| name | helical-gears |
| description | Helical gear design — helix angle, virtual number of teeth, AGMA strength, Lewis form factor, dynamic factor, geometry, axial thrust, shaft loads. |
| metadata | {"priority":7,"promptSignals":{"phrases":["helical gear","helix angle","axial thrust","normal module","virtual teeth"],"minScore":3}} |
Helical Gears — Complete Skill
Geometry
Basic Parameters
ψ = helix angle [°] (typically 15-30°)
m_n = normal module [mm] (standard module)
m_t = transverse module = m_n / cos(ψ)
p_n = normal circular pitch = π m_n
p_t = transverse circular pitch = π m_t
Pressure angles:
φ_n = normal pressure angle (typically 20°)
φ_t = transverse pressure angle: tan(φ_t) = tan(φ_n) / cos(ψ)
Pitch diameter:
d = m_t × z = m_n × z / cos(ψ)
z = number of teeth
Virtual (formative) number of teeth:
z_v = z / cos³(ψ) [used for tooth strength calculations via equivalent spur gear]
Lead: L = π d / tan(ψ) (axial distance for one full helix wrap)
Face width b: typically b = 8-16 × m_n (wider is smoother but more axial load)
Contact ratio: ε_α (transverse) + ε_β (overlap) = ε_total > 1.2
Overlap ratio: ε_β = b × tan(ψ) / (π × m_n) — must be > 1 for quiet operation
Forces on Helical Gear
Transmitted Load
W_t = 2T/d [tangential force, N, d in m, T in N·m]
W_r = W_t × tan(φ_t) [radial separating force]
W_a = W_t × tan(ψ) [axial thrust force — must be reacted by thrust bearing]
W_total = W_t / (cos(φ_n) × cos(ψ)) [resultant tooth force]
Sign of axial thrust:
Depends on hand of helix AND direction of rotation
Right-hand helix, driving gear, CW from pinion end → axial thrust toward left
Use herringbone gears to cancel axial thrust
AGMA Strength Analysis
Bending Stress (AGMA 2101)
σ = W_t × K_o × K_v × K_s × (P_d/b) × (K_H × K_B / Y_J)
P_d = diametral pitch = 1/m_t [in⁻¹], or use: σ = W_t × K_o × K_v × K_s / (b × m_t) × (K_H/Y_J)
Form factor Y_J (AGMA J-factor): accounts for tooth shape + stress concentration
From AGMA charts: Y_J depends on z, addendum modification x, and virtual tooth count z_v
Contact Stress
σ_H = Z_E × √(W_t × K_o × K_v × K_s × K_H / (d_p × b × Z_I))
Z_E = elastic coefficient = √(E/(2(1-ν²) × π)) [MPa^0.5]
Z_I = geometry factor for contact: Z_I = sin(φ_t)cos(φ_t) / 2 × m_N × (u/(u+1))
m_N = load-sharing ratio (= 1 for spur, < 1 for helical due to overlap)
u = gear ratio = z_g / z_p
Dynamic Factor K_v
K_v depends on pitch line velocity V and quality number Q_v:
At V = 5 m/s, Q_v = 6: K_v ≈ 1.35; Q_v = 9: K_v ≈ 1.12; Q_v = 12: K_v ≈ 1.05
Allowable Stresses (AGMA)
σ_F,allow = S_t × Y_N / (K_T × K_R) × S_F (bending)
σ_H,allow = S_c × Z_N × Z_W / (K_T × K_R) × S_H (contact)
Y_N, Z_N = life factors (from AGMA charts, function of cycles N)
K_R = reliability factor (K_R=1.00 @ 99%, 1.25 @ 99.9%)
S_F, S_H = safety factors (S_F ≥ 1.2, S_H ≥ 1.1 typical)
Material Selection
Grade 1 (S_t = 170-310 MPa): through-hardened steel
Grade 2 (S_t = 200-380 MPa): carburized case-hardened, Rc 55-64
Grade 3 (S_t up to 450 MPa): same, premium quality
S_c (contact allowable) ≈ 2.4-4.5× S_t for case-hardened steels
Output
Provide: m_n [mm], z_p and z_g, ψ [°], d_p and d_g [mm], W_t/W_r/W_a [N], bending σ vs. σ_allow, contact σ_H vs. σ_H,allow, safety factors S_F and S_H, face width b [mm], overlap ratio ε_β.
