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


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


2000 MR Subject Classification



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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

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