Sommerfeld Number (Bearing Characteristic Number)

Arnold Sommerfeld's 1904 analysis of Reynolds' equation produced the group that every journal bearing is now designed against: $S = (r/c)^2 (\mu N / P)$. It is dimensionless, and that is the entire point. Two journal bearings with the same Sommerfeld number and the same length-to-diameter ratio behave identically — same eccentricity ratio, same dimensionless minimum film thickness, same friction variable — regardless of whether one is a wristwatch pivot and the other a turbine bearing a metre across. One family of charts, drawn once by Raimondi and Boyd in 1958, covers all of them.

Read the group as a contest. The numerator is what holds the surfaces apart: viscosity and speed, the two things that drag oil into the converging wedge and pressurise it. The denominator is what pushes them together: the unit load. And the clearance ratio squared, typically around $10^6$ since $r/c$ is usually near 1000, is the geometric leverage that makes a film only microns thick capable of carrying megapascals. A large $S$ means the film is winning, the journal runs nearly concentric, and Petroff's concentric analysis is nearly right. A small $S$ means the load is winning, the journal is pushed far over toward the bore, and the film at the bottom is getting thin.

Both ends have their own failure mode, which is the useful thing to know. At low $S$ — heavily loaded, slow, or thin oil — the minimum film approaches the roughness, the lambda ratio falls below 3 and then below 1, and the bearing moves into mixed and boundary lubrication where actual wear happens. At high $S$ — lightly loaded and fast — the bearing has the opposite problem: with little load to hold the journal in position, the shaft can go into OIL WHIRL, an instability in which it orbits the bore at about half shaft speed, and from there into oil whip when the whirl frequency meets a shaft natural frequency. This is not a hypothetical: it is why lightly loaded high-speed bearings are given lobes, pressure dams, elliptical bores or tilting pads, all of which exist to break up the circumferential symmetry the whirl depends on.

Two conventions have to be right or the number means nothing. $P$ is the load over the PROJECTED area, $2rL$ — diameter times length, the shadow of the journal — not the wrapped area of the bore, which is $\pi$ times larger. And $N$ is a rotational frequency, revolutions per second on this site, so 1800 rpm is 30. Older references frequently state $S$ with $N$ in rev/min and $P$ in psi, and the resulting numbers are not comparable with these — when a chart's axis and your calculation disagree by a factor near 60, that is why.

The loop that makes journal-bearing design iterative is worth spelling out, because it catches everyone once. $S$ depends on viscosity. Viscosity depends on the oil's temperature. The oil's temperature depends on the friction, and the friction is what $S$ is being used to find. So the calculation goes round: assume an operating temperature, read the viscosity off the oil's chart, compute $S$, get the friction variable and the flow from the charts, work out the heat balance between friction power in and oil flow plus housing losses out, find the temperature that results, and start again with that. It converges quickly, and skipping it is how a bearing gets specified with a VG 68 oil on the strength of a viscosity that only exists at 40 °C.

Finally, remember what $S$ does not include. It says nothing about the bearing MATERIAL, which has its own pressure limit — soft babbitt fatigues above a few megapascals however happy the oil film may be — nothing about misalignment, nothing about dirt, and nothing about the seconds at every start and stop when the film has not formed and the whole hydrodynamic theory does not apply.