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Mode-locking in nonlinear rotordynamics

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Summary

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.

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Author notes

  1. G. H. M. van der Heijden

    Present address: Centre for Nonlinear Dynamics, Civil Engineering Building, University College London, Gower Street, WC1E 6BT, London, UK

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  1. Mathematics Institute, University of Utrecht, Utrecht, The Netherlands

    G. H. M. van der Heijden

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Communicated by Stephen Wiggins

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van der Heijden, G.H.M. Mode-locking in nonlinear rotordynamics. J Nonlinear Sci 5, 257–283 (1995). https://doi.org/10.1007/BF01212957

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  • DOI: https://doi.org/10.1007/BF01212957

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