Journal of Nonlinear Science

, Volume 5, Issue 3, pp 257-283

First online:

Mode-locking in nonlinear rotordynamics

  • G. H. M. van der HeijdenAffiliated withMathematics Institute, University of Utrecht

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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 dynamics bearing clearance mode-locking resonance