickness stretch and thickness twist, is observed. Weak coupling between the third order thickness shear displacement and the zeroth-, first-, and second-order displacements is noted. The magnitudes of the lower order displacements are found to be about two orders smaller than that of the third overtone thickness shear displacement.<<ETX>>

A multi-mesh, preconditioned conjugate gradient solver for eigenvalue problems in finite element models

A multi-mesh, preconditioned conjugate gradient solver is proposed to solve large finite element eigenvalue problems in engineering applications. A generalized eigenvalue problem is solved by a conjugate gradient iteration method with a preconditioner matrix which is a partially factorized stiffness matrix. Initial trial eigenvectors for the proposed solver are obtained by interpolation using the eigenvectors obtained from a coarser mesh with a much smaller number of degrees of freedom. The employment of these trial eigenpairs was found to significantly increase the rate of convergence of the solver and also to prevent slow convergence/convergence failure in problems with closely spaced eigenvalues and repeated eigenvalues. Hence, the proposed solver presents a significant performance improvement over the existing preconditioned conjugate gradient method. Finite element eigenvalue problems for plates and shells are evaluated in this study and the proposed methods are shown to provide savings in both memory and computational time for large size problems. In the examples conducted, an eigenvalue problem for a square plate with about 50,000 degrees of freedom was solved two times faster than the direct (Lanczos) method, along with a memory storage requirement that is about 12 times smaller. However, very thin plate examples (length to thickness ratio > 150) show slow convergence and sometimes convergence failure. The convergence failure for very thin plates can be controlled by allowing more fill-ins in the incomplete Cholesky factorization of the preconditioner matrix.

NUMERICAL ALGORITHMS FOR SOLUTIONS OF LARGE EIGENVALUE PROBLEMS IN PIEZOELECTRIC RESONATORS

Two algorithms for eigenvalue problems in piezoelectric finite element analyses are introduced. The first algorithm involves the use of Lanczos method with a new matrix storage scheme, while the second algorithm uses a Rayleigh quotient iteration scheme. In both solution methods, schemes are implemented to reduce storage requirements and solution time. Both solution methods also seek to preserve the sparsity structure of the stiffness matrix to realize major savings in memory. In the Lanczos method with the new storage scheme, the bandwidth of the stiffness matrix is optimized by mixing the electrical degree of freedom with the mechanical degrees of freedom. The unique structural pattern of the consistent mass matrix is exploited to reduce storage requirements. These major reductions in memory requirements for both the stiffness and mass matrices also provided large savings in computational time. In the Rayleigh quotient iteration method, an algorithm for generating good initial eigenpairs is employed to improve its overall convergence rate, and its convergence stability in the regions of closely spaced eigenvalues and repeated eigenvalues. The initial eigenvectors are obtained by interpolation from a coarse mesh. In order for this multi-mesh iterative method to be effective, an eigenvector of interest in the fine mesh must resemble an eigenvector in the coarse mesh. Hence, the method is effective for finding the set of eigenpairs in the low-frequency range, while the Lanczos method with a mixed electromechanical matrix can be used for any frequency range. Results of example problems are presented to show the savings in solution time and storage requirements of the proposed algorithms when compared with the existing algorithms in the literature.

Accuracy of crystal plate theories for high-frequency vibrations in the range of the fundamental thickness shear mode

The first-order plate theories with correction factors are generally assumed to predict accurately the plate modes which have half wavelengths greater than the plate thickness, and at frequencies up to 20% higher than the fundamental thickness shear frequency. This assumption is assessed by comparing the straight crested wave solutions of the plate theories with those of the three-dimensional elastic equations of motion. The frequency spectra for bandwidths of resonant frequencies versus the aspect ratio of length to thickness of plate are compared for three sets of plate equations: the first-order Mindlin plate equations, the third-order Mindlin plate equations, and the third-order Lee and Nikodem plate equations. The finite element results for a quartz SC-cut strip with free edges show that Mindlins first-order plate equations, and Lee and Nikodems third-order plate equations yield less accurate frequency spectra of the modes in the vicinity of the fundamental thickness shear mode than the third-order Mindlin plate equations without correction factors. The degree of inaccuracy increases with the ratio of plate length to thickness, and the slope of the modal branches in the frequency spectra. For a plate length to thickness ratio of 31 to 33, the first-order plate theory is found to yield accurate frequency spectra for normalized frequencies less than 0.3, which is lower than previously assumed. At normalized frequencies greater than 0.3, deviations are seen in the frequency spectra, starting with the modal branches which are more steeply inclined.