Non-smooth Hopf-type bifurcations arising from impact–friction contact events in rotating machinery

Karin Mora, Christopher Budd, Paul Glendinning, Patrick Keogh

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We analyse the novel dynamics arising in a nonlinear rotor dynamic system by investigating the discontinuity-induced bifurcations corresponding to collisions with the rotor housing (touchdown bearing surface interactions). The simplified Föppl/Jeffcott rotor with clearance and mass unbalance is modelled by a two degree of freedom impact–friction oscillator, as appropriate for a rigid rotor levitated by magnetic bearings. Two types of motion observed in experiments are of interest in this paper: no contact and repeated instantaneous contact. We study how these are affected by damping and stiffness present in the system using analytical and numerical piecewise-smooth dynamical systems methods. By studying the impact map, we show that these types of motion arise at a novel non-smooth Hopf-type bifurcation from a boundary equilibrium bifurcation point for certain parameter values. A local analysis of this bifurcation point allows us a complete understanding of this behaviour in a general setting. The analysis identifies criteria for the existence of such smooth and non-smooth bifurcations, which is an essential step towards achieving reliable and robust controllers that can take compensating action.
Original languageEnglish
Pages (from-to)20140490
Number of pages23
JournalProceedings of the Royal Society A: Mathematical Physical and Engineering Sciences
Issue number2171
Early online date17 Sept 2014
Publication statusPublished - 8 Nov 2014

Bibliographical note

The research undertaken involved collaboration between the Departments of Mechanical Engineering and Mathematical Sciences (Bath) and the Centre for Interdisciplinary Computational and Dynamical Analysis (CICADA) and School of Mathematics (Manchester). The cross-disciplinary nature was necessary to reveal the complex dynamics associated with rotor-stator contact events in magnetic bearing systems. This forms a critical step in the ability to control the system dynamics.


  • non-smooth bifurcation,
  • rotordynamics
  • impact with friction,
  • magnetic bearing system
  • ,Hopf-type bifurcation


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