Invariant manifolds and the parameterization method in coupled energy harvesting piezoelectric oscillators

TL;DR

This paper explores how invariant manifold theory and the parameterization method can be used to analyze and potentially enhance energy transfer in coupled piezoelectric oscillators for energy harvesting, considering perturbations like damping and forcing.

Contribution

It demonstrates the existence of invariant manifolds in coupled piezoelectric oscillators and applies the parameterization method to study their dynamics under perturbations.

Findings

Existence of a 4D Normally Hyperbolic Invariant Manifold in the unperturbed system.

Numerical computation of perturbed invariant manifolds and their dynamics.

Evidence of homoclinic connections indicating complex energy transfer pathways.

Abstract

Energy harvesting systems based on oscillators aim to capture energy from mechanical oscillations and convert it into electrical energy. Widely extended are those based on piezoelectric materials, whose dynamics are Hamiltonian submitted to different sources of dissipation: damping and coupling. These dissipations bring the system to low energy regimes, which is not desired in long term as it diminishes the absorbed energy. To avoid or to minimize such situations, we propose that the coupling of two oscillators could benefit from theory of Arnold diffusion. Such phenomenon studies $O(1)$ energy variations in Hamiltonian systems and hence could be very useful in energy harvesting applications. This article is a first step towards this goal. We consider two piezoelectric beams submitted to a small forcing and coupled through an electric circuit. By considering the coupling, damping and forcing as perturbations, we prove that the unperturbed system possesses a $4$-dimensional Normally Hyperbolic Invariant Manifold with $5$ and $4$-dimensional stable and unstable manifolds, respectively. These are locally unique after the perturbation. By means of the parameterization method, we numerically compute parameterizations of the perturbed manifold, its stable and unstable manifolds and study its inner dynamics. We show evidence of homoclinic connections when the perturbation is switched on.