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# Petroff's Law - Torque required to shear the lubricant film (for small radial loads)

## Description

In the design of fluid bearings, the Sommerfeld number (S), or bearing characteristic number, is a dimensionless quantity used extensively in hydrodynamic lubrication analysis. The Sommerfeld number is very important in lubrication analysis because it contains all the variables normally specified by the designer.

The Sommerfeld number is named after Arnold Sommerfeld (1868–1951).

Derivation:

Petroff’s Law

Petroff’s method of lubrication analysis, which assumes a concentric shaft and bearing, was the first to explain the phenomenon of bearing friction. This method, which ultimately produces the equation known as Petroff’s Law, is useful because it defines groups of relevant dimensionless parameters, and predicts a fairly accurate coefficient of friction, even when the shaft is not concentric.

Considering a vertical shaft rotating inside a bearing, it can be assumed that the bearing is subjected to a negligible load, the radial clearance space is completely filled with lubricant, and that leakage is negligible. The surface velocity of the shaft is: U = 2*π*r*N where N is the rotational speed of the shaft in rev/s.

The torque required to shear the film is shown here for small radial loads where the frictional drag force can be considered equal to the product fW.

Related formulas

## Variables

 T torque required to shear the lubricant film (small radial loads) (N*m) r shaft radius (m) f coefficient of friction (dimensionless) l bearing length (m) P radial load per unit of project bearing area (pressure) (pascal)