Nonlinear Piezoelectric Energy Harvester Research Articles

Dynamic Characteristics Analysis of Tri-Stable Cantilever Piezoelectric Energy Harvester with a Novel-type Dynamic Amplifier

In this paper, a non-linear tri-stable piezoelectric cantilever energy harvester with a novel-type dynamic magnifier was proposed to achieve more effective broadband energy harvesting under low-level ambient excitations. According to the generalized Hamilton principle, a mathematical distributed parameter model of the piezoelectric energy harvester was proposed. The novel-type dynamic magnifier is a system consisting of two spring masses, one placed between the fixed end of the piezoelectric beam and the L-shaped frame, and the other, between the L-shaped frame and the base. The harmonic balance method was adopted to work out the analytical expressions of the steady-state displacement, steady-state output voltage and power amplitude of the energy harvester system. The effects of the distance between the magnets, the spring stiffness of the dynamic magnifier, and the load resistance on the performance of the system were also investigated. The results show that different from that of the conventional tri-stable piezoelectric energy harvester, the frequency response curve of the proposed novel-type energy harvester system with a two-spring-mass dynamic magnifier exhibits two peaks as a result of the interactions of the coupled elastic system, where the left peak stands for the resonant value of the tri-stable piezoelectric energy harvester, while the right one the resonant value of the dynamic magnifier. It is able to achieve higher output power over a broader frequency band under low-level environmental excitations, and the harvested power can be significantly strengthened if the mass and stiffness of the dynamic magnifier are selected properly.

An Impedance Matching Solution to Increase the Harvested Power and Efficiency of Nonlinear Piezoelectric Energy Harvesters

Circuit theory and nonlinear dynamics are instrumental to design efficient energy harvesters for ambient mechanical vibrations. In this work, we show that an impedance matching networks can be designed that maximizes the harvested power, and improves the power efficiency. The proposed matching network achieves impedance matching at a single frequency, that can be chosen at will by the designer, and does not need to coincide with the resonant frequency of the harvester. Moreover, the matching network also increases the harvested power over a wide frequency bandwidth. According to our numerical simulations, the matching network increases the maximum harvested power by a factor greater than 3, and the power harvested over the whole frequency spectrum by a factor of 6. The frequency bandwidth can be further extended considering nonlinear energy harvesters. Even using the matching network designed for the linear case, performance is significantly nonetheless improved for the nonlinear harvester.

Investigation of a nonlinear piezoelectric energy harvester with advanced electric circuits with the finite element method

In this contribution, a system simulation method based on the finite element method (FEM) is applied to simulate a strongly coupled bimorph piezoelectric vibration-based energy harvester (PVEH) with various nonlinear, non-ideal and active circuits: The standard circuit, the synchronized switch harvesting on inductor circuit and the synchronized electric charge extraction circuit are considered. Furthermore, nonlinear elastic behavior of the piezoelectric material is taken into account and harmonic base excitations of different magnitudes at a fixed frequency are applied. The holistic FEM-based system simulation approach solves the complete set of piezoelectric equations together with the equation of the electric circuit such that all electromechanical coupling phenomena are taken into account. This fully coupled numerical analysis enables the detailed evaluation of the influences of the electric circuits on the vibrational behavior and the harvested energy of the PVEH with respect to the magnitude of base excitation. Results from literature on the efficiency of electric circuits are confirmed and interactions between mechanical and electrical nonlinearities of PVEHs are revealed.Article highlightsSystem simulations of a mechanically and electrically nonlinear piezoelectric energy harvester are performed using only one software tool.The infuence of electric circuits on the vibration behavior and the effciency of an energy harvester are investigated in detail.Interactions between mechanical and electrical nonlinearities of an energy harvester are revealed.

An enhanced nonlinear piezoelectric energy harvester with multiple rotating square unit cells

In this paper, a new method is proposed to enhance the performance of auxetic nonlinear piezoelectric energy harvesting by increasing the unit cell number of auxetic structure. This method is validated through an auxetic nonlinear piezoelectric energy harvester, which combines a clamped–clamped beam with multiple rotating square unit cells. On one hand, the key structural parameter of the square rotating unit cell is adjusted to obtain desired broad operating bandwidth. On the other hand, the number of the unit cell is increased to improve the output power of the proposed design with minor influence on the operating bandwidth. Therefore, by tuning the parameter and unit cell number, the proposed harvester can obtain both broad operating bandwidth and high output power, to reduce the trade-off between these two aspects in the previous auxetic nonlinear energy harvesters. FEM simulation is conducted to investigate the characteristics of the proposed design. A dynamic model is built to theoretically analyse the proposed harvester, which matches with the testing results well. In the experiments, under 0.3 g base excitation, the output power of the proposed harvester is enhanced by 77% as the unit cell number increases from single to 4 × 4, without obvious reduction on the operating bandwidth. In addition, the bandwidth and output power of the proposed harvester are increased by 14% and 268%, respectively, compared with the conventional nonlinear energy harvester.

Design, development, and theoretical and experimental tests of a nonlinear energy harvester via piezoelectric arrays and motion limiters

This paper presents the design, development and test of a nonlinear piezoelectric energy harvesting array with wideband performance under directly and parametrically excited conditions; theoretical verifications of equation-type as well as finite-element-methods are also provided. The array, as the core element, consists of four cantilever beams with attached piezoelectric layers and individual tip masses; two pairs of motion limiters were designed to limit the cantilever motions. By introducing 1-pair motion limiters, a strong hardening frequency response extended the system resonance regime and generated an enhanced bandwidth; when the monotonically increased system response triggered engagement with 2-pair motion limiters, the operational bandwidth was even further extended and formed a fully merged resonance regime for energy harvesting purposes. With motion limiters, the directly excited device achieved a continuous operational frequency bandwidth from 5.7 Hz to 12.3 Hz, which is a frequency bandwidth increase of 240%; by introducing the nonlinear frequency response in parametric resonance to the proposed device, the operational frequency bandwidth is increased by 579%. The corresponding piezoelectric voltage output of the proposed device is compared with conventional no limiter and one limiter counterparts. Theoretical investigations using an equation for the motion of the system along with a time-integration solution, as well as a finite element method using ANSYS, have been carried out to verify the results, showing good agreement. The results reveal that the proposed device has potential for dealing with different excitation levels and low frequency applications while broadening the frequency bandwidth.

Misalignment-induced bending-torsional coupling vibrations of doubly-clamped nonlinear piezoelectric energy harvesters

This article presents a theoretical and experimental study on the unique bending-torsional coupling vibration of a doubly-clamped nonlinear piezoelectric energy harvester due to the misalignment of the centroid and the torsional axis in vibration environments. Based on the Euler-Bernoulli beam theory, the extended Hamilton’s principle, and the Galerkin discretization, a distributed-parameter model considering the misalignment is developed, and the corresponding governing equations are formulated. The developed model is validated against experimental data and Finite Element Analysis, where good agreements are achieved. Effects of the key parameters on the system’s responses, and the characteristics and the causes of the bending-torsional coupling vibration are studied numerically based on the developed model. The results show that the torsion-induced chaotic motion can activate the jump-down phenomenon, largely reducing the electrical output of the harvester. Additionally, the occurrence of the torsion-induced jump-down phenomenon is sensitive to the system’s initial state in some cases and can be reduced or even avoided by effectively tuning parameters that have slight effects on the bending resonance, such as the preloaded axial force and the distance between the torsional axis and the centroid.

Research and analysis of an energy harvester of piezoelectric cantilever beam based on nonlinear magnetic action.

In this paper, a nonlinear piezoelectric energy harvester is developed based on rotational motion applications. It consists of the pedestal, the piezoelectric beam, the connection mass, the tip magnetic mass, the revolving host, the support frame, and the bolts. This device drives the intermittent magnetic vibration between the magnet and the tip magnetic mass to generate electric energy, avoids mechanical collision and wear, and extends the service life of the device. The working principle and vibration model of the proposed energy harvester are studied theoretically. The displacement state of the piezoelectric beam under a magnetic force is simulated and analyzed. In addition, a series of experiments verify the simulation results. With two driving magnets, 5g tip magnetic mass, and 10mm radial excitation distances, a piezoelectric energy harvester can capture energy efficiently. The results demonstrate that the piezoelectric energy harvester produces four resonance frequencies of 4, 11, 15, and 19Hz. When the rotation frequency is 4Hz, the maximum open-circuit voltage of the piezoelectric energy harvester is 96.87V. The piezoelectric energy harvester gets the maximum average power of 8.97 mW when the external resistance is 300 kΩ. At this time, the voltage across the resistance is 51.87V.

Numerical analysis and experiments of an underwater magnetic nonlinear energy harvester based on vortex-induced vibration

This paper designs an underwater magnetic nonlinear piezoelectric energy harvester (PEH) based on the principle of vortex-induced vibration (VIV). The equivalent stiffness of the magnetic nonlinear PEH is significantly changed by varying the relative position of the magnet. It is observed that the proposed PEH displays hardening behavior and softening behavior due to the impact of magnetic forces, which greatly increase the performance of the magnetic nonlinear PEH. The analysis of the results shows that the open-circuit voltage (VRMS) of the hard stiffness system is increased from 9.43 V to 10.31 V, the frequency band is widened by 36.8%, and the efficiency is higher at high flow rates. The initial vibration velocity of the soft stiffness system is reduced. When the water velocity is 0.2 m/s, its VRMS is 10 times that of the classical configuration. Both numerical simulations and experiments prove that the soft stiffness system is more suitable for applications in low-speed water environments. This proposed design can provide the groundwork to promote the performance of the conventional PEH, further enabling to realize self-powered systems.

Design and analysis of a double-acting nonlinear wideband piezoelectric energy harvester under plucking and collision

This paper proposes a double-acting nonlinear wideband piezoelectric energy harvester under plucking and collision(W-PEH) for energy during rotation. The device is mainly composed of a base, rectangular piezoelectric patches, circular piezoelectric generating units and permanent magnets. The circular power generation units are constructed by using the nonlinear magnetic coupling mechanism and the contact frequency conversion principle. The nonlinearity caused by the magnetic force makes the circular piezoelectric power generation units work stably and efficiently at low speeds. And the contact-type collision excitation enables the circular piezoelectric patches inside the circular piezoelectric generating units to produce greater deformation and output a higher voltage. The rectangular piezoelectric patches are excited by the plucking, and the vibration amplitude is greater at high speeds, and it also produces a higher output voltage. Two different excitations make W-PEH's working frequency band wider, and the overall output voltage is also higher. The electromechanical coupling dynamics model of the system was constructed from Hamilton's principle and the magnetic dipole theory. The experimental results indicate that W-PEH can work effectively in different frequency bands. Under the load of 10 KΩ, the peak-to-peak power of the 150r/min integrated device is 0.350W, and that of the 450r/min integrated device is 0.107W.

Modal reduction-based finite element method for nonlinear FG piezoelectric energy harvesters

In this study, harvesting energy from a nonlinear functionally graded (FG) piezoelectric cantilever beam under harmonic excitation is investigated. The material properties of the piezoelectric are assumed to be a combination of piezo-ceramic and piezo-polymer materials for high performance and flexibility. The neutral axis is obtained in order to eliminate bending–stretching coupling. The geometrical nonlinearity and electromechanical coupling are incorporated in the coupled nonlinear equations that are derived using the generalized Hamilton’s principle and solved using the combination of modal reduction and finite element methods. The shooting method is employed to obtain steady-state periodic response of an FG nonlinear harvester with appropriate initial conditions. Also it is shown that at least two-mode approximation is required for accurate estimation of nonlinear response and harvested power. Using the method of nonlinear modal reduction, the unstable branches for frequency domain solution are estimated and the computational time is reduced considerably compared to full finite element method. A case study is also accomplished in detail to analyze the effects of base amplitude values and material distribution on harvested power and bandwidth.

Efficiency of mono-stable piezoelectric Duffing energy harvester in the secondary resonances by averaging method, Part 2: Super-harmonic resonance

In the paper, the potential of harvesting vibratory energy via a mono-stable Duffing energy harvester with base excitation in super-harmonic resonance is investigated. The averaging method is exploited to provide an approximate analytical solution for the system. The displacement and voltage responses are determined depending explicitly on the displacement amplitude and excitation frequency. The explicit approximate analytical expressions of the peak amplitude and corresponding base excitation frequency are also presented. Next, a detailed power flow analysis in super-harmonic resonance is made, then formulations for input mechanical power and energy, useful electrical power and energy, and efficiency are obtained, respectively. Numerical simulations are carried out to show qualitative comparisons between sup- and sub-harmonic cases and linear cases at corresponding base excitation frequencies. It is obtained that super- and sub-harmonic resonances are effective solutions to enlarge the performance of nonlinear piezoelectric energy harvester in wide-band excitation, especially compared to linear one that only works at a single frequency.

Nonlinear dual action piezoelectric energy harvester for collecting wind energy from the environment

This paper studies a nonlinear dual-action piezoelectric energy harvester for collecting wind energy from the environment. The components of the device are fan blade, rotating shaft, rotator, rotating body, piezoelectric cantilever beam, exciting magnet, mass magnet. Under the excitation of wind energy in the environment, the rotating shaft rotates, the magnetic force and the strike force act on the piezoelectric cantilever beam respectively. According to the piezoelectric effect, the piezoelectric cantilever beam is deformed to produce electric energy. Through theoretical analysis, the formulas of magnetic motion, bending moment, elastic strain, and moment of inertia are obtained. In this experiment, some discussions are made to make it clear whether the magnetic pole of the exciting magnet is the same as the magnetic pole of the mass magnet, the number of exciting magnets, and the distance between the exciting and the mass magnet to ensure the best performance parameters are obtained. The experimental results show that when the working voltage of the blower is 150 V, the exciting magnet and the mass magnet attract each other, the number of exciting magnets is 1, and the distance between the exciting magnet and the mass magnet is 5 mm, the output voltage of the device is the highest and the output voltage is 56 V. If the external resistance is 1500 Ω, the device realizes the maximum output of power. Therefore, small windmills have the potential to collect wind energy as generators and can be widely developed and applied in urban areas.

Nonlinear broadband piezoelectric vibration energy harvesting enhanced by inter-well modulation

Broadening the working bandwidth of piezoelectric energy harvesting devices is of great importance owing to the wide frequency spectrum of environmental vibrations. Efforts, including stiffness nonlinearity and multimodal optimization, have been proposed to improve harvesters’ responses towards the broadband frequency region. This paper designs a spring-based bistable energy harvester (SBEH) and proposes its two configurations (SBEH-sup and SBEHsub) to achieve high working efficiency under different excitation levels through inter-well modulation. The nonlinear force produced by the compressed springs generates nonlinear dynamical responses, and thus broadens the effective bandwidth of the SBEH. Compared to the SBEH-sup configuration, the SBEH-sub configuration further utilizes magnetic force to dynamically alter the potential energy threshold in the nonlinear system, which contributes to the SBEH maintaining inter-well motion and enhancing the working performance under lower intensity excitation. The working mechanisms of the SBEH-sup configuration and the SBEH-sub configuration are revealed by derived mathematical models. The feasibility of the SBEH with two proposed configurations is verified by experimental tests. Compared with its linear counterpart, the significant advantages in the SBEH-sup configuration show 673% increment in effective bandwidth and 179% increment in working efficiency under suprathreshold-level excitation. Compared with the SBEH-sup configuration, the SBEH-sub configuration further broadens the operating bandwidth by 838% and enhances the working efficiency by 269% under subthreshold-level excitation. This spring-based bistable energy harvester provides a new design philosophy for nonlinear energy harvesters, its two configurations with inter-well modulation achieve satisfactory performance for scavenging energy from different-level vibrations.

On the Efficiency of a Piezoelectric Energy Harvester under Combined Aeroelastic and Base Excitation.

A flutter-type, nonlinear piezoelectric energy harvester was tested in various combinations of aerodynamic and harmonic base excitation to study its power output and efficiency. The commercial polyvinylidene fluoride film transducer LDT1-028K was used in 33 excitation mode. The aerodynamic excitation was created by a centrifugal fan and the base excitation by a cone speaker. The excitations were produced by varying independently the mean airflow velocity and the frequency of base vibration. A capacitive load was used to store the harvested energy. A line laser was employed along with long exposure photography and high-speed video, for the visualization of the piezo film’s mode shapes and the measurement of maximum tip deflection. The harvested power was mapped along with the maximum tip deflection of the piezo-film, and a process of optimally combining the two excitation sources for maximum power harvesting is demonstrated. The energy conversion efficiency is defined by means of electrical power output divided by the elastic strain energy rate of change during oscillations. The efficiency was mapped and correlated with resonance conditions and results from other studies. It was observed that the conversion efficiency is related to the phase difference between excitation and response and tends to decrease as the excitation frequency rises.

An auxetic nonlinear piezoelectric energy harvester for enhancing efficiency and bandwidth

In this paper, we design and experimentally validate an auxetic nonlinear piezoelectric energy harvester. This harvester adopts a clamp-clamp beam with auxetic structures, which can improve the efficiency and bandwidth of the energy harvesting based on a pure mechanical structure without outer magnets or stoppers. Finite element analysis is performed to analyse the effects of the auxetic structures on the efficiency and nonlinearity of the energy harvester. The lumped parameter model is utilized to predict the performance of the energy harvester, which well matches the experimental results. In the experimental validation, at 0.1 g base acceleration, the power output of the two types of auxetic energy harvesters is 173% and 94% higher than the conventional nonlinear energy harvester. Besides, the bandwidths of the two types of auxetic energy harvesters are broadened by 1556% and 2142% compared with the linear systems.

Performance of a novel dual-magnet tri-stable piezoelectric energy harvester subjected to random excitation

Usually conventional nonlinear tri-stable piezoelectric energy harvesters (TPEHs) contain two external magnets, which make both the system structure and dynamic analysis more complex. Herein, a novel dual-magnet TPEH with only one external magnet is proposed. This TPEH consists of a rectangular magnet, a piezoelectric cantilever beam, a ring magnet and a base. Next, the lumped-parameter model of the TPEH is established to obtain the dynamic equation, circuit equation, magnetic force expression and potential function. The analysis of the magnetic force expression and potential function finds that the system degenerates from a tri-stable state to a bi-stable state, and then to a mono-stable state with the increase of magnet distance due to the changes of magnetic force. Subsequently, Runge-Kutta method is used to simulate the nonlinear dynamics and energy harvesting performance of the energy harvester under the filtered Gaussian white noise over the frequency range of 0–120 Hz. Simulation results shows that the harvester changes from intra-well movement to inter-well movement with a few transitions, and then to inter-well movement with many transitions as excitation level increases. Additionally, the output performance in the bi-stable state is better than that in the tri-stable state when the excitation is small, and the output performance in the tri-stable state is better than that in the bi-stable state when the excitation is large enough. The reason is that the depth and width of the outer potential wells in the tri-stable state are deeper and narrower, while the distance between the two outer wells in the tri-stable state is larger, when compared to the bi-stable state. Finally, designed experiments are carried out to determine the parameters in the dynamic equation and measure the energy harvesting performance of the TPEH. Experimental results are consistent with simulation results. This study suggests that it is practicable to use the dual-magnet structure for constructing a multi-stable piezoelectric energy harvester.

Development of a Non-Linear Bi-Directional Vortex-Induced Piezoelectric Energy Harvester with Magnetic Interaction.

In this study, magnetic force is introduced to the design of a bi-directional U-shaped piezoelectric energy harvester for vortex-induced vibrations. The theoretical model of the beam structure is derived based on the Euler–Bernoulli beam theory. The vortex-induced vibration and the non-linear magnetic force are modeled according to the Rayleigh oscillator and the charge model, respectively. A prototype is fabricated and tested in two orthogonal directions under vortex-induced vibrations in a wind tunnel. Up and down wind-speed sweeps are carried out to investigate the non-linear responses of the harvester. The distance between the magnets and the length of the side beams are adjusted to examine the influence of the magnetic force on the lock-in region and voltage output of the harvester. Overall, the harvester shows strong non-linearity in the horizontal excitations. After adding magnets to the system, significant improvement of the lock-in region and the peak voltage is noticed in the horizontal mode under both up and down sweeps.

A nonlinear M-shaped tri-directional piezoelectric energy harvester

In this paper, we design and experimentally validate a geometric nonlinear M-shaped tri-directional piezoelectric energy harvester. This harvester adopts a single structural design, which consists of a proof mass and a beam bent into an M shape with both ends fixed. Through structural parameter modification, the vibration directions of the first three resonant modes can be nearly orthogonal, which is beneficial to achieve tri-directional energy harvesting. Finite element analysis is performed to guide the design and analyse the working effectiveness of the proposed harvester. The influence of the geometric parameters on the nonlinear stiffness is investigated to broaden the bandwidth of energy harvesting. In the experimental validation, an obvious hardening effect can be observed in vertical excitation at 0.3 m s−2 base acceleration. On the other hand, the power output of the harvester in three directions is in similar level, indicating the capability of tri-directional energy harvesting.

Piezoelectric energy harvesting from nonlinear dynamics of functionally graded plate under multi-moving loads and masses

In this study, energy harvesting from nonlinear vibrations of a functionally graded plate with bonded piezoelectric patch under moving loads and moving masses is investigated. The coupled nonlinear system of equations are derived using the generalized Hamilton’s principle considering both the geometrical nonlinearity and the nonlinear electromechanical coupling. Then, the equations are solved using finite element method and Newmark time integration scheme. For a case study, the effects of different parameters such as piezoelectric patch thickness, location of piezoelectric patch, different boundary conditions, and different width of plate on harvested power and the plate’s nonlinear vibration are investigated. This article is motivated by the lack of published studies on nonlinear distributed piezoelectric energy harvesters under moving loads (masses).

The effects of nonlinear energy sink and piezoelectric energy harvester on aeroelastic instability of an airfoil

The potential of absorbing and harvesting energy from a two-degree-of-freedom airfoil using an attachment of a nonlinear energy sink and a piezoelectric energy harvester is investigated. The equations of motion of the airfoil coupled with the attachment are solved using the harmonic balance method. Solutions obtained by this method are compared to the numerical ones of the pseudo-arclength continuation method. The effects of parameters of the integrated nonlinear energy sink-piezoelectric attachment, namely, the attachment location, nonlinear energy sink mass, nonlinear energy sink damping, and nonlinear energy sink stiffness on the dynamical behavior of the airfoil system are studied for both subcritical and supercritical Hopf bifurcation cases. Analyses demonstrate that absorbing vibration and harvesting energy are profoundly affected by the nonlinear energy sink parameters and the location of the attachment.