An analytical method for determining stochastic response and survival probability of nonlinear oscillators endowed with fractional element and subjected to evolutionary excitation is developed in this paper. This is achieved by the variational formulation of the recently developed analytical Wiener path integral (WPI) technique. Specifically, the stochastic average/linearization treatment of the fractional-order non-linear equation of motion yields an equivalent linear time-varying substitute with integer-order derivative. Next, relying on the path integral technique, a closed-form analytical approximation of the response joint transition probability density function (PDF) for small intervals is obtained. Further, a combination of the derived joint transition PDF and the discrete version of Chapman–Kolmogorov (C–K) equation, leads to analytical solution of the non-stationary response and survival probability of non-linear oscillator under the evolutionary excitation. Finally, pertinent numerical examples, including a hardening Duffing and a bi-linear hysteretic oscillator, are considered to demonstrate the reliability of the proposed technique.

本文开发了一种确定具有分数元的非线性振荡器的随机响应和生存概率的分析方法，该方法受到演化激励。这是通过最近开发的分析维纳路径积分 (WPI) 技术的变分公式实现的。具体而言，分数阶非线性运动方程的随机平均/线性化处理产生具有整数阶导数的等效线性时变替代。接下来，依靠路径积分技术，获得了小区间响应联合转移概率密度函数 (PDF) 的封闭形式解析近似。此外，导出的联合转换 PDF 和 Chapman-Kolmogorov (C-K) 方程的离散版本的组合，导致进化激励下非线性振荡器的非平稳响应和生存概率的解析解。最后，考虑了相关的数值示例，包括硬化 Duffing 和双线性滞后振荡器，以证明所提出技术的可靠性。