Ultra-broadband piezoelectric energy harvesting via bistable multi-hardening and multi-softening

An electromechanical coupled distributed parameter model is derived for a broadband piezoelectric energy harvester with nonlinear magnetic interaction and inductive–resistive interface circuit in the framework of the Hamilton’s principle and Gauss law. The approximate analytical solutions of the responses are obtained based on the equivalent mechanical representation and harmonic balance method. They are validated by experiment data and numerical simulations. The cubic-function discriminant of the analytical solution is introduced to determine the nonlinear boundaries of multiple solutions and the bandwidth with high harvested power. The stability of the multiple solutions is analyzed through Jacobi matrix of the modulation equation. The upward and downward sweep experiments exhibit the bistable and jump phenomena in the hardening range. The state plane of the modulation equation is used to show and explain why different initial conditions yield different stable dynamic motions and exhibit jump phenomenon. Multi-hardening and multi-softening nonlinearities are noted due to the multiply resonances by the inductance in the circuit and nonlinear characteristics of magnetic interaction in the structure. The analytical expression of the determinant of the nonlinear magnetic coefficient with double root of the response is derived to effectively characterize the observed phenomena. Different nonlinear types, e.g., typical nonlinear hardening and softening with two stable and one unstable solutions, and special nonlinear hardening and softening with one stable and one unstable solutions, are noted and investigated. The inductance and cubic magnetic coefficient affect the number and type of the nonlinearities. Multi-hardening or multi-softening nonlinearities enhance the performance of the piezoelectric energy harvester since its bandwidth is significantly broadened to cover up to 40 Hz in the low-frequency range.