Probabilistic solution of non-linear vibration energy harvesters driven by Poisson impulses

This paper develops a new solution procedure for obtaining the joint probability density function (PDF) of non-linear energy harvesters under Poisson impulses using the state-space-split method and the exponential-polynomial closure method. In the beginning, a generic electromechanical energy harvester is introduced and its governing equations are transformed into non-dimensional ones. According to the non-dimensional governing equations, a Duffing-type oscillator with piezoelectric conversion mechanism is further considered in this study. The proposed solution procedure includes three steps. First the joint PDF of this system is governed by the generalized Fokker-Plank-Kolmogorov (FPK) equation. The state-space-split method is used to reduce this generalized FPK equation to a low-dimensional one only about displacement and velocity. After that, the exponential-polynomial closure method is further adopted to solve the reduced FPK equation. Finally, the joint PDF of displacement, velocity and voltage can be approximated by the product of the obtained PDF and the conditional Gaussian PDF of voltage. Four cases are investigated considering the effects of non-linearity coefficient, impulse arrival rate and a bistable Duffing oscillator. The PDFs of these state variables and the harvested power are compared with the simulation results, respectively. The good agreement between the obtained PDFs and the simulated results is achieved. Compared with the results of the equivalent linearization method, the strong non-linearity in displacement and lower impulse rate lead the tail PDF of the harvested power to exhibiting hardening and softening behaviors, respectively. For the bistable Duffing oscillator, the PDF of the harvested power differs significantly from the result of the equivalent linearization method in the tail region.