Revised method of multiple scales for 1:2 internal resonance piezoelectric vibration energy harvester considering the coupled frequency

The piezoelectric vibration energy harvester based on the 1:2 internal resonance possesses the advantages of wide-band and high-efficiency energy harvesting. The method of multiple scales (MMS) has been applied to derive the approximate analytical solution of the output response. However, the coupled frequency caused by internal resonance has strong effect on the output voltage response, which cannot be determined by traditional MMS. To overcome this shortage, the method of equivalent forced load is proposed to decouple the voltage frequency, and then obtain the revised approximate analytical solution (MMS-New) of the output responses. The responses of the electromechanical-coupled governing equations of the system are verified by experiments. Compared with the approximate solutions of the traditional MMS, the accuracy of the revised approximate solutions are significantly improved. In the internal resonance region, the maximum accuracy of output voltage response can be improved by 40.64%, and the maximum accuracy can be increased to 10 times outside the internal resonance region. In the whole excitation frequency range, the amplitude–frequency response curve exhibits nonlinear hardening and softening when the forward and reverse sweep frequency analysis are respectively performed. The Jacobi matrix and Lyapunov stability theory of the modulation equations are derived to determine the stability of the approximate solutions. The dynamic behavior in unstable region are investigated using means of time history curve, spectrum, phase orbit and Poincare section, the quasi periodic and chaotic motions are observed.