Nonlinear analysis of galloping piezoelectric energy harvesters with inductive-resistive circuits for boundaries of analytical solutions

For nonlinear characterization of the galloping piezoelectric energy harvester with the inductive-resistive circuit, the electromechanical coupled distributed parameter model is concisely retrospected. A general electromechanical decoupled model is proposed for the series and parallel circuits with different algebraic expressions of the electrical damping and modified frequency. The electrical damping corresponding to Hopf bifurcation (EDHB) is derived and noticed to be linearly and positively varied with the wind speed. Galloping occurs when the electrical damping is smaller than EDHB. To analyze the effects of the electrical circuits on Hopf bifurcation, the inductance corresponding to Hopf bifurcation (IHB) is proposed as a function of the load resistance and EDHB. The stable and galloping regions of the inductance and modified frequency varied with the wind speed are determined and found to be strongly dependent on the load resistance. Hopf bifurcation for the small load resistance of the series connection is similar as that for the large load resistance of the parallel case, and vice versa. The time history, phase portrait and power spectrum are introduced to show the differences between the results obtained with the small and large initial conditions. It is found that the large initial condition corresponds to the large electrical damping and the small initial condition relates to the small electrical damping. This is expected to be the reason that the large electrical damping is more difficult to be excited. Different from our common concept, the tip displacement with the small initial condition is larger than that with the large initial condition in some situations. With the galloping region determined by the presented nonlinear analyses, the analytical solutions agree well with the numerical results using the small and large initial conditions.