gearbox-design

name	gearbox-design
description	Gearbox design — gear ratio selection, spur/helical/bevel/planetary layout, shaft design, bearing selection, housing design, lubrication system, power density, AGMA 6014/API 613, center distance, efficiency, thermal rating.
metadata	{"priority":7,"promptSignals":{"phrases":["gearbox design","gearbox layout","planetary gearbox","bevel gearbox","AGMA 6014","API 613 gearbox"],"minScore":3}}

Gearbox Design — Complete Skill

Gear Ratio Selection and Layout

Single-Stage Reduction Limits

Spur/helical single stage: i = 1–6 (practical); up to 8 with special design Bevel single stage: i = 1–5 (shaft intersection angle 90°) Worm single stage: i = 5–80 (high ratio in one stage; low efficiency 50–90%) Planetary single stage: i = 3–12 (compact; high power density) Epicyclic double-ring: i up to 20

Multi-Stage Layout

Total ratio: i_total = i₁ × i₂ × i₃ (cascaded stages) Optimal stage ratio distribution: For minimum gearbox size: i₁ ≈ i₂ ≈ √i_total (2-stage); i₁ ≈ i₂ ≈ i₃ ≈ ∛i_total (3-stage) High-speed stage → low torque → smaller gears; low-speed stage → high torque → larger gears

Center distance: a = m × (z₁ + z₂) / 2 [mm; m = module; z = number of teeth] Larger a → heavier but stronger; smaller → more compact but limited power

Planetary Gearbox

Kinematic relationship: i = 1 + z_ring / z_sun [with carrier as output; sun as input; ring fixed] i = z_ring / z_planet [with ring as output; carrier fixed] (not common)

Power split: sun → N_planets (typically 3–5); each planet carries T_total / N_planets → compact, high power density; shared load → lower tooth stress

Planet carrier design: Floating planet: planet pins self-aligns; equal load sharing possible Fixed carrier (integrated): rigid; manufacturing critical for equal planet loading Load sharing factor (K_Hα): accounts for non-uniform load distribution → AGMA 2001

Double planetary (compound): Two sets of planets meshing with two ring gears → very high ratio in small space; used in automotive automatic transmissions

Shaft Design for Gearboxes

Torque from power: T = P / ω = P × 60 / (2π × N) [N·m; P in W; N in RPM]

Tangential force at pitch circle: F_t = 2T / d_pitch [N; d_pitch = pitch diameter] Radial force: F_r = F_t × tan(φ_t) [φ_t = transverse pressure angle; 20° typical → F_r = 0.364 F_t] Axial force (helical): F_a = F_t × tan(β) [β = helix angle]

Shaft sizing (combined bending + torsion): σ_eq = √(σ_b² + 3τ²) [von Mises; σ_b = 32M/(πd³); τ = 16T/(πd³)] d_shaft ≥ [32/(π) × √(M² + T²) / σ_allow]^(1/3) [simplified; combined Tresca criterion]

Critical speed (first lateral): f_c = π/(2L²) × √(EI/m_eff) [Hz; simplified for uniform shaft; L = span between bearings] Design criterion: f_c > 1.3 × maximum operating speed (for subcritical) or f_c < 0.7 × min speed (supercritical)

Bearing Selection for Gearboxes

Bearing types and typical applications:

• Cylindrical roller (NU, NJ): radial loads; allows axial freedom; high-speed; input shaft
• Tapered roller: combined radial + axial; face-to-face pair for pinion shaft; automotive differentials
• Deep groove ball: moderate radial + axial; high-speed; low-torque positions
• Four-point contact ball: bi-directional axial + radial; limited space; planet pin bearings
• Spherical roller: misalignment tolerance; heavy radial; output shaft of large gearboxes

L10h bearing life: L10h = 10⁶/(60n) × (C/P)^k [hours; C = dynamic load rating; P = equivalent dynamic load; k = 3 for ball, 10/3 for roller] Target life: industrial: L10h ≥ 20,000 h; API 613: ≥ 100,000 h for critical machinery; automotive: ≥ 10 years / 200,000 km

Equivalent dynamic load: P = X × F_r + Y × F_a [X, Y = bearing-specific factors from SKF/FAG catalog] For tapered roller: P = 0.4F_r + 0.4F_a (typical at moderate axial ratio)

SKF a_SKF / a_ISO 281 modified rating: L10h_modified = a₁ × a_SKF × L10h [a₁ = reliability factor; a_SKF = ISO life modification factor including lubrication] L10h_modified used for design verification at required reliability and lubrication conditions

Lubrication System

Splash (bath) lubrication: Suitable: peripheral velocity < 15 m/s; bottom gear dips in oil; simplest; automotive final drives Oil level: 1/3 of lowest gear submerged (spur); ≤ 1 module depth (helical, bevel) Heat removal: limited; requires external cooler if high duty cycle

Forced (pressure) lubrication: Required: > 15 m/s pitch line velocity or continuous high duty; API 614 / API 613 requirements Oil pump (gear or vane): pressure 1.5–5 bar; directed oil jets to gear mesh and bearings Filter: 10–25 μm absolute; 5 μm for precision bearings Oil cooler: shell-and-tube or plate; maintain T_oil ≤ 80°C supply; 120°C maximum

API 614 / API 613 oil system: Dual filters: change filter without shutdown Dual oil coolers: redundancy Rundown tank: gravity oil supply for 1–2 min after emergency shutdown Pressure gauges and temperature points: per API 614

Gear oil selection (API grades): ISO VG 46–68: high-speed lightly loaded gears (turbines); low viscosity → EHD film; low friction ISO VG 100–150: medium-speed industrial gearboxes ISO VG 220–460: slow-speed high-load applications (worm gears, heavy industrial) API GL-4: manual transmissions; moderate EP additives API GL-5: rear axles; high EP additives (may damage yellow metals)

Minimum film thickness (EHD): h_min = 3.63 × U^0.68 × G^0.49 × W^(-0.073) × R [Hamrock-Dowson; U = speed parameter; G = materials parameter; W = load parameter; R = radius] Specific film thickness: Λ = h_min / √(Ra₁² + Ra₂²) > 2.0 → full film lubrication

Housing Design

Cast iron housing: most common; good vibration damping; FE analysis for deflection < 0.01 module at bearing seats Ductile iron (GGG50): higher strength than gray iron; preferred for high-load housings Aluminum housing: weight savings 60%; for automotive; lower stiffness → need thicker walls or ribs Welded steel: custom heavy industrial; high strength; poor damping

Bearing bore tolerance: Tolerance: H7 (housing bore) / j6 or k5 (shaft); provides light interference for rotating inner ring For outer ring (rotating load): M7 interference in housing bore

Thermal expansion: Housing CTE × ΔT × L_center_distance → center distance change → affects backlash Steel gearbox: Δa = 12 × 10⁻⁶ × ΔT × a [μm; a in mm; ΔT in °C]

Gearbox Thermal Rating

Heat generated: Q_generated = P_input × (1 - η) [W; η = gearbox efficiency]

Gearbox efficiency (approximate): Per stage: spur = 0.98–0.99; helical = 0.98–0.99; bevel = 0.97–0.99; worm = 0.50–0.90 Bearings: 0.99–0.995 per stage; seals: 0.995 per seal

Thermal equilibrium: Q_generated = Q_radiated + Q_forced_cooling For splash-lubricated gearbox: Q_radiated = h_conv × A_housing × (T_oil - T_ambient) h_conv ≈ 10–15 W/(m²·K) natural convection; 30–50 W/(m²·K) with fan

Maximum continuous power (thermally limited): P_thermal = h_conv × A_housing × ΔT_max / (1 - η) [W; ΔT_max = maximum allowable oil temperature rise]

Standards

Standard	Scope
AGMA 6014-A06	Gear power rating for cylindrical shell and trunnion gearboxes
AGMA 2001-D04	Fundamental rating (bending/contact)
API 613	Special purpose gear units for petroleum industry
API 614	Lubrication systems for petroleum industry
ISO 6336	Load capacity calculation for spur/helical gears
AGMA 6009	Specifications for enclosed helical gearboxes

Output

Provide: gear arrangement (spur/helical/planetary/bevel), stage ratios and total ratio, module and number of teeth for each stage, center distance [mm], pitch line velocity [m/s], tangential force F_t [kN], shaft diameters [mm] and critical speed [RPM], bearing type and L10h life [hours], lubrication type (splash/forced), oil ISO VG grade, specific film thickness Λ, gearbox efficiency per stage and total, power loss Q [kW], thermal rating (max continuous P [kW] without overcooling), housing material, and applicable standard (AGMA 6014, API 613, ISO 6336).