bevel-gears

name	bevel-gears
description	Bevel gear design — straight, spiral, and Zerol bevel gears, pitch cone, back cone, virtual number of teeth, AGMA analysis, forces, Gleason system.
metadata	{"priority":7,"promptSignals":{"phrases":["bevel gear","miter gear","spiral bevel","pitch cone","Gleason"],"minScore":3}}

Bevel Gears — Complete Skill

Geometry

Pitch Cone

Shaft angle Σ = γ_p + γ_g (usually 90° for right-angle drive) Pitch cone angles: tan(γ_p) = N_p / N_g (for Σ = 90°) γ_g = 90° - γ_p

Mean cone distance: A_m = A_0 (1 - F/(2A_0)) where F = face width Outer pitch diameter: d_e = m × N (m = module at large end) A_0 = outer cone distance = d_e,p / (2 sin γ_p)

Face width: F ≤ A_0/3 AND F ≤ 10m (smaller governs)

Back cone radius (Tregold's approximation): r_b = d_e / (2 cos γ) Virtual number of teeth: N_v = N / cos γ → use for Lewis form factor Y

Tooth Proportions (Gleason System)

For straight bevel (standard full-depth, Gleason): Working depth h_k = 2m Addendum (pinion): a_p = m × (1 + 0.460(1-N_p/N_g))/(1 + N_p/N_g) Addendum (gear): a_g = 2m - a_p Dedendum = 1.188m + clearance

Spiral bevel: spiral angle ψ = 35° (Gleason standard)

Forces — Straight Bevel (Σ = 90°)

W_t = 2T / d_m [at mean pitch diameter d_m = d_e - F sin γ] W_r = W_t tan(φ) cos(γ) [radial, separating pinion from gear] W_a,p = W_t tan(φ) sin(γ_p) [axial on pinion = radial on gear] W_a,g = W_t tan(φ) cos(γ_p) [axial on gear] φ = pressure angle (usually 20°)

For spiral bevel (ψ = spiral angle): Tooth normal force: W_N = W_t / (cos φ_n cos ψ_m) W_r (pinion) = W_t [tan φ_n sin γ_p - sin ψ_m cos γ_p] / cos(ψ_m) W_a (pinion) = W_t [tan φ_n cos γ_p + sin ψ_m sin γ_p] / cos(ψ_m) Note: direction reverses with hand of spiral and rotation

AGMA Strength (Bevel Gears)

Bending Stress

σ = W_t × K_o × K_v × K_s × P_d/(F × J) [same form as spur, different J factor] J = Lewis form factor modified for bevel (use virtual teeth N_v in AGMA charts) K_s = 1 for small bevel; K_mb for overhanging mount

Contact Stress

σ_H = C_p √(W_t × C_o × C_v × C_s × C_mc/(d_p × F × I)) I = (sin φ cos φ / 2) × (m_N × u/(u+1)) where u = N_g/N_p C_p = elastic coefficient (same as spur gear)

Overhanging mount factor K_mb: Straddle-mounted: K_mb = 1.0; one-side overhang: K_mb = 1.25-1.50

Spiral Bevel Advantages

• Smoother, quieter operation (gradual tooth engagement)
• Higher load capacity (spiral contact ratio added)
• Standard: ψ = 35°, φ_n = 20° (Gleason)
• Used in automotive differentials, aerospace gearboxes

Output

Provide: γ_p and γ_g [°], d_m,p and d_m,g [mm], W_t, W_r, W_a [N], bending SF and contact SH, back cone radii [mm], virtual tooth counts N_vp and N_vg.