Multi-stable Energy Harvester Research Articles

Chaotic responses across the potential barrier of bistable vortex-induced vibration energy harvesters

Large-amplitude interwell limit cycle oscillations (LCOs) are regarded as an ideal high-output state of multistable vibration energy harvesters. However, chaotic responses, which result in lower output voltages than interwell LCOs, can be observed in bistable vortex-induced vibration energy harvesters (VIVEHs). In this paper, the chaos in bistable VIVEHs is investigated through numerical simulation, theoretical analyses, and experimental validation. The distribution of the periodic solutions obtained by the incremental harmonic balance (IHB) method and chaos judgment using Lyapunov exponents are elucidated. It is found that the bistable VIVEH shows intrawell periodic solutions at high reduced wind speeds (RWSs) and period-doubling bifurcation appears with the periodic solutions turning into chaos with decreasing RWS. Analyses of the influence of nonlinear parameters on the chaotic responses identify the nonlinear stiffness as the dominant factor contributing to the chaos. Meanwhile, the chaotic responses and interwell LCOs would exist at the same RWS with different initial states. Finally, wind tunnel experiments confirm the occurrence of homologous chaotic responses of the bistable VIVEH which feature aperiodic jumping back and forth motion between two potential wells, in agreement with numerical simulations. Overall, this paper provides a framework for analyzing the chaotic responses of bistable VIVEHs, offering valuable insights for complex response mechanism understanding.

Geometrical uncertainties effects on the dynamics and effectiveness of a multi-stable vibratory energy harvester

Geometrical uncertainties effects on the dynamics and effectiveness of a multi-stable vibratory energy harvester

A multi-stable rotational energy harvester for arbitrary bi-directional horizontal excitation at ultra-low frequencies for self-powered sensing

A rotational multi-stable energy harvester has been presented in this paper for harnessing broadband ultra-low frequency vibrations. The novel design adopts a toroidal-shaped housing to contain a rolling sphere magnet which absorbs mechanical energy from bidirectional base excitations and performs continuous rotational movement to transfer the energy using electromagnetic transduction. Eight alternating tethering magnets are placed underneath its rolling path to induce multi-stable nonlinearity in the system, to capture low-frequency broadband vibrations. Electromagnetic transduction mechanism has been employed by mounting eight series connected coils aligned with the stable regions in the rolling path of the sphere magnet, aiming to achieve greater power generation due to optimized rate of change of magnetic flux. A theoretical model has been established to explore the multi-stable dynamics under varying low-frequency excitation up to 5 Hz and 3 g acceleration amplitudes. An experimental prototype has been fabricated and tested under low frequency excitation conditions. The harvester is capable of operating in intra-well, cross-well, and continuous rotation mode depending on the input excitation, and the validated physical device can generate a peak power of 5.78 mW with 1.4 Hz and 0.8 g sinusoidal base excitation when connected to a 405 Ω external load. The physical prototype is also employed as a part of a self-powered sensing node and it can power a temperature sensor to get readings every 13 s on average from human motion, successfully demonstrating its effectiveness in practical wireless sensing applications.

Modeling and Evaluation of a Multi-Stable Hybrid Energy Harvester

This article develops a multi-stable hybrid energy harvester (MSHEH) which consists of a piezoelectric energy harvester (PEH) and an electromagnetic energy harvester (EMEH). By tuning two parameters, the MSHEH can achieve a mono-stable, bi-stable, and tri-stable state, respectively. A numerical procedure is developed to compute the EMEH’s transduction factor. The obtained result is validated experimentally. Using the equivalent magnetic 2-point dipole theory, the restoring force model of the magnetic spring is established. The obtained model is verified experimentally. The energy harvesting performances of the MSHEH under the four different configurations (linear, mono-stable, bi-stable and tri-stable) subjected to frequency sweep excitations are evaluated by simulation and validated by experiment. The comparative analysis focuses on power output, accumulated harvested energy, and effective energy-harvesting bandwidth. The optimum load resistances are investigated by Pareto front optimizations. The following key findings are obtained. When subjected to high-level frequency sweep excitation, the tri-stable configuration exhibits the widest frequency bandwidth and the highest total accumulated harvested energy. When subjected to low-level frequency sweep excitation, the bi-stable configuration is more efficient in energy harvesting. The best performance trade-off between the PEH and EMEH can be achieved by selecting the optimum load resistances properly.

Nonlinear dynamic characteristics of the multistable wake-galloping energy harvester

Nonlinear dynamic characteristics of the multistable wake-galloping energy harvester

Data-driven modelling and dynamic analysis of the multistable energy harvester with non-Gaussian Lévy noise

In engineering, due to the complex structural characteristics of system and the non-Gaussian properties of random excitation, it is difficult to establish an accurate stochastic dynamic model for the strongly nonlinear multistable vibration energy harvester (VEH), especially for these driven by non-Gaussian Lévy noise. From the view of machine learning, a data-driven model identification method is devised to extract the non-Gaussian governing laws of the multistable VEH with the aid of the observed sample trajectory data. Based on the Nonlocal Kramers-Moyal formulas, the Lévy, drift and diffusion terms can be approximately expressed by the sample trajectories of the system. By implementing the least square method and the stepwise sparse regressor algorithm, the optimal drift and diffusion coefficients can be identified, and then the non-Gaussian stochastic differential equation of VEH is extracted. Two examples are utilized to verify the feasibility and effectiveness of the data-driven modelling method in VEH, which indicates that the identified results agree well with the original system. Finally, the stochastic dynamic behaviors induced by non-Gaussian Lévy noise are explored based on the data-driven penta-stable VEH. The proposed method can provide the theoretical guidance for the modelling and dynamics research of VEH in engineering.

Nonlinear dynamics of a compact and multistable mechanical energy harvester

The use of smart materials as transducers in mechanical energy harvesting systems has gained significant attention in recent years. Despite the numerous proposed solutions in the literature, challenges still exist in terms of their implementation within limited spaces while maintaining optimal performance. This paper addresses these challenges through the concepts of compactness and space-efficient design, as well as the incorporation of nonlinear characteristics and additional degrees-of-freedom. A multistable dual beam nonlinear structure featuring two magnetic interactions and two piezoelectric transducers is presented. A reduced order model with 2-degrees-of-freedom is established based on the harvester structure in order to capture the essential qualitative characteristics of the system. Stability analysis demonstrates that the combination of two nonlinear magnetic interactions furnish unprecedented multistable characteristics to this type of harvester. A framework using a nonlinear dynamics perspective is established to analyze multistable systems based on energy harvesting purposes. Different dynamical and stability characteristics are determined by the differences in the system stiffness ratio. Parametric analyses are carried out classifying regions of high performance in the external excitation parameter space. These regions are associated with rich and complex dynamics. Finally, a comprehensive comparison is conducted between the proposed harvester and the classical bistable harvester, revealing improvements in performance across nearly all relevant conditions. These findings highlight the enhanced capabilities of the proposed harvester design, solidifying its potential of application in diverse energy harvesting scenarios.

Double-Versus Triple-Potential Well Energy Harvesters: Dynamics and Power Output.

The basic types of multi-stable energy harvesters are bistable energy harvesting systems (BEH) and tristable energy harvesting systems (TEH). The present investigations focus on the analysis of BEH and TEH systems, where the corresponding depth of the potential well and the width of their characteristics are the same. The efficiency of energy harvesting for TEH and BEH systems assuming similar potential parameters is provided. Providing such parameters allows for reliable formulation of conclusions about the efficiency in both types of systems. These energy harvesting systems are based on permanent magnets and a cantilever beam designed to obtain energy from vibrations. Starting from the bond graphs, we derived the nonlinear equations of motion. Then, we followed the bifurcations along the increasing frequency for both configurations. To identify the character of particular solutions, we estimated their corresponding phase portraits, Poincare sections, and Lyapunov exponents. The selected solutions are associated with their voltage output. The results in this numerical study clearly show that the bistable potential is more efficient for energy harvesting provided the corresponding excitation amplitude is large enough. However, the tristable potential could work better in the limits of low-level and low-frequency excitations.

Force and stability mechanism analysis of two types of nonlinear mono-stable and multi-stable piezoelectric energy harvesters using cantilever structure and magnetic interaction

Nonlinear mono-stable and multi-stable piezoelectric energy harvesters have attracted a lot of attention owing to their broadband frequency spectra and excellent energy harvesting performance. Herein, two types of nonlinear mono-stable, bi-stable, tri-stable, and quad-stable piezoelectric energy harvesters using cantilever structure and magnetic interaction are compared and analyzed. Based on the magnetizing current method, the magnetic force equations are obtained. Calculation results demonstrate that the stability of these harvesters is dependent on the equivalent linear elastic force and the vertical magnetic force. The equilibrium point occurs when the equivalent linear elastic force equals to the vertical magnetic force. The relationship between the number of stable equilibrium points E S and the number of the intersections of the two force curves N I is that E S= (N I + 1)/2. Experiments are carried out to verify the equivalent linear elastic force, vertical magnetic force, and the number of stable equilibrium points of the fabricated prototypes. The experimental results are consistent with the calculated results, which verifies the correctness of the stability mechanism. Moreover, it is found that the stability mechanism is also applicable to the harvesters with more stable equilibrium points, such as penta-stable and hexa-stable harvesters. This work reveals the stability mechanism of nonlinear mono-stable and multi-stable energy harvesters using cantilever structure and magnetic interaction, and provides technical methods for the design of multi-stable energy harvesters.

Validation and optimization of two models for the magnetic restoring forces using a multi-stable piezoelectric energy harvester.

This article presents a tunable multi-stable piezoelectric energy harvester. The apparatus consists of a stationary magnet and a cantilever beam whose free end is attached by an assembly of two cylindrical magnets that can be moved along the beam and a small cylindrical magnet that is fixed at the beam tip. By varying two parameters, the system can assume three stability states: tri-stable, bi-stable, and mono-stable, respectively. The developed apparatus is used to validate two models for the magnetic restoring force: the equivalent magnetic point dipole approach and the equivalent magnetic 2-point dipole approach. The study focuses on comparing the accuracy of the two models for a wide range of the tuning parameters. The restoring forces of the apparatus are determined dynamically and compared with their analytical counterparts based on each of the models. To improve the model accuracy, a model optimization is carried out by using the multi-population genetic algorithm. With the optimum models, the parametric sensitivity of each of the models is investigated. The stability state region is generated by using the optimum second model.

Dynamically synergistic transition mechanism and modified nonlinear magnetic force modeling for multistable rotation energy harvester

Multistable energy harvesters with magnetic interaction have received increasing attention. However, the synergistic transition mechanism of the different oscillation states for rotation harvesters remains uninvestigated, and there is lack of precise magnetic force model for investigating the effect of the external magnetic force on dynamic characteristics and harvesting performance. This study proposes a synergistic transition and regulation mechanism for rotation energy harvesting and a modified model of the magnetic force is derived by taking the relative displacement and rotational angle into consideration. The transition mechanism of different oscillation states for the rotation system is studied under the guidance of static bifurcation, potential diagram, and dynamic behaviors, respectively. Based on the extended Hamilton’s principle, the theoretical model is established and the complexity dynamic characteristics are simulated. A prototype of multistable energy harvesters with magnetic interaction was fabricated and the effectiveness of the theoretical model was validated by experiments. The experimental results exhibit good agreement with the numerical results. The proposed harvester can harvest energy effectively in a wide rotation speed range of 280–500 rpm, and the output voltage and power can reach 6.4 V and 4 mW.

Bifurcation analysis and nonlinear dynamics of a rolling magnet multistable electromagnetic energy harvester

The objective of this study is to provide a rolling magnet multistable electromagnetic-induction energy harvester (RM-MEH) for scavenging energy from a wide frequency range. The RM-MEH designed based on a magnetic levitation rolling magnet possesses the merit of small damping. By clarifying the mechanics of the RM-MEH, the static bifurcation behavior is investigated and results demonstrate that monostable, bistable, and tristable configurations with different types of potentials can be obtained with proper parameters. Numerical simulations under sweep frequency excitation indicate that the monostable, bistable, and tristable configurations with proper potential functions achieve large-amplitude oscillation in a wide frequency range. Under constant frequency excitation, the nonlinear dynamics are analyzed utilizing the bifurcation diagram, phase orbit, and Poincaré map, and the multiple vibrational patterns are verified. Finally, the vibration isolation characteristics of the system are briefly discussed considering the electromagnetic force of the RM-MEH. In summary, the demonstration of the novel RM-MEH expands the approach to achieving multistable oscillation and provides a new way of thinking for designing nonlinear energy harvesters.

Applicability of magnetic force models for multi-stable energy harvesters

Multi-stable piezoelectric energy harvesters have been exploited to enhance performance for extracting ambient vibrational energy from a broadband energy source. Since magnetic force plays a significant role in enhancing the dynamic behavior of harvesters, it is necessary to model and understand the significant influencing of structural parameters on magnetic force. Recently, several theoretical modeling methods, including magnetic dipole, improved dipole, magnetic current, and magnetic charge models, have been developed to calculate the magnetic force in multi-stable energy harvesters. However, the influence of structural parameters and magnet dimensions on the accuracy of magnetic force calculation for these methods has not been analyzed. Therefore, it is necessary to investigate the applicability of these methods under a range of operating conditions. New insights into the accuracy and application constraints of these methods are presented in this paper to calculate the impact of magnetic force on multi-stable energy harvesters. From the theoretical derivation of models and numerical results obtained, a quantitative assessment of errors under different structural parameters and magnet sizes is presented and compared to evaluate the application constraints. Moreover, experimental measurements are performed to verify the applicability of these modeling methods for bi-stable and tri-stable energy harvesters with different structural parameters.

Dynamic Design of a Quad-Stable Piezoelectric Energy Harvester via Bifurcation Theory

The parameter tuning of a multi-stable energy harvester is crucial to enhancing harvesting efficiency. In this paper, the bifurcation theory is applied to quantitatively reveal the effects of structural parameters on the statics and dynamics of a quad-stable energy harvester (QEH). Firstly, a novel QEH system utilizing the geometric nonlinearity of springs is proposed. Static bifurcation analysis is carried out to design quad-stable working conditions. To investigate the cross-well and high-energy vibration, the complex dynamic frequency (CDF) method, suitable for both weakly and strongly nonlinear dynamic problems, is then applied to deduce the primary response solution. By using the unfolding analysis in singularity theory, four steady-state properties and dozens of primary resonance modes are demonstrated. Based on the transition set, the effective bandwidth for energy harvesting can be customized to adapt well to various vibration environments by parametric adjustment. Finally, the experimental tests verify that the output power can reach up to 1 mW. The proposed QEH and its mechanics optimization can guide energy supply for next-generation wireless systems and low-power sensors under magnetic forbidding environments.

Energy Harvesting Performance of a Novel Nonlinear Quad-Stable Piezoelectric Energy Harvester with Only One External Magnet

Nonlinear multi-stable piezoelectric energy harvesters show broadband frequency spectra and excellent energy harvesting performance, owing to their high output power related to inter-well transitions. However, existing quad-stable piezoelectric energy harvesters contain too many structural parameters, which makes the systems clumsy, and increases the difficulties of dynamic analysis and structural optimization. Herein, a nonlinear quad-stable piezoelectric energy harvester, with only one external magnet, is proposed based on the magnetic force characteristics between a ring magnet and a rectangular magnet. Under selected structural parameters, as the magnet spacing increases, the stability characteristic of the harvester changes from quad-stability to bi-stability, and then to mono-stability. The transformation of the stability characteristic results from the changes in the variation rate of the vertical magnetic force. Subsequently, under the filtered Gaussian white noise within the frequency range of 0–120 Hz, the energy harvesting performance of the harvester is simulated by the classic fourth-order Runge-Kutta method. Simulation results show that the performance of the harvester under the quad-stable structural parameters is better than that under the bi-stable structural parameters, independent of whether the excitation acceleration is small or large. This result is related to the potential well characteristics under the quad-stable and bi-stable structural parameters. More specifically, the potential well depths under the quad-stable and bi-stable structural parameters are almost the same, but the distance between the two outer potential wells under the quad-stable structural parameters is larger than that under the bi-stable structural parameters. Finally, a fabricated prototype is used to measure the experimental performance of the harvester. The experimental data and the estimated data share the same trend. This study provides a new conception and technical method for the design, optimization, and application of quad-stable piezoelectric energy harvesters.

Research on multi-valued response and bursting oscillation of series multi-stable piezoelectric energy harvester

Research on multi-valued response and bursting oscillation of series multi-stable piezoelectric energy harvester

Sensitivity-based nonlinear restoring force identification of multistable piezoelectric energy harvesters

Sensitivity-based nonlinear restoring force identification of multistable piezoelectric energy harvesters

Large stroke tri-stable vibration energy harvester: Modelling and experimental validation

An important issue in vibration energy harvesting is to increase the effective operation bandwidth. One solution is to develop multi-stable energy harvesters which can execute inter-well motion under certain conditions. However, almost all the n-stable harvesters are based on the cantilever-beam-magnets configuration if n > 2, namely, to arrange several fixed magnets near the free end of a piezoelectric cantilever beam with tip magnet, which usually has relatively small stroke due to the deformation limitation of the beam. In this paper, we propose a new tri-stable mechanism using four mechanical springs with well-designed configuration, which has large stroke due to the structural property. Thereafter a tri-stable electromagnetic energy harvester is constructed based on the proposed tri-stable mechanism. The dynamic response and electrical output characteristics are theoretically analyzed by numerical simulations using the dimensionless electromechanical coupled equations. It is shown that the inter-well oscillation can be activated in broadband low-frequency range, which generates considerable induced current. Experiments are also conducted to validate the theoretical model and to demonstrate the low-frequency broadband energy harvesting capability. The output power reaches the level of several hundred milliwatts, which is much larger than that of the previous tri-stable harvesters at milliwatt level.

Discovering governing equation from data for multi-stable energy harvester under white noise

Discovering governing equation from data for multi-stable energy harvester under white noise

An annularly-distributed poly-stable array for broadband vibrational energy

Harvesting the ambient broadband vibrational energy is the ultimate objective of multi-stable vibration energy harvesters. The potential energy of a multi-stable system has more potential wells corresponding stable equilibrium states than a linear energy harvester, and the lower excitation energy is required to overcome the potential barriers to broaden the working bandwidth. However, for traditional multi-stable harvesters, increasing the number of stable equilibrium states to improve energy harvesting efficiency in random frequency bands is difficult. Without the need of external magnets generating static magnetic field, we propose an annularly-distributed poly-stable array which makes stable equilibrium states grow exponentially in maximum number. The experiments indicate that, comparing with a corresponding linear energy harvester, the working frequency bandwidth widened to 283%, the power density increased by a factor of 577 and the electric field energy raised to 133%. The proposed array shows the superiority in broadband vibrational energy harvesting and provides a solution collecting the abundant energy renewable in nature.