Multiple Solutions of the Tristable Energy Harvester

Abstract

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 w, 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.

On Theoretical and Numerical Aspects of Bifurcations and Hysteresis Effects in Kinetic Energy Harvesters

“…The space dimension is a multiple of the number of degrees of freedom. Considering the research problem formulated in this way, it is possible to conduct complex numerical calculations, the results of which can be illustrated in the form of a diagram of solutions (DS) [ 27 , 28 ]. In this approach, the information about the efficiency of harvesting energy from coexisting solutions is neglected.…”

Energy Harvesting in a System with a Two-Stage Flexible Cantilever Beam

“…At this point, the focus was on the identification of periodic solutions that occur in the tested structure of the energy harvesting system. Periodic solutions gain significant importance because when the response of the system changes from periodic to chaotic, a reduction in the efficiency of energy harvesting is observed [ 53 , 56 ]. In our model tests, a numerical procedure based on the estimation of the distinct Poincaré points was used.…”

Nonlinear Dynamics of a Star-Shaped Structure and Variable Configuration of Elastic Elements for Energy Harvesting Applications

“…Considering the construction of energy harvesters to recover energy from mechanical vibrations, most of them are based on a flexible cantilever beam with piezoelectric transducers glued to this beam [ 12 , 13 ]. When the beam is excited to vibrate, it deforms, which allows it to harvest energy.…”