Petroff's Equation (Journal Bearing Friction Torque)

Nikolai Petroff published this in 1883, three years before Osborne Reynolds explained why journal bearings carry load at all, and it remains the right first calculation for one. The model is as simple as a bearing model can be: assume the journal runs CONCENTRIC in the bore, so the oil film has the same thickness $c$ all the way round. Then the film is just a sheared layer. The journal surface moves at $U = 2\pi r N$, the bore stands still, the velocity gradient is $U/c$, and Newton's viscosity law gives a shear stress $\tau = \mu U/c$ everywhere. Multiply by the rubbing area $2\pi r L$ to get the friction force, multiply by $r$ to get a torque, and out comes $T_f = 4\pi^2 \mu N r^3 L / c$.

The assumption is false and the answer is useful anyway, which is a combination worth understanding. A loaded journal does not run concentric — it is pushed sideways until the film forms a converging wedge, and the pressure generated in that wedge is what supports the load. Reynolds' equation describes it, Petroff's does not, and a bearing with zero eccentricity would carry zero load. So Petroff is exactly right for a bearing with no load and progressively optimistic as the load grows: the friction it predicts runs perhaps 10 to 20 % below a moderately loaded bearing and further below a heavily loaded one. It is a floor.

What makes it valuable is not the number but the STRUCTURE. Rearrange Petroff's result as a coefficient of friction and you get $f = 2\pi^2 (\mu N/P)(r/c)$, where $P$ is the unit load. Everything on the right is grouped into two dimensionless factors — the clearance ratio $r/c$, and the duty parameter $\mu N/P$ — and their product is what the whole of journal-bearing design turns out to depend on. That is where the Sommerfeld number comes from, and it is why an 1883 result derived from a wrong assumption still frames the subject.

Three input traps, in the order they are usually fallen into. The clearance is RADIAL. It is half the diametral clearance, which is what a machinist measures with a bore gauge and a micrometer, and using the diametral figure halves every answer on this page. The viscosity must be at the running temperature. An ISO VG grade is quoted at 40 °C, oil viscosity roughly halves for every 20 °C rise, and a bearing sitting at 80 °C is shearing something like a quarter of the viscosity printed on the drum. Since the friction this equation predicts is what heats the oil in the first place, the honest calculation is iterative: guess a temperature, find the friction, find the heat, check the temperature, go round again. And $r$ is a radius, not a diameter — it appears cubed, so that error is a factor of eight.

That cube is itself the design lesson. Friction torque goes as $r^3$, so a journal 10 % larger drags 33 % harder for the same oil, speed, length and clearance. Multiply the torque by $2\pi N$ for the power, and the loss goes as the SQUARE of speed: double the shaft speed and the bearing puts four times the heat into the oil. High-speed journal bearings are therefore designed around their cooling before anything else, and oil flow through them is sized to carry heat away rather than merely to keep the film supplied.

A last practical note on clearance, which is the one variable a designer genuinely gets to choose. Opening it up cuts the drag and lets more oil through to remove heat; closing it down thickens the film's load-carrying capacity and controls the shaft position better. Neither direction is free, and a drawing specifies a tolerance BAND rather than a value — so the bearing has to behave at both ends of it, hottest at the tight limit and thinnest-filmed at the loose one.