Mechanical Machine Design, Gearboxes & EV Powertrain Kinematics
Epicyclic Planetary Gear Willis Speed Ratio & Torque Formula
Calculate Coaxial Mesh Planet Tooth Count N_p, Fixed-Ring Sun-to-Carrier Speed Reduction Ratio (1 + N_r / N_s), Carrier Output Speed (RPM), and Carrier Output Torque (N·m).
Why fixed-ring planetary reduction equals 1 + N_r / N_s—NOT N_r / N_s—and requires even (N_r - N_s)!
One of the most frequent kinematics mistakes when sizing EV hub-drive or servo planetary gearheads (stationary internal Ring Gear N_r, input Sun Gear N_s, output Planet Carrier) is writing Speed_Ratio = N_r / N_s as if it were a simple countershaft pair! Applying Willis tabular relative-velocity equation (omega_ring - omega_carrier) / (omega_sun - omega_carrier) = -N_s / N_r with omega_ring = 0 proves that Sun-to-Carrier Speed Reduction is EXACTLY i_sc = 1 + N_r / N_s—giving +1 extra unit of torque multiplication from the carrier orbiting in the same direction as the sun! Furthermore, coaxial radial geometry pitch_r = pitch_s + 2 * pitch_p forces Planet Tooth Count N_p = (N_r - N_s) / 2 to be a positive integer >= 12 (meaning N_r - N_s MUST be even)!
Try this 3,000 RPM Servo Planetary Stage first—Sun_Teeth N_s = 24 (B2), Ring_Teeth N_r = 96 (B3 -> Coaxial Planet Teeth N_p = (96 - 24) / 2 = 36 teeth in E2), Sun_Input_Speed = 3,000.0 RPM (B4), Sun_Input_Torque = 120.0 N·m (B5), and Stage_Mesh_Efficiency = 0.97 (97.0%, B6): Willis Speed Reduction Ratio hits 1 + 96 / 24 = 5.00 (E3, versus 4.00 if you forgot the +1!), Carrier Output Speed drops to 3,000 / 5.00 = 600.00 RPM (E4), and Carrier Output Torque reaches 120 * 5.00 * 0.97 = 582.00 N·m (E5)! In a shared Arte Sheets drivetrain workbook, you can ask the metered in-app Pro AI Assistant for a hint ("Why must (N_r - N_s) be divisible by 2 for coaxial planet gears to mesh without pitch-circle interference?") before testing alternate tooth counts.
| Planetary Gearbox KPI (Ns = 24, Nr = 96, 3,000 RPM, 120 N·m, 97% Eff) | Formula | Verified Output |
|---|---|---|
| Required Planet Teeth N_p | (Ring_Teeth_Nr - Sun_Teeth_Ns) / 2 | 36 (24 @ Nr = 72) |
| Willis Speed Reduction Ratio | 1 + (Ring_Teeth_Nr / Sun_Teeth_Ns) | 5.00 (4.00 @ Nr = 72) |
| Carrier Output Speed (RPM) | Sun_Input_RPM / (1 + Nr / Ns) | 600.00 RPM (750.00 @ Nr = 72) |
| Carrier Output Torque (N·m) | Sun_Torque * (1 + Nr / Ns) * Efficiency | 582.00 N·m (465.60 @ Nr = 72) |
Coaxial Mesh Guard & Willis Planetary Reduction Formula
Validate integer N_s >= 12, integer N_r >= N_s + 24, MOD(N_r - N_s, 2) = 0 (coaxial integer planet teeth!), RPM > 0, Torque > 0, and Efficiency in (0.70, 1.0]!
=LET(ns_t, B2, nr_t, B3, rp_s, B4, tq_s, B5, ef_m, B6, IF(OR(ns_t < 12, INT(ns_t) <> ns_t, nr_t < ns_t + 24, INT(nr_t) <> nr_t, MOD(nr_t - ns_t, 2) <> 0, rp_s <= 0, tq_s <= 0, ef_m <= 0.7, ef_m > 1), "Invalid Planetary Mesh", ROUND(1 + (nr_t / ns_t), 2)))Smaller Ring Gear 4:1 Ratio & Odd-Tooth Mesh Rejection Check (Set N_r = 72 vs 95)
Change Ring_Teeth B3 from 96 to 72: Planet Teeth E2 adjusts to 24, Speed Ratio E3 becomes 4.00, Output Speed E4 rises to 750.00 RPM, and Torque E5 becomes 465.60 N·m—whereas entering odd N_r = 95 immediately flags "Invalid Planetary Mesh" because N_p = 35.5 half-teeth cannot exist!
=IF(ISTEXT(E3), E3, ROUND(B5 * (1 + (B3 / B2)) * B6, 2))Try the example
Use the original values or change them to test your understanding.