| name | worm-gears |
| description | Worm gear design — lead angle, velocity ratio, efficiency, self-locking, AGMA stress, thermal rating, bronze gear materials, center distance. |
| metadata | {"priority":7,"promptSignals":{"phrases":["worm gear","worm wheel","lead angle","self-locking","worm drive"],"minScore":3}} |
Worm Gears — Complete Skill
Geometry
Basic Parameters
m_x = axial module of worm = m_n (normal module at worm pitch cylinder)
d_w = worm pitch diameter [mm]
z_w = number of worm starts (1, 2, 4 typical)
z_g = number of gear teeth
Velocity ratio: i = z_g / z_w (typically 5:1 to 80:1 in single stage)
Lead: L = z_w × m_x × π [axial distance per revolution of gear = helical pitch × starts]
Lead angle (worm): tan(λ) = L / (π d_w) = z_w × m_x / d_w
λ = 3-30° (higher λ = higher efficiency; lower λ = self-locking tendency)
Center distance:
a = (d_w + d_g) / 2
d_g = m_x × z_g (gear pitch diameter)
AGMA center distance recommendation:
d_w ≈ (0.875 to 1.75) × a^0.875 [mm, a in mm] — for best efficiency
Worm geometry ratio: z_w² + (a/m_x - z_w/2)² → usually 1-4 starts
Efficiency and Self-Locking
Efficiency
η = tan(λ) / tan(λ + φ')
φ' = arctan(μ / cos(φ_n)) [virtual friction angle]
μ = coefficient of friction (depends on sliding velocity V_s)
Friction coefficient vs. sliding speed:
V_s = V_w / cos(λ) [m/s]; V_w = π d_w n_w / 60000 [m/s]
V_s < 0.05 m/s: μ = 0.15; 0.5 m/s: μ = 0.05; 5 m/s: μ = 0.025; >10 m/s: μ = 0.015
Typical efficiency:
Single-start (z_w=1): η = 40-60% (low λ, low efficiency)
Four-start (z_w=4): η = 70-90% (high λ)
Self-locking condition: λ < φ' → no back-driving possible
For steel worm / bronze gear: self-locking when λ < ~5-6° (μ ≈ 0.1)
Forces
W_t,g = 2T_g / d_g [tangential force on gear = output torque]
W_a,g = W_t,g × tan(λ + φ') / tan(λ) [NOT simple — includes friction]
Exact force analysis:
W_N = W_t,g / (cos(φ_n) × cos(λ)) [resultant tooth force]
W_t,w = W_N × (cos(φ_n) sin(λ) + μ cos(λ)) [tangential on worm = input]
W_r,w = W_N sin(φ_n) [radial on worm]
W_a,w = W_t,g [axial on worm = tangential on gear]
AGMA Strength Rating (Worm Gears)
Gear tooth bending: F_t = C_s × d_g^0.8 × F_e × m_n [allowable tangential force]
C_s = strength factor for material (bronze: 60-70; hardened steel: 80-90)
F_e = effective face width = min(F_actual, 0.67 d_w)
Contact durability:
Limited by gear material (bronze wears against hardened steel worm)
Bronze: max Hertz stress ~690 MPa (tin bronze), ~825 MPa (aluminum bronze)
Thermal Rating
Power lost as heat: P_loss = P_input × (1 - η)
Heat dissipation (natural convection): P_allowed = k_s × A_housing × (T_oil - T_ambient) [W]
k_s ≈ 15 W/m²K (natural), 30 W/m²K (forced)
T_oil,max = 120°C (synthetic), 90°C (mineral); T_ambient = 40°C
If P_loss > P_allowed: add cooling fan, oil bath cooling, or external cooler
Materials
Worm: hardened alloy steel (case-hardened Rc 58-62) OR through-hardened (Bhn 250-300)
Gear: phosphor tin bronze (C90700) for high loads; aluminum bronze (C95400) for corrosion; centrifugal cast for best grain structure
Output
Provide: λ [°], i (velocity ratio), η [%], self-locking check, W_t,w and W_t,g [N], T_output [N·m], thermal rating P_thermal [kW] vs. P_input, center distance a [mm], d_w and d_g [mm].
