Advertisement

Applied Mathematics and Mechanics

, Volume 24, Issue 6, pp 638–645 | Cite as

Bifurcation of a shaft with hysteretic-type internal friction force of material

  • Ding Qian
  • Chen Yu-shu
Article

Abstract

The bifurcation of a shaft with hysteretic internal friction of material was analysed. Firstly, the differential motion equation in complex form was deduced using Hamilton principle. Then averaged equations in primary resonances were obtained using the averaging method. The stability of steady-state responses was also determined. Lastly, the bifurcations of both normal motion (synchronous whirl) and self-excited motion (nonsynchronous whirl) were investigated using the method of singularity. The study shows that by a rather large disturbance, the stability of the shaft can be lost through Hopf bifurcation in case the stability condition is not satisfied. The averaged self-excited response appears as a type of unsymmetrical bifurcation with high orders of co-dimension. The second Hopf bifurcation, which corresponds to double amplitude-modulated response, can occur as the speed of the shaft increases. Balancing the shaft carefully to decrease its unbalance level and increasing the external damping are two effective methods to avoid the appearance of the self-sustained whirl induced by the hysteretic internal friction of material.

Key words

shaft hysteretic-type internal friction of material Hopf bifurcation nonsynchronous whirl 

Chinese Library Classification

O322 

2000 MR Subject Classification

74D10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Den Hartog J P.Mechanical Vibration[M]. New York: McGraw-Hill, 1965.Google Scholar
  2. [2]
    Tondl A.Some Problems of Rotor Dynamics[M]. London: Chapmann & Hall, 1965.Google Scholar
  3. [3]
    Vance J M, Lee J. Stability of high speed rotors with internal damping[J].J Engineering for Industry, 1974,96(4):960–968.Google Scholar
  4. [4]
    Zhang W, Ling F H. Dynamic stability of the rotating shaft mode of Boltzmann visco-elastic solid [J].J Appl Mech, 1986,53(2):424–429.zbMATHCrossRefGoogle Scholar
  5. [5]
    Shaw J, Shaw S W. Instability and kifurcation in a rotating shaft[J].J Sound Vibration, 1989,132 (2):227–244.MathSciNetCrossRefGoogle Scholar
  6. [6]
    Chang C O, Cheng J W. Non-linear dynamics and instability of a rotating shaft-disk system[J].J Sound Vibration, 1993,160(3):433–454.zbMATHCrossRefGoogle Scholar
  7. [7]
    Shaw J, Shaw S W. Non-linear resonance of an unbalanced rotating shaft with internal damping[J].J Sound Vibration, 1991,147(3):435–451.MathSciNetCrossRefGoogle Scholar
  8. [8]
    ZHONG Yi-e, HE Yan-zong, WANG Zheng,et al.Rotor Dynamics[M]. Beijing: Tsinghua University Press, 1987. (in Chinese)Google Scholar
  9. [9]
    CHEN Yi-shu.Nonlinear Vibration[M]. Tianijn: Tianjin Science and Technology Press, 1983. (in Chinese)Google Scholar
  10. [10]
    DING Qian, CHEN Yu-shu. Non-stationary analysis of rotor/casing rubbing[J].J Aerospace Power, 2000,15(2):191–195 (in Chinese)Google Scholar
  11. [11]
    Golubitsky M, Schaeffer D G.Singularities and Groups in Bifurcation Theory, Vol 1[M]. New York: Springer-Verlag, 1985.zbMATHGoogle Scholar
  12. [12]
    CHEN Fang-qi, WU Zhi-qiang, CHEN Yu-shu. Universal unfolding and bifurcation problem with high order co-dimensions of visco-elastic shell dynamics[J].Acta mechanica Sinica, 2001,33(5): 661–668. (in Chinese)MathSciNetGoogle Scholar

Copyright information

© Editorial Committee of Applied Mathematics and Mechanics 2003

Authors and Affiliations

  • Ding Qian
    • 1
  • Chen Yu-shu
    • 1
  1. 1.Department of MechanicsTianjin UniversityTianjinP. R. China

Personalised recommendations