| name | planetary-gears |
| description | Planetary gear systems — velocity ratio (tabular method), torque split, gear forces, efficiency, Ravigneaux, compound planetary, design rules. |
| metadata | {"priority":7,"promptSignals":{"phrases":["planetary gear","epicyclic","sun gear","ring gear","planet carrier","Ravigneaux"],"minScore":3}} |
Planetary Gear Systems — Complete Skill
Notation
S = sun gear (teeth z_s), P = planet gear (z_p), R = ring gear (z_r), C = carrier
z_r = z_s + 2z_p (geometric constraint for simple planetary)
Velocity Ratio — Tabular Method
Steps
- Lock carrier (C fixed): sun rotates +1 rev → planet rotates -(z_s/z_p) → ring rotates +(z_s/z_r) [same direction as sun]
- Unlock: add x revolutions to entire system (uniform rotation)
Speed equations:
n_s = x + 1; n_p = x - z_s/z_p; n_r = x + z_s/z_r; n_c = x
Common Configurations
Sun input, carrier output, ring fixed:
n_s × z_s = n_c × (z_s + z_r) (ring fixed: n_r = 0)
i = n_s/n_c = 1 + z_r/z_s
Example: z_s=20, z_r=80 → i = 5 (5:1 reduction)
Ring input, carrier output, sun fixed:
i = n_r/n_c = 1 + z_s/z_r = (z_r+z_s)/z_r
Carrier input, ring output, sun fixed:
i = n_c/n_r = z_r/(z_r+z_s) < 1 (speed increase)
Two inputs (differential):
n_c = (n_s × z_s + n_r × z_r) / (z_s + z_r)
Planetary Train Formula (Willis Equation)
(n_s - n_c) / (n_r - n_c) = -z_r/z_s [fundamental]
Sign: negative because sun and ring rotate opposite directions when carrier is fixed
Torque Distribution
For simple planetary (steady state, no friction):
T_s + T_r + T_c = 0 (torque balance on system)
T_r / T_s = z_r / z_s (gear ratio)
T_c = T_s × (1 + z_r/z_s) = T_r × (1 + z_s/z_r)
Number of planets N_p: Torque split equally: each planet carries T_total / N_p
Typical: 3 planets (most common); 4 or 6 for heavy loads
Assembly condition: (z_s + z_r) / N_p must be integer for equal spacing
Tooth Loads
W_t (on sun tooth) = 2T_s / d_s
Each planet shares load: W_t,each = W_t / N_p (for N_p planets)
W_t (ring-planet contact) = W_t,each × d_s/d_r (force balance)
Efficiency
Simple planetary: η ≈ 1 - (1-η_pair) × |e|
e = fundamental ratio = (n_s - n_c)/(n_r - n_c) = -z_r/z_s
η_pair = single mesh efficiency ≈ 0.98-0.99
For sun-input, ring-fixed, carrier-output: η ≈ 0.97-0.99 (high efficiency)
For self-locking planets: back-driving possible or locked based on e and η_pair
Design Rules
- z_s ≥ 12 teeth (avoid undercutting)
- Contact ratio ε_α > 1.2 for each mesh
- Planet bearing selection: high-speed, loaded from both sides → use needle or cylindrical roller
- Housing and carrier stiffness critical for load sharing among planets
- Profile shift recommended: positive shift on sun (strength), negative on ring
Compound and Multi-Stage Planetary
Two-stage: overall ratio = i₁ × i₂ (can achieve 20:1 to 100:1+)
Ravigneaux: two sun gears + long/short planets + shared ring (automatic transmissions)
Wolfrom (3-stage in one): very high ratio in compact package
Output
Provide: velocity ratio i, n_output [rpm], T_output [N·m], T_sun/T_ring/T_carrier [N·m], tooth loads [N] per planet mesh, number of planets N_p with assembly check, efficiency η [%].
