Lambda Ratio⚠ unverified
Mechanical / Lubrication · Compute the lambda ratio of film thickness to surface roughness
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| h_min | hmin | m | 1.0 | Minimum film thickness |
| Ra | Ra | m | 1.0 | Composite surface roughness |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | λ | — | Lambda ratio (dimensionless). Returns positive infinity when ``Ra`` is non-positive |
The science & history
Understanding the Parameters
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Minimum film thickness $h_{min}$ — the thinnest point of the lubricant film, from hydrodynamic (Minimum Film Thickness) or elastohydrodynamic (Elastohydrodynamic Film) analysis. It is what the lubricant provides.
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Composite roughness $R_a$ — the surface finish of the two contacting parts combined, what the film must overcome. Smoother surfaces (smaller $R_a$) raise $\lambda$ for the same film — which is why precision-ground and superfinished races extend bearing and gear life.
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The regime thresholds — $\lambda$ maps directly to the Stribeck regimes:
- $\lambda < 1$ — boundary: asperities carry the load; high friction, adhesive/abrasive wear.
- $1 < \lambda < 3$ — mixed: partial contact; the surfaces wear in but life is finite.
- $3 < \lambda < 10$ — full film (hydrodynamic / EHL): surfaces fully separated; long, wear-free life.
- $\lambda > 10$ — thick film; no wear but higher viscous drag. Rolling bearings and gears are typically designed for $\lambda \gtrsim 2$–$4$ to ensure surface-fatigue life.
Derivation (Approaching a Proof)
The lambda ratio is a definition, but a physically motivated one: whether two rough surfaces touch through a lubricant film depends on how the film thickness compares to the height of the asperities. Model each surface as a random rough profile with RMS roughness $\sigma_1$, $\sigma_2$; the combined roughness is
$$\sigma = \sqrt{\sigma_1^2 + \sigma_2^2}$$
(variances add for independent random surfaces). The fraction of the load carried by asperity contact versus the fluid film depends on the ratio of the mean separation ($h_{min}$) to this combined roughness:
$$\lambda = \frac{h_{min}}{\sigma}.$$
When $\lambda \gg 1$ the film is many roughness-heights thick and asperity contact is statistically negligible (full film). When $\lambda \lesssim 1$ the separation is comparable to the asperity heights and a significant number of peaks penetrate the film (boundary/mixed). Greenwood–Williamson rough-surface contact theory makes this statistical statement precise; the lambda ratio is its practical distillation into one number. It ties directly to bearing/gear surface-fatigue life, which drops sharply below $\lambda \approx 1$.
Dimensional check. $[\lambda] = \text{m}/\text{m} = 1$ (dimensionless). ✓
History and Development
The specific film thickness (lambda ratio) as a lubrication-regime and bearing-life indicator was developed by Tallian and others at SKF in the 1960s from rolling-contact-fatigue data, building on Greenwood–Williamson (1966) rough-surface contact theory. It underpins the lubrication life-adjustment factors in ISO 281 bearing rating (Bearing Life Modifier) and is standard in tribology and gear/bearing design.
Related Concepts: Stribeck Curve, Minimum Film Thickness, Elastohydrodynamic Film, Friction Coefficient Boundary, Viscosity Required, Bearing Life Modifier
Notes: Use the composite roughness $\sqrt{R_{a1}^2 + R_{a2}^2}$ (see note). Regimes: $<1$ boundary, $1$–$3$ mixed, $>3$ full film. Design rolling contacts for $\lambda \gtrsim 2$–$4$.