Response statistics of nonlinear systems with fractional derivative elements subject to nonstationary excitations

In this paper nonlinear systems/structures endowed with fractional derivative elements are analyzed. The excitation considered can be either stationary or nonstationary, and either white or non-white. As an analysis tool, the stochastic averaging technique involving a widely-used approximation of fr...

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Veröffentlicht in:Nonlinear dynamics Jg. 113; H. 25; S. 34371 - 34388
Hauptverfasser: Luo, Yi, Spanos, Pol D.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Dordrecht Springer Nature B.V 01.12.2025
Schlagworte:
Approximation
Derivatives
Differential equations
Excitation
Galerkin method
Mathematical analysis
Monte Carlo simulation
Nonlinear systems
Power spectral density
Random vibration
ISSN:0924-090X, 1573-269X
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Zusammenfassung:In this paper nonlinear systems/structures endowed with fractional derivative elements are analyzed. The excitation considered can be either stationary or nonstationary, and either white or non-white. As an analysis tool, the stochastic averaging technique involving a widely-used approximation of fractional derivatives is employed. This leads to a one-dimensional integer-order stochastic differential equation governing the evolution of the response amplitude. Further, the derived averaged Fokker–Planck-Kolmogorov equation is tackled by two techniques: the Galerkin technique and numerical path integration. Notably, significant modifications are needed to extend the Galerkin scheme to systems under nonstationary and nonwhite excitation with inseparable power spectral density. Numerical examples including a Van der Pol system, and a Duffing system are considered. The reliability and accuracy of the proposed scheme are demonstrated by juxtaposing the results derived by the proposed approaches with pertinent Monte Carlo simulation results.
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ISSN:0924-090X
1573-269X
DOI:10.1007/s11071-025-11648-5