Mode coupling of a flexible rotor supported by a superconducting magnetic bearing due to the nonlinearity of electromagnetic force

Recently magnetic bearings have been applied to many engineering products, such as flywheels and turbo-chargers. Now development of flywheel energy storage systems using superconducting magnetic bearings has reached its validation phase. In most of these systems, active magnetic bearings, as auxiliary bearings, are used to reduce vibration of the rotor. However, because of its high rotational speed, the spill-over effect of elastic deformation of the rotor has become a problem. This complicated phenomenon can be partly caused by nonlinear coupling between plural vibration modes. This study investigates nonlinear mode coupling of a flexible rotor system. We developed an essential model of a rotor with an elastic shaft supported by a superconducting magnetic bearing. The equations of motion of the rotor were derived by taking into account the nonlinearity of the electromagnetic force and also by modeling the elastic shaft as a continuum body. The obtained equations show nonlinear coupling between the first mode and second mode of elastic vibration, and also suggest that mutual resonance can occur between these modes due to a cubic nonlinearity if the ratio of their natural frequencies is one to three. This prediction was consistent with our numerical results. The above findings may be useful for considering avoidance of the spill-over effect.