Nonlinear Harvester Research Articles

Stochastic stability of a piezoelectric vibration energy harvester under a parametric excitation and noise-induced stabilization

Vibration energy harvesters seek to convert the energy of ambient, random vibrations into electrical power, often using piezoelectric transduction. The stochastic dynamics of a piezoelectric harvester subjected to a parametric (state-dependent) excitation has not been comprehensively investigated in the nonequilibrium regime. Motivated by mathematical results that establish the stabilization of dynamic response using noise, we investigate the stochastic stability of a generic harvester in the linear and the monostable nonlinear regimes excited by a multiplicative noise process characterized by both Brownian and Lévy distributions. Stability is characterized in each case by studying the approach of the harvester response towards equilibrium in the time domain, longer term proximity (or divergence) of two solutions starting with nearby initial conditions in the phase plane as well as the Lyapunov exponent. In the linear case, we analytically obtain a lower bound on the magnitude of the noise intensity that guarantees stability. The bound is derived as an inequality in terms of the system parameters. This analytic result is validated numerically using the Euler-Maruyama scheme by: (1) computing the harvester response and its approach to equilibrium in terms of its displacement, velocity and voltage, (2) computing the trajectories of two solutions with nearby initial conditions in the phase plane and (3) the sign of the maximal Lyapunov exponent in terms of energy. We find that noise-induced stabilization occurs consistently for the noise intensities greater than the lower bound in linear and weakly nonlinear harvesters, for both Brownian and Lévy excitation. Instabilities emerge for the noise intensities lower than the bound. The results lead to the interesting conclusion that noise of appropriate strength can induce stabilization and are expected to be useful in the design of energy harvesters. In addition, the results are expected to be significant in the study of phenomena such as noise-induced transport in Brownian rotors where stability is an important aspect of the stochastic dynamics.

Positive pseudo almost periodic solutions to a class of hematopoiesis model: oscillations and dynamics

This paper presents a new generalized Mackey-Glass model with a non-linear harvesting term and mixed delays. The main purpose of this work is to study the existence and the exponential stability of the pseudo almost periodic solution for the considered model. By using fixed point theorem and under suitable Lyapunov function, sufficient conditions are given to study the pseudo almost periodic solution for the considered model. Moreover, an illustrative example is given to demonstrate the effectiveness of the obtained results.

Dynamics in a Plankton Model with Toxic Substances and Phytoplankton Harvesting

In this paper, a phytoplankton–zooplankton model incorporating toxic substances and nonlinear phytoplankton harvesting is established. The existence and stability of the equilibrium of this model are first investigated. The occurrence of transcritical, saddle-node, Hopf and Bautin bifurcations at different equilibria is then verified. In addition, the properties of Hopf bifurcation and Bautin bifurcation are discussed by using normal form method. These results demonstrate that phytoplankton and zooplankton populations will oscillate periodically when the harvesting level is high. More interestingly, it is found that the oscillations are always unstable for small phytoplankton carrying capacity, while the dynamics have close relations with the initial population densities for a large environmental capacity. The existence of Bautin bifurcation theoretically indicates that toxic phytoplankton can cause extinction once there exist harmful algal blooms for some time. These results are numerically illustrated for the model with spatial diffusion, which shows that local phytoplankton blooms will lead to global populations extinction.

Global dynamics of a Holling Type-III two prey–one predator discrete model with optimal harvest strategy

This paper deals with a discrete-time two prey–one predator system with Holling Type-III functional response, along with inter-specific competition between the prey and intra-specific competition among the predators, with nonlinear harvesting of a prey population. The conditions for existence and local stability of the equilibrium points and the sufficient conditions for global asymptotic stability of the interior equilibrium point are derived. Furthermore, optimal harvesting policy is obtained by extension of Pontryagin’s maximum principle to discrete system. Meanwhile, some numerical simulations are provided to corroborate the analytical results.

Pattern Formation Scenario through Turing Instability in Interacting Reaction Diffusion Systems with Both Refuge and Nonlinear Harvesting

Of concern in the present theoretical study is to carry out the complex dynamics of a reaction-diffusion predator-prey model incorporating constant proportion of prey refuge and nonlinear prey harvesting with zero-flux boundary conditions. By the method of Lyapunov function, the global stability of the feasible interior equilibrium point for nonspatial model was established. The conditions of diffusion-driven instability were obtained and the Turing space in the parameters space was given as well. Consequently, we present the evolutionary procedure that occupies organism distribution and their interaction of spatially distributed species with diffusion and locate that the model dynamics reveals a diffusion-controlled formation growth to hole patterns or labyrinthine patterns or hole-stripe patterns replication over the whole spatial domain. The analytical results are then authenticated with the help of numerical simulations. Our results points out that the diffusion has an immense impact on the prey refuge as well as prey harvesting and extend well the findings of spatiotemporal dynamics in the reaction-diffusion model.

Broadband energy harvesting from time-delayed nonlinear oscillations of magnetic levitation

This article investigates quasiperiodic energy harvesting in a nonlinear vibration-based harvester, consisting of delayed nonlinear vibrations from magnetic levitation subjected to harmonic base acceleration, in which time-delay is introduced. For energy harvesting, a coil made up of seven layers of 36 gauges wound around the outer casing is utilized. The first- and second-order perturbation approximations are used to obtain the frequency responses of the power and vibration amplitudes in the vicinity of primary resonance. Numerical results are presented to verify the correctness of the analytical solutions. The results show that there is a good agreement between the second approximation and the numerical method; thus, we used the second-order perturbation to present the results. The impact of several parameters on the power and vibration amplitudes in the absence and presence of delay is studied. It was observed that using the nonlinear vibrations due to magnetic levitation in the absence and presence of the delay enables energy harvesting over broadband frequencies distant from primary resonance, which benefits preventing the instability and hysteresis regions near resonance. Dynamic behaviors of the motion are shown in the form of time histories and phase portraits. The results indicate periodic and quasiperiodic motions for the different values of the delay parameters. Also, the effects of the separation distance between magnets [Formula: see text], without and with time-delay, on the vibration and power amplitudes were examined. It was observed that the [Formula: see text], for both points chosen from stable periodic solution and unstable periodic solution, has a significant influence on the system behavior. Thus, choosing proper values for these parameters enables the device to extract more power in lower and broadband excitation frequencies. Besides, we illustrate that for specific values of [Formula: see text] and delay time parameters, the maximum system power output is not necessarily accompanied by maximum vibration amplitude.

On Broadband Nonlinear Piezoelectric Energy Harvesting Techniques

With the development of portable electronic devices, internet of things, micro-electromechanical system (MEMS) and wireless sensor networks, mirco sources and self-powered devices have been attracted great attention. It is an effective way to supply mirco power or realize the miniaturization of the device via the piezoelectric energy harvesting effect. Therefore, research on piezoelectric energy harvesting techniques has been highly valued. To enhance adaptability and improve harvesting efficiency, the broadband nonlinear piezoelectric energy harvesting technique turns to be a research hotspot. The current status of broadband nonlinear piezoelectric energy harvesting technique with nonlinear stiffness was reviewed according to methods of realizing nonlinearity. The characteristics and existed problems of each kind of nonlinear harvester were then concluded. The nonlinear piezoelectric energy harvester with nonlinear stiffness has a good prospect if its advantages can be given full use.

Hopf Bifurcation and Singularity Induced Bifurcation in a Leslie-Gower Predator-Prey System with Nonlinear Harvesting

Hopf Bifurcation and Singularity Induced Bifurcation in a Leslie-Gower Predator-Prey System with Nonlinear Harvesting

Stability and bifurcation of a delayed diffusive predator-prey system with food-limited and nonlinear harvesting.

Based on ecological significance, a delayed diffusive predator-prey system with food-limited and nonlinear harvesting subject to the Neumann boundary conditions is investigated in this paper. Firstly, the sufficient conditions of the stability of nonnegative constant steady state solutions of system are derived. The existence of Hopf bifurcation is obtained by analyzing the associated characteristic equation and the conditions of Turing instability are derived when the system has no delay. Furthermore, the occurrence conditions the Hopf bifurcation are discussed by regarding delay expressing the gestation time of the predator as the bifurcation parameter. Secondly, by using upper-lower solution method, the global asymptotical stability of a unique positive constant steady state solution of system is investigated. Moreover, we also give the detailed formulas to determine the direction, stability of Hopf bifurcation by applying the normal form theory and center manifold reduction. Finally, numerical simulations are carried out to demonstrate our theoretical results.

The Dynamics of an Eco-Epidemiological Model with Allee Effect and Harvesting in the Predator

The aim of this study was to propose and evaluate an eco-epidemiological model with Allee effect and nonlinear harvesting in predators. It was assumed that there is an SI-type of disease in prey, and only portion of the prey would be attacked by the predator due to the fleeing of the remainder of the prey to a safe area. It was also assumed that the predator consumed the prey according to modified Holling type-II functional response. All possible equilibrium points were determined, and the local and global stabilities were investigated. The possibility of occurrence of local bifurcation was also studied. Numerical simulation was used to further evaluate the global dynamics and the effects of varying parameters on the asymptotic behavior of the system.

Method for controlling vibration by exploiting piecewise-linear nonlinearity in energy harvesters.

Vibration energy is becoming a significant alternative solution for energy generation. Recently, a great deal of research has been conducted on how to harvest energy from vibration sources ranging from ocean waves to human motion to microsystems. In this paper, a theoretical model of a piecewise-linear (PWL) nonlinear vibration harvester that has potential applications in variety of fields is proposed and numerically investigated. This new technique enables automatic frequency tunability in the energy harvester by controlling the gap size in the PWL oscillator so that it is able to adapt to changes in excitations. To optimize the performance of the proposed system, a control method combining the response prediction, signal measurement and gap adjustment mechanism is proposed in this paper. This new energy harvester not only overcomes the limitation of traditional linear energy harvesters that can only provide the maximum power generation efficiency over a narrow frequency range but also improves the performance of current nonlinear energy harvesters that are not as efficient as linear energy harvesters at resonance. The proposed system is demonstrated in several case studies to illustrate its effectiveness for a number of different excitations.

Analysis of nonlinear vibration energy harvesters using a complex dynamic frequency method

To understand the complicated dynamic behavior of a Nonlinear Piezoelectric Energy Harvester (NPEH), this paper develops an improved Complex Dynamic Frequency (CDF) method based on complex normal form. CDF introduces a dynamic frequency factor and establishes a set of algebraic equations in handling the effect of higher-order nonlinear terms in a wide frequency band to obtain periodic responses of NPEH. Numerical and experimental studies verify that the proposed CDF gives consistent and accurate predictions of the systems with both weak and strong nonlinearity. Furthermore, through an implicit relationship between magnet arrangement and output performance, one may effectively control the sweep frequency with softening and hardening characteristics. That is a major breakthrough toward the further nonlinear design for broad bandwidth harvesters. As the application, the experimental results reveal the high response profiles can be in a wide frequency range from 10.8 Hz to 24.5 Hz for the NPEH developed that allows an output power of 9 times higher than the conventional linear structure.

Energy and Phase Space Analysis of Response Stabilization Control for Nonlinear Wideband Vibration Energy Harvesting

Energy and Phase Space Analysis of Response Stabilization Control for Nonlinear Wideband Vibration Energy Harvesting

Near-Limit Kinetic Energy Harvesting From Arbitrary Acceleration Waveforms: Feasibility Study by the Example of Human Motion

Kinetic energy harvesters have become a common power source for autonomous sensors operating at micro- and meso-scales. The conventional approach to kinetic energy harvesting is to assume that the proof mass of the mechanical component in an energy harvester is actuated by external motion produced by the sensor’s environment. This approach, dominant since the beginning of micro-scale energy harvesting, has now resulted in the design of advanced, nonlinear harvesters suitable for non-harmonic vibrations produced by many systems of interest. In this paper, we present a feasibility study of an alternative approach to kinetic energy harvesting, where the motion of the proof mass is actively synthesized.

An Analysis of A Fishing Model with Nonlinear Harvesting Function

In this study, considering the importance of how to exploit renewable natural resources, we analyze a fishing model with nonlinear harvesting function in which the players at the equilibrium point do a static game with complete information that, according to the calculations, will cause a waste of energy for both players and so the selection of cooperative strategies along with the agreement between the players is the result of this research.

Analysis of a plankton–fish model with external toxicity and nonlinear harvesting

In this paper, we consider a three species plankton–fish system that incorporates external toxicity and nonlinear harvesting. We consider that the growth of species are affected directly or indirectly by an external toxic substance and the feeding of the predator on the affected prey is considered as Holling type II functional response. All the possible biological feasible equilibrium points are determined analytically as well as numerically and performed stability analysis around these equilibrium points. It is shown that the system undergoes for Hopf bifurcation when the growth rate of prey passes some threshold value. Furthermore, Pontryagin’s maximum principle has been applied to obtain optimal control of harvesting to maximize the benefit as well as the conservation of the ecosystem. We perform numerical simulations to justify and illustrate our analytical results. Some numerical tools such as phase portraits, time evaluation and bifurcation diagrams are presented to ensure the complex dynamics in the system. Period doubling cascade route to chaos is examined and validated through the numerical calculation of Lyapunov exponents and sensitivity analysis.

Almost periodic solutions of a commensalism system with Michaelis-Menten type harvesting on time scales

Abstract In this paper, we consider an almost periodic commensal symbiosis model with nonlinear harvesting on time scales. We establish a criterion for the existence and uniformly asymptotic stability of unique positive almost periodic solution of the system. Our results show that the continuous system and discrete system can be unify well. Examples and their numerical simulations are carried out to illustrate the feasibility of our main results.

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.

Qualitative analysis of a diffusive Crowley–Martin predator–prey model: the role of nonlinear predator harvesting

In this paper, we discuss a diffusive predator–prey system with mutually interfering predator and nonlinear harvesting in predator with Crowley–Martin functional response. The mathematical analysis of the system starts with the existence and uniqueness of solution of the system using $$C_0$$ semigroup. The analysis reflects that the upper bound of rate of predator harvesting for the coexistence of the species can be guaranteed. In addition, we establish the existence and nonexistence of non-constant positive steady state. Explicit conditions on predator harvesting are obtained for local and global stability of interior equilibrium and also for the existence and nonexistence of non-constant steady-state solution. We also investigate conditions for Turing instabilities of the diffusive system analytically. Our results show that the effort of harvesting (g) provides a threshold value for existence of non-constant positive stationary solution. Furthermore, we illustrate the spatial patterns via numerical simulations, which show that the system exhibits interesting patterns. Some biological implications of obtained theoretical results have also been discussed.

Dynamics of a fishery system in a patchy environment with nonlinear harvesting

In this paper, a stock‐effort dynamical model with two fishing zones is discussed. The nonlinear harvesting function is assumed depending upon stock size as well as fishing effort. The migration of fish is considered between two zones. The harvesting vessels also move between zones to increase their revenue. The movements of fish and fishing vessels between zones are assumed to take place at a faster time scale as compared with processes involving growth and harvesting occurring at a slow time scale. The aggregated model is obtained for total fish stock and fishing effort. This aggregated (reduced) model is analyzed analytically as well as numerically. Biological and bionomic equilibria of the system are obtained, and criteria for local stability or instability of the system are derived. The impact of levels of taxation T on the fish population and on the revenue earned by the fishery is investigated. An optimal harvesting policy is also discussed using the Pontryagin's maximum principle. The aggregated model also exhibits Hopf and transcritical bifurcation with respect to the bifurcation parameter tax T. Numerical simulations are presented to illustrate the results.