Nonlinear dynamic and energetic characteristics of piezoelectric energy harvester with two rotatable external magnets

• A multi-stable energy harvester with rotatable magnets is analytically studied.
• The effect of the inclination angle of two external magnets is investigated.
• Bifurcation characteristics are depicted with respect to the inclination angle.
• Frequency response curves of the system with bi- or tri-stability are investigated.

This study investigated the nonlinear dynamic and energetic characteristics of an energy harvester system composed of a cantilever beam with two fully deposited piezoelectric layers and a tip magnet attached to the beam׳s free end. In this system, the tip magnet interacts with two external rotatable magnets fixed in free space, in which the number of equilibrium points of the tip magnet changes with respect to the inclination angle of the rotatable magnets. The field equations of the beam and magnetic force are sophisticatedly derived using the modified Hamilton׳s principle and magnetic current model, respectively. The derived equations are qualitatively validated by comparison with the experimental results reported in Zhou et al. (2013) [31] and by conducting experiments, it is shown that the mathematical model can predict large-amplitude motion well. The linear terms of the magnetic force and torque are properly reflected in deriving the accurate eigen-modes, and the field equation of the cantilever beam coupled with the electronic circuit is discretized into a single mode oscillator using the Galerkin method. Finally, the characteristics of the system bifurcating into a mono-, bi- or tri-stable system with respect to the inclination angle of the external magnet and separation distance between the tip magnet and external magnets are investigated with the help of both the bifurcation and potential energy diagrams. It is also shown that the broadband and power-generating characteristics of the system can be thoroughly examined by using the harmonic balance method as a simple, manipulable tool.