Vibrations Of Piezoelectric Plates Research Articles

A nonlinear analysis of electrically forced vibrations of piezoelectric plates with viscous damping near the thickness-shear mode

Based on the theory of nonlinear piezoelectricity, an approximate solution for electrically forced nonlinear thickness-shear vibrations of piezoelectric plates is introduced. The model considers an infinite piezoelectric plate subjected to an alternative voltage, incorporating 3rd-order and 4th-order elastic constants and viscous damping under finite deformation. The system's differential equations for steady-state forced vibrations are derived from the nonlinear pure thickness-shear vibration problem, and transformed so that the new electrical boundary conditions are free. After the application of Galerkin's method, the transformed differential equations lead to a cubic equation, capturing the nonlinear current-frequency curves of AT-cut quartz plates without the need of the quality factor Q, due to the introduced viscous damping. When the damping effect is ignored, the frequency response curves of the AT-cut quartz plate under a specific voltage are confirmed with the existing literature. The study emphasizes the presence of the critical voltage and the importance of the fourth-order elastic constant (c6666) in the stability, revealing lower c6666 values correspond to improved stability. Additionally, a proposed algorithm efficiently determines critical voltages for resonator stability.

In-plane-dominated vibration characteristics of piezoelectric thick circular plates based on higher-order plate theories

ABSTRACT Numerous engineering applications exist for the piezoelectric effect, which results from the electromechanical coupling between electrical and mechanical fields. In-plane vibrations of piezoelectric plates’ resonance frequencies and associated mode shapes have been thoroughly investigated. However, analytical solutions for in-plane-dominated vibrations of thick piezoelectric circular plates are limited. In this paper, higher-order plate theories for the in-plane-dominated vibration characteristics of piezoelectric circular thick plates under fully clamped and completely free boundary conditions are presented. The resonant frequencies and associated mode shapes were investigated based on two higher-order plate theories: second-order shear deformation plate theory and third-order shear deformation plate theory, as well as simplified third-order linear piezoelectric theory. Hamilton's principle was applied to derive equations of motion and boundary conditions. In the theoretical analysis, the resonant frequencies, associated mode shapes and distribution of electric displacements for various radius-to-thickness ratios were calculated. The numerical results obtained by the finite element method were compared with those obtained from theoretical analysis. Excellent agreement was found between the theoretical and numerical results for the thick piezoelectric circular plates.

Transient processes in thin film bulk acoustic wave piezoelectric resonators

ABSTRACTWe studied electrically forced, transient thickness-extensional vibration of piezoelectric plates of crystals of class 6 mm as thin film bulk acoustic wave resonators. A generalized Fourier series solution was obtained and verified by comparison with finite element numerical analysis. The transient processes of the startup or shutting down of a resonator was examined. It was found that for a typical thin film resonator of ZnO or AlN with a 2 µm thickness, a fundamental thickness-extensional frequency of 1–2 GHz, and a damping described by a material quality factor of 105, the characteristic startup time from rest to time-harmonic motion is about 0.1 millisecond.

Forced vibrations of SC-cut quartz crystal rectangular plates with partial electrodes by the Lee plate equations

Lee plate equations for high frequency vibrations of piezoelectric plates have been established and perfected over decades with the sole objective of obtaining accurate predictions of frequency and mode shapes to aid the analysis and design of quartz crystal resonators. The latest improvement includes extra terms related to derivatives of the flexural displacement to provide much accurate solutions for vibrations of the thickness-shear mode, which is the functioning mode of resonators and has much higher frequency than the flexural mode. The improved Lee plate equations have been used in the analysis of high frequency vibrations of quartz crystal plates as an essential step for analysis of AT- and SC-cut quartz crystal resonators after validations with fully electrode quartz crystal piezoelectric plates. In this study, closed-form solutions of free and forced vibrations of SC-cut quartz plates with partial electrodes are obtained. A procedure has been established for the calculation of dispersion relations, frequency spectra, selected vibration modes, and capacitance ratios of forced vibrations. The vibration solutions obtained with the first-order Lee plate equations are proven to be close to solutions from the Mindlin plate equations. It is now clear that both the Mindlin and Lee plate equations can be used in the analysis and design of quartz crystal resonators.

Droplets merging through wireless ultrasonic actuation

A new technique of droplets merging through wireless ultrasonic actuation has been proposed and experimentally investigated in this work. The proposed method is based on the principle of resonant inductive coupling and piezoelectric resonance. When a mechanical vibration is excited in a piezoelectric plate, the ultrasonic vibration transmitted to the droplets placed on its surface and induces merging. It has been observed that the merging rate of water droplets depends on the operating frequency, mechanical vibration of piezoelectric plate, separation distance between the droplets, and volume of droplets. The investigated technique of droplets merging through piezoelectric actuation is quite useful for microfluidics, chemical and biomedical engineering applications.

Variational analysis of thickness-shear vibrations of a quartz piezoelectric plate with two pairs of electrodes as an acoustic wave filter

We study thickness-shear vibrations of a piezoelectric plate of AT-cut quartz with two pairs of electrodes. The plate represents a monolithic, two-pole acoustic wave filter. The scalar differential equations by Tiersten and Smythe for thickness- shear vibrations of electroded and unelectroded quartz plates are employed. Based on the variational formulation of the scalar differential equations established in a previous paper and the Ritz method with trigonometric functions as basis functions, free vibration resonant frequencies and thickness-shear modes of the plate are obtained. For a structurally symmetric filter, the modes can be separated into symmetric and antisymmetric ones. Trapped modes with vibrations mainly under the electrodes are presented. The effects of the electrode inertia and the distance between the two pairs of electrodes are examined. It is also found that the classical frequency prediction given by Tiersten from an approximate analysis has an inaccuracy of tens of parts per million, significant in filter design and application.

Equations for high-frequency vibrations of piezoelectric plates derived from a semi-mixed variational principle and applications in resonators

We derive two-dimensional equations for high-frequency vibrations of piezoelectric plates from a semi-mixed vari- ational principle with the mechanical displacement vector, the electric potential, and the electric displacement vector as inde- pendent fields. This variational formulation allows independent expansions of the three fields into power series of the plate thickness coordinate. Similar to the shear correction factors used in Mindlin's equations for piezoelectric plates to adjust the thickness-shear frequencies predicted by the plate equations slightly so that they become the same as the exact frequencies from the three-dimensional equations, a charge correction factor is conveniently introduced in the plate equations in the present paper so that the inaccuracy of the plate equations in the literature in predicting the charge or current on the electrodes of an electroded plate is corrected. The final equations obtained are convenient and accurate to use for both electroded and unelectroded plates, and for calculating both the voltage and the charge on the electrodes at the plate surfaces in the case of an electroded plate.

Localized bending vibrations of piezoelectric plates

This paper investigates the existence and propagation of electro-elastic bending waves localized at the free edge of a piezoelectric plate. The problem is considered within the framework of the high-order refined plate theory introduced by Ambartsumian. The condition for existence of a localized bending wave is obtained, and the dispersion equation solved with respect to a dimensionless frequency. It is shown that the piezoelectric effect can increase the attenuation coefficient for a localized wave by up to 70% compared with that for a purely elastic plate, thus significantly decreasing the depth of penetration. The problem is also solved within the classical Kirchhoff theory. A comparison of results is carried out between two theories.

High-frequency vibrations of piezoelectric plates driven by lateral electric fields

We study coupled face-shear and thickness-twist motions of piezoelectric plates of monoclinic crystals driven by lateral electric fields. The first-order theory of piezoelectric plates is used. Pure thickness modes and propagating waves in unbounded plates as well as vibrations of finite plates are studied. Both free vibrations and electrically forced vibrations are considered. Basic vibration characteristics including resonant frequencies, dispersion relations, frequency spectra and motional capacitance are obtained. Numerical results are presented for AT-cut quartz plates. The results are expected to be useful for the understanding and design of resonant piezoelectric devices using lateral field excitation.

Energy trapping in high-frequency vibrations of piezoelectric plates with partial mass layers under lateral electric field excitation

We study coupled face-shear (FS) and thickness-twist (TT) motions of a piezoelectric plate of monoclinic crystals with mass layers on the central parts of the plate surfaces. The plate is driven by a lateral electric field. Mindlin’s first-order theory of piezoelectric plates is used. An analytical solution is obtained. Numerical results are presented for an AT-cut quartz plate, including the motional capacitance of the plate as a resonator and the vibration modes trapped under the mass layers in the central portion of the plate. The relationship between the dimension of the mass layers and the number of trapped modes is examined.

Thickness vibration of piezoelectric plates of 6 mm crystals with tilted six-fold axis and two-layered thick electrodes

We perform a theoretical analysis of thickness vibrations in piezoelectric plates of crystals with 6 mm symmetry. The six-fold axis is tilted with respect to the plate surfaces. The major surfaces of the plate are covered with two layers of electrodes of different metals. The equations of linear piezoelectricity are used for the crystal plate. The electrodes are modeled by the equations of elasticity. Thickness vibrations frequencies and modes as well as impedance are calculated and examined.

Lee plate equations for electroded quartz crystal plates with the consideration of electrode density and stiffness

Lee plate equations for high-frequency vibrations of piezoelectric plates have been established and improved over the last decades with the sole objective of obtaining the accurate prediction of frequency and mode shapes to aid crystal resonator design. The latest improvement includes extra terms related to derivatives of the flexural displacement to adjust the accuracy and consider electrodes for practical applications. As part of the efforts to make the equations applicable for resonator design with the improved frequency accuracy and consideration of electrodes, we derived Lee plate equations for electroded plates by changing the integration limits in the dimension reduction procedure to signify the dominant role of the crystal plate. By modifying the density terms in plate equations to include the contribution of both electrode stiffness and density, the accuracy of the thickness-shear vibration frequency of electroded plates is improved for commonly used electrode materials.

Elastic characterizations of α-GaPO4 single crystals grown by the flux method

Transparent gallium orthophosphate single crystals with theα-quartz-typestructure, α-GaPO4, were obtained using the high temperature solution growth technique in aLi2O–3MoO3 flux. A first measurement of several elastic constantsCijkl of themillimeter-size α-GaPO4 piezoelectric single crystals obtained is reported. The elastic constants were computed fromthe resonance frequencies of the thickness vibration modes measured, at room temperature,in plates polished in these crystals. These resonances were excited either by an electric fieldnormal to the plates (conventional thickness excitation) or by a field parallel tothe surface of the plates (lateral field excitation). As usual, the elastic constantswere extracted using the formulae given by the corresponding one-dimensionaltheories of thickness vibration of piezoelectric plates. The measured elastic constantsCijkl of theflux-grown α-GaPO4 were generally found to be higher than those measured withα-GaPO4 crystals grown using the hydrothermal technique. This is most probably related to theextremely weak concentration of OH impurities existing in the crystals obtained using thisflux-growth method.

The determination of electrical parameters of quartz crystal resonators with the consideration of dissipation

Recently, as the dissipation of quartz crystal through material viscosity is being considered in vibrations of piezoelectric plates, we have the opportunity to obtain electrical parameters from vibration solutions of a crystal plate representing an ideal resonator. Since the solutions are readily available with complex elastic constants from Mindlin plate equations for thickness-shear vibrations, the calculation of resistance and other parameters related to both mechanical deformation and electrical potential is straightforward. We start with the first-order Mindlin plate equations of a piezoelectric plate for the thickness-shear vibration analysis of a simple resonator model. The electrical parameters are derived with emphasis on the resistance that is related to the imaginary part of complex elastic constants, or the viscosity. All the electrical parameters are frequency dependent, enabling the study of the frequency behavior of crystal resonators with a direct formulation. Through the full consideration of complications like partial electrodes and supporting structures, we should be able obtain electrical parameters for practical applications in resonator design.

Thickness vibrations of a piezoelectric plate under biasing fields

Thickness vibrations of a piezoelectric plate under uniform initial thermoelectromechanical fields are investigated in this article. An exact solution is obtained for materials with general anisotropy. A plate of langasite, a relatively new piezoelectric material with strong piezoelectric coupling, is analyzed with imposed uniform acceleration as the biasing field as an example. The results show that the biasing fields have remarkable effect on the free vibrations of piezoelectric plate.

On the thickness expansion of the electric potential in the determination of two-dimensional equations for the vibration of electroded piezoelectric plates

In the derivation of two-dimensional equations for the vibration of piezoelectric plates from variational equations, expansions of the mechanical and electrical variables in the thickness coordinate are employed. If the major surfaces of the plate are electroded and the electric potential is expanded in functions of the thickness coordinate which do not vanish at the electrodes, the variations of the different orders of the expansion potentials are not independent because the electric potential must satisfy constraint conditions at the electrodes where it is independent of position. In this work, the electric potential is expanded in functions of the thickness coordinate which do not vanish at the surface electrodes and the constraint conditions are included by means of the method of Lagrange multipliers. The resulting piezoelectric plate equations are obtained along with an integral condition on the Lagrange multipliers over the electrodes, which results in the equation for the current through the electrodes. It is shown that the elimination of the Lagrange multipliers results in a reduced system of electrostatic plate equations and associated edge conditions, which is easier to use.

Analysis of the natural vibrations of circular piezoceramic plates with partial electrodes

The finite-element method is used to analyze the thickness-symmetric vibrations of piezoelectric plates with partial electrodes. The spectra of the natural vibrations at resonance and antiresonance, the dynamic electromechanical coupling coefficient, and the vibration modes of these plates are studied for a wide range of geometric dimensions of both the plates and the partial electrodes. The optimal dimensions of the plates and electrodes, which correspond to the maximal values of the coupling coefficient, are determined. The increase in the coupling coefficient due to the utilization of the partial electrodes is considered for piezoelectric plates made of ceramics of various compositions. It is shown that all piezoceramic compositions can be divided into two groups. For the first group, the utilization of the partial electrodes can increase the coupling coefficient of the thickness vibrations by 7–23%, depending on the vibration mode. For the second group of piezoceramics, the coupling coefficient cannot be increased in this way; in other words, complete electrodes are optimal for the thickness vibrations of plates made of piezoceramics that belongs to the second group.

Ultrasonic surface vibration investigation by atomic force microscopy

Abstract This study presents a novel method to study ultrasonic surface vibrations and ultrasound influence on the surface topography image obtained by atomic force microscope (AFM). The vibrations of piezoelectric plate in the frequency range of 10–200 kHz were investigated. The experiments show the possibility of using the AFM for the detection of acoustic waves and the measurement of the amplitude of vibrations and the distribution of electric field in a piezoplate. Influence of the ultrasonic surface vibrations on AFM tip-surface interaction was noticed. Lateral surface vibrations reduced the tip-surface adhesion down to zero. Experimental results show that the AFM cantilever in the presence of vibrations has a lateral vibration mode coupled with a normal vibration mode. It increased the contrast of an image obtained in the lateral force mode.

Electromechanical coupling factors and fundamental material constants of thickness vibrating piezoelectric plates

The vibrations of piezoelectric plates are governed by the linear piezoelectric equations which may contain a rather large number of elastic, piezoelectric and dielectric constants. But under certain very restrictive but practically important circumstances, piezoelectric coupling coefficients occur naturally in the solutions of the equations, a different coefficient arising in each particular case. Under more general circumstances many of the material constants remain in the solution, and the utility or significance of any of the coupling coefficients appears questionable. The solution of the problem of the thickness vibrations of an arbitrarily anisotropic plate is used in demonstrating the significance and limitations of the coupling coefficient for various types of anisotropy. In many of the instances when the coupling coefficient turns out to be useful, the frequency equation shows that the overtone resonances are not integral multiples of the fundamental. The departure from an integral multiple relationship depends on the coupling coefficient only, and consequently, the coupling coefficient can be determined from simple resonance measurements.

Vibration of piezoelectric plates with a small non-zero potential across their electrodes

The problem of an infinite piezoelectric plate vibrating between two electrodes of a plane-parallel condenser is studied in this paper for the case of a small d.c. potential maintained across the condenser electrodes. The general solution, including the displacement vector and the vector of electric field intensity inside the plate, is presented together with the complete set of associated conditions that must be fulfilled simultaneously. The results are expressed with reference to an arbitrary set of reference axes in the crystalline material and given in rationalized MKS units. Being based on a paper by Lawson the results obtained in this paper hold within the limits of validity of his work (Lawson A. W.: Phys. Rev.62 (1942), 71).