Chaos in the imbalance response of a rigid rotor in active magnetic bearings

Abstract

Active magnetic bearings exhibit highly nonlinear characteristics that can be detrimental to the performance of the rotating machinery supported by them. There are several sources of nonlinearity in an active magnetic bearing system, of which the most prominent is the relationship between the forces generated in the electromagnetic actuator and the coil current and the air gap between the rotor and the stator. Cross-coupling between the electromagnetic forces acting in two orthogonal directions that arise due to the geometry of the actuators is also a source of nonlinearity in a magnetic bearing system. This work reports on a numerical study undertaken to investigate the response of an imbalanced rigid rotor supported by active magnetic bearings. The mathematical model of the rotor-bearing system used in this study incorporates nonlinearity arising from the electromagnetic force - coil current - air gap relationship, and the effects of geometrical cross-coupling. The response of the rotor is observed to exhibit a rich variety of dynamical behavior including synchronous, sub-synchronous, quasi-periodic and chaotic vibrations. The transition from synchronous rotor response to chaos is via the torus breakdown route. As the rotor imbalance magnitude is increased, the synchronous rotor response undergoes a secondary Hopf bifurcation resulting in quasi-periodic vibration, which is characterized by a torus attractor. With further increase in the rotor imbalance magnitude, this attractor is seen to develop wrinkles and becomes unstable resulting in a fractal torus attractor. The fractal torus is eventually destroyed as the rotor imbalance magnitude is further increased. Quasi-periodic and frequency-locked sub-synchronous vibrations are seen to appear and disappear alternately before the emergence of chaos in the response of the rotor. The magnitude of rotor imbalance where sub-synchronous, quasi-periodic and chaotic vibrations are observed in this study, albeit being higher than the specified imbalance level for rotating machinery, may possibly occur with eroded rotors or in the event of a partial or entire blade failure.