This paper presents the modeling and analysis of a nonlinear wideband vibration energy harvester (VEH) with an asymmetric restoring force. It is commonly recognized that a VEH based on a nonlinear resonator having an odd-symmetric hardening (or softening) restoring force can show wideband frequency characteristics due to its bent resonance peak while keeping its maximum power performance. In practice, however, it often happens that the restoring force has some asymmetry, for instance, due to a bias force (e.g. gravity), or irregular asymmetry in the geometry. In this paper, a hardening resonator with a constant bias force is particularly focused on, and its approximate steady-state solution is studied based on a newly proposed averaging method combined with harmonic balancing. The validity of the approximate solutions are verified by comparing them with numerical solutions. As a result of the approximate and numerical analyses, it is shown that the frequency response displays a resonance peak climbing along an S-shaped backbone curve which is because of the softening effect due to the quadratic nonlinearity stemming from the asymmetry, followed by the hardening nature of the restoring force. Consequently, the frequency response yields the coexistence of multiple stable steady-state solutions on both sides of the resonance peak, and the highest-energy orbit exhibits a well-defined wideband behavior.
This paper presents a design study of a miniaturized nonlinear vibration energy harvester based on a mechanically and magnetically-sprung resonator for a low-frequency application. The resonator to be investigated consists of a moving magnet composite as a mass, which is sprung by two planar springs and two fixed ring magnets. The planar springs with spiral-like shape are respectively connected to the both ends of the magnet composite so that they can provide a linear stiffness in a compact size. Mechanical stoppers installed to constrain the deformation of the spring give the resonator hardening characteristics which effectively widen the resonance band. The magnet composite is comprised of two repelling cylindrical magnets and a steel disk between them, all encapsulated in a thin stainless steel cylinder whose outer diameter is smaller than the diameter of the ring magnets. The pole arrangement of the ring magnets is repelling so that they can suspend the magnet composite between them. This configuration of the magnets yields a local minimum in the magnetic attractive force between the magnet composite and the single ring magnet. Consequently, it can show either monostable or bistable property depending on the distance between two ring magnets. If the distance is adjusted so that the bistability emerges, it can cancel the linear stiffness of the planar springs, so that the overall bandwidth can be extended lower which is suitable for low frequency application. In this paper, the hardening effect of the proposed mechanical stopper arrangement is examined by an initial prototype of a miniaturized electromagnetic harvester designed and fabricated without ring magnets. The performance of the harvester in terms of the frequency responses demonstrates a pronounced band widening effect due to the proposed stopper arrangement.