Limit Cycle Vibration Analysis of a Long Rotating Cylinder Partially Filled With Liquid

Abstract

The dynamics of a long, stiff cylinder, flexibly suspended at the ends, partially filled with liquid, is analyzed for varying rotating speed. It appears from the analysis that in the absence of external damping, two distinct speed ranges with unstable whirl are present, as opposed to one instability region for the short cylinder, which has been analyzed by a number of authors (Kuipers, 1964; Wolf, 1968). This is in agreement with field experience with centrifuges, where several regions of instability are often encountered, each corresponding to a particular vibration mode. The results should also apply to a jet engine with oil accidentally trapped in the rotor, or any hollow rotor with liquid trapped in the cavity. When external damping is applied the linear theory predicts the rotor to be unstable at all speeds (Kuipers, 1964). This is clearly not in accordance with field experience, and other authors have suggested different types of nonlinear analysis that can give finite amplitude stable whirl or pulsating whirl (Berman et al., 1985). In the present analysis a simplified nonlinear analysis known as the hydraulic jump approximation is applied in the two unstable speed ranges predicted by the linear theory, and a stable whirl finite amplitude, dependent on the external damping, follows. It is argued that the amplitudes found this way should always be higher than those predicted by a more sophisticated analysis, and also higher than the amplitudes measured by other authors, so that the procedure described should give a safe worst case prediction of rotor whirl amplitudes for a given external damping. Finally, an experimental setup intended to verify the analysis in a quantitative way is presented.