Transition mechanism and dynamic behaviors of a multi-stable piezoelectric energy harvester with magnetic interaction

•A multi-stable piezoelectric energy harvester (MPEH) is modeled and analyzed.•Geometric nonlinearity (GN) is considered in the model to reveal the transition mechanism.•The MPEH has five transition modes from the mono-stability to the multi-stability.•The GN is beneficial for energy harvesting enhancement.•The GN has little effect on equilibrium bifurcations but great effect on the dynamic response. Multi-stable piezoelectric energy harvesters (MPEH) with magnetic interaction are more prone to oscillate into an inter-well motion at a lower excitation level to achieve high power generation than the bi-stable ones. However, the geometric nonlinearity induced by the large deformation of inter-well motion is ignored by most of the derived mathematic models, resulting in the transition mechanism and complicated dynamics of the MPEH are not well illustrated. This paper focuses on a MPEH configuration that is composed of a piezoelectric cantilever beam with a tip magnet and three external magnets. The magnetic interaction between the tip and external magnets is modeled as point dipole with the path integral approach, and the mathematic model of the MPEH is derived by taking into account the geometric nonlinearity of the beam. The transition mechanism from mono-stability to multi-stability is deeply investigated by means of equilibrium bifurcation, potential well diagrams and dynamic bifurcation, respectively. The complicated dynamic responses of the MPEH are then simulated by means of frequency sweeping method. A prototype is subsequently developed and the theoretical results are validated by experiments. The results show that experiments and simulations are in qualitative agreement. The MPEH has five transition modes from the mono-stability to the quad-stability and has three different potential-well shapes in quad-stable period state. The geometric nonlinearity of the cantilever beam introduces additional third-order and fifth-order nonlinear stiffness, which is conducive to generate inter-well motions and to broad frequency bandwidth of the MPEH, as well as to low the requirement of the excitation intensity that jumping from intra-well motion to inter-well motion.