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The response of stochastic vibro-impact system calculated by a new path integration algorithm

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Abstract

In this paper, a new path integration algorithm is proposed specifically for the stochastic vibro-impact system. By introducing the concept of absorption surface and the impact completion condition, this new path integration algorithm can be directly used to study the stochastic response of vibro-impact systems without any non-smooth approximation. The algorithm is suitable for arbitrary recovery coefficients and can retain the non-smooth characteristics of the vibro-impact system. Linear and nonlinear vibro-impact systems excited by the Gaussian white noise are used as examples to show the unique results obtained by our method. Finally, the Monte Carlo method is used to verify the efficiency and accuracy of this new algorithm.

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References

  1. Leine, R.I., Glocker, C., Campen, D.H.V.: Nonlinear dynamics of the Woodpecker Toy. In: ASME 2001 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (2001)

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

    Article  MathSciNet  Google Scholar 

  3. Li, T., Lamarque, C.H., Seguy, S., et al.: Chaotic characteristic of a linear oscillator coupled with vibro-impact nonlinear energy sink. Nonlinear Dyn. 91, 2319–2330 (2017)

    Article  Google Scholar 

  4. Zhang, H., Zhang, Y., Luo, G.: Basins of coexisting multi-dimensional tori in a vibro-impact system. Nonlinear Dyn. 79(3), 2177–2185 (2015)

    Article  Google Scholar 

  5. Li, G., Ding, W.: Global behavior of a vibro-impact system with asymmetric clearances. J. Sound Vib. 423, 180–194 (2018)

    Article  Google Scholar 

  6. Luo, G., Xie, J.: Bifurcations and chaos in a system with impacts. Physica D 148(3–4), 183–200 (2001)

    Article  MathSciNet  Google Scholar 

  7. Brogliato, B.: Piecewise-smooth dynamical systems: theory and applications (M. di Bernardo, C.J. Budd, A.R. Champneys & P. Kowalczyk; 2008) [Book Review]. IEEE Control Syst. Mag. 28(5), 141–143 (2008)

    Article  Google Scholar 

  8. Nordmark, A.B.: Non-periodic motion caused by grazing incidence in an impact oscillator. J. Sound Vib. 145(2), 279–297 (1991)

    Article  Google Scholar 

  9. Holmes, P.J.: The dynamics of repeated impacts with a sinusoidally vibrating table. J. Sound Vib. 84(2), 173–189 (1982)

    Article  MathSciNet  Google Scholar 

  10. Shaw, S.W., Rand, R.H.: The transition to chaos in a simple mechanical system. Int. J. Non-Linear Mech. 24(1), 41–56 (1989)

    Article  MathSciNet  Google Scholar 

  11. Shaw, S.W., Holmes, P.J.: A periodically forced piecewise linear oscillator. J. Sound Vib. 90(1), 129–155 (1983)

    Article  MathSciNet  Google Scholar 

  12. Whiston, G.S.: Global dynamics of a vibro-impacting linear oscillator. J. Sound Vib. 118(3), 395–424 (1987)

    Article  MathSciNet  Google Scholar 

  13. 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 

  14. Wu, Y., Zhu, W.Q.: Stationary response of multi-degree-of-freedom vibro-impact systems to Poisson white noises. Phys. Lett. A 372(5), 623–630 (2008)

    Article  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  Google Scholar 

  16. Ma, S.C., Wang, L., Ning, X., et al.: Probabilistic responses of three-dimensional stochastic vibro-impact systems. Chaos Solitons Fractals 126, 308–314 (2019)

    Article  MathSciNet  Google Scholar 

  17. Zhuravlev, V.F.: A method for analyzing vibration-impact systems by means of special functions. Mech. Solids 11(2), 23–27 (1976)

    MathSciNet  Google Scholar 

  18. Ivanov, A.P.: Impact oscillations: linear theory of stability and bifurcations. J. Sound Vib. 178(3), 361–378 (1994)

    Article  MathSciNet  Google Scholar 

  19. Feng, J., Xu, W., Rong, H., et al.: 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  Google Scholar 

  20. Gu, X., Zhu, W.: 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 

  21. Zhao, X., Xu, W., Yang, Y., et al.: Stochastic responses of a viscoelastic-impact system under additive and multiplicative random excitations. Commun. Nonlinear Sci. Numer. Simul. 35(Jun.), 166–176 (2016)

    Article  MathSciNet  Google Scholar 

  22. Xu, M., Wang, Y., Jin, X.L., et al.: Random response of vibro-impact systems with inelastic contact. Int. J. Non-Linear Mech. 52(Jun.), 26–31 (2013)

    Article  Google Scholar 

  23. Su, M., Xu, W., Yang, G.: Stochastic response and stability of system with friction and a rigid barrier. Mech. Syst. Signal Process. 132, 748–761 (2019)

    Article  Google Scholar 

  24. Wang, D.L., Xu, W., Gu, X.D., et al.: Response analysis of nonlinear vibro-impact system coupled with viscoelastic force under colored noise excitations Int. J Non-Linear Mech 86, 55–65 (2016)

    Article  Google Scholar 

  25. Li, C.: Stochastic response of a vibro-impact system with variable mass. Physica A Stat. Mech. Appl. 516, 151–160 (2019)

    Article  MathSciNet  Google Scholar 

  26. Wang, D., Xu, W., Gu, X., et al.: Stationary response analysis of vibro-impact system with a unilateral nonzero offset barrier and viscoelastic damping under random excitations. Nonlinear Dyn. 86(2), 1–19 (2016)

    Google Scholar 

  27. Su, M., Xu, W., Yang, G.: Response analysis of van der Pol vibro-impact system with Coulomb friction under Gaussian White noise. Int. J. Bifurc. Chaos 28(13), 1830043 (2018)

    Article  MathSciNet  Google Scholar 

  28. Zhu, H.T.: Stochastic response of vibro-impact Duffing oscillators under external and parametric Gaussian white noises. J. Sound Vib. 333(3), 954–961 (2014)

    Article  Google Scholar 

  29. Zhu, H.T.: Probabilistic solution of vibro-impact stochastic Duffing systems with a unilateral non-zero offset barrier. Physica A Stat. Mech. Appl. 410, 335–344 (2014)

    Article  MathSciNet  Google Scholar 

  30. 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  Google Scholar 

  31. Di Paola, M., Bucher, C.: Ideal and physical barrier problems for non-linear systems driven by normal and Poissonian white noise via path integration method. Int. J. Non-Linear Mech. 81, 274–282 (2016)

    Article  Google Scholar 

  32. Ren, Z.X., Xu, W., Wang, D.: Dynamic and first passage analysis of ship roll motion with inelastic impacts via path integration method. Nonlinear Dyn. 97(4), 391–402 (2019)

    Article  Google Scholar 

  33. Wang, L., Ma, S.C.: The stochastic response of a class of impact systems calculated by a new strategy based on generalized cell mapping method. J. Appl. Mech. Trans. ASME (2018)

  34. Wehner, M.F., Wolfer, W.G.: Numerical evaluation of path-integral solutions to Fokker–Planck equations. Phys. Rev. A 27, 2663–2670 (1983)

    Article  Google Scholar 

  35. Naess, A., Johnsen, J.M.: Response statistics of nonlinear dynamic systems by path integration. In: Bellomo, N., Casciatti, F. (eds.) Nonlinear Stochastic Mechanics, pp. 401–414. Springer, Berlin (1992)

    Chapter  Google Scholar 

  36. Naess, A., Moe, V.: Efficient path integration methods for nonlinear dynamic systems. Probab. Eng. Mech. 15(2), 221–231 (2000)

    Article  Google Scholar 

  37. Hertz, J.A., Roudi, Y., Sollich, P.: Path integral methods for the dynamics of stochastic and disordered systems. J. Phys. A Math. Theor. 50(3), 033001 (2017)

    Article  MathSciNet  Google Scholar 

  38. Naess, A., Kolnes, F.E., Mo, E.: Stochastic spur gear dynamics by numerical path integration. J. Sound Vib. 302(4–5), 936–950 (2007)

    Article  Google Scholar 

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11972289, 11672232).

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Authors and Affiliations

  1. Department of Applied Probability and Statistics, School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710129, People’s Republic of China

    Liang Wang, Jiahui Peng, Bochen Wang & Wei Xu

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  1. Liang Wang
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  2. Jiahui Peng
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  3. Bochen Wang
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  4. Wei Xu
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These authors contributed equally to this work.

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Correspondence to Jiahui Peng.

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Wang, L., Peng, J., Wang, B. et al. The response of stochastic vibro-impact system calculated by a new path integration algorithm. Nonlinear Dyn 104, 289–296 (2021). https://doi.org/10.1007/s11071-021-06305-6

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