Plain bearing relative clearance

What does this quantity describe?

The relative clearance ψ = (dL−dW)/dL relates the absolute clearance between bearing bore and shaft diameter to the diameter itself. It is dimensionless and, for most hydrodynamic plain bearings, ranges from 0.5‰ (heavily loaded, slow) to 3‰ (lightly loaded, fast-running).

Think of ψ as the relative play between a shaft and its bore relative to overall size — similar to the clearance between a piston and its cylinder, but expressed relative to diameter rather than as an absolute dimension.

Formula and variables

ψ = (dL − dW) / dL

• ψ = (dL − dW) / dL
• dW = dL · (1 − ψ)
• dL = dW / (1 − ψ)

Choose the inputs correctly

dL is the bore diameter of the bearing shell, dW the outside diameter of the shaft journal in the bearing.

How to use the calculator

Enter dL and dW to get ψ. For sizing, instead enter ψ as a target value (e.g. from TB 15-8) and one of the two diameters to find the other.

Worked example

dL = 50 mm and dW = 49.95 mm give ψ = 0.05 mm / 50 mm = 0.001 = 1‰.

Understand the result and units

ψ = 1‰ falls in the mid-range of typical plain bearing designs. Too small a ψ increases seizing risk under thermal expansion; too large a ψ degrades the load-carrying capacity of the lubricant film.

dL and dW are given in mm, ψ is dimensionless and often expressed in ‰ (per mille).

Cold installation clearance versus operating clearance

The ψ computed here describes clearance at installation temperature (usually room temperature). In operation, shaft and bearing shell heat up at different rates depending on material and thermal path. The actual operating clearance ψB follows approximately from ψB = ψE + Δψ, with Δψ = (αL−αW)·(Jeff−20°C), where αL and αW are the expansion coefficients of bearing shell and shaft. For large temperature differences or dissimilar materials, operating clearance can differ substantially from the cold installation clearance.

Typical applications

Relative clearance is a core input for calculating the Sommerfeld number and minimum film thickness, and it directly influences the ISO fit selection for shaft and bearing bore.

Assumptions, limits and common mistakes

The ψ computed here is the cold installation clearance. Actual operating clearance differs due to differential thermal expansion of shaft and bearing shell and must be corrected separately (ψE or ψB) for operating temperature.

Common mistake: A common mistake is confusing the absolute clearance s = dL−dW with the relative clearance ψ without dividing by dL — the two quantities differ by several orders of magnitude.