Explanation of the onset of bouncing cycles in isotropic rotor dynamics; a grazing bifurcation analysis

• Published in: Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences Vol. 476; no. 2237; pp. 1 - 22
• Main Authors: Mora, Karin, Champneys, Alan R., Shaw, Alexander D., Friswell, Michael I.
• Format: Journal Article
• Language: English
• Published: England Royal Society 01-05-2020; The Royal Society Publishing
• Subjects: bifurcation; grazing; resonance; rotordynamics; bouncing
• ISSN: 1364-5021; 1471-2946
• DOI: 10.1098/rspa.2019.0549

The dynamics associated with bouncing-type partial contact cycles are considered for a 2 degree-of-freedom unbalanced rotor in the rigid-stator limit. Specifically, analytical explanation is provided for a previously proposed criterion for the onset upon increasing the rotor speed Ω of single-bounce-per-period periodic motion, namely internal resonance between forward and backward whirling modes. Focusing on the cases of 2 : 1 and 3 : 2 resonances, detailed numerical results for small rotor damping reveal that stable bouncing periodic orbits, which coexist with non-contacting motion, arise just beyond the resonance speed Ωp:q
. The theory of discontinuity maps is used to analyse the problem as a codimension-two degenerate grazing bifurcation in the limit of zero rotor damping and Ω = Ωp:q
. An analytic unfolding of the map explains all the features of the bouncing orbits locally. In particular, for non-zero damping ζ, stable bouncing motion bifurcates in the direction of increasing Ω speed in a smooth fold bifurcation point that is at rotor speed 𝓞(ζ) beyond Ωp:q
. The results provide the first analytic explanation of partial-contact bouncing orbits and has implications for prediction and avoidance of unwanted machine vibrations in a number of different industrial settings.