Approximate Fokker-Planck-Kolmogorov equation analysis for asymmetric multistable energy harvesters excited by white noise

Due to the broadband response characteristics at low levels of excitations, nonlinear multistable systems have garnered a great deal of attention in the area of energy harvesting. Moreover, various performance enhancement strategies of multistable harvesters have been proposed and discussed extensively for systems with perfectly symmetric potentials. However, it is very difficult or even impossible in practice to modulate a nonlinear system with completely symmetric potentials. Therefore, this paper investigates the stochastic response characteristics of asymmetric potential multistable harvesters excited by Gaussian white noise. Approximate Fokker-Planck-Kolmogorov equation for multistable harvester is provided, and solved by the method of detailed balance. Numerical simulations are carried out to characterize the probability distribution and power output of the response of the harvesters. Results indicate that the theoretical method could well predict the response probability distribution of the systems under white noise excitations. To be more specific, the probability distribution of velocity response at a given excitation level is not affected by the asymmetry of potentials, while the probability density function of displacement is very dependent on the shape of potentials. In terms of the power output, the numerical results agree well with the theoretical results. However, the error between numerical and theoretical results increases with the increase in the number of the steady states and the degree of asymmetry of the harvesters.