Variational formulation of the Stevens-Tiersten equation and application in the analysis of rectangular trapped-energy quartz resonators.

The two-dimensional scalar differential equation for transversely varying thickness modes in quartz crystal resonators operating with thickness-shear modes is formulated into variational form for trapped-energy resonators with both electroded and unelectroded regions. A theoretical analysis of rectangular trapped-energy resonators of singly rotated quartz is performed using the Ritz method based on the variational formulation. Free vibration resonant frequencies and modes are obtained. The results show the existence of trapped modes under the electrodes. The effects of various geometric and physical parameters on the trapped modes are examined. It is also found that the classical frequency prediction given by Tiersten and Smythe from an approximate analysis using the scalar differential equation has an inaccuracy on the order of 100 ppm for the fundamental mode, significant in resonator design.

Effects of Semiconduction on Thickness-Extensional Modes of Piezoelectric Resonators.

We study thickness-extensional vibrations of a piezoelectric semiconductor plate for resonator application. A perturbation integral for the semiconduction-induced frequency shift is obtained, which shows that the first-order frequency shift represents a damping effect due to semiconduction. Numerical results for ZnO and AlN plates are presented.

Vibrating Flexoelectric Micro-Beams as Angular Rate Sensors.

We studied flexoelectrically excited/detected bending vibrations in perpendicular directions of a micro-beam spinning about its axis. A set of one-dimensional equations was derived and used in a theoretical analysis. It is shown that the Coriolis effect associated with the spin produces an electrical output proportional to the angular rate of the spin when it is small. Thus, the beam can be used as a gyroscope for angular rate sensing. Compared to conventional piezoelectric beam gyroscopes, the flexoelectric beam proposed and analyzed has a simpler structure.

Magnetically Induced Carrier Distribution in a Composite Rod of Piezoelectric Semiconductors and Piezomagnetics.

In this work, we study the behavior of a composite rod consisting of a piezoelectric semiconductor layer and two piezomagnetic layers under an applied axial magnetic field. Based on the phenomenological theories of piezoelectric semiconductors and piezomagnetics, a one-dimensional model is developed from which an analytical solution is obtained. The explicit expressions of the coupled fields and the numerical results show that an axially applied magnetic field produces extensional deformation through piezomagnetic coupling, the extension then produces polarization through piezoelectric coupling, and the polarization then causes the redistribution of mobile charges. Thus, the composite rod exhibits a coupling between the applied magnetic field and carrier distribution through combined piezomagnetic and piezoelectric effects. The results have potential applications in piezotronics when magnetic fields are relevant.

Collective buckling of a two-dimensional array of nanoscale columns.

Periodic soft nanostructures are building blocks for small devices. However, mechanical failure in the form of structure buckling or distortion from their original shape is often reported when the dimension of these soft structures were reduced to below submicron scale. Such a phenomenon seriously limits a reliable impact of these nanostructures to greater applications. Current understandings of buckling of soft 2-D nanostructures are limited. The substrate for these soft nanostructures is usually very compliant. Neighboring nanostructures could interact through the deformation of the substrate. We analyze the collective buckling of a two-dimensional array of nanoscale columns with their lower ends built into an elastic substrate. Buckling of these nanostructures is mathematically described by an eigenvalue problem. Numerical analyses show patterned collapse for these 2-D nanostructures, qualitatively matching reported experimental findings. Our efforts are useful toward the understanding and manufacturing of many two-dimensional nanoscale features.

Coupling to extension in a thickness-shear resonator due to relatively large thickness-shear deformation.

We show that in a plate thickness-shear mode resonator of rotated Y-cut quartz coupling to extension occurs when the shear deformation is no longer infinitesimal. A set of coupled equations is derived with which the effect of coupling to extension on the thickness-shear operating mode is examined.

Vibration of a thickness-twist mode piezoelectric resonator with asymmetric, nonuniform electrodes.

We studied the effect of electrodes with varying thicknesses on thickness-twist modes in a piezoelectric plate resonator of crystals of 6mm symmetry. The focus is on the effects of asymmetric electrodes, which do not seem to have been examined before. A theoretical analysis is performed using the theory of linear piezoelectricity. A trigonometric series solution is obtained that is relatively rare from the equations of piezoelectricity. Numerical calculations are made based on the series solution. Results show that asymmetric, nonuniform electrodes have a strong effect on vibration mode shapes. This effect offers the possibility of using nonuniform electrodes in design to achieve various goals. The quantitative results in this paper are useful for the purpose.

On the eigenvalue problem for free vibrations of a piezoelectric/piezomagnetic body.

We present a systematic analysis of the eigenvalue problem associated with free, small-amplitude vibrations in an elastic body possessing piezoelectric, piezomagnetic, and magnetoelectric couplings. 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 vibration modes are proved concisely. A variational formulation and a perturbation analysis of the eigenvalue problem are also given, based on the abstract formulation.

Trapped-Energy Thickness-Extensional Mode of a Partially Electroded ZnO Thin-Film Resonator.

Free and forced vibration analysis of ZnO thin-film resonator operating with trapped-energy thickness-extensional (TE) mode is performed based on the dispersion curves for both the unbounded fully electroded and unelectroded films. The thickness solutions to the free vibration problem consist of the mode branches of the dispersion curves for the electroded and the unelectroded regions, respectively. The mode branches in the unelectroded region that carry energy away from the vibration zone are neglected because this effective energy loss corresponds to a complex frequency in the frequency spectrum. Since the thickness solutions in each region have satisfied the differential equations and the boundary conditions on the major surfaces exactly, the substitution of the thickness solutions into the modified Hamilton variational principle derived by Tiersten gives an approximate continuity condition in the form of integral over the thickness of the resonator at the interface between the electroded and unelectroded regions. The stationary condition of the integral continuity condition leads to a system of homogeneous linear equations, which determines the frequency spectrum ranging from the first TE cutoff frequency of the fully electroded film to that of the unelectroded film where the conventional trapped-energy vibration occurs. Forced vibration analysis of ZnO thin film driven into TE mode by applying a voltage to the top and bottom electrodes is also performed, which further verifies the validity of the obtained results from the free vibration analysis.

Comment on "modeling of elastic nonlinearities in ferroelectric materials".

This letter discusses certain aspects in the modeling of nonlinear mechanical effects that were not correctly considered in a recent article in this journal.

Thickness-twist edge modes in a semi-infinite piezoelectric plate of crystals with 6mm symmetry.

It is shown that simple and exact thickness-twist edge modes exist in a semi-infinite piezoelectric plate of crystals with 6mm symmetry. This suggests the possibility of new thickness-twist plate edge mode resonators and acoustic wave sensors with certain advantages.

Comment on "admittance matrix of asymmetric piezoelectric bimorph with two separate electrical ports under general distributed load".

We show that, in general, the neutral axis cannot be used as a reference axis in the modeling of asymmetrically laminated beams or plates with elastic and piezoelectric layers.

Piezoelectric transformer structural modeling--a review.

A review on piezoelectric transformer structural modeling is presented. The operating principle and the basic behavior of piezoelectric transformers as governed by the linear theory of piezoelectricity are shown by a simple, theoretical analysis on a Rosen transformer based on extensional modes of a nonhomogeneous ceramic rod. Various transformers are classified according to their structural shapes, operating modes, and voltage transforming capability. Theoretical and numerical modeling results from the theory of piezoelectricity are reviewed. More advances modeling on thermal and nonlinear effects also are discussed. The article contains 167 references.

Nonlinear torsional vibration of a circular cylindrical piezoelectric rod with relatively large shear deformation.

We show that, in a circular cylindrical rod, torsional modes are coupling to extension when the shear deformation associated with the torsional modes is no longer infinitesimal. A set of a couple equations is derived with which the effect of extension on the torsional frequency is examined. The results are useful to the understanding and design of devices operating with torsional modes.

On the notations of the nonlinear theory of electroelasticity.

We summarize four existing notation systems for the nonlinear theory of electroelasticity. It is hoped that this will be helpful to researchers when reading the literature in different notations.

Thickness vibration of an electroelastic plate under biasing fields.

We study thickness vibrations of an electroelastic plate under uniform but otherwise arbitrary thermoelectromechanical biasing fields in a manner more general than previous studies of thickness vibrations of plates. An exact solution is obtained for materials with general anisotropy. Special attention is paid to the case of thickness-shear vibrations of plates of monoclinic crystals. The special cases of unidirectional extension and free thermal expansion are discussed in detail. Simple and useful formulas for frequency shifts and capacitance change versus biasing fields are obtained.

One-dimensional equations for planar piezoelectric curved bars.

A set of one-dimensional equations for coupled extension, flexure, and shear of a planar piezoelectric curved bar is obtained from the three-dimensional equations of piezoelectricity by expanding the mechanical displacement vector and the electric potential into power series in the cross-sectional coordinates and retaining lower-order terms. The equations are useful in the analysis and design of one-dimensional piezoelectric devices.

An analysis of partially electroded, contoured, quartz resonators with beveled cylindrical edges.

We report the last set of data from a series of theoretical studies of contoured resonators with beveled cylindrical edges. What is new in this report is that the contoured region is only partially electroded. The equation for transversely varying thickness modes in doubly-rotated quartz resonators is used. Resonant frequency, mode shape, and motional capacitance are calculated.

Comment on Y.-H. Hsu et al., "electrical and mechanical fully coupled theory and experimental verification of Rosen-type piezoelectric transformers" [see reference [1]].

This letter discusses the difference between piezoelectric constitutive relations for the case of one-dimensional stress and the case of one-dimensional strain, and its implications in the modeling of Rosen piezoelectric transformers.

Analysis of a rectangular ceramic plate in electrically forced thickness-twist vibration as a piezoelectric transformer.

A rectangular ceramic plate with appropriate electrical load and operating mode is analyzed for piezoelectric transformer application. An exact solution from the three-dimensional equations of linear piezoelectricity is obtained. The solution simulates the real operating situation of a transformer as a vibrating piezoelectric body connected to a circuit. Transforming ratio, input admittance, and efficiency of the transformer are obtained.