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

This study is concerned with a solution procedure to obtain the probability density function (PDF) of a vibro-impact Duffing oscillator under fully con-elated 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 bather. 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.