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.     

非线性磁式压电振动能量采集系统建模与分析

为便于评价、优化磁式压电振动能量采集系统的性能,系统研究了该类系统的建模与分析方法,建立了非线性的分布参数模型用于描述系统的非线性动力学行为,并采用谐波平衡法给出了谐波响应的解析解.随后利用仿真模型分析了磁铁间距、加速度幅值、负载阻抗对输出功率的影响,比较了不同激励频率和加速度幅值下的最优阻抗.结果表明:双稳态特性适用于低强度的振动环境,且愈接近临界区域,输出功率愈高,而单稳态渐硬特性适用于高强度振动环境,其最优间距并不靠近临界区域;阱间大幅运动和阱内小幅运动均存在高低能量态共存的现象,愈接近临界区域,现象愈明显;激振频率是影响最优负载阻抗的决定性因素.     

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.     

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.     

Mechanics of flexible and stretchable piezoelectrics for energy harvesting

As rapid development in wearable/implantable electronic devices benefit human life in daily health monitoring and disease treatment medically, all kinds of flexible and/or stretchable electronic devices are booming, together with which is the demanding of energy supply with similar mechanical property. Due to its ability in converting mechanical energy lying in human body into electric energy, energy harvesters based on piezoelectric materials are promising for applications in wearable/implantable device's energy supply in a renewable, clean and life-long way. Here the mechanics of traditional piezoelectrics in energy harvesting is reviewed, including why piezoelectricity is the choice for minor energy harvesting to power the implantable/wearable electronics and how. Different kinds of up to date flexible piezoelectric devices for energy harvesting are introduced, such as nanogenerators based on Zn O and thin and conformal energy harvester based on PZT. A detailed theoretical model of the flexible thin film energy harvester based on PZT nanoribbons is summarized, together with the in vivo demonstration of energy harvesting by integrating it with swine heart. Then the initial researches on stretchable energy harvesters based on piezoelectric material in wavy or serpentine configuration are introduced as well.     

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.     

Performance of a cantilever piezoelectric energy harvester impacting a bump stop

Piezoelectric cantilever beam energy harvesters are commonly used to convert ambient vibration into electrical energy. In practical applications, energy harvesters are subjected to large shocks which can shorten the service life by causing mechanical failure. In this work, a bump stop is introduced into the design of a piezoelectric cantilever beam energy harvester to limit the maximum displacement of the cantilever and prevent excessively high bending stresses developing as a result of shocks. In addition to limiting the maximum displacement of the beam, it is inevitable that the deflected shape of the beam and the electrical output are modified. A theoretical model for a piezoelectric cantilever beam harvester impacting against a stop is derived, which aims to develop an understanding of the vibration characteristics of the cantilever and quantify how the electrical output of the harvester is affected by the stop. An experiment is set up to measure the dynamics and the electrical output of a bimorph energy harvester and to validate the theoretical model. Numerical simulation results are presented for energy harvesters with different initial gaps and different stop locations, and it is found that the reduction in maximum bending stress is at the expense of the electrical power of the harvester.     

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.     

Theoretical investigations of energy harvesting efficiency from structural vibrations using piezoelectric and electromagnetic oscillators

Conversion of ambient vibrational energy into electric power has been the impetus of much modern research. The traditional analysis has focused on absolute electrical power output from the harvesting devices and efficiency defined as the convertibility of an infinite resource of vibration excitation into power. This perspective has limited extensibility when applying resonant harvesters to host resonant structures when the inertial influence of the harvester is more significant. Instead, this work pursues a fundamental understanding of the coupled dynamics of a main mass-spring-damper system to which an electromagnetic or piezoelectric mass-spring-damper is attached. The governing equations are derived, a metric of efficiency is presented, and analysis is undertaken. It is found that electromagnetic energy harvesting efficiency and maximum power output is limited by the strength of the coupling such that no split system resonances are induced for a given mass ratio. For piezoelectric harvesters, only the coupling strength and certain design requirements dictate maximum power and efficiency achievable. Since the harvesting circuitry must "follow" the split resonances as the piezoelectric harvesters become more massive, the optimum design of piezoelectric harvesters appears to be more involved than for electromagnetic devices.     

A low frequency piezoelectric power harvester using a spiral-shaped bimorph

We propose a spiral-shaped piezoelectric bimorph power harvester operating with coupled flexural and extensional vibration modes for applications to low frequency energy sources. A theoretical analysis is performed and the computational results show that the spiral structure has relatively low operating frequency compared to beam power harvesters of the same size. It is found that to optimize the performance of a piezoelectric spiral-shaped harvester careful design is needed.     

Response of duffing-type harvesters to band-limited noise

This paper aims to experimentally investigate the influence of stiffness-type nonlinearities on the transduction of vibratory energy harvesters (VEHs) under band-limited noise. For the purpose of the study, 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. The axial load permits the harvester to operate with different potential energy characteristics, namely, the mono-stable (pre-buckling) and bi-stable (post-buckling) configurations. The performance of the harvester in both the configurations is investigated and compared by tuning the harvester's oscillation frequencies around the static equilibria such that they have equal values in both scenarios. The harvester is then subjected to random base excitations of different levels, bandwidths, and center frequencies. The variance of the output voltage is measured across an arbitrary, purely resistive load and used for the purpose of performance comparison. Critical conclusions pertinent to the influence of the nonlinearity and relative performance in both configurations are presented and discussed.     

Output response identification in a multistable system for piezoelectric energy harvesting

In this paper we examine in detail the multiple responses of a novel vibrational energy harvester composed of a vertical bistable beam whose complex non-linear behavior is tuned via magnetic interaction. The beam was excited horizontally by a harmonic inertial force while mechanical vibrational energy is converted to electrical power through a piezoelectric element. The bistable laminate beam coupled to the piezoelectric transducer showed a variety of complex responses in terms of the beam displacement and harvested power output. The range of vibration patterns in this non-linear system included single-well oscillations and snap-through vibrations of periodic and chaotic character. Harvested power was found to be strongly dependent on the vibration pattern with nonlinearities providing a broadband response for energy harvesting. Wavelet analysis of measured voltage, displacement and velocity time histories indicated the presence of a variety of nonlinear periodic and also chaotic phenomena. To measure the complexity of response time series we applied phase portraits and determine stroboscopic points and multiscale entropy. It is demonstrated that by changing parameters such as the magnetic interaction, the characteristics of the bistable laminate harvester, such as the natural frequency, bandwidth, vibration response and peak power can be readily tailored for harvesting applications.     

A comparison between nonlinear cantilever and buckled beam for energy harvesting

Nonlinear dynamics has become one of the key aspect to improve the efficiency of kinetic energy harvesters working in the real environment. Different methods based on the exploitation of the dynamical features of stochastic nonlinear oscillators using bi-stable piezoelectric cantilevers or buckled beams have been proposed in the past years. Such methods are shown to outperform standard linear oscillators and to overcome some of the most severe limitations of present approaches once applied to ambient vibrations. This work presents simulation results comparing the two methods. The same piezoelectric element subjected to a fixed vibrating body in a cantilever or bridge configuration has been simulated. The kinetic excitation considered is a zero mean exponentially correlated gaussian noise with different amplitudes. The piezoelectric oscillator output response has been obtained as a function of a nonlinear parameter. This work is intended to help designing the most performing energy harvester for real world applications starting from the same piezoelectric element.     

采用Helmholtz共鸣器与悬臂梁压电换能器的声能采集器研究

针对环境中广泛存在的声能,提出了一种采用Helmholtz共鸣器和悬臂梁压电换能器的声能采集器。Helmholtz共鸣器对入射声压进行放大,放大后的声压引起共鸣器弹性薄壁振动,薄壁的振动传递到压电换能器产生电能输出。建立了带弹性壁的立方形共鸣器的等效集中参数理论模型,并与压电换能器的机电特性结合,分析了声能采集器的声-机-电转换原理,研究了声压、声波频率和负载阻抗对输出功率的影响,研究结果为此类声能采集器的优化设计及工程应用提供了一种可行的方法。实验中,声源通过声波导管输出声能,当共鸣器管口处的声压级为94 dB时,系统实测最大输出功率达240μW。该采集器不仅可作为声能自供能采集器,还可在较远距离为低能耗电子装置进行有源声供能。      

Response of uni-modal duffing-type harvesters to random forced excitations

Linear energy harvesters have a narrow frequency bandwidth and hence operate efficiently only when the excitation frequency is very close to the fundamental frequency of the harvester. Consequently, small variations of the excitation frequency around the harvester's fundamental frequency drops its small energy output even further making the energy harvesting process inefficient. To extend the harvester's bandwidth, some recent solutions call for utilizing energy harvesters with stiffness-type nonlinearities. From a steady-state perspective, this hardening-type nonlinearity can extend the coupling between the excitation and the harvester to a wider range of frequencies. In this effort, we investigate the response of such harvesters, which can be modeled as a uni-modal duffing-type oscillator, to White Gaussian and Colored excitations. For White excitations, we solve the Fokker-Plank-Kolmogorov equation for the exact joint probability density function of the response. We show that the expected value of the output power is not even a function of the nonlinearity. As such, under White excitations, nonlinearities in the stiffness do not provide any enhancement over the typical linear harvesters. We also demonstrate that nonlinearities in the damping and inertia may be used to enhance the expected value of the output power. For Colored excitations, we use the Van Kampen expansion and long-time numerical integration to investigate the influence of the nonlinearity on the expected value of the output power. We demonstrate that, regardless of the bandwidth or the center frequency of the excitation, the expected value of the output power decreases with the nonlinearity. With such findings, we conclude that energy harvesters modeled as uni-modal duffing-type oscillators are not good candidates for harvesting energy under forced random excitations. Using a linear transformation, results can be extended to the base excitation case.     

Sustained high-frequency energy harvesting through a strongly nonlinear electromechanical system under single and repeated impulsive excitations

This work investigates a vibration-based energy harvesting system composed of two oscillators coupled with essential (nonlinearizable) stiffness nonlinearity and subject to impulsive loading of the mechanical component. The oscillators in the system consist of one grounded, weakly damped linear oscillator mass (primary system), which is coupled to a second light-weight, weakly damped oscillating mass attachment (the harvesting element) through a piezoelastic cable. Due to geometric/kinematic mechanical effects the piezoelastic cable generates a nonlinearizable cubic stiffness nonlinearity, whereas electromechanical coupling simply sees a resistive load. Under single and repeated impulsive inputs the transient damped dynamics of this system exhibit transient resonance captures (TRCs) causing high-frequency ‘bursts’ or instabilities in the response of the harvesting element. In turn, these high-frequency dynamic instabilities result in strong and sustained energy transfers from the directly excited primary system to the lightweight harvester, which, through the piezoelastic element, are harvested by the electrical component of the system or, in the present case, dissipated across a resistive element in the circuit. The primary goal of this work is to demonstrate the efficacy of employing this type of high-frequency dynamic instability to achieve enhanced nonlinear vibration energy harvesting under impulsive excitations.     

Nonlinear dynamics of galloping-based piezoaeroelastic energy harvesters

The normal form is proposed as a tool to analyze the performance and reliability of galloping-based piezoaeroelastic energy harvesters. Two different harvesting systems are considered. The first system consists of a tip mass prismatic structure (isosceles 30° or square cross-section geometry) attached to a multilayered cantilever beam. The only source of nonlinearity in this system is the aerodynamic nonlinearity. The second system consists of an equilateral triangle cross-section bar attached to two cantilever beams. This system is designed to have structural and aerodynamic nonlinearities. The coupled governing equations for the structure’s transverse displacement and the generated voltage are derived and analyzed for both systems. The effects of the electrical load resistance and the type of harvester on the onset speed of galloping are quantified. The results show that the onset speed of galloping is strongly affected by the load resistance for both types of harvesters. The normal form of the dynamic system near the onset of galloping (Hopf bifurcation) is then derived. Based on the nonlinear normal form, it is demonstrated that smaller levels of generated voltage or power are obtained for higher absolute values of the effective nonlinearity. For the first harvesting system, the results show a supercritical Hopf bifurcation for both isosceles 30° or square cross-section geometries. The nonlinear normal form shows that the isosceles triangle section (30°) is more efficient than the square section. For the second harvesting system, the normal form is used to identify the values of the nonlinear torsional spring which changes the harvester’s instability. It is demonstrated that this critical value of the nonlinear torsional spring depends strongly on the load resistance.     

Coupled magneto-electro-mechanical lumped parameter model for a novel vibration-based magneto-electro-elastic energy harvesting systems

The objective of this paper is to present a coupled magneto-electro-mechanical (MEM) lumped parameter model for the response of the proposed magneto-electro-elastic (MEE) energy harvesting systems under base excitation. The proposed model can be used to create self-powering systems, which are not limited to a finite battery energy. As a novel approach, the MEE composites are used instead of the conventional piezoelectric materials in order to enhance the harvested electrical power. The considered structure consists of a MEE layer deposited on a layer of non-MEE material, in the framework of unimorph cantilever bars (longitudinal displacement) and beams (transverse displacement). To use the generated electrical potential, two electrodes are connected to the top and bottom surfaces of the MEE layer. Additionally, a stationary external coil is wrapped around the vibrating structure to induce a voltage in the coil by the magnetic field generated in the MEE layer. In order to simplify the design procedure of the proposed energy harvester and obtain closed form solutions, a lumped parameter model is prepared. As a first step in modeling process, the governing constitutive equations, Gauss's and Faraday's laws, are used to derive the coupled MEM differential equations. The derived equations are then solved analytically to obtain the dynamic behavior and the harvested voltages and powers of the proposed energy harvesting systems. Finally, the influences of the parameters that affect the performance of the MEE energy harvesters such as excitation frequency, external resistive loads and number of coil turns are discussed in detail. The results clearly show the benefit of the coil circuit implementation, whereby significant increases in the total useful harvested power as much as 38% and 36% are obtained for the beam and bar systems, respectively.     

弹性支撑双稳压电悬臂梁振动响应及能量采集研究

在分析了常规刚性支撑非线性能量采集系统的研究基础上,提出外部磁铁弹性支撑的结构设想,保证系统在低强度激励条件下也能处于双稳态振荡,提高机电能量转换效率.研究表明,对于强度变化的随机激励历程,弹性支撑非线性能量采集系统不需要实时调整磁铁间距,能够更好地迎合强度时刻变化的随机激励源,实现高效的机电能量转换.     

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.