When a fluid, either liquid or gas, is constrained within the space between two concentric cylinders, one rotating and one stationary, the fluid in the clearance between the two cylinders will be set into circumferential motion. This can happen in rotating machinery within the fluid-lubricated bearings, seals, around pump impellers, or in any fluid-filled gap between the rotor and the stator.
In a hydrodynamic, fluid film bearing, the fluid velocity at the surface of the bearing is zero, while the fluid velocity at the surface of the rotor is equal to the rotor surface speed. The fluid near the rotor surface moves at a slightly slower velocity that continues to decrease with the distance from the rotor and reaches zero at the bearing surface. It is easy to see that the fluid must have some overall average velocity which is less than the rotor speed, and that the faster the rotor turns, the faster the average fluid velocity must be. Previous work has shown that, in the absence of any form of tangential pre-swirl from fluid injection or other forces, that the average fluid velocity in the annular space will typically vary from 39% to 48% of operating speed; i.e. from 0.39X to 0.48X.
If a radial load is applied to a rotor operating in the center of a hydrodynamic bearing, the rotor will be displaced from the center of the bearing. The reduced clearance on one side will create a flow restriction with a corresponding increase in fluid pressure which exerts a force on the rotor.
This fluid force can be separated into a radial part, which points back toward the center of the bearing, and a tangential part, which acts at 90° to the radial force. Both the radial and tangential fluid forces are related to the displacement of the rotor from the center of the bearing. Thus, the fluid force acts like a spring with a complex Dynamic Stiffness.
The radial part
The radial part of the Dynamic Stiffness is referred to as the Direct Stiffness Term, and the tangential part (which acts at 90° to the Direct Stiffness Term) is referred to as theQuadrature Stiffness Term.
The Quadrature Stiffness Term is comprised of both a stabilizing component proportional to the vibration precession frequency and damping; i.e. the shock absorber effect (rotor pushing on the fluid) and a destabilizing component proportional to the average fluid velocity (cross-coupled effect; i.e. fluid pushing on the rotor).
It is this complicated spring effect that provides the primary support for the rotor. The rotor and the fluid pressure "wedge" move until the fluid forces in the bearing exactly balance the summation of radial loads applied to the rotor and the rotor settles into an equilibrium position. Any disruption of the force balance within this pure circular system will allow the minimum oil film to circumnavigate the bearing in a forward manner. (Refer to page 423 of reference ii.)
Fluid Induced Instabilities (oil whirl / oil whip) are a form of self-excited vibration where the inherent force balance within the bearing, seal, etc., has been disrupted thus allowing the tangential force to push the rotor in a forward circular direction.
Oil whirl is characterized as a rigid body mode with maximum displacement occurring at the bearing or seal where the fluid induced instability is occurring. As rotative speed changes, the frequency of the oil whirl instability tracks at a constant percentage of rotative speed, i.e. typically from 0.39X to 0.48X.
Although theoretically the vibration amplitude would grow unbounded, in reality it will increase until a limit cycle vibration is reached where the shaft is restrained by the clearance of the bearing, seal, or other stator parts. The tangential force will continue to push the rotor circumferentially in the forward direction at a frequency equal to the average fluid velocity ratio with a peak to peak displacement amplitude essentially equal to the diametral clearance of the bearing or seal within which the fluid induced instability is occurring.
Oil whip is characterized as a flexible body mode due to re-excitation of the 1st critical speed with maximum displacement, for a "beam style" rotor, occurring in the mid-span of the rotor. Oil whip can be a very destructive mechanism due to the much larger vibration amplitudes in the mid-span of the rotor. Oil whip can occur on machinery operating above twice the 1st critical speed and usually begins as oil whirl and then transitions to oil whip as rotative speed increases and the frequency of the instability locks on to the 1st critical speed and re-excites that mode.
Main vibration characteristics of Fluid Induced Instability:
(The author is Principal Engineer, Metrix SETPOINT Contition Monitoring Group, Minden, NV)
For additional information on the topic, the reader is referred to the following texts: