Bimorph Beam Research Articles

Analytical modeling and experimental verification of vibration-based piezoelectric bimorph beam with a tip-mass for power harvesting

Power harvesting techniques that convert vibration energy into electrical energy through piezoelectric transducers show strong potential for powering smart wireless sensor devices in applications of structural health monitoring. This paper presents an analytical model of the dynamic behavior of an electromechanical piezoelectric bimorph cantilever harvester connected with an AC–DC circuit based on the Euler–Bernoulli beam theory and Hamiltonian theorem. A new cantilevered piezoelectric bimorph structure is proposed in which the plug-type connection between support layer and tip-mass ensures that the gravity center of the tip-mass is collinear with the gravity center of the beam so that the brittle fracture of piezoelectric layers can also be avoided while vibrating with large amplitude. The tip-mass is equated by the inertial force and inertial moment acting at the end of the piezoelectric bimorph beam based on D'Alembert's principle. An AC–DC converting circuit soldered with the piezoelectric elements is also taken into account. A completely new analytic expression of the global behavior of the electromechanical piezoelectric bimorph harvesting system with AC–DC circuit under input base transverse excitation is derived. Moreover, an experimental energy harvester is fabricated and the theoretical analysis and experimental results of the piezoelectric harvester under the input base transverse displacement excitation are validated by using measurements of the absolute tip displacement, electric voltage response, electric current response and electric power harvesting.

Analytical techniques for broadband multielectromechanical piezoelectric bimorph beams with multifrequency power harvesting

This paper presents the multifrequency responses of multielectromechanical piezoelectric bimorph beams using a novel analytical model based on the closed-form boundary value method reduced from the strong form of Hamiltonian¿s principle. The reduced constitutive multielectromechanical dynamic equations for the multiple bimorph beams connected in series, parallel, and mixed series-parallel connections can be further formulated using Laplace transformation to give new formulas for power harvesting multifrequency response functions. The parametric case studies based on the change in geometrical structures of the multiple bimorphs with and without tip masses are discussed to analyze the trend of multifrequency power harvesting optimization under resistive load. Nyquist responses based on varying geometrical structures and load resistances were used to analyze the multifrequency power amplitudes in the complex domain. Overall, the trend of system response using multiple tiers consisting of multiple bimorphs was found to significantly widen the multifrequency band followed by increasing the power amplitudes.

A scalable piezoelectric impulse-excited energy harvester for human body excitation

Harvesting energy from low-frequency and non-harmonic excitations typical of human motion presents specific challenges. While resonant devices do have an advantage in environments where the excitation frequency is constant, and while they can make use of the entire proof mass travel range in the case of excitation amplitudes that are smaller than the internal displacement limit, they are not suitable for body applications since the frequencies are random and the amplitudes tend to be larger than the device size. In this paper a piezoelectric, impulse-excited approach is presented. A cylindrical proof mass actuates an array of piezoelectric bi-morph beams through magnetic attraction. After the initial excitation these transducers are left to vibrate at their natural frequency. This increases the operational frequency range as well as the electromechanical coupling. The principle of impulse excitation is discussed and a centimetre-scale functional model is introduced as a proof of concept. The obtained data show the influence of varying the frequency, acceleration and proof mass. Finally, a commercially available integrated circuit for voltage regulation is tested. At a frequency of 2 Hz and an acceleration of 2.7 m s−2 a maximal power output of 2.1 mW was achieved.

Comparison of Finite Element Models for Piezoelectric Materials

Abstract The presented work is focused on simulation of piezoelectric effect in thin ceramic layers using finite element method. The layers are the primary components of piezoelectric patches – flexural transducers used e.g. in vibration control or structural health monitoring. At first the model of piezoelectric bimorph beam was investigated. The beam is composed of two actuators loaded by opposite electric potentials which cause the beam to bend. Further, the influence of approximation of electric potential through the thickness was tested on a simple piezoelectric cantilever beam loaded by external tip force. The numerical results obtained using several finite element types are compared mutually and with simplified analytical solution.

Experimental Study of A Frequency-Adjustable Piezoelectric Bimorph Energy Harvester

Piezoelectric energy harvesters are ideal for capturing energy from mechanical vibrations in ambient environments. The operating and source frequencies must match for the energy harvester to produce useful power. In this article, a new, broadband piezoelectric energy harvester that is implemented as a series of connected vibrating piezoelectric bimorph beams is designed and tested. The experimental results show that the operating frequency can be adjusted by changing the stiffness of a connecting spring and/or a small mass mounted on the bimorph. The results provide a foundation for the design of frequency-self tuning piezoelectric energy harvesters capable of maximizing power efficiency.

Impact-Enhanced Multi-Beam Piezoelectric Converter for Energy Harvesting in Autonomous Sensors

This work proposes and experimentally validates a piezoelectric vibration energy harvester, which exploits the impact of a central compliant driving beam onto two piezoelectric parallel bimorph beams on flexible steel. At suitable mechanical excitation conditions, the central driving beam impacts the piezo beams and triggers a nonlinear frequency-up conversion mechanism that improves the overall effectiveness, i.e. increases the overall rms output voltage and widens the equivalent bandwidth of the converter with respect to the condition of the noninteracting linear converters.

Analytical modeling of self-powered electromechanical piezoelectric bimorph beams with multidirectional excitation

Unused mechanical energies can be found in numerous ambient vibration sources in industry including rotating equipment, vehicles, aircraft, piping systems, fluid flow, and even external movement of the human body. A portion of the vibrational energy can be recovered using piezoelectric transduction and stored for subsequent smart system utilization for applications including powering wireless sensor devices for health condition monitoring of rotating machines and defence communication technology. The vibration environment in the considered application areas is varied and often changes over time and can have components in three perpendicular directions, simultaneously or singularly. This paper presents the development of analytical methods for modeling of self-powered cantilevered piezoelectric bimorph beams with tip mass under simultaneous longitudinal and transverse input base motions utilizing the weak and strong forms of Hamiltonian's principle and space- and time-dependent eigenfunction series which were further formulated using orthonormalization. The reduced constitutive electromechanical equations of the cantilevered piezoelectric bimorph were subsequently analyzed using Laplace transforms and frequency analysis to give multi-mode frequency response functions (FRFs). The validation between theoretical and experimental results at the single mode of eigenfunction solutions reduced from multi-mode FRFs is also given.

Analytical and experimental comparisons of electromechanical vibration response of a piezoelectric bimorph beam for power harvesting

Power harvesters that extract energy from vibrating systems via piezoelectric transduction show strong potential for powering smart wireless sensor devices in applications of health condition monitoring of rotating machinery and structures. This paper presents an analytical method for modelling an electromechanical piezoelectric bimorph beam with tip mass under two input base transverse and longitudinal excitations. The Euler–Bernoulli beam equations were used to model the piezoelectric bimorph beam. The polarity-electric field of the piezoelectric element is excited by the strain field caused by base input excitation, resulting in electrical charge. The governing electromechanical dynamic equations were derived analytically using the weak form of the Hamiltonian principle to obtain the constitutive equations. Three constitutive electromechanical dynamic equations based on independent coefficients of virtual displacement vectors were formulated and then further modelled using the normalised Ritz eigenfunction series. The electromechanical formulations include both the series and parallel connections of the piezoelectric bimorph. The multi-mode frequency response functions (FRFs) under varying electrical load resistance were formulated using Laplace transformation for the multi-input mechanical vibrations to provide the multi-output dynamic displacement, velocity, voltage, current and power. The experimental and theoretical validations reduced for the single mode system were shown to provide reasonable predictions. The model results from polar base excitation for off-axis input motions were validated with experimental results showing the change to the electrical power frequency response amplitude as a function of excitation angle, with relevance for practical implementation.

Relative performance of a vibratory energy harvester in mono- and bi-stable potentials

Motivated by the need for broadband vibratory energy harvesting, many research studies have recently proposed energy harvesters with nonlinear characteristics. Based on the shape of their potential function, such devices are classified as either mono- or bi-stable energy harvesters. This paper aims to investigate the relative performance of these two classes under similar excitations and electric loading conditions. To achieve this goal, an energy harvester consisting of a clamped–clamped piezoelectric beam bi-morph is considered. The shape of the harvester's potential function is altered by applying a static compressive axial load at one end of the beam. This permits operation in the mono-stable (pre-buckling) and bi-stable (post-buckling) configurations. For the purpose of performance comparison, the axial load is used to tune the harvester's oscillation frequencies around the static equilibria such that they have equal values in the mono- and bi-stable configurations. The harvester is subjected to harmonic base excitations of different magnitudes and a slowly varying frequency spanning a wide band around the tuned oscillation frequency. The output voltage measured across a purely resistive load is compared over the frequency range considered. Two cases are discussed; the first compares the performance when the bi-stable harvester has deep potential wells, while the second treats a bi-stable harvester with shallow wells. Both numerical and experimental results demonstrate the essential role that the potential shape plays in conjunction with the base acceleration to determine whether the bi-stable harvester can outperform the mono-stable one and for what range of frequencies. Results also illustrate that, for a bi-stable harvester with shallow potential wells, super-harmonic resonances can activate the inter-well dynamics even for a small base acceleration, thereby producing large voltages in the low frequency range.

Parameter identification for resonant piezoelectric energy harvesters in the low- and high-coupling regimes

We demonstrate a method for identifying the parameters of piezoelectric harvesters by simple electrical measurements. The basic equations are derived from a simplified yet accurate model. The accuracy of the model is proven by comparison with the full model. Two beam harvesters, one with a unimorph beam and one with a bimorph beam, have been fabricated and characterized. The results coincide well with proposed theory and demonstrate the differences between harvesters with low and high coupling constants. It turns out that four parameters are sufficient to completely describe the electrical behavior of a harvester with a low coupling constant. For harvesters with high coupling constants, five parameters are needed.

Determining the shock load on an electroelastic bimorph beam with split conductive coatings

The external nonstationary load acting on an electroelastic bimorph (metal–piezoceramics) beam with split conductive coatings is identified. Use is made of the Laplace transform. The original functions are recovered analytically. The unknown quantities are determined by solving a system of Volterra equations

Active damping of the nonstationary flexural vibrations of a bimorph beam

A two-layer (metal–piezoceramic) beam with hinged ends under a nonstationary distributed load is considered. A method for solving the problem of active damping of the nonstationary flexural vibrations of this structure is proposed. The shape of the electric signal applied to the electrode of the electroelastic layer to keep the beam almost undeformed is identified. The solution is obtained using the Laplace transform with respect to time. Numerical results are presented and analyzed

Theoretical and experimental studies of beam bimorph piezoelectric power harvesters

This paper presents a theoretical model for simulating a piezoelectric beam bimorph power harvester consisting of a laminated piezoelectric beam, a proof mass, and an electrical load. The vertical offset of the proof mass center from the beam centroid couples the bending and longitudinal motions, which makes it necessary to consider both longitudinal and lateral vibrations simultaneously. Experiments were carried out on a beam bimorph prototype mounted on a shaker to measure the electrical output. Numerical results obtained using the proposed procedure for piezoelectric bimorph power harvesters are in good agreement with the experimental data.

The analysis of a piezoelectric bimorph beam with two input base motions for power harvesting

The exploitation of usable power from vibration environments shows potential benefit for recharging batteries and powering wireless transmission. In this paper, we present a novel technique for simulating the electromechanical cantilevered piezoelectric bimorph beam system with two input base transverse and longitudinal motions for predicting power harvesting. The piezoelectric bimorph beam with tip mass was modelled using the Euler-Bernoulli beam assumptions. The strain fields from transverse bending and longitudinal forms can affect the physical behaviour of the polarity and electric field in terms of the series and parallel connections of the piezoelectric bimorph, in such way that each connection has two vector configurations of X-poling and Y-poling due to input base motions. This situation must be correctly identified to form the piezoelectric couplings. The piezoelectric couplings can create the electrical force and moment of each piezoelectric layer in the mechanical domain. At this point, we introduce a new method of modelling the piezoelectric bimorph beam under two input base-motions using coupling superposition of the elastic-polarity field for predicting power harvesting. The constitutive dynamic equations were derived using the weak form from the Hamiltonian theorem, with Laplace transforms being used to obtain the multi-mode frequency response functions (FRFs) relating the input mechanical vibrations with the output dynamic displacement, velocity and power harvesting. The power harvesting predictions under parallel connection at frequencies close to the fundamental bending frequency demonstrate a possibility of being able to produce around 0.4 mW per unit input base transverse acceleration of 3 m/s2. Furthermore, it is shown that varying the load resistance from 20 kΩ to 200 kΩ affects the amplitude of power harvesting as well as resulting in a shift of the first natural frequency from 76 Hz to 79 Hz.

Modeling and characterization of piezoelectric transducers by means of scattering parameters. Part I: Theory

This paper introduces an electromechanical model suitable for most piezoelectric devices, including the piezoelectric bimorph beams. The model provides an improved approach to design and to analyze the performance of piezoelectric transducers, especially in the case where piezoelectric devices are embedded in compound electronic systems. The main objectives of this work are: (i) to develop a versatile and easy-to-use electromechanical model of a piezoelectric transducer. The proposed model is based on the physical properties of the device and can be easily adapted to different operating conditions using experimental data analysis; (ii) to introduce an approach based on scattering ( S-)parameters for better characterizing the frequency response of systems that exhibit resonant phenomena. To our knowledge, this is the first time that this approach has been applied to electromechanical transducers; (iii) to show the functionality of the model on a specific application. Results prove that although the model does not take into account second order effects such as nonlinear behavior and hysteresis, it fairly fits experimental data so as to be used for proposing and precisely describing new applications of piezoelectric sensors and actuators.

“Equivalent” Electromechanical Coefficient for IPMC Actuator Design Based on Equivalent Bimorph Beam Theory

This paper addresses an “equivalent” electromechanical coupling coefficient that may be used in designing Ionic Polymer Metal Composite (IPMC) actuators. The coefficient is not a material constant and derived from equivalent bimorph beam model. The collective effect of the membrane thickness and operating voltage on the coefficient is demonstrated by using a design of experiment (DOE) of three and five levels of the two factors, respectively. Experiments and linear finite element analyses with MD.NASTRAN at DOE points are performed. The tip displacement and the coupling coefficient are reported and their response surface (RS) approximations as function of the thickness and voltage are constructed. Experiments and RS predictions indicate that actuator thickness and applied voltage are two interacting major factors for maximum tip displacement. The equivalent coupling coefficient is primarily driven by the thickness of actuator moreover voltage appears to contribute as the thickness increases. The initial curvature of the strips before electrical excitation is also shown to be a factor for “equivalent” coupling coefficient, it is not, however sufficient to explain the variation in the experimental data. A correction factor approach is proposed and applied to the straight beam tip displacement RS that filters out experimental variation. A corrected RS enables including the pre-imposed initial curvature as design parameter along with the actuator thickness and the operating peak voltage when predicting the tip displacement and the equivalent coupling coefficient.

Vibration energy scavenging via piezoelectric bimorphs of optimized shapes

Compact autonomous power sources are one of the prerequisites for the development of wireless sensor networks. In this work vibration energy harvesting via piezoelectric resonant bimorph beams is studied. The available analytical approaches for the modeling of the coupled electromechanical behavior are critically evaluated and compared with a finite element (FEM) numerical model. The latter is applied to analyze thoroughly the stress and strain states, as well as to evaluate the resulting voltage and charge distributions in the piezoelectric layers. The aim of increasing the specific power generated per unit of scavenger volume is pursued by optimizing the shape of the scavengers. Two optimized trapezoidal configurations are hence identified and analyzed. An experimental set-up for the validation of the proposed numerical model and of the obtained optimized structures is developed. Results of a preliminary experimental assessment, confirming the improved performances of optimized scavengers, are finally given.

Verification of Beam Models for Ionic Polymer-Metal Composite Actuator

Ionic Polymer-Metal Composite (IPMC) can work as an actuator by applying a few voltages. A thick IPMC actuator, where Nafion-117 membrane was synthesized with polypyrrole/alumina composite filler, was analyzed to verify the equivalent beam and equivalent bimorph beam models. The blocking force and tip displacement of the IPMC actuator were measured with a DC power supply and Young's modulus of the IPMC strip was measured by bending and tensile tests respectively. The calculated maximum tip displacement and the Young's modulus by the equivalent beam model were almost identical to the corresponding measured data. Finite element analysis with thermal analogy technique was utilized in the equivalent bimorph beam model to numerically reproduce the force-displacement relationship of the IPMC actuator. The results by the equivalent bimorph beam model agreed well with the force-displacement relationship acquired by the measured data. It is confirmed that the equivalent beam and equivalent bimorph beam models are practically and effectively suitable for predicting the tip displacement, blocking force and Young's modulus of IPMC actuators with different thickness and different composite of ionic polymer membrane.

Constitutive equations for 0-3 polarized PLZT actuators

This article presents constitutive equations of transparent lead lanthanum zirconate titanate (PLZT) materials poling along thickness direction or with 0-3 polarization. Discussed first are the Beer’s law of light transmission in transparent solid and the photovoltaic current density in PLZT materials. Formulas for field strength and electrical potential between two electrodes of a PLZT wafer are then derived from the basic photovoltaic equations. By superposing the photovoltaic and the converse piezoelectric effects, we formulate a novel expression for photo-induced strain that is nonlinearly dependent of incident light intensity and varies with light penetration depth in through-thickness direction. On the basis of the derived photo-induced strain, linear and nonlinear constitutive equations of 0-3 polarized PLZT actuators are formulated. The present model is validated by using the available experimental data in the literature. By using the present model with an exponential variation of strain through the thickness, closed-form solutions for the equivalent loads acting on a PLZT unimorph, bimorph and intelligent beam are obtained and then compared with numerical results available in the literature.

Piezoelectric aluminum nitride nanoelectromechanical actuators

This letter reports the implementation of ultrathin (100 nm) aluminum nitride (AlN) piezoelectric layers for the fabrication of vertically deflecting nanoactuators. The films exhibit an average piezoelectric coefficient (d31∼−1.9 pC/N), which is comparable to its microscale counterpart. This allows vertical deflections as large as 40 nm from 18 μm long and 350 nm thick multilayer cantilever bimorph beams with 2 V actuation. Furthermore, in-plane stress and stress gradients have been simultaneously controlled. The films exhibit leakage currents lower than 2 nA/cm2 at 1 V, and have an average relative dielectric constant of approximately 9.2 (as in thicker films). These material characteristics and actuation results make the AlN nanofilms ideal candidates for the realization of nanoelectromechanical switches for low power logic applications.