Explanation of the Onset of Bouncing Cycles in Isotropic Rotor Dynamics; A Grazing Bifurcation Analysis

K Mora (Paderborn University, Germany), AR Champneys (University of Bristol), AD Shaw & MI Friswell (Swansea University)

Proceedings of the Royal Society A, Vol. 476, Issue 2237, 1 May 2020, paper 20190549

Abstract

The dynamics associated with bouncing-type partial contact cycles are considered for a two 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 co-exist 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.

Paper Availability

This material has been published in Proceedings of the Royal Society A, Vol. 476, Issue 2237, 1 May 2020, paper 20190549, the only definitive repository of the content that has been certified and accepted after peer review.


Link to paper using doi: 10.1098/rspa.2019.0549

Proceedings of the Royal Society A