Plain bearing Sommerfeld number
Calculates the dimensionless Sommerfeld number from specific load, relative clearance, viscosity and speed -- the central figure of hydrodynamic plain-bearing theory, here in an explicitly named convention.
Formula
So = pL * psi^2 / (eta * omega), omega = 2*pi*n
The Sommerfeld number combines load, geometry and lubrication state into a dimensionless figure that, together with the width-to-diameter ratio b/d, determines eccentricity and minimum film thickness of a hydrodynamic plain bearing via characteristic functions.
What this model does not show
IMPORTANT: NormCalc explicitly uses the convention So = pL*psi^2/(eta*omega) here (Roloff/Matek style). Other sources -- including some ISO 7902 editions -- define the Sommerfeld number as the reciprocal or with different factors. A quoted Sommerfeld number value is comparable only within the same convention. This number alone also gives no pass/fail verdict: eccentricity and minimum film thickness require the full ISO 7902-2 characteristic functions, which additionally depend on the width-to-diameter ratio.
Worked example
p = 2.5 MPa = 2.5e6 Pa, psi = 0.001, eta = 30 mPa*s = 0.03 Pa*s, n = 1,500 rpm. Angular velocity omega = 2*pi*1,500/60 ~ 157.08 rad/s. So = 2.5e6 Pa x 0.001^2 / (0.03 Pa*s x 157.08 rad/s) = 2.5e6 x 1e-6 / 4.712 = 2.5 / 4.712 ~ 0.531. This value is usable for further evaluation per ISO 7902-2 only together with the stated convention and the b/d ratio.
Common mistakes
- Comparing a Sommerfeld number quoted from another source directly with the value computed here, without checking the convention used.
- Using eta at room temperature instead of at the actual operating film temperature.
- Concluding directly from a single Sommerfeld number that a bearing is 'safe' or 'unsafe' without consulting the characteristic functions (which also depend on b/d).
Frequently asked questions
Which Sommerfeld number convention does NormCalc use?
So = pL*psi^2/(eta*omega) with pL in Pa, psi dimensionless, eta in Pa*s and omega = 2*pi*n in rad/s -- this matches the definition common in Roloff/Matek. Other sources may use the reciprocal or different reference quantities.
Why do Sommerfeld number definitions differ between sources?
Historically, different authors and standard series introduced the dimensionless figure with different normalization (e.g. taking the reciprocal or a different reference pressure). It is the same underlying physical idea but different mathematical conventions.
What does a high or low Sommerfeld number mean?
Roughly, a low Sommerfeld number (in this convention) tends to indicate a thicker film for a given b/d ratio, a high one a thinner film -- but the precise meaning requires the ISO 7902-2 characteristic functions, not the number alone.
Is the Sommerfeld number alone sufficient for approval?
No. It is an intermediate quantity. Actual eccentricity epsilon and minimum film thickness h_min only follow from the full ISO 7902-2 characteristic functions, which additionally depend on the width-to-diameter ratio b/d.
Which viscosity should I use?
The dynamic viscosity at the actual operating film temperature, not at room temperature or the datasheet reference temperature. This can be determined via calculators 51 and 52.
What comes after the Sommerfeld number?
With an eccentricity epsilon determined from the ISO 7902-2 characteristic functions, minimum film thickness (calculator 59) can be calculated.