Sommerfeld number of a hydrodynamic journal bearing

What does this quantity describe?

The Sommerfeld number So = pL·ψ²/(ηeff·ωeff) combines all key operating parameters of a hydrodynamic plain bearing into a single characteristic number. Bearings with the same Sommerfeld number are hydrodynamically similar, provided the width ratio and oil feed arrangement also match. Using TB 15-11, the relative eccentricity ε — needed for minimum film thickness — can be read from So.

The Sommerfeld number works much like the Reynolds number in fluid mechanics: a single dimensionless number combines several physical effects and lets very differently sized bearings be compared based on the same operating state.

Formula and variables

So = pL · ψ² / (ηeff · ωeff)

• So = pL · ψ² / (ηeff · ωeff)
• pL = So · ηeff · ωeff / ψ²
• ηeff = pL · ψ² / (So · ωeff)

Choose the inputs correctly

pL is the specific bearing load (from the corresponding calculator), ψ the relative clearance, ηeff the dynamic viscosity of the lubricating oil at effective operating temperature, and n the shaft speed, converted internally to the angular velocity ωeff.

How to use the calculator

First compute pL and ψ with their respective calculators, choose ηeff from the oil's viscosity-temperature chart at the expected operating temperature, and enter n to get So.

Worked example

pL = 2.5 MPa, ψ = 1‰, ηeff = 20 mPa·s and n = 1,500 rpm (ωeff ≈ 157.1 s⁻¹) give So ≈ 0.80.

Understand the result and units

So ≥ 1 with a relative eccentricity ε between 0.6 and 0.95 indicates trouble-free hydrodynamic operation. So < 1 together with ε < 0.6 points to a risk of unstable shaft position, while very high So values (So ≥ 10) with ε near 1 can indicate wear risk.

pL in MPa (N/mm²), ψ dimensionless, ηeff in mPa·s (or Pa·s), n in rpm (converted internally to s⁻¹); So itself is dimensionless.

From Sommerfeld number to relative eccentricity

The Sommerfeld number alone says nothing yet about the shaft's actual position within the bearing. Using a chart or table such as TB 15-11, the relative eccentricity ε — describing how close the shaft centre comes to the bearing shell — is read from So (together with the width ratio b/dL). Only with ε can the minimum film thickness h0 = 0.5·dL·ψ·(1−ε) be calculated — the Sommerfeld number is thus a necessary intermediate value, not an end result in itself.

Typical applications

The Sommerfeld number is the key figure for judging whether a plain bearing operates in the hydrodynamic regime, and serves as an intermediate value for determining minimum film thickness via the relative eccentricity.

Assumptions, limits and common mistakes

The formula applies to steadily loaded, full (360°) journal bearings with constant load magnitude and direction. ηeff must be used at the actual operating temperature, not at 20°C, since viscosity is strongly temperature-dependent.

Common mistake: A common mistake is plugging the shaft speed n directly into the formula instead of the derived angular velocity ωeff = 2π·n, and using a viscosity determined at room temperature instead of operating temperature.