This paper presents a technique based on Wiener path integrals (WPI) for computing the survival probability of nonlinear oscillators under combined nonstationary stochastic and periodic excitation. Specifically, the equation of motion is equivalently decomposed into a correlated combination of a nonlinear stochastic differential equation (SDE) and a nonlinear deterministic differential equation. The nonstationary response probability density function (PDF) of the stochastic component is obtained using the Wiener path integral concept in conjunction with stochastic averaging/linearization. The approach involves utilizing stochastic averaging/linearization techniques to transform the nonlinear SDE into a linear one with varying equivalent stiffness and damping. Afterwards, using the WPI method and the concept of the most probable path, a closed-form approximate analytical expression of the joint transition probability density function (PDF) is obtained for small intervals corresponding to the equivalent linear SDE. Additionally, the survival probability of the original nonlinear oscillator subject to combined excitation with fixed barriers can be approximated equivalently by using the survival probability with time-varying barriers of the equivalent linear system with time-varying stiffness and damping subject only to the nonstationary stochastic excitation. Finally, by using the analytical expression of the transition PDF of the equivalent linear system and a discrete version of the Chapman–Kolmogorov (C–K) equation, it is possible to determine the response and survival probability of the original oscillator under combined excitation step by step. Examples involving a Duffing oscillator and a vibro-impact oscillator are analyzed to showcase the precision and efficacy of the suggested technique in contrast to relevant Monte Carlo simulations.

本文介绍了一种基于维纳路径积分 (WPI) 的技术，用于计算非平稳随机和周期性激励组合下非线性振荡器的生存概率。具体来说，运动方程等​​效地分解为非线性随机微分方程（SDE）和非线性确定性微分方程的相关组合。随机分量的非平稳响应概率密度函数 (PDF) 是使用维纳路径积分概念结合随机平均/线性化获得的。该方法涉及利用随机平均/线性化技术将非线性 SDE 转换为具有不同等效刚度和阻尼的线性 SDE。之后，使用WPI方法和最可能路径的概念，对于对应于等效线性 SDE 的小区间，获得了联合转移概率密度函数 (PDF) 的封闭形式近似解析表达式。此外，原始非线性振荡器在固定势垒联合激励下的生存概率可以等效地近似为具有时变刚度和阻尼的等效线性系统的时变势垒生存概率仅受非平稳随机激励的影响. 最后，通过使用等效线性系统的过渡 PDF 的解析表达式和 Chapman-Kolmogorov (C-K) 方程的离散版本，可以确定原始振荡器在组合激励步长下的响应和生存概率循序渐进。