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Approximate stochastic response of hysteretic system with fractional element and subjected to combined stochastic and periodic excitation

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Abstract

A method based on statistical linearization is proposed, for determining the response of single-degree-of-freedom hysteretic systems endowed with fractional derivative element and subjected to combined periodic and white/colored excitation. The method is developed by decomposing the system response into a combination of a periodic and of a zero-mean stochastic components. In this regard, first, the equation of motion is cast into two coupled fractional-order nonlinear differential equations with unknown deterministic and stochastic response components. Next, the harmonic balance method for the fractional-order deterministic equation and the statistical linearization for the stochastic equation are used, to obtain the Fourier coefficients of the deterministic response component and the variance of the stochastic response component, respectively. This yields two sets of coupled nonlinear algebraic equations which can be solved by appropriate standard numerical method. Pertinent numerical examples, including both softening and hardening Bouc–Wen hysteretic system endowed with different fractional-orders, are used to demonstrate the applicability and accuracy of the proposed method.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 52078399) and by the Fundamental Research Funds for the Central Universities(WUT:212274016). The first author would like to thank the Chinese Scholarship Council (CSC) for financial support (File No. 201706955030) during his visit to Rice University as a visiting scholar. The first author would like to greatly thank Professor Pol D. Spanos at Rice University for the discussing the statistical linearization method for fractional-order systems.

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All sources of funding for the research reported have been declared.

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Authors and Affiliations

  1. School of Civil Engineering and Architecture, Wuhan University of Technology, 122 Luoshi Road, Wuhan, 430070, Hubei, China

    Fan Kong & Renjie Han

  2. School of Safety Science and Emergency Management, Wuhan University of Technology, 122 Luoshi Road, Wuhan, 430070, China

    Yuanjin Zhang

  3. School of Civil Engineering, Hefei University of Technology, 193 Tunxi Road, Hefei, 230009, China

    Fan Kong

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  1. Fan Kong
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Contributions

KF conceived the study, derived the theoretical formulation and finalized the manuscript. HRJ composed the relevant codes, drafted the manuscript. ZYJ participated in coding, and drafting the manuscript.

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Correspondence to Yuanjin Zhang.

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All codes relevant to the present work are available upon request.

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Kong, F., Han, R. & Zhang, Y. Approximate stochastic response of hysteretic system with fractional element and subjected to combined stochastic and periodic excitation. Nonlinear Dyn 107, 375–390 (2022). https://doi.org/10.1007/s11071-021-07014-w

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  • DOI: https://doi.org/10.1007/s11071-021-07014-w

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