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<p>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.</p>

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

  • 期刊论文

  • 发表日期：

    2025年8月23日

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  • Yi Luo,
  • Pol D. Spanos

摘要

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.