Nonlinear Piezoelectric 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.

A stochastic averaging mathematical framework for design and optimization of nonlinear energy harvesters with several electrical DOFs

Energy harvesters for mechanical vibrations are electro-mechanical systems designed to capture ambient dispersed kinetic energy, and to convert it into usable electrical power. The random nature of mechanical vibrations, combined with the intrinsic non-linearity of the harvester, implies that long, time domain Monte-Carlo simulations are required to assess the device performance, making the analysis burdensome when a large parameter space must be explored. Therefore a simplified, albeit approximate, semi-analytical analysis technique is of paramount importance. In this work we present a methodology for the analysis and design of nonlinear piezoelectric energy harvesters for random mechanical vibrations. The methodology is based on the combined application of model order reduction, to project the dynamics onto a lower dimensional space, and of stochastic averaging, to calculate the stationary probability density function of the reduced variables. The probability distribution is used to calculate expectations of the most relevant quantities, like output voltage, harvested power and power efficiency. Based on our previous works, we consider an energy harvester with a matching network, interposed between the harvester and the load, that reduces the impedance mismatch between the two stages. The methodology is applied to the optimization of the matching network, allowing to maximize the global harvested power and the conversion efficiency. We show that the proposed methodology gives accurate predictions of the harvester’s performance, and that it can be used to significantly simplify the analysis, design and optimization of the device.

A quad-stable nonlinear piezoelectric energy harvester with piecewise stiffness for broadband energy harvesting

A quad-stable nonlinear piezoelectric energy harvester with piecewise stiffness for broadband energy harvesting

Dynamics of a 3D Piezo-Magneto-Elastic Energy Harvester with Axisymmetric Multi-Stability.

In this investigation, a three-dimensional (3D) axisymmetric potential well-based nonlinear piezoelectric energy harvester is proposed to increase the broadband frequency response under low-strength planar external excitation. Here, a two-dimensional (2D) planar bi-stable Duffing potential is generalized into three dimensions by utilizing axial symmetry. The resulting axisymmetric potential well has infinitely many stable equilibria and one unstable equilibria at the highest point of the potential barrier for this cantilevered oscillator. Dynamics of such a 3D piezoelectric harvester with axisymmetric multi-stability are studied under planar circular excitation motion. Bifurcations of average power harvested from the two pairs of piezoelectric patches are presented against the frequency variation. The results show the presence of several branches of large-amplitude cross-well type period-1 and subharmonic solutions. Subharmonics involved in such responses are verified from the Fourier spectra of the solutions. The identified subharmonic solutions perform interesting patterns of curvilinear oscillations, which do not cross the potential barrier through its highest point. These solutions can completely or partially avoid the climbing of the potential barrier, thereby requiring low input excitation energy for barrier crossing. The influence of excitation amplitude on the bifurcations of normalized power is also investigated. Through multiple solution branches of subharmonic solutions, producing comparable power to the period-1 branch, broadband frequency response characteristics of such a 3D axisymmetically multi-stable harvester are highlighted.

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.

Piezoelectric energy harvester for wind turbine blades based on bistable response of a composite beam in post-buckling

A bistable piezoelectric energy harvester (PEH) is presented for harvesting power from vibrations occurring at low frequencies, as is the case of wind turbine blades. The axial compressive prestress of a piezoelectric composite beam at post-buckling serves as the bistability source, leading to high mechanical to electric power conversion. An in-house harvesting circuit connected to the piezoelectric terminals is used for demonstration of its harvesting power capabilities. A finite element (FE) model is used to analyze and optimize the coupled nonlinear electromechanical response of the PEH, including structure and circuit. A physical prototype has been manufactured and tested for validation of the electromechanical design and the FE modeling approach. Predictions and measurements indicate an increase of harvested power with applied prestress up to a transition point, where a sudden drop in power occurs. Good comparison between numerical and experimental results verified the modeling approach, whereas deviations related to physical boundary conditions at large compressive forces affected the prediction of the transition point in harvested power. The harvester produced 1.32 mW of electrical power under tonal base excitation of 1 g at 8 Hz. Hence, the nonlinear PEH has demonstrated its capability to harvest energy at frequencies much lower than its first linear modal frequency and could thus serve as a promising solution for powering IoT devices and sensors in large vibrating structures.

Optimized multi-frequency nonlinear broadband piezoelectric energy harvester designs

Many electrical devices can be powered and operated by harvesting the wasted energy of the surroundings. This research aims to overcome the challenges of output power with a sharp peak, small bandwidth, and the huge dimensions of the piezoelectric energy harvesters relative to the output power. The aforementioned challenges motivated us to investigate the effect of nonlinearity in the shape (tapered and straight cross-section area) as well as the fixation method (the number of fastened ends) to determine the optimal design with high output power and wide working frequency. This research proposes a novel piezoelectric energy harvester array, where each beam is made up of three fixed beams that are joined together by a center mass. The proposed design produces an output power of 35 mW between 25 and 40 Hz. The output power of the proposed design is 3.24 times more than the conventional designs. The recommended approach is simulated utilizing finite element analysis FEA. Analytical and experimental methods validate the proposed FEA, which exhibits excellent agreement.

Experimental and simulation of a tension-like nonlinear piezoelectric energy harvester

This article proposes a low-frequency and efficient tension-like nonlinear piezoelectric energy harvester, which increases the harvesting performance of the harvester by swinging the support frame. Firstly, design the structure of the harvester and draw a physical model. Secondly, finite element models are used to simulate and analyze the structure’s dynamic characteristics. Finally, make an experimental prototype of the harvester and explore the power generation characteristics of the device through experiments. The results indicate that the harvester has two natural frequencies under low-frequency conditions. The energy harvester has the optimal external resistance to achieve peak output power. The greater the excitation acceleration, the greater the total output power value of the energy harvester. When the mass of the mass block is 3.7 g and the excitation acceleration is 0.4 g, the total output power value of the energy harvester is 3.88 mW, and the harvesting frequency band is wide, indicating excellent output performance of the harvester. The experimental results of the energy harvester device are consistent with the simulation results, fully verifying the correctness of the experiment and simulation. This piezoelectric energy harvester can efficiently harvest vibration energy at low frequencies, providing a good prospect for supplying power to microelectronic devices.

Bayesian model class selection of nonlinear constitutive relationships for piezoelectric energy harvesters with small set of observations

The demand for efficient energy harvesting from vibratory environments has notably increased with the proliferation of piezoelectric devices. However, mismatches between model predictions and experimental observations underpin a critical challenge: accurately modeling piezoelectric energy harvesters (PEHs) while accounting for the inherent uncertainties in their electromechanical properties. A comprehensive Bayesian inference framework is posited for identifying electromechanical properties and facilitating model class selection of nonlinear PEHs. This approach addresses uncertainties by integrating prior knowledge and experimental data to update model parameters, offering the potential to balance model complexity and predictive precision systematically. The influence of different excitation amplitudes and the number of observations employed in the framework is studied. The results demonstrated the ability to capture both linear and nonlinear behaviors, while the model class selection effectively determined the simplest constitutive models sufficient for accurate predictions, validated against varying excitation intensities. By elucidating the trade-off between model simplicity and accuracy, this method not only refines the understanding of PEH behavior under diverse operating conditions but also equips designers with a robust predictive tool. Consequently, this methodology serves to optimize the deployment of PEHs in real-world applications, such as structural health monitoring, ensuring reliable energy supply with quantifiable confidence levels.

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.

The design, fabrication and analysis of a cantilever-based tensile-mode nonlinear piezoelectric energy harvester

In this paper, a cantilever-based tensile-mode piezoelectric energy harvester is proposed. In contrast to a bending mode, the tensile mode enables a uniform strain distribution within the piezoelectric material. This stands in contrast to the strain concentration at the fixed end of a typical cantilevered design. The proposed design comprises a two-segment cantilever beam and an elastic PVDF film that are interconnected. The PVDF film undergoes pre-stretching to guarantee that the harvester maintains a state of tension rather than compression during excitations, as the film is not capable of withstanding compressive forces. The theoretical model of the proposed harvester is derived and validated via experiments. Due to the geometric nonlinearity of this mechanism, a highly nonlinear hardening effect is observed in the simulation and experiments. The effects of different parameters, including preload, dimensions of the elastic beam, tip mass, and excitation accelerations, are explored. The optimal configuration can provide a maximum peak power output of 1827 μW.

Theoretical and experimental research on a Quasi-Zero-Stiffness-Enabled nonlinear piezoelectric energy harvester

Piezoelectric material provides a convenient and efficient avenue to convert ambient vibration energy into electrical energy to power wireless sensor network nodes of Internet of Things. However, the design of effective ultralow-frequency and low-amplitude harvesters using piezoelectric materials remains a challenge. To address this challenging problem, this paper devises a quasi-zero-stiffness-enabled (QZSE) piezoelectric energy harvester based on a geometrical nonlinear structure. The basic configuration and operation principles of the QZSE piezoelectric energy harvester ware elaborated firstly, then the nonlinear electromechanical coupling equation is derived, whose solution informs on both the dynamic and electrical responses of the harvester. The electromechanical-coupled equations are solved with the aid of the fourth-order Runge-Kutta algorithm. The energy harvesting performance is then investigated with respect to the parameter effects of excitation, damping and proof mass. For validating numerical results and demonstrating the advantages of the proposed design, a prototype is fabricated and experimentally tested. The results confirm the ability of the QZSE piezoelectric energy harvester in producing effective vibration-to-electricity energy conversion in ultralow frequency range. This study provides new impetus to energy harvesting from ultralow frequency ambient vibration using piezoelectric materials.

Multistability Mechanisms for Improving the Performance of a Piezoelectric Energy Harvester with Geometric Nonlinearities

This study presents multistability mechanisms that can enhance the energy harvesting performance of a piezoelectric energy harvester (PEH) with geometrical nonlinearities. To configure triple potential wells, static bifurcation diagrams in the structural parameter plane are depicted. On this basis, the key structural parameter is considered, of which three reasonable values are then chosen for comparing and evaluating the performances of the triple-well PEH under them. Then, intra-well responses and the corresponding voltages of the system are investigated qualitatively. A preliminary analysis of the suitable energy-harvesting conditions is carried out, which is then validated by numerical simulations of the evolution of coexisting attractors and their basins of attraction with variations in the excitation level and frequency. It follows that, under a low-level ambient excitation, the intra-well responses around the trivial equilibrium dominate the energy-harvesting performance. When the level of the environmental excitation is very low, which one of the three values of the key structural parameter is the best for improving the performance of the PEH system depends on the range of the excitation frequency; when the excitation level increases sufficiently to induce inter-well responses, the maximum one is the best for improving the performance of the PEH. The findings provide valuable insights for researchers working in the structure optimization and practical applications of geometrically nonlinear PEHs.

Investigation of nonlinear magnetic stiffness based thin-layer stacked piezoelectric generators with a force-amplification structure

This paper investigates a nonlinear piezoelectric stack energy harvester incorporating a magnetic spring (NPEH) that utilizes impact-induced frequency modulation to augment the harvested power. The proposed configuration comprises a force amplifier, a piezoelectric stack, a spring-mass system, and a limiter. We formulate a two-degree-of-freedom (2DoF) theoretical model that accounts for the nonlinear characteristics of the system and corroborate it with experimental data. We examine the effects of system parameters on the output performance of the NPEH and demonstrate that the nonlinear magnetic spring facilitates a wideband energy harvesting. For instance, when the surface magnetic field intensities of the upper and lower magnets are 156 mT and 256 mT, respectively, the operational bandwidth reaches 5.23 Hz, which is 154 % greater than that of a linear spring system. Additionally, the nonlinear magnetic spring mitigates the voltage decay during the collision period due to its hysteresis response, thereby increasing the average power. At an excitation frequency of 9 Hz and with surface field strengths of the upper and lower magnets set at 156 mT and 256 mT respectively, the NPEH generates an RMS power of 2.2 mW (with an instantaneous power peak of 1.20 W). We provide design guidelines for optimizing the energy harvesting performance based on the parametric analysis.

Magnetic-linkage nonlinear piezoelectric energy harvester with time-varying potential wells: Theoretical and experimental investigations

Energy harvesting has emerged as a prospective solution for powering wireless sensors on the Internet of Things (IoT). However, the limited energy conversion efficiency remains a significant challenge, thereby hindering the feasibility of self-powered sensing in IoT. To address this issue, this paper proposes a magnetic-linkage nonlinear piezoelectric energy harvester (PEH) capable of effectively operating over a wide frequency bandwidth, particularly under low-level base excitations. The proposed PEH comprises one vertical and two horizontal piezoelectric beams, which are coupled by the magnetic force to realize the magnetic-linkage effect and create time-varying potential wells. A comprehensive theoretical model has been developed to reveal the working mechanisms of the magnetic-linkage effect and corresponding time-varying potential wells. Based on these foundations, theoretical results indicate that the magnetic configuration parameters have a significant impact on the magnetic-linkage effect and potential barriers. In addition, experimental investigations indicate that the magnetic-linkage effect broadens the effective frequency bandwidth, leading to a significant improvement in energy harvesting efficiency of up to 400% compared to the PEH without magnetic-linkage effect. Furthermore, the proposed PEH enables the realization of self-powered sensing in experiments. In summary, by exploiting the magnetic-linkage effect to create the time-varying potential wells, this innovative approach holds a great promising application for achieving high-efficiency energy harvesting from low-level environmental vibrations.

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.

On exploiting the transition of nonlinear characteristics of a broadband piezoelectric energy harvester for tuning it to low and high-frequency applications

In this work, the applicability of nonlinear piezoelectric energy harvesters is demonstrated for high and low-frequency applications. An axially loaded piezoelectric patched beam model is derived considering in-plane stretching. The harvester setup is developed in the laboratory, and the energy-based numerical model is validated with experimental results. Optimum dimensions of the harvester are obtained. The broadband nature of the harvester is explored, considering nonlinear effects and axial loads on the harvester’s performance. The applicability of the proposed harvester is demonstrated meaningfully by exploiting bifurcation between hardening and softening types of non-linearity by adjusting axial loads with the help of a simple tunable harvester setup.

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.

Nonlinear energy harvesting via an axially moving piezoelectric beam with both d 31 and d 33 modes

Piezoelectric energy harvesters (PEHs) in the literature typically operate with a single conversion mechanism (either d 31 or d 33); the output power, therefore, is limited, and not sufficient to sustainably energize low-power electronics. In this study, a nonlinear PEH with coupled d 31 and d 33 modes is designed and evaluated. An axially moving piezoelectric beam (AMPB) was applied to investigate the contribution of d 31 and d 33 to the output, and the critical parameters of the configuration were determined. A distributed parametric electromechanical model was established to characterize the non-linear dynamics of AMPB with d 31 and d 33 modes. The Galerkin approach and the harmonic-balance approach were employed conjointly to investigate the forced response of the energy harvesting system. The axial velocity’s effects upon energy harvesting were as well discussed. Comparison of the frequency response functions (FRFs) for voltage and power output between energy structures of d 31 and d 33 modes revealed several discrepancies. For instance, the voltage and power output of the d 33 mode were greater than those of d 31 mode for low frequencies, and the difference between the two modes decreased as the frequency increased. For the composite mode d 31 and d 33, under the same parameter conditions, the voltage and power output were greater than the output of any single mode. The analytical results were supported by a numerical method through the finite difference method. Both analytical and numerical results indicated the FRF could be increased by increasing the excitation amplitude, reducing the damping coefficient, or increasing the electrode spacing. The present study showed the efficiency of the use of the FRF using nonlinear transverse vibration of AMPB for d 31 and d 33 modes.

A novel two-degree-of-freedom nonlinear piezoelectric energy harvester with a stopper for broadband low-frequency vibration

Energy conversion efficiency and operation bandwidth are two main concerns in piezoelectric energy harvesting in low-frequency vibration environments. This paper presents a compact two-degree-of-freedom nonlinear piezoelectric energy harvester (2DOF NPEH) with a stopper. The proposed harvester consists of an elastic beam, a piezoelectric beam with tip mass, a pair of repulsive magnets, and a mechanical stopper. A bistable configuration is established through magnetic interaction. This study investigates the generating characteristics of the 2DOF NPEH under frequency-sweep excitation, utilizing a rigid stopper (RS) and different elastic stoppers (ESs). The experimental results indicate that the implementation of a mechanical stopper in a NPEH contributes to widening the operation bandwidth and enhancing the peak voltage. As the stiffness of the stopper increases, the bandwidth gradually shrinks while the peak voltage increases. The optimal energy conversion efficiency depends on the stiffness of the ES, and it occurs when the wire diameter of the spring is 1.2 mm. Compared with the 2DOF NPEH without a stopper, the utilization of a RS results in a 13.85-fold increase in the bandwidth of the generator at the first resonance. Moreover, the maximum absolute voltage is 12.82 times and 3.47 times the output voltage of the 2DOF NPEH at the first and second resonance, respectively. This work offers valuable insights into the design and optimization of piezoelectric energy harvesters in practical applications.