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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. [ABSTRACT FROM AUTHOR]

The subject of the research contained in this paper is a new design solution for an energy harvesting system resulting from the combination of a quasi-zero-stiffness energy harvester and a two-stage flexible cantilever beam. Numerical tests were divided into two main parts-analysis of the dynamics of the system due to periodic, quasiperiodic, and chaotic solutions and the efficiency of energy generation. The results of numerical simulations were limited to zero initial conditions as they are the natural position of the static equilibrium. The article compares the energy efficiency for the selected range of the dimensionless excitation frequency. For this purpose, three cases of piezoelectric mounting were analyzed-only on the first stage of the beam, on the second and both stages. The analysis has been carried out with the use of diagrams showing difference of the effective values of the voltage induced on the piezoelectric electrodes. The results indicate that for effective energy harvesting, it is advisable to attach piezoelectric energy transducers to each step of the beam despite possible asynchronous vibrations. [ABSTRACT FROM AUTHOR]

In this paper, a study on modelling energy harvesting efficiency of a quasi-zero stiffness system is presented. Mechanical characteristics of the system are identified, and the effect of its stiffness and geometry on the function describing energy potential barrier is determined. It has been shown numerically that an increase in equivalent stiffness of the quasi-zero stiffness system limits the potential barrier width. On the other hand, increased the spacing between compensating springs results in increased barrier width. Simulation results of the quasi-zero stiffness system are compared with those obtained for a triple-well system with permanent magnets. Based on mathematical models, multi-color diagrams depicting the largest Lyapunov exponent are plotted. The effect of selected values of external excitation frequency and amplitude on the efficiency of energy harvesting is determined. The rms value of time sequence is taken as a measure of the energy harvesting efficiency. Obtained numerical results are plotted as phase trajectories. [ABSTRACT FROM AUTHOR]

The subject of the model research contained in this paper is a new design solution of the energy harvesting system with a star-shaped structure of elastic elements and variable configuration. Numerical experiments focused mainly on the assessment of the configuration of elastic elements in the context of energy harvesting efficiency. The results of computer simulations were limited to zero initial conditions as it is the natural position of the static equilibrium. The article compares the energy efficiency for the selected range of the dimensionless excitation frequency. For this purpose, four cases of elastic element configurations were compared. The results are visualized based on the diagram of RMS voltage induced on piezoelectric electrodes, bifurcation diagrams, Lyapunov exponents, and Poincaré maps, showing the impact of individual solutions on the efficiency of energy harvesting. The results of the simulations show that the harvester's efficiency ranges from 4 V to 20 V depending on the configuration and the frequency range of the excitation, but the design allows for a smooth adjustment to the given conditions. [ABSTRACT FROM AUTHOR]

The paper presents the results of numerical investigations of the overhead travelling cranes load motion. The model studies assume that the load is suspended on the inextensible rope. Conversely, its motion is triggered by an external moment. In addition, energy losses in the construction node connecting the rope to the drum are included. At the same time these losses were mapped through a linear viscous damper. The main objective was to evaluate the impact of individual mathematical model parameters on the dynamics of the transported load. The results were compared between two models: with/without crane structure vibrations included. The results were illustrated by multi-colored maps of the largest Lyapunov exponent, bifurcation diagrams, and Poincare cross-sections. [ABSTRACT FROM AUTHOR]

This paper presents the results of computer simulations of a gear model, where the variable stiffness of the meshing and backlash are considered. The outcome of such assumptions is a non-linear mathematical model in which chaotic phenomena can occur. During model studies, attention was paid to the identification of areas limited by the physical parameters, for which the analysed system behaved chaotically. To determine the ranges of irregular gear behaviour, numerical procedures were used to plot the bifurcation diagram, the Lyapunov exponent, the amplitude-frequency distribution and the Poincaré cross section. [ABSTRACT FROM AUTHOR]

• A new energy harvesting system has been designed QZEH. • The areas where the best energy harvesting is possible have been identified. • The method of impulse excitation on the solution was proposed. • The probability of the solution with the best effectivity was determined. This paper presents the results of modelling a new nonlinear multi-stable QZEH (quasi-zero energy harvester) system for harvesting energy from vibrating mechanical devices. Detailed tests were carried out on the system model, which consisted of a beam and a system of springs, which were used to determine the potential of a quasi-flat well. Two-dimensional distributions of the Lyapunov exponent were output from the numerical model, using the assumed range of variability within the control parameters, and plotted as a map in multi-color. These maps are related to diagrams of the RMS values of the voltage that is induced on the piezoelectric electrodes. To identify the optimal conditions for harvesting energy from mechanical vibrations, a multi-colored map of the RMS voltage values was produced. Its reference to the Lyapunov distribution map, showed that in the chaotic motion zones, energy harvesting is reduced. Based on the established sections of the Lyapunov exponent, diagrams of solutions (DS) showing the number of coexisting solutions and their periodicity were drawn. Multiple solutions and basins of attraction have been identified. On their basis it was possible to estimate the probability of obtaining a solution with the greatest energy harvesting efficiency. Moreover, a method of acting on the solution trajectory, by means of an impulse initiated at a specific moment in time, has been proposed. The results of the model tests were visualized as multi-colored maps of impulse excitations. The direct reference of the results of QZEH model tests to the tristable energy harvesting (TEH) system clearly indicates the advisability of using the QZEH system in terms of higher excitation amplitudes. The QZEH system also shows an improved ability to harvest energy in the low range of values of excitation frequencies. [ABSTRACT FROM AUTHOR]

The piezoelectric energy-harvesting system with double-well characteristics and hysteresis in the restoring force is studied. The proposed system consists of a bistable oscillator based on a cantilever beam structure. The elastic force potential is modified by magnets. The hysteresis is an additional effect of the composite beam considered in this system, and it effects the modal solution with specific mass distribution. Consequently, the modal response is a compromise between two overlapping, competing shapes. The simulation results show evolution in the single potential well solution, and bifurcations into double-well solutions with the hysteretic effect. The maximal Lyapunov exponent indicated the appearance of chaotic solutions. Inclusion of the shape branch overlap parameter reduces the distance between the external potential barriers and leads to a large-amplitude solution and simultaneously higher voltage output with smaller excitation force. The overlap parameter works in the other direction: the larger the overlap value, the smaller the voltage output. Presumably, the successful jump though the potential barrier is accompanied by an additional switch between the corresponding shapes. [ABSTRACT FROM AUTHOR]

This paper presents the results of numerical simulations of a non-linear, tristable system for harvesting energy from vibrating mechanical devices. Detailed model tests were carried out in relation to the system consisting of a beam and three permanent magnets. Based on the derived mathematical model and assuming a range of control parameter variability, a three-dimensional image of the distribution of the largest Lyapunov exponent was plotted. On its basis, the regions of chaotic and predictable movement of the considered system exist have been established. With reference to selected plane of the largest Lyapunov exponent cross-sections, possible co-existing solutions were identified. To identify multiple solutions, a diagram of solutions (DS) diagram was used to illustrate the number of existing solutions and their periodicity. The proposed calculation tool is based on the so-called fixed points of Poincaré cross-section. In relation to selected values of the control parameter ω, coexisting periodic solutions were identified for which phase trajectories and basins of attraction were presented. Based on the model tests carried out, it was found that in order to efficiently harvest energy, appropriate transducer adjustment is required. Calibration of the transducer is necessary to obtain the greatest amplitude of vibration of the beam, which corresponds to the phase trajectory limited by external energy potential barriers. As expected, the average voltage induced on the electrodes of the piezoelectric transducer and the average electrical power recorded on the resistive element are directly proportional to the amplitude and average kinetic energy of the beam. [ABSTRACT FROM AUTHOR]