Application of Hopf bifurcation theory to rotor-bearing systems with consideration of turbulent effects J.K. Wang, M.M. Khonsari * Department of Mechanical Engineering, Louisiana State University, 2508 CEBA Building, Baton Rouge, LA 70803, USA Received 24 January 2005; accepted 4 July 2005 Available online 31 August 2005 Abstract This paper shows that the steady state and the dynamic characteristics of a rotor-bearing system with turbulent effects can be conveniently evaluated by applying the Hopf bifurcation theory to the equations of motion of a rigid rotor symmetrically supported by two identical fluid- film journal bearings. Results are presented for stiffness and damping coefficients and stability characteristics of a journal bearing, which take turbulence into consideration. q 2005 Elsevier Ltd. All rights reserved. Keywords: Hopf bifurcation theory; Rotor-bearing system; Turbulent effects 1. Introduction The application of Hopf bifurcation theory (HBT) for the stability analysis of a rigid rotor symmetrically supported by two identical, infinitely long journal bearings was presented by Myers [1] in 1984. He identified the existence of three separate regions in the parameter space of the steady state eccentricity ratio. These three regions, from low to high values are: subcritical bifurcation, supercritical bifurcation, and again subcritical bifurcation. A similar analysis, but using the infinitely short bearing theory formulated in Cartesian coordinates, was published by Hollis and Taylor [2]. They reported that, in short journal bearings, the subcritical bifurcation exists if the modified Sommereld number is greater than a critical value. Otherwise, the supercritical bifurcation exists. Sundararajan and Noah [3,4] analyzed a more general case of finite journal bearings using HBT and discussed the effects of different length/diameter ratios on the subcritical bifur- cation and supercritical bifurcation regimes. Deepak and Noah [5] provided experimental evidence that verified the subcritical bifurcation of a single disk rotor supported on a short journal bearing. All of the abovementioned papers were based on the assertion that oil flow remains laminar. Research dealing with the effect of turbluence on performance of bearings has been primarily limited to steady-state, with the exception of a series of notable contributions by Hashimoto et al. [6–10] who investigated dynamic effects using the traditional analytical methods. The purpose of this paper is to provide additional insight into the application of HBT to the stability analysis of a rotor-bearing system with consideration of the turbulent effects. The analysis pertains to a rotor-bearing system, which consists of a rigid rotor symmetrically supported by two identical short journal bearings. The outline of this paper is as follows. First, a brief description of Hopf bifurcation theory is presented. Then, HBT is applied to the equations of motion of the rotor- bearing system with provision for turbulence. Appropriate equations are derived. These equations describe the relationships between the eccentricity ratio 3, the Sommer- feld number S and the attitude angle, the linearized dynamic coefficients  k ij and  b ij (i,jZ3,f), the threshold speed  u st , and the whirl frequency ratio U. A series of results is presented that illustrates how the shape, size and stability of the periodic solutions of the journal orbit with consideration of turbulent effects can be easily predicted using HBT. The paper is concluded with the presentation of the general Tribology International 39 (2006) 701–714 www.elsevier.com/locate/triboint 0301-679X/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.triboint.2005.07.031 * Corresponding author. Tel.: C1 225 578 9192; fax: C1 225 578 5924. E-mail address: khonsari@me.lsu.edu (M.M. Khonsari).