A generic double-curvature piezoelectric shell energy harvester: Linear/nonlinear theory and applications

Converting vibration energy to useful electric energy has attracted much attention in recent years. Based on the electromechanical coupling of piezoelectricity, distributed piezoelectric zero-curvature type (e.g., beams and plates) energy harvesters have been proposed and evaluated. The objective of this study is to develop a generic linear and nonlinear piezoelectric shell energy harvesting theory based on a double-curvature shell. The generic piezoelectric shell energy harvester consists of an elastic double-curvature shell and piezoelectric patches laminated on its surface(s). With a current model in the closed-circuit condition, output voltages and energies across a resistive load are evaluated when the shell is subjected to harmonic excitations. Steady-state voltage and power outputs across the resistive load are calculated at resonance for each shell mode. The piezoelectric shell energy harvesting mechanism can be simplified to shell (e.g., cylindrical, conical, spherical, paraboloidal, etc.) and non-shell (beam, plate, ring, arch, etc.) distributed harvesters using two Lamé parameters and two curvature radii of the selected harvester geometry. To demonstrate the utility and simplification procedures, the generic linear/nonlinear shell energy harvester mechanism is simplified to three specific structures, i.e., a cantilever beam case, a circular ring case and a conical shell case. Results show the versatility of the generic linear/nonlinear shell energy harvesting mechanism and the validity of the simplification procedures.     

Investigations of a nonlinear energy harvester with a bistable potential well

This paper investigates a nonlinear energy harvester that uses magnetic interactions to create an inertial generator with a bistable potential well. The motivating hypothesis for this work was that nonlinear behavior could be used to improve the performance of an energy harvester by broadening its frequency response. Theoretical investigations study the harvester's response when directly powering an electrical load. Both theoretical and experimental tests show that the potential well escape phenomenon can be used to broaden the frequency response of an energy harvester.     

带碰撞双稳态压电俘能系统的俘能特性研究

双稳态俘能系统的运动常常会陷入单个势能阱中, 导致俘能效率降低. 为了解决这个问题, 本文提出了一类带碰撞的磁斥力双稳态压电振动能量采集系统. 建立了该碰撞双稳态系统的机电耦合方程, 分析了碰撞对双稳态系统动力学特性的影响. 研究了确定性激励和低强度随机激励下碰撞对系统响应特性和俘能效率的影响. 结果表明: 简谐激励下, 碰撞能够使得原双稳态系统的单阱小幅周期运动转变为双阱间的大幅运动, 从而有效地提高输出功率. 得到了低强度随机激励下, 不同碰撞间隙对系统动力响应特性和输出功率的影响规律. 对一个给定的随机激励, 存在一个最优的碰撞间隙, 此时碰撞能够将原双稳态系统单阱内的随机运动转化为频繁的双阱跳跃, 出现大幅值运动, 从而大幅提高了系统的俘能效率.     

抗磁悬浮振动能量采集器动力学响应的仿真分析

分析了微型抗磁悬浮振动能量采集器中悬浮磁体的受力特性,发现了能量采集器的单稳态和双稳态现象,研究了能量采集器在不同工作状态下该两种稳态类型时的动力学响应特性.当能量采集器处于非工作的单稳态状态时,其动力学响应是在线性系统的基础上加入非线性扰动、幅频响应曲线向右偏转;热解石墨板间距越大,非线性扰动越强烈,右偏现象则越显著.当能量采集器处于非工作的双稳态状态时,其动力学响应比较复杂,出现倍周期、4倍周期以及混沌等非线性系统特有的现象.当能量采集器处于工作状态的双稳态状态时,其振动频率和外界激励频率保持一致,进行周期振动.该研究对抗磁悬浮振动能量采集器的结构设计具有重要的参考价值,为提高能量采集器的响应特性和输出性能提供了理论指导.     

Broadband energy harvesting via magnetic coupling between two movable magnets

Harvesting energy from ambient mechanical vibrations by the piezoelectric effect has been proposed for powering microelectromechanical systems and replacing batteries that have a finite life span. A conventional piezoelectric energy harvester （PEH） is usually designed as a linear resonator, and suffers from a narrow operating bandwidth. To achieve broadband energy harvesting, in this paper we introduce a concept and describe the realization of a novel nonlinear PEH. The proposed PEH consists of a primary piezoelectric cantilever beam coupled to an auxiliary piezoelectric cantilever beam through two movable magnets. For predicting the nonlinear response from the proposed PEH, lumped parameter models are established for the two beams. Both simulation and experiment reveal that for the primary beam, the introduction of magnetic coupling can expand the operating bandwidth as well as improve the output voltage. For the auxiliary beam, the magnitude of the output voltage is slightly reduced, but additional output is observed at off-resonance frequencies. Therefore, broadband energy harvesting can be obtained from both the primary beam and the auxiliary beam.     

Broadband piezoelectric power generation on high-energy orbits of the bistable Duffing oscillator with electromechanical coupling

An important issue in resonant vibration energy harvesters is that the best performance of the device is limited to a very narrow bandwidth around the fundamental resonance frequency. If the excitation frequency deviates slightly from the resonance condition, the power out is drastically reduced. In order to overcome this issue of the conventional resonant cantilever configuration, a non-resonant piezomagnetoelastic energy harvester has been introduced by the authors. This paper presents theoretical and experimental investigations of high-energy orbits in the piezomagnetoelastic energy harvester over a range of excitation frequencies. Lumped-parameter nonlinear equations (electromechanical form of the bistable Duffing oscillator with piezoelectric coupling) can successfully describe the large-amplitude broadband voltage response of the piezomagnetoelastic configuration. Following the comparison of the electromechanical trajectories obtained from the theory, it is experimentally verified that the piezomagnetoelastic configuration can generate an order of magnitude larger power compared to the commonly employed piezoelastic counterpart at several frequencies. Chaotic response of the piezomagnetoelastic configuration is also compared against the periodic response of the piezoelastic configuration theoretically and experimentally. Overcoming the bias caused by the gravity in vertical excitation of the piezomagnetoelastic energy harvester is discussed and utilization of high-energy orbits in the bistable structural configuration for electrostatic, electromagnetic and magnetostrictive transduction mechanisms is summarized.     

Equivalent damping and frequency change for linear and nonlinear hybrid vibrational energy harvesting systems

A unified approximation method is derived to illustrate the effect of electro-mechanical coupling on vibration-based energy harvesting systems caused by variations in damping ratio and excitation frequency of the mechanical subsystem. Vibrational energy harvesters are electro-mechanical systems that generate power from the ambient oscillations. Typically vibration-based energy harvesters employ a mechanical subsystem tuned to resonate with ambient oscillations. The piezoelectric or electromagnetic coupling mechanisms utilized in energy harvesters, transfers some energy from the mechanical subsystem and converts it to an electric energy. Recently the focus of energy harvesting community has shifted toward nonlinear energy harvesters that are less sensitive to the frequency of ambient vibrations. We consider the general class of hybrid energy harvesters that use both piezoelectric and electromagnetic energy harvesting mechanisms. Through using perturbation methods for low amplitude oscillations and numerical integration for large amplitude vibrations we establish a unified approximation method for linear, softly nonlinear, and bi-stable nonlinear energy harvesters. The method quantifies equivalent changes in damping and excitation frequency of the mechanical subsystem that resembles the backward coupling from energy harvesting. We investigate a novel nonlinear hybrid energy harvester as a case study of the proposed method. The approximation method is accurate, provides an intuitive explanation for backward coupling effects and in some cases reduces the computational efforts by an order of magnitude.     

On the performance and resonant frequency of electromagnetic induction energy harvesters

This paper investigates the linear response of an archetypal energy harvester that uses electromagnetic induction to convert ambient vibration into electrical energy. In contrast with most prior works, the influence of the circuit inductance is not assumed negligible. Instead, we highlight parameter regimes where the inductance can alter resonance and derive an expression for the resonant frequency.The governing equations consider the case of a vibratory generator directly powering a resistive load. These equations are non-dimensionalized and analytical solutions are obtained for the system's response to single harmonic, periodic, and stochastic environmental excitations. The presented analytical solutions are then used to study the power delivered to an electrical load.     

An autoparametric energy harvester

This paper presents a numerical study of an autoparametric system composed of two elements: a pendulum and an excited nonlinear oscillator. Owing to an inertial coupling between the two elements, different types of motion are possible, from periodic to chaotic. This study examines a linear induction of an energy harvester depending on the pendulum motion. The harvester consists of a cylindrical permanent magnet mounted on a rotor and of four windings fixed to the housing as a stator. When the pendulum is rotating or swinging, the converter is generating energy due to magnetic induction. In this paper, a method utilizing parametrical resonance for harvesting energy from low frequency vibrations is studied. The authors compare energy induced by different types of pendulum motion: swinging, rotation and chaotic dynamics. Additionally, voltage values for different parameters of excitation are estimated.     

一种采用双换能器和摆式结构的宽频振动能量采集器

针对悬臂梁振动能量采集器在大振幅振动下梁容易断裂的缺点,本文提出了一种基于摆式结构的具有宽频和倍频特性的振动能量采集器,该采集器由两个Terfenol-D/PMN-PT/Terfenol-D磁电换能器和嵌有六个磁铁的旋转摆构成.文中建立了摆式结构的摆动方程,分析了采集器的频率响应特性以及谐振时的机-磁-电转换特性,并对采集器输出电压波形进行了频谱分析.理论和实验研究表明:该采集器具有宽频和倍频特性,采集器样机在1 g(1 g=9.8 m/s~2)有效值加速度振动下,向下扫频时的半功率带宽达到4.8 Hz,且能在f=16.9 Hz的振动下获得3.569 mW的负载功率.利用双换能器以及采集器的倍频和宽频特性,能有效地提高低频时采集器的输出功率.     

Snap-through piezoelectric energy harvesting

Snap-through mechanism is employed to harvest electricity from random vibration through piezoelectricity. The random excitation is assumed to be Gaussian white noise. The snap-through piezoelectric energy harvester possesses the bistability. For small-amplitude vibration in a potential well, the Ito stochastic differential equation of the electromechanical coupling system is derived from the Taylor approximation at a stable equilibrium point. The method of the moment differential equations is applied to determine the statistical moments of the displacement response and the output voltage. The effects of the system parameters on the output voltage and the output power are examined. The approximate analytical outcomes are qualitatively and quantitatively supported by the numerical simulations. For large-amplitude interwell motion, the effects of the parameters on the output voltage and the output power are numerically investigated. Nonlinearity produced by the snap-through improves energy harvesting so that the snap-through piezoelectric energy harvester can outperform the linear energy harvester in the similar size under Gaussian white noise excitations.     

Experimental model validation for a nonlinear energy harvester incorporating a bump stop

In some practical applications, cantilever beam piezoelectric energy harvesters are subjected to large amplitude base excitations which induce nonlinear behaviour in the harvester that affects their performance. In this paper, a cantilever piezoelectric energy harvester model is developed which takes account of geometric nonlinearity arising through the inextensible beam condition and material nonlinearity arising in the piezoelectric layers of the harvester. The model is validated against experimental measurements for different base accelerations and load resistances, and an investigation into the nonlinear behaviour indicates that nonlinear softening is caused predominantly by material nonlinearity. To reduce the beam amplitude and the resulting bending stress in the cantilever harvester, a bump stop is incorporated into the harvester design and the influence of the bump stop is modelled. Comparisons of theoretical predictions with experimental measurements indicate that taking account of the nonlinear behaviour improves the prediction significantly in some cases. Parameter studies are also conducted to investigate how the stop location and initial gap size between the harvester and stop affect the performance of the nonlinear energy harvester.     

Prompt efficiency of energy harvesting by magnetic coupling of an improved bi-stable system

In order to improve the transform efficiency of bi-stable energy harvester(BEH),this paper proposes an advanced bi-stable energy harvester(ABEH),which is composed of two bi-stable beams coupling through their magnets.Theoretical analyzes and simulations for the ABEH are carried out.First,the mathematical model is established and its dynamical equations are derived.The formulas of magnetic force in two directions are given.The potential energy barrier of ABEH is reduced and the snap-through is liable to occur between potential wells.To demonstrate the ABEH's advantage in harvesting energy,comparisons between the ABEH and the BEH are carried out for both harmonic and stochastic excitations.Our results reveal that the ABEH's inter-well response can be elicited by a low-frequency excitation and the harvester can attain frequent jumping between potential wells at fairly weak random excitations.Thus,it can generate a higher output power.The present findings prove that the ABEH is preferable in harvesting energy and can be optimally designed such that it attains the best harvesting performance.     

Extending the dynamic range of an energy harvester using nonlinear damping

This paper introduces the use of nonlinear damping for extending the dynamic range of vibration energy harvesters. A cubic nonlinear damper is initially considered and the average harvested power and the throw are obtained for different sinusoidal base excitation amplitudes and frequencies, both numerically and analytically. It is demonstrated that when excited at resonance, at an amplitude below its maximum operational limit, the harvested power using a nonlinear damper can be significantly larger than that of a linear energy harvester, therefore expanding its dynamic range. A potential approach to implementing cubic nonlinearity using a shunted electromagnetic device is also presented.     

Nonlinear size-dependent longitudinal vibration of carbon nanotubes embedded in an elastic medium

In this paper, we study the longitudinal linear and nonlinear free vibration response of a single walled carbon nanotube (CNT) embedded in an elastic medium subjected to different boundary conditions. This formulation is based on a large deformation analysis in which the linear and nonlinear von Kármán strains and their gradient are included in the expression of the strain energy and the velocity and its gradient are taken into account in the expression of the kinetic energy. Therefore, static and kinetic length scales associated with both energies are introduced to model size effects. The governing motion equation along with the boundary conditions are derived using Hamilton's principle. Closed-form solutions for the linear free vibration problem of the embedded CNT rod are first obtained. Then, the nonlinear free vibration response is investigated for various values of length scales using the method of multiple scales.     

Modeling of a nanoscale flexoelectric energy harvester with surface effects

This work presents the modeling of a beam energy harvester scavenging energy from ambient vibration based on the phenomenon of flexoelectricity. By considering surface elasticity, residual surface stress, surface piezoelectricity and bulk flexoelectricity, a modified Euler-Bernoulli beam model for the energy harvester is developed. After deriving the requisite energy expressions, the extended Hamilton's principle and the assumed-modes method are employed to obtain the discrete electromechanical Euler-Lagrange's equations. Then, the expressions of the steady-state electromechanical responses are given for harmonic base excitation. Numerical simulations are conducted to show the output voltage and the output power of the flexoelectric energy harvesters with different materials and sizes. Particular emphasis is given to the surface effects on the performance of the energy harvesters. It is found that the surface effects are sensitive to the beam geometries and the surface material constants, and the effect of residual surface stress is more significant than that of the surface elasticity and the surface piezoelectricity. The axial deformation of the beam is also considered in the model to account for the electromechanical coupling due to piezoelectricity, and results indicate that piezoelectricity will diminish the output electrical quantities for the case investigated. This work could lead to the development of flexoelectric energy harvesters that can make the micro- and nanoscale sensor systems autonomous.     

Scaling of electromagnetic transducers for shunt damping and energy harvesting

In order for an electromagnetic transducer to operate well as either a mechanical shunt damper or as a vibration energy harvester, it must have good electromechanical coupling. A simple two-port analysis is used to derive a non-dimensional measure of electromechanical coupling, which must be large compared with unity for efficient operation in both of these applications. The two-port parameters for an inertial electromagnetic transducer are derived, from which this non-dimensional coupling parameter can be evaluated. The largest value that this parameter takes is approximately equal to the square of the magnetic flux density times the length of wire in the field, divided by the mechanical damping times the electrical resistance. This parameter is found to be only of the order of one for voice coil devices that weigh approximately 1 kg, and so such devices are generally not efficient, within the definition used here, in either of these applications. The non-dimensional coupling parameter is found to scale in approximate proportion to the device's characteristic length, however, and so although miniaturised devices are less efficient, greater efficiency can be obtained with large devices, such as those used to control civil engineering structures.     

Modification on static responses of a nano-oscillator by quadratic optomechanical couplings

A quadratic coupling enabled parametric oscillation in an optomechanical system is used to modify the nonlinear static responses of a mechanical oscillator with a normal linear coupling.The mean value study showed that the modification of the static response on a mechanical oscillator is extremely sensitive and useful,which can readily enhance or suppress the nonlinear displacement response from a bistability case to singlet or triplet well case,freely bifurcating the equilibrium position from one to two or three.The static equilibria structure and the stability regions for mean-value controls on nano-oscillator were analyzed under the possible modification parameters.     

Potential of a piezoelectric energy harvester from sea waves

A sea wave energy harvester from the longitudinal wave motion of water particles is developed. The harvester consisting of a cantilever substrate attached by piezoelectric patches and a proof mass is used to collect electrical energy owing to the electromechanical coupling effect of the piezoelectric patches from the longitudinal wave motion. To describe the energy harvesting process, a mathematical model is developed to calculate the output charge and voltage from the piezoelectric patches according to the Airy linear wave theory and classical elastic beam model. Results show that the mean value of the generated power increases with the increase in the ratio of the width to the thickness of the cantilever, the wave height, the sea depth (which equals to the cantilever height in this study), the ratio of the proof mass to the cantilever mass, and the ratio of the sea depth to the wave length. A value of the power up to 55 W can be realized for a practical sea wave with the values of the sea depth, wave height and wave length to be 3 m, 2 m, and 15 m, respectively. The collected power harvesting with respect to different categories of the sea waves are provided. Our simulations also show the generated electric power can be further increased by an increase in dimensions of the harvester considering the scale effect. This research develops a new technique for energy harvesting from sea waves by piezoelectric energy harvesters.