Fokker–Planck equation analysis of randomly excited nonlinear energy harvester

•A coupled nonlinear electromechanical system for energy harvesting is considered.•The input excitation is assumed to be stationary Gaussian white noise.•Fokker–Planck equation describing the evolution of the PDF is formulated explicitly.•The finite element method is used to solve the 3-dimensional FP equation.•Results are validated using direct Monte-Carlo simulations.

The probability structure of the response and energy harvested from a nonlinear oscillator subjected to white noise excitation is investigated by solution of the corresponding Fokker–Planck (FP) equation. The nonlinear oscillator is the classical double well potential Duffing oscillator corresponding to the first mode vibration of a cantilever beam suspended between permanent magnets and with bonded piezoelectric patches for purposes of energy harvesting. The FP equation of the coupled electromechanical system of equations is derived. The finite element method is used to solve the FP equation giving the joint probability density functions of the response as well as the voltage generated from the piezoelectric patches. The FE method is also applied to the nonlinear inductive energy harvester of Daqaq and the results are compared. The mean square response and voltage are obtained for different white noise intensities. The effects of the system parameters on the mean square voltage are studied. It is observed that the energy harvested can be enhanced by suitable choice of the excitation intensity and the parameters. The results of the FP approach agree very well with Monte Carlo Simulation (MCS) results.

Graphical abstractA novel Fokker–Plank equation based approach is proposed for nonlinear energy harvesting under stochastic excitation.Figure optionsDownload full-size imageDownload as PowerPoint slide