Enhanced modeling of nonlinear restoring force in multi-stable energy harvesters

It is acknowledged that traditional linear piezoelectric cantilever energy harvesters are not effective to extract ambient energy due to the narrow resonant bandwidth and low efficiency. A popular strategy to overcome these disadvantages is applying external magnets to realize multi-stable energy harvesters. Under the external excitation, the inter-well motion occurs between different potential wells in multi-stable energy harvesters, so as to bring out larger displacement and higher voltage output than intra-well motion. The performance of multi-stable energy harvesters has a strong relationship with the nonlinear restoring force, but there is lack of models for accurately describing nonlinear restoring force. This paper proposes a new enhanced model of the nonlinear restoring force to improve the calculating accuracy by taking the influence of rotational angle and axial displacement into consideration. The rotational magnetic charge is presented to calculate the magnetic force exerted by the external magnets. Then, the trajectory method of cantilever beams is employed to reduce the calculation error caused by axial displacement during oscillation. The accuracy of the proposed model for magnetic force calculation along horizontal and vertical directions is verified by the results of numerical simulation. Additionally, experimental results of the nonlinear restoring force for multi-stable energy harvesters show a good agreement with theoretical results. The influence of system parameters on equilibrium points, nonlinear restoring force and dynamic responses of multi-stable energy harvesters has been investigated based on the proposed model.