Global Dynamics of a Vibro-Impact Energy Harvester

Zhenbang Cao,Jianliang Shi,Xuegang Yu,Hu Yang,Ge Ren,Haotong Ma,Yi Tan

doi:10.3390/math10030472

Abstract

In this paper, we consider a two-sided vibro-impact energy harvester described as a forced cylindrical capsule inclined at a horizontal angle, and the motion of the ball inside the capsule follows from the impacts with the capsule ends and gravity. Two distinct cases of dynamical behavior are investigated: the nondissipative and dissipative cases, where the dissipation is given by a restitution coefficient of impacts. We show that the dynamics of the system are described by the use of a 2D implicit map written in terms of the variables’ energy and time when the ball leaves the moving capsule ends. More precisely, in the nondissipative case, we analytically show that this map is area-preserving and the existence of invariant curves for some rotation number with Markoff constant type is proved according to Moser’s twist theorem in high energy. The existence of invariant curves implies that the kinetic energy of the ball is always bounded, and hence, the structure of system is not destroyed by the impacts of the ball. Furthermore, by numerical analysis we also show that the dynamical behavior of this system is regular, mainly containing periodic points, invariant curves and Aubry–Mather sets. After introducing dissipation, the dissipation destroys the regular dynamical behavior of the nondissipative case, and a periodic point with low energy is generated.

Highlights

Energy harvesting (EH) refers to the process of converting various renewable energy sources such as wind, solar, wave, and vibration into electrical energy, which usually can be used to complement and substitute other sources of energy [1]
The vibro-impact energy harvester (VI-EH) system can be modeled as a small ball of unitary mass rolling inside a forced cylindrical capsule that is inclined at a horizontal angle of β
We first discuss the dynamical behavior of the system in the nondissipative case, and discuss the dissipative case

Summary

Introduction

Energy harvesting (EH) refers to the process of converting various renewable energy sources such as wind, solar, wave, and vibration into electrical energy, which usually can be used to complement and substitute other sources of energy [1]. Linear systems are generally not suitable as EH devices under excitation with varying frequencies, since the high-power output required by the system can only be achieved by near resonance, leading to reliability and fatigue issues [6,7]. The second is that a proper approach to studying VI systems is to attempt to establish its discrete Poincaré map (sometimes called the first return map). This map usually cannot be solved explicitly, thereby presenting the main difficulty with obtaining further analytical results. The main contribution of this paper is that we prove that the structure of the system is not destroyed by the impact of the ball.

Statement of the Problem

The Existence of Invariant Curves

Numerical Simulations

Conclusions