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A memory-free formulation for determining the non-stationary response of fractional nonlinear oscillators subjected to combined deterministic and stochastic excitations

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Abstract

A memory-free formulation is proposed for determining the non-stationary stochastic response of fractional single-degree-of-freedom nonlinear/hysteretic dynamic systems subjected to combined periodic and non-stationary stochastic excitations. This is achieved by decomposing the system response in a periodic and a zero-mean stochastic component, while utilizing the memory-free formulation to treat the fractional derivative terms. Specifically, first, the response decomposition leads to a system of coupled differential sub-equations of fractional order governing the deterministic and the stochastic response components. Then, invoking the memory-free formulation, the coupled system of equations is transformed into a system of deterministic and stochastic differential equations with integer-order derivatives. Next, a statistical linearization method-based framework is proposed for treating the stochastic sub-equation. This leads to the determination of the equivalent linear stochastic dynamic system, as well as of the related Lyapunov differential equation. Finally, the Lyapunov differential equation and the deterministic sub-equation with integer-order derivative are solved simultaneously using standard numerical algorithms. The applicability and accuracy of the proposed semi-analytical method is demonstrated by pertinent numerical examples.

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All data, models or code that supports the findings of this study is available from the corresponding author upon reasonable request.

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Acknowledgements

The corresponding author would like to express his gratitude to professor Fan Kong at Hefei University of Technology and doctor Vasileios C. Fragkoulis at the University of Liverpool for their insightful advice. This work was supported by the National Natural Science Foundation of China (Grant no. 52078399).

Funding

This work was supported by the National Natural Science Foundation of China (Grant no. 52078399).

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  1. College of Civil Engineering, Tongji University, 1239 Siping Road, Shanghai, 200092, China

    Renjie Han

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Han, R. A memory-free formulation for determining the non-stationary response of fractional nonlinear oscillators subjected to combined deterministic and stochastic excitations. Nonlinear Dyn 111, 22363–22379 (2023). https://doi.org/10.1007/s11071-023-08984-9

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