Mechanical Engineering & Tribology
Tallian Specific Film Thickness Lambda and Stribeck Regime Formula
Calculate composite RMS surface roughness SQRT(Rq_1^2 + Rq_2^2), Tallian’s specific oil-film ratio Lambda, and Stribeck lubrication regime (Boundary, Mixed, or Full Film).
Add surface roughnesses in quadrature — never by linear addition!
When evaluating rolling-element bearings, cam followers, and involute gear teeth, engineers compare minimum elastohydrodynamic lubricant film thickness h_min (in micrometers) against the combined RMS surface roughness of raceway 1 (Rq_1) and rolling element 2 (Rq_2). Because independent Gaussian surface asperity heights add in variance rather than linearly, composite RMS roughness is sigma_comp = SQRT(Rq_1^2 + Rq_2^2), and Tallian’s (1967) specific film parameter is Lambda = h_min / SQRT(Rq_1^2 + Rq_2^2)!
Check by hand with a classic 3-4-5 Pythagorean roughness pair: ground gear flank Rq_1 = 0.30 um (B3), honed pinion flank Rq_2 = 0.40 um (B4), and EHL minimum oil film h_min = 1.25 um (B2). Naive linear sum 0.30 + 0.40 = 0.70 um understates Lambda at 1.7857, whereas true composite roughness is SQRT(0.09 + 0.16) = 0.5000 um, giving exact Tallian Lambda = 1.25 / 0.50 = 2.5000 (Mixed Lubrication regime 1 <= Lambda < 3; superfinishing both surfaces to 0.15 um and 0.20 um cuts sigma_comp to 0.25 um and lifts Lambda to 5.0000 -> Full Hydrodynamic / EHL separation!).
| Tallian Parameter Range | Stribeck Tribological Regime | Asperity & Wear Consequence |
|---|---|---|
| Lambda < 1.0 | Boundary Lubrication | Direct metal-on-metal asperity load; anti-wear additives essential |
| 1.0 <= Lambda < 3.0 | Mixed / Partial EHL Lubrication | Intermittent asperity contact; micro-pitting risk |
| Lambda >= 3.0 | Full-Film EHL / Hydrodynamic | Complete fluid film separation; maximum fatigue life |
Composite RMS Surface Roughness & Tallian Lambda Formula
Keep h_min, Rq_1, and Rq_2 in matching length units (micrometers um or nanometers nm) and guard non-positive roughness inputs.
=LET(h_oil, B2, rq_a, B3, rq_b, B4, IF(OR(h_oil < 0, rq_a <= 0, rq_b <= 0), "Invalid Surface Input", ROUND(h_oil / SQRT((rq_a^2) + (rq_b^2)), 4)))Stribeck Lubrication Regime Classification Formula
Classify Lambda < 1 as Boundary, 1 <= Lambda < 3 as Mixed, and Lambda >= 3 as Full-Film Hydrodynamic. Test thermal oil thinning where h_min drops from 1.25 um to 0.40 um (set B2 = 0.40): Lambda falls to 0.8000, triggering an immediate Boundary Lubrication alert!
=IF(ISNUMBER(E3), IF(E3 < 1, "Boundary Lubrication", IF(E3 < 3, "Mixed Lubrication", "Full-Film EHL")), E3)Try the example
Use the original values or change them to test your understanding.