Non-linear stability analysis of floating ring bearings using Hopf bifurcation theory

Abstract

Floating ring bearings are used to support and guide rotors in several high-speed rotating machinery applications. They are usually credited for lower heat generation and higher vibration suppressing ability. Similar to conventional hydrodynamic bearings, floating ring bearings may exhibit unstable behaviour above a certain stability critical speed. Linear stability analysis is usually applied to predict the stability threshold speed. Non-linear stability analysis, however, is needed to predict the presence and the size of stable limit cycles above the stability threshold speed or unstable limit cycles below the stability critical speed. The prediction of limit cycles is an important step in bearing stability analysis. In this article, a non-linear dynamic model is derived and used to investigate the stability of a perfectly balanced symmetric rigid rotor supported by two identical floating ring bearings near the critical stability boundaries. The fluid film hydrodynamic reactions of the floating ring bearings are modelled by applying the short bearing theory and the half Sommerfeld solution. Hopf bifurcation theory is then utilized to determine the existence and the approximate size of stable and unstable limit cycles in the neighbourhood of the stability critical speed depending on the bearing design parameters. Numerical integration of the non-linear equations of motion is then carried out in order to compare the trajectories obtained by numerical integration to those obtained analytically using Hopf bifurcation analysis. Stability boundary curves for typical bearing design parameters have been decomposed into boundaries with supercritical stable limit cycles and boundaries with subcritical unstable limit cycles. The shape and size of the limit cycles for selected bearing parameters are presented using both analytical and numerical approaches. This article shows that floating ring stability boundaries may exhibit either stable supercritical limit cycles or unstable subcritical limit cycles predictable by Hopf bifurcation.

Nonlinear stability analysis and numerical continuation of bifurcations of a rotor supported by floating ring bearings

“…This means that lower lubricant viscosity and larger journal mass lead to an increase in the stability threshold speed value as seen in references. 4,23…”

“…Rotors in floating ring bearings modeling presented in the literature consider either linear 2,20 or nonlinear models. 1,4,11,17,19 Since these latter are more suitable for the evaluation of the rotors supported by FRBs dynamics, a nonlinear model of a perfectly balanced symmetric rigid rotor supported by two identical floating ring bearings is considered in this study. Both bearings have the same motions, thus only one bearing is examined.…”

“…To set up the motion equations of the considered rotor-bearing system, the formulation used by references 1 and 4 is closely adopted. Applying Newton’s second law, the nondimensional form of the system of equations of motion established in polar coordinates may be given as 4 …”

“…The approximate expression corresponding to the asymptotic value of the floating ring angular speed ratio at high rotor speeds is used 4 …”

“…Amamou and Chouchane 4 used analytical methods to determine the nonlinear stability of a rotor-bearing system supported by floating ring bearings. They applied the Hopf bifurcation theory to identify the existence and the approximate size of stable and unstable limit cycles in the neighborhood of the stability threshold speed depending on the bearing design parameters.…”