Transient response of the time-delay system excited by Gaussian noise based on complex fractional moments

Niu, Lizhi; Xu, Weiya; Guo, Qin

doi:10.1063/5.0033593

Cited by 10 publications

(2 citation statements)

References 24 publications

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“…Herein, it is assumed that for any time t, the amplitude PDF p A (a, t) does not diverge as a → 0, and p A (a, t) and ∂p A (a, t)/∂a converge to zero faster than power functions of a as a → ∞. These assumptions have been used successfully in CFM-based analyses for various integer-order nonlinear systems [50][51][52][53]. Furthermore, it was recently showed that the stationary amplitude PDF for a fractional linear oscillator under Gaussian white noise (i.e., the special case of Eq.…”

Section: Stochastic Averagingmentioning

confidence: 99%

“…The CFM method combined with stochastic averaging yielded transient and stationary response PDFs with high accuracy, and its computational time was significantly reduced compared to the corresponding MCS. In the subsequent studies, this method has been extended to a wide class of integer-order nonlinear oscillators, including systems with nonlinearities in both stiffness and damping [51], nonlinear vibro-impact systems [52] and nonlinear systems with time delay [53].…”

Section: Introductionmentioning

confidence: 99%

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Transient response analysis of nonlinear oscillators with fractional derivative elements under Gaussian white noise using complex fractional moments

Tsuchida

Itoh

Eguchi

2023

Preprint

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An approximate analytical approach using complex fractional moments (CFMs) is developed for determining the transient response　probability density of nonlinear oscillators with fractional derivative elements under Gaussian white noise. The CFM is defined as the Mellin transform of a probability density function. The system response is represented in the form of a pseudo-harmonic oscillation with amplitude and phase slowly varying with time. Equivalent linearization is first implemented to obtain an equivalent natural frequency and an equivalent damping coefficient of the oscillator. In this regard, simple formulas for determining them are proposed, in which both the nonlinear elements and the elastic and viscous contributions of the fractional elements are taken into account. Next, applying stochastic averaging, the response amplitude is described by a one-dimensional stochastic differential equation, and the corresponding Fokker-Planck equation is derived. The Mellin transform then converts the Fokker-Planck equation into coupled linear ordinary differential equations governing the evolution of amplitude CFMs, which are solved with a constraint corresponding to the normalization condition for a probability density. Finally, the inverse Mellin transform of the CFMs yields the amplitude probability density. The joint probability density of displacement and velocity is also obtained from the amplitude probability density in conjunction with the Jacobian of the pseudo-harmonic variable transformation for the response. Three linear and nonlinear oscillators with fractional derivatives are considered in numerical examples. For all cases, the analytical results obtained by the proposed method are in good agreement with the results of the pertinent Monte Carlo simulation.

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Section: Stochastic Averagingmentioning

confidence: 99%