Lubrication of Journal Bearings

Journal bearings (or sleeve bearings) are used to support rotating shafts. Hydrodynamic lubrication occurs when the rotating shaft pulls a wedge of lubricant into the converging space between the shaft and the bearing, creating sufficient fluid pressure to completely separate the surfaces.

1. Hydrodynamic Lubrication Principles

In full-film hydrodynamic lubrication, there is no metal-to-metal contact during normal operation. The load is supported entirely by the pressure within the fluid film.

Key parameters include:

• Clearance ($c$): The radial difference between the bearing and the journal.
• Viscosity ($\mu$): The dynamic viscosity of the lubricant.
• Rotational Speed ($N$): Typically expressed in revolutions per second (rev/s).
• Bearing Pressure ($P$): The load $W$ divided by the projected area (length $L$ $\times$ diameter $D$).

2. Petroff's Equation

Petroff's equation provides a simplified model for the coefficient of friction in a lightly loaded journal bearing. It assumes concentricity, which is an idealization since a loaded shaft always operates eccentrically.

The coefficient of friction $f$ according to Petroff is:

$f = 2\pi^2 \left( \frac{\mu N}{P} \right) \left( \frac{r}{c} \right)$

Where:

• $\mu$ is the dynamic viscosity (Pa·s).
• $N$ is the rotational speed (rev/s).
• $P$ is the nominal bearing pressure ($P = W / 2rL$) (Pa).
• $r$ is the journal radius (m).
• $c$ is the radial clearance (m).

3. The Bearing Characteristic Number (Sommerfeld Number)

The term $(\mu N / P)$ is highly significant and forms the basis of the Sommerfeld number ($S$), a dimensionless parameter used extensively in bearing design:

$S = \left( \frac{r}{c} \right)^2 \left( \frac{\mu N}{P} \right)$

The Sommerfeld number is used alongside design charts to determine key operating characteristics such as minimum film thickness, power loss, and required lubricant flow rate.