Dynamics, Bifurcations and Normal Forms in Arrays of Magnetostrictive Energy Harvesters with All-to-All Coupling

Modeling and bifurcation analysis of an energy harvesting system composed of coupled resonators using the Galfenol-based magnetostrictive material are presented. The analysis in this work should be broad enough to be applicable to a large class of vibratory-based energy harvesting systems since various types of vibratory harvesters share the same normal forms, e.g. magnetostrictive and piezoelectric materials. A combined model of the mechanical and electrical domains of a single energy harvester is discussed first. Building on this model, the governing equations of the coupled system are derived, leading to a system of differential equations with an all-to-all coupling between the resonators. A bifurcation analysis of the system equations reveals different patterns of collective oscillations. Among the many different patterns, a synchronous state exists and it is stable over a broad region of parameter space. This pattern has the potential to yield significant increases in power output and it will be used as a starting point to guide future experimental work. A Hamiltonian approach is employed to study analytically the nature of the bifurcations and to calculate an expression for the onset of synchronization valid for any number of harvesters.