Journal of Nonlinear Science

, Volume 5, Issue 3, pp 257–283

Mode-locking in nonlinear rotordynamics

  • G. H. M. van der Heijden

DOI: 10.1007/BF01212957

Cite this article as:
van der Heijden, G.H.M. J Nonlinear Sci (1995) 5: 257. doi:10.1007/BF01212957


We present a computer-assisted study of the dynamics of two nonlinearly coupled driven oscillators with rotational symmetry which arise in rotordynamics (the nonlinearity coming from bearing clearance). The nonlinearity causes a splitting of the twofold degenerate natural frequency of the associated linear model, leading to three interacting frequencies in the system. Partial mode-locking then yields a biinfinite series of attracting invariant 2-tori carrying (quasi-) periodic motion.

Due to the resonance nature, the (quasi-) periodic solutions become periodic in a corotating coordinate system. They can be viewed as entrainments of periodic solutions of the associated linear problem. One presumably infinite family is generated by (scaled) driving frequencies ω = 1+2/n,n = 1,2,3,...; another one is generated by frequencies ω =m,m = 4,5,6,... Both integersn andm can be related to discrete symmetry properties of the particular periodic solutions.

Under a perturbation that breaks the rotational symmetry, more complicated behavior is possible. In particular, a second rational relation between the frequencies can be established, resulting in fully mode-locked periodic motion.

Key words

rotor dynamicsbearing clearancemode-lockingresonance

Copyright information

© Springer-Verlag New York Inc. 1995

Authors and Affiliations

  • G. H. M. van der Heijden
    • 1
  1. 1.Mathematics InstituteUniversity of UtrechtUtrechtThe Netherlands
  2. 2.Centre for Nonlinear Dynamics, Civil Engineering BuildingUniversity College LondonLondonUK