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Mora K, Champneys AR, Shaw AD, Friswell MI. Explanation of the onset of bouncing cycles in isotropic rotor dynamics; a grazing bifurcation analysis. Proc Math Phys Eng Sci 2020; 476:20190549. [DOI: 10.1098/rspa.2019.0549] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/25/2019] [Accepted: 04/09/2020] [Indexed: 11/12/2022] Open
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.
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Affiliation(s)
- Karin Mora
- Department of Mathematics, Paderborn University, 33098 Paderborn, Germany
| | - Alan R. Champneys
- Department of Engineering Mathematics, University of Bristol, BS8 1UB Bristol, UK
| | - Alexander D. Shaw
- College of Engineering, Swansea University, Bay Campus, Swansea SA1 8EN, UK
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