Simultaneous determination of stochastic dynamic responses and reliabilities for bilateral vibro-impact systems under colored noise excitation

Simultaneous determination of stochastic dynamic responses and reliabilities for bilateral vibro-impact systems holds significant theoretical and practical importance. Non-smooth transformation is usually used to handle the discontinuities in vibro-impact systems, which also eliminates the velocity jump feature that is present in non-smooth systems. To address the boundary discontinuity challenge of impact behavior and stochastic dynamics analysis of bilateral vibro-impact systems, this study proposes the boundary correction based direct probability integral method (DPIM) without using non-smooth transformation. By devising a reflection-based boundary correction approach, the missing probability mass in the truncated region is compensated to reconstruct complete probability density function (PDF). The PDFs of stochastic dynamic responses and first-passage reliabilities of vibro-impact systems with two symmetric stoppers under colored noise are calculated by utilizing the proposed method. Comparisons with Monte Carlo simulation (MCS), quasi-MCS and the results in references demonstrate high efficiency and accuracy of DPIM with boundary correction. Determining the asymptotic stability with probability one of system by calculating the largest Lyapunov exponent. Finally, the influences of the stopper position, restitution coefficient, colored noise intensity and damping coefficient on stochastic dynamic behavior of bilateral vibro-impact systems are scrutinized. Parametric studies reveal that the changes of damping coefficient induce the occurrence of P-bifurcation, while the variations in restitution coefficient and colored noise intensity cause the occurrence of D-bifurcation. In addition, colored noise intensity significantly affects the reliability and stochastic stability of system.