Oil-whirl instability in an automotive turbocharger

This thesis is concerned with a theoretical investigation into the nonlinear dynamic behaviour of a turbocharger. Specifically the instabilities due to oil-whirl are examined. These are self-excited vibrations existing in the form of an in-phase whirl mode and a conical whirl mode. Waterfall plots were provided by Cummins Turbo-Technologies Ltd., Huddersfield, UK, based on test data using two different unbalance levels on a turbocharger. The test with the high unbalance indicated that there was shift in the sub-synchronous frequency to synchronous frequency between about 80,000 rpm to 130,000 rpm. The literature suggests that this self-excited vibration can be suppressed using forced excitation. Moreover, it is well known that the existence of limit cycles enables successful operation of a turbocharger. This limit cycle is a periodic motion attributed to the nonlinearity of the oil-film, other than the stable and the unstable equilibrium states predicted by the linear analysis. Hence, a nonlinear analysis is required to analyse the limit cycle and to determine the effect of synchronous excitation on it. In the literature a variety of parameters has been shown to influence the dynamic behaviour of a rotor-bearing system. To avoid over-complicated mathematical modeling, the influence of two such parameters: gyroscopic moment and shaft flexibility are first investigated in this thesis using linear stability theory to determine their significance. Effects of gyroscopic action are investigated using symmetric and asymmetric rigid rotors supported on short journal bearings with full-film using rigid and damped supports. In this thesis, the damper supported journal bearing is used to simulate the floating ring bearings that are commonly used in automotive turbochargers. The outer film of the floating ring bearing is treated as an external damper, since the ring is assumed not to rotate but only wobble giving the damping effect from the squeezing action. A gyroscopic coefficient, which is defined as the ratio of the polar to the transverse moment of inertia of the rotor, is introduced. The threshold value of this coefficient is determined to be 1 for the suppression of the conical whirl instability. The stability of the in-phase whirl mode is unaffected by this parameter. A flexible rotor mounted in floating ring bearings with full-film, is analysed to confirm that it behaves as a rigid body up to a speed of 100,000 rpm. Prior to the unbalance response study, a perfectly balanced rigid rotor supported by rigidly supported bearings is first analysed to determine the nonlinear behaviour of the in-phase whirl. To include the stiffness-like radial restoring force, an oscillating 2 ? -film cavitation model for the hydrodynamic bearings is used. The effect of a static load on the rotor is analysed to determine the nonlinear behaviour for a wide range of steady-state eccentricity ratios. A parameter plane separating the region of instability from that of stability is presented using linear analysis to determine the stability threshold at which the oil-whirl is initiated. The onset of oil-whirl phenomenon is shown to be the Hopf bifurcation. Particular emphasis is placed on examining the limit cycles (periodic oscillations) around the stability threshold. Reducing the nonlinear equation of motion to Poincare? normal form, the first Lyapunov coefficients are evaluated to show the change in the type of bifurcation from sub-critical bifurcation (disappearance of an unstable limit cycle) to super-critical bifurcation (appearance of a stable limit cycle) around the stability threshold. Such bifurcations are demonstrated through plots of orbits using numerical integration by the Runge-Kutta method. With some unbalance added to the rotor-system, waterfall plots are generated to simulate the response characteristics observed in the test data, by running-up the speed. After the Centre Manifold reduction, the equations of motions are averaged for analysis. Using a numerical and an analytical procedure, it is shown that the unbalance is more effective in the transient motion than in the steady-state condition. Unbalance introduces a reduction in the growth rate of whirl amplitude upto a certain optimum unbalance value, above which the effect is reversed. The mechanism behind this behaviour is shown to be the shift in phase caused by the unbalance at the start of whirling, when the dynamic forces are comparable with the unbalance force. This is due to the coupling effect of amplitude and phase in an unbalanced rotor system