Explanation of the onset of bouncing cycles in isotropic rotor dynamics; a grazing bifurcation analysis. (27th May 2020)

• Record Type: Journal Article
• Authors: Mora, Karin; Champneys, Alan R.; Shaw, Alexander D.; Friswell, Michael I.
• Is Part Of: Proceedings. Volume 476:Number 2237(2020)
• Publication Date: 2020-05-27
• Subjects: rotordynamics -- resonance -- bouncing -- grazing -- bifurcation; Physical sciences -- Periodicals; Engineering -- Periodicals; Mathematics -- Periodicals; 500
• DOI: 10.1098/rspa.2019.0549
• Languages: English
• ISSNs: 1364-5021

Abstract

Abstract : 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 Ω p : 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 = Ω p : 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 Ω p : 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.