Governing equations for a piezoelectric plate with graded properties across the thickness

Experimental and numerical investigations of vibration characteristics for parallel-type and series-type triple-layered piezoceramic bimorphs

The dynamic characteristics of parallel-type and series-type piezoelectric bimorphs are analyzed in this study. The transverse (out-of-plane) and planar (in-plane) vibrations for piezoceramic bimorphs in normal and abnormal connections are investigated experimentally by 2 noncontact optical techniques and impedance analyzer. Electronic speckle pattern interferometry (ESPI) is the major experimental technique for measuring the resonant frequency and corresponding vibration mode shape. Out-of-plane and in-plane vibrations of piezoelectric bimorphs at resonance are obtained by a self-arranged ESPI optical setup. The laser Doppler vibrometer (LDV) is a point-wise measurement technique for out-of-plane displacement and is used to determine the out-of-plane resonant frequencies. The impedance analyzer is used to measure the resonant frequencies for in-plane motions. It is noted from the experimental results that the out-of-plane modes are the dominant motion for the normal connection and only symmetric vibration mode shapes can be excited. The in-plane motions are large enough to be measured using the ESPI method for normal connections. The in-plane resonant modes are observed for parallel-type piezoelectric bimorph in parallel connections; however, the in-plane mode shapes are similar to the out-of-plane mode shapes for the series-type piezoelectric bimorph in series connection. Hence, the particle motions of the piezoelectric bimorph at resonance are essentially 3-D for the normal connection. It is interesting to note that both symmetric and asymmetric out-of-plane vibration mode shapes can be excited with large applied voltage but no in-plane motion is observed for the abnormal connection. In addition to experimental methods, numerical computations based on the finite element method are used to verify the experimental results. Good agreements of the resonant frequencies and mode shapes are obtained for experimental and numerical results.

Scattering of anti-plane shear waves by an interface crack between two bonded dissimilar functionally graded piezoelectric materials

In the present paper, the dynamic behaviour of a Griffith crack situated at the interface of two bonded dissimilar functionally graded piezoelectric materials (FGPMs) is considered. It is assumed that the elastic stiffness, piezoelectric constant, dielectric permittivity and mass density of the FGPMs vary continuously as an exponential function of the x and y coordinates, and that the FGPMs are under anti-plane mechanical loading and in-plane electrical loading. By using an integral transform technique the problem is reduced to four pairs of dual integral equations, which are transformed into four simultaneous Fredholm integral equations with four unknown functions. By solving the four simultaneous Fredholm integral equations numerically the effects of the material properties on the stress and electric displacement intensity factors are calculated and displayed graphically.

DEPP Co-Extruded Functionally Graded Piezoceramics

Functionally Graded Piezoceramics (FGP) offer performance similar to conventional piezoceramic actuators while reducing the problems associated with their bonded construction (high stress levels, large stress discontinuities, delamination, etc.). This paper presents the Dual Electro/Piezo Property (DEPP) gradient method and the tools necessary for designing, modeling, and producing DEPP FGP actuators including: material property gradient maps, a Micro-Fabrication by Co-eXtrusion (MFCX) process, and experimentally validated analytic and numeric performance and stress modeling methodologies that account for continuous and layered material gradients and complex electric field profiles. These models predict a dramatic internal stress reduction achieved by the DEPP method. Preliminary reliability testing confirm this with an increase in piezoelectric actuator lifetimes over 1010 cycles, improvement of almost four orders of magnitude compared to conventional piezoceramic actuation.

An asymptotic theory for vibrations of inhomogeneous/laminated piezoelectric plates

An asymptotic theory for the vibration analysis of inhomogeneous monoclinic piezoelectric plates is developed by using small parameter expansion. The theory includes the important special case of a laminated plate in which each layer is homogeneous and orthotropic, but distinct from the other layers by having a different material or a different orientation. A hierarchy of two-dimensional equations of the same homogeneous operator for each order is reduced from the three-dimensional framework of linear piezoelectricity. The elasticity version of the leading-order equation is the same as that of the classical Kirchhoff inhomogeneous plate theory and, therefore, is easily solvable. By contrast, it is not straightforward to find solutions of the higher-order equations. The solvability condition is thus established for this purpose, by which higher-order frequency parameters are derived. The present theoretical formulation is examined by comparing the present asymptotic results with an exact three-dimensional solution for a piezoelectric bimorph strip, and excellent agreement is reached. Some new results are presented.

Mechanical effects of electrodes on the vibrations of quartz crystal plates

A system of approximate first-order equations is extracted from an infinite system of 2-D equations for piezoelectric crystal plates with thickness-graded material properties, which is deduced from the 3-D equations of linear piezoelectricity. These equations are used to study mechanical effects on the thickness-shear (TS), flexural (F), and face-shear (FS) vibrations of an AT-cut quartz plated with two identical electrodes. Dispersion curves are calculated from the present 2-D equations as well as the 3-D equations. The comparison of these curves shows that the agreement is very close for all three frequency branches of TS, F, and FS modes in a range up to the 1.5 times the fundamental TS frequency and for gold and aluminum electrodes with R, the ratio of the mass of the electrodes to that of the plate, equal to 0.05. without introducing any correction factors. In order to assess electrode effects, spectra of omega vs. a/bq (length-to-thickness ratio of the quartz) are computed for plates with gold and aluminum electrodes and different R ratios. And the spectrum of omega vs. R is computed for plates with aluminum electrodes and a given a/bq ratio. For a plate with gold electrodes, the frequencies of predominant TS, F, and FS modes are decreasing as R increases, but the amount of frequency changes for the TS mode is much greater than those for the other two modes. However, for a plate with aluminum electrodes, the frequencies of the TS and FS modes are decreasing, but those of the F modes are increasing as R increases.

Impedance and admittance matrices of symmetric piezoelectric annular bimorphs and their applications

The 5 x 5 impedance and admittance matrices of a symmetric triple-layer piezoelectric annular bimorph (PAB) are presented and the PAB with the piezoelectric layers partially covering the shim layer in harmonic motion is analyzed using the matrices. The electromechanical flow vectors are defined as the mechanical displacements and rotations at the inner and outer boundaries and the electric current of the piezoelectric layer; the conjugate parameters, i.e., the electromechanical effort vectors, are accordingly defined. The impedance matrix which relates the flow vector with the effort vector is divided into three matrices, the mechanical, electrical, and electromechanical coupling impedance matrices; each matrix is represented by a block network with five ports. The resonance and the antiresonance frequencies and the effective electromechanical coupling factors of the PAB excited by the partially covering piezoelectric layers are then calculated for various boundary conditions imposed at the inner and outer surfaces. The numerical results by the impedance matrix are compared with those by the finite element methods; they are in excellent agreement with each other. It is also shown that the matrix representations can be easily applied to the piezoelectric circular bimorph (PCB) with the extended shim layer.

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.

Piezoelectric ceramic disks with thickness-graded material properties

A system of two-dimensional, first-order equations for electroded piezoelectric crystal plates with general symmetry and thickness-graded material properties recently was deduced from the three-dimensional equations of linear piezoelectricity. These equations are simplified for the two limiting cases of thickness-graded piezoelectric properties, i.e., the homogeneous plate and bimorph of piezoelectric ceramics. Closed-form solutions are obtained from these reduced equations for the flexural and thickness-shear vibrations and static response of bimorph disks as well as for the extensional and thickness-stretch vibrations of homogeneous disks. Frequency spectra and modes are computed and examined. Resonance frequencies for both homogeneous and bimorph disks of PZT-857 are computed and measured. The comparison of the results shows that the agreement is close.