Applied Mathematics and Mechanics

, Volume 40, Issue 4, pp 579–590 | Cite as

Path integral solution of vibratory energy harvesting systems

  • Wenan JiangEmail author
  • Peng Sun
  • Gangling Zhao
  • Liqun Chen


A transition Fokker-Planck-Kolmogorov (FPK) equation describes the procedure of the probability density evolution whereby the dynamic response and reliability evaluation of mechanical systems could be carried out. The transition FPK equation of vibratory energy harvesting systems is a four-dimensional nonlinear partial differential equation. Therefore, it is often very challenging to obtain an exact probability density. This paper aims to investigate the stochastic response of vibration energy harvesters (VEHs) under the Gaussian white noise excitation. The numerical path integration method is applied to different types of nonlinear VEHs. The probability density function (PDF) from the transition FPK equation of energy harvesting systems is calculated using the path integration method. The path integration process is introduced by using the Gauss-Legendre integration scheme, and the short-time transition PDF is formulated with the short-time Gaussian approximation. The stationary probability densities of the transition FPK equation for vibratory energy harvesters are determined. The procedure is applied to three different types of nonlinear VEHs under Gaussian white excitations. The approximately numerical outcomes are qualitatively and quantitatively supported by the Monte Carlo simulation (MCS).

Key words

nonlinear energy harvester path integration probability density function (PDF) 

Chinese Library Classification

O324 O316 

2010 Mathematics Subject Classification



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Copyright information

© Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Wenan Jiang
    • 1
    Email author
  • Peng Sun
    • 1
  • Gangling Zhao
    • 2
  • Liqun Chen
    • 3
  1. 1.Department of Engineering MechanicsJiangsu University of Science and TechnologyZhenjiang, Jiangsu ProvinceChina
  2. 2.School of Electronic and Electrical EngineeringShangqiu Normal UniversityShangqiu, Henan ProvinceChina
  3. 3.Department of MechanicsShanghai UniversityShanghaiChina

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