Two-dimensional equations for electroelastic plates with relatively large shear deformations

A set of two-dimensional, nonlinear equations for electroelastic plates in moderately large thickness-shear deformations is obtained from the variational formulation of the three-dimensional equations of nonlinear electroelasticity by expanding the mechanical displacement vector and the electric potential into power series in the plate thickness coordinate. As an example, the equations are used to study nonlinear thickness-shear vibrations of a quartz plate driven by an electrical voltage. Nonlinear electrical current amplitude-frequency behavior near resonance is obtained. The equations and results are useful in the study and design of piezoelectric crystal resonators and the measurement of nonlinear material constants of electroelastic materials.     

Analysis of nonlinear transient responses of piezoelectric resonators

The electric transient response method is an effective technique to evaluate material constants of piezoelectric ceramics under high-power driving. In this study, we tried to incorporate nonlinear piezoelectric behaviors in the analysis of transient responses. As a base for handling the nonlinear piezoelectric responses, we proposed an assumption that the electric displacement is proportional to the strain without phase lag, which could be described by a real and constant piezoelectric e-coefficient. Piezoelectric constitutive equations including nonlinear responses were proposed to calculate transient responses of a piezoelectric resonator. The envelopes and waveforms of current and vibration velocity in transient responses observed in some piezoelectric ceramics could be fitted with the calculation including nonlinear responses. The procedure for calculation of mechanical quality factor Q(m) for piezoelectric resonators with nonlinear behaviors was also proposed.     

Optimal cuts to extract the third-order elastic constants of langasite single crystals

Optimal cuts to determine the third-order elastic constants of langasite single crystals by the resonator method are proposed. By designing a small number of langasite resonators with optimal cut angles and measuring their force-frequency effects, the third-order elastic constants of langasite single crystals may be extracted separately. The numerical method to search for these optimal cut angles is presented. All 14 third-order elastic constants may be determined through a series of experimental measurements. This method will simplify traditional methods used to determine the third-order elastic constants and could potentially produce more accurate results.     

计及压电效应时矩形振子的三维振动研究

在计及压电效应的情况下，本文利用解析方法对有限尺寸压电陶瓷矩形振子的三维振动进行了研究，推出了振子耦合振动的频率方程，并对振子的振动模式进行了分析。理论研究表明，利用本文中的解析法研究振子的耦合振动，计算简单、物理意义明显。与传统的一维理论分析方法及数值方法相比，由于本研究考虑了振子的压电效应以及不同振动模式不同的相互耦合，因此，振子的理论计算频率与测量值更加符合。     

Coupled mode theory for nonlinear piezoelectric plate vibrations

Coupled-mode equations were developed for thickness-longitudinal vibrations of piezoelectric plate resonators. The equations describe the resonator as a parametrically excited nonlinear vibrational system with two degrees of freedom. Thus one can examine the behavior of piezoelectric resonators from a nonlinear dynamic point of view. In the present paper the derivation of the equations is given. As an example of the application of the equations, the behavior of the resonator in the vicinity of primary resonance is discussed. The method of multiple scales is used to solve the coupled-mode equations. An experiment which verises the amplitude-driving voltage relation predicted by the theory is conducted for LiNbO(3) resonators.     

Acoustic, piezoelectric, and dielectric nonlinearities of AlN in coupled resonator filters for high RF power levels

Coupled resonator filters (CRFs) are the new generation of BAW filters recently designed for the front-end modules of mobile transmission systems. Looking for designers' requirements, CRF devices have been characterized and modeled. The model based on equivalent circuits relies on material constants such as stiffness and electro-coupling coefficients, and works only for linear-mode propagation. Because of their positions between antennas and power amplifiers, they often work under high RF power, inducing nonlinear response in the AlN piezoelectric layer. In this work, we analyze for the first time the nonlinear behavior of AlN material particularly for coupled BAW resonators. To characterize the nonlinear effects in CRFs, we measure the 1-dB gain compression point (P1dB) and the intercept point (IP(3)). Then, we develop a nonlinear model of CRFs using harmonic balance (HB) simulation in commercially available software. The HB environment allows fitting simulations to measurements in terms of P(1dB) and IP(3). We find that a high RF power induces nonlinear changes in the material constants' real parts: elastic stiffness c(33) (4.9%), piezoelectric e(33) (17.4%), and permittivity ?(33) (5.2%). These nonlinear variations of material constants describe the nonlinear behavior of CRF devices using the same deposit process for AlN material.     

Generalized self-consistent electroelastic estimation of piezoelectric nanocomposites accounting for fiber section shape under antiplane shear

Amicromechanicsmodel composed of a three-phase confocal elliptical cylinder of the nanocompositeswithinterface effect is proposed.Ageneralized self-consistent method for the piezoelectric nanocompositesaccounting for fiber section shape under far-field mechanical–electrical loads is presented based on the model.By using the theory of Gurtin–Murdoch surface/interface and the conformal mapping technique, a closedformsolution of the effective electroelastic constants is obtained. The present solution can be degenerated intothe existing solution. The results show that the effective electroelastic constants are dramatically size dependentwhen the size of the fiber is on the order of nanometers. The effective electroelastic constants decreasemonotonically with the fiber section aspect ratio \({\gamma}\) increasing from 0 to 1. The elastic constant and dielectricconstant obtained by the present solution are very similar to the results obtained by the classical electroelastictheory, whereas the piezoelectric coupling constant obtained by the present solution is very different from theresults obtained by the classical electroelastic theory.     

Pure-mode measurements of Li2B4O7material properties

Recent measurements of doubly rotated plate resonators have highlighted the need for more accurate material constants for use in dilithium tetraborate resonator design. In this paper, we report on the room-temperature determination of the elastic, piezoelectric, and dielectric constants of dilithium tetraborate using frequency domain measurements of primarily pure-mode vibrations in thin, flat plates     

Neutral-coated circular piezoelectric inclusions

This study is concerned with the neutrality of a coated circular piezoelectric inclusion when the matrix is subjected to remote uniform electroelastic loadings. Two design schemes are proposed to make the coated inclusion neutral. In the first scheme, the thickness of the coating can be designed for given electroelastic constants of the composite so as to make the inclusion neutral to certain remote uniform loadings. In the second scheme, the electroelastic constants of the coating can be designed for given electroelastic constants of the inclusion and the matrix, and given coating thickness so as to make the inclusion neutral to any remote uniform loadings. The analysis indicates that even in the absence of the coating, the piezoelectric inclusion can be made neutral if the electroelastic constants of the two-phase composite satisfy a restriction.     

A new method of determining the equivalent circuit parameters of piezoelectric resonators and analysis of the piezoelectric loading effect

An inductive method of piezoelectric resonance detection is applied to the determination of equivalent circuit parameters of piezoelectric resonators. Using this method one can measure the resonance frequency and mechanical Q-factor of a resonator directly as well as their dependences on the electrical impedance which is connected to the resonator. From the equivalent circuit analysis the changes in resonance frequency and Q-factor due to the piezoelectric loading effects are determined. Measurements on two typical commercial piezoelectric resonators, an AT-cut quartz crystal and a PZT ceramic resonator, are in good agreement with the analysis.     

Finite element simulations of thin-film composite BAW resonators

A finite element method (FEM) formulation is presented for the numerical solution of the electroelastic equations that govern the linear forced vibrations of piezoelectric media. A harmonic time dependence is assumed. Both of the approaches, that of solving the field problem (harmonic analysis) and that of solving the corresponding eigenvalue problem (modal analysis), are described. A FEM software package has been created from scratch. Important aspects central to the efficient implementation of FEM are explained, such as memory management and solving the generalized piezoelectric eigenvalue problem. Algorithms for reducing the required computer memory through optimization of the matrix profile, as well as Lanczos algorithm for the solution of the eigenvalue problem are linked into the software from external numerical libraries. Our FEM software is applied to detailed numerical modeling of thin-film bulk acoustic wave (BAW) composite resonators. Comparison of results from 2D and full 3D simulations of a resonator are presented. In particular, 3D simulations are used to investigate the effect of the top electrode shape on the resonator electrical response. The validity of the modeling technique is demonstrated by comparing the simulated and measured displacement profiles at several frequencies. The results show that useful information on the performance of the thin-film resonators can be obtained even with relatively coarse meshes and, consequently, moderate computational resources     

圆环形截面压电纤维复合材料有效电弹性性能研究

研究含周期分布压电纤维的压电复合材料的有效电弹性性能。通过在材料代表性体积单元边界上施加位移和电势周期边界条件,利用有限元法求得了代表性体积单元内的电弹性场。由平均电弹性场和压电复合材料有效电弹性性能定义,预测了圆环形截面压电纤维复合材料的有效电弹性系数。通过算例,比较了相同压电材料体积分数下圆环形截面压电纤维复合材料与圆截面压电纤维复合材料有效电弹性性能的差异,讨论了圆环形截面压电纤维内部非压电填充物的力学性质对有效压电系数的影响。该文结论可为高灵敏度压电复合材料设计提供 参考。     

Electroelastic properties of cracked piezoelectric materials under longitudinal shear

Analytical solutions are obtained to quantify the influence of cracks on electroelastic properties of piezoelectric materials containing doubly-periodic arrays of cracks. Both the rectangular and diamond-shaped arrays of cracks are considered. Solutions are obtained for the case of an antiplane shear load coupled with an in-plane electrical load. This study makes it possible to understand the multicrack interactions in piezoelectric solids and their effects on the fracture and electroelastic properties. The crack tip field intensity factors and the change in stored electroelastic energy due to the presence of many microcracks are calculated. These calculations enable the prediction of the effective elastic, piezoelectric and dielectric constants of a damaged piezoelectric material. The results of this work can be useful in developing a technique to determine the state of mechanical and electrical damage in piezoelectric materials.     

Investigation of loading assumptions on the effective electroelastic constants for PFRC materials

Single- and multiple-loading assumptions were previously used to derive the closed-form formulas for the effective constants of composite materials. This paper aims at investigating the effects of various loading assumptions, i.e., single- and multiple-loading assumptions, on the effective electroelastic constants for piezoelectric fibre reinforced composite materials. The required closed-form formulas under various loading assumptions are derived using the linear piezoelectric theory and iso-field assumptions. A numerical study reveals that the predicted electroelastic constants under various loading assumptions are generally bounded by those obtained under the single- and multiple-loading assumptions. The effective constants along the fibre direction obtained under the single-loading assumption are the same as those evaluated using the rule of mixtures, and those predicted under the multiple-loading assumption are close to the finite element analysis and testing results available in the literature.     

Contribution to the electromechanical and nonlinear properties of GaPO4 crystals

In the last decade, much attention has been given to piezoelectric crystals with large electromechanical coupling coefficient. The quartz homeotypes berlinite and gallium orthophosphate (GaPO/sub 4/), along with the calcium gallo-germanates such as langasite are representative of these crystals. The coupling coefficient k/sub 26/ associated with thickness-shear mode resonators is two times greater than that of quartz, increasing the spacing between the series and parallel resonance frequencies of resonators suitable for the frequency range from 1 to 100 MHz. This is important for some types of crystal oscillators and monolithic filters. The large electromechanical coupling coefficient also increases the difference between the temperature dependencies of the fundamental resonance frequency and its harmonics. In this paper, measured resonance frequency-temperature characteristics of the fundamental and third harmonics of selected rotated Y-cut GaPO/sub 4/ resonators vibrating in the thickness-shear mode are presented. Further attention is given to the measurement of some nonlinear properties of rotated Y-cut GaPO/sub 4/ resonators. Knowledge of such nonlinear interactions is important for the analysis of intermodulation phenomena in resonators, and for the application of GaPO/sub 4/ resonators in crystal oscillators, filters and other electronic devices.     

材料固有损耗对压电振子谐振特性及参数测量的影响

本从考虑损耗时压电振子的阻抗（或导纳）议程出发,详细分析了材料固有损耗对压电振子的频率特性及参数测量的影响。分析结果表明：谐振（或反谐振）频率与理想无损时的相应值的偏差随损耗的增大而增大、随谐序的增大而减小。中还进一步分析了机械损耗对压电振子几种常用参数测量结果的影响。     

Modification of piezoelectric vibratory gyroscope resonator parameters by feedback control

A method for analyzing the effect of feedback control on the dynamics of piezoelectric resonators used in vibratory gyroscopes has been developed. This method can be used to determine the feasibility of replacing the traditional mechanical balancing operations, used to adjust the resonant frequency, by displacement feedback and for determining the velocity feedback required to produce a particular bandwidth. Experiments were performed on a cylindrical resonator with discrete piezoelectric actuation and sensing elements to demonstrate the principles. Good agreement between analysis and experiment was obtained, and it was shown that this type of resonator could be balanced by displacement feedback. The analysis method presented also is applicable to micromachined piezoelectric gyroscopes.     

Langasite as a piezoelectric material for near-field microscopy resonant cantilevers

Quartz length-extension resonators have already been used to obtain atomically-resolved images by frequency-modulation atomic force microscopy. Other piezoelectric materials such as gallium orthophosphate (GaPO(4)), langatate (LGT), and langasite (LGS) could be appropriate for this application. In this paper, the advantages of langasite crystal are presented and the fabrication of similar microsensors in langasite temperature-compensated cuts by chemical etching is proved. A monolithic length extension resonator, with a tip at its end, is obtained which constitutes a real advantage in regard to the existing quartz devices.     

Analysis of aging of piezoelectric crystal resonators

Aging of piezoelectric (quartz crystal) resonator has been identified as one of the most important quality control problems of quartz crystal products. Aging is defined as frequency change with time. Aging in quartz resonators can be due to several sources: mass transfer due to contamination inside the resonator enclosure, stress-strain in the resonator blank, quartz defect, etc. In this study, the stress-strain effect, which has been believed as a dominant factor contributing to aging, is studied. The stress-strain effect is caused mainly by the long-term viscoelastic properties of bonding adhesive that attach quartz crystal plate to the ceramic base package. With the available accelerating testing method under elevated temperatures, the stress-strain induced aging in the quartz crystal resonators can be investigated. Because of the miniaturized size of the resonator, a digital image analysis method called image intensity matching technique (IIMT) is applied to obtain deformation patterns in the quartz blank due to thermal load. Our preliminary results showed that the unsymmetric thermal deformations may be a dominant contributing factor to aging. For simulation purposes, finite-element analysis is used to investigate the deformation patterns (i.e., stress-strain distributions) and corresponding natural frequency shift in the piezoelectric resonators. The viscoelastic behavior of mounting adhesives is incorporated into the analysis to show the dominant effect of long-term behavior of stress-strain developed in the crystal resonators. Also, some geometrical aspects-such as uneven mounting supports due to distances, volumes and heights of the adhesives-are simulated in the model.     

Advances in high-Q piezoelectric resonator materials and devices

In order to compare piezoelectric materials and devices, an intrinsic parameter, the motional time constant tau(1)/sup (m/)=(omega/sub m/Q/sub m/)(-1) for a particular mode m is employed. The use of tau(1)/sup (m/) follows from the accommodation of acoustic loss in the elastic compliance/stiffness and the establishment of material coefficients that are elements of viscosity matrices. Alternative and fully equivalent definitions of tau(1) are given based on the RC time constant derived from the equivalent circuit representation of a crystal resonator, acoustic attenuation, logarithmic decrement, and viscosity or damping. For quartz devices, the variation of tau(1): for any simple thickness mode, for the Y'X shear mode for rotated Y-cuts, and with diameter-thickness ratio for AT-cuts is discussed. Other factors such as mounting loss and loss caused by crystal inhomogeneities (dislocations, defect positions in the resonator, and impurity migration under vibrational stress) are briefly considered with quartz devices as the model. Some new piezoelectric materials/material constants/devices are reviewed and their motional time constants are compared. A physical parameter, composed of acoustic velocity, piezoelectric coupling, and tau(1) is identified which aids in understanding the maximum frequency limitations of plate resonators.