Stochastic averaging of energy harvesting systems |
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Affiliation: | 1. School of Mathematics, Shanxi University, Taiyuan, 030006, PR China;2. Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an 710072, PR China;3. School of Applied Science, Taiyuan University of Science and Technology, Taiyuan, 030024, PR China;4. School of Mathematics and Information Science, Beifang University of Nationalities, Yinchuan 750021, PR China;5. Potsdam Institute for Climate Impact Research, Potsdam 14412, Germany |
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Abstract: | A stochastic averaging method is proposed for nonlinear energy harvesters subjected to external white Gaussian noise and parametric excitations. The Fokker–Planck–Kolmogorov equation of the coupled electromechanical system of energy harvesting is a three variables nonlinear parabolic partial differential equation whose exact stationary solutions are generally hard to find. In order to overcome difficulties in solving higher dimensional nonlinear partial differential equations, a transformation scheme is applied to decouple the electromechanical equations. The averaged Itô equations are derived via the standard stochastic averaging method, then the FPK equations of the decoupled system are obtained. The exact stationary solution of the averaged FPK equation is used to determine the probability densities of the displacement, the velocity, the amplitude, the joint probability densities of the displacement and velocity, and the power of the stationary response. The effects of the system parameters on the output power are examined. The approximate analytical outcomes are qualitatively and quantitatively supported by the Monte Carlo simulations. |
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Keywords: | Energy harvesting Stochastic averaging method Gaussian white noise Parametric excitation Monte Carlo simulation |
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