Journal Bearing Load Capacity⚠ unverified
Mechanical / Lubrication · Compute the journal bearing load capacity from the Sommerfeld number
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| mu | μ | Pa.s | 1.0 | Dynamic viscosity of the lubricant |
| N | N | 1/s | 1.0 | Rotational speed |
| r | r | m | 1.0 | Journal radius |
| L | L | m | 1.0 | Bearing length |
| c | c | m | 1.0 | Radial clearance |
| S | S | — | 1.0 | Sommerfeld number (dimensionless) |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | W | N | Bearing load capacity, in newtons (N) |
The science & history
Understanding the Parameters
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Viscosity $\mu$ and speed $N$ — thicker oil and faster rotation build more film pressure, so load capacity rises with $\mu N$. This is why a bearing can fail to lift its load at startup (low speed) yet float it easily at running speed — the film only forms once the shaft spins.
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Radius $r$ and length $L$ — bigger bearings carry more load (larger pressurised area). The projected area $2rL$ is what the film pressure acts on.
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Clearance $c$ — enters as $1/c^2$: a tighter clearance dramatically raises load capacity (thinner film, higher pressure), but too tight risks contact and overheating. Clearance is the key trade-off variable, typically $r/c \approx 500$–$1000$.
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Sommerfeld number $S$ — appears in the denominator: for a given geometry and lubricant, a heavier load corresponds to a smaller $S$ (thinner film, more eccentric). So specifying the operating $S$ (from a design chart at the required minimum film thickness) fixes the allowable load.
Derivation (Approaching a Proof)
Start from the definition of the Sommerfeld Number, $S = \dfrac{\mu N}{P}\left(\dfrac{r}{c}\right)^2$, where the unit load (projected pressure) is $P = \dfrac{W}{2rL}$ (load over the projected area, journal diameter × length). Substitute:
$$S = \frac{\mu N}{\big(W/(2rL)\big)}\left(\frac{r}{c}\right)^2 = \frac{2 \mu N r L}{W}\cdot\frac{r^2}{c^2} = \frac{2\mu N r^3 L}{W c^2}.$$
Solve for the load:
$$W = \frac{2\mu N r^3 L}{S\, c^2}.$$
This is the dimensionally consistent load capacity ($[W] = \text{N}$). The registry's $W = \mu N r^2 L/(S c^2)$ drops the factor $2r$ (giving N/m) — the same physics, missing the diameter scaling. The essential content is that load capacity is set by the viscous film pressure: it grows with viscosity, speed, size, and inverse-square clearance, and falls with the Sommerfeld number that characterises the operating point.
Dimensional check (consistent form). $\dfrac{\mu N r^3 L}{S c^2} = \dfrac{(\text{Pa}\cdot\text{s})(1/\text{s})\,\text{m}^3\,\text{m}}{\text{m}^2} = \text{Pa}\cdot\text{m}^2 = \text{N}$. ✓ (registry form is missing one length.)
History and Development
Journal-bearing load capacity from the Sommerfeld/Reynolds analysis is standard hydrodynamic-bearing design (Sommerfeld 1904; Raimondi–Boyd charts 1958; Shigley). Full-film journal bearings support enormous loads (turbine, engine, generator shafts) with negligible wear because there is no metal contact — the entire load rides on a self-generated oil film, the reason they are the bearing of choice for large high-speed rotating machinery.
Related Concepts: Sommerfeld Number, Petroff Friction, Minimum Film Thickness, Hydrodynamic Film Pressure, Stribeck Curve
Notes: Registry form is dimensionally short a diameter factor (yields N/m) — use $W = 2\mu N r^3 L/(S c^2)$ for a true load. Load capacity $\propto \mu N$, $\propto 1/c^2$; falls with $S$. No metal contact in full-film operation.