Nonlinear 1:2 internal resonance response of L-shaped piezoelectric energy harvester under the influence of electrical damping

The internal resonance mechanism is widely used to improve the output performance of vibration energy harvester owing to its modal interaction between modes. The electromechanical-coupled theoretical model has been verified by experiments in our previous work (Nie et al., 2019). The approximate analytical solutions of the electromechanical-coupled governing equations of the harvester are derived using the method of multiple scales, and verified by the numerical method. The equilibrium stability of the output responses are determined by the eigenvalues of Jacobian matrix of the modulation equations. The effects of load resistance on electrical damping, natural frequency and primary resonance responses of the harvester are investigated. The results show that the harvested power is associated with the electrical damping caused by the load resistance. The threshold of the excitation amplitude that triggers the internal resonance from the perspective of load resistance are investigated. Under the influence of load resistance and excitation amplitude, the saturation phenomenon, jumping phenomenon, softening phenomenon, Hopf bifurcation and saddle–node bifurcation are explored. The results indicate that the internal resonance responses of the harvester vary with the excitation amplitude and load resistance, and jump between the upper and lower branches depending on initial displacement.