Probabilistic failure analysis of stochastically excited nonlinear structural systems with fractional derivative elements

In this paper, the application of the relaxed power spectral density (PSD) framework is developed for quantifying uncertainties in dynamical systems with fractional derivative elements. The proposed methodology offers a systematic treatment of uncertainties in spectrum-based stochastic simulation and their propagation for response determination of systems with memory-dependent or viscoelastic behavior. A key advantage of the framework lies in its ability to model the variability of estimated PSD functions using a non-parametric probabilistic representation, while explicitly accounting for frequency-domain correlations that are typically overlooked in conventional PSD-based estimates. First, a “relaxed” version of the power spectral density is derived by extracting statistical moments across ensembles of discretized PSD estimates. Next, frequency-dependent truncated normal distributions are employed to capture PSD uncertainties. Statistically compatible realizations are generated using three distinct sampling strategies: a single-variable inverse cumulative distribution function-based method for efficient sampling of marginal probability density functions, a multivariate Gaussian approach that incorporates cross-frequency covariance to capture global correlation structure, and an Ornstein–Uhlenbeck Markov process model, which reconstructs smoothly correlated PSD trajectories. The efficiency of the proposed approach is demonstrated by considering three representative case studies. These are a Duffing nonlinear oscillator with fractional damping, a tuned mass-damper-inerter system with nonlinear coupling characteristics, and a nonlinear vibration energy harvester under stochastic excitation. It is shown that by accounting for a comprehensive probabilistic treatment of the PSD, the proposed framework yields enhanced reliability analysis results of dynamical systems under spectral uncertainty.