Viscosity Required⚠ unverified

Understanding the Parameters

• Mean diameter $d_m$ — the pitch diameter through the rolling-element centres, approximated as the average of bore $d$ and outside diameter $D$. It sets the surface speed at the raceway: a larger bearing sweeps its contacts faster for the same rpm, which builds film more readily, so larger bearings need less viscosity — hence $d_m$ in the denominator.

• Rotational speed $n$ — the shaft speed. Faster rotation entrains oil into the contact more vigorously and builds a thicker film, so faster bearings also need less viscosity — $n$ is in the denominator too, with the larger exponent (0.7) because speed dominates film formation. This is the key counter-intuition: high-speed bearings are lubricated with thinner oils, not thicker.

• Required viscosity $\nu_1$ — the reference viscosity that would give a viscosity ratio $\kappa = 1$ (a just-adequate film). You then check that the chosen oil, at its actual operating temperature (viscosity falls steeply with temperature — see Viscosity Temperature), still meets or beats $\nu_1$. Aim for $\kappa \ge 1$–$4$ for good bearing life.

Registry note: the input $n$ is labelled dimensionless in the registry but physically must be rev/min for the empirical constant 4500 to be correct. Minor unit-label bug; flagged in Known Issues. Also, SKF publishes speed-range-dependent constants/exponents (e.g. a different form below/above ~1000 rpm); treat this single expression as a nomogram approximation.

Derivation (Approaching a Proof)

The formula is an empirical curve-fit to the classic SKF $\nu_1$ nomogram, which itself compresses elastohydrodynamic film theory into an engineering chart. The physical chain behind it:

1. EHL film thickness (Dowson–Higginson). The minimum film thickness in a rolling contact scales as $h_{\min} \propto (\eta_0\,u)^{0.7}\,\alpha^{0.54}\,R^{0.43}\,\dots$, i.e. it grows strongly with the product of viscosity $\eta_0$ and entrainment speed $u$.

2. Entrainment speed. The surface speed at the raceway is proportional to $n\,d_m$, so film thickness rises with $(n\,d_m)^{0.7}$ through the viscosity–speed product.

3. Adequate-film condition. Requiring the film to reach a fixed adequacy (the $\lambda$ ratio of film to surface roughness $\approx 1$) means the required viscosity must fall as speed and size rise, roughly as $\nu_1 \propto (n\,d_m)^{-0.7}\dots$. Fitting the actual SKF chart across the catalogue range yields the split exponents $n^{-0.7} d_m^{-0.5}$ and the constant $4500$.

So the exponents are not arbitrary — the $0.7$ echoes the Dowson–Higginson speed/viscosity exponent, and the whole expression is the inverse of "how much viscosity does it take to hold a film at this speed and size." It is a heuristic fit, accurate within the chart's intended range, not a first-principles identity.

Dimensional check (as an empirical fit). The constant $4500$ carries the units that make $\nu_1$ come out in mm²/s when $n$ is in rev/min and $d_m$ in mm; the expression is dimensionally consistent only with those specific units, as is normal for a fitted nomogram.

History and Development

The $\nu_1$ reference-viscosity method and its nomogram were developed by SKF and popularised through their General Catalogue and Bearing Maintenance Handbook from the 1970s onward, building on the elastohydrodynamic lubrication theory of Dowson and Higginson (1966) and Grubin's earlier film analysis. The viscosity ratio $\kappa = \nu/\nu_1$ became the standard input to the ISO 281:2007 $a_{\text{ISO}}$ life-modification factor, linking lubricant selection directly to predicted bearing life.

Related Concepts: Film Thickness Parameter, Viscosity Temperature, Elastohydrodynamic Film, Minimum Film Thickness, Bearing Life Modifier, Stribeck Curve

Notes: Compute $\nu_1$, then compare against the lubricant's actual viscosity at operating temperature to form $\kappa = \nu/\nu_1$. Empirical nomogram fit — use SKF's speed-range-specific formulas for precision, especially far from ~1000 rpm.