Investigations on the presence of electrical frequency on the characteristics of energy harvesters under base and galloping excitations

The characteristics and performance of piezoelectric energy harvesters concurrently subjected to galloping and base excitations when using a complex electrical circuit are studied. The considered energy harvester is composed of a bilayered cantilever beam with a square cylindrical structure at its tip. Euler–Bernoulli beam theory, nonlinear quasi-steady hypothesis, and Galerkin method are used to develop a reduced order model of this system. The electrical circuitry of the harvester consists of a load resistance, a capacitance, and an inductance. The impacts of the electrical components of the harvester’s circuitry, the wind speed, and the base excitation frequency and acceleration on the broadband characteristics of the harvester, quenching phenomenon, and appearance of new nonlinear behaviors are deeply investigated and discussed. Different methods were used to characterize the new nonlinear phenomena that took place due to mechanical electrical interaction like the power spectrum and time history. When both coupled frequencies of electrical and mechanical types exist and are far from each other, it is shown that the quenching phenomenon is only related to the coupled frequency of mechanical type. It was also proven that this configuration results on high harvested power with low displacement near the electrical frequency. On the other hand, for a well-defined choice of the electrical components, the results show that a broadband configuration of the harvester can be designed. It is also indicated that the quenching phenomenon interacts with the appearance of hysteresis regions that depends on the value of the base acceleration and initial conditions. Moreover, it was shown that the presence of this inductance may result in broadband system harvesting more power from both galloping and base excitation.     

Nonlinear characterization of concurrent energy harvesting from galloping and base excitations

An energy harvester is proposed to concurrently harness energy from base and galloping excitations. This harvester consists of a triangular cross-sectional tip mass attached to a multilayered piezoelectric cantilever beam and placed in an incompressible flow and subjected to a harmonic base excitation in the cross-flow direction. A coupled nonlinear-distributed-parameter model is developed representing the dynamics of the transverse degree of freedom and the generated voltage. The galloping force and moment are modeled by using a nonlinear quasi-steady approximation. Under combined loadings and when the excitation frequency is away from the global natural frequency of the harvester, the response of the harvester mainly contains these two harmonic frequencies. Thus, the harvester’s response is generally aperiodic and is either periodic with large period (i.e., period- \(n\) ), or quasi-periodic, or chaotic. To characterize the harvester’s response under a combination of vibratory base excitations and aerodynamic loading, we use modern methods of nonlinear dynamics, such as phase portraits, power spectra, and Poincaré sections. A further analysis is then performed to determine the effects of the wind speed, frequency excitation, base acceleration, and electrical load resistance on the performance of the harvester under separate loadings.     

Global nonlinear distributed-parameter model of parametrically excited piezoelectric energy harvesters

A global nonlinear distributed-parameter model for a piezoelectric energy harvester under parametric excitation is developed. The harvester consists of a unimorph piezoelectric cantilever beam with a tip mass. The derived model accounts for geometric, inertia, piezoelectric, and fluid drag nonlinearities. A reduced-order model is derived by using the Euler–Lagrange principle and Gauss law and implementing a Galerkin discretization. The method of multiple scales is used to obtain analytical expressions for the tip deflection, output voltage, and harvested power near the first principal parametric resonance. The effects of the nonlinear piezoelectric coefficients, the quadratic damping, and the excitation amplitude on the output voltage and harvested electrical power are quantified. The results show that a one-mode approximation in the Galerkin approach is not sufficient to evaluate the performance of the harvester. Furthermore, the nonlinear piezoelectric coefficients have an important influence on the harvester’s behavior in terms of softening or hardening. Depending on the excitation frequency, it is determined that, for small values of the quadratic damping, there is an overhang associated with a subcritical pitchfork bifurcation.     

Performance enhancement of wing-based piezoaeroelastic energy harvesting through freeplay nonlinearity

We investigate experimentally how controlled freeplay nonlinearity affects harvesting energy from a wing-based piezoaeroelastic energy harvesting system. This system consisits of a rigid airfoil which is supported by a nonlinear torsional spring (freeplay) in the pitch degree of freedom and a linear flexural spring in the plunge degree of freedom. By attaching a piezoelectric material (PSI-5A4E) to the plunge degree of freedom, we can convert aeroelastic vibrations to electrical energy. The focus of this study is placed on the effects of the freeplay nonlinearity gap on the behavior of the harvester in terms of cut-in speed and level of harvested power. Although the freeplay nonlinearity may result in subcritical Hopf bifurcations (catastrophic for real aircrafts), harvesting energy at low wind speeds is beneficial for designing piezoaeroelastic systems. It is demonstrated that increasing the freeplay nonlinearity gap can decrease the cut-in speed through a subcritical instability and gives the possibility to harvest energy at low wind speeds. The results also demonstrate that an optimum value of the load resistance exists, at which the level of the harvested power is maximized.     

Nonlinear dynamic analysis of cantilevered piezoelectric energy harvesters under simultaneous parametric and external excitations

The nonlinear dynamics of cantilevered piezoelectric beams is investigated under simultaneous parametric and external excitations. The beam is composed of a substrate and two piezoelectric layers and assumed as an Euler–Bernoulli model with inextensible deformation. A nonlinear distributed parameter model of cantilevered piezoelectric energy harvesters is proposed using the generalized Hamilton's principle. The proposed model includes geometric and inertia nonlinearity, but neglects the material nonlinearity. Using the Galerkin decomposition method and harmonic balance method, analytical expressions of the frequency–response curves are presented when the first bending mode of the beam plays a dominant role. Using these expressions, we investigate the effects of the damping, load resistance, electromechanical coupling, and excitation amplitude on the frequency–response curves. We also study the difference between the nonlinear lumped-parameter and distributedparameter model for predicting the performance of the energy harvesting system. Only in the case of parametric excitation, we demonstrate that the energy harvesting system has an initiation excitation threshold below which no energy can be harvested.We also illustrate that the damping and load resistance affect the initiation excitation threshold.     

面内压电振动能量采集动力学设计与性能研究

压电振动能量采集将环境中普遍存在的机械能转换为电能，可以实现自供能传感、控制与驱动，具备灵活、节能环保、可持续的优势，具有广阔的应用前景。为了促进压电振动能量采集器件的集成与融合，提出面内压电振动能量采集，将压电振动能量采集器进行扁平化设计，使其在二维平面内采集振动能量，在保证较大功率输出下能够显著减小器件所需三维空间。为了提高输出功率与工作频宽，设计了具有双稳态与力放大机制的面内压电振动能量采集器。考虑弯张小变形，通过能量法建立了面内压电振动能量采集器的机电耦合动力学模型。分析了关键设计参数对面内压电振动能量采集器性能的影响。数值仿真了面内压电振动能量采集器在简谐激励下的俘能性能，结果表明，通过合理的设计，面内压电振动能量采集器可以低频、宽频弱激励下有效俘获能量。面内压电振动能量采集设计方法有利于推动便携式、可穿戴式自供能等方面的应用和产业化。     

Dynamic analysis of piezoelectric energy harvester under combination parametric and internal resonance: a theoretical and experimental study

In this work, theoretical and experimental analysis of a piezoelectric energy harvester with parametric base excitation is presented under combination parametric resonance condition. The harvester consists of a cantilever beam with a piezoelectric patch and an attached mass, which is positioned in such a way that the system exhibits 1:3 internal resonance. The generalized Galerkin’s method up to two modes is used to obtain the temporal form of the nonlinear electromechanical governing equation of motion. The method of multiple scales is used to reduce the equations of motion into a set of first-order differential equations. The fixed-point response and the stability of the system under combination parametric resonance are studied. The multi-branched non-trivial response exhibits bifurcations such as turning point and Hopf bifurcations. Experiments are performed under various resonance conditions. This study on the parametric excitation along with combination and internal resonances will help to harvest energy for a wider frequency range from ambient vibrations.

     

曲梁压电俘能器强迫振动的格林函数解

本文运用格林函数法求解了曲梁压电俘能器在强迫振动下的解析解.运用微分法分析了压电层合曲梁结构面内各内力,根据曲梁压电 俘能器的动力学方程组,基于压电本构关系,建立了包含径向阻尼但不考虑俘能器曲梁结构部分的轴向力以及轴向惯性项的Prescott力 电耦合模型. 采用Laplace变换法求得了耦合振动方程的格林函数解.根据叠加原理和格林函数的物理意义,对耦合的系统方程解耦进而 求得强迫振动下曲梁压电俘能器的输出电压. 数值计算中,通过与现有文献的解析解进行对比,验证了本文解析解的有效性,并研究了阻 尼、电阻等重要物理参数对压电函数和谐振频率的影响.通过与有关传统直梁压电俘能器研究成果的对比,体现了曲梁压 电俘能器Prescott模型的高效集能特性. 数值分析研究表明:(1)使得曲梁俘能器达到最大输出电压时连接的最优负载电 阻为1 M$\Omega$;(2)通过更换适当的基底材料,降低材料的弹性模量,可以改变曲梁俘能器的高基频现象,以使结构适应 更复杂的工作环境,但这会导致俘能器的工作效率降低.      

Design of piezoaeroelastic energy harvesters

We design a piezoaeroelastic energy harvester consisting of a rigid airfoil that is constrained to pitch and plunge and supported by linear and nonlinear torsional and flexural springs with a piezoelectric coupling attached to the plunge degree of freedom. We choose the linear springs to produce the minimum flutter speed and then implement a linear velocity feedback to reduce the flutter speed to any desired value and hence produce limit-cycle oscillations at low wind speeds. Then, we use the center-manifold theorem to derive the normal form of the Hopf bifurcation near the flutter onset, which, in turn, is used to choose the nonlinear spring coefficients that produce supercritical Hopf bifurcations and increase the amplitudes of the ensuing limit cycles and hence the harvested power. For given gains and hence reduced flutter speeds, the harvested power is observed to increase, achieve a maximum, and then decrease as the wind speed increases. Furthermore, the response undergoes a secondary supercritical Hopf bifurcation, resulting in either a quasiperiodic motion or a periodic motion with a large period. As the wind speed is increased further, the response becomes eventually chaotic. These complex responses may result in a reduction in the generated power. To overcome this adverse effect, we propose to adjust the gains to increase the flutter speed and hence push the secondary Hopf bifurcation to higher wind speeds.     

Effects of nonlinear piezoelectric coupling on energy harvesters under direct excitation

A nonlinear analysis of an energy harvester consisting of a multilayered cantilever beam with a tip mass is performed. The model takes into account geometric, inertia, and piezoelectric nonlinearities. A combination of the Galerkin technique, the extended Hamilton principle, and the Gauss law is used to derive a reduced-order model of the harvester. The method of multiple scales is used to determine analytical expressions for the tip deflection, output voltage, and harvested power near the first global natural frequency. The results show that one- or two-mode approximations are not sufficient to produce accurate estimates of the voltage and harvested power. A parametric study is performed to investigate the effects of the nonlinear piezoelectric coefficients and the excitation amplitude on the system response. The effective nonlinearity may be of the hardening or softening type, depending on the relative magnitudes of the different nonlinearities.     

Enhanced galloping energy harvester with cooperative mode of vibration and collision

The low power and narrow speed range remain bottlenecks that constrain the application of small-scale wind energy harvesting. This paper proposes a simple, low-cost, and reliable method to address these critical issues. A galloping energy harvester with the cooperative mode of vibration and collision (GEH-VC) is presented. A pair of curved boundaries attached with functional materials are introduced, which not only improve the performance of the vibration energy harvesting system, but also convert more mechanical energy into electrical energy during collision. The beam deforms and the piezoelectric energy harvester (PEH) generates electricity during the flow-induced vibration. In addition, the beam contacts and separates from the boundaries, and the triboelectric nanogenerator (TENG) generates electricity during the collision. In order to reduce the influence of the boundaries on the aerodynamic performance and the feasibility of increasing the working area of the TENG, a vertical structure is designed. When the wind speed is high, the curved boundaries maintain a stable amplitude of the vibration system and increase the frequency of the vibration system, thereby avoiding damage to the piezoelectric sheet and improving the electromechanical conversion efficiency, and the TENG works with the PEH to generate electricity. Since the boundaries can protect the PEH at high wind speeds, its stiffness can be designed to be low to start working at low wind speeds. The electromechanical coupling dynamic model is established according to the GEH-VC operating principle and is verified experimentally. The results show that the GEH-VC has a wide range of operating wind speeds, and the average power can be increased by 180% compared with the traditional galloping PEH. The GEH-VC prototype is demonstrated to power a commercial temperature sensor. This study provides a novel perspective on the design of hybrid electromechanical conversion mechanisms, that is, to combine and collaborate based on their respective characteristics.

     

附磁压电悬臂梁流致振动俘能特性分析

流致振动蕴含巨大的能量, 本文基于流致振动理论,设计了一种附加磁力激励的压电悬臂梁流致振动俘能器,并通过理论和实验研究其振动俘能特性.该俘能器由压电悬臂梁、圆柱绕流体和磁铁组成;首先基于Euler-Bernoulli梁理论,推导了流致振动附磁压电俘能器的能量函数,利用Hamilton原理建立了流致振动附磁压电俘能器的机电耦合方程;利用数值方法研究详细分析了流速、圆柱绕流体直径和长度、磁间距、磁极和外接电阻等系统参数对压电俘能器振动特性和输出电压的影响.分析结果表明, 该型压电俘能器的振动幅值在低流速条件下产生涡激振动,并产生最大的输出电压;磁力可以降低压电俘能器的共振频率并能够拓宽压电俘能器频带带宽,因此,附磁压电俘能器具有相比没有附磁的压电俘能器更适用于低速层流环境;实验结果与数值结果吻合较好,验证了附磁压电悬臂梁流致振动俘能器的理论分析的正确性.      

参数和直接激励下非线性压电俘能器性能分析的多尺度法

考虑几何非线性、阻尼非线性和梁的轴向不可伸长条件,利用Hamilton变分原理,建立了参数激励和直接激励下压电俘能器的非线性力电耦合的运动微分方程;利用Galerkin法,将所建立的动力学偏微分方程降阶为力电耦合的Mathieu-Duffing型方程;采用多尺度法获得了梁的位移和输出电压的解析表达式,给出了解的稳定性条件;利用解析表达式研究了单独参数激励以及参数激励和直接激励共同作用下阻尼系数对压电俘能器性能的影响。结果表明,在参数激励情况下,线性阻尼会显著影响超临界分岔点的位置,非线性二次阻尼不会影响超临界分岔点的位置。参数激励和直接激励的结合可以作为提升压电能量俘获器性能的解决方案。     

Hopf bifurcation analysis of reaction–diffusion neural oscillator system with excitatory-to-inhibitory connection and time delay

The active control approach generally requires power input to suppress vibrations of structures, while the conventional passive manner often causes waste of energy after transferring vibrations of the primary structure to the auxiliary system. In this work, an innovative control strategy based on energy harvesting for efficiently suppressing the cross-flow-induced vibrations such as galloping is proposed. The novel design facilitates the harvester of not only alleviating the oscillation of the primary structure but also seizing the transferred vibrational energy. An analytical model for the coupled nonlinear dynamical system is established by utilizing the Euler–Lagrange principle and implementing the Galerkin discretization. The impacts of the electrical load resistance and tip mass of the energy harvester on the coupled frequency, damping, and the onset speed of instability of the coupled multi-mode system are investigated in details. The results show that there exists an optimal load resistance for each tip mass which maximizes the onset speed of galloping. For control purposes, it is found that there is a well-defined tip mass of the energy harvester at which the coupled system has the highest onset speed of instability, and hence, the bluff body has the lowest vibration amplitude for all considered load resistances. However, to efficiently harvest energy and control the bluff body, both the tip mass of the energy harvester and electrical load resistance can be accurately determined.     

Enhancement of power harvesting from piezoaeroelastic systems

We investigate the effects of varying the eccentricity between the gravity axis and the elastic axis on the level of energy harvested from a piezoaeroelastic energy harvester consisting of a pitching and plunging rigid airfoil supported by nonlinear springs. The normal form of the dynamics of the harvester near the Hopf bifurcation is used to determine the critical nonlinear coefficients of the springs and maximize the harvested power for different eccentricities. Two configurations are evaluated in terms of the power generated from limit cycle oscillations and a range of operating wind speeds. The impact of the load resistance on the harvested power is also assessed.     

Phenomena and modeling of piezoelectric energy harvesting from freely oscillating cylinders

We investigate energy harvesting from vortex-induced vibrations of a freely moving rigid circular cylinder with a piezoelectric transducer attached to its transverse degree of freedom. The power levels that can be generated from these vibrations and variations of these levels with the freestream velocity are determined. A mathematical model that accounts for the coupled lift force, cylinder motion, and harvested voltage is presented. Linear analysis is performed to determine the effect of the electrical load resistance of the transducer on the natural frequency of the cylinder and the onset of synchronization (the shedding frequency is equal to the cylinder oscillating frequency) region. The impact of the nonlinearities on the cylinder response and harvested energy is investigated. The results show that the load resistance shifts the onset of synchronization to higher freestream velocities. For two different system parameters, the results show that the nonlinearities result in a hardening behavior for some values of the load resistance.     

压电俘能器结构及其力/电耦合作用分析

介绍了一种新的切实可行的能量俘获方法,即通过压电结构的力/电转换功能从环境振动中提取能量,实现微电子器件的无线供能。这种由压电结构制作的能从环境振动中提取能量的声波器件称为压电俘能器,可分为两类：一类是压电结构在环境振动激励下所输出的功率直接供给微电子器件工作,不进行能量储存,这类俘能器称为第一类压电俘能器（piezoelectric power harvester）;另一类是环境振动较弱,压电结构的输出功率低于器件的瞬时耗能,考虑到某些器件大部分时间处于休眠状态,而俘能器却随时可以从环境振动中提取能量,因此经过一段时间的能量累积后,由俘能器所储存的能量仍能满足器件的短期工作耗能。显然,这类俘能器需要具备能量储存功能,称为第二类压电俘能器（piezoelectric energy harvester）。本文详细介绍了压电俘能器结构以及两类俘能器的不同分析方法,阐述了提高俘能效率的有效措施并揭示了相关的改进机理,对压电俘能器的设计和应用具有重要意义。     

等效刚度对非线性压电俘能系统性能影响研究

基于Hamilton原理，考虑几何非线性和梁的不可伸长条件，建立了五层压电双晶片叠合梁俘能器在直接和参数激励作用下的运动微分方程。利用Galerkin法和谐波平衡法获得了俘能器的位移、输出电压和输出功率的解析解。引入随时间变化的扰动，提出了非线性方程解的稳定性条件。为了对压电俘能器的结构-性能关系进行综合分析，研究了被动层的配置形式、被动层与主动层的厚度比和弹性模量对压电俘能系统性能的影响。结果表明，在叠合梁厚度不变的情况下，采用五层的压电双晶片叠合结构，选择合理的被动层与主动层厚度比、被动层弹性模量、被动层厚度比和负载电阻，可以有效提高能量俘获的效率。     

附磁阶梯变厚度悬臂梁压电俘能器的理论建模及分析

论文建立了一种附磁阶梯变厚度压电悬臂梁的动力学模型并分析了系统的俘能特性。基于Euler-Bernoulli梁理论分段建立系统能量函数并引入非线性磁势能，利用Lagrange方程建立了系统机电耦合动力学方程；利用数值方法分析了磁间距对系统振动特性的影响，此外还研究了系统单稳态和双稳态响应，探讨了厚度比、长度比、磁间距和外激励幅值对系统动力学响应和俘能特性的影响。结果表明，磁间距是影响系统势能的主要因素，调节磁间距可使系统产生单稳态和双稳态响应，从而有效提高俘能器俘能特性；与传统等截面悬臂梁压电俘能器相比，通过优化结构参数，附磁阶梯变厚度悬臂梁压电俘能器能够发生明显的非线性振动现象，实现宽频带振动能量采集。     

动态多稳态压电风能俘能结构设计与实验验证

传统的线性颤振式风能俘能结构在变风速环境下转换效率不高。针对此问题，本文提出了一种动态多稳态颤振式压电俘能结构，可在较宽的风速范围内保持较高的电压输出。该结构由矩形平板、压电悬臂梁与3块永磁体组成，并通过磁吸力实现结构的多稳态。通过在不同风速下开展的能量转换特性实验研究，发现当风速较小时，该结构具有双稳态特性；而当风速较大时，会出现新的稳定位置，结构变为三稳态系统。这样的动态稳定位置，可以保证结构在很宽的风速范围内出现阱间跳跃，进而保持大的电能输出。实验结果表明，这种动态多稳态结构，风速在2.0m/s—7.5m/s区间变化时，能够激发并保持阱间跳跃，甚至相干共振，产生较大的电能输出。