Efficient survival probability determination of nonlinear multi-degree-of-freedom oscillators with fractional derivatives and subject to non-stationary excitation

Determining the survival probability of nonlinear dynamical systems subject to random excitation is a persistent challenge in engineering dynamics, and addressing this challenge requires efficient and accurate mathematical and numerical methodologies. To this end, we propose a semi-analytical technique for estimating the survival probability of a nonlinear/hysteretic multi-degree-of-freedom (MDOF) oscillator endowed with fractional derivatives, subject to non-stationary excitation. In this technique, $n$ single-degree-of-freedom (SDOF) oscillators with time-dependent effective damping and stiffness terms are determined based on the variances of the system response displacement and velocity, approximated using statistical linearization. These oscillators govern the dynamics of the response of each degree of freedom (DOF) of an $n$-DOF nonlinear/hysteretic oscillator. Novel expressions for the effective properties of the SDOF oscillators are proposed, incorporating an appropriate approximation for Caputo’s fractional derivative using hypergeometric functions. Additionally, approximated closed-form expressions are derived for the transition probability density function of the response amplitude process, enabling the estimation of conditional probabilities along the time domain at minimal computational cost, which is necessary for approximating the survival probability. To assess the accuracy and computational performance of the proposed methodology, we consider numerical examples involving a hardening Duffing, a softening stiffness, and a Bouc–Wen MDOF oscillator with fractional derivatives and subject to a non-stationary excitation with a non-separable evolutionary power spectrum. Comparisons with Monte Carlo simulation data are included to evaluate the accuracy and computational performance of the proposed approach.