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Stationary response probability density of nonlinear random vibrating systems: a data-driven method

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Abstract

A data-driven method is established to derive the (approximately) analytical expression of the stationary response probability density of nonlinear random vibrating system, which explicitly includes system features and intensity of excitation. The stationary response probability density is first assumed as an exponential form by using the principle of maximum entropy. Through the rule of dimensional consistency, the power of exponential function is expressed as a linear combination of a set of nondimensional parameter clusters which are constituted by system features, intensity of excitation, and state variables. By comparing the power of exponential function with the approximate logarithm probability density evaluated from simulated data statistically, the determination of unknown coefficients comes down to the solution of (overdetermined) simultaneous linear algebraic equations. The data-driven method rediscovers the exact stationary response probability density of random-excited Duffing oscillator and derives an approximately analytical expression of stationary response probability density of van der Pol system from the simulated data of six cases with different values of system features and intensity of excitation. This data-driven method is a unique method which can explicitly include the information of system and excitation in the analytical expression of stationary response probability density. It avoids the solution of simultaneous nonlinear algebraic equations encountered in the maximum entropy method and closure methods and, in the meanwhile, avoids the sophisticated selection of weighting functions in closure methods.

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Funding

This study was supported by the National Natural Science Foundation of China under Grant Nos. 11872328, 11532011, 11972317, and 11621062 and the Fundamental Research Funds for the Central Universities under Grant No. 2018FZA4025.

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Authors and Affiliations

  1. College of Mechanical Engineering, Hangzhou Dianzi University, Hangzhou, 310018, China

    Yanping Tian

  2. Department of Engineering Mechanics, Zhejiang University, Hangzhou, 310027, China

    Yong Wang & Zhilong Huang

  3. School for Engineering of Matter, Transport and Energy, Arizona State University, Tempe, AZ, 85287, USA

    Hanqing Jiang

  4. Department of Ocean and Mechanical Engineering, Florida Atlantic University, Boca Raton, FL, 33431, USA

    Isaac Elishakoff & Guoqiang Cai

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  1. Yanping Tian
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  2. Yong Wang
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  4. Zhilong Huang
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  5. Isaac Elishakoff
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Correspondence to Yong Wang.

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Tian, Y., Wang, Y., Jiang, H. et al. Stationary response probability density of nonlinear random vibrating systems: a data-driven method. Nonlinear Dyn 100, 2337–2352 (2020). https://doi.org/10.1007/s11071-020-05632-4

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  • DOI: https://doi.org/10.1007/s11071-020-05632-4

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