Bearing Analysis · DWG LS-Λ01 · hidden

EHL film thickness & lambda (Λ) calculator

Central and minimum elastohydrodynamic (EHL) oil-film thickness at the most heavily loaded ball of a radial ball bearing, from the NASA Hamrock–Dowson equations (TP-1342), divided by the composite surface roughness to give the film parameter Λ. Shown next to the ISO 281 viscosity ratio κ for the same bearing and oil.

Published method (NASA Hamrock–Dowson)Last reviewed

Source: NASA TP-1342 (NTRS) · NASA RP-1255 (NTRS)

Ball size, ball count, pitch diameter and groove conformity are estimated when unknown, as marked.

Validation
  • NASA RP-1255 §24.12.2 radial ball-bearing example: minimum film 0.557 / 0.666 µm (book 0.557 / 0.665 µm), Λ 2.99 / 3.58 (book 3.00 / 3.58).
  • κ matches the Minimum Required Viscosity calculator (ν₁ 12.46 cSt for 45 × 100 mm at 1,800 rpm).

How calculators are checked: editorial standards.

1 · Bearing & internal geometry
Looks up bore, OD, width and type from the LubeSuite bearing data (same data as Bearing & Housing Lookup).
For reference only; the film calculation does not use B.
Estimate Dw ≈ 0.3·(D − d): middle of the 0.15–0.45·(D − d) design range (Rao & Tiwari 2007, restated in Rana et al. 2019). Use the real value if you know it.
Estimate: the largest Z that fits the Conrad assembly angle φ0, Z ≤ 1 + φ0 / (2·sin⁻¹(Dw/dm)) (Rao & Tiwari 2007).
Estimate dm ≈ 0.5·(d + D), as ISO 281:2007 9.3.3.3.1 allows for Dpw.
Default 0.52 and 0.53 (outer): the limits ISO 281:2007 5.1.1 assumes for radial ball bearings.
2 · Speed, load & oil
One ring turns, the other is stationary. Pure rolling.
Most heavily loaded ball Qmax = 5·Fr/Z (Stribeck's 4.37 raised to 5 for clearance, as in NASA RP-1255 §24).
Converts ν to dynamic viscosity η0 = ν·ρ. Typical 15 °C density by oil type, less about 0.65 kg/m³ per °C. Enter the data-sheet value if you have it.
3 · Materials & surface finish
Defaults: through-hardened bearing steel, E 208 GPa, ν 0.3 (both editable).
Ra ranges are the "average application" bands for each process from the ASME B46.1 chart (as printed in Machinery's Handbook). The pick is a bearing-grade value inside the band. Rq = 1.25·Ra (√(π/2), exact for a Gaussian height distribution). Composite roughness σ = √(Rqring² + Rqball²).
Λ inner (min film / σ)
—
Λ outer
—
κ = ν / ν₁ (ISO 281)
—
Most heavily loaded ballInner raceOuter race
Ball load Qmax—
Rx (rolling direction)——
Ry (across the groove)——
Ellipticity k = 1.03·(Ry/Rx)0.64——
Speed parameter U——
Load parameter W——
Materials parameter G = αE′—
Central film hc——
Minimum film hmin——
Composite roughness σ—
Λ = hmin/σ——
Regime——
    Equations & sources
    • Hamrock & Dowson, NASA TP-1342 (1978), equations re-checked against the scanned page images: H = h/Rx, 1/Rx = 1/rAx + 1/rBx (1, 2); U = η0u/(E′Rx), u = (uA+uB)/2 (3, 4); E′ = 2/[(1−νA²)/EA + (1−νB²)/EB] (5); W = F/(E′Rx²) (6); G = αE′ (7); k = 1.03·(Ry/Rx)0.64 (9).
    • Minimum film, viscous-elastic (EHL) regime, eq (26): Hmin = 3.63·U0.68·G0.49·W−0.073·(1 − e−0.68k). The same equation is eq (22-18) in NASA RP-1255.
    • Central film, TP-1342 eq (31), in reduced form: Ĥc = 3.61·gV0.53·gE0.13·(1 − 0.61e−0.73k), with Ĥ = H(W/U)², gV = GW³/U², gE = W8/3/U² (eqs 11–13). Rewritten in U, G and W, this is Hc = 3.61·U0.68·G0.53·W−0.0633·(1 − 0.61e−0.73k). The cross-check value is the direct fit in RP-1255 eq (22-20), Hc = 2.69·U0.67·G0.53·W−0.067·(1 − 0.61e−0.73k).
    • Ball-bearing contacts (pure rolling, contact angle 0): inner Rx = Dw(dm−Dw)/(2dm), Ry = fiDw/(2fi−1); outer Rx = Dw(dm+Dw)/(2dm), Ry = foDw/(2fo−1); u = ω(dm² − Dw²)/(4dm). All as in the RP-1255 §24.12.2 worked example, which this page reproduces (inner 0.557 µm, outer 0.665 µm).
    • Fit range: k = 1 to 8 (TP-1342, 34 cases). In RP-1255 Table 22-1 those cases span U 0.084–5.05×10⁻¹¹, W 0.11–1.29×10⁻⁶ and G 2310–6785. Isothermal, fully flooded, Newtonian. No inlet-shear heating or starvation correction, so at high speed h is likely overestimated.
    • Λ = hmin/√(Rq,a² + Rq,b²): NASA RP-1255 eq (3-20). Bands: full film Λ > 3 and partial (mixed) 1 < Λ < 3 (Tripp & Hamrock, NASA TP-2488, 1985); boundary Λ < 1 (RP-1255 §3.8, where Hamrock calls the bands "rough estimates").
    • κ = ν/ν₁, with ν₁ from ISO 281:2007 eqs (28)–(29) and dm = (d+D)/2. This is the same code as the minimum-viscosity calculator. ISO VG to ν at temperature uses the Walther / ASTM D341 line through KV40 and a KV100 solved from the VI (ASTM D2270), the same method as the drum-oil-temperature calculator.
    • α defaults (GPa⁻¹, editable): mineral 20 (paraffinic about 18–28, naphthenic up to about 32), PAO 15, ester 14, PAG 14. Sources: Bair-type high-pressure measurements in Lubricants 2023, 11(2), 55 (paraffinic, naphthenic, PAO and ester base oils, about 13–15 GPa⁻¹ for PAO16 and esters at 50–80 °C); J. Chem. Eng. Data (2010), PAG about 14–17 GPa⁻¹ at 40–80 °C; Roelands (1966) values as tabulated by Dr. ir. A. van Beek, tribology-abc.com (piezo-viscosity table). α falls with temperature, so use a measured value when you have one.
    • Geometry estimates: Dw and Z from the design-variable ranges and the assembly-angle constraint of Rao & Tiwari (Mech. Mach. Theory 42, 2007), restated in Rana, Patel & Lalwani (2019); dm ≈ 0.5(d+D) and fi 0.52, fo 0.53 from ISO 281:2007. Qmax = 5Fr/Z from NASA RP-1255 §24.12.2.

    NASA TP-1342 · NASA RP-1255 (Hamrock, Fundamentals of Fluid Film Lubrication, 1991) · NASA TP-2488 · ISO 281:2007

    An engineering estimate. The film depends strongly on the real ball size, conformity, finish and α. Verify with the bearing maker before acting.