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Stochastic stability of variable-mass Duffing oscillator with mass disturbance modeled as Gaussian white noise

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Abstract

The stochastic asymptotic stability with probability one of the stochastic variable- mass Duffing oscillator under weakly parametric excitation is investigated by using the largest Lyapunov exponent, in which the small disturbance of mass is modeled as Gaussian white noise. First, the original system is replaced by an approximate system without mass disturbance based on an equivalent transformation. Then the averaged Itô equation of the approximate system is derived by applying the stochastic averaging method based on the generalized harmonic function. Finally, the Lyapunov exponent of the linearized averaged Itô equation is derived and a necessary and sufficient condition for the stability of the trivial solution is obtained approximately by letting this Lyapunov exponent be negative. The effects of mass disturbance on system stability are explored adequately. It turns out that mass disturbance deteriorates system stability remarkably. It is remarkable that the influences of linear stiffness on the stability of a variable-mass system are significantly different from that of an associated invariable-mass system.

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Acknowledgements

This study was supported by the National Natural Science Foundation of China (Grant Nos. 11472212, 11532011, 11502199 and 11502160) and the Seed Foundation of Innovation and Creation for Graduate Students in Northwestern Polytechnical University (Grant No. Z2016161).

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Authors and Affiliations

  1. Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, 710072, People’s Republic of China

    Yan Qiao, Wei Xu, Wantao Jia & Weiyan Liu

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  1. Yan Qiao
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  2. Wei Xu
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  3. Wantao Jia
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  4. Weiyan Liu
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Correspondence to Wei Xu.

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Qiao, Y., Xu, W., Jia, W. et al. Stochastic stability of variable-mass Duffing oscillator with mass disturbance modeled as Gaussian white noise. Nonlinear Dyn 89, 607–616 (2017). https://doi.org/10.1007/s11071-017-3474-5

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  • DOI: https://doi.org/10.1007/s11071-017-3474-5

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