Stochastic averaging based on generalized harmonic functions for energy harvesting systems

A stochastic averaging method is proposed for nonlinear vibration energy harvesters subject to Gaussian white noise excitation. The generalized harmonic transformation scheme is applied to decouple the electromechanical equations, and then obtained an equivalent nonlinear system which is uncoupled to an electric circuit. The frequency function is given through the equivalent potential energy which is independent of the total energy. The stochastic averaging method is developed by using the generalized harmonic functions. The averaged Itô equations are derived via the proposed procedure, and the Fokker-Planck-Kolmogorov (FPK) equations of the decoupled system are established. The exact stationary solution of the averaged FPK equation is used to determine the probability densities of the amplitude and the power of the stationary response. The procedure is applied to three different type Duffing vibration energy harvesters under Gaussian white excitations. The effects of the system parameters on the mean-square voltage and the output power are examined. It is demonstrated that quadratic nonlinearity only and quadratic combined with properly cubic nonlinearities can increase the mean-square voltage and the output power, respectively. The approximate analytical outcomes are qualitatively and quantitatively supported by the Monte Carlo simulations.