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A Solution Procedure for a Vibro-Impact Problem under Fully Correlated Gaussian White Noises

H.T. Zhu 1
State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 300072, P.R. China. E-mail:

Computer Modeling in Engineering & Sciences 2014, 97(3), 281-298.


This study is concerned with a solution procedure to obtain the probability density function (PDF) of a vibro-impact Duffing oscillator under fully correlated external and parametric Gaussian white noises. The proposed solution procedure consists of three steps. In the first step, the Zhuravlev non-smooth coordinate transformation is adopted to introduce an additional impulsive damping term, in which the original vibro-impact oscillator is converted into a new oscillator without any barrier. After that, the PDF of the new oscillator is obtained by solving the Fokker-Planck equation with the exponential-polynomial closure method. Last, the PDF of the original oscillator is formulated in terms of the methodology on seeking the PDF of a function of random variables. A numerical analysis on four different cases is conducted to examine the effectiveness of the proposed solution procedure. Comparison with the simulated result shows that the proposed solution procedure can provide a satisfactory PDF solution for the four cases. The tail region of the PDF solution is also approximated well. The numerical analysis also shows that the change of parametric excitation has a significant effect on the PDF distributions of displacement and velocity.


vibro-impact, random vibration, probability density function, correlated white noises, Duffing oscillator.

Cite This Article

Zhu, H. (2014). A Solution Procedure for a Vibro-Impact Problem under Fully Correlated Gaussian White Noises. CMES-Computer Modeling in Engineering & Sciences, 97(3), 281–298.

This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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