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Research Papers

# Probabilistic Solution of a Duffing-Type Energy Harvester System Under Gaussian White Noise

[+] Author and Article Information
H. T. Zhu

Associate Professor Mem. ASME State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 300072, China e-mail: htzhu@tju.edu.cn

Manuscript received July 31, 2014; final manuscript received November 28, 2014; published online February 27, 2015. Assoc. Editor: Bilal M. Ayyub.

ASME J. Risk Uncertainty Part B 1(1), 011005 (Feb 27, 2015) (8 pages) Paper No: RISK-14-1035; doi: 10.1115/1.4029143 History: Received July 31, 2014; Accepted December 09, 2014; Online February 27, 2015

## Abstract

This paper proposes a solution procedure to formulate an approximate joint probability density function (PDF) of a Duffing-type energy harvester system under Gaussian white noise. The joint PDF solution of displacement, velocity, and an electrical variable is governed by the Fokker-Planck (FP) equation. First, the FP equation is reduced to a lower-dimensional FP equation only about displacement and velocity by a state-space-split (SSS) method. The stationary joint PDF of displacement and velocity can be solved exactly. Then, the joint PDF of displacement, velocity, and the electrical variable can be approximated by the product of the obtained exact PDF and the conditional Gaussian PDF of the electrical variable. A parametric study is further conducted to show the effectiveness of the proposed solution procedure. The study considers weak nonlinearity, strong nonlinearity, high excitation level, and a bistable oscillator. Comparison with the simulated results shows that the proposed solution procedure is effective in obtaining the joint PDF of the energy harvester system in the examined examples.

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## Figures

Fig. 1

Comparison of PDFs in Case 1 (weak nonlinearity): (a) PDFs of displacement; (b) logarithmic PDFs of displacement; (c) PDFs of velocity; (d) logarithmic PDFs of velocity; (e) PDFs of an electrical variable; and (f) logarithmic PDFs of an electrical variable

Fig. 2

Comparison of PDFs in Case 2 (strong nonlinearity): (a) PDFs of displacement; (b) logarithmic PDFs of displacement; (c) PDFs of velocity; (d) logarithmic PDFs of velocity; (e) PDFs of an electrical variable; and (f) logarithmic PDFs of an electrical variable

Fig. 3

Comparison of PDFs in Case 3 (high excitation level): (a) PDFs of displacement; (b) logarithmic PDFs of displacement; (c) PDFs of velocity; (d) logarithmic PDFs of velocity; (e) PDFs of an electrical variable; and (f) logarithmic PDFs of an electrical variable

Fig. 4

Comparison of PDFs in Case 4 (a bistable oscillator): (a) PDFs of displacement; (b) logarithmic PDFs of displacement; (c) PDFs of velocity; (d) logarithmic PDFs of velocity; (e) PDFs of an electrical variable; and (f) logarithmic PDFs of an electrical variable

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