gear-design

name	gear-design
description	Complete gear design — spur, helical, bevel, worm. Lewis bending equation, AGMA stress, pitting resistance, gear ratio, center distance, module/diametral pitch sizing, efficiency, lubrication.
metadata	{"priority":8,"promptSignals":{"phrases":["gear","spur gear","helical gear","bevel gear","worm gear","gear ratio","gear design","Lewis","AGMA","diametral pitch"],"minScore":4}}

Gear Design — Complete Skill

Geometry Fundamentals

Diametral pitch (Imperial): P_d = N/d = N/d [teeth/inch] Module (SI): m = d/N [mm/tooth] → m = 25.4/P_d Standard modules (mm): 1, 1.25, 1.5, 2, 2.5, 3, 4, 5, 6, 8, 10, 12, 16, 20

Pitch diameter: d = N/P_d (Imperial), d = mN (SI) Center distance: C = (d_p + d_g)/2 = m(N_p + N_g)/2 Gear ratio: i = N_g/N_p = d_g/d_p = ω_p/ω_g Standard pressure angle: φ = 20° (most common), 14.5° (old), 25° (high load) Addendum: a = 1/P_d (Imp.), a = m (SI) Dedendum: b = 1.25/P_d (Imp.), b = 1.25m (SI) Whole depth: h = 2.25/P_d (Imp.), h = 2.25m (SI) Clearance: c = 0.25/P_d (Imp.), c = 0.25m (SI)

Helical gear — helix angle ψ (15-30° typical): Normal module: m_n = m_t·cos(ψ) Normal pressure angle: tan(φ_n) = tan(φ_t)·cos(ψ) Axial pitch: p_x = π·m_t/tan(ψ) Face width: b ≥ 2p_x (for smooth overlap), typically b = 8-16m

Transmitted Force

W_t = 60000·P_kW/(π·d_mm·n_rpm) [N] or W_t = 33000·HP/(V_fpm) [lbf] W_r = W_t·tan(φ) (spur) For helical: W_r = W_t·tan(φ_n)/cos(ψ), W_a = W_t·tan(ψ)

AGMA Bending Stress (J-factor method)

σ_b = W_t · K_o · K_v · K_s · (P_d/F) · (K_H/J)

Where:

• K_o = overload factor (1.0-2.75 depending on shock)
• K_v = dynamic factor = (A+√V)² / A² (A = 50+56(1-B), B depends on quality number Qv)
• K_s = size factor = 1.192(F√Y/P_d)^0.0535
• K_H = load distribution factor = 1 + C_mc(C_pf·C_pm + C_ma·C_e)
• J = geometry factor for bending (from AGMA charts, function of N, φ)
• F = face width

Allowable bending stress: σ_b,all = S_t · Y_N / (K_T · K_R · S_F) S_t = allowable bending stress number (from AGMA 2001, depends on material/hardness) Y_N = stress cycle factor for bending S_F = safety factor (≥1.2 design, ≥1.5 conservative)

AGMA Contact (Pitting) Stress

σ_c = -C_p · √(W_t · K_o · K_v · K_s · K_H / (d_p · F · Z_R · I))

Where:

• C_p = elastic coefficient = √(1/(π·((1-ν_p²)/E_p + (1-ν_g²)/E_g))) For steel-steel: C_p = 191 MPa^0.5 (2290 psi^0.5)
• I = geometry factor for contact = sin(φ)cos(φ)/(2m_N) · m_G/(m_G+1)
• m_N = load sharing ratio (1.0 for spur, varies for helical)
• Z_R = surface condition factor

Allowable contact stress: σ_c,all = S_c · Z_N · C_H / (K_T · K_R · S_H) S_H = safety factor for pitting (≥1.0 design)

Spur Gear Design Procedure

1. Determine gear ratio i, power P, input speed n
2. Select material: steel, cast iron, bronze
3. Estimate face width: F = 8m to 16m
4. Choose Qv (quality number 6-12): higher = quieter
5. Calculate W_t from power equation
6. Apply AGMA factors, compute σ_b and σ_c
7. Compare to allowables, iterate on m (module) if needed
8. Check interference: N_p > 2k/((1+2m)(sin²φ-1)) [no undercutting] Minimum teeth (20° PA): N_min = 17 (spur), 14 (helical)

Bevel Gears (Straight)

Virtual number of teeth: N'_p = N_p/cos(Γ_p), N'_g = N_g/cos(Γ_g) Pitch cone angle: tan(Γ_p) = N_p/N_g (for 90° shaft angle) Mean pitch diameter: d_m = d - F·sin(Γ) Use AGMA 2003 for bevel gear rating

Worm Gears

Lead angle: λ = arctan(L/πd_w) [L = lead = N_w·p_x] Gear ratio: i = N_g/N_w (N_w = number of worm threads, 1,2,4) Efficiency: η = tan(λ)/tan(λ+φ') where φ' = friction angle = arctan(μ) Self-locking: λ < φ' (η < 50%) Thermal rating critical — worm gears generate heat; check oil temperature

Material Selection

Application	Pinion	Gear
General	4140 HRC 55-60	4140 HRC 50-55
High load	9310 case-hardened	8620 case-hardened
Low noise	1020 case-hardened	1020 case-hardened
Corrosive	316SS	316SS
Low cost	Cast iron Gr.30	Cast iron Gr.20

Lubrication

AGMA 9005: viscosity grade by pitch line velocity V < 5 m/s: ISO VG 320 V = 5-15 m/s: ISO VG 220 V > 15 m/s: ISO VG 150

Output

Provide: m or P_d selected, d_p, d_g, F, σ_b, σ_c vs allowables, S_F, S_H, center distance C.