Stability and Andronov-Hopf Bifurcation of Steady-State Motion of Rotor System Partly Filled With Liquid: Continuous and Discrete Models

Abstract

In this paper, a new method for investigation of dynamics of fluid-filled rotor systems is presented. The method consists of development of finite degrees-of-freedom (discrete) models for the rotor systems. The discrete models are physically justified and demonstrative. Being described by the system of ordinary differential equations, they allow one to employ powerful tools of the theoretical mechanics and oscillation theory. The method is applied to the case of the plane model of the rotor system partly filled with incompressible liquid. Both the continuous and discrete models are considered. The main attention is paid to the latter model. The discrete model consists of a disk symmetrically fixed on the shaft (Laval scheme), the ends of which are in viscoelastic bearings, and a ring sliding over the disk with friction. The centers of the disk and ring are elastically connected. The disk models the rotor, while the ring describes the liquid filling. When the ring is sliding over the disk surface, an interaction force arises that is diverted from the direction of the relative velocity at the contact points. It is demonstrated that an appropriate choice of the parameters of the discrete model allows one to determine the stability domain of the steady-state rotation of the rotor in the plane of the parameters of the shaft bearings with an excellent accuracy. It is found out that when the parameters overstep the limits of the stability domain, the Andronov-Hopf bifurcation occurs: a periodic motion of a kind of a circular precession arises from the steady-state rotation regime either “softly” or “hardly.”