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TECHNICAL PAPERS

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

[+] Author and Article Information
N. V. Derendyaev

Nizhny Novgorod State University, Faculty of Computational Mathematics and Cybernetics, Gagarin Avenue 23, 603022 Nizhny Novgorod, Russiader@tudm.unn.ac.ru

A. V. Vostrukhov

Mechanical Engineering Institute of Russian Academy of Sciences, Nizhny Novgorod branch, Belinskogo 85, 603024 Nizhny Novgorod, Russiavostroukhov@yahoo.co.uk

I. N. Soldatov

Mechanical Engineering Institute of Russian Academy of Sciences, Nizhny Novgorod branch, Belinskogo 85, 603024 Nizhny Novgorod, Russiawvs@dynamo.nnov.ru

J. Appl. Mech 73(4), 580-589 (Nov 04, 2005) (10 pages) doi:10.1115/1.2164514 History: Received April 29, 2004; Revised November 04, 2005

## 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.”

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## Figures

Figure 1

Cross section of rotor partly filled with liquid

Figure 2

Noninertial reference system

Figure 3

D-decomposition of the plane of the parameters of the bearings

Figure 4

D-decomposition of the plane of the parameters of the bearings: expanded scale

Figure 5

Effect of the Ekman number and the fluid-fill ratio on D-decomposition of the plane of the parameters of the bearings

Figure 6

Effect of the Ekman number and the fluid-fill ratio on the boundaries of D2(0) domain

Figure 7

Discrete model of the rotor with liquid

Figure 8

D-decomposition of the plane of the parameters of the bearings for the continuous (solid line) and the discrete (dashed line) models: (a) boundaries of D2(0) (expanded scale); (b) D1(0) domains

Figure 9

General view of D-decomposition of the plane of the parameters of the bearings for the continuous (solid line) and the discrete (dashed line) models

Figure 10

“Dangerous” and “safe” intervals of the D2(0) stability domain

Figure 11

“Dangerous” and “safe” intervals of the D1(0) stability domain

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