Nonlinear Vibration Energy Harvester Research Articles

Experimentally validated passive nonlinear capacitor in piezoelectric vibration applications

Abstract Piezoelectric vibration isolation and energy harvesting applications have been extensively studied in the literature. The studies include linear and nonlinear approaches. Linear methods are simpler but possess inherent limitations. On the other hand, nonlinear ones could perform better over a broader operating frequency range. Nonlinearity can be introduced in the mechanical domain or electrical domain actively or passively. Since electrical components can be on smaller scales compared to mechanical counterparts, inducing nonlinearity on the mechanical system through the electrical domain can be more practical. Moreover, passive structures require no energy supply and controller therefore they are simpler and more reliable than active ones. In this paper, a novel way to attain passive hardening stiffness was suggested by introducing an electrical component in a shunt circuit for passive nonlinear piezoelectric vibration isolation or energy harvesting applications and the induced structural non-linearity is demonstrated experimentally. A passive nonlinear component is suggested to be a hardening capacitor obtained by the P–N junction. An analytic model is derived for parallel connected macro-fiber composite (MFC) piezoelectric material attached bimorph configuration on a cantilever beam and the model is solved numerically. MFC integrated bimorph model, and P–N junction approximate model are presented. The frequency response of the coupled system is obtained by using numerical models and experiments. Both numerical analysis and experiments validated the hardening stiffness effect of the P–N junction. To the best of the authors’ knowledge, this study is the first study to demonstrate that nonlinear capacitance of P–N junctions can be used to attain nonlinearity in a mechanical system.

Role of inertial nonlinearity and coupling stiffness on a series of coupled harvesters

This study investigates the effect of coupling configuration on a state-of-the-art harvesting system consisting of a series of spring-coupled piezoelectric nonlinear monostable tip-loaded cantilever-based vibration energy harvesters (VEHs). The investigation finds favorable coupling configurations that exploit complex nonlinear interaction for improving system performance. The system model incorporates the beam's geometric nonlinearities in the governing equation using inertial and stiffness nonlinear terms. The study first analytically investigates the effect of grounding stiffness on a single harvester's performance. Then, it numerically investigates the effect of increasing array size and changing stiffness distribution. The study discovers a novel hardening/softening transition phenomenon at a critical grounding stiffness value induced by the inertial nonlinear effect. This critical event also maximizes peak output power near the largest eigenfrequency at an optimum grounding stiffness value. At larger arrays, the peak power shows a complex bifurcation behavior, hinting at the presence of multiple attractors. The numerical study of different stiffness configurations and linearized eigenanalysis reveals the necessity of a multilevel optimization strategy focused on both the stiffness ratio between the coupling springs' and their overall stiffness level. This study bridges the gap between previous studies on spring-coupled linear and bistable arrays, offering new insights into coupled nonlinear interactions and proposing novel optimization strategies.

Nonlinear vibration energy harvesting via parametric excitation: Snap-through with time-varying potential wells

This study introduces a novel method to improve snap-through transitions in a parametrically excited cantilever-based energy harvester featuring repulsive magnetic coupling. The aim is to enhance the efficiency of a piezoelectric energy harvester while broadening its operational frequency range. The design incorporates an oscillating substructure that magnetically interacts with the beam, promoting easier transitions. Additionally, this setup capitalizes on the resonant energy of the substructure to optimize energy capture. This study addresses a previously unexplored limitation in the directly excited counterpart by optimizing the use of substructure’s resonant energy to significantly improve the energy harvesting device’s efficiency. We present a theoretical model based on the Hamilton principle, which is solved numerically. Comprehensive experimental studies are conducted to validate the theoretical results, showing a strong correlation between the experimental outcomes and theoretical predictions. The findings indicate that an oscillating substructure with a tailored natural frequency close to twice that of the parametrically excited piezoelectric beam can induce time-varying potential wells to facilitate snap-through transitions. Despite the damping coefficient of the piezoelectric beam being three times higher than that of its simple beam counterpart, periodic perturbations still effectively expand its instability region. This enlarged instability region, resulting from such perturbations, not only exhibits broadband characteristics but also facilitates efficient energy harvesting at lower excitation levels.

Response and reliability analysis of a nonlinear VEH systems with FOPID controller by improved stochastic averaging method and LBFNN algorithm

In engineering, it is an important issue to guarantee the normal working of a vibration energy harvesting (VEH) device to harvest electricity. However, the determination of the system dynamics is a huge challenge due to random excitation, nonlinear structure, and control force. This paper aims to study the response and reliability of a kind of nonlinear and coupled VEH system with an FOPID controller. The stationary response is analyzed by an improved stochastic averaging method, in which an amplitude-dependent frequency is taken into account during the averaging procedure in the case of the strongly nonlinear system. Time-varying reliability in the VEH system is analyzed by using a new LBFNN algorithm, and weight coefficients at each time in this algorithm is obtained through an iteration procedure. Numerical results show that there are essential differences between the weakly nonlinear system and the strongly nonlinear one in terms of averaging procedure and response probability. The fractional orders in the FOPID controller are critical parameters to bring stochastic bifurcations. In addition, strongly nonlinear structure together with lower-order fractional derivative and higher-order fractional integration in FOPID controller are helpful to enhance the reliability performance of the VEH system.

Response analysis of asymmetric monostable energy harvester with an uncertain parameter

The nonlinear piezoelectric vibration energy harvesting system offers a solution to the power supply challenge for low-power sensors. However, In practical engineering, structural parameter uncertainties arise from manufacturing errors, environmental temperature fluctuations, and changes in material properties. Besides, due to inherent flaws in the structure and materials, asymmetry may occur in the potential well, which leads to the complexity of the dynamic behavior exhibited in energy harvesting system. In this paper, in order to investigate the dynamics of the asymmetric monostable piezoelectric energy harvesting system under an uncertain parameter, the orthogonal polynomial approximation method is applied to transform the stochastic system into an equivalent deterministic system. Then, the responses of the equivalent deterministic system are used to reveal the stochastic behavior of the uncertainty system. The results indicate that, when the uncertainty intensity of the cubic nonlinear coefficient is 0, an increase in the asymmetry level of the potential well leads to an increase in the output voltage. Increasing the other parameters, the output voltage is reduced. In addition, as the uncertainty intensity increases, the parameter intervals of chaotic response become wider and the energy harvesting efficiency decreases. Moreover, as the potential well asymmetry increases, the sensitivity of energy harvesting system to uncertain parameter increases.

Stochastic dynamics of a nonlinear vibration energy harvester subjected to a combined parametric and external random excitation: The distinct cases of Itô and Stratonovich stochastic integration

The Stratonovich and Itô interpretations of a stochastic integral are mathematically consistent, distinct representations that yield contrasting response characteristics for stochastic dynamical systems subjected to multiplicative noise, such as parametric random excitation. The contrasting dynamics from the two interpretations currently remain unexplored in vibration energy harvester dynamics. We analytically investigate the dynamics of a generic piezoelectric, nonlinear vibration energy harvester simultaneously subjected to stochastic damping and external random excitation, focusing on the contrasting dynamics engendered by the two interpretations. Numerically solving the Stratonovich and Itô stochastic differential equations in the nonequilibrium regime of harvester dynamics, we find that the Stratonovich version yields significantly higher root mean square values of the harvester voltage output, for lower values of the mechanical damping coefficient. Furthermore, we find positive values for the leading Lyapunov exponent indicating instabilities in the harvester response unique to the Stratonovich interpretation. Additionally, comparing the results for the harvester electrical output between the cases of external excitation represented by the white noise and the Ornstein–Uhlenbeck noise processes, we find lower root mean square values of the voltage output in the latter case. Moreover, the results indicate that the averaged voltage output becomes progressively lower with increasing correlation time of the Ornstein–Uhlenbeck process. Studying the equilibrium dynamics by deriving Fokker–Planck equations for the response amplitude of the harvester using the harmonic balance method and stochastic averaging techniques, we solve for the stationary probability densities to find that the harvester is more likely to attain a well-defined equilibrium state under the Itô interpretation. In summary, the results contribute to the theoretical understanding of the fundamental contrast between the Stratonovich and Itô interpretations in vibration energy harvester dynamics and are expected to motivate advances in harvester design for various applications.

Nonlinear electromagnetic vibration energy harvester comprising dual helical-plane springs and multiple Halbach arrays for low-frequency and small-amplitude vibrations

Electromagnetic vibration energy harvesters are widely investigated for self-powered wireless sensors and nonlinearity has been introduced for low-frequency and broadband vibration energy harvesting. But how to realize the nonlinearity with small amplitudes, high reliability and few complexities is still a challenge in real-world applications. In this paper, a novel electromagnetic vibration energy harvester comprising dual helical-plane springs and multiple Halbach arrays is proposed. The structural characteristics are analyzed, which indicate that the helical-plane springs can have nonlinear stiffness under small-amplitude vibrations and multiple Halbach arrays can greatly enhance the magnetic field. Then a magnet-electro-mechanical model is built by combining mechanical dynamics &electrodynamics, which is numerically solved by using the Runge-Kutta algorithm. Finally, the feasibility of the nonlinear electromagnetic vibration energy harvester is validated both numerically and experimentally. The results show that it has a nonlinear stiffness, a resonance bandwidth of 3 Hz and a peak power of 14 mW when the excitation amplitude is only 0.5 g. In particular, the resonance frequency range depends on the excitation amplitude. Furthermore, the prototype of a self-powered wireless temperature sensor is constructed and testified. The results of this study indicate that the proposed structure can be utilized and extended to build compact, reliable and nonlinear electromagnetic vibration energy harvesters for low-frequency, small-amplitude and broadband vibrations in real-world applications.

Optimized and robust orbit jump for nonlinear vibration energy harvesters

Optimized and robust orbit jump for nonlinear vibration energy harvesters

Investigative Study of the Effect of Damping and Stiffness Nonlinearities on an Electromagnetic Energy Harvester at Low-Frequency Excitations

Ambient vibration energy is widely being harnessed as a source of electrical energy to drive low-power devices. The vibration energy harvester (VEH) of interest employs an electromagnetic transduction mechanism, whereby ambient mechanical vibration is converted to electrical energy. The limitations affecting the performance of VEHs, with an electromagnetic transduction structure, include its operational bandwidth as well as the enclosure-size constraint. In this study, an analysis and design of a nonlinear VEH system is conducted using the Output Frequency Response Function (OFRF) representations of the actual system model. However, the OFRF representations are determined from the Generalised Associated Linear Equation (GALE) decompositions of the system of interest. The effect of both nonlinear damping and stiffness characteristics, to, respectively, extend the average power and operational bandwidth of the VEH device, is demonstrated.

Combining orbit jump and potential wells optimizations for nonlinear vibration energy harvesters

Nonlinear vibration energy harvesters (VEHs) are widely used for scavenging vibrational energy due to their broadband behaviors. However, they exhibit multiple orbits of different powers for a given excitation, including low-power orbits that might limit their performance. To address this issue and enhance nonlinear VEHs performance, various studies have defined orbit jump strategies to transition from low-power to high-power orbits. Another way to maximize the power of nonlinear VEHs is to optimize their geometry by finely engineering their potential wells (PWs). In this letter, we propose an orbit jump strategy for bistable VEHs that combines the two latter approaches, i.e. that simultaneously optimizes their PWs while jumping from low-power to high-power orbits. This orbit jump strategy is optimized using a numerical criterion that takes into account the robustness of the jumps and the invested energy. The proposed orbit jump strategy has been experimentally validated for vibration frequencies between 30 and 60 Hz. It is shown that the proposed approach can increase the power by an average of 121 times over the considered frequency range. Compared to traditional orbit jump strategies, the proposed approach, which combines orbit jumping and PWs optimizations, increases by up to three times the harvested power.

Frequency Response for Sweep-Up and Sweep-Down of Piezoelectric Energy Harvester

PurposeThe aim of this work is to study the frequency response of linear and nonlinear piezoelectric vibration energy harvester (pVEH) based on Duffing-type nonlinearity and Kirchhoff’s law, and the effect of the effective electromechanical coupling on the frequency response of nonlinear pVEH.MethodsThe analytical technique based on the harmonic balance method (HBM) is used to obtain the mechanical and electrical frequency responses of linear and nonlinear pVEH, and the Jacobian matrix is used to analyze the solutions’ stability. The Runge–Kutta fourth-order method as a numerical technique is used to solve the ordinary differential equations of pVEH.ResultsThe analytical calculations are completed, and the frequency response solution of displacement of pVEH is contained in real and complex values and those of electrical responses are presented. The harmonic balance results are compared with numerical simulations for various hardening nonlinear springs. Simple analytical expressions for the jump points are suggested for the sweep-up and sweep-down frequencies of pVEH.ConclusionsThe analytical analysis shows that the jump-down frequency ratios have definite behaviors for each low and high effective electromechanical coupling of hardening-nonlinear pVEHs. Moreover, the jump-up frequency ratio is directly proportional to the effective electromechanical coupling of hardening nonlinear pVEHs.

Theoretical and experimental investigation on the advantages of auxetic nonlinear vortex-induced vibration energy harvesting

Due to its wide applicability, vortex-induced vibration (VIV) from wind and water flow has been explored for piezoelectric energy harvesting. However, the broadband VIV energy harvesting at low wind speeds is still a great challenge. In this paper, an auxetic nonlinear VIV energy harvester (ANVEH) is proposed. For the first time, the advantages of monostable softening behavior using magnetic attraction are investigated. It is found theoretically and experimentally that the monostable softening can broaden the working wind speed range whereas decreasing the peak energy output. Thus, the auxetic structure is proposed to compensate the decrease of peak energy output with the monostable softening due to its negative Poisson’s ratio and high stress distribution. Utilizing the experimentally validated theoretical model, parametric investigations are undertaken to examine the effects of various factors, including the magnetic separation distance, the stiffness and mass of the harvester, the dimension of the bluff body, the connected resistance, and the electromechanical coupling coefficient. Results show that no matter how the system parameters vary, the ANVEH has the superior performance than the plain nonlinear VIV energy harvester (PNVEH).

Piezomagnetic vibration energy harvester with an amplifier

We study the effect of an amplification mechanism in a nonlinear vibration energy harvesting system where a ferromagnetic beam resonator is attached to the vibration source through an additional linear spring with a damper. The beam moves in the nonlinear double-well potential caused by interaction with two magnets. The piezoelectric patches with electrodes attached to the electrical circuit support mechanical energy transduction into electrical power. The results show that the additional spring can improve energy harvesting. By changing its stiffness, we observed various solutions. At the point of the optimal stiffness of the additional spring, the power output is amplified a few times depending on the excitation amplitude.

Data-driven modelling and dynamic analysis of the multistable energy harvester with non-Gaussian Lévy noise

In engineering, due to the complex structural characteristics of system and the non-Gaussian properties of random excitation, it is difficult to establish an accurate stochastic dynamic model for the strongly nonlinear multistable vibration energy harvester (VEH), especially for these driven by non-Gaussian Lévy noise. From the view of machine learning, a data-driven model identification method is devised to extract the non-Gaussian governing laws of the multistable VEH with the aid of the observed sample trajectory data. Based on the Nonlocal Kramers-Moyal formulas, the Lévy, drift and diffusion terms can be approximately expressed by the sample trajectories of the system. By implementing the least square method and the stepwise sparse regressor algorithm, the optimal drift and diffusion coefficients can be identified, and then the non-Gaussian stochastic differential equation of VEH is extracted. Two examples are utilized to verify the feasibility and effectiveness of the data-driven modelling method in VEH, which indicates that the identified results agree well with the original system. Finally, the stochastic dynamic behaviors induced by non-Gaussian Lévy noise are explored based on the data-driven penta-stable VEH. The proposed method can provide the theoretical guidance for the modelling and dynamics research of VEH in engineering.

Nonlinear energy harvesting from rotating system

A simple electromagnetic energy harvester with a cubic nonlinear stiffness is proposed and dynamic responses as well as the potential power harvested are studied in this paper. The proposed nonlinear electromagnetic vibration energy harvester can be installed on a host base that is rotating at a constant speed and vibrating vertically. Considering the combination of the vertically vibrating and rotating frequency of the host base, the 1:1 primary resonance is investigated by the method of multiple scales. The dynamic characteristics of responses in terms of system parameters are studied and bifurcation curves are constructed. In addition to possessing the resonance characteristics, some saddle–node bifurcations occur for the steady-state solutions under certain conditions. Moreover, the effects of system parameters on the output average power are investigated as well. The results of analysis show that multiple frequency bands and parameter regions exist where a large power can be harvested due to the nonlinear resonances. Introducing the nonlinearity not only can increase the power output but also expand the frequency bands and parameter regions to harvest more power. A carefully selected system parameters can help to optimize the average power harvested in the design.

Frequency tunable electromagnetic vibration energy harvester using piecewise linear nonlinearity

Vibration energy harvesting is increasingly being seen as a viable energy source to provide for our energy-dependent society. There has been great interest in scavenging previously unused or wasted energy in a large variety of systems including vibrating machinery, ocean waves and human motion. In this work, a bench-top system of a piecewise-linear nonlinear vibration energy harvester is studied. A similar idealized model of the system had previously been studied numerically, and in this work the method is adjusted to better account for the physical system. This new design is able to actively tune the system’s resonant frequency to match the current excitation through the adjustment of the gap size between the oscillator and mechanical stopper; thus maximizing the system response over a broad frequency range. This design shows an increased effective frequency bandwidth compared with traditional linear systems and improves upon current nonlinear designs that are less effective than linear harvesters at resonance. In this paper, the physical system is tested at various excitation conditions and gap sizes to showcase the new harvester design’s effectiveness.

A nonlinear electromagnetic vibration energy harvester lubricated by magnetic fluid for low-frequency vibration

Using vibration energy harvesters (VEHs) to achieve self-power is an effective method to ensure long-term use of sensor networks. This paper proposes a nonlinear electromagnetic VEH lubricated by magnetic fluid (MF) for low-frequency vibration energy harvesting. The VEH uses fixed small magnets to reduce natural frequency and prevent moving a magnet stack from flipping, MF for lubrication to improve output voltage under weak vibration, and elastomers to prevent magnet damage under strong vibration. In addition, MF and elastomers can increase the nonlinearity of restoring force, broadening the response frequency of VEHs. The motion state of the VEH is analyzed, and a crank-slider mechanism is used to test the output performance of VEHs. The influence of shell conductivity, the number of fixed small magnets, and length of VEH on output voltage is studied, and the results show that the VEH with an insulated shell, fewer fixed small magnets, and suitable length generates higher output voltage. The MF mainly plays a damping role in strong vibration but mainly plays a lubricating role in weak vibration. For experiments with 15 mm amplitude at 7.9 Hz, the output power density of the VEH without MF and lubricated by MF are 2.436 and 1.862 mW/cm3, respectively. MF damping reduces the output power density by 23.56%. However, for experiments with 7.5 mm amplitude at 3 Hz, the output power for VEHs without MF and lubricated by MF are 0.065 and 0.254 mW, respectively. The output power is increased by 291% with MF lubrication.

Electromechanical model and simple numerical analysis for a piezoelectric vibration energy harvester considering nonlinear piezoelectricity, nonlinear damping, and self-powered synchronized switch circuit

Establishment of an analytical technique that considers nonlinear piezoelectricity, nonlinear damping, and the connections to the self-powered synchronized switch circuit represents a challenging but practical issue in the development of piezoelectric vibration energy harvesters. The two-way coupled analysis method, which combines numerical software with a circuit simulator, can simulate the performance of the harvester, but it imposes a high computational load. We develop a numerical analysis technique that is more readily implemented to allow us to produce accurate predictions of the output power, displacement, and frequency response characteristics of this harvester with the switch circuit. First, we derive the electromechanical equation required for the miniature harvester, and we then determine the parameters in this equation using the harmonic balance method. Finally, the governing equation coupling the piezoelectric vibration energy harvester with the self-powered synchronized switch circuit is simplified, taking the form of a quartic equation with respect to the displacement. The simulation replicated a reduction in the displacement at higher resistive loads, wider variations in the resonance frequency, and the nonlinear stiffness softening characteristics that were observed experimentally. This paper provides a practical method for predicting the performance of the nonlinear piezoelectric vibration energy harvester with the synchronized switch circuit.

Nonlinear dynamics of asymmetric bistable energy harvesters

The paper investigates asymmetries effects over a nonlinear vibration energy harvester dynamics. The asymmetric system performance is compared with symmetric ones. Different asymmetry levels on restoring force and gravity action are investigated from a system-sloping angle variation. Bifurcation diagrams and basins of attraction are used to examine the local and global characteristics underlying dynamical systems under different excitation energy. The results show the adverse effects of asymmetries on system dynamics. They also reveal ways to overcome them by canceling asymmetric influence from optimal sloping angle values and improving asymmetric system performance over symmetrical ones. This comprehensive numerical study provides novel valuable insights into asymmetrical energy harvester dynamics, a wide and still less explored topic.

Bursting Oscillations in General Coupled Systems: A Review

In this paper, the bursting oscillation phenomenon in coupled systems with two time scales is introduced. Firstly, several types of bifurcation are briefly introduced: fold bifurcation, Hopf bifurcation, fold limit cycle bifurcation, homoclinic bifurcation, etc. The bursting oscillations of the system with two excitation terms and the bifurcation delay in the bursting oscillations are considered. Secondly, some simple bursting oscillations are introduced, such as fold/fold bursting, fold/supHopf bursting, subHopf/subHopf bursting, fold/LPC bursting, Hopf/LPC bursting, fold/homoclinic bursting, Hopf/homoclinic bursting, etc. At the same time, the system also has some complex bursting oscillations, such as asymmetric bursting, delayed bursting, bursting with hysteresis loop, etc. Finally, the practical applications of bursting oscillations, such as dynamic vibration absorbers and nonlinear vibration energy harvesting technology, are introduced.