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Response of strongly nonlinear vibro-impact system with fractional derivative damping under Gaussian white noise excitation

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Abstract

This paper aims to investigate the strongly nonlinear vibro-impact dynamic system with fractional derivative damping under Gaussian white noise excitation, and this system considers both nonlinear factors and non-smooth factors. With the help of non-smooth transformation, the original system is rewritten as a smooth system, which is an easier handled form. For the altered function, we obtain the approximate stationary solutions analytically by generalized stochastic averaging method. To prove the validity of the approximate analytical methods, an efficient scheme with high accuracy is adopt to simulate the fractional derivative, then the fourth-order Runge–Kutta approach is used to obtain the numerically response statistics. Meanwhile the analytical solutions are verified by the numerical simulation solutions. At last, we research the responses of this system. In this context, we can consider the influences to this system caused by the fractional order, the restitution coefficient and the noise intensity through changing the values of corresponding parameters. The observation and investigation state that fractional derivative term, impact conditions and stochastic excitation can influence the responses of the fractional vibro-impact dynamic system.

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References

  1. Debnath, L.: Recent applications of fractional calculus to science and engineering. Int. J. Math. Math. Sci. 2003(54), 3413–3442 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Dalir, M., Bashour, M.: Applications of fractional calculus. Appl. Math. Sci. 4(21), 1021–1032 (2010)

    MathSciNet  MATH  Google Scholar 

  3. Galucio, A., Deü, J.-F., Ohayon, R.: Finite element formulation of viscoelastic sandwich beams using fractional derivative operators. Comput. Mech. 33(4), 282–291 (2004)

    Article  MATH  Google Scholar 

  4. Deü, J.-F., Matignon, D.: Simulation of fractionally damped mechanical systems by means of a Newmark-diffusive scheme. Comput. Math. Appl. 59(5), 1745–1753 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Shen, Y., Wei, P., Sui, C., Yang, S.: Subharmonic resonance of van der Pol oscillator with fractional-order derivative. Math. Probl. Eng. 2014(1), 688–706 (2014)

    MathSciNet  Google Scholar 

  6. Padovan, J., Sawicki, J.T.: Nonlinear vibrations of fractionally damped systems. Nonlinear Dyn. 16(4), 321–336 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, L., Zhu, W.: Stochastic averaging of strongly nonlinear oscillators with small fractional derivative damping under combined harmonic and white noise excitations. Nonlinear Dyn. 56(3), 231–241 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Huang, Z., Jin, X.: Response and stability of a SDOF strongly nonlinear stochastic system with light damping modeled by a fractional derivative. J. Sound Vib. 319(3), 1121–1135 (2009)

    Article  Google Scholar 

  9. Chen, L., Zhu, W.: First passage failure of SDOF nonlinear oscillator with lightly fractional derivative damping under real noise excitations. Probab. Eng. Mech. 26(2), 208–214 (2011)

    Article  Google Scholar 

  10. Yang, Y., Xu, W., Jia, W., Han, Q.: Stationary response of nonlinear system with Caputo-type fractional derivative damping under Gaussian white noise excitation. Nonlinear Dyn. 79(1), 139–146 (2015)

    Article  MathSciNet  Google Scholar 

  11. Xu, Y., Li, Y., Liu, D., Jia, W., Huang, H.: Responses of Duffing oscillator with fractional damping and random phase. Nonlinear Dyn. 74(3), 745–753 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Liu, D., Li, J., Xu, Y.: Principal resonance responses of SDOF systems with small fractional derivative damping under narrow-band random parametric excitation. Commun. Nonlinear Sci. Numer. Simul. 19(10), 3642–3652 (2014)

    Article  MathSciNet  Google Scholar 

  13. Diethelm, K., Ford, N.J., Freed, A.D.: A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29(1–4), 3–22 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Di Paola, M., Failla, G., Pirrotta, A.: Stationary and non-stationary stochastic response of linear fractional viscoelastic systems. Probab. Eng. Mech. 28, 85–90 (2012)

    Article  Google Scholar 

  15. Chen, L., Zhao, T., Li, W., Zhao, J.: Bifurcation control of bounded noise excited Duffing oscillator by a weakly fractional-order PID feedback controller. Nonlinear Dyn. 83(1–2), 529–539 (2016)

    Article  MathSciNet  Google Scholar 

  16. Acary, V., Brogliato, B.: Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics. Springer Science & Business Media, New York (2008)

    MATH  Google Scholar 

  17. de Weger, J., Binks, D., Molenaar, J., van de Water, W.: Generic behavior of grazing impact oscillators. Phys. Rev. Lett. 76(21), 3951 (1996)

    Article  Google Scholar 

  18. Toulemonde, C., Gontier, C.: Sticking motions of impact oscillators. Eur. J. Mech. A Solids 17(2), 339–366 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  20. Budd, C., Dux, F.: Chattering and related behaviour in impact oscillators. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 347(1683), 365–389 (1994)

    Google Scholar 

  21. 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 

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

    MathSciNet  Google Scholar 

  23. Feng, J., Xu, W., Rong, H., 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 

  24. 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. Phys. A Stat. Mech. its Appl. 392(6), 1269–1279 (2013)

  25. Rong, H., Wang, X., Xu, W., Fang, T.: Subharmonic response of a single-degree-of-freedom nonlinear vibroimpact system to a randomly disordered periodic excitation. J. Sound Vib. 327(1), 173–182 (2009)

    Article  MathSciNet  Google Scholar 

  26. Namachchivaya, N.S., Park, J.H.: Stochastic dynamics of impact oscillators. J. Appl. Mech. 72(6), 862–870 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Feng, Q., He, H.: Modeling of the mean Poincaré map on a class of random impact oscillators. Eur. J. Mech A Solids 22(2), 267–281 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  29. Zhu, H.: 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 

Download references

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant Nos, 11472212, 11532011).

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Authors and Affiliations

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

    Yanwen Xiao, Wei Xu & Yongge Yang

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  1. Yanwen Xiao
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  2. Wei Xu
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  3. Yongge Yang
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Correspondence to Wei Xu.

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Xiao, Y., Xu, W. & Yang, Y. Response of strongly nonlinear vibro-impact system with fractional derivative damping under Gaussian white noise excitation. Nonlinear Dyn 85, 1955–1964 (2016). https://doi.org/10.1007/s11071-016-2808-z

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  • DOI: https://doi.org/10.1007/s11071-016-2808-z

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