Interwell Motion Research Articles

Utilizing Hertzian contact model for a vibro-impact unimorph bistable energy harvester to increase working frequency bandwidth and harvested power

The typical bistable energy harvester (TBEH) systems usually have a limited working frequency bandwidth and low performance in energy conversion. This paper proposed a unilateral vibro-impact piezoelectric cantilever beam energy harvester. The novel vibro-impact bistable energy harvester (VIBEH) improves the drawbacks of a TBEH system. Despite previous research, in this paper, the collision process is modeled by nonlinear Hertzian contact theory. Nonlinear contact law adds strong nonlinearity to the harvester. By using the extended Hamilton principle and Euler-Lagrange equation, the governing equations of the energy harvester are derived and solved numerically. The existing nonlinearities cause frequency leaks and affect the type of motion (inter-well or intra-well motion) by changing the equilibrium points of the system. By applying nonlinear tools like Poincare diagrams, bifurcation plots, and phase portraits, the effects of periodic or chaotic oscillations will be studied on the performance of the VIBEH system. Finally, the effects of barrier stiffness, initial gap, barrier location, and electrical load are determined on the performance of the VIBEH system. The obtained results showed that using the VIBEH system could broaden the working frequency bandwidth by about 71.9%. Furthermore, the optimized values of impacting properties and electrical load are examined.

Inertial amplification as a performance enhancement method for snap-through vibration energy harvester

In recent years, there has been a lot of interest in exploiting the bistable behavior of snap-through systems to harvest energy from vibration sources. The efficient operation of any bistable VEH depends on its ability to exhibit large-amplitude interwell motion. Under weak ambient excitation, bistable VEH performs marginally because of the confinement of motion to a single well. Frequency up-conversion, multi-stability, and adaptive techniques are some of the performance enhancement strategies suggested for bistable-VEH. Considering the VEH's space constraints, the above designs are hard to implement in practical cases. This study introduces an inertial amplification mechanism (IAM) as a simple passive strategy to enhance the performance of a snap-through VEH, a concept not explored in previous studies. The addition of IAM increases the effective mass without increasing the physical mass and thereby enhances energy harvesting, especially from weak ambient excitation sources. The dynamics and performance of the enhanced snap-through VEH are investigated analytically and numerically under harmonic and random excitations. The harmonic balance method (HBM) derives the frequency-amplitude relationship, which shows a hardening behavior and an increase in bandwidth. The effective potential method provides a closed-form expression for the joint probability density function (Joint PDF), which is governed by the Fokker-Planck equation. The joint PDF shows a transition from bimodal to unimodal with an increase in the value of the geometrical parameter. The stochastic averaging method is employed to obtain the stationary probability density function, which defines the long-term dynamics of the VEH. The effects of noise intensity, mass ratio, and inertial amplifier angle on the dynamics are investigated. Finally, the performance of the proposed VEH is compared with a conventional snap-through VEH, an equivalent linear VEH, and a multistable VEH under harmonic and random excitation conditions. The findings suggest that the snap-through VEH with the IAM has advantages over the linear and multistable nonlinear VEH in terms of extracting energy from low-intensity harmonic and random excitation sources. This simple augmentation strategy preserves the original system's bistability, eliminating the need for the complex design of a multistable VEH.

A broadband bistable energy harvester with self-decreasing potential barrier effect: Design, modeling and experiments

The conventional bistable energy harvester (CBEH) is difficult to break through the potential barrier for large-amplitude inter-well motion under weak excitation environments, which results in a narrow effective bandwidth. This paper proposes a bistable energy harvester with self-decreasing potential barrier effect (BEH-DPBE) for harvesting low-frequency vibrational energy. A spring is introduced to transform the fixed end of the S-type power generating beam into a movable end, while its oscillator has a dual function of dynamically reducing the potential barrier in the return trip and enhancing the potential energy in the outward trip. The characteristics of the proposed energy harvester are evaluated and analyzed by combining finite element simulations, numerical studies, and experiments. Finally, the effects of key parameters, such as the spring pre-compression, proof mass, and load resistance, are studied. In low-frequency conditions, the effective bandwidth and peak-to-peak voltage of BEH-DPBE are 6.35 Hz (5.98–12.33 Hz) and 10.1 V, respectively. The proposed BEH-DPBE can effectively capture energy in low-frequency ambient vibrations.

Design, modeling and experiments of bistable piezoelectric energy harvester with self-decreasing potential energy barrier effect

It is difficult for conventional bistable energy harvesters to undergo interwell motion in weak excitation environment. In this study, a bistable energy harvester with self-decreasing potential energy barrier effect is proposed, which innovatively turns a fixed end of S-shaped generator beam into the movable end with a spring oscillator, which effectively decreases the potential energy barrier and makes the harvester susceptible to interwell motion. Furthermore, a theoretical analysis of the nonlinear driving magnetic force and the system governing equations is carried out, then a numerical simulation of potential energy surface of harvester is performed to demonstrate the effect of the self-decreasing potential energy barrier in detail. The effect of spring pre-compression and stiffness of the moveable end on the system potential energy is analyzed by numerical simulation. Finally, the key parameters affecting the output performance, such as spring pre-compression, magnets distance, and external load resistance, are analyzed. The harvester has a maximum effective bandwidth of 6.62Hz (4.61–11.23Hz), a maximum peak-to-peak voltage of 10.2V, an RMS voltage of 1.3V, and a maximum output power of 2.91 μW. The proposed harvester achieves efficient broadband energy harvesting in low frequency environments.

Enhancing low-orbit vibration energy harvesting by a tri-stable piezoelectric energy harvester with an innovative dynamic amplifier

Piezoelectric energy harvesting faces a primary challenge in effectively capturing low-orbit vibration energy across a broad frequency range. In this paper, we present a tri-stable piezoelectric energy harvester that incorporates a dynamic amplifier (TPEH + DM), specifically designed for efficient collection of low-orbit vibration energy. The TPEH + DM comprises a piezoelectric cantilever beam connected to an innovative dynamic amplifier at its restrained end, which enhances both the rotational and lateral displacement of the piezoelectric cantilever beam simultaneously. The governing coupled differential equations of motion for the system is derived based on the Lagrange equation, and analytical expressions for its steady-state response are obtained using the multi-scale method. The influence of factors such as the mass and the stiffness ratio of the dynamic amplifier on the steady-state dynamic output characteristics of the system is investigated using the theoretical analysis and numerical simulation. The results indicate that TPEH + DM exhibits significantly improved energy harvesting performance compared to TPEH under low-orbit external excitations. The bandwidth of inter-well motion and the TPEH + DM power output may be further increased by suitably modifying the relative stiffness between the cantilever beam and the dynamic amplifier. In addition, we analyze the time-domain behavior of the system’s output voltage using the ode45 solver under various external excitation frequencies and intensities. The results demonstrate that with appropriate adjustments to the mass of the tip magnet and the stiffness ratio of the dynamic amplifier, the proposed TPEH + DM system can harvest energy efficiently across a broad frequency range, even under low-orbit excitations.

A novel M-shaped 2-DOF piezoelectric energy harvester with built-in outer-inner magnetic tri-stable oscillators for energy converting enhancement and applications

Tri-stable oscillators have been widely used to develop various nonlinear broadband energy harvesters. Focusing on further achieving high-performance piezoelectric energy harvester under broadband vibrations, a novel M-shaped 2-DOF piezoelectric energy harvester (M-PEH) with built-in outer-inner magnetic tri-stable oscillators is proposed, in which an inner magnetic tri-stable oscillator is built-in into a U-shaped outer magnetic tri-stable oscillator using its cut-out cantilever beam, assisted by the coupling interface to further enhance the dynamic responses. A complete physical model by considering the nonlinear coupling interaction between the outer-inner magnetic tri-stable oscillators is established to describe the dynamic performances under different system parameters. The theoretical model is well validated by experiments, which show that the proposed M-PEH has more compact size, wider operating frequency and higher power generation, offering an effective bandwidth of 20.1 Hz (5.1–25.2 Hz) at base excitation A = 5 m/s2, and achieving maximum output voltage 8.5 V and total peak power 26.7 μW. Compared to the original magnetic-coupling nonlinear tri-stable PEH, the effective bandwidth and total power have increased by 300 % and 662 % respectively, and the average power density is increased by 917 %. In addition, four main coupling interaction modes between the outer and inner built-in tri-stable oscillator are observed, in which the nonlinear synchronous coupling interaction is conducive to generating inter-well motion, leading to more mechanical energy being transferred from the outer tri-stable oscillator to the inner built-in tri-stable oscillator, thereby improving power generation capacity. Besides, the coupling interaction between these two tri-stable oscillators produces two close resonances which can be easily tuned to form a wide operating bandwidth. Practical applications prove that the proposed M-PEH has the ability to power low-powered electronic products, meeting the power supplying requirements of engineering community.

Dynamic Behavior of Bistable Shallow Arches: From Intrawell to Chaotic Motion

Abstract Bistable shallow arches are ubiquitous in many engineering systems ranging from compliant mechanisms and biomedical stents to energy harvesters and passive fluidic controllers. In all these scenarios, the bistable states of the arch and the sudden transitions between them via snap-through instability are harnessed. However, bistable arches have been traditionally studied and characterized by triggering snap-through instability using quasi-static forces. Here, we analytically examine the effect of oscillatory loads on bistable arches and investigate the dynamic behaviors ranging from intrawell motion to periodic and chaotic interwell motion. The linear and nonlinear dynamic responses of both elastically and plastically deformed shallow arches are presented. Introducing an energy potential criterion, we classify the structure’s behavior within the parameter space. This energy-based approach allows us to explore the parameter space for high-dimensional models of the arch by varying the force amplitude and excitation frequency. Bifurcation diagrams, Lyapunov exponents, and maximum critical energy plots are presented to characterize the dynamic response of the system. Our results reveal that unstable solutions admitted through higher modes govern the critical energy required for interwell motion. This study investigates the rich nonlinear dynamic behavior of the arch element and it introduces an energy potential criterion that can scale easily to classify motion of arrays of bistable arches for future developments of multistable mechanical metamaterials.

Time-delayed Duffing oscillator in an active bath.

During recent decades active particles have attracted an incipient attention as they have been observed in a broad class of scenarios, ranging from bacterial suspension in living systems to artificial swimmers in nonequilibirum systems. The main feature of these particles is that they are able to gain kinetic energy from the environment, which is widely modeled by a stochastic process due to both (Gaussian) white and Ornstein-Uhlenbeck noises. In the present work, we study the nonlinear dynamics of the forced, time-delayed Duffing oscillator subject to these noises, paying special attention to their impact upon the maximum oscillations amplitude and characteristic frequency of the steady state for different values of the time delay and the driving force. Overall, our results indicate that the role of the time delay is substantially modified with respect to the situation without noise. For instance, we show that the oscillations amplitude grows with increasing noise strength when the time delay acts as a damping term in absence of noise, whereas the trajectories eventually become aperiodic when the oscillations are sustained by the time delay. In short, the interplay among the noises, forcing, and time delay gives rise to a rich dynamics: a regular and periodic motion is destroyed or restored owing to the competition between the noise and the driving force depending on time delay values, whereas an erratic motion insensitive to the driving force emerges when the time delay makes the motion aperiodic. Interestingly, we also show that, for a sufficient noise strength and forcing amplitude, an approximately periodic interwell motion is promoted by means of stochastic resonance.

Dynamical analysis and design theory for active bistable vibration isolators considering delay effect

Bistable vibration isolators (BVIs), which can undergo intra- and inter-well motions, have recently gained much attention as different motions can lead to different low-frequency isolation properties. To fully exploit the strength of BVIs, feedback actuation is injected into a classic three-spring-two-link bistable structure, and the classic PD control is applied to tune system's stiffness and damping. Moreover, the effect of the inevitable loop delay on BVI performance is considered. Operable regions of control parameters to maintain bistable characteristics and robustness regions where active BVIs are operable regardless of delay perturbations are determined. Besides, bistable dynamics, complicated by the transcendentality caused by delay, are investigated. By taking the BVI as phase zero, the number of variables to be solved is reduced, and an efficient frequency-sweeping-based calculation procedure is proposed. The BVI is evaluated by force and displacement transmissibility, showing that both intra- and inter-well motions benefit vibration isolation. For the former, the BVI yields broadband vibration isolation similar to conventional high-static-low-dynamics-stiffness vibration isolators. For the latter, an isolation valley can occur, resulting in exclusive ultra-low-frequency vibration isolation. Furthermore, properly tuning feedback actuation enhances the vibration isolation performance in both cases, and the often-overlooked loop delay can play a very positive role. Finally, the mechanism of isolation valley is qualitatively elucidated. The presented theories, including control stability analysis, delay-coupled nonlinear dynamical analysis, and beneficial parameter tuning, establish a basic design framework for active BVIs.

Energy capture performance enhancement of point absorber wave energy converter using magnetic tristable and quadstable mechanisms

This paper proposes two magnetic multistable (tristable and quadstable) mechanisms to enhance the energy capture performance of point absorber wave energy converters (PAWECs) for low-amplitude waves. Accordingly, two mechanisms consisting of several coaxial magnetic rings are designed based on the superposition effect of magnetic fields. Compared with conventional magnetic bistable mechanisms, the proposed tristable and quadstable mechanisms feature lower potential barriers and greater distances between two outermost potential wells, facilitating the occurrence of large-amplitude interwell motions in low-amplitude waves. A mathematical model is established using the Cummins equation and equivalent magnetic charge approach to depict the dynamic behavior and performance of PAWECs. The energy capture performance and average annual output power of PAWECs with tristable and quadstable mechanisms are evaluated for regular and irregular waves. The findings demonstrate that the two mechanisms significantly improve the energy conversion efficiency, enhancing their robustness against low-amplitude waves and damping detuning. The design concept of nonlinear multistable PAWECs using the superposition effect of magnetic fields provides a promising alternative for exploiting wave energy resources with low energy densities.

Bio-inspired quad-stable piezoelectric energy harvester for low-frequency vibration scavenging

Due to the unique and diverse nonlinear characteristics, bionic structures can effectively broaden the acquisition bandwidth and improve the performance of harvesters. Inspired by dipteran flight motion, a novel bionic quad-stable piezoelectric energy harvester (BQPEH) is designed to collect low-frequency ambient vibration. The novelty of this bionic energy harvester lies in the combination of the “click” mechanism and “snap-through” motion, thus broadening the application scenarios and improving adaptability for the harvesters. A mass block and a post-buckled piezoelectric beam are connected by a rigid bar to create quadruple-well potentials. The static bifurcation analysis is performed to identify the quad-stable region in the parameter space. The dynamic responses of the system are investigated theoretically and numerically under constant and swept-frequency excitations and then verified experimentally. Broadband low-frequency and high-power energy harvesting are obtained through the inter-well motion induced by the combined nonlinearity of the BQPEH. The results indicate the potential of the bionic strategy for powering wireless devices in low-frequency environments and provide a novel concept for the innovative development of energy harvesters.

Coupling nonlinearities investigation and dynamic modeling of a tristable combined beam rotational energy harvesting system

Energy harvesting system is regarded as major units in self-powering of massive sensors of Artificial Internet of Things, whose mechanical dynamics play a crucial role in energy conversion efficiency and overall service performance. During rotational motion, periodic disturbances near resonant regions may lead to a large-amplitude oscillation of the harvesters. Due to the coupling effect of rotation and vibration, the distributions of mass and stiffness are related with local deformation and nonlinear rotational frequency in harvesting system, which are yet to be well understood and it is still an open issue. In this paper, a tristable combined beam rotational energy harvester is proposed to obtain higher energy conversion efficiency. Based on the extended Hamilton’s principle, a comprehensive coupling nonlinearities dynamic model is established with consideration of strain distributions, geometric nonlinearity, centrifugal effect, gravitation effect, and nonlinear rotational frequency. The effect of geometric nonlinearity and gravity on the synergistic transition and form mechanism of the tristable harvesting system is explored under the guidance of the static bifurcation, and the influence mechanism of the geometric nonlinearity and nonlinear rotational frequency on the mechanic and energetic characteristics of the system are thoroughly revealed. Numerical and experimental studies are performed to validate the effectiveness of the established model. Results show that geometric nonlinearity and nonlinear rotational frequency lead to a wider frequency bandwidth and a more complex inter-well snap-through motion. Meanwhile, the established model provides a theoretical fundamental framework of high-performance harvester’s design in large-amplitude complex excitation scenarios, which is conducive to control harvester in actual engineering applications.

Improving energy harvesting from low-frequency excitations by a hybrid tri-stable piezoelectric energy harvester

This paper presents the design of a novel hybrid tri-stable piezoelectric energy harvester (HTPEH) that improves the energy harvesting efficiency of piezoelectric energy harvesters operating at low frequencies. The proposed HTPEH is an extension of a conventional tri-stable cantilever piezoelectric energy harvester with an elastic base (TPEH + EB), and it incorporates vertical and rotational elastic amplifiers that are installed between the mass block at the left end of a cantilever beam and the left side of a U-shaped frame to enhance the system's nonlinear characteristics. By utilizing Hamilton's principle, a HTPEH's nonlinear electromechanical equation is formulated, and the harmonic balance method is used to obtain analytical solutions for the displacement amplitude, voltage amplitude, and power amplitude. The study investigates the effects of various parameters, including the elastic amplifier-to-base stiffness ratio and elastic amplifier-to-piezoelectric cantilever beam mass ratio on the energy harvesting performance of the HTPEH. The results indicate that the displacement and voltage amplitude of the HTPEH exhibit two peaks as the external excitation frequency increases, and adjusting the relative mass and stiffness between the elastic amplifier and the piezoelectric cantilever beam can alter the frequency band interval where the two response peaks of the system are located. This allows the HTPEH to enter inter-well motion at low-level external excitation, resulting in high output power. The HTPEH outperforms the TPEH + EB in terms of energy harvesting efficiency under low-frequency environmental excitation.

Enhanced signal response in globally coupled networks of bistable oscillators: Effects of mean field density and signal shape.

This paper studies a set of globally coupled bistable oscillators, all subjected to the same weak periodic signal and identical coupling. The effect of mean field density (MFD) on global dynamics is analyzed. The oscillators switch from intra- to interwell motion as MFD increases, clearly demonstrating MFD-enhanced signal amplification. A maximum amplification also occurs at a moderate level of MFD, indicating that the response exhibits a nonmonotonic sensitivity to MFD. The MFD-enhanced response depends mainly on the signal intensity but not on the signal frequency or the network topology. The analytical investigation provides a simplified model to study the mechanism underlying this resonancelike behavior. It is shown that by modifying the bistability nature of the potential energy, the mean field density can promote well-to-well oscillations and larger amplitude motions. Finally, the robustness of this phenomenon to various signal waveforms is examined. It can therefore be used alternatively to efficiently amplify weak signals in practical situations with large network sizes.

A mechanical-free designing method for tailoring nonlinearity in bi-stable piezoelectric energy harvesters

This paper presents a mechanical-free method for providing and tailoring the nonlinear force in bistable piezoelectric energy harvesters (BPEHs). The nonlinear force can be tailored to obtain a lower threshold for inter-well motions, or for the harvester to operate at various excitation levels and frequencies without changing the mechanical structure or the overall assembly. In BPEHs, the nonlinear force is tailored to match a specific excitation level and frequency, and the mechanical structure is designed to achieve higher strain (and thus higher output power). The design of nonlinearity can be separated from the design of the mechanical structure by using magnetic interactions. Hence, the design of nonlinearity is the arrangement of the external magnetic field of the harvester. In this paper, arranging the external magnetic field is achieved by arranging the magnetization distribution of one external magnet. With the locally demagnetizing technique, a uniformly magnetized permanent magnet can be locally demagnetized with desired patterns. The external magnetic field is provided by a locally demagnetized permanent magnet (LDPM). The nonlinear force can be tailored by simply altering the properties of the LDPM. This method converts the design of providing and tailoring the nonlinear force into the design of the LDPMs. For demonstration, we show that without increasing the distance between magnets, the potential barrier of the bistable system is dramatically reduced by using LDPMs. Melnikov’s method is utilized to show that the energy harvesters with LDPMs possess a lower threshold for homoclinic tangency than energy harvesters with a normal magnet. The influence of the parameters of the LDPMs on the energy harvesting performance is studied via simulations and experiments. Results show that without violating the mechanical part, changing the locally demagnetizing patterns can effectively change the harvester’s working frequency and excitation threshold.

Dynamic Characteristic Analysis of Tri-Stable Piezoelectric Energy Harvester with Double Elastic Amplifiers

In order to further improve the vibration energy harvesting efficiency of piezoelectric energy harvester under low frequency environmental excitation, this paper, based on the traditional magnetic tri-stable piezoelectric energy collector model, proposes a tri-stable piezoelectric energy harvester (TPEH+DEM) model with two elastic amplifiers which are installed between the U-shaped frame and the base and between the fixed end of the piezoelectric cantilever beam and the U-shaped frame respectively. Based on Hamilton principle, the motion equation of electromechanical coupling of TPEH+DEM system is established, and the analytical solutions of displacement, output voltage and power of the system are obtained by harmonic balance method. The effects of the mass of elastic amplifier, spring stiffness, magnet spacing and load resistance on the dynamic characteristics of energy harvesting of TPEH+DEM system are analyzed. The result shows that there are two peaks in the response output power of TPEH+DEM system in the operating frequency range. By adjusting the mass and stiffness of the elastic amplifier reasonably, the system can move into the inter-well motion under low external excitation intensity, and produce high output power. Compared with the traditional model which only has an elastic amplifier on the base of piezoelectric energy harvester, TPEH+DEM model has better energy harvesting performance under low frequency and low intensity external excitation.

Exploring nonlinear degradation benefit of bio-inspired oscillator for engineering applications

Nonlinear structure design is an indispensable link in the development of vibration isolation, energy harvesting and fault diagnosis technologies. It is also one of the most effective technical means to improve the performance of structures in engineering applications. In this study, a bio-inspired nonlinear oscillator is proposed and the benefits of nonlinear degradation in engineering applications are explored. First, the degradation bistable oscillator and high-order quasi-zero stiffness oscillator can be obtained after the degradation of the bio-inspired nonlinear oscillator. Through the amplitude-frequency analysis, it is found that the nonlinear degradation can reduce the unstable solution in the ultra-low frequency region and reduce the resonant frequency of the bio-inspired nonlinear oscillator. Therefore, the degeneration nonlinear oscillator can realize the large-amplitude inter-well motion at ultra-low frequency or low random excitation intensity. Secondly, the theory and method of vibration isolation and energy harvesting based on the nonlinear degradation mechanism are proposed, and the vibration isolation and energy harvesting performance of the oscillator under harmonic and stochastic excitations are analyzed. Finally, the use of the stochastic resonance of the degeneration bistable oscillator can greatly improve the signal-to-noise ratio, enhance the weak fault signal, and realize early fault diagnosis. The related results help to deepen the understanding of nonlinear degenerate systems and provide new technical approaches for engineering applications.

A bistable rotary-translational energy harvester from ultra-low-frequency motions for self-powered wireless sensing

This paper presents the design, theoretical modelling and experimental study of a bi-stable energy harvester (EH) using rotary-translation motion for ultra-low frequency and low excitation amplitude energy sources. A spherical magnet is adopted to produce the rotary-translational motion to convert ultralow-frequency kinetic energy into electricity over a wide frequency range. The bi-stable mechanism is realized by introducing two tethering magnets underneath the sphere magnet’s oscillating path, significantly enhancing the operating range of the harvester. A theoretical model including the impact dynamics, magnetic interaction and electromagnetic conversion has been established to explore the electromechanical behaviours of the harvester under different operating conditions. The results illustrate that the EH operates in intra-well or inter-well motion depending on whether the input excitation is adequate to conquer the potential barrier depth. A prototype is developed to illustrate the design and to validate the theoretical model. The prototype generates sufficient power (mW) at frequencies lower than 2 Hz with excitation amplitudes as low as 0.1 g. A peak output power of 9 mW (1.53 mW RMS) is obtained at 2 Hz and 0.7 g with 750 Ω external load. The developed EH is integrated with an off-the-shelf power management solution to power a wireless sensing system to successfully record real-time temperature variation in the environment.

Design and investigation of a quad-stable piezoelectric vibration energy harvester by using geometric nonlinearity of springs

In this paper, a quad-stable piezoelectric vibration energy harvester induced by the geometric nonlinearity of springs is designed and investigated. The novelty is that by replacing the permanent magnets with a combination of springs, the system can be effectively protected from electromagnetic interference in the working environment. The innovative design of three hinged springs attached to the tip of the cantilever beam makes the stiffness of the structure inherently nonlinear, which can induce the multi-stability of the system. The Euler-Lagrange equations are applied to establish the governing equation of the energy harvester. The static bifurcation analysis reveals the evolution process of multi-stability. The complex dynamic frequency (CDF) method, an efficient approach to solving strongly nonlinear dynamic problems with multi-well systems, is then applied to deduce the system's amplitude-frequency response. Combinations of theoretical and numerical results show that among a wide frequency bandwidth, the system can exhibit such as single-, double- and quadruple-well periodic motions. Finally, experiments are set up to verify the advantage of inter-well motion on harvesting power. Result shows that geometric nonlinearity can effectively make the energy harvester exhibit multi-stability, generate different movement patterns, and obtain high harvesting power.

Parametric Analysis of Nonlinear Bi-Stable Piezoelectric Energy Harvester Based on Multi-Scale Method

Given the geometric nonlinearity of the piezoelectric cantilever beam, this study establishes a distributed parameter model of the nonlinear bi-stable cantilever piezoelectric energy harvester, following the generalized Hamilton variational principle. The analytical expressions of the dynamic response were obtained for the energy harvesting system using Galerkin modal decomposition and the multi-scale method. The investigation focuses on how the performance of the energy harvesting system is influenced by the gap distance between magnets, external excited amplitude, mechanical damping ratio and external load resistance. The calculation results were compared with those obtained neglecting the geometric nonlinearity of the beam. The results show that the system responses contain jump and multiple solutions. The consideration of the geometrical nonlinearity significantly amplified the peak displacement and peak output power of the intra-well and inter-well motions. There is an evident hardening effect of the inter-well motion frequency response curve. By reasonable adjusting the parameters, it is possible to improve the output power of the piezoelectric energy harvesting system and broaden the operating frequency of the system.