L-shaped piezoelectric motor--part I: design and experimental analysis

This paper proposes an L-shaped piezoelectric motor consisting of two piezoelectric bimorphs of different lengths arranged perpendicularly to each other. The coupling of the bending vibration mode of the bimorphs results in an elliptical motion at the tip. A detailed finite element model was developed to optimize the dimensions of bimorph to achieve an effective coupling at the resonance frequency of 246 Hz. The motor was characterized by developing rotational and linear stages. The linear stage was tested with different friction contact surfaces and the maximum velocity was measured to be 12 mm/s. The rotational stage was used to obtain additional performance characteristics from the motor: maximum velocity of 120 rad/s, mechanical torque of 4.7 × 10-(5) N·m, and efficiency of 8.55%.     

双迭片压电振子的振动研究

利用压电方程和小挠度弯振理论,推导了边界自由压电陶瓷双迭片薄板小挠度条件下弯曲振动位移的一般解及振型频率方程。由频率方程计算得到的各阶振型频率与用有限元方法计算的结果基本相符,并基本得到实验结果的验证。所得结论对边界自由双迭片压电振子弯曲振动的设计提供了理论参考。     

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.

     

Approximate equations for the flexure of thin,incomplete, piezoelectric bimorphs

Summary In this paper the linear, three-dimensional, piezoelectric equations for a body in equilibrium are reduced to approximate, two-dimensional ones, treating the flexure of thin bimorphs, partly coated by electrodes (incomplete bimorphs). For that purpose two-dimensional equations are derived for piezoelectric plates and for bimorphs with completely coated faces. An assumption about the charge distribution on the inner electrode is given, stating that the charge vanishes on those parts where the outer faces are free of electrodes. This assumption allows the application of the mentioned, approximate equations for plates and bimorphs to the parts of incomplete bimorphs. By stating edge and continuity conditions, the approximate theory ls completed. The solution for a circular, incomplete, piezoceramic bimorph, loaded by a singular force in the centre, is given and compared with experimental results.     

Efficiency of excitation of piezoceramic transducers at antiresonance frequency

The efficiency of piezoceramic transducers excited at both the resonance and antiresonance frequency was investigated. Losses in piezoceramics are phenomenologically considered to have three coupled mechanisms: dielectric, mechanical, and piezoelectric losses. Expressions for the resonance and antiresonance quality factors, which ultimately determine transducer efficiency, have been received on the basis of complex material constants for both stiffened and unstiffened vibration modes. Comparison of electric and mechanical fields, thermal and electrical losses of power supply, and their distribution in the transducer volume have been made. For a given constant mechanical displacement of the transducer top, the required electric voltage applied to the transducer at the antiresonance frequency is proportional to the resonance quality factor, but the changes in the intrinsic electric and mechanical field characteristics in the common case are not too essential. The requirements on the piezoceramic parameters, types of transducer vibration, and especially on the factor of piezoelectric losses in a range of physically valid values were established to provide maximal quality factors at the antiresonance frequency.     

Electric energy generator

This study reports an extremely cost-effective mechanism for converting wind energy into electric energy using piezoelectric bimorph actuators at small scale. The total dimensions of the electric energy generator are 5.08 x 11.6 x 7.7 cm3. The rectangular, box-shaped body of the overall structure is made using 3.2-mm thick plastic. Slits are made on two opposite faces of the box so that two columns and six rows of bimorph actuators can be inserted. Each row of bimorph actuators is separated from each other by a gap of 6 mm, and the two columns of bimorphs are separated from each other by a gap of 6.35 mm. In between the two columns, a cylindrical rod is inserted consisting of six rectangular hooks. The hooks are positioned in such a way that each of them just touches the two bimorphs on either side in a particular row. As the wind flows across the generator, it creates a rotary motion on the attached fan that is converted into vertical motion of the cylindrical rod using the cam-shaft mechanism. This vertical motion of the cylindrical rod creates oscillating stress on the bimorphs due to attached hooks. The bimorphs produce output voltage proportional to the applied oscillating stress through piezoelectric effect. The prototype fabricated in this study was found to generate 1.2 mW power at a wind speed of 12 mph across the load of 1.7 komega.     

High-frequency resonant characteristics of triple-layered piezoceramic bimorphs determined using experimental measurements and theoretical analysis

This is an experimental, theoretical, and numerical investigation of vibration characteristics in high-frequency resonance, which are studied for parallel- and series-type piezoelectric bimorphs. In the experimental measurements, the full-field optical technique known as electronic speckle pattern interferometry (ESPI) is used to measure the transverse (out-of-plane) and planar (in-plane) resonant frequencies and corresponding mode shapes for piezoelectric bimorphs. In addition, in-plane resonant frequencies are obtained from impedance analysis and the response curves of the frequency spectra show different vibration characteristics of the piezoelectric bimorphs with different electrical connections. Piezoelectric bimorphs with normal connections have three-dimensional coupled vibration characteristics and the out-of-plane vibration dominates the motion. However, only in-plane vibration motions can be excited in the high-frequency range for abnormal connections, and the resonant characteristics are similar to the single-layered piezoelectric plate. The triple-layered piezoelectric bimorphs with abnormal connection are also analyzed using theoretical analysis. The resonant frequencies, mode shapes, and normalized displacements are calculated based on the analytical solution. The experimental results and the theoretical analysis are in good agreement with the numerical calculations using the finite element method. From the discussion of the results for the parallel- and series-type piezoelectric bimorphs with normal and abnormal connections, the vibration characteristics at high frequencies are completely analyzed in this study.     

直线式压电垂直驱动振动给料机设计

目的 为了满足现代电子工业领域中对微小电子元器件高速送料的需要,采用环形压电双晶片,设计一种直线式压电垂直驱动振动给料机。方法 首先对给料机结构进行设计,阐述其工作原理,通过对环形压电双晶片进行模态分析,确定双晶片工作振型。然后利用小挠度的弹性薄板理论,推导压电双晶片输出力的表达式,利用Matlab软件,探究压电陶瓷和基板的厚度与出力系数的关系,以及压电陶瓷和基板的弹性模量与出力系数的关系。最后制作振动给料机样机,并对其进行实验测试。结果 测试结果表明,系统有效工作频率为184～190 Hz,当驱动电源电压为220 V,频率186.8 Hz时,对3.2 mm×2.8 mm的发光二极管(LED)给料速度可达到22.6个/s,比市场上同型号的矩形压电双晶片驱动给料机速度提升了14.7%。结论 在精密电子元件高速输送的自动化包装或自动化检测线上具有良好的应用前景。     

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.     

Experimental measurement and numerical analysis on resonant characteristics of cantilever plates for piezoceramic bimorphs

Three experimental techniques are used in this study to access the resonant characteristics of piezoceramic bimorphs in parallel and series connections. These experimental methods, including the amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI), laser Doppler vibrometer-dynamic signal analyzer (LDV-DSA), and impedance analysis, are based on the measurement of full-field displacement, point-wise displacement, and electric impedance, respectively. Because the clear fringe patterns will be shown only at resonant frequencies, both the resonant frequencies and the corresponding vibration mode shapes are successfully obtained at the same time by the AF-ESPI method. LDV-DSA is used to determine the resonant frequencies of the vibration mode for out-of-plane motion. The impedance analysis is used to measure the resonant and antiresonant frequencies for in-plane motion. Although the out-of-plane mode is the dominant motion of piezoceramic bimorphs, it is found in this study that the amount of displacement for the in-plane motion in parallel connection is large enough to be measured by AF-ESPI and impedance. It is interesting to note that resonant frequencies of the specimen in parallel connection for the out-of-plane motion determined by LDV-DSA are the same as that for the in-plane motion obtained by impedance. Furthermore, both in-plane and out-of-plane mode shapes for the specimen in parallel connection are obtained in the same resonant frequency from the AF-ESPI method. It is concluded in this study that the particle motions of piezoceramic bimorphs for parallel connection in resonance are essentially three-dimensional. However, it is found that only out-of-plane vibration modes can be excited for the specimen in series connection. Numerical computations based on the finite-element method are presented, and the theoretical predicted results are compared with the experimental measurements. Good agreements between the experimental measured data and numerical calculated results are found for resonant frequencies and mode shapes of the piezoceramic bimorph.     

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.     

Modeling for temperature compensation and temperature characterizations of BAW resonators at GHz frequencies

This paper deals with the temperature dependence of electrical and physical features of various kinds of solidly mounted resonators (SMR). The presented bulk acoustic wave (BAW) devices are designed for the 2 GHz application. The temperature coefficient of frequency (TCF) is determined from measurements well above the temperature range defined for wireless telecommunication system specifications. Therefore, evolution of electromechanical coupling factors and quality factors at resonance and antiresonance are also monitored. Results of characterizations show the trend for a subsequent theoretical temperature compensation study by using analytical modeling. To improve accuracy of modeling, an attempt to extract temperature dependence of dielectric permittivity epsilon(33) and piezoelectric coefficient e(33) is made. Finally, a well-known analytical model is modified to take into account the temperature dependence of length, density, stiffness coefficient, dielectric permittivity, and piezoelectric coefficient. Modeling highlights the need to deposit a material with positive temperature coefficient of stiffness on the top electrode. Realistic thickness of such a layer is determined. At the same time, it is necessary to adjust piezoelectric and electrode thin film thicknesses for simultaneously keeping a constant antiresonance frequency, reaching a zero temperature coefficient of frequency for antiresonance, and minimizing the decrease in the coupling factor because of the mass-loading deposition.     

Time and frequency domain analysis of piezoelectric energy harvesters by monolithic finite element modeling

The successful design of piezoelectric energy harvesting devices relies upon the identification of optimal geometrical and material configurations to maximize the power output for a specific band of excitation frequencies. Extendable predictive models and associated approximate solution methods are essential for analysis of a wide variety of future advanced energy harvesting devices involving more complex geometries and material distributions. Based on a holistic continuum mechanics modeling approach to the multi‐physics energy harvesting problem, this article proposes a monolithic numerical solution scheme using a mixed‐hybrid 3‐dimensional finite element formulation of the coupled governing equations for analysis in time and frequency domain. The weak form of the electromechanical/circuit system uses velocities and potential rate within the piezoelectric structure, free boundary charge on the electrodes, and potential at the level of the generic electric circuit as global degrees of freedom. The approximation of stress and dielectric displacement follows the work by Pian, Sze, and Pan. Results obtained with the proposed model are compared with analytical results for the reduced‐order model of a cantilevered bimorph harvester with tip mass reported in the literature. The flexibility of the method is demonstrated by studying the influence of partial electrode coverage on the generated power output.     

Admittance matrix of asymmetric piezoelectric bimorph with two separate electrical ports under general distributed loads

The dynamic admittance matrix of the asymmetric triple-layer piezoelectric bimorph subjected to the general distributed harmonic loads as well as the flexural moments and the vertical loads at the tip are presented. The top and bottom piezoelectric layers have two separate electrical ports such that each layer can be used as either a sensor or an actuator. The variation principle is used for deriving the motion equations and the conjugate parameters that maintain the symmetry of the admittance matrix. The mechanical displacements and forces at the tip are expressed in a matrix form, which, together with the reciprocal condition, greatly simplify the analysis procedure. The derived admittance matrix under the cantilevered condition is presented by a five-by-five matrix, each row representing the relationships of the displacement and rotation at the tip, the volume averaged displacement, the separate electrical charges with the flexural moment and vertical load at the tip, the magnitude of the distributed load, and the voltages. The matrices, which reduce to simpler forms for several special cases, are then used to determine the two-port electrical admittance. It is shown that the derived admittance matrix covers the various boundary conditions, the electrical parallel and series connections, and the arbitrary lay-up, including the unimorph, used as both sensors and actuators     

L-shaped piezoelectric motor--part II: analytical modeling

This paper develops an analytical model for an L-shaped piezoelectric motor. The motor structure has been described in detail in Part I of this study. The coupling of the bending vibration mode of the bimorphs results in an elliptical motion at the tip. The emphasis of this paper is on the development of a precise analytical model which can predict the dynamic behavior of the motor based on its geometry. The motor was first modeled mechanically to identify the natural frequencies and mode shapes of the structure. Next, an electromechanical model of the motor was developed to take into account the piezoelectric effect, and dynamics of L-shaped piezoelectric motor were obtained as a function of voltage and frequency. Finally, the analytical model was validated by comparing it to experiment results and the finite element method (FEM).     

Multi-objective optimization of a triple layer piezoelectric bender with a flexible extension using genetic algorithm

This article presents an electromechanical analysis for a piezoelectric bimorph actuator with a flexible extension, which is used to increase the tip deflection. The performance measuring attributes of such an actuator are derived, and a genetic algorithm is used for multi-objective optimization. The analysis reveals that for a thick flexible extension, the length of the extension provides Pareto optimal solutions for multi-objective optimization. The analysis also shows that as the thickness of the flexible extension decreases, the Pareto optimal solutions converge to a single solution for multi-objective optimization. We have considered nonlinear deflection behavior of piezoelectric materials at high electric fields, and series and parallel electrical connections in the analysis.     

Low-frequency meandering piezoelectric vibration energy harvester

The design, fabrication, and characterization of a novel low-frequency meandering piezoelectric vibration energy harvester is presented. The energy harvester is designed for sensor node applications where the node targets a width-to-length aspect ratio close to 1:1 while simultaneously achieving a low resonant frequency. The measured power output and normalized power density are 118 μW and 5.02 μW/mm(3)/g(2), respectively, when excited by an acceleration magnitude of 0.2 g at 49.7 Hz. The energy harvester consists of a laser-machined meandering PZT bimorph. Two methods, strain-matched electrode (SME) and strain-matched polarization (SMP), are utilized to mitigate the voltage cancellation caused by having both positive and negative strains in the piezoelectric layer during operation at the meander's first resonant frequency. We have performed finite element analysis and experimentally demonstrated a prototype harvester with a footprint of 27 x 23 mm and a height of 6.5 mm including the tip mass. The device achieves a low resonant frequency while maintaining a form factor suitable for sensor node applications. The meandering design enables energy harvesters to harvest energy from vibration sources with frequencies less than 100 Hz within a compact footprint.     

Efficient charge recovery method for driving piezoelectric actuators with quasi-square waves

In this paper, an efficient charge recovery method for driving piezoelectric actuators with low frequency square waves in low-power applications such as mobile microrobots is investigated. Efficiency issues related to periodic mechanical work of the actuators and the relationship among the driving electronics efficiency, the piezoelectric coupling factor, and the actuator energy transmission coefficient are discussed. The proposed charge recovery method exploiting the energy transfer between an inductor and a general capacitive load is compared with existing techniques that lead to inherent inefficiencies. A charge recovery method is then applied to piezoelectric actuators, especially to bimorph ones. Unitary efficiency can be obtained theoretically for purely capacitive loads while intrinsic losses such as hysteresis necessarily lower the efficiency. In order to show the validity of the method, a prototype driving electronics consisting of an extended H-bridge is constructed and tested by experiments and simulations. Preliminary results show that 75% of charge (i.e., more than 56% of energy) can be recovered for bending actuators such as bimorphs without any component optimization at low fields.     

Bimorph piezoelectric vibration energy harvester with flexible 3D meshed-core structure for low frequency vibration

This paper proposes a bimorph piezoelectric vibration energy harvester (PVEH) with a flexible 3D meshed-core elastic layer for improving the output power while lowering the resonance frequency. Owing to the high void ratio of the 3D meshed-core structure, the bending stiffness of the cantilever can be lowered. Thus, the deflection of the harvester and the strain in the piezoelectric layer increase. According to vibration tests, the resonance frequency is 15.8% lower and the output power is 68% higher than in the conventional solid-core PVEH. Compared to the solid-core PVEH, the proposed meshed-core PVEH (10 mm × 20 mm × 280 μm) has 1.3 times larger tip deflection and the maximum output power is 24.6 μW under resonance condition at 18.7 Hz and 0.2G acceleration. Hence it can be used as a power supply for low-power-consumption sensor nodes in wireless sensor networks.     

Piezoelectric bimorph optical-fiber sensor

We propose and demonstrate a novel high-voltage optical-fiber sensor. This sensor consists of an emitting fiber, a receiving fiber, and a piezoelectric bimorph transducer. The emitting fiber is fixed in a base, whereas the receiving fiber is mounted on the free end of the piezoelectric bimorph transducer. When a voltage is applied to the piezoelectric bimorph transducer, its free end is displaced over a distance delta. The displacement induces a loss in the optical coupling between the emitting and the receiving fiber. The voltage can be measured by monitoring the coupling loss.