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

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. * The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme (FP7/2007-2013. This work has also been partially supported by MINECO MTM2015-65715-P Spanish grant. We acknowledge the use of the UPC Dynamical Systems group's cluster for research computing (https://dynamicalsystems.upc.edu/en/computing/)

A General Mechanism of Diffusion in Hamiltonian Systems: Qualitative Results

“…Other geometric methods have been applied in [16,28,33,38,51,67,75,95]. These geometric methods also work for systems with magnetic fields [73] and for dissipative systems [61].…”

A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold

“…Of course, the abundance of the hypotheses and the methods to check them are different in the Hamiltonian systems from the non-Hamiltonian ones. An interesting class of systems is that of non-Hamiltonian perturbations of Hamiltonian systems [38,34].…”

“…There are several mechanisms of Arnold diffusion based on NHIM's. Since NHIM's exist even in non-Hamiltonian systems, these mechanisms apply to more general systems (e.g., magnetic fields [44] or dissipative systems [38]). The mechanisms based on NHIM's can accommodate also general time-dependent perturbations (e.g., quasiperiodic [19], or time-recurrent [32]).…”

“…The general shadowing theorem can be used even for non-Hamiltonian systems. In the paper, [38], the mechanism is applied to show existence of orbits transferring energy in some models of piezoelectric energy harvesters. These systems consist of coupled piezolectric oscillators, and they can convert ambient vibrations into electrical energy.…”

Multiple Solutions of the Tristable Energy Harvester

“…Wang et al concluded that their system design allows for an almost 35-fold increase in energy harvesting compared to a conventional method. There are also solutions where the movement of two flexible beams is coupled via a spring [38]. Another concept for energy harvesting is presented in [39], where two beams with glued transducers are inertially coupled, as a result of which piezoelectrics are affected by components resulting from bending of beams and fluctuations of the inertial coupling mass.…”