TY - JOUR

T1 - Probabilistic failure analysis of stochastically excited nonlinear structural systems with fractional derivative elements

AU - Behrendt, Marco

AU - Fragkoulis, Vasileios C.

AU - Pasparakis, George D.

AU - Beer, Michael

N1 - Publisher Copyright: © 2025 Elsevier Ltd

PY - 2026/2

Y1 - 2026/2

N2 - 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.

AB - 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.

KW - Fractional derivatives

KW - Power spectral density function

KW - Stochastic dynamics

KW - Structural reliability analysis

KW - Uncertainty quantification

UR - http://www.scopus.com/inward/record.url?scp=105018044623&partnerID=8YFLogxK

U2 - 10.1016/j.ress.2025.111647

DO - 10.1016/j.ress.2025.111647

M3 - Article

AN - SCOPUS:105018044623

VL - 266

JO - Reliability Engineering and System Safety

JF - Reliability Engineering and System Safety

SN - 0951-8320

M1 - 111647

ER -