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Transient response analysis of a system with nonlinear stiffness and nonlinear damping excited by Gaussian white noise based on complex fractional moments

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Abstract

An analytical method based on complex fractional moments (CFMs) is presented to obtain the transient response probability density of a dynamic system with nonlinearity in both stiffness and damping excited by Gaussian white noise. The CFM is a new kind of statistical moment developed in recent years and is related to the Mellin transform of a probability density function. In the present method, first, the equivalent linearization technique is applied to determine the equivalent natural frequency of the system, which is a function of the response amplitude. Then, using the equivalent natural frequency and the stochastic averaging procedure, the stochastic differential equation with respect to the response amplitude and the associated Fokker–Planck equation are derived. The Mellin transform of the Fokker–Planck equation yields the governing equations of the amplitude CFMs. The governing equations are given by simultaneous linear ordinary differential equations. Finally, the inverse Mellin transform of the CFMs obtained from the above equations leads to the response probability density function. In numerical examples, three types of nonlinear stochastic systems are considered. It is shown that the results of the transient response probability densities obtained by the present method agree well with the corresponding Monte Carlo simulation results, including their tail region.

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Acknowledgements

This work was supported by JSPS KAKENHI Grant Number JP18K13712. The authors are deeply grateful to Prof. Emeritus Koji Kimura for his comment in improving the manuscript.

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  1. Department of Systems and Control Engineering, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo, 1528552, Japan

    Daizoh Itoh & Takahiro Tsuchida

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  1. Daizoh Itoh
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Correspondence to Takahiro Tsuchida.

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Itoh, D., Tsuchida, T. Transient response analysis of a system with nonlinear stiffness and nonlinear damping excited by Gaussian white noise based on complex fractional moments. Acta Mech 233, 2781–2796 (2022). https://doi.org/10.1007/s00707-022-03264-w

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