Probabilistic solution of the vibratory energy harvester excited by Gaussian white noise

  • Wen-An Jiang
  • Peng Sun
  • Zhao-Wang Xia


This paper aims to investigate the statistical characteristics of strongly nonlinear vibratory energy harvesters under Gaussian white noise excitation. The high-dimensional Fokker–Planck–Kolmogorov (FPK) equation of the coupled electromechanical system is reduced to a low-dimensional equation by using the state-space-split method. The conditional moment given by the equivalent linearization method is employed to decouple the FPK equations of coupled system, and then obtained an equivalent nonlinear uncoupled subsystem. The exact stationary solution of the reduced FPK equation of the subsystem is established. The mean output power is derived by the second order conditional moment from the associated approximate probability density function of mechanical subsystem. The procedure is applied to mono- and bi-stable energy harvesters. Effectiveness of the probability density function of the proposed approach is examined via comparison with equivalent linearization method and Monte Carlo simulation. The effects of the system parameters on the mean-square displacement and the mean output power are discussed. The approximate analytical outcomes are qualitatively and quantitatively supported by the numerical simulations.


Nonlinear State-space-split method Conditional moment Energy harvesting Fokker–Planck–Kolmogorov equation 



This work was supported by the National Natural Science of China (Nos. 11702119 and 11502071), the Natural Science Foundation of Jiangsu Province (No. BK20170565).


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Engineering MechanicsJiangsu University of Science and TechnologyZhenjiangChina
  2. 2.College of Energy and Power EngineeringJiangsu University of Science and TechnologyZhenjiangChina

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