Hopf Bifurcation to Limit Cycles in Fluid Film Bearings

Abstract

The nonlinear response of a cylindrical journal bearing operating close to the critical speed stability boundary is studied in this paper. Using linear stability theory, the value of the critical variable (usually speed) at the point of loss of stability is obtained and shown to agree with results of previous researchers. Using Hopf bifurcation analysis, parameters for determining the behavior close to this point are obtained. Analytically, these parameters prove that the system can exhibit stable limit cycles for speeds above the critical speed. Such supercritical limit cycles only exist for a narrow range of values of modified Sommerfeld number. In other cases, subcritical limit cycles are predicted. The results are supported by numerical simulation. The results show why it may be difficult to observe supercritical limit cycles in test rigs.

The effects of unbalance on oil whirl

“…Figures 4,5,7, and 8, are obtained from "time 2,-7" maps of equation (10) while Figurcs 9 and 11 are obtained from "time 4rr" maps. The center manifold theorem assures that the rotor inotions governed by equation (4) in the neighborhood of the threshold speed are topologically equivalent to those oi1 the two dimensional center manifold governed bv equation (10). The stability types of steady state solutions between the Poincard maps and equation (10) and between equation (10) and equation (4) also correspond.…”

Dynamic analysis of short and long journal bearings in laminar and turbulent regimes, application in critical shaft stiffness determination

“…Here, the non-dimensional running speed ¯ is considered as the system parameter when all other parameters of the rotor-bearing system are fixed. According to HBT, if the parameter ¯ in the equations of motion ( 24 ) and ( 25 ) becomes greater than some critical value ¯ cr , an isolated stationary point x s ( ¯ ) will lose its linear stability by having a complex conjugate pair of eigenvalues of the linearized system crossing the imaginary axis of the complex plane [27] . Hence, this critical value ¯ cr is the non-dimensional form of the threshold speed of instability of a rotor-bearing system.…”

“…The Hopf bifurcation theory describes the birth of a periodic solution of a system whose behavior is defined by the ordinary differential equations ˙ x = f ( x , μ) , ( x ∈ R n ) . A Hopf bifurcation occurs, when as the system parameter μ varies, a single complex conjugate pair of eigenvalues of the linearized system equations become purely imaginary (in the process of crossing the imaginary axis) [27] . The conditions that must be met for the Hopf bifurcation theory are [28] : 1 , .…”

Analytical bifurcation analysis of a rotor supported by floating ring bearings

“…At n c1 the center position of the rotor loses its stability and a stable limit-cycle is born ("oil whirl"). The stability loss of the equilibrium position of a rigid rotor at the speed n c1 has been widely investigated both analytically and numerically by many authors [3,7,9]. As n trespasses n c2 , this first limit cycle becomes unstable and the system jumps to a second limit-cycle of higher amplitude ("oil whip") which is limited only by the bearing clearance.…”