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.     

Thickness-shear vibrations of a quartz plate under time-dependent biasing deformations

Incremental thickness-shear vibrations of a Y-cut quartz crystal plate under time-harmonic biasing extensional deformations are studied using the two-dimensional equations for small fields superposed on finite biasing fields in an electroelastic plate. It is shown that the incremental thickness-shear vibrations are governed by the well-known Mathieu's equation with a time-dependent coefficient. Both free and electrically forced vibrations are studied. Approximate analytical solutions are obtained when the frequency of the biasing deformation is much lower than that of the incremental thickness-shear vibration. The incremental thickness-shear free vibration mode is shown to be both frequency and amplitude modulated, with the frequency modulation as a first-order effect and the amplitude modulation a second-order effect. The forced vibration solutions show that both the static and motional capacitances become time-dependent due to the time-harmonic biasing deformations.     

Equations for small fields superposed on finite biasing fields in a thermoelectroelastic body

Equations for infinitesimal incremental fields superposed on finite biasing fields in a thermoelectroelastic body are derived from the nonlinear equations of thermoelectroelasticity. The equations are general in the sense that no assumptions are made on the biasing fields. The general equations obtained are reduced to various special cases under different approximations.     

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

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

Vibrations and static responses of asymmetric bimorph disks of piezoelectric ceramics

In an earlier article, the flexural vibrations in bimorph disks and extensional vibrations in homogeneous disks of piezoelectric ceramics were studied. In the present paper, the coupled flexural and extensional vibrations and static responses in an asymmetric bimorph disk, which is formed by bonding together two piezoelectric ceramic disks of unequal thickness and opposite polarization, are investigated. Governing equations of coupled motions for asymmetric bimorphs are deduced from the recently derived 2-D, first-order equations for piezoelectric crystal plates with thickness-graded material properties. Then, closed form solutions of these equations for circular disks are obtained for free vibrations, piezoelectrically forced vibrations, and responses under static voltage difference. Resonance frequencies, distribution of displacements and surface charges, impedances, and static responses are calculated for asymmetric bimorph disks of various thickness ratios and diameter-to-thickness ratios. Experimental data on resonances and impedances are obtained for asymmetric bimorph disks of PZT-857 for different thickness ratios. Comparisons of predicted and measured results show that the agreements are close.     

Thickness vibrations of a piezoelectric plate with dissipation

The three-dimensional (3-D) equations of linear piezoelectricity with quasi-electrostatic approximation are extended to include losses attributed to the acoustic viscosity and electrical conductivity. These equations are used to investigate effects of dissipation on the propagation of plane waves in an infinite solid and forced thickness vibrations in an infinite piezoelectric plate with general symmetry. For a harmonic plane wave propagating in an arbitrary direction in an unbounded solid, the complex eigenvalue problem is solved from which the effective elastic stiffness, viscosity, and conductivity are computed. For the forced thickness vibrations of an infinite plate, the complex coupling factor K*, input admittance Y are derived and an explicit, approximate expression for K* is obtained in terms of material properties. Effects of the viscosity and conductivity on the resonance frequency, modes, admittance, attenuation coefficient, dynamic time constant, coupling factor, and quality factor are calculated and examined for quartz and ceramic barium titanate plates.     

Torsional vibrations of circular elastic plates with thickness steps

This paper presents a theoretical study of torsional vibrations in isotropic elastic plates. The exact solutions for torsional vibrations in circular and annular plates are first reviewed. Then, an approximate method is developed to analyze torsional vibrations of circular plates with thickness steps. The method is based on an approximate plate theory for torsional vibrations derived from the variational principle following Mindlin's series expansion method. Approximate solutions for the zeroth- and first-order torsional modes in the circular plate with one thickness step are presented. It is found that, within a narrow frequency range, the first-order torsional modes can be trapped in the inner region where the thickness exceeds that of the outer region. The mode shapes clearly show that both the displacement and the stress amplitudes decay exponentially away from the thickness step. The existence and the number of the trapped first-order torsional modes in a circular mesa on an infinite plate are determined as functions of the normalized geometric parameters, which may serve as a guide for designing distributed torsional-mode resonators for sensing applications. Comparisons between the theoretical predictions and experimental measurements show close agreements in the resonance frequencies of trapped torsional modes.     

Thickness-shear vibration of rotated Y-cut quartz plates with relatively thick electrodes of unequal thickness

An exact solution is obtained from the three-dimensional equations of linear piezoelectricity for thickness-shear vibrations of rotated Y-cut quartz plates with relatively thick electrodes of unequal thickness at its major faces. Effects of the shear stiffness of the electrodes on resonant frequencies are examined.     

Effects of a liquid layer on thickness-shear vibrations of rectangular AT-cut quartz plates

Thickness-shear vibrations of rectangular AT-cut quartz with one face in contact with a layer of Newtonian (linearly viscous and compressible) fluid are studied. The two-dimensional (2D) governing equations for vibrations of piezoelectric crystal plates given previously are used in the present study. The solutions for 1D shear wave and compressional wave in a liquid layer are obtained, and the stresses at the bottom of the liquid layer are used as approximations to the stresses exerted on the crystal surface in the plate equations. Closed form solutions are obtained for both free and piezoelectrically forced thickness-shear vibrations of a finite, rectangular AT-cut quartz plate in contact with a liquid layer of finite thickness. From the present solutions, a simple and explicit formula is deduced for the resonance frequency of the fundamental thickness-shear mode, which includes the effects of both shear and compressional waves in the liquid layer and the effect of the thickness-to-length ratio of the crystal plate. The formula reduces to the widely used frequency equation obtained by many previous investigators for infinite plates. The resonance frequency of a rectangular AT-cut quartz, computed as a function of the thickness of the adjacent liquid layer, agrees closely with the experimental data measured by Schneider and Martin (Anal. Chem., vol. 67, pp. 3324-3335, 1995)     

Perturbation analysis of frequency shifts in an electroelastic body under biasing fields

We analyze the eigenvalue problem associated with small-amplitude vibrations superposed on finite-biasing fields in an electroelastic body. The widely used first-order perturbation integral by Tiersten is generalized in two different ways: a second-order perturbation analysis is given when the biasing fields are not infinitesimal and their second order effects need to be considered; a first-order perturbation analysis is given when an eigenvalue is associated with more than one eigenvector (a degenerate eigenvalue).     

Three-dimensional vibration analyses of functionally graded material rectangular plates with through internal cracks

The free vibrations of rectangular FGM plates with through internal cracks are investigated using the Ritz method. Three-dimensional elasticity theory is employed, and new sets of admissible functions for the displacement fields are proposed to enhance the effectiveness of the Ritz method in modeling the behaviors of cracked plates. The proposed admissible functions accurately describe the stress singularities at the fronts of the crack and display displacement discontinuities across the crack. The correctness and validity of the present approach are established through comprehensive convergence studies and comparisons with published results for homogeneous cracked plates, based on various plate theories. The locally effective material properties of FGM in the thickness direction are estimated by a simple power law. The effects of the volume fraction of the constituents of FGM and the thickness-to-length ratio on the frequencies are investigated. Frequency data for FGM square plates with three types of boundary conditions along the four side faces and with internal cracks of various crack lengths, positions and orientations are tabulated for the first time.     

Free vibrations of an electroelastic body under biasing fields

We present a systematic analysis of the eigen-value problem associated with free, small-amplitude vibrations superposed on finite biasing fields in an electroelastic body. An abstract formulation is introduced. The operators in the abstract formulation are shown to be self-adjoint, from which a series of fundamental properties of resonant frequencies and modes are proved concisely. A variational formulation and a perturbation analysis of the eigenvalue problem also are given based on the abstract formulation.     

A Reissner-Mindlin-type plate theory including the direct piezoelectric and the pyroelectric effect

Summary This paper is concerned with flexural vibrations of composite plates, where piezoelastic layers are used to generate distributed actuation or to perform distributed sensing of strains in the plate. Special emphasis is given to the coupling between mechanical, electrical and thermal fields due to the direct piezoelectric effect and the pyroelectric effect. Moderately thick plates are considered, where the influence of shear and rotatory inertia is taken into account according to the kinematic approximations introduced by Mindlin. An equivalent single-layer theory is thus derived for the composite plates. It is shown that coupling can be taken into account by means of effective stiffness parameters and an effective thermal loading. Polygonal plates with simply supported edges are treated in some detail, where quasi-static thermal bending as well as free, forced and actuated vibrations are studied.     

A corrected modal representation of thickness vibrations in quartz plates and its influence on the transversely varying case

A modal representation of the thickness vibrations of rotated Y-cut quartz plates, which was used in the treatment of driven transversely varying thickness modes, is shown to be defective in certain respects. The differential equations and edge conditions for transversely varying thickness modes have been used in the accurate treatment of trapped energy resonators, monolithic crystal filters, and contoured quartz resonators, even though those defects were present. In this work those defects in the thickness solution are corrected along with the influence on the differential equations and edge conditions in the transversely varying case. The corrected modal representation shows that, because in practical applications to the above mentioned devices, the driving frequency is always near a thickness resonant frequency, essentially the same results will be obtained with the corrected representation as were obtained with the defective one, which explains why the results obtained with the defective equations were so accurate.     

On the use of Levy's method for symmetrically laminated composite plates

For rectangular plates with two opposite edges simply supported and no bending twisting coupling, Levy's method can yield exact solutions for static deflections, free vibrations and buckling loads. However, three distinct solutions are obtained, depending on the relative magnitude of the plate rigidities, which in turn are functions of material properties, laminate thickness and lay-up. In this communication a stiffness invariant formulation is used to identify those laminates which fall under the three cases defined.     

Fundamental frequency and design of the CFCF composite sandwich plate

The design efficiency of sandwich panels is often associated with the value of fundamental frequency. This paper investigates the free vibrations of rectangular sandwich plates having two adjacent edges fully clamped and the remaining two edges free (CFCF). The vibration analysis is performed by applying Hamilton’s principle in conjunction with the first-order shear deformation theory. The analytical solution determining the fundamental frequency of the plate is obtained using the generalised Galerkin method and verified by comparison with the results of finite element modal analysis. The approach developed in the paper and equations obtained are applied to the design of sandwich plates having composite facings and orthotropic core. Design charts representing the effects of the thickness of the facings and core on the mass of composite sandwich panel for a given value of the fundamental frequency are obtained.     

Asymmetric free vibrations of laminated annular cross-ply circular plates including the effects of shear deformation and rotary inertia: spline method

Asymmetric free vibrations of annular cross-ply circular plates are studied using spline function approximation. The governing equations are formulated including the effects of shear deformation and rotary inertia. Assumptions are made to study the cross-ply layered plates. A system of coupled differential equations are obtained in terms of displacement functions and rotational functions. These functions are approximated using Bickley- type spline functions of suitable order. Then the system is converted into the eigenvalue problem by applying the point collocation technique and suitable boundary conditions. Parametric studies have been made to investigate the effect of transverse shear deformation and rotary inertia on frequency parameter with respect to the circumferential node number, radii ratio and thickness to radius ratio for both symmetric and anti-symmetric cross-ply plates using various types of material properties.     

Rapid heating of FGM rectangular plates

Thermally induced vibrations of functionally graded material rectangular plates are investigatedin this research. The thermomechanical properties of the plate are assumed to be temperature and positiondependent. Dependency on temperature is expressed based on theTouloukian formula, and position dependencyis written as a power-law function. The ceramic-rich surface of the plate is subjected to temperature rise orheat flux, whereas the metal rich surface is kept at reference temperature or thermally insulated. Temporalevolution of the temperature profile across the plate thickness is obtained by the solution of one-dimensionalheat conduction equation. This equation is originally nonlinear since temperature dependency of thermalconductivity is taken into account. The solution of this equation is obtained by means of the generalizeddifferential quadrature (GDQ) accompanied with the successive Runge–Kutta algorithm in time domain. Themotion equations of the plate are obtained based on the first-order shear deformation theory of plates under smallstrains and small deformations assumptions. Hamilton’s principle is used to establish the motion equations.These equations are discreted in the plate domain bymeans of the two-dimensional GDQ method. The resultingequations are linear time-dependent coupled equations which are traced in time by means of the Newmarktime-marching method. Conducting comparison studies to assure the validity and accuracy of the proposedmodel, parametric studies are carried out to examine the influences of temperature dependency, thermal andmechanical boundary conditions, power-law index, plate geometry and boundary conditions. It is shown thatthermally induced vibrations exist for thin plates.     

Extensional vibrations of piezoelectric plates

Summary We consider piezoelectric plates with a thickness small with respect to the lateral dimensions. The surfaces of these plates are partly coated with electrodes. Equations are derived which describe the lateral vibrations of these plates approximately. The first resonance-frequency of a circular plate as a function of the radius of the electrodes is computed and compared with measured values.     

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.