rotating machineries, the unbalance is the most common source of excitation, which comes from rotors. Similarly, oil-film bearings are possibly the most common source of instability in such rotating machineries. The oil-whirl comes basically from the non-linear behaviour of bearing forces and to analyze the behaviour a non-linear analysis is essential. Oil-whirl instability is also known as ‘half-speed whirl' because the frequency of whirl (vibrations) which are set up is often just below half the shaft rotational frequency (typically 0.46-0.48 times of shaft rotational frequency). This instability tends to occur only in lightly loaded oil-film bearings

operating at a very small eccentricity ratio (  , where er is the eccentricity of the rotor centre with respect to the bearing centre and cr is the radial clearance). It is an extremely dangerous condition because bearing load-carrying capacity decreases and results in very high whirl amplitude and consequently the destruction of the bearings is a possibility. Let us consider a fluid-film bearing in which journal is rotating with frequency, ω , and whirls around the bearing clearance (i.e., er) at frequency as shown in Figure 11.4. Because the bearing is lightly loaded (so operates at only a small eccentricity er in the first instance) the variation in fluid pressure around the bearing circumference may be considered to be negligible so that the only fluid flows around the bearing is that which is induced by the rotation of the journal. The lubricant flow rate into, and out of, the dotted wedge-shaped are as shown in Figure 11.4(a), is given by

(a) Journal and lubricant in a bearing during oil whirl (b) The whirl orbit of the shaft centre

Figure 11.4 Oil whirl in fluid film bearings

Since journal is whirling within the bearing clearance with some frequency  , the

tangential velocity of the journal center will be   as shown in Figure 11.4(b). So the volume of dotted area, per unit length of the bearing must be increasing at a rate given by

where  is the tangential velocity of journal centre and is the shaded area of journal per unit length of bearing. The volume flow rate must be provided by the net lubricant flow into the dotted area under consideration, so that we may write

or

(11.8)

which gives  . So the frequency of whirl is half the rotational frequency of journal. The deviation from the actual case (0.46 ω to 0.48 ω ) is due to the assumption made in the analysis regarding no flow due to pressure variation across the circumference of the bearing.

Example 11.1 Let us consider two identical bearings which are symmetrically supporting a light symmetrical rotor at its ends. Through measurement the following data were found: the

bore of the journal bearing is 3 cm with the radial clearance of 5  , and the rotor spin speed

is 3000 rpm. The flow measurement were  m . If the rotor is under the half-speed whirl, obtain the eccentricity ratio of the rotor centre in the journal.

Solution : We have the following data

Hence the eccentricity ratio can be written as

Hence, it has very low eccentricity ratio.