Approximate solutions and their stability of a broadband piezoelectric energy harvester with a tunable potential function

•A novel nonlinear piezoelectric energy harvester with a tunable potential function is proposed.•The harvester is theoretically modeled and the governing equations are derived with coupled higher-order terms.•The approximate solutions are derived by harmonic balance analysis.•An approach is proposed for the stability analysis of the solutions.•The harvester has a wide frequency bandwidth over 0–91 Hz at the excitation level of 0.5 g. A broadband piezoelectric energy harvester (PEH) with a mechanically tunable potential function is modeled and analytically analyzed. The harvester consisting of a beam and a pre-compression spring at one end can be tuned to both monostable and bistable configurations. The axial motion of the beam resulting from the transverse vibration and spring load induces two coupled higher-order terms of displacement, velocity and acceleration into the governing equations. This significantly complicates the theoretical analysis, especially the stability analysis of solutions. Harmonic balance method is employed to investigate the dynamic characteristics of the nonlinear energy harvester. An effective approach is developed to solve the entries of the Jacobian matrix for determining the stability of analytical solutions. This approach offers a criterion for solution stability analysis of congeneric nonlinear systems with coupled higher-order terms. The energy harvesting performance and the nonlinear dynamic characteristics of the proposed PEH are explored for various base excitation levels, electrical resistive loads and pre-deformations of the spring. The approximate analytical solutions are validated by numerical simulations. Results demonstrate that the energy harvesting performance of the proposed PEH could be effectively tuned by the pre-deformation of the spring. The proposed PEH could harvest vibration energy in a wide frequency range of 0–91 Hz at the excitation level of 0.5 g.