Dynamics of a coupled quad-stable hybrid vibration energy harvesting system under dual-frequency excitation

Motivated by enhancing energy harvesting under low-frequency excitation, a coupled quad-stable hybrid vibration energy harvesting system driven by dual-frequency force is considered in this paper. By combining the harmonic balance method with the method of variable separation, analytical expressions for the steady-state response and harvested power are derived. The response amplitude is then defined and formulated to characterize the vibration resonance (VR). Results reveal that adopting a piezoelectric–electromagnetic hybrid design, together with appropriate choices of the time constant ratio and electromechanical coupling coefficient, can significantly improve harvesting performance. Multi-solution behavior is observed in the quad-stable system, with up to seven solutions (including four stable and three unstable states). This feature enables the system to maintain a relatively high output level even under low input amplitude or frequency. Variation of the high-frequency excitation and stiffness coefficient can adjust the equivalent linear stiffness, thus changing the natural frequency and inducing a cascade of vibration resonance. Appropriate combinations of high-frequency excitations ( F , Ω) can induce VR, during which the energy harvesting performance is significantly enhanced compared with the non-resonant case. By adjusting the position of the magnets to modify system parameters, the optimal VR can be induced to improve energy harvesting. Numerical simulations are conducted to validate the effectiveness of the theoretical results.