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Probabilistic response and analysis for a vibro-impact system driven by real noise

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Abstract

In this paper, a new stochastic averaging method is proposed to derive the stationary probabilistic response of vibro-impact system subject to real noise excitation, and the effects of real noise and the restitution coefficient on the response are discussed in detail. According to the size relationship between the system energy and the potential energy of the impact position \(\varDelta \), the motions of the unperturbed vibro-impact system are divided into two types: the free motion with no energy loss and the impact motion with energy loss. Based on those relationships, a new method that combines a transformation and the Fourier expansion scheme by introducing, the mean drift and diffusion coefficients of the averaged Itô stochastic differential equation for the two types of motions are obtained, which are depended on the bandwidth and the noise density of real noise. The probability density function of stationary response of vibro-impact system is derived through solving the associated Fokker–Planck–Kolmogorov equation. Finally, the effects of the restitution coefficient and the noise intensity of real noise in different bandwidths are detailedly analyzed through an example. The main results on probabilistic response analysis of vibro-impact Duffing system are obtained to illustrate the proposed stochastic averaging method, and Monte Carlo simulation method is also conducted to show the effectiveness of the proposed method.

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References

  1. Dimentberg, M.F., Iourtchenko, D.V.: Random vibrations with impacts: a review. Nonlinear Dyn. 36(2–4), 229–254 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ibrahim, R.A.: Vibro-impact dynamics: modeling, mapping and applications. Springer, Berlin (2009)

    Book  MATH  Google Scholar 

  3. Makarenkov, O., Lamb, J.S.W.: Dynamics and bifurcations of nonsmooth systems: a survey. Physica D 241(22), 1826–1844 (2012)

    Article  MathSciNet  Google Scholar 

  4. Ibrahim, R.A.: Recent advances in vibro-impact dynamics and collision of ocean vessels. J. Sound Vib. 333(23), 5900–5916 (2014)

    Article  Google Scholar 

  5. Shaw, S.W., Holmes, P.J.: A periodically forced impact oscillator with large dissipation. ASME J. Appl. Mech. 50(4a), 849–857 (1983)

    Article  MATH  Google Scholar 

  6. Luo, G.W.: Period-doubling bifurcations and routes to chaos of the vibratory systems contacting stops. Phys. Lett. A 323(3–4), 210–217 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Wagg, D.J.: Rising phenomena and the multi-sliding bifurcation in a two-degree of freedom impact oscillator. Chaos Soliton Fract. 22(3), 541–548 (2004)

    Article  MATH  Google Scholar 

  8. Sushko, I., Gardini, L.: Degenerate bifurcations and border collisions in piecewise smooth 1D and 2D maps. Int. J. Bifurc. Chaos 20(7), 2045–2070 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Yue, Y., Miao, P., Xie, J.: Coexistence of strange nonchaotic attractors and a special mixed attractor caused by a new intermittency in a periodically driven vibro-impact system. Nonlinear Dyn. 87(2), 1187–1207 (2017)

    Article  MATH  Google Scholar 

  10. Xu, Y., Guo, R., Jia, W., Li, J.: Stochastic averaging for a class of single degree of freedom systems with combined Gaussian noises. Acta Mech. 225(9), 2611–2620 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Wang, Z.Q., Xu, Y., Yang, H.: L\(\grave{e}\)vy noise induced stochastic resonance in an FHN model. Sci. China Technol. SC 59(3), 371–375 (2016)

    Article  Google Scholar 

  12. Xu, Y., Li, H., Wang, H., Jia, W., Yue, X., Kurths, J.: The estimates of the mean first exit time of a bistable system excited by Poisson white noise. ASME J. Appl. Mech. 84(9), 091004 (2017)

    Article  Google Scholar 

  13. Gan, C., Lei, H.: Stochastic dynamical analysis of a kind of vibro-impact system under multiple harmonic and random excitations. J. Sound Vib. 330(10), 2174–2184 (2011)

    Article  Google Scholar 

  14. Yue, X.L., Xu, W., Wang, L.: Global analysis of boundary and interior crises in an elastic impact oscillator. Commun. Nonlinear Sci. Numer. Simul. 18(12), 3567–3574 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Li, C., Xu, W., Yue, X.L.: Stochastic response of a vibro-impact system by path integration based on generalized cell mapping method. Int. J. Bifurc. Chaos 24(10), 1450129 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jing, H.S., Sheu, K.C.: Exact stationary solutions of the random response of a single-degree-of-freedom vibro-impact system. J. Sound Vib. 141(3), 363–373 (1990)

    Article  MathSciNet  Google Scholar 

  17. Huang, Z.L., Liu, Z.H., Zhu, W.Q.: Stationary response of multi-degree-of-freedom vibro-impact systems under white noise excitations. J. Sound Vib. 275(1–2), 223–240 (2004)

    Article  Google Scholar 

  18. Dimentberg, M.F., Gaidai, O., Naess, A.: Random vibrations with strongly inelastic impacts: response PDF by the path integration method. Int. J. Non Linear Mech. 44(7), 791–796 (2009)

    Article  MATH  Google Scholar 

  19. Feng, J., Xu, W., Wang, R.: Stochastic responses of vibro-impact duffing oscillator excited by additive Gaussian noise. J. Sound Vib. 309(3–5), 730–738 (2008)

    Article  Google Scholar 

  20. Feng, J.Q., Xu, W., Rong, H.W., Wang, R.: Stochastic responses of Duffing-Van der Pol vibro-impact system under additive and multiplicative random excitations. Int. J. Non Linear Mech. 44(1), 51–57 (2009)

    Article  MATH  Google Scholar 

  21. Li, C., Xu, W., Feng, J., Wang, L.: Response probability density functions of Duffing-Van der Pol vibro-impact system under correlated Gaussian white noise excitations. Physica A 392(6), 1269–1279 (2013)

    Article  MathSciNet  Google Scholar 

  22. Gu, X.G., Zhu, W.Q.: A stochastic averaging method for analyzing vibro-impact systems under Gaussian white noise excitations. J. Sound Vib. 333(9), 2632–2642 (2014)

    Article  Google Scholar 

  23. Yang, G., Xu, W., Feng, J., Gu, X.: Response analysis of Rayleigh-Van der Pol vibroimpact system with inelastic impact under two parametric white-noise excitations. Nonlinear Dyn. 82(4), 1797–1810 (2015)

    Article  MathSciNet  Google Scholar 

  24. Yang, G., Xu, W., Gu, X., Huang, D.: Response analysis for a vibroimpact Duffing system with bilateral barriers under external and parametric Gaussian white noises. Chaos Soliton Fract. 87, 125–135 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhu, H.T.: Response of a vibro-impact Duffing system with a randomly varying damping term. Int. J. Non Linear Mech. 65, 53–62 (2014)

    Article  Google Scholar 

  26. Zhu, H.T.: Stochastic response of a vibro-impact Duffing system under external Poisson impulses. Nonlinear Dyn. 82(1–2), 1001–1013 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Xie, X.F., Li, J.L., Liu, D., Guo, R.: Transient response of nonlinear vibro-impact system under Gaussian white noise excitation through complex fractional moments. Acta Mech. 228(3), 1–11 (2016)

    MathSciNet  Google Scholar 

  28. Namachchivaya, N.S., Roessel, H.J.V.: Maximal Lyapunov exponent and rotation numbers for two coupled oscillators driven by real noise. J. Stat. Phys. 71(3–4), 549–567 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wu, Y.J., Luo, M., Zhu, W.Q.: First-passage failure of strongly nonlinear oscillators under combined harmonic and real noise excitations. Arch. Appl. Mech. 78(7), 501–515 (2008)

    Article  MATH  Google Scholar 

  30. Xu, Y., Gu, R.C., Zhang, H.Q., Xu, W., Duan, J.Q.: Stochastic bifurcations in a bistable Duffing-Van der Pol oscillator with colored noise. Phys. Rev. E 83(5), 056215 (2011)

    Article  Google Scholar 

  31. Liu, D., Xu, Y., Li, J.L.: Probabilistic response analysis of nonlinear vibration energy harvesting system driven by Gaussian colored noise. Chaos Soliton Fract. 104, 806–812 (2017)

    Article  Google Scholar 

Download references

Acknowledgements

This work is supported by the National Nature Science Foundation of China (No. 11402139) and the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (No. 2016114).

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Authors and Affiliations

  1. School of Mathematics, Shanxi University, Taiyuan, 030006, China

    Di Liu & Mei Li

  2. School of Applied Science, Taiyuan University of Science and Technology, Taiyuan, 030024, China

    Junlin Li

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  1. Di Liu
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  2. Mei Li
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Correspondence to Di Liu.

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Liu, D., Li, M. & Li, J. Probabilistic response and analysis for a vibro-impact system driven by real noise. Nonlinear Dyn 91, 1261–1273 (2018). https://doi.org/10.1007/s11071-017-3943-x

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  • DOI: https://doi.org/10.1007/s11071-017-3943-x

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