Lubrication and viscosity

Why the film matters

A rolling contact runs on an oil film a fraction of a micrometre thick. Where the film separates the surfaces, the steel sees only the Hertz pressure; where roughness peaks break through it, surface distress and local stress raisers shorten the life. ISO 281 captures this through the viscosity ratio κ in aISO (see Rating life (ISO 281)). The app also computes the film itself and the film parameter λ, which tell you how close the contacts are to metal contact.

All numbers here come from the bearing-6207 template: ν40 = 100 mm²/s, ν100 = 11 mm²/s, 70 °C, 2 900 rpm, default film details (pressure–viscosity 20 1/GPa, density 880 kg/m³ at 15 °C, raceway Rq 0.15 µm, ball Rq 0.05 µm, oil bath).

Viscosity at the operating temperature

Oil viscosity falls steeply with temperature. The Walther equation, the basis of ASTM D341 viscosity charts, makes the relation a straight line on double-logarithmic paper:

$\log \log (\nu +0.7)=A-B\cdot \log T$, with ν in mm²/s and T in kelvin.

The two data-sheet values ν40 and ν100 fix A and B, and the line gives ν at any temperature. For the template oil that is 27.0 mm²/s at 70 °C. The dynamic viscosity $\eta =\nu \cdot \rho$ uses the density corrected to the temperature: 22.8 mPa·s.

In the inspector, Viscosity at 40 °C and Viscosity at 100 °C take the data, and Viscosity at operating temperature (for κ) chooses between "Viscosity curve (Walther, ASTM D341)" (default) and "Linear between 40 and 100 °C (simpler, older method)". The straight line overestimates the viscosity between the two points, and with it κ and the life. ν40 must exceed ν100, both within 0.5 to 100 000 mm²/s, and the bearing temperature must lie within −60 to 300 °C; a bearing whose κ cannot be computed gets no life.

The operating temperature is the bearing's Temperature or, with "Compute the temperature from a heat balance", the temperature at which friction power and heat removal balance (see Bearing friction and thermal speeds).

Rated viscosity ν1 and the viscosity ratio κ

ISO 281 compares the operating viscosity with the reference viscosity ν1 a bearing needs for an adequate film. ν1 falls with speed and with size, because both help to build the film (9.3.3.3.1, Figure 2):

• ${\nu }_1=45000\cdot n^{-0.83}\cdot D_{pw}^{-0.5}$ for n < 1 000 rpm (Eq. (28)),
• ${\nu }_1=4500\cdot n^{-0.5}\cdot D_{pw}^{-0.5}$ for n ≥ 1 000 rpm (Eq. (29)),

with n in rpm and Dpw in mm (the mean diameter (d + D)/2 may stand in). The viscosity ratio is $\kappa =\nu /{\nu }_1$ (Eq. (27)). For grease the base oil viscosity counts (9.3.3.3.3). The template has ν1 = 11.4 mm²/s and κ = 2.36. At 30 rpm the same oil gives κ = 0.074, below the range of ISO 281; see Rating life (ISO 281) for how the app handles that.

The relation rests on mineral oils and well-finished raceways. ISO 281 allows the film parameter λ to be used instead for unusual surfaces or oils, with κ roughly equal to ${\lambda }^{1.3}$ (Eq. (30)). The app keeps κ from the viscosities for the life and reports λ alongside.

Elastohydrodynamic film

In the inlet of a rolling contact the oil is dragged into a converging gap, and inside the contact the pressure of 1 000 MPa or more raises its viscosity by orders of magnitude. The pressure–viscosity coefficient α describes this rise, $\eta (p)\approx {\eta }_0\cdot e^{\alpha \cdot p}$: about 20 1/GPa for mineral oil, 13 to 16 1/GPa for synthetic PAO (Pressure–viscosity α). The film forms in the inlet and stays almost parallel through the contact, with a constriction near the outlet where it is thinnest.

The film is written with dimensionless groups: speed $U={\eta }_0\cdot u/(E^{\prime }\cdot R_x)$, material $G=\alpha \cdot E^{\prime }$, load $W=Q/(E^{\prime }\cdot R_x^2)$ (point contact) or $W^{\prime }=w/(E^{\prime }\cdot R_x)$ per unit length (line contact). Here u is the entrainment speed (the mean rolling speed of the two surfaces), Rx the effective radius in the rolling direction, k the ellipse ratio and E' the reduced modulus.

Point contact (balls), Hamrock–Dowson:

Line contact (rollers), Pan–Hamrock, with the mean load per unit length:

Speed and viscosity dominate (exponent about 0.7), the load hardly matters (exponent about −0.07 for balls). Doubling the load thins a ball's film by only 5 %; doubling the speed thickens it by 60 %.

Inlet shear heating

At high speed the oil heats up by shear in the inlet and the film is thinner than the isothermal formulas say. The app multiplies the film by

${\phi }_T=3.94/(3.94+L^{0.62})$, with the thermal loading $L={\eta }_0\cdot \beta \cdot u^2/k$ (Murch and Wilson),

where β is the temperature–viscosity coefficient (taken from the Walther curve) and k the oil's thermal conductivity (Oil conductivity, 0.14 W/(m·K) by default). For the template φT = 0.947. The checkbox "Reduce the film for inlet shear heating" switches the factor off.

Starvation

A fully flooded inlet is not always there: grease after run-in or minimal oil leaves only a thin layer on the raceway ahead of each element. With an Oil supply layer (µm) the film is reduced by

${\phi }_S=R/{(1+R^{1.7})}^{1/1.7}$, with R the supply layer over the fully flooded central film (Chevalier et al., point contacts).

A layer equal to the flooded film gives φS = 0.66. Without a supply layer the contact is fully flooded (φS = 1).

Film parameter λ and regimes

The film is set against the composite roughness of the two surfaces:

$\lambda =h_{\min }/\sigma$, with $\sigma ={(R_{q1}^2+R_{q2}^2)}^{1/2}$.

The app reports a regime from λ: boundary below 1, mixed from 1 to 3, full film from 3. Raise λ with a more viscous oil, a cooler bearing or smoother surfaces (Raceway roughness Rq, Element roughness Rq).

With σ = 0.158 µm, the thinner inner film gives λ = 1.59, mixed lubrication. The same bearing at 30 rpm has hmin = 0.012 µm and λ = 0.075 (boundary). With a heat balance instead of the fixed 70 °C, the template settles at 46.2 °C, where the oil is thicker: λ = 2.96 and κ = 6.40.

Grease relubrication interval

With Method (for friction) set to "Grease or oil mist", the results include a relubrication interval after the bearing makers' rule of thumb:

in hours, with n in rpm, d the bore in mm and k = 10 for radial ball bearings, 5 for cylindrical and needle roller bearings and 1 for spherical, tapered and thrust bearings. The interval halves for every 15 K above 70 °C. d is the bore from the bearing's mounting, else estimated as dm − 1.7·Dw. An interval of 0 means that the speed is too high for grease by this rule.

For the template on grease, d is estimated as 34.6 mm and tf = 10 · (14·10⁶/(2 900 · √34.6) − 4 · 34.6) = 6 821 h at 70 °C. The result reports baseInterval, typeFactor, temperatureFactor and interval.

What is not covered

• Non-Newtonian behaviour and the limiting shear stress of the oil, which affect traction rather than film thickness.
• Roughness effects on the film itself (micro-EHL); λ treats the surfaces statistically.
• Grease ageing and bleeding apart from the relubrication rule and the supply layer you enter.
• The EP-additive allowance of ISO 281, 9.3.3.3.4.
• Water in the oil.

From the command line

axm bearing update p.axm.json BR1 --patch '{"lubricationDetails":{"method":"grease","supplyLayer":0.3}}' --json
axm solve p.axm.json --json

bearings[].lubrication holds viscosity, dynamicViscosity, entrainmentSpeed, inner and outer films with thermalFactor and starvationFactor, compositeRoughness, lambda and regime; maintenance.relubrication the interval.

Related

• Rating life (ISO 281)
• Bearing friction and thermal speeds
• Hertz contact and subsurface stress
• Bearings
• Open the bearing-6207 example