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

Abstract

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 piecewisesmooth 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.

Asynchronous partial contact motion due to internal resonance in multiple degree-of-freedom rotordynamics

“…Alternatively, in the rigid stator limit, the condition for the linear undamped system to have a periodic solution can be considered as the conditions for a grazing bifurcation (e.g. [ 49 ]). Further discussion on this point will be made in § 5 .…”

“…The same authors use a similar assumption to solve synchronous responses in [ 48 ]. It is interesting to note the recent work of Mora et al [ 49 ] who use recent techniques from piecewise-smooth dynamical systems theory to analyse the bifurcations that can occur at such impacts at primary resonances.…”

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

“…The analysis of this contacting behaviour that initiates at the primary resonance is not the main subject of this paper, but see e.g. [7] and references therein for analyses of this relatively well-understood phenomenon.…”

“…Periodic orbits with one bounce per period correspond to fixed points of P I . To find fixed points of P I , we use the complex form of the dynamical system (2.10); using the same approach as in [7], for which we merely sketch the main ideas. The map P I : Σ → Σ, maps a state z(0 − ) ∈ Σ through R to a post-bounce state z(0 + ) ∈ Σ which is used as an initial condition for the flow Φ until we experience our impact, at a point z(T − ) ∈ Σ, after a time t = T > 0.…”

The study of non-smooth dynamics of a class of multi-degree-of-freedom vibro-impact system with clearance

“…Ding and Xie [3] studied the path from torus bifurcations to chaos in a threedegree-of-freedom vibro-impact system. Karin Mora [4] analysed the novel dynamics arising in a nonlinear rotor dynamic system by investigating the discontinuity-induced bifurcations corresponding to collisions with the rotor housing. Keogh & Cole [5] show that a rotor-stator system with damping and friction can exhibit various forms of stable and unstable synchronous single impact limit cycles.…”