tribology

name	tribology
description	Tribology — lubrication regimes (Stribeck curve), hydrodynamic bearing theory (Reynolds equation), EHD lubrication, film thickness, wear models (Archard), friction, lubricant viscosity selection.
metadata	{"priority":7,"promptSignals":{"phrases":["tribology","lubrication","friction","wear","Stribeck","hydrodynamic bearing","EHD","lubricant","viscosity","bearing film"],"minScore":4}}

Tribology — Complete Skill

Lubrication Regimes (Stribeck Curve)

Plot: friction coefficient μ vs. Hersey number (ηN/P = viscosity × speed / load)

Boundary Lubrication (low ηN/P)

Surfaces in contact; asperity contact dominates μ = 0.05-0.15 (with boundary additives), 0.3-0.5 (dry) Wear rate high; controlled by surface chemistry (EP/AW additives)

Mixed Lubrication

Partial fluid film + asperity contact μ = 0.01-0.1 Transition region; running-in phase of new machinery

Hydrodynamic (HD) Lubrication (high ηN/P)

Full fluid film separates surfaces; no solid contact μ = 0.001-0.005 (viscous drag only) Film thickness >> surface roughness (h_min/R_q > 3: λ ratio = film parameter) Governed by Reynolds equation

Elastohydrodynamic (EHD) Lubrication

HD film with elastic deformation of surfaces + pressure-viscosity effect Occurs in: rolling element bearings, gears (Hertz contact + thin film) Film thickness much thinner than HD (μm scale)

Lambda (Λ) Ratio (Film Parameter)

Λ = h_min / √(R_q1² + R_q2²) Λ > 3: full film (EHD regime), negligible wear 1 < Λ < 3: mixed lubrication, some asperity contact Λ < 1: boundary lubrication, significant wear

R_q = RMS roughness of each surface [μm]

Reynolds Equation (Hydrodynamic Theory)

∂/∂x(h³/η × ∂p/∂x) + ∂/∂z(h³/η × ∂p/∂z) = 6U × ∂h/∂x + 12 × ∂h/∂t

Where: h(x) = film thickness [m] η = dynamic viscosity [Pa·s] p = pressure [Pa] U = entraining velocity = (U₁ + U₂)/2

Simplified (1D, Rigid Surfaces)

For inclined slider: p(x) from ∂/∂x(h³∂p/∂x) = 6ηU × ∂h/∂x Sommerfeld solution for journal bearings (full 360° film)

Journal Bearing Theory

Sommerfeld Number (S)

S = (r/c)² × η × N / P r = journal radius, c = radial clearance, N = speed [rev/s], P = load/projected area

Short bearing approximation (L/D < 0.5): ε = eccentricity ratio = e/c (e = journal offset from center) From load → S → ε (eccentricity curve)

Minimum film thickness: h_min = c(1 - ε) Design requirement: h_min > 5 × R_q_surface (λ > 5 for safe operation)

Typical values: Clearance ratio c/r: 0.001-0.003 (1-3‰ of radius) Sommerfeld number for full film: S > 0.1

Petroff's equation (lightly loaded, concentric): μ = 2π²η × r/c × N/P (friction coefficient) Friction power: P_fric = μ × W × r × ω

Oil film temperature rise: ΔT ≈ P_fric / (ṁ_oil × c_p_oil) [≤ 20-30°C rise acceptable]

EHD Film Thickness (Rolling Contacts)

Dowson-Higginson Formula (Line Contact — Gears, Cylindrical Bearings)

h_min = 2.65 × α^0.54 × (η₀U)^0.7 × R'^0.43 / (E'^0.03 × W'^0.13)

Dimensionless groups: U = η₀U_s/(E'R') [speed parameter] W = F/(E'R'L) [load parameter] G = αE' [material parameter] α = pressure-viscosity coefficient [Pa⁻¹], η₀ = viscosity at ambient [Pa·s]

Elliptic contact (ball bearings — Hamrock-Dowson): h_min = 3.63 U^0.68 G^0.49 W^(-0.073) (1 - e^(-0.68k)) × R_x k = ellipticity ratio = a/b (long/short axis of contact ellipse)

Typical EHD film thickness: 0.1-2 μm (gears), 0.05-1 μm (ball bearings)

Viscosity and Lubricants

Dynamic vs. Kinematic Viscosity

η [Pa·s = mPa·s = cP] = dynamic (absolute) ν [m²/s = mm²/s = cSt] = kinematic = η/ρ Conversion: ν [cSt] = η [cP] / ρ [g/cm³]

Viscosity-temperature (Walther equation): log log(ν + 0.7) = A - B × log(T) [ASTM D341, T in Rankine or Kelvin] Viscosity Index (VI): higher VI → less viscosity change with temperature VI > 100: multigrade oil; VI < 0: highly temperature-sensitive

ISO Viscosity Grades (ISO VG — at 40°C)

VG 10, 15, 22, 32, 46, 68, 100, 150, 220, 320, 460, 680... ±10% tolerance around grade center

AGMA Gear Oil Grades

AGMA 1 ≈ ISO VG 46; AGMA 2 ≈ VG 68; AGMA 4 ≈ VG 150; AGMA 6 ≈ VG 320; AGMA 8 ≈ VG 460 R&O (rust & oxidation inhibited): plain gearboxes EP (extreme pressure): bevel, worm, hypoid, high-load

SAE Engine Oil Grades

Monograde: SAE 30, 40, 50 (kinematic at 100°C) Multigrade: SAE 5W-30 (5W = cold, 30 = hot) — W = winter SAE 5W-30: η cold = 3,500 mPa·s at -25°C; ν = 9.3-12.5 cSt at 100°C

Pressure-Viscosity Effect

η = η₀ × exp(α × P) α ≈ 10-30 GPa⁻¹ for mineral oils At 1 GPa contact pressure: η/η₀ = e^(20×10⁻⁹ × 10⁹) = e^20 = 5×10⁸ ← huge increase! This is why EHD works — viscosity soars under Hertzian contact pressure

Wear Models

Archard Wear Law

Q = k × P × L / H Q = wear volume [m³] k = dimensionless wear coefficient (10⁻⁸ to 10⁻³, material/lubricant dependent) P = normal load [N], L = sliding distance [m], H = hardness [Pa]

Wear rate: dV/ds = k × P/H Specific wear rate k_s = k/H [m²/N] = archard coefficient / hardness

Typical k values: Steel on steel (dry): k = 10⁻² to 10⁻³ Steel on steel (lubricated): k = 10⁻⁵ to 10⁻⁷ PTFE on steel: k = 10⁻⁸ to 10⁻⁶ DLC on steel (dry): k = 10⁻⁹

Wear Regimes

Mild wear: oxide/protective layer intact; k ≈ 10⁻⁶ Severe wear: metallic contact, delamination; k ≈ 10⁻³ Transition: depends on PV value (pressure × velocity)

PV limit: max pressure × velocity before transition to severe wear PTFE: PV_max ≈ 0.03 MPa·m/s (dry); nylon: 0.07; bronze: 0.5; steel (oil): 3.0

Friction Coefficient Reference Table

Pair	Dry	Lubricated
Steel - Steel	0.5-0.8	0.05-0.15
Steel - Cast Iron	0.3-0.5	0.05-0.10
Steel - Bronze	0.15-0.3	0.05-0.10
Steel - PTFE	0.04-0.20	0.04-0.10
Steel - Rubber	0.5-1.5	0.2-0.5
Bronze - Bronze	0.2-0.3	0.04-0.08
DLC - Steel	0.05-0.15	0.01-0.05

Output

Provide: lubrication regime (HD/EHD/mixed/boundary), Λ ratio, minimum film thickness h_min [μm], Sommerfeld number S, friction coefficient μ, wear rate Q [mm³/hr], lubricant viscosity grade (ISO VG), oil film temperature rise ΔT [°C].