Nonlinear Energy Harvesting System Research Articles

Improved energy harvesting by enhanced nonlinearities: New phenomena and experimental demonstration

This paper investigates a nonlinear piezoelectric energy harvesting system that enhances conversion of mechanical energy to electrical energy by integrating piezoelectric materials with a nonlinear system. The research simplifies a flexible beam with macro fiber composite (MFC) piezoelectric patches using a lumped-parameter approach and derives the dynamic equations of the system using the Lagrange equations. The incremental harmonic balance method (IHBM) is then employed to solve these equations, simulating the nonlinear dynamics involved in the harvested voltage. Key findings include the initial discovery of multiple voltage islands which are voltage jumps that improve energy conversion efficiency, confirmed by experimental data validating the IHBM simulations. The study reveals that these phenomena are due to the enhanced nonlinearities of the system, particularly the geometric nonlinearity caused by the spring k1 and the nonlinear magnetic force between magnets. Adjusting k1 allows the observation of multiple voltage islands, and increasing d0 results in a broad-frequency constant voltage, offering new insights. The research concludes that the nonlinear piezoelectric energy harvester can effectively convert vibration energy into electrical one, as demonstrated by successfully powering multiple light-emitting diodes (LEDs). Bifurcation diagrams illustrate the adaptability of the system, showing transitions among periodic, quasi-periodic, and chaotic states for different parameters.

Nonlinear energy harvesting system with multiple stability

The nonlinear energy harvesting systems of the forced vibration with an electron-mechanical coupling are widely used to capture ambient vibration energy and convert mechanical energy into electrical energy. However, the nonlinear response mechanism of the friction induced vibration (FIV) energy harvesting system with multiple stability and stick-slip motion is still unclear. In the current paper, a novel nonlinear energy harvesting model with multiple stability of single-, double- and triple-well potential is proposed based on V-shaped structure spring and the belt conveying system. The dynamic equations for the energy harvesting system with multiple stability and self-excited friction are established by using Euler-Lagrangian equations. Secondly, the static characteristics of the nonlinear restoring force, the friction force, and the potential energy surfaces are obtained to show the nonlinear stiffness, multiple equilibrium points, discontinuous behaviors and multiple well responses. Then, the equilibrium surface of bifurcation sets for the autonomous system is given to show the third-order quasi zero stiffness (QZS3), fifth-order quasi zero stiffness (QZS5), double well (DW) and triple well (TW). The co-dimension bifurcation sets of the self-excited vibration system are analyzed and the corresponding phase portraits for the coexistent of multiple limit cycles are obtained. Furthermore, the analytical formula of amplitude frequency response of the approximated system are obtained by the complex harmonic method. The response amplitudes of charge, current, voltage and power of the forced electron-mechanical coupled vibration system for QZS3, QZS5, DW and TW are analyzed by using the numerically solution. Finally, a prototype of FIV energy harvesting system is manufactured and the experimental system is setup. The experimental works of static restoring forces, damping forcse and the electrical outputs are well agreeable with the numerical results, which testified the proposed FIV energy harvesting model.

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.

Numerical analysis and experimental validation of nonlinear broadband monostable and bistable energy harvesters

Vibration based piezoelectric energy harvesting has been receiving more and more attention for supplying usable electrical energy to low-power electronic devices. This type of energy can produce the desired voltage to power any low electronic device or wireless sensor. But most of them provide lower voltage and insufficient power. In this paper, the nonlinear magnetic field is introduced to broaden the effective resonant bandwidth for overcoming this issue. The dynamic linear and nonlinear model of magnetic coupling piezoelectric vibration energy harvester is established based on the Hamilton principle, Euler-Bernoulli beam theory, piezoelectric theory, and Kirchhoff's law and law of mechanics. Numerical solutions of nonlinear energy harvesting systems are studied by using the Runge-Kutta method. The bifurcation diagram, Poincare map, power spectrum, and phase trajectory of voltage output are investigated with different excitation amplitudes and frequencies. Simulation of linear and nonlinear model were illustrated at given excitation levels. Simulation results show the harvesting performance can be improved by proper external magnetic coupling and potential energy function. The theoretical harmonic balance analysis results verify the numerical solution and influence the mechanism of excitation levels and interface resistances. Experimental verification is carried out to measure the nonlinear restoring force and examine the performance of nonlinear bistable and monostable energy harvesters. The experiment results show the nonlinear bistable and monostable energy harvesters can provide larger voltage and more power.

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.

Safety design study for energy harvesters

Modern offshore and onshore green energy engineering includes energy harvesting—as a result, extensive experimental investigations, as well as safety and reliability analysis are crucial for design and engineering. For this study, several wind-tunnel experiments under realistic in situ wind speed conditions have been conducted to examine the performance of galloping energy harvester. Next, a novel structural reliability approach is presented here that is especially well suited for multi-dimensional energy harvesting systems that have been either numerically simulated or analog observed during the representative time lapse, yielding an ergodic system time record. As demonstrated in this study, the advocated methodology may be used for risk assessment of dynamic system structural damage or failure. Furthermore, traditional reliability methodologies dealing with time series do not easily cope with the system’s high dimensionality, along with nonlinear cross-correlations between the system’s components. This study’s objective was to assess state-of-the-art reliability method, allowing efficient extraction of relevant statistical information, even from a limited underlying dataset. The methodology described in this study aims to assist designers when assessing nonlinear multidimensional dynamic energy harvesting system’s failure and hazard risks.

Performance analysis of nonlinear piezoelectric energy harvesting system under bidirectional excitations

In this paper, a cantilever beam piezoelectric energy harvester model of nonlinear partial differential equation based on Hamilton's Principle is established. A piezoelectric device using a bimorph cantilever beam with a fixed is subjected to horizontal and vertical excitation. Based on the Euler-Bernoulli thin beam model assumptions and inextensible deformation, the analysis further includes the effects of geometric nonlinearity and damping nonlinearity. The reduced-order nonlinear partial differential equations of motion with electro-mechanical coupling distributed parameter are derived by using Galerkin method. By using the method of multiple scales and solving the equations of motion, the energy harvester in its fundamental first-order resonance response is analyzed and the amplitudes of fundamental transverse deflection, output voltage and output power are determined. The major influence aspects of excitation amplitude, linear damping coefficient, nonlinear damping coefficient, impedance, nonlinear electro-mechanical coupling coefficient on transverse deflection, voltage and power are analyzed. The result shows that the linear damping coefficient and resistance affect the initial threshold of parametric excitation. The combination of parametric excitation and direct excitation gives full play to the advantages of parametric excitation and improve the energy conversion efficiency of the energy harvesting system.

Time-sharing orbit jump and energy harvesting in nonlinear piezoelectric energy harvesters using a synchronous switch circuit

Nonlinearity has enabled energy harvesting to advance toward higher power output and broader bandwidth in monostable, bistable, and multistable systems. However, operating in the preferable high-energy orbit (HEO) rather than the low-energy orbit (LEO) for making such advancement has restricted their applications. Based on a monostable nonlinear system, this paper proposes a self-contained solution for time-sharing orbit jump and energy harvesting. The joint dynamics of an electromechanical assembly consisting of a nonlinear energy harvester and a switched-mode piezoelectric interface circuit for high-capability energy harvesting is studied. The proposed solution is carried out by utilizing a cutting-edge switched-mode bidirectional energy conversion circuit (BECC), which enables time-sharing dual functions of energy harvesting and vibration exciting. A theoretical model is established based on impedance analysis and multiple time scales method to analyze the stability, frequency response, and phase evolution of the autonomous and nonautonomous nonlinear energy harvesting systems. In particular, the detailed dynamics for the orbit jumps with the vibration exciting mode of BECC are studied. Experiments are performed to validate the full-hysteresis-range orbit jumps with the monostable nonlinear energy harvester. The harvested power after orbit jumps yields a nine-fold increase, compensating for the energy consumption under vibration exciting mode quickly. The proposed solution also refrains the system from extra mechanical or electrical energy sources for orbit jumps, which leads to the first self-contained solution for simultaneous energy harvesting and orbit jump in nonlinear piezoelectric energy harvesting. This work enhances the practical utility of nonlinear energy harvesting technologies toward engineering applications.

A parametric frequency domain approach to analysis and design of critical design parameters of nonlinear energy harvesting systems: Parametric output spectrum and power generation functions

Nonlinear energy harvesting systems have been extensively studied which usually involve quite a few critical design parameters to determine. However, a straightforward parametric design method for such a purpose is still far from developed. Traditional methods with harmonic balance or numerical simulation usually cannot reveal a straightforward relationship between the energy harvesting performance and system parameters, and in most cases face relatively large computational costs during parametric studies. To this aim, a novel Volterra series-based frequency domain method is established in this study with respect to a typical nonlinear harvesting system. A concise parametric characteristic approach, namely the linear and nonlinear characteristic output spectrum (lnCOS) method, is thus proposed to achieve parametric characteristic expression for the nonlinear output frequency response and power generation function (PGF) of the system, referred to as the lnCOS function and the Parametric-PGF respectively. Compared with the harmonic balance method, the vibration responses/power generation can be obtained in a more straightforward way without solving complex algebraic equations, and the lnCOS method directly links the power outputs to critical system parameters in a concise parametric way. It is for the first time to provide explicit expressions for vibration responses and power generations with respect to structural parameters of a nonlinear energy harvesting system, which cannot be achieved in a traditional way. Based on the obtained explicit expressions of the lnCOS function/Parametric-PGF, the sensitivity analysis of vibration response and power generation with respect to structural parameters can be directly performed. Our work provides an efficient way to characterize both the linear and nonlinear effects on a nonlinear energy harvesting system and demonstrates an efficient approach to the analysis and design of nonlinear energy harvesting systems.

Nonlinear multimodal energy harvesting system: A dual-cantilever beam structure for combining piezoelectric and electromagnetic mechanisms

A nonlinear piezoelectric-electromagnetic composite energy harvester (NEPH) is proposed. The dynamic equations of its electromechanical system are established and numerical methods are used to simulate its voltage output characteristics if the system parameters are changed. The experimental setup is built up to measure the output characteristics of the prototype. Experimental results show at the first resonant frequency (22 Hz), the total power of the PH generator under a load impedance of 50 kΩ is 5.51 mW, and the total power of the EH generator under a load impedance of 200 Ω is 72.48 μW; at the second resonant frequency (30 Hz), the total power of the PH generator under load impedance of 50 kΩ is 1.79 mW, and the total power of the EH generator under load impedance of 200 Ω is 27.42 μW. In addition, the two electromagnetic generators have the same amplitude, frequency, and phase, so they can be directly connected in series or parallel without rectification. Finally, a practical application in driving electronic loads proves the potential value in supplying power to some low-power electronic devices.

Controlling chaos for energy harvesting via digital extended time-delay feedback

Chaotic vibrations may appear in nonlinear energy harvesting systems, which can be problematic when using the recovered power, as it may require an extra expenditure of energy to rectify the voltage signal or reduce the harvesting process efficiency when charging the battery. Both cases can derail the energy harvester’s functionality. An alternative in this situation is to explore chaos control to stabilize the system dynamics so that the recovered voltage signal is regular and more suitable for use in the applications of interest. This paper address this problem employing an extended delayed feedback method that combines a displacement actuator and a digital controller to implement the control mechanism. The control strategy is mathematically formulated and tested in a bistable energy harvesting system that often operates in a chaotic regime. The controller shows itself capable of stabilizing the chaotic dynamics at a very low energetic cost.

Performance analysis of nonlinear vibration energy harvesting system with inelastic barrier under colored noise excitation

The vibro-impact structures are introduced into the piezoelectric vibration energy harvesting (VEH) systems, which can broaden the operating bandwidth of the piezoelectric VEH systems and thus improve the harvesting efficiency. The process of vibro-impact not only causes energy loss during the impact process but also causes the local deformation between two impact bodies. Therefore, this manuscript proposes a class of stochastic non-classical inelastic impact VEH models, whose non-classical impact processes are described by Hertz contact forces. Firstly, the relationship between the total energy of the unperturbed system and the potential energy of the system at the barrier position is used to determine whether the impact phenomena have occurred or not, then used to derive the period and frequency of the system. Secondly, using the quasi-conservative stochastic averaging method, the averaged drift term and the averaged diffusion term of the piecewise smooth averaged Ito^ equation are derived. The stationary probability density functions (PDFs), the mean square output (MSO) voltage and the root mean square (RMS) voltage of the system response can be obtained from solving the corresponding Fokker-Planck-Kolmogorov (FPK) equation for averaged Ito^ equation. Finally, an example is provided to verify the effectiveness of the proposed method, and the effects of the model parameters on the MSO voltage and power conversion efficiency (PCE) are analyzed. The relationship between the MSO voltage and the PCE is also analyzed. It is of great significance for the design and optimization of stochastic VEH systems.

Investigations on energy harvesting systems incorporating mechanical and electrical nonlinearities to charge reusable batteries

This report analyzes the electromechanical coupling evident in vibration energy harvesting systems that charge reusable batteries. A model of a vibration energy harvesting system with far-from-equilibrium nonlinearities is considered for sake of capitalizing on high-energy forced vibrations. The piezoelectric structure is coupled with nonlinear interface circuits that leverage rectification and control stages to enhance power delivery to the battery. Three circuits, namely standard energy harvesting (SEH) circuit, parallel synchronized switching harvesting on inductor (P-SSHI) circuit, and synchronized charge extraction (SCE) are considered as interfaces between the nonlinear energy harvester and a battery. A simplified power calculation is numerically verified to accurately estimate the harvested power respectively delivered from three interface circuits to the battery. The investigations reveal that increasing required battery voltage may inhibit means to achieve high-energy vibrations and thus prevent battery charging. By a conservation of energy, it is shown that the phase lag between beam velocity and the base acceleration is the primary indicator of interference between dynamic energy flows in the electromechanical sub-systems. This research sheds light on the influences of energy storage components on the functioning of nonlinear vibration energy harvesting systems. Moreover, the principles uncovered for the inhibition of high-energy oscillation may guide future nonlinear energy harvesting system design to extract the best power out from the ambient vibration environments.

STONEHENGE — Suite for nonlinear analysis of energy harvesting systems

STONEHENGE is a toolbox designed to evaluate nonlinear vibration-based energy harvesting systems, which demand careful studies regarding their nontrivial behavior. It is composed of an ensemble of codes to study and characterize the dynamic behavior, as well as deal with varieties of physical parameters and excitation. For this, it has six modules, initial value problem, dynamic animation, nonlinear tools, sensitivity analysis, stochastic simulation, and chaos control. A bistable oscillator is used as a benchmark for a vibration harvester. We hope this toolbox can contribute to the development and improvement of old and new generations of nonlinear vibration-based energy harvesting systems.

Design strategy for optimizing the bandwidth of the hardening vibration generator with customized response

For popular nonlinear vibration energy harvesting systems, such as the Duffing type or the piecewise type, the stiffness profile generally follows a specific explicit expression. The response is thus determined by the coefficients of the stiffness expression and the performance trade-off between high- and low- frequency ranges always exists. To break the restriction, the non-typical nonlinear system whose stiffness profile is arbitrarily variable without an explicit expression is a promising approach. This paper proposes a novel design strategy for the hardening system to optimize the bandwidth. A customized response composed of the starting segment and the flat transition segment is firstly sketched for a target frequency range. The first segment is adopted from a linear oscillator with the rapidly increasing response while the second segment is a straight line connecting the response peak at the high cut-off frequency to the starting segment at the low-end frequency. By finding out the flattest straight line for the transition segment, the high-frequency response is ensured and the low-frequency response is maximized at the same time. Experimental tests on an elaborated prototype show that the designed generator can achieve the optimized response as desired. The superior performance due to the performance enhancement in the low-frequency range is confirmed by comparison with literature generators.

Response Analysis of the Tristable Energy Harvester with an Uncertain Parameter

In the stage of modelling, measuring, mechanical processing and manufacturing of the nonlinear energy harvesting system, deviations and errors of system parameters are inevitable. Even slight variation of key parameters may have a significant influence on the output voltages, especially for the multi-stable nonlinear case. Therefore, the investigation of dynamic behaviors for the tristable energy harvesting system with uncertain parameters is of important value both for research and application. In this paper, the uncertainty of a tristable piezoelectric vibration energy harvester with a random coefficient ahead of the nonlinear term is studied. By using the Chebyshev polynomial approximation, this tristable energy harvesting system is first reduced into an equivalent deterministic form, the ensemble mean responses of which are derived to exhibit the stochastic behaviors. The periodic and chaotic motions, bifurcations and crises under different conditions are analyzed. The results show that the output voltage is sensitive to the uncertainty of the nonlinear coefficient, which leads to unstable behavior around the bifurcation and crisis points particularly. Exploring the influence pattern of uncertain parameters on the output voltage and avoiding the unstable parameter intervals are essential for optimizing the structure. It can further improve the efficiency of the nonlinear energy harvesting system.

Nonlinear vibration energy harvesting and vibration suppression technologies: Designs, analysis, and applications

Limited by the structure, the high-efficiency vibration energy harvesting and vibration suppression have always been a theoretical bottleneck and technical challenge in this field. The nonlinear design of the new vibration structure is an indispensable link in the development of vibration energy harvesting and vibration suppression technologies. Nonlinear technologies not only have the potential to improve the efficiency of the energy harvesters by increasing the useful frequency bandwidth and output power but also have the potential to improve the efficiency of vibration suppressors by reducing the transmission rate and transfer energy. Nonlinear vibration energy harvesting and vibration suppression technologies have been salient topics in the literature and have attracted widespread attention from researchers. The present work provides a comprehensive review on the recent advances in nonlinear vibration energy harvesting and vibration suppression technologies. In particular, the latest developments in multifunctional hybrid technologies are proposed. Various key aspects to improve the performance of nonlinear vibration energy harvesting and vibration suppression systems are discussed, including implementations and configuration designs, nonlinear dynamics mechanisms, various optimizations, multifunctional hybrid, application prospects, and future outlooks.

A machine learning approach for maximizing direct current power of nonlinear energy harvesting systems subjected to periodic impulse excitation

Vibration energy harvesting is a thoroughly tested approach to support the growing number of Internet-of-Things devices by providing a local and sustainable electrical power resource. Yet, the discovery of best practices for deploying vibration energy harvesters in environments is limited by inability to decipher high-dimensional data associated with design and implementation details. The study aims to devise one such approach to conclusively identify conditions under which an impulse-excited nonlinear energy harvesting system delivers peak electrical power. To accomplish this goal, physics-based and machine learning algorithms are integrated to uncover the underlying relationships between the impulse-induced nonlinear dynamics and the result converted electrical energy. The accuracy of the predictions from a machine learning model are confirmed through experiments and against cross-validation simulation data. The parameters that result in snap-through, as determined through machine learning methods, are then probed through a first-principles model to confirm that optimal circuit design conditions agree with the principle of impedance matching. The findings guide attention to previously unseen nuances of exploiting nonlinear vibration energy harvesting systems in impulse excitation environments by revealing parameter sensitivities that inhibit or promote the realization of snap-through vibration for peak power generation. This research demonstrates a successful synthesis of machine learning algorithms and first principles models to guide attention to optimal designs of nonlinear energy harvesting systems subjected to periodic impulse excitation.

Resonance response analysis of nonlinear vibration energy harvesting system under bounded noise excitation

In this paper, the resonance response of piezoelectric vibration energy harvester (VEH) driven by bounded noise is discussed through the quasi-conservative stochastic averaging method. A nonlinear transformation based on the total energy is firstly established to transform piezoelectric VEH system from an electromechanical coupled nonlinear system into a single-degree-of-freedom (SDOF) system. Then the SDOF system is rewritten as It\^o stochastic system about the energy and residual phase under the case of p:q resonance through the quasi-conservative stochastic averaging method. And the joint probability density function (JPDF) of the stationary response is obtained by solving the corresponding two-dimensional Fokker-Planck-Kolmogorov (FPK) equation using the finite difference method. Meanwhile, the mean-square electric voltage and the mean output power are further analytically given through the JPDF. Finally, the resonance response of piezoelectric VEH system is analyzed in detail in case of the primary resonance, and the Monte Carlo (MC) simulation technique is adopted to validate the effectiveness of the finite difference method.

Piezoelectric autoparametric vibration energy harvesting with chaos control feature

To extend the working bandwidth for piezoelectric vibration energy harvesting, autoparametric internal resonant system is proposed. Unstable quasiperiodic motion and chaos, however, are accompanied by broadband characteristic in nonlinear energy conversion. In order to control unwanted irregular motions for power management, we presented a new autoparametric vibration energy harvester coupled by nonlinear energy sink. Mathematical model considering the electromechanical coupling and modal coupling for the high-order nonlinear energy harvesting system is established. Nonlinear analyses via power spectrum, phase portrait and Poincaré map illustrate the effectiveness of chaos control using nonlinear energy sink. In addition, average power of the lower boundary of the harvesting band is enhanced by increasing the stiffness and damping of nonlinear energy sink. Average power of the upper boundary of the harvesting band is improved by increase of the nonlinear energy sink mass and substructural natural frequency ratio. The maximum average power of the proposed piezoelectric autoparametric vibration energy harvester is enhanced 200%, compared to the power harvested by the system without the nonlinear energy sink. This study provides a chaos control method for stable broadband vibration energy harvesting. With simultaneous energy harvesting and chaos control, such system can fit in with more applications.