Nonlinear Piezoelectric Energy Harvester Research Articles

Modeling and Simulation Analysis of a Two-degree-of-Freedom Nonlinear Piezoelectric Energy Harvester

In order to improve energy harvesting efficiency, this paper shows the design of a two-degree-of-freedom (2-DOF) nonlinear piezoelectric energy harvester. The proposed harvest is realized by introducing magnetic attraction to a 2-DOF piezoelectric cantilever beam in order to broaden the energy harvesting bandwidth. Electromechanical equations describing this nonlinear system are presented. The simulation results confirmed the advantage of this new 2-DOF nonlinear harvester. Performance comparison shows that the 2-DOF nonlinear harvester has much higher energy harvesting efficiency than a conventional one-degree-of-freedom (1-DOF) nonlinear one.

Experimental study on band merging of non-linear multi-band piezoelectric energy harvester into single broadband using magnetic coupling

The operational bandwidth of Vibration Energy Harvesters (VEH) is area of concern due to stochastic, time-varying, random and multi-frequency nature of available environmental vibrating sources. Most of the VEH have narrow bandwidth providing usable power at specific frequencies. Efforts have been made to increase the frequency range by introducing non-linear structures and techniques. In this paper, multi-band output of the non-linear Piezoelectric Energy Harvester (PEH) is transformed into single wider band output using additional non-linear phenomenon. Dual region operation of PEH results into two separate band output. First region is the outcome of beam resonance and Centre of Gravity (CoG) shift whereas second region is due to the non-linear behaviour of cylinders. In this work, these separate bands are merged to form a single wider band. For merging these two bands and enhancing the bandwidth of PEH, additional phenomenon is introduced using two permanent magnets. A varying magnetic field by changing the distance between magnets changes stiffness of the cantilever beam and that leads to a change in the resonant frequency of band-I. Thus, the overall process shifts band-I towards band-II. In this work, the two separate bands are merged to have one wider band providing 53.22% more frequency coverage than our previous work with a bandwidth of 47.5 Hz. This band includes vibrational frequency range of 25.65–73.15 Hz at 1.4 g acceleration. Cylinder material and its effect with magnetic interaction is also studied. The magnetic force between two permanent magnets is measured experimentally. Effect of magnetic force on centre resonant frequency of beam is compared with experimental and simulated results. Effect of magnetic force on bandwidth of the device is studied.

Mechanically and electrically nonlinear non-ideal piezoelectric energy harvesting framework with experimental validations

In the literature of vibration energy harvesting, mechanically nonlinear frameworks have mostly employed linear electrical circuitry to formulate AC input–AC output problems, while the existing efforts on nonlinear power conditioning circuits have assumed linear mechanical behavior. However, even for the simplest case of a stiff (geometrically linear) piezoelectric cantilever, material softening and dissipative nonlinearities in the mechanical domain have to be accomodated to accurately predict the response for moderate to high excitation levels, and likewise a stable DC signal must be obtained to charge a storage component in realistic energy harvesting applications. Furthermore, often times the voltage output is not large enough to assume ideal diode behavior to reduce diodes to switches in AC–DC conversion modeling. Therefore, a relatively complete representation of piezoelectric energy harvesting requires accounting for the mechanical (e.g., material and dissipative) nonlinearities as well as the nonlinear process of AC–DC conversion with non-ideal circuit elements, such as real diodes. To this end, we present and experimentally validate a multiphysics framework and harmonic balance analysis that combines these mechanical and electrical nonlinear non-ideal effects to predict the DC electrical output (DC voltage across the load) in terms of the AC mechanical input (base vibration) for arbitrary vibration and voltage levels. The focus is placed on a bimorph cantilever with piezoelectric laminates under base excitation. The terminals of the piezoelectric layers are combined in series and connected to a bridge rectifier with non-ideal diodes and a filter capacitor. The multi-term harmonic balance framework can capture the ripple in the DC signal as well as amplitude-dependent nonlinear dynamics accounting for realistic diode behavior. In addition to quantitative comparisons and validations by comparing the experimental data and model simulations, important qualitative trends are unveiled for mechanically and electrically nonlinear non-ideal dynamics of piezoelectric energy harvesting.

Nonlinear arbitrary-directional broadband piezoelectric vibration energy harvester using 3-DOF parallel mechanism

Overcoming the disadvantages of direction selectivity and narrow bandwidth are two key problems to improve the efficiency of energy harvesting from ambient vibration, and also the premises of the practical application of the piezoelectric vibration energy harvester. In this paper, utilizing a 3-DOF parallel mechanism and nonlinear magnetic forces generated by polar opposing magnets, a nonlinear piezoelectric vibration energy harvester is proposed and explored. The kinematic equation, the governing equation of motion, and the piezoelectric coupling equation of the vibration energy harvester are established. On this basis, through numerical simulation, the normal modes of the vibration energy harvester in linear case are explored, the effect of the polar opposing magnets and the nonlinear characteristics are analyzed, the 8 potential wells in three-dimensional space of the vibration energy harvester are investigated, and the motion trajectories of the octuplet stable system are estimated. Then, the developed vibration energy harvester is tested with vibration excitation along different directions and sweep excitations. The results show that the developed vibration energy harvester can harvest vibration energy along any directions, and its bandwidth is extended by introducing magnets. The maximum average power of the developed vibration energy harvester is 22.3 mw for 1 g excitation, and the bandwidth is about 5.5 Hz with its center frequency around 11 Hz. This work proves that using parallel mechanism with nonlinearity is a promising solution to realize arbitrary-directional vibration energy harvester with wide bandwidth.

Performance analysis of parametrically and directly excited nonlinear piezoelectric energy harvester

The performance of bimorph cantilever energy harvester subjected to horizontal and vertical excitations is investigated. The energy harvester is simulated as an inextensible piezoelectric beam with the Euler–Bernoulli assumptions. A horizontal base excitation along the axis of the beam is converted into the parametric excitation. The governing equations include geometric, inertia and electromechanical coupling nonlinearities. Using the Galerkin method, the electromechanical coupling Mathieu–Duffing equation is developed. Analytical solutions of the frequency response curves are presented by using the method of multiple scales. Some analytical results are obtained, which reveal the influence of different parameters such as the damping, load resistance and excitation amplitude on the output power of the energy harvester. In the case of parametric excitation, the effect of mechanical damping and load resistance on the initiation excitation threshold is studied. In the case of combination of parametric and direct excitations, the dynamic characteristics and performance of the nonlinear piezoelectric energy harvesters are studied. Our studies revealed that the bending deformation generated by direct excitation pushes the system out of axial deformation and overcomes the limitation of initial threshold of parametric excitation system. The combination of parametric and direct excitations, which compensates and complements each other, can be served as a better solution which enhances performance of energy harvesters.

Simulation and experiment on bridge-shaped nonlinear piezoelectric vibration energy harvester

There is abundant of vibration energy in environment. Using piezoelectric energy harvesters to power ultra-low power wireless sensors has become a hot topic in recent years. Most piezoelectric energy harvesters have satisfactory output only at one resonant frequency, while once the external excitation frequency deviates from the resonant frequency, the voltage output will drop significantly. Therefore, this paper proposes a bridge-shaped piezoelectric energy harvester to broaden the frequency band for energy harvesting, and the harvester can adaptively adjust the natural frequency of the system to match the external excitation. Based on the two-dimensional nonlinear Euler–Bernoulli beam hypothesis and the Hamilton principle, the distributed-parameter dynamic model of the nonlinear energy harvesting system is set up. By use of variation operator and nondimensionalization, we derive the dimensionless electromechanical coupling equations. With the harmonic balance method, the approximate equations for calculating voltage amplitude and average power are derived. And then the model is numerically solved and analyzed by considering the influence of excitation frequency, excitation amplitude, mass position and load on the output performance of the bridge-type piezoelectric energy harvester. Finally, various comprehensive experiments are performed to verify the results of simulation, and the dynamic effects of the moving mass on the output characteristics are studied. It can be noted that the moving mass can improve the voltage output of the bridge type energy harvester by changing the resonant frequency of the bridge-type piezoelectric energy harvester and impacting the piezoelectric beam. The proposed vibration energy collection device can adaptively tune the resonant frequency and has practical value in engineering.

Nonlinear piezoelectric vibration energy harvester with frequency-tuned impacting resonators for improving broadband performance at low frequencies

The paper reports on a nonlinear frequency up-converting vibration energy harvester with enhanced broadband performance below 50 Hz, which is attained by leveraging vibro-impact interaction between two high-frequency piezoelectric generators and two low-frequency resonators with commensurate natural frequencies in the 2:1 ratio. Finite element model was implemented to assess the proposed concept by treating the harvester as a vibro-impact coupled piezoelectric-circuit system. The model accommodates multiple mechanical contact pairs defined by nonlinear dissipative Hunt–Crossley contact force formulation. Effective design principles were applied to maximize power output including near-optimal positioning of the neutral plane in the generators and rational adjustment of response settling time. The fabricated proof-of-concept device was tested in different clearance configurations and under varying harmonic excitation conditions to reveal rich nonlinear dynamics behavior. The measured hardening-type power frequency responses were examined to gain better insight into nonlinear resonance characteristics (instability jumps, frequency hysteresis, etc), which determine effectiveness of micro-power generation. It was demonstrated that the proposed dual-resonator energy harvester favorably harnesses nonlinear resonance amplification to deliver larger average power output (up to 37 μW) and useable bandwidth (up to 9.5 Hz) in comparison to the conventional single-resonator counterparts (delivering up to 26 μW and up to 7 Hz, respectively). The maximum normalized power density of the device is 41 μW cm−3 g−2, while volume figure of merit and 3 dB or 1 dB bandwidth-based figures of merit reach 0.18%, 11 μW cm−3 g−2 and 1.5%, respectively. The latter value indicates superior broadband performance of the device. Adoption of mean performance index and coefficient of variation as complementary figures of merit proved to be valuable in identifying device configuration that yields the most effective and stable broadband operation in a targeted frequency range.

Design and performance evaluation of a piezoelectric aeroelastic energy harvester based on the limit cycle oscillation phenomenon

In the present era, the demand for self-powered electronic instruments is increasing and their energy consumption is decreasing. The ability to extract energy from the operating environment is of great importance in advanced industrial applications particularly in the field of aerospace. In this research, a nonlinear piezoelectric aeroelastic energy harvester (PAEH) is modeled based on fluid-structure interaction (FSI) that represents an important area of research for the development of innovative energy harvesting solution. This PAEH operates on post critical aeroelastic behavior i.e., Limit Cycle Oscillations (LCOs) that arise after the flutter velocity. Moreover, it is emphasized that the determination of aerodynamic model is necessary for correct prediction of PAEH performance. The design of an aeroelastic harvester based on the use of two different types of piezoelectric materials is presented. The designed model is suitable for energy harvesting and can be utilized to drive nano and microelectronics. Maximum output power obtained for the designed PAEH is found to be 0.55 mW.

Soft magneto-sensitive elastomer and polyvinylidene fluoride polymer based nonlinear piezoelectric energy harvesting: design, modelling and experiment

In this paper, a broadband vibrational energy harvester (VEH) is developed based on a soft magneto-sensitive elastomer (SMSE). The utilization of SMSE provides the VEH with a strong softening effect when it is subjected to the magnetic field of a permanent magnet. A polyvinylidene fluoride (PVDF) layer is attached to the SMSE to convert vibrational energy into electricity. A magneto-electro-mechanical model based on finite element method is developed and is further reduced using assumed modes. Two prototypes are fabricated with different SMSE thicknesses (3 mm and 5 mm respectively). Frequency sweep experiments are conducted to investigate their broadband behavior. The difference between jumping frequencies are 2.39 and 4.34 Hz at a low acceleration level of 0.3 g. With the broadband characteristics of SMSE, the two prototypes are capable of providing power over a wide frequency range and generating an average power of 0.096 μW and 0.11 μW respectively with a load of 4.7 MΩ.

Low-cost orbit jump in nonlinear energy harvesters through energy-efficient stiffness modulation

This paper aims at exposing a nonlinear piezoelectric vibration energy harvester which can achieve low-cost orbit jump and therefore improve its energy harvesting capability through energy-efficient transient stiffness modulation using Synchronized Switch Stiffness Control (SSSC) technique. The proposed orbit jump concept is numerically and experimentally studied with a softening harvester. Corresponding experimental results validate the jump orbit concept and confirm the significant enhancement of the harvested energy. While the presented softening Duffing system is used as demonstration example, the proposed approach can be implemented to hardening Duffing oscillator as well.

Distributed Parameter Modelling of Cutout 2-DOF Cantilevered Piezo-Magneto-Elastic Energy Harvester

Energy harvesters play the vital role in replacing/ recharging the electrochemical batteries in low-power electronic devices. Non-linear piezoelectric energy harvesters generate electrical power from broadband frequency spectrum using ambient vibrational sources. Non-linear cutout 2- degree of freedom (DOF) piezo-magneto-elastic (PME) energy harvester has been proposed to enhance the bandwidth where two close resonant peaks are achieved using magnetic interaction. In this paper, the novel theoretical distributed parametric model of cutout 2-DOF energy harvester is presented which can predict the dynamic behavior of the linear (without magnetic force) and non-linear monostable configurations accurately. The closed-form analytical solutions are derived for the harvested electrical responses and displacements using forcing function, forward and backward coupling factors through Extended Hamilton principle, Galerkin approach, and Euler–Lagrange equations. Specifically, the static displacements and tuned resonant frequencies are obtained as a function of the initial distance between the two magnets in attractive and repulsive configurations. The theoretical findings are validated with COMSOL simulation results and experimental data. The comparison among theoretical, finite-element, and experimental results show that the developed model of cutout 2-DOF piezo-magneto-elastic energy harvester can accurately predict the response for linear and non-linear monostable configurations. [2018-0068]

Theoretical analysis and experimental study of a nonlinear U-shaped bi-directional piezoelectric energy harvester

Narrow bandwidth and uni-directional acquisition restrict the application of a conventional cantilever piezoelectric energy harvester. In this study, a magnet-induced nonlinear U-shaped bi-directional piezoelectric harvester is proposed to overcome the shortcomings of the conventional harvester. The U-shaped harvester consists of a beam bent into a U shape with both ends fixed. A pair of piezoelectric plates is attached to the beam for power generation. To introduce nonlinear magnetic force, a magnet is attached to the center of the beam while a pair of magnets are fixed to the base. The theoretical model of the proposed harvester is developed and validated with experimental results. The proposed harvester is investigated in two orthogonal directions of vibration. The proposed harvester shows linear responses in vertical excitations. However, a strong hardening effect can be observed in horizontal excitations especially when the magnet gap is small. It is shown that the hardening effect can improve both the harvesting bandwidth and the output power in horizontal excitations. On the other hand, the resonant frequency of the vertical mode can be increased while that of the horizontal mode is decreased by reducing the magnet gap. This indicates that the magnet gap can be utilized to tune the ratio between the vertical and horizontal resonant frequencies to match the frequencies of vibration sources. Overall, the proposed nonlinear U-shaped harvester demonstrates the capability of bi-directional harvesting and improved performance with the aid of nonlinear magnetic force.

A harmonic balance approach with alternating frequency/time domain progress for piezoelectric mechanical systems

Abstract A semi-analytical approach based on harmonic balance (HB) method is proposed to predict steady-state responses of nonlinear piezoelectric mechanical systems. In order to handle complicated non-polynomial forces of the systems, the alternating frequency/time domain (AFT) progress is applied to establish a set of implicit algebraic equations of HB method. The stabilities of periodic responses are analyzed via Floquet theory. The proposed approach is successfully implemented a nonlinear piezoelectric energy harvester with a magnetic oscillator and a Duffing energy harvester with strong nonlinear electric terms. The HB solutions are well supported by the simulation results of the Runge-Kutta method. Besides, an experiment piezoelectric device is developed by a circular laminated plate, and an experimental model is established by system identifications. The model includes both the nonlinear stiffness and nonlinear damping which are characterized by the combined functions of polynomials and hyperbolic tangents. The model is analyzed via the proposed approach. The numerical simulations under different excitations show good agreements with the harmonic solutions. The frequency sweeps experimentally pursue the system has both softening and hardening nonlinearities which are verified by HB results. The proposed approach requires less analysis during the solving processes, and shows high accuracy to the strongly nonlinear piezoelectric mechanical systems, even the system with complicated nonlinearities.

Fabrication and characterization of miniature nonlinear piezoelectric harvester applied for low frequency and weak vibration

Abstract In order to collect energy from human activities, piezoelectric energy harvesters should have a flexible structure, low resonant frequency and wide frequency bandwidth. In this paper, the fabrication and characterization of a metal-based Pb(Zr,Ti)O3 (PZT) bimorph thick film energy harvester. PZT thick films were deposited on cantilever-shaped titanium (Ti) substrate via a dip-coating process. The electrical properties of the PZT films sintered at different temperatures were investigated. The nonlinear piezoelectric energy harvester has the following advantages: it is suitable for low frequency, weak vibration, and wide frequency bandwidth. The maximum open-circuit voltage of the piezoelectric energy harvester is 1.6 V and the output power is 0.85 μW with 150 kΩ load resistance at low resonant frequency of 17.6 Hz when the input vibration acceleration was 0.12 g. In addition, the frequency bandwidth is broadened from 0 to 8.4 Hz as the input acceleration increases from 0.05 g to 0.12 g.

Numerical and experimental study of bistable piezoelectric energy harvester

Piezoelectric energy harvesting from mechanical vibrations is a reliable technology to charge low power electronic equipment. It has been reported in the literature that low frequency broadband vibrations cannot be harvested effectively using linear piezoelectric energy harvester (PEH). In this paper a bistable nonlinear PEH with two distinct energy wells generated using repulsive magnetic interactions between a cantilever magnetic proof mass and an external magnet is considered. It has been modeled using finite element method and validated with experimental results. Experimental results verify that, for our study, distance between magnetic proof mass and external magnet of 1.32 T flux density should be less than or equal to 8.5 mm for maintaining bistability. This distance also affects to the resonance frequency of harvester. Maximum efficiency of the harvester has been noticed at 8.5mm. Further it has been shown experimentally that the bistable nonlinear piezoelectric energy harvester takes almost half of the time taken by its linear counterpart to charge a 20 mAh battery.

Equivalent electrical circuit framework for nonlinear and high quality factor piezoelectric structures

Equivalent electrical circuits are useful simulation tools to emulate and investigate the behavior of electromechanically coupled systems and structures as well as to develop energy harvesting or control circuits, among other configurations with two-way coupling. The existing efforts in this context have mostly considered linear structural (mechanical) domain, occasionally with nonlinear circuit for signal process and control. Typically, resonant circuits are employed to represent the mechanical domain of single- or multi-degree-of-freedom systems, while ideal transformers or current- and voltage-dependent sources are employed to model the electromechanical coupling. However, practical limitations of ideal transformers and dependent sources are challenges for experimental implementations of equivalent circuits. Furthermore, the internal resistance of equivalent resonant circuits limits the representation of high quality factor systems. This paper introduces equivalent electrical circuits for linear and nonlinear electromechanically coupled systems with high quality factor and various types of nonlinearities. The focus is placed on piezoelectric structures that exhibit stiffness and damping nonlinearities. An alternative to the existing models for the electromechanical coupling is also presented for convenient simulation of coupled system dynamics using standard electronic simulation programs. The equivalent circuit framework given here for high quality factor electromechanical systems is validated against both linear and nonlinear case studies, including published data for a nonlinear piezoelectric energy harvester. The proposed framework paves the way for the design of circuits emulating nonlinear structures, such as nonlinear vibration absorbers and sinks.

A seesaw-type approach for enhancing nonlinear energy harvesting

Harvesting sustainable mechanical energy is the ultimate objective of nonlinear energy harvesters. However, overcoming potential barriers, especially without the use of extra excitations, poses a great challenge for the development of nonlinear generators. In contrast to the existing methods, which typically modify the barrier height or utilize additional excitations, this letter proposes a seesaw-type approach to facilitate escape from potential wells by transfer of internal energy, even under low-intensity excitation. This approach is adopted in the design of a seesaw-type nonlinear piezoelectric energy harvester and the energy transfer process is analyzed by deriving expressions for the energy to reveal the working mechanism. Comparison experiments demonstrate that this approach improves energy harvesting in terms of an increase in the working frequency bandwidth by a factor of 60.14 and an increase in the maximum output voltage by a factor of 5.1. Moreover, the output power is increased by a factor of 51.3, which indicates that this approach significantly improves energy collection efficiency. This seesaw-type approach provides a welcome boost to the development of renewable energy collection methods by improving the efficiency of harvesting of low-intensity ambient mechanical energy.

A Novel Nonlinear Piezoelectric Energy Harvesting System Based on Linear-Element Coupling: Design, Modeling and Dynamic Analysis.

This paper presents a novel nonlinear piezoelectric energy harvesting system which consists of linear piezoelectric energy harvesters connected by linear springs. In principle, the presented nonlinear system can improve broadband energy harvesting efficiency where magnets are forbidden. The linear spring inevitably produces the nonlinear spring force on the connected harvesters, because of the geometrical relationship and the time-varying relative displacement between two adjacent harvesters. Therefore, the presented nonlinear system has strong nonlinear characteristics. A theoretical model of the presented nonlinear system is deduced, based on Euler-Bernoulli beam theory, Kirchhoff’s law, piezoelectric theory and the relevant geometrical relationship. The energy harvesting enhancement of the presented nonlinear system (when n = 2, 3) is numerically verified by comparing with its linear counterparts. In the case study, the output power area of the presented nonlinear system with two and three energy harvesters is 268.8% and 339.8% of their linear counterparts, respectively. In addition, the nonlinear dynamic response characteristics are analyzed via bifurcation diagrams, Poincare maps of the phase trajectory, and the spectrum of the output voltage.

Nonlinear Piezoelectric Energy Harvester with Ball Tip Mass

In this study, a nonlinear piezoelectric energy harvester (PEH) with a ball tip mass was designed and fabricated for broadband energy harvesting. The proposed nonlinear PEH exhibits two resonant frequencies (5 and 15 Hz) and can harvest a considerable amount of electrical energy, whereas a conventional PEH with a rigid tip mass only exhibits one (15 Hz). The minimum acceleration that can induce nonlinearity in the proposed PEH was determined to be 3 m s–2. In order to maximize the electrical output of the proposed PEH, 0.1 mm was selected as the optimal vibration amplitude of the ball among three options (0.1, 0.5, and 1.0 mm). The maximum output power of the proposed PEH was measured as 13.5 mW at 15 Hz and at a load resistance of 30 kΩ, which is the matching load resistance calculated and verified with the experimental result. The output power of the proposed PEH was measured to be 1.8 mW at 5 Hz and 45 kΩ, whereas that of the conventional PEH at 5 Hz was 0.03 mW, implying that the proposed PEH possesses one additional resonant frequency over the conventional PEH. In addition, the proposed PEH performs significantly better than the conventional PEH in terms of broadband energy harvesting. At a power level of 100 μW, the proposed PEH at 3 m s–2 provides a bandwidth of 20 Hz, which is more than 133% wider than the 15 Hz provided by the conventional PEH. By utilizing the nonlinearity with a ball tip mass, the PEH vibration is capable of harvesting considerable electrical energy from more than two resonant frequencies.

Nonlinear Hybrid Piezoelectric and Electromagnetic Energy Harvesting Driven by Colored Excitation

It is well known that when excited by a stochastic base acceleration, the power absorbed by a nonlinear Duffing-type piezoelectric (PE) or electromagnetic (EM) energy harvester might not be increased effectively compared to a linear one. When intentionally introducing nonlinear magnetic forces into a doubly-clamped hybrid PE and EM energy harvester subjected to narrow-band (colored) excitation however, the power output could be improved to a much higher value. Also, in comparison with the typical nonlinear PE or EM generator, the influence of load and excitation parameters on the performance of a nonlinear hybrid energy harvester under colored excitation has been proven rather different as well. These results are derived analytically by solving the Fokker-Planck (FP) equation, and numerically by Monte Carlo (MC) simulations for validation. Besides, for a nonlinear hybrid configuration excited by colored noise approaching white Gaussian excitation, theoretical output characteristics are discussed and compared with results from a reported theory for white Gaussian excited case, which again verifies the feasibility of the theoretical analysis.