Three-dimensional analysis of an antiparallel piezoelectric bimorph

Summary Three-dimensional electromechanical responses of a piezoelectric bimorph are studied. The bimorph is antiparallel in the sense that it consists of two identical, plate-like piezoelectric elements with opposite poling directions. Both the top and bottom surfaces of the bimorph are fully covered with negligibly thin conductive electrodes. By introducing a small parameter and using the transfer matrix method it is shown that a three-dimensional solution of the problem can be readily constructed, provided the solution to a set of two-dimensional equations very similar to those in the classic plate theory is obtainable. The three-dimensional solution satisfies all the field equations as well as the boundary conditions on the major surfaces and at the interface between the two piezoelectric plates. In many special cases, the electric edge condition can be fulfilled point by point, and thus the solution is exact in Saint-Venant's sense. The formulation and new analytical results for a strip-shaped cantilever bimorph under the action of applied voltage and end moment are presented.     

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.     

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.     

热电耦合场中压电粘弹性微梁的静力坍塌分析

基于粘弹性理论、欧拉-伯努利梁理论以及压电理论,在考虑微结构蠕变现象的情况下,建立了热电耦合场中压电粘弹性微梁的静力坍塌模型及控制方程,研究了压电粘弹性微梁的静力坍塌临界条件,计算和分析了压电层控制电压和温度等参数对微梁结构静力坍塌的影响。研究结果表明:持久坍塌电压可作为压电粘弹性微梁静力坍塌失稳的临界条件,并且温度和压电层控制电压均会对坍塌失稳的临界值造成影响。该文的研究可为MEMS中梁式微结构的设计与优化提供理论参考。     

Two-dimensional contaminant transport modeling using meshfree point collocation method (PCM)

Groundwater contamination is a severe problem in many parts of the world including India. The complex problem of groundwater flow and contaminant transport is studied generally by solving the governing equations of flow and transport using numerical models such as finite difference method (FDM) or finite element method (FEM). Meshfree (MFree) method is an alternative numerical approach to solve these governing equations in simple and accurate manner. MFree method does not require any grid and only makes the use of a set of scattered collocation points, regardless of the connectivity information between them. Kansa (1990) [9] developed a multi-quadratic (MQ) based MFree method for the solution of partial differential equations. Based on the Kansa’s method, the present study proposes a MFree point collocation method (PCM) with multi-quadric radial basis function (MQ-RBF) for the two-dimensional coupled groundwater flow and transport simulation in unconfined conditions. The accuracy of the developed model is verified with available analytical solutions in literature. The coupled model developed is further applied to a field problem to compute the groundwater head and concentration distribution and the results are compared with available finite element based simulation. The outcomes of the model results showed the applicability of the present approach.     

Meshless Local Petrov-Galerkin Method for Plane Piezoelectricity

Piezoelectric materials have wide range engineering applications in smart structures and devices. They have usually anisotropic properties. Except this complication electric and mechanical fields are coupled each other and the governing equations are much more complex than that in the classical elasticity. Thus, efficient computational methods to solve the boundary or the initial-boundary value problems for piezoelectric solids are required. In this paper, the Meshless local Petrov-Galerkin (MLPG) method with a Heaviside step function as the test functions is applied to solve two-dimensional (2-D) piezoelectric problems. The mechanical fields are described by the equations of motion with an inertial term. To eliminate the time-dependence in the governing partial differential equations the Laplace-transform technique is applied to the governing equations, which are satisfied in the Laplace-transformed domain in a weak-form on small subdomains. Nodal points are spread on the analyzed domain and each node is surrounded by a small circle for simplicity. The spatial variation of the displacements and the electric potential are approximated by the Moving Least-Squares (MLS) scheme. After performing the spatial integrations, one obtains a system of linear algebraic equations for unknown nodal values. The boundary conditions on the global boundary are satisfied by the collocation of the MLS-approximation expressions for the displacements and the electric potential at the boundary nodal points. The Stehfest's inversion method is applied to obtain the final time-dependent solutions.     

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

Two-dimensional first-order governing equations for electroded piezoelectric crystal plates with general symmetry and thickness-graded material properties are deduced from the three-dimensional equations of linear piezoelectricity by Mindlin's general procedure of series expansion. Mechanical displacements and thickness-graded material properties, i.e., the elastic stiffnesses, piezoelectric coefficients, dielectric permittivities, and mass density, are expanded in powers of the thickness coordinate, while electric potential is expanded in a special series in order to accommodate the specified electric potentials at electroded faces of the plate. The effects of graded material properties on the piezoelectrically induced stresses or deformations by the applied surface potentials are clearly exhibited in these newly derived equations which reduce to Mindlin's first-order equations of elastic anisotropic plates when the material properties are homogeneous. Closed form solutions are obtained from the three-dimensional equations of piezoelectricity and from the present two-dimensional equations for both homogeneous plates and bimorphs of piezoelectric ceramics. Dispersion curves for homogeneous plates and bimorphs and resonance frequencies for bimorph strips with finite width are computed from the solutions of three-dimensional and two-dimensional equations. Comparison of the results shows that predictions from the two-dimensional equations are very close to those from the three-dimensional equations.     

Gradient piezoelectricity for cracks under an impact load

The flexoelectric effect on elastic waves is investigated in nano-sized cracked structures. The strain gradients are considered in the constitutive equations of a piezoelectric solid for electric displacements and the higher-order stress tensor. The governing equations with the corresponding boundary conditions are derived from the variational principle. The finite element method (FEM) is developed from the principle of virtual work. It is equivalent to the weak-form of derived governing equations in gradient elasticity. The computational method can be applied to analyze general 2D boundary value problems in size-dependent piezoelectric elastic solids with cracks under a dynamic load. The FEM formulation is implemented for strain-gradient piezoelectricity under a dynamic load.     

Electromechanical modeling and analytical investigation of nonlinearities in energy harvesting piezoelectric beams

Piezoelectric materials are extensively applied for vibrational energy harvesting especially in micro-scale devices where other energy conversion mechanisms such as electromagnetic and electrostatic methods encounter fabrication limitations. A cantilevered piezoelectric bimorph beam with an attached proof (tip) mass for the sake of resonance frequency reduction is the most common structure in vibrational harvesters. According to the amplitude and frequency of applied excitations and physical parameters of the harvester, the system may be pushed into a nonlinear regime which arises from material or geometric nonlinearities. In this study nonlinear dynamics of a piezoelectric bimorph harvester implementing constitutive relations of nonlinear piezoelectricity together with nonlinear curvature and shortening effect relations, is investigated. To achieve this goal first of all a comprehensive fully-coupled electromechanical nonlinear model is presented through a variational approach. The governing nonlinear partial differential equations of the proposed model are order reduced and solved by means of the perturbation method of multiple scales. Results are presented for a PZT/Silicon/PZT laminated beam as a case study. Findings indicate that material nonlinearities of the PZT layer has the dominant effect leading to softening behavior of the frequency response. At the primary resonance, different frequency responses of the extracted power can be distinguished according to the excitation amplitude, which is due to harmonic generation as a result of piezoelectric nonlinearity. The extracted power is analytically computed and validated with a good agreement by a numerical solution.     

Adjusting the resonant frequency of a PVDF bimorph power harvester through a corrugation-shaped harvesting structure

We propose a corrugated polyvinylidene fluoride (PVDF) bimorph power harvester with the harvesting structure fixed at the two edges in the corrugation direction and free at the other edges. The resonant frequency of a corrugated PVDF bimorph is readily adjusted through changing either its geometrical configuration or the span length, which can keep the harvester operating at the optimal state in environments with different ambient vibrations. The governing equations of a PVDF bimorph with a corrugation shape are derived from the transfer-matrix technique. Statistical results show that the adaptability of a harvester to the operating environment can be improved greatly by designing the harvesting structure with adjustable resonant frequency.     

Resonance and antiresonance of symmetric and asymmetric cantilevered piezoelectric flexors

The resonances of dynamically excited symmetric piezoelectric bimorphs have been determined from the equations of state. Under the effect of sinusoidal stimuli: a moment exerted at the tip M, a force exerted perpendicular to the plane of the bimorph also applied at the tip F, a uniformly applied pressure p, and an electrode voltage V, they respond with a sinusoidal tip rotation alpha, tip deflection delta, volume displacement nu, and electrode charge Q. All of the former are related to all of the latter through a dynamic admittance matrix B. The antiresonance frequency of the capacitance C have been found while also antiresonance in off-diagonal elements have been determined. The latter indicate that at these frequencies the bimorph does not work as an actuator or sensor in the particular domain of the off-diagonal. The mode shape at these antiresonance frequencies has been determined. The antiresonance of b(14) determines that for this frequency the tip has deflection but no rotation, while the antiresonance of b(24 ) indicates that the tip has rotation but no deflection. No antiresonance in the volume displacement is found, indicating that the bimorph is a pressure converter (microphone) at all frequencies. Micromachined piezoelectric heterogeneous bimorphs have been fabricated using the techniques of I.C. fabrication. Their deflections have been measured as a function of frequency and applied voltage, while these have been compared with the theoretical predictions. An anomalously large quadratic deflection has been found, superimposed on the linear piezoelectric behavior. The agreement between the linear part of the experimental deflection and the theory was quite good.     

Simulation of groundwater flow in unconfined aquifer using meshfree point collocation method

For appropriate management of available groundwater, the flow behavior in the porous media has to be analyzed. The complex problem of groundwater flow can be studied by solving the governing equations analytically or by using numerical methods. As the analytical solutions are available only for simple idealized cases, numerical methods such as finite difference method (FDM) and finite element method (FEM) are generally used for field problems. Meshfree (MFree) method is an alternative numerical approach to solve complex groundwater problems in simple manner. MFree method eliminates the drawback of meshing and remeshing as in FDM and FEM which can translate to substantial cost and time savings in modeling. In this paper, a model using MFree point collocation method (PCM) with multi-quadric radial basis function (MQ-RBF) is proposed for 2D groundwater flow simulation. The accuracy of the developed model is verified with available analytical solution in literature. The developed model is applied initially for a hypothetical problem and further for a field problem to compute head distribution. The PCM model results for the hypothetical problem are compared with FEM simulations while that of field problem are compared with boundary element based model results. The PCM model results are found to be satisfactory showing the applicability of the present approach.     

Benchmark analysis of piezoelectric bimorph energy harvesters composed of laminated composite beam substrates

This paper is concerned with the derivation of exact solutions for the responses of piezoelectric bimorph energy harvesters composed of laminated composite beam substrates. An electro-elastic finite element model is also developed based on the layer wise first order shear deformation theory for computing the responses of the bimorphs under general boundary and loading conditions. Both series and parallel connections of the piezoelectric layers of the bimorphs are considered. The responses computed by the finite element model excellently match with that obtained by the exact solutions. The induced electric potential in case of the bimorph in which the piezoelectric layers are connected in series is significantly larger than that in case of the bimorph with piezoelectric layers connected in parallel. If the thickness of the piezoelectric layers and the substrate remain same, the piezoelectric bimorph composed of antisymmetric angle-ply substrate beam is capable of inducing more electric potential than the bimorphs with cross-ply substrate beams. Also, if the bimorph is cantilever, it induces significantly more electric potential than when it is simply supported. Optimum thickness of the piezoelectric layers of the bimorph and unimorph harvesters has been determined. Most importantly, it is found that the bimorph with its piezoelectric layers connected in series performs significantly better than the unimorph if the mass and volume of the piezoelectric layers and the substrates remain same. The results presented here may serve as the benchmark results for verifying experimental and numerical models.

     

On the anisotropic piezoelastic contact problem for an elliptical punch

Summary This paper firstly conducts a systematic investigation of the problem of a rigid punch indenting an anisotropic piezoelectric half-space. The Fourier transform method is employed to the mixed boundary value problem. Using the principle of linear superposition, the resulting transformed (algebraic) equations, whose right-hand sides contain both pressure and electric displacement terms, can be solved by superposing the solutions of two sets of algebraic equations, one containing pressure and another containing electric displacement. For an arbitrarily shaped punch, two governing equations are derived, which can be solved numerically. In the case of transversely isotropic piezoelectric media, the two governing equations are corresponding with that given by others using potential theory. Particularly, when the punch has elliptic cross-section, and the pressure and electric displacement are given by some certain forms of polynomial functions, then the displacement and electric potential are prescribed by polynomial functions in the contact area. The parameters contained in it satisfy a set of linear algebraic equations, whose coefficients involve contour integrals. The problem of indentation by a smooth flat punch is examined for special orthotropic piezoelectric media, and some results obtained can be degenerated to the case of transversely isotropic piezoelectric media.     

Micromachining of a bimorph Pb(Zr,Ti)O3 (PZT) cantilever using a micro-electromechanical systems (MEMS) process for energy harvesting application

We designed and fabricated a bimorph Pb(Zr,Ti)O3 (PZT) cantilever with an integrated Si proof mass to obtain a low resonant frequency for an energy harvesting application. The cantilevers were fabricated on the micro-electromechanical systems (MEMS) scale. A mode of piezoelectric conversions were d31 and d33 mode in cantilever vibration Therefore, we designed and fabricated a single cantilever with d31 unimorph, d31 bimorph, d33 unimorph, and d33 bimorph modes. Finally, we fabricated a device with beam dimensions of about 5,400 microm x 480 microm x 14 microm (< +/- 5%), and an integrated Si proof mass with dimensions of about 1,481 microm x 988 microm x 450 microm (< +/- 5%). In order to measure the d31 and d33 modes, we fabricated top and bottom electrodes. The distance between the top electrodes was 50 microm and the resonant frequency was 89.4 Hz. The average powers of the d31 unimorph, d31 bimorph, d33 unimorph, and d33 bimorph modes were 3.90, 9.60, 21.42, and 22.47 nW at 0.8 g (g = 9.8 m/s2) and optimal resistance, respectively.     

Electromechanical coupling and output efficiency of piezoelectric bending actuators

Electromechanical coupling mechanisms in piezoelectric bending actuators are discussed in this paper based on the constitutive equations of cantilever bimorph and unimorph actuators. Three actuator characteristic parameters, (e.g., electromechanical coupling coefficient, maximum energy transmission coefficient, and maximum mechanical output energy) are discussed for cantilever bimorph and unimorph actuators. In the case of the bimorph actuator, if the effect of the bonding layer is negligible, these parameters are directly related to the transverse coupling factor lest. In the case of the unimorph actuator, these parameters also depend on the Young's modulus and the thickness of the elastic layer. Maximum values for these parameters can be obtained by choosing proper thickness ratio and Young's modulus ratio of elastic and piezoelectric layers. Calculation results on four unimorph actuators indicate that the use of stiffer elastic material is preferred to increase electromechanical coupling and output mechanical energy in unimorph actuators.     

Flexural vibration behavior of piezoelectric heterogeneous bimorph beams

Bending oscillations of piezoelectric bimorph beams are effective sound sources in gases or fluids, and, therefore, of practical interest. On the basis of the piezoelectric constitutive relations and the elastodynamic equations, the differential equation of flexural vibrations of thin rectangular piezoelectric heterogeneous bimorph beams, consisting of a piezoelectric layer glued onto an elastic substrate, is derived. The piezoelectric layer is polarized in thickness direction and can be excited to thickness vibrations by an electric alternating current voltage applied to electrodes covering the upper and lower surface of the layer. This causes an oscillating transverse contraction in the piezoelectric layer but not in the substrate, and, thus, generates flexural vibrations of the beam. The differential equation is solved analytically for beams of finite length with both ends free, one clamped and one free end, as well as for both ends clamped. Their vibration behavior in viscous fluids is considered. For a piezo-ceramic composite layer joined to a steel plate vibrating in air and in water, the analytical results are evaluated numerically as function of frequency     

Nonlinear vibration of piezoelectric nanoplates using nonlocal Mindlin plate theory

ABSTRACT

This article investigates the nonlinear vibration of piezoelectric nanoplate with combined thermo-electric loads under various boundary conditions. The piezoelectric nanoplate model is developed by using the Mindlin plate theory and nonlocal theory. The von Karman type nonlinearity and nonlocal constitutive relationships are employed to derive governing equations through Hamilton's principle. The differential quadrature method is used to discretize the governing equations, which are then solved through a direct iterative method. A detailed parametric study is conducted to examine the effects of the nonlocal parameter, external electric voltage, and temperature rise on the nonlinear vibration characteristics of piezoelectric nanoplates.     

A comparative study of the behaviour of a direct solver and a preconditioned iterative solver for the equations arising from the discretization by Chebyshev collocation of a second-order partial differential equation on a square

The behaviour is compared of two solvers for the discrete equations arising from the discretization using Chebyshev collocation of a second-order linear partial differential equation on a square. The alternative solvers considered are a direct solver and an iterative solver based on preconditioning with the matrix arising from finite-difference discretization of the governing equation. The total error of the collocation derivatives and the separate contributions from round-off and discretization error are examined. The efficiency of the two solvers is compared. The iterative solver is more efficient than the direct solver on fine grids for equations similar to the Poisson equation, provided that there are Dirichlet boundary conditions on at least three of the sides of the square.     

Theoretical modeling and equivalent electric circuit of a bimorph piezoelectric micromachined ultrasonic transducer

An electric circuit model for a circular bimorph piezoelectric micromachined ultrasonic transducer (PMUT) was developed for the first time. The model was made up of an electric mesh, which was coupled to a mechanical mesh via a transformer element. The bimorph PMUT consisted of two piezoelectric layers of the same material, having equal thicknesses, and sandwiched between three thin electrodes. The piezoelectric layers, having the same poling axis, were biased with electric potentials of the same magnitude but opposite polarity. The strain mismatches between the two layers created by the converse piezoelectric effect caused the membrane to vibrate and, hence, transmit a pressure wave. Upon receiving the echo of the acoustic wave, the membrane deformation led to the generation of electric charges as a result of the direct piezoelectric phenomenon. The membrane angular velocity and electric current were related to the applied electric field, the impinging acoustic pressure, and the moment at the edge of the membrane using two canonical equations. The transduction coefficients from the electrical to the mechanical domain and vice-versa were shown to be bilateral and the system was shown to be reversible. The circuit parameters of the derived model were extracted, including the transformer ratio, the clamped electric impedance, the spring-softening impedance, and the open-circuit mechanical impedance. The theoretical model was fully examined by generating the electrical input impedance and average plate displacement curves versus frequency under both air and water loading conditions. A PMUT composed of piezoelectric material with a lossy dielectric was also investigated and the maximum possible electroacoustical conversion efficiency was calculated.