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.     

Fokker–Planck equation analysis of randomly excited nonlinear energy harvester

The probability structure of the response and energy harvested from a nonlinear oscillator subjected to white noise excitation is investigated by solution of the corresponding Fokker–Planck (FP) equation. The nonlinear oscillator is the classical double well potential Duffing oscillator corresponding to the first mode vibration of a cantilever beam suspended between permanent magnets and with bonded piezoelectric patches for purposes of energy harvesting. The FP equation of the coupled electromechanical system of equations is derived. The finite element method is used to solve the FP equation giving the joint probability density functions of the response as well as the voltage generated from the piezoelectric patches. The FE method is also applied to the nonlinear inductive energy harvester of Daqaq and the results are compared. The mean square response and voltage are obtained for different white noise intensities. The effects of the system parameters on the mean square voltage are studied. It is observed that the energy harvested can be enhanced by suitable choice of the excitation intensity and the parameters. The results of the FP approach agree very well with Monte Carlo Simulation (MCS) results.     

Nonlinear dynamics of a bistable piezoelectric-composite energy harvester for broadband application

The continuing need for reduced power requirements for small electronic components, such as wireless sensor networks, has prompted renewed interest in recent years for energy harvesting technologies capable of capturing energy from ambient vibrations. A particular focus has been placed on piezoelectric materials and devices due to the simplicity of the mechanical to electrical energy conversion and their high strain energy densities compared to electrostatic and electromagnetic equivalents. In this paper an arrangement of piezoelectric layers attached to a bistable asymmetric laminate is investigated experimentally to understand the dynamic response of the structure and power generation characteristics. The inherent bistability of the underlying structure is exploited for energy harvesting since a transition from one stable configuration to another, or “snap-through”, is used to repeatedly strain the surface bonded piezoelectric and generate electrical energy. This approach has been shown to exhibit high levels of power extraction over a wide range of vibrational frequencies. Using high speed digital image correlation, a variety of dynamic modes of oscillation are identified in the harvester. The sensitivity of such modes to changes in vibration frequency and amplitude are investigated. Power outputs are measured for repeatable snap-through events of the device and are correlated with the measured modes of oscillation. The typical power generated is approximately 3.2?mW, comparing well with the needs of typical wireless senor node applications.     

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.     

Nonlinear vibration control and energy harvesting of a beam using a nonlinear energy sink and a piezoelectric device

This paper presents an optimal design for a system comprising a nonlinear energy sink (NES) and a piezoelectric-based vibration energy harvester attached to a free–free beam under shock excitation. The energy harvester is used for scavenging vibration energy dissipated by the NES. Grounded and ungrounded configurations are examined and the systems parameters are optimized globally to both maximize the dissipated energy by the NES and increase the harvested energy by piezoelectric element. A satisfactory amount of energy has been harvested as electric power in both configurations. The realization of nonlinear vibration control through one-way irreversible nonlinear energy pumping and optimizing the system parameters result in acquiring up to 78 percent dissipation of the grounded system energy.     

The dynamic characteristics of harvesting energy from mechanical vibration via piezoelectric conversion

As an alternative power solution for low-power devices, harvesting energy from the ambient mechanical vibration has received increasing research interest in recent years. In this paper we study the transient dynamic characteristics of a piezoelectric energy harvesting system including a piezoelectric energy harvester, a bridge rectifier, and a storage capacitor. To accomplish this, this energy harvesting system is modeled, and the charging process of the storage capacitor is investigated by employing the in-phase assumption The results indicate that the charging voltage across the storage capacitor and the gathered power increase gradually as the charging process proceeds, whereas the charging rate slows down over time as the charging voltage approaches to the peak value of the piezoelectric voltage across the piezoelectric materials. In addition, due to the added electrical damping and the change of the system natural frequency when the charging process is initiated, a sudden drop in the vibration amplitude is observed, which in turn affects the charging rate. However, the vibration amplitude begins to increase as the charging process continues, which is caused by the decrease in the electrical damping （i.e., the decrease in the energy removed from the mechanical vibration）. This electromechanical coupling characteristic is also revealed by the variation of the vibration amplitude with the charging voltage.     

带有分数阶阻尼的压电能量采集系统相干共振

研究了含分数阶阻尼的双稳态能量采集系统的相干共振. 建立了带有分数阶阻尼的轴向受压梁压电能量采集系统动力学模型. 对于分数阶方程, 采用Euler-Maruyama-Leipnik方法进行求解, 计算了不同阻尼阶数下的能量采集系统的信噪比、响应均值、跃迁数目等统计物理量. 结果表明: 此压电能量采集系统在随机激励下可以实现相干共振, 阻尼阶数对相干共振的临界噪声强度和相干共振幅值有很大影响. 关键词： 分数阶阻尼 随机激励 能量采集系统 相干共振     

Enhanced vibration energy harvesting using dual-mass systems

A type of dual-mass vibration energy harvester, where two masses are connected in series with the energy transducer and spring, is proposed and analyzed in this paper. The dual-mass vibration energy harvester is proved to be able to harvest more energy than the traditional single degree-of-freedom (dof) one when subjected to harmonic force or base displacement excitations. The optimal parameters for maximizing the power output in both the traditional and the new configurations are discussed in analytical form while taking the parasitic mechanical damping of the system into account. Consistent of the previous literature, we find that the optimal condition for maximum power output of the single dof vibration energy harvester is when the excitation frequency equals to the natural frequency of the mechanical system and the electrical damping due to the energy harvesting circuit is the same as the mechanical damping. However, the optimal conditions are quite different for the dual-mass vibration energy harvester. It is found that two local optimums exist, where the optimal excitation frequency and electrical damping are analytically obtained. The local maximum power of the dual-mass vibration energy harvester is larger than the global maximum power of single dof one. Moreover, at certain frequency range between the two natural frequencies of the dual-mass system, the harvesting power always increases with the electrical damping ratio. This suggests that we can obtain higher energy harvesting rate using dual-mass harvester. The sensitivity of the power to parameters, such as mass ratio and tuning ratio, is also investigated.     

随机激励下双稳态压电俘能系统的相干共振及实验验证

随着压电晶体材料的迅速发展, 基于压电效应的能量采集系统是俘获环境中的宽带随机振动能量的一种有效途径. 研究了有限宽带随机激励作用下, 磁斥力双稳态压电俘能系统的相干共振俘能机理, 并进行了实验验证. 运用Euler-Maruyama方法求解了随机非线性压电振动耦合方程, 比较分析了相干共振发生前后系统的动力学特性和俘能效率, 然后基于Kramers逃逸速率解释了相干共振. 最后的随机振动实验结果验证了双稳态压电俘能系统的相干共振俘能机理. 并且观察到: 当相干共振发生时, 系统会在两个势能阱之间剧烈运动, 此时宽带随机振动能量会被转化为大幅值窄带低频振动响应, 从而极大地提高了宽带随机振动能量的俘获效率.     

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.     

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.     

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.     

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.     

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.     

Study of broad bandwidth vibrational energy harvesting system with optimum thickness of PET substrate

In this paper, we present the development of a flexible PET-based (polyethylene terephthalate; PET) vibrational energy harvesting system with broad bandwidth. This broad bandwidth harvesting system comprises of four units of individual ZnO (zinc oxide) piezoelectric harvester in the form of a cantilever structure connected in parallel, and rectifying circuit with storage module. This system has ability to convert mechanical energy into electrical energy from the varying ambient vibration. The design and simulation of a piezoelectric cantilever plate was described by using commercial software ANSYS FEA (Finite Element Analysis) to determine the optimum thickness of PET substrate, internal stress distribution, operation frequency and electric potential. With the optimum thickness predicted by developed accurate analytical formula analysis, the one-way mechanical strain that is efficient to enhance the induced electric potential can be controlled within the piezoelectric ZnO layer. In addition, the relationship among the model solution of piezoelectric cantilever plate equation, vibration-induced electric potential and electric power was realized. An individual piezoelectric harvester consists of flexible PET substrate, piezoelectric ZnO thin film with (002) c-axis preferred orientation, and selectively deposited UV-curable resin lump structure which is used to change the resonant frequency of the harvester. In combination with multi-harvesters and rectifying with storage module together, an energy harvesting system with broad bandwidth can be fabricated. One individual harvester achieves a maximum OCV (open-circuit voltage) up to 4 V with power density of 1.247 μW/cm³. So far, we succeeded in accomplishing a broad bandwidth system with operating frequency range within 100 Hz-450 Hz to enhance powering efficiency. When the DC voltage (direct current voltage) across a storage module is charged up to 1.55 V after rectification, a flash LED (light emitting diode) is driven.     

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.     

Energy harvesting under excitations of time-varying frequency

The design and optimization of energy harvesters capable of scavenging energy efficiently from realistic environments require a deep understanding of their transduction under non-stationary and random excitations. Otherwise, their small energy outputs can be further decreased lowering their efficiency and rendering many critical and possibly life saving technologies inefficient. As a first step towards this critical understanding, this effort investigates the response of energy harvesters to harmonic excitations of time-varying frequency. Such excitations can be used to represent the behavior of realistic vibratory environments whose frequency varies or drifts with time. Specifically, we consider a piezoelectric stack-type harvester subjected to a harmonic excitation of constant amplitude and a sinusoidally varying frequency. We analyze the response of the harvester in the fixed-frequency scenario then use the Jacobi-Anger's expansion to analyze the response in the time-varying case. We obtain analytical expressions for the harvester's response, output voltage, and power. In-depth analysis of the attained results reveals that the solution to the more complex time-varying frequency can be understood through a process which “samples” the fixed-frequency response curve at a discrete and fixed frequency interval then multiplies the response by proper weights. Extensive discussions addressing the effect of the excitation parameters on the output power is presented leading to some initial suggestions pertinent to the harvester's design and optimization in the sinusoidally varying frequency case.     

Influence of potential function asymmetries on the performance of nonlinear energy harvesters under white noise

To improve the broadband transduction capabilities of vibratory energy harvesters (VEHs) under random and non-stationary excitations, many researchers have resorted to purposefully introducing nonlinearities into the restoring force of the harvester. While performing this task, it is often very challenging to maintain a perfectly symmetric restoring force which yields a VEH with an asymmetric potential energy function. This paper investigates the influence of potential function asymmetries on the performance of nonlinear VEHs under white noise inputs. To that end, a quadratic nonlinearity is introduced into the restoring force and its influence on the mean output power of the harvester for mono- and bi-stable quartic potentials is investigated. It is shown that, for VEHs with a mono-stable quartic potential function, the mean output power increases with the degree of potential function asymmetry. On the other hand, for energy harvesters with a bi-stable quartic potential function, asymmetries in the restoring force appear to worsen performance especially for low to moderate noise intensities. When the noise intensity becomes sufficiently large, the influence of the potential function?s asymmetry on the mean power diminishes. Results also reveal that a VEH with a symmetric bi-stable quartic potential function produces higher mean power levels than the one with the most asymmetric mono-stable potential. As such, it is concluded that a VEH with a symmetric bi-stable potential is most desirable to improve performance under white noise.     

Higher-order stochastic averaging to study stability of a fractional viscoelastic column

Stochastic stability of a fractional viscoelastic column axially loaded by a wideband random force is investigated by using the method of higher-order stochastic averaging. By modelling the wideband random excitation as Gaussian white noise and real noise and assuming the viscoelastic material to follow the fractional Kelvin–Voigt constitutive relation, the motion of the column is governed by a fractional stochastic differential equation, which is justifiably and uniformly approximated by an averaged system of Itô stochastic differential equations. Analytical expressions are obtained for the moment Lyapunov exponent and the Lyapunov exponent of the fractional system with small damping and weak random fluctuation. The effects of various parameters on the stochastic stability of the system are discussed.     

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.