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

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.

Effect of gear topology discontinuities on the nonlinear dynamic response of a multi-degree-of-freedom gear train

“…This happens because we consider damped dynamics. In the conservative case, the transition between impacting and non-impacting orbits is expected to take place through a SN bifurcation [35,36].…”

Continuation analysis of a nonlinear rotor system

“…Contrary to the systems with impact definitions in the contact [8,23], the smoothness of the trajectories of the cubic stiffness model is not broken. Also the effect of contact dissipation cannot be seen in the present model, on which Shaw et al [34] noted that stiffer, dissipative contact definitions eliminated most of the sustained intermittent bouncing cycles.…”

“…The study gave upper and lower limits of unbalance to achieve the periodic contacting motions. Mora et al [23] employed a rigid impact model for the contacting interactions, and explained the transition into bouncing motions as a grazing bifurcation.…”

Constrained Control of Coexisting Attractors in Impact Oscillator with Delay

“…In Jeffcott rotors, the impact of the rotor and the snubber ring results in the degradation of the system, and it is desired to be avoided. To this purpose, the exchange between impacting to non-impacting attractor jumps the system from the impacting to the non-impacting attractor [9,10]. Similarly, the impact should be avoided in the railway wheelset to increase the system's lifetime.…”