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Codimension two bifurcation and chaos of a vibro-impact forming machine associated with 1:2 resonance case

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Abstract

A vibro-impact forming machine with double masses is considered. The components of the vibrating system collide with each other. Such models play an important role in the studies of dynamics of mechanical systems with impacting components. The Poincaré section associated with the state of the impact-forming system, just immediately after the impact, is chosen, and the period n single-impact motion and its disturbed map are derived analytically. A center manifold theorem technique is applied to reduce the Poincaré map to a two-dimensional map, and the normal form map associated with codimension two bifurcation of 1:2 resonance is obtained. Unfolding of the normal form map is analyzed. Dynamical behavior of the impact-forming system, near the point of codimension two bifurcation, is investigated by using qualitative analyses and numerical simulation. Near the point of codimension two bifurcation there exists not only Neimark-Sacker bifurcation associated with period one single-impact motion, but also Neimark-Sacker bifurcation of period two double-impact motion. Transition of different forms of fixed points of single-impact periodic orbits, near the bifurcation point, is demonstrated, and different routes from periodic impact motions to chaos are also discussed.

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References

  1. Aidanpää J.O., Gupta B.R.: Periodic and chaotic behaviour of a threshold-limited two-degree- of-freedom system. Journal of Sound and Vibration, 165(2): 305–327 (1993)

    Google Scholar 

  2. Jin D.P., Hu H.Y.: Periodic vibro-impacts and their stability of a dual component system. Acta Mechanica Sinica, 13(4): 366–376 (1997)

    Google Scholar 

  3. Li Q.H., Lu Q.S.: Coexisting periodic orbits in vibro- impacting dynamical systems. Applied Mathematics and Mechanics, 24(3): 261–273 (2003)

    Google Scholar 

  4. Whiston G.S.: Singularities in virbo-impact dynamics. Journal of Sound and Vibration, 152(3): 427–460 (1992)

    Google Scholar 

  5. Ivanov A.P.: Bifurcation in impact systems. Chaos, Solitons & Fractal, 7(10): 1615–1634 (1996)

  6. Peterka F.: Bifurcation and transition phenomena in an impact oscillator. Chaos, Solitons & Fractals, 7(10): 1635–1647 (1996)

  7. Hu H.Y.: Detection of grazing orbits and incident bifurcations of a forced continuous, piecewise-linear oscillator. Journal of Sound and Vibration, 187(3): 485–493 (1994)

    Google Scholar 

  8. Bernardo D., Feigin M.I., Hogan S.J., Homer M.E.: Local analysis of C-bifurcations in N-dimensional piecewise- smooth dynamical systems. Chaos, Solitons & Fractals, 10(11): 1881–1908 (1999)

    Google Scholar 

  9. Nguyen D.T., Noah S.T., Kettleborough C.F.: Impact behaviour of an oscillator with limiting stops, part I: a parametric study. Journal of Sound and Vibration, 109(2): 293–307 (1986)

    Google Scholar 

  10. Natsiavas S.: Dynamics of multiple-degree-of-freedom oscillators with colliding components. Journal of Sound and Vibration, 165(3): 439–453 (1993)

    Google Scholar 

  11. Chatterjee S., Mallik A.K.: Bifurcations and chaos in autonomous self-excited oscillators with impact damping. Journal of Sound and Vibration, 191(4): 539–562 (1996)

    Google Scholar 

  12. Luo G.W., Xie J.H.: Hopf bifurcation of a two-degree-of- freedom vibro-impact system. Journal of Sound and Vibration, 213(3): 391–408 (1998)

    Google Scholar 

  13. Hu H.Y.: Controlling chaos of a periodically forced nonsmooth mechanical system. Acta Mechanica Sinica, 11(3): 251–258 (1995)

    Google Scholar 

  14. Hu H.Y.: Active control of chaos of mechanical systems. Advances in Mechanics, 26(4): 453–463 (1996) (in Chinese)

  15. Hu H.Y.: Controlling chaos of a dynamical system with discontinuous vector field. Physica D, 106(1): 1–8 (1997)

    Google Scholar 

  16. Jin D.P., Hu H.Y.: An experimental study on possible types of vibro-impacts between two elastic beams. Journal of Experimental Meachanics, 14(3): 129–135 (1999) (in Chinese)

    Google Scholar 

  17. Mejaard J.P., De Pater A.D.: Railway vehicle systems dynamics and chaotic vibrations. International Journal of Non-Linear Mechanics, 24(1): 1–17 (1989)

    Google Scholar 

  18. Zeng J., Hu S.: Study on frictional impact and derailment for wheel and rail. Journal of Vibration Engineering, 14(1): 1–6 (2001) (in Chinese)

    Google Scholar 

  19. Yu K., Luo Albert C.J.: The periodic impact responses and stability of a human body in a vehicle traveling on rough terrain. Journal of Sound and Vibration, 272(1–2): 267–286 (2004)

    Google Scholar 

  20. Feng Q.: Modeling of the mean Poincare map on a class of random impact oscillators. European Journal of Mechanics, A/Solids, 22(2): 267–281 (2003)

    Google Scholar 

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

    Google Scholar 

  22. Shu Z.Z., Shen X.Z.: Theoretical analysis of complete stability and automatic vibration isolation of impacting and vibrating systems with double masses. Chinese Journal of Mechanical Engineering, 26(3): 50–57 (1990) (in Chinese)

    Google Scholar 

  23. Xie J.H.: The mathematical model for the impact hammer and global bifurcations. Acta Mechanica Sinica, 29(4): 456–463 (1997) (in Chinese)

    Google Scholar 

  24. Yang Z.Z.: Optimization of impact position and simulation of small vibratory pile driver. Journal of Hydraulic Engineering, 8: 72–76 (1999) (in Chinese)

    Google Scholar 

  25. Luo G.W.: Stability and bifurcations of periodic motions of a vibro-impact forming machinery with double masses. Engineering Mechanics, 21(1): 118–124 (2004) (in Chinese)

    Google Scholar 

  26. Han P.R.S., Luo A.C.J.: Chaotic motion of a horizontal impact pair. Journal of Sound and Vibration, 181(2): 231–250 (1995)

    Google Scholar 

  27. Bapat C.N.: The general motion of an inclined impact damper with friction. Journal of Sound and Vibration, 184(3): 417–427 (1995)

    Google Scholar 

  28. Lin T.J., Li R.F., Tao Z.G.: Numerical simulation of 3-D gap type nonlinear dynamic contact-impact characteristics for gear transmission. Chinese Journal of Mechanical Engineering, 36(6): 55–58 (2000) (in Chinese)

    Google Scholar 

  29. Dong H.J., Shen Y.W., Liu M.J., Zhang S.H.: Research on the dynamical behaviors of rattling in gear system. Chinese Journal of Mechanical Engineering, 40(1): 136–141 (2004) (in Chinese)

    Google Scholar 

  30. Wen B.C., Liu F.Q.: Theory and Application of Vibratory Mechanism. Beijing: Mechanism industry Press, 1982

  31. Kuznetsov Y.A.: Elements of Applied Bifurcation Theory, Second Edition. New York: Springer-Verlage, 1998

  32. Carr J.: Application of center manifold theory. In: Applied Math. Sciences, 35. New York: Springer-Verlag, 1981

  33. Guckenheimer J., Holmes P.J.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Second Printing. New York, Berlin, Heidelberg, Tokyo: Springer-Verlag, 1986

  34. Wen B.C., Li Y.L., Han Q.K.: Theory and Application of Nonlinear Oscillation. Shenyang: Northeast University Press, 2001(in Chinese)

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Authors and Affiliations

  1. School of Mathematics, Physics and Software Engineering, Lanzhou Jiaotong University, Lanzhou, 730070, China

    Guanwei Luo & Jianning Yu

  2. Department of Engineering Machanics, Southwest Jiaotong University, Chengdu, 610031, China

    Jianhua Xie

Authors
  1. Guanwei Luo
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  2. Jianning Yu
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  3. Jianhua Xie
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Correspondence to Guanwei Luo.

Additional information

The project supported by the National Natural Science Foundation of China (10572055, 50475109) and the Natural Science Foundation of Gansu Province Government of China (3ZS051-A25-030(key item)) The English text was polished by Keren Wang.

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Luo, G., Yu, J. & Xie, J. Codimension two bifurcation and chaos of a vibro-impact forming machine associated with 1:2 resonance case. ACTA MECH SINICA 22, 185–198 (2006). https://doi.org/10.1007/s10409-006-0105-7

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  • DOI: https://doi.org/10.1007/s10409-006-0105-7

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