Engineering
Sommerfeld Journal-Bearing Number (S) & Petroff Friction Torque
Calculate dimensionless hydrodynamic bearing Sommerfeld Number S = ((r / c)^2) * ((mu * N) / P) for Raimondi-Boyd eccentricity charts and Petroff's concentric viscous friction coefficient & shaft torque.
Design hydrodynamic crankshaft and turbine oil-film bearings with S!
Physicist Arnold Sommerfeld (1904) collapsed every geometric and operating variable of a hydrodynamic journal bearing—journal radius r (m), radial clearance c (m, typically r/c ~ 500–1,000), oil dynamic viscosity mu (Pa*s), shaft rotational speed N (in rev/s = RPM / 60), and projected bearing pressure P = Load / (2 * r * L) (Pa)—into a single dimensionless Bearing Characteristic Number:
S = ((r / c) ^ 2) * ((mu * N) / P)! Entering Raimondi-Boyd design charts with S determines minimum oil-film thickness h_0 and eccentricity ratio! And for lightly loaded / high-speed bearings (large S -> concentric shaft), Nikolay Petroff's (1883) analytical law gives Friction Coefficient f = 2 * (PI() ^ 2) * ((mu * N) / P) * (r / c) and Viscous Drag Torque T = (4 * (PI() ^ 2) * mu * N * L * (r ^ 3)) / c (N*m)!
| Journal Bearing Metric (r=40 mm, c=80 um, L=60 mm, N=25 rps, P=1.5 MPa) | Formula | mu = 0.03 Pa*s (Hot Oil) | mu = 0.09 Pa*s (Cold Oil) |
|---|---|---|---|
| Clearance Ratio (r / c) | 0.040 / 0.00008 | 500 | 500 |
| Sommerfeld Number S | ((r/c)^2) * (mu * N / P) | 0.1250 | 0.3750 (Thicker film!) |
| Petroff Friction Torque (N*m) | (4 * PI()^2 * mu * N * L * r^3) / c | 1.4212 N*m (223 W loss) | 4.2637 N*m (3x drag!) |
Dimensionless Hydrodynamic Sommerfeld Bearing Number S
Evaluates ((r / c)^2) * ((mu * N_rps) / P_Pa) from radius B2 (m), radial clearance B3 (m), viscosity B4 (Pa*s), speed B5 (rev/s), and unit load B6 (Pa).
=IF(OR(B3 <= 0, B6 <= 0), "Invalid Clearance or Bearing Load", ROUND(((B2 / B3) ^ 2) * ((B4 * B5) / B6), 4))Petroff Concentric Journal-Bearing Drag Torque (N*m)
Computes viscous oil-shear torque across bearing length B7 (m).
=ROUND((4 * (PI() ^ 2) * B4 * B5 * B7 * (B2 ^ 3)) / B3, 4)Try the example
Use the original values or change them to test your understanding.