Validation and optimization of two models for the magnetic restoring forces using a multi-stable piezoelectric energy harvester

This article presents a tunable multi-stable piezoelectric energy harvester. The apparatus consists of a stationary magnet and a cantilever beam whose free end is attached by an assembly of two cylindrical magnets that can be moved along the beam and a small cylindrical magnet that is fixed at the beam tip. By varying two parameters, the system can assume three stability states: tri-stable, bi-stable, and mono-stable, respectively. The developed apparatus is used to validate two models for the magnetic restoring force: the equivalent magnetic point dipole approach and the equivalent magnetic 2-point dipole approach. The study focuses on comparing the accuracy of the two models for a wide range of the tuning parameters. The restoring forces of the apparatus are determined dynamically and compared with their analytical counterparts based on each of the models. To improve the model accuracy, a model optimization is carried out by using the multi-population genetic algorithm. With the optimum models, the parametric sensitivity of each of the models is investigated. The stability state region is generated by using the optimum second model.