| 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
- Determine gear ratio i, power P, input speed n
- Select material: steel, cast iron, bronze
- Estimate face width: F = 8m to 16m
- Choose Qv (quality number 6-12): higher = quieter
- Calculate W_t from power equation
- Apply AGMA factors, compute σ_b and σ_c
- Compare to allowables, iterate on m (module) if needed
- 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.
