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Path integral solution of vibratory energy harvesting systems

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Abstract

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).

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Authors and Affiliations

  1. Department of Engineering Mechanics, Jiangsu University of Science and Technology, Zhenjiang, Jiangsu Province, 212003, China

    Wenan Jiang & Peng Sun

  2. School of Electronic and Electrical Engineering, Shangqiu Normal University, Shangqiu, Henan Province, 476000, China

    Gangling Zhao

  3. Department of Mechanics, Shanghai University, Shanghai, 200072, China

    Liqun Chen

Authors
  1. Wenan Jiang
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  2. Peng Sun
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  3. Gangling Zhao
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  4. Liqun Chen
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Corresponding author

Correspondence to Wenan Jiang.

Additional information

Citation: JIANG, W. A., SUN, P., ZHAO, G. L., and CHEN, L. Q. Path integral solution of vibratory energy harvesting systems. Applied Mathematics and Mechanics (English Edition), 40(4), 579–590 (2019) https://doi.org/10.1007/s10483-019-2467-8

Project supported by the National Natural Science Foundation of China (Nos. 11702119 and 51779111) and the Natural Science Foundation of Jiangsu Province of China (Nos.BK20170565 and BK20170581)

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Jiang, W., Sun, P., Zhao, G. et al. Path integral solution of vibratory energy harvesting systems. Appl. Math. Mech.-Engl. Ed. 40, 579–590 (2019). https://doi.org/10.1007/s10483-019-2467-8

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  • DOI: https://doi.org/10.1007/s10483-019-2467-8

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