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Discrete Dynamics in Nature and Society
Volume 2017, Article ID 7064590, 11 pages
https://doi.org/10.1155/2017/7064590
Research Article

The Nonsmooth Vibration of a Relative Rotation System with Backlash and Dry Friction

1Liren College of Yanshan University, Qinhuangdao 066004, China
2College of Electrical Engineering, Yanshan University, Qinhuangdao 066004, China

Correspondence should be addressed to Shuo Li; nc.ude.usy.liamuts@ilouhs

Received 2 May 2017; Accepted 14 August 2017; Published 7 November 2017

Academic Editor: Viktor Avrutin

Copyright © 2017 Minjia He et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. M. Carmeli, “Field theory on R × S3 topology. I. The Klein-Gordon and Schrödinger equations,” Foundations of Physics. An International Journal Devoted to the Conceptual Bases and Fundamental Theories of Modern Physics, vol. 15, no. 2, pp. 175–184, 1985. View at Google Scholar · View at MathSciNet
  2. M. Carmeli, “Rotational relativity theory,” International Journal of Theoretical Physics, vol. 25, no. 1, pp. 89–94, 1986. View at Publisher · View at Google Scholar · View at MathSciNet
  3. S. K. Luo, J. L. Fu, and X. W. Chen, “Basic theory of relativistic birkhoffian dynamics of rotational system,” Acta Physica Sinica, vol. 50, no. 3, pp. 383–389, 2001. View at Google Scholar
  4. S. K. Luo, “The theory of relativistic analytical mechanics of the rotational systems,” Applied Mathematics and Mechanics, vol. 19, no. 1, pp. 45–57, 1998. View at Publisher · View at Google Scholar
  5. S. Liu, B. Liu, and P. M. Shi, “Nonlinear feedback control of Hopf bifurcation in a relative rotation dynamical system,” Acta Physica Sinica, vol. 58, no. 7, pp. 4383–4389, 2009. View at Google Scholar · View at MathSciNet
  6. B. Liu, Y. K. Zhang, S. Liu, and Y. Wen, “Hopf bifurcation and stability of periodic solutions in a nonlinear relative rotation dynamical system with time delay,” Acta Physica Sinica, vol. 59, no. 1, pp. 38–43, 2010. View at Google Scholar
  7. S. Pei-Ming, L. Ji-Zhao, L. Bin, and H. Dong-Ying, “Stability and time-delayed feedback control of a relative-rotation nonlinear dynamical system under quasic-periodic parametric excitation,” Acta Physica Sinica, vol. 60, no. 9, p. 094501, 2011. View at Google Scholar
  8. M. Zong, F. Li-Yuan, and S. Ming-Hou, “Bifurcation of a kind of nonlinear-relative rotational system with combined harmonic excitation,” Acta Physica Sinica, vol. 62, no. 5, p. 054501, 2013. View at Google Scholar
  9. L. Hai-Bin, W. Bo-Hua, Z. Zhi-Qiang, L. Shuang, and L. Yan-Shu, “Combination resonance bifurcations and chaos of some nonlinear relative rotation system,” Acta Physica Sinica, vol. 61, no. 9, p. 094501, 2012. View at Google Scholar
  10. P. Shi, B. Liu, and D. Hou, “Global dynamic characteristic of nonlinear torsional vibration system under harmonically excitation,” Chinese Journal of Mechanical Engineering, vol. 22, no. 1, pp. 132–139, 2009. View at Publisher · View at Google Scholar · View at Scopus
  11. P. M. Shi, D. Y. Han, and B. Liu, “Chaos and chaotic control in a relative rotation nonlinear dynamical system under parametric excitation,” Chinese Physics B, vol. 19, no. 9, p. 090306, 2010. View at Google Scholar
  12. J. Hartog and S. Mikina, “Forced vibrations with non-linear spring constants,” ASME Journal of Applied Mechanics, vol. 58, pp. 157–164, 1932. View at Google Scholar
  13. S. W. Shaw and P. J. Holmes, “A periodically forced piecewise linear oscillator,” Journal of Sound and Vibration, vol. 90, no. 1, pp. 129–155, 1983. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. M. Kleczka, E. Kreuzer, and W. Schiehlen, “Local and global stability of a piecewise linear oscillator,” Philosophical Transactions of the Royal Society of London. Series A. Mathematical, Physical Sciences and Engineering, vol. 338, no. 1651, pp. 533–546, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  15. A. C. J. Luo, Analytical modeling of bifurcation, chaos and multifractals in nonlinear dynamics, University of Manitoba, Manitoba, Canada, 1995.
  16. A. C. J. Luo and S. Menon, “Global chaos in a periodically forced, linear system with a dead-zone restoring force,” Chaos, Solitons & Fractals, vol. 19, no. 5, pp. 1189–1199, 2004. View at Publisher · View at Google Scholar · View at Scopus
  17. A. B. Nordmark, “Non-periodic motion caused by grazing incidence in an impact oscillator,” Journal of Sound and Vibration, vol. 145, no. 2, pp. 279–297, 1991. View at Publisher · View at Google Scholar · View at Scopus
  18. B. Błazejczyk, T. Kapitaniak, J. Wojewoda, and R. Barron, “Experimental Observation Of Intermittent Chaos In A Mechanical System With Impacts,” Journal of Sound and Vibration, vol. 178, no. 2, pp. 272–275, 1994. View at Publisher · View at Google Scholar · View at Scopus
  19. B. Blazejczyk-Okolewska, K. Czolczynski, and T. Kapitaniak, “Dynamics of a two-degree-of-freedom cantilever beam with impacts,” Chaos, Solitons & Fractals, vol. 40, no. 4, pp. 1991–2006, 2009. View at Publisher · View at Google Scholar · View at Scopus
  20. G. W. Luo, X. H. Lv, and Y. Q. Shi, “Vibro-impact dynamics of a two-degree-of freedom periodically-forced system with a clearance: diversity and parameter matching of periodic-impact motions,” International Journal of Non-Linear Mechanics, vol. 65, no. 4, pp. 173–195, 2014. View at Publisher · View at Google Scholar
  21. M. di Bernardo, C. J. Budd, and A. R. Champneys, “Normal form maps for grazing bifurcations in n-dimensional piecewise-smooth dynamical systems,” Physica D: Nonlinear Phenomena, vol. 160, no. 3-4, pp. 222–254, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  22. M. di Bernardo, A. Nordmark, and G. Olivar, “Discontinuity-induced bifurcations of equilibria in piecewise-smooth and impacting dynamical systems,” Physica D: Nonlinear Phenomena, vol. 237, no. 1, pp. 119–136, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. A. C. Luo, “A theory for non-smooth dynamic systems on the connectable domains,” Communications in Nonlinear Science and Numerical Simulation, vol. 10, no. 1, pp. 1–55, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. C. Zhang, X. Han, and Q. Bi, “Dynamical behaviors of the periodic parameter-switching system,” Nonlinear Dynamics, vol. 73, no. 1-2, pp. 29–37, 2013. View at Publisher · View at Google Scholar · View at Scopus
  25. A. C. Luo, “A periodically forced, piecewise linear system. I. Local singularity and grazing bifurcation,” Communications in Nonlinear Science and Numerical Simulation, vol. 12, no. 3, pp. 379–396, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. A. C. J. Luo and L. Chen, “Periodic motions and grazing in a harmonically forced, piecewise, linear oscillator with impacts,” Chaos, Solitons & Fractals, vol. 24, no. 2, pp. 567–578, 2005. View at Publisher · View at Google Scholar · View at Scopus
  27. J. P. D. Hartog, “Forced vibrations with coulomb and viscous damping,” Transactions of the American Society of Mechanical Engineers, vol. 53, pp. 107–115, 1931. View at Google Scholar
  28. E. S. Levitan, “Forced oscillation of a spring-mass system having combined Coulomb and viscous damping,” The Journal of the Acoustical Society of America, vol. 32, pp. 1265–1269, 1960. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  29. S. W. Shaw, “On the dynamic response of a system with dry friction,” Journal of Sound and Vibration, vol. 108, no. 2, pp. 305–325, 1986. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  30. B. Feeny, “A nonsmooth Coulomb friction oscillator,” Physica D: Nonlinear Phenomena, vol. 59, no. 1-3, pp. 25–38, 1992. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  31. B. Feeny and F. C. Moon, “Chaos in a Forced Dry-Friction Oscillator: Experiments and Numerical Modelling,” Journal of Sound and Vibration, vol. 170, no. 3, pp. 303–323, 1994. View at Publisher · View at Google Scholar · View at Scopus
  32. N. Hinrichs, M. Oestreich, and K. Popp, “Dynamics of oscillators with impact and friction,” Chaos, Solitons & Fractals, vol. 8, no. 4, pp. 535–558, 1997. View at Publisher · View at Google Scholar · View at Scopus
  33. N. Hinrichs, M. Obstreich, and K. Popp, “On the modelling of friction oscillators,” Journal of Sound and Vibration, vol. 216, no. 3, pp. 435–459, 1998. View at Publisher · View at Google Scholar · View at Scopus
  34. J. Bastien, F. Bernardin, C.-H. Lamarque, and N. Challamel, “Non-smooth Deterministic or Stochastic Discrete Dynamical Systems: Applications to Models with Friction or Impact,” ISTE Ltd-Wiley, vol. 496, 2013. View at Publisher · View at Google Scholar · View at Scopus