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With large accelerations, the mechanical vibrations of device structures can be driven to nonlinear states yielding strong variations of vibration characteristics such as frequency change and modes...

Mode-coupled vibrations in a thickness-shear (TSh) mode and laterally finite film bulk acoustic resonator (FBAR) with one face in contact with Newtonian (linearly viscous and compressional) liquid are investigated. With boundary conditions and interface continuity conditions, exact dispersion curves in FBAR sensors contacting with two kinds of liquids are obtained, and they are compared with the dispersion curves in a bare sensor without liquid contact. Frequency spectra, describing mode couplings between the main TSh modal branch and undesirable modal branches, are calculated by employing weak boundary conditions at lateral free edges constructed based on the variational principle. Mode shapes of mechanical displacements in both the sensor and liquid layer are presented, and mode transformations are observed due to the liquid contact and lateral edge effect. The effect of liquid thickness on frequency spectra is also studied. Numerical results reveal that the generation of shear wave in the liquid layer results in the shifts of spectrum curves along the frequency axis and hence it is the main factor of frequency shifts of FBAR sensors. The compressional wave causes the shifts of spectrum curves along the lateral aspect ratio axis. Then for a given FBAR sensor, the liquid thickness changes could also cause frequency shifts. Therefore, desirable vibration modes should be chosen based on the frequency spectra to avoid strong mode couplings and to eliminate frequency shifts induced by the liquid thickness changes in real applications.

Mode-coupled vibrations in an ultra-high frequency (UHF) ZnO thin film bulk acoustic resonator (FBAR) operating at thickness-extensional (TE) mode are studied by employing weak boundary conditions (WBCs), constructed based on Saint-Venant’s principle and mixed variational principle in the piezoelectric theory. The frequency spectra, describing the lateral size-dependence of mode couplings between the main mode (TE) and undesirable eigen-modes, for clamped lateral edges are compared with the existing frequency spectra for free lateral edges to illustrate the boundary influence. The displacement and stress variations in FBAR volume are also presented to intuitionally understand and distinguish the difference of frequency spectra between these two different lateral edges, and then we discuss how to select outstanding lateral sizes to weaken the mounting effect. The frequency spectra predicted from our approximate WBCs are also compared with and agree well with those predicted by the finite element method (FEM) using COMSOL, which proves the correctness and accuracy of our theoretical method. These results indicate that the WBCs could have potentials in the valid predictions of lateral size-dependence of mode couplings in piezoelectric acoustic wave devices.

Forced mode coupling vibration of ZnO thin film resonator with its c-axis along thickness direction operating at thickness-extensional mode is studied in this paper. The zone dispersion curve is obtained since the mechanical damping of ZnO thin film is considered in this case. The displacement solutions are constructed based on the zone dispersion curve and the Mason model solution, which are then substituted into weak boundary conditions at the left and right edges determining the undetermined constants. The effect of lateral electrode size on suppressing spurious modes is particularly studied. The results show that with lateral electrode size increasing, more spurious modes occurs in the band between fs and fp but their amplitude strength become weaker. Therefore, in realistic resonators a larger lateral boundary is more desirable and then the band between fs and fp would become smoother.

The thickness-extensional mode in layered piezoelectric plates can be inevitably interfered by some undesirable eigen-modes due to the lateral edge effect. A reported theoretical method to predict high-frequency coupled vibrations is the 2D high-order plate theory derived based on approximate dispersion relations. In this paper, based on accurate dispersion relations a novel approach called Frequency Spectrum Quantitative Prediction (FSQP) is developed to quantitatively investigate coupled vibration behaviors in piezoelectric multilayered plates operating at ultra-high frequency. Two significant sub-goals need to be achieved: one is the accurate dispersion relations of the layered structure; the other is the variational formulation for the layered plates with piezoelectric and/or elastic phases. Lastly, the objective equation, i.e., frequency spectra, describing coupling strengths between the thickness-extensional mode and unwanted eigen-modes, is derived. A reported numerical example of a piezoelectric thin-film resonator is considered to demonstrate the correctness and superiority of the proposed methodology. Mode shapes of mechanical displacements are investigated in detail to illustrate the application of frequency spectra to suppress the undesirable eigen-modes. Numerical results show that the proposed approach FSQP is efficient and more accurate than the existing 2D high-order plate theory in quantitative predictions of high-frequency coupled vibrations in layered piezoelectric plates.

Incremental piezoelectric equations for small vibrations superposed on initial deformations are presented. The equations are implemented in COMSOL finite element models (FEA). Equations are validated by comparing the results for the force sensitivity coefficient Kf of a circular quartz plate subjected to a pair of diametrical forces with measured data. The model results show a consistent trend with the experimental results, and the relative difference between our FEA results and Ballato's measured result is about 13%. A detailed study of the acceleration sensitivity of a rectangular AT-cut quartz plate is presented. The plate resonator is fixed along one edge as a cantilever. For AT-cut quartz resonators with the crystal digonal X-axis perpendicular to plate X-axis, the in-plane acceleration sensitivity is found to be negligible compared with the out-of-plane (Y-axis) acceleration sensitivity. For AT-cut quartz resonators with the crystal digonal X-axis parallel to plate X-axis, the Y-axis acceleration sensitivity is found to be rectified, that is the fractional change in frequency is positive with respect to both positive and negative Y-axis accelerations. The Y-axis acceleration sensitivity is small in comparison with the in-plane acceleration sensitivity for small body forces. However, for large body forces, the Y-axis acceleration sensitivity dominates because it increases nonlinearly with the Y-axis acceleration. The resonator rectified acceleration sensitivity is confirmed by phase noise measurements. For reduced acceleration sensitivity, two pairs of electrodes along the plate edges reduce the bending of the plate resonator and subsequently reduce acceleration sensitivity. We present a new method using these edge electrodes in which a dc bias field is employed to control the resonant frequency of resonator subjected to g body forces. A dc bias field with an appropriate dc bias voltage could potentially yield a reduction of acceleration sensitivity in Y-axis direction of about two orders of magnitude.

The langasite crystal has the capability of no phase change at temperatures exceeding 600°C for high temperature applications. Since the material properties of langasite were measured usually at 25 0C as the reference temperature they are no longer accurate at high temperature environments. We employ a Lagrangian formulation to investigate the temperature behavior of thickness vibrations of langasite in a wide temperature range from 25°C to 600°C. The analysis only requires the elastic constants and their effective temperature derivatives of elastic constants and thermal expansion coefficients to predict the material constants at any given temperature. Based on the new values of the elastic constants and thermal expansion coefficients, the zero-valued temperature coefficients of frequency are investigated. The presently identified temperature compensated crystal cuts of the thickness-shear modes of langasite at different temperatures were compared with those at reference temperature and also with some cuts found in the literature.

The frequency–temperature characteristics of quartz crystal resonators, particularly the frequency stability in a specific temperature range in which the vibration modes are strongly coupled, has been an important requirement in most applications. The analytical work on frequency–temperature relations has been done over the last decades in many aspects, ranging from the fundamental theory for the thermal effect on vibrations of elastic solids to simplified plate equations of a few strongly coupled vibration modes. However, it has been clearly observed that due to complications of the resonator structures such as the presence of a mounting structure and electrodes, simple and analytical solutions will not be able to consider all the factors which will have inevitable and noticeable effects. In this paper, we incorporate the frequency–temperature theory for crystal plates based on the incremental thermal field formulation by Lee and Yong into our finite element analysis implementation, which is then used to analyze the free vibrations of crystal plates with the higher-order Mindlin plate theory. The effect of electrodes on the frequency–temperature relation is also considered. The computational results are compared with experimental ones from actual products. The satisfactory agreement demonstrates the precise prediction of the frequency–temperature behavior and practical applications of the finite element analysis in product modeling and development.

A novel design method for high Q piezoelectric resonators was presented and proposed using the 3-D equations of linear piezoelectricity with quasi-electrostatic approximation which include losses attributed to mechanical damping in solid and resistance in current conduction. There is currently no finite element software for estimating the Q of a resonator without apriori assumptions of the resonator im- pedance or damping. There is a necessity for better and more realistic modeling of resonators and filters due to miniaturization and the rapid advances in frequency ranges in telecommunication. We presented new three-dimensional finite element models of quartz and barium titanate resonators with mechanical damping and resistance in current conduction. Lee, Liu and Ballato's 3-D equations of linear piezoelectricity with quasi-electro- static approximation which include losses attributed to mechanical damping in solid and resistance in current conduction were formulated in a weak form and implemented in COMSOL. The resulting finite element model could predict the Q and other electrical parameters for any piezoelectric resonator without apriori as- sumptions of damping or resistance. Forced and free vibration analyses were per- formed and the results for the Q and other electrical parameters were obtained. Comparisons of the Q and other electrical parameters obtained from the free vibra- tion analysis with their corresponding values from the forced vibration analysis were found to be in excellent agreement. Hence, the frequency spectra obtained from the free vibration analysis could be used for designing high Q resonators. Results for quartz thickness shear AT-cut and SC-cut resonators and thickness stretch poled barium titanate resonators were presented. An unexpected benefit of the model was the prediction of resonator Q with energy losses via the mounting supports.

In this paper, we introduce a novel design length extension vibratory gyroscope to detect the angular velocity rotation about z-axis. The proposed gyroscope is a new type of a gyroscope which utilizes a length extension mode as a driving mode and a flexure mode as a sensing mode to detect the Coriolis force generated by the rotation of the system. The gyroscope was designed and gyro-characteristics were simulated using COMSOL, finite element method (FEM) software. Quartz and langatate crystals are used for gyroscopes and compared. The driving frequencies and sensing frequencies of each gyroscope are obtained by optimizing the geometries of the each gyroscope using eigenfrequency analyses. Frequency response analyses were performed to simulate the gyro-characteristics of the gyroscopes which subjected to the angular velocity about z-axis. The results show that the length extension gyroscope can be used as a gyro-sensor. Moreover, we find that langatate crystals are more suitable materials for higher precision piezoelectric gyro-sensors than quartz crystal.

The Mindlin plate equations with the consideration of thickness-shear deformation as an independent variable have been used for the analysis of vibrations of quartz crystal resonators of both rectangular and circular types. The Mindlin or Lee plate theories that treat thickness-shear deformation as an independent higher-order vibration mode in a coupled system of two-dimensional variables are the choice of theory for analysis. For circular plates, we derived the Mindlin plate equations in a systematic manner as demonstrated by Mindlin and others and obtained the truncated two-dimensional equations of closely coupled modes in polar coordinates. We simplified the equations for vibration modes in the vicinity of fundamental thickness-shear frequency and validated the equations and method. To explore newer structures of quartz crystal resonators, we utilized the Mindlin plate equations for the analysis of annular plates with fixed inner and free outer edges for frequency spectra. The detailed analysis of vibrations of circular plates for the normalized frequency versus dimensional parameters provide references for optimal selection of parameters based on the principle of strong thickness-shear mode and minimal presence of other modes to enhance energy trapping through maintaining the strong and pure thickness-shear vibrations insensitive to some complication factors such as thermal and initial stresses., Comment: Paper to be presented to the 2015 IEEE International Frequency Control Symposium and European Frequency and Time Forum, Denver, CO, USA. April 12-16, 2015

The Mindlin plate theory was developed to provide accurate solutions of vibrations in the vicinity of the fundamental thickness-shear mode, which has a very high frequency compared to flexural vibrations. The most important application of the theory is the high frequency vibrations of crystal plates although it has been applied to many problems beyond the original purpose. Recent studies found that, to improve the frequency solutions for plates with larger aspect ratios, the third-order plate based on Mindlin's power series expansion has to be used. It was shown through comparisons with three-dimensional elasticity solutions that the fundamental thickness-shear frequency is almost exact. The third-order theory was applied to frequency, mode shape, and other related analyses. In this study, we reconfirm that the third-order plate theory is very accurate because it has an almost exact cut-off frequency for the fundamental thickness-shear mode. By adopting a procedure developed by Mindlin, we find the inaccuracies in cut-off frequencies of the fundamental thickness modes and their overtones can be improved through the introduction of new correction factors. Corrections can be made with either the natural or symmetric procedure. Correction factors for natural and symmetric procedures based on stresses will be given.

A finite element formulation of the piezoelectric vibrations of quartz resonators based on Mindlin plate theory is derived. The higher-order plate theory is employed for the development of a collection of successively higher-order plate elements which can be effective for a broad frequency range including the fundamental and overtone modes of thickness-shear vibrations. The presence of electrodes is also considered for their mechanical effects. The mechanical displacements and electric potential are combined into a generalized displacement field, and the subsequent derivations are carried out with all the generalized equations. Through the standard finite element procedure, the vibration frequency, the vibration mode shapes and the electric potential distribution are obtained. The frequency spectra are compared with some well-known experimental results with good agreement. Our previous experience with finite element analysis of high-frequency quartz plate vibrations leads us to believe that memory and computing time will always remain as key issues despite the advances in computers. Hence, the use of sparse matrix techniques, efficient eigenvalue solvers, and other reduction procedures are explored.

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.

Finite element matrix equations, derived from two-dimensional, piezoelectric high frequency plate theory proposed by Lee, Syngellakis, and Hou (1987), 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 is employed to calculate the piezoelectric resonant frequencies which reduces the memory requirements and computational time significantly. 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 rectangular and circular plates are calculated. A pure third overtone thickness shear displacement, coupled with the third overtone of thickness stretch and thickness twist, is observed. Weak coupling between the zeroth, first and second order displacements and the third order thickness shear displacement 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

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...

A novel analytical/numerical method for calculating the resonator Q, and its equivalent electrical parameters due to viscoelastic, conductivity and mounting supports losses was presented. The method presented will be quite useful for designing new resonators, and reducing their time and costs of prototyping. There was also a necessity for better and more realistic modeling of the resonators due to miniaturizations, and the rapid advances in the frequency ranges of telecommunication. We present new three-dimensional finite elements models of quartz resonators with viscoelasticity, conductivity, and mounting support losses. For quartz the materials losses attributed to electrical conductivity and acoustic viscosity were obtained from Lee, Liu and Ballato, and Lamb and Richter, respectively. The losses at the mounting supports were modeled by perfectly matched layers (PML's). The theory for dissipative anisotropic piezoelectric solids given by Lee, Liu and Ballato was formulated in a weak form for finite element applications. PML's were placed at the base of the mounting supports to simulate the energy losses to a semi-infinite base substrate. FE simulations were carried out for free vibrations and forced vibrations of quartz tuning fork and AT-cut resonators. Results for quartz tuning fork and thickness shear AT-cut resonators were presented and compared with experimental data. Results for the resonator Q and the equivalent electrical parameters were compared with their measured values. Good comparisons were found. Results for both low and high Q AT-cut quartz resonators compared well with their experimental values. A method for estimating the Q directly from the frequency spectrum obtained for free vibrations was also presented. An important determinant of the quality factor Q of a quartz resonator is the loss of energy from the electrode area to the base via the mountings. The acoustical characteristics of the plate resonator are changed when the plate is mounted onto a base substrate. The base affects the frequency spectra of the plate resonator. A resonator with a high Q may not have a similarly high Q when mounted on a base. Hence, the base is an energy sink and the Q will be affected by the shape and size of this base. A lower bound Q will be obtained if the base is a semi-infinite base since it will absorb all acoustical energies radiated from the resonator.

Finite-element solutions for the fundamental thickness shear mode and the second-anharmonic overtone of a circular, 1.87-MHz AT-cut quartz plate with no electrodes are presented and compared with previously obtained results for a rectangular plate of similar properties. The edge flexural mode in circular plates, a vibration mode not seen in the rectangular plate is also presented. A 5-MHz circular and electroded AT-cut quartz plate is studied. A portion of the frequency spectrum is constructed in the neighborhood of the fundamental thickness-shear mode. A convergence study is also presented for the electroded 5-MHz plate. A new two-dimensional (2-D) technique for visualizing the vibration mode solutions is presented. This method departs substantially from the three-dimensional (3-D) 'wire-frame' plots presented in the previous analysis. The 2-D images can be manipulated to produce nodal line diagrams and can be color coded to illustrate mode shapes and energy trapping phenomenon. A contour plot of the mass-frequency influence surface for the plated 5-MHz resonator is presented. The mass-frequency influence surface is defined as a surface giving the frequency change due to a small localized mass applied to the resonator surface. >

Equations for the calculation of the equivalent circuit parameters of the piezoelectric resonator are presented. The results from both eigen-frequency and frequency response analyses could be used for calculating the electrical parameters of the Butterworth Van Dyke model of the piezoelectric resonator. The values of the electrical parameters C 1 , and L 1 obtained from both types of analyses were found to be similar, and were found to compare reasonably well with the measured values of two 12 MHz AT-cut quartz resonators. The values of R 1 , and Q were less well predicted when the resonator has a lower Q. The static capacitance C 0 could be calculated accurately by its frequency response at 1 Hz. The eigen-frequency analysis is better suited than the frequency response analysis for the study of electrode and plate geometry designs, hence the eigen-frequency equations for calculating the equivalent circuit parameters are more useful for parametric designs.

The finite element analysis of quartz crystal resonators has been gradually adopted in the design and product improvement for the advantage in predicting the vibration frequency, energy trapping, and calculation of device properties. The analysis can be done with 3D theory of elasticity or piezoelectricity or the Mindlin plate theory by general purpose software or custom development, which has the advantage of reducing the size of analysis significantly for typical thickness-shear vibrations of quartz crystal plates. While linear finite element analysis are adequate for the vibration frequency and frequency-temperature relations, further analysis on important phenomena and electrical parameters require the consideration of nonlinear material properties of quartz crystal. The finite element analysis of quartz crystal resonators has been making great contribution to the design and improvement facing the fast shrinkage of resonator size and elevated precision requirements. The full advantage of the finite element analysis can be taken if electrical parameters and performance behavior can be predicted with the improved analytical model and consideration of nonlinear material properties. The current approach based on the nonlinear theory will meet these objectives since the advantage of the finite element analysis on parallel platforms have been well understood and widely implemented. Our nonlinear analysis will utilize the nonlinear Mindlin plate theory and provide important and essential details on the nonlinear behavior of quartz crystal resonators.

In this study, we analyze the thickness-shear vibration of a simple resonator model by means of complex material constants with consideration of the electrode inertia. The effects of viscosity on frequency behavior of piezoelectric plate are presented and discussed. The results can be useful for design of the relevant resonators.

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.

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. Mindlin's (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. >

The accuracy of the finite element analysis for thickness shear quartz resonators is a function of the mesh resolution; the finer the mesh resolution, the more accurate the finite element solution. A certain minimum number of elements are required in each direction for the solution to converge. This places a high demand on memory for computation, and often the available memory is insufficient. Typically the thickness of the electrode films is very small compared with the thickness of the resonator itself; as a result, electrode elements have very poor aspect ratios, and this is detrimental to the accuracy of the result. In this paper, we propose special methods to model the electrodes at the crystal interface of an AT cut crystal. This reduces the overall problem size and eliminates electrode elements having poor aspect ratios. First, experimental data are presented to demonstrate the effects of electrode film boundary conditions on the frequency-temperature curves of an AT cut plate. Finite element analysis is performed on a mesh representing the resonator, and the results are compared for testing the accuracy of the analysis itself and thus validating the results of analysis. Approximations such as lumping and Guyan reduction are then used to model the electrode thin films at the electrode interface and their results are studied. In addition, a new approximation called merging is proposed to model electrodes at the electrode interface.

The feasibility of a new angular velocity sensor using the temperature stable AT-cut quartz was presented. The sensor consisted of one pair of electrodes for driving the fundamental thickness shear mode, and another pair of electrodes for sensing the angular velocity. Two tines extended outward from resonator, and in the plane of the plate. The tines were designed to be sensitive to angular velocity of the resonator. The Coriolis body force caused by the cross product of the angular velocity with the linear momentum of the vibrating tines changes the their mode shapes that in turn perturbed the thickness shear mode, and changed the voltage at the sensing electrodes. A vibratory gyroscope with a trapped energy thickness mode as the main driving mode offers good improvements in terms of frequency stability and less dependence on the mounting supports and lead installation of the sensing element. Since the AT-cut resonators were known to have good f-T curves and long term aging, such a gyroscope would have the advantages of a stable quartz AT-cut resonator. Results for two angular velocity sensors were presented: (1) a 5 MHz sensor with a sensitivity of 5.8 mV/deg./s angular velocity about the X-axis, and (2) a 37 MHz sensor with a sensitivity of 0.38 mV/deg./s angular velocity about the Z-axis.

A novel AT-cut quartz vibratory gyroscope with a sensitivity to angular velocity of 1.3 mV/deg/s was presented. The gyroscope employed the thickness shear mode as the driving mode and the flexure mode of tines as the sensing mode. The tines are placed normally on the AT-cut quartz plate surface. These tines were tuned to resonate with the harmonic Coriolis body force caused by the angular rotation of the plate about its plate thickness axis. The flexure mode of the tines changed with the angular velocity about the thickness axis of the plate. The change in the flexure mode of the tines could be detected by the sensing electrode separated from the driving electrode by an acoustic gap. Since the AT-cut resonators are known to have good f-T curves and long term aging, the new AT-cut gyroscope would have the advantages of a highly stable quartz AT-cut resonator.

A new thickness shear mode vibratory gyroscope which employed the thickness shear mode at both the driving and sensing electrodes was presented. The thickness shear mode shape changed slightly with the Coriolis force, and this change could be detected by the sensing electrode separated from the driving electrode by an acoustic gap. Vibrating tines normal to the plate surface were placed along the acoustic gap. These tines were tuned to resonate with the harmonic Coriolis force caused by the angular rotation of the plate about its plate thickness axis. The change in vibration of the tines with the Coriolis force caused a change in charge or voltage at the sensing electrodes. Since the AT-cut resonators are known to have good f-T curves and long term aging, the new AT-cut gyroscope would have the advantages of a highly stable quartz AT-cut resonator.

In this paper, we analyze two piezoelectric crystals using this direct analysis of the CAD data and compare the results of analysis to experimental results to demonstrate the accuracy of this new method. We also compare the memory requirements of this method with the minimum memory requirements in conventional FEA to demonstrate that lesser memory is required in this method for convergence of the solutions.

A new method which accurately predicts the frequency-temperature (f-T) behavior of quartz resonators affected by transient and steady state temperature changes is presented. Most practical resonators are subjected to thermal stresses. Conventional finite element analytical tools such as ANSYS cannot provide a sufficiently accurate model for the f-T curves of quartz resonators under thermal stresses. In our previous paper [Yong, Y-K, et al., 2005], we had proposed a method which employed a superposition of two f-T curves: one due to a stress-free, homogeneous thermal strain field, and the other due to the nonhomogeneous thermal stresses. The assumptions underlying the two f-T curves were not satisfactorily consistent. This paper presents a consistent and superior method for the problem: the constitutive equations for the Lagrangean, incremental displacements (small vibrational displacements) incorporate the temperature derivatives of the material constants. The incremental equations of small vibrations superposed on initial thermal stresses and strains are solved, and there was no need for the superposition of two f-T curves. Numerical results are compared with experimental results for a 50 MHz AT-cut quartz resonator mounted on a glass package. Good comparison between the experimental results and numerical results from our new method is shown. The difference between the thermal expansion coefficients of glass and quartz give rise to the thermal stresses that have an adverse effect on the f-T curves of AT-cut resonators. Different optimal crystal cut angles of quartz, and resonator geometry were found to achieve stable frequency-temperature behavior of the resonator in a glass package

The equivalent electrical parameters C/sub 0/, C/sub 1/, L/sub 1/ and R/sub 1/ can be calculated in an eigenvalue analysis. For high Q resonators, the motional capacitance C/sub 1/ and inductance L/sub 1/ can be calculated from the short and open circuit resonant frequencies. The prediction of Q due to power loss via the mounting supports is demonstrated. For accurate prediction of the energy loss via the mounting supports it is important to provide an energy sink such as a large base. The Q estimated from an eigenvalue analysis can be used in a forced vibration model to calculate equivalent electrical parameters C/sub 0/, C/sub 1/, L/sub 1/ and R/sub 1/.

There is currently a need to miniaturize AT-cut resonators. However, the size reduction sometimes sacrifices the resonator's Q-values and crystal impedances. Hence, there is a requirement to develop new resonator designs by changing the cut angles or electrode configurations. For these purposes, the 3-D finite element method is employed as a promising design/analysis and prototyping tool for new quartz resonators. There are generally two types of analyses: Eigenvalue analysis and forced vibration analysis. The eigenvalue analysis of resonant modes in AT-cut resonators has been shown to be accurate in predicting the resonant frequencies as a function of the resonator and electrode geometry. The forced vibration (steady state) analysis is used to calculate the motional impedance and capacitance in the resonator. In this paper we propose using the Eigenvalue analysis for comparing the impedance and energy trapping of the fundamental thickness shear mode of an AT-cut resonator as a function of the resonator and electrode geometries. By comparing and calibrating the numerical results with experimental data, the eigenvalue analysis can efficiently estimate Q-values, crystal impedances, strength of activity dips and frequency- temperature stabilities. The eigenvalue analysis could further generate information useful for choosing resonator and electrode geometry that have higher dimensional tolerance in a fabrication process. The proposed method could be employed to develop a new resonator with a different cut angle and with different electrode configurations. The forced vibration analysis is more cumbersome for these types of analyses.

An analytical technique for designing high Q, thickness shear micro electromechanical, ring electroded mesa quartz resonators is proposed. The method is demonstrated using two-dimensional straight crested wave analysis. The design method is based on the two characteristics of a stable resonator: (a) The mode is energy trapped and relatively isolated from its supports, and (b) the motional impedance of the mode is low. The root mean squares of vibration displacements are employed to characterize the modes of vibration, and the thickness shear mode has a large rms u/sub 1/ displacement in the x/sub 1/ direction (diagonal axis). The rms displacement is used to compare the energy trapping of the thickness shear mode as a function of the electrode and plate geometry. For each mode of vibration, the electric flux density D/sub 2/ is calculated at the quartz to electrode interface to yield the electric current at the electrodes. Given a constant driving voltage, the magnitude of the electric current is inversely proportional to the motional impedance. Hence the electric current for a mode as a function of the electrode and plate geometry is employed as a further means for comparing the merits of different resonator designs. Results are shown for a 1 GHz inverted mesa AT-cut resonator.

Electrodes on a crystal resonator has been traditionally considered as mass addition to the crystal plate, thus resulting the neglect of their stiffness. This assumption is considered reasonable if electrodes are thin in comparison with the crystal in terms of the mass ratio, the relative mass of electrodes. For thicker electrodes of high frequency resonators, this assumption has to be reexamined for better prediction on their effects on the resonance frequency and the frequency-temperature characteristics. In this study, the electrodes are considered as layers of plates with their own stiffness and mass properties. As a result, the deformation of electrodes also involve independent ones in addition to the crystal deformation. The theory employed also enables us to consider the deformation to a higher order of degree. The layerwise plate theory is derived and implemented for the finite element method. Numerical results are compared with the other computing schemes with and without electrodes considerations.

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. >

A finite-element program incorporating R.D. Mindlin's first-order plate equations (1955) is developed. Calculations are performed for a miniature, third-overtone extensional, Z-cut resonator. Its frequency spectrum as a function of the mounting length and displacement mode shapes is discussed. The resonator motional resistances for different mounting lengths are measured. Normalized strain-energy ratios in the base area and time area are computed and compared with the normalized motional resistance at various lengths. Good correlations are found: the peaks in strain energy ratio correspond to peaks in motional resistance. Since the motional resistance is inversely proportional to the quality factor, Q, of the resonator, the strain-energy ratio in the base and tines (the supporting structure) could be used as a relative criterion for judging the Q of a certain resonator design. >

Finite element matrix equations employing high frequency, piezoelectric plate equations are derived. The equations may be used for modeling third harmonic overtone of thickness-shear vibrations. A perturbation technique is developed to account for piezoelectric stiffening in the mechanical stiffness matrix. Results from the perturbation method compare well with the direct solution of the piezoelectric finite element equations. The technique will result in significant savings in computer memory and computational time. Numerical results for straight crested waves in a third overtone SC-cut quartz strip, with and without electrodes are presented. Steady state response to an electrical excitation is calculated. >

The frequency-temperature behavior of straight crested wave modes in AT- and SC-cut quartz plates are studied using Mindlin first- and third-order plate equations. The first order Mindlin plate theory with shear correction factors was previously found to yield an 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. Correction factors for the flexure and face-shear branches may be needed. Hence, a total of five correction factors may be needed for the first-order plate theory to yield accurate frequency spectra and frequency-temperature curves.

For crystal resonators, it is always desirable to calculate the electric properties accurately for application purposes. As an extension of the Mindlin plate theory based finite element analysis of crystal resonators, a new theory for the electroded plates is derived and the piezoelectrically forced vibrations are formulated and implemented in this paper in a manner similar to our previous work. The effect of the electrodes and the electric boundary conditions are taken into considerations through the modification of the higher-order plate equations by changing the expansion function of the electric potential for this particular problem. Through the conventional discretization of the new plate theory, the linear equations for the piezoelectric plate under thickness excitation are constructed and solved with efficient numerical computation techniques such as the sparse matrix handling. Numerical examples showing good predictions of the resonance frequency and capacitance ratio of electroded crystal plates of AT-cut quartz are presented with experimental data.

The frequency-temperature characteristics of quartz crystal resonators, particularly the frequency stability in a specific temperature range in which the vibration modes could be strongly coupled, has been an important requirement in most applications. The analytical work on the frequency-temperature relations has been done over last decades in many aspects, ranging from fundamental theory of the thermal effect to the simplified plate equations of a few strongly coupled vibration modes. However, it has been clearly observed that due to the complication of resonator structures, such as the presence of mounting structure and asymmetric electrodes, simple and analytical solutions will not be able to consider all the factors which will have inevitable and noticeable effects on the resonators. In this paper, we incorporate the frequency-temperature theory for crystal plates based on incremental thermal field theory by Lee and Yong (1986) into our finite element analysis program, which can analyze the free vibrations of crystal plates with higher-order plate theory. The effect of the electrodes on the frequency-temperature relations is also studied in detail. The computational results are pared with experimental ones from real products. Satisfactory agreement demonstrates the precise prediction of the frequency-temperature behavior and practical applications of the current finite element analysis in product modeling and development.

The advantages of contouring in crystal resonator, such as energy trapping and reducing the displacements in the edges, have been well observed and utilized. Analytical efforts for thorough understanding and precise prediction of these effects have been made through the simplified equations with few strongly coupled vibration modes and prescribed thickness variations by solving the differential equations for solutions in infinite series. These solutions have been useful in revealing and explaining some well-known phenomena such as the weakening of the couplings, but many real devices have contours which cannot be effectively expressed in simple functions and this has made it is impossible to solve the equations. In this paper, we started with the derivation of power series based Mindlin plate theory specifically for plates with variable thickness, finding that the effect of thickness variation is only limited to the face-traction terms of the two-dimensional equations of motion. The equations are further implemented in the finite element analysis by taking into consideration of the variation of the thickness through the integration of each element over the plate. Consequently, the finite element analysis is formulated in a manner similar to uniform plates with the exception that the thickness is no longer a constant. The numerical results from several thickness variation cases are presented and analyzed to show the effects of the contours.

A finite element formulation of the vibrations of piezoelectric quartz resonators based on Mindlin plate theory is derived. The higher-order plate theory is employed for the development of a collection of successively higher-order plate elements which can be effective for a broad frequency range including the fundamental and overtone modes of thickness-shear vibrations. The presence of electrodes is also considered for its mechanical effects. The mechanical displacements and electric potential are combined into a generalized displacement field, and the subsequent derivations are carried out with all the generalized equations. Through standard finite element procedure, the vibration frequency and vibration mode shapes including the electric potential distribution are obtained. The frequency spectra is compared with some well-known experimental results with good agreement. Our previous experience with finite element analysis of high frequency quartz plate vibrations leads us to believe that memory and computing time will always remain as key issues despite the advances in computers. Hence, the use of sparse matrix techniques, efficient eigenvalue solvers, and other reduction procedures are explored.

Equations for calculating the Lagrangean temperature derivatives of the elastic and piezoelectric constants of quartz using their classical temperature derivatives are derived and presented. In the classical formulation, the resonator geometry and hence the reference frame changes with temperature, while in the Lagrangean formulation the reference frame is fixed at a certain temperature, say 25/spl deg/C. The immediate consequence of changing the reference frame in the classical formulation would be that the temperature coefficients of the material constants are referred to a reference frame which is itself a function of temperature. Another consequence is the difficulty in maintaining the conservation of mass at all temperatures. Hence the theoretical foundation of the classical method is unsound. For certain crystal symmetries there are similarities between the two formulations; however, in general there are significant differences between them, and going forward the Lagrangean formulation should be employed. The Lagrangean-classical relationships presented here will allow us to calculate the Lagrangean temperature derivatives of material constants such as the elastic and piezoelectric constants from the existing and published classical temperature coefficients of the said constants. Results are shown for the temperature derivatives of the elastic and piezoelectric constants of alpha quartz. Simple one-dimensional vibration problems are used to illustrate the similarities and differences between the two formulations.

Piezoelectric, Lagrangian equations for the frequency-temperature behavior of quartz are presented. From the solutions of the third order temperature perturbations of these Lagrangian equations for the thickness resonances of infinite quartz plates with air-gap electrodes, regression equations for determining the temperature derivatives of elastic constants, piezoelectric constants and dielectric permittivities am developed. By using these regression equations, and the measured data on temperature coefficients of frequency by Bechmann, Ballato and Lukaszek [1962] for doubly rotated cuts, the first, second, and third temperature derivatives of elastic constants, piezoelectric constants and dielectric permittivities are obtained. The second and third temperature derivatives of the piezoelectric constants and dielectric permittivities were not available in the literature and are published here for the first time. These temperature derivatives will provide a more accurate map of the temperature stable cuts for bulk wave and surface acoustic wave quartz resonators.

We developed the method to analyze the effect of the supporting wire on frequency of a quartz crystal tuning fork independent of the FEM. In order to estimate the influence of the supporting wire of tuning fork, we approximated the right half section of tuning fork to an L-shaped bar, of which the bars at both the base and the arm are in bending vibration. Furthermore, we approximated the supporting wire to elastic foundation with spring constant K. A comparison was made of frequency changes calculated by this theory with those by FEM, and the influence of spring constant K and the length of the supporting wire on frequency change calculated by this theory is discussed.

Linear field equations for small vibrations superposed on thermally‐induced deformations by steady and uniform temperature changes are derived from the nonlinear field equations of thermoelasticity in Lagrangian formulation. From the solutions of these equations for the thickness vibrations, the temperature derivatives of elastic stiffness are related analytically to the known or measured properties such as the second‐ and third‐order elastic stiffnesses, thermal expansion coefficients, and temperature coefficients of frequency of quartz plates. Six values of the first temperature derivative C(1)pq and six values of the effective second temperature derivative C(2)pq are calculated from the temperature coefficients of frequency measured by Bechmann, Ballato, and Lukaszek for various doubly‐rotated quartz plates. The presently calculated values are compared with the first temperature derivatives obtained by Sinha and Tiersten. In the incremental stress‐strain‐temperature relations, certain expressions involving the elastic stiffnesses, temperature derivatives, and thermal expansion coefficients can be identified as having similar significance as the temperature coefficients of Cpq defined by Bechmann. Values of these expressions are calculated and compared with the existing values. The loci of the zeros of the first and second order temperature coefficients of frequency for thickness shear (B and C) modes and the frequency‐temperature characteristics of the LC cut are studied and compared with experimental values.

Three‐dimensional linear equations of motion for small vibrations superposed on thermal deformations induced by steady, uniform temperature change in quartz are obtained. The material properties of quartz, such as the elastic stiffnesses and thermal expansion coefficients, are assumed temperature dependent and expressible by third‐degree polynomials in temperature change. From the solutions of third‐order perturbations of these equations for the thickness resonances of infinite quartz plates, six values of the effective third temperature derivatives of elastic stiffnesses C(3)pq are calculated by the use of the measured temperature coefficients of frequency by Bechmann, Ballato, and Lukaszek [Proc. IRE 50, 1812 (1962)] for various doubly rotated cuts and the values of the first temperature derivatives C(1)pq and the effective second temperature derivatives C(2)pq obtained in a previous study. An infinite system of two‐dimensional equations of motion is derived by Mindlin’s method of power‐series expansi...

Finite-element matrix equations based on the Lagrangian, first-order, incremental plate equations of motion superposed on homogeneous thermal strains were formulated using virtual work principles. A program for an isoparametric, four-node quadrilateral element was written and applied to the study of the frequency-temperature (FT) behavior of flexure-mode quartz resonators. The lumped-mass and consistent-mass matrices were found to yield practically the same FT curves. For simple prismatic resonators, two schemes, reduced/selective integration and incompatible modes, produced relatively similar FT curves. The incompatible modes scheme yielded better results for resonators of more complex shapes, such as the tuning fork. It is concluded that the six-degree-of-freedom per node element is needed for the analysis of the FT behavior of a fully anisotropic flexure-mode resonator. >

Three‐dimensional linear equations of motion for small vibrations superposed on thermal deformations induced by steady, uniform temperature change in quartz are obtained. The material properties of quartz, such as the elastic stiffnesses and thermal expansion coefficients, are assumed temperature dependent and expressible by third‐degree polynomials in temperature change. From the solutions of third‐order perturbations of these equations for the thickness resonances of infinite quartz plates, six values of the effective third temperature derivatives of elastic stiffnesses C(3)pq are calculated by the use of the measured temperature coefficients of frequency by Bechmann, Ballato, and Lukaszek [Proc. IRE 50, 1812 (1962)] for various doubly rotated cuts and the values of the first temperature derivatives C(1)pq and the effective second temperature derivatives C(2)pq obtained in a previous study. An infinite system of two‐dimensional equations of motion is derived by Mindlin’s method of power‐series expansi...