Mechanical Power Transmission, Worm Drives & Hoist Safety
Worm Gear Lead Angle, Sliding Efficiency & Self-Locking Formula
Calculate Worm Lead Angle lambda (deg), Worm-Driving-Gear Mechanical Efficiency (%), Speed Reduction Ratio (N_g / N_w), and Static Self-Locking Status.
Self-locking worm drives sacrifice efficiency (<50% statically)—multi-start high-efficiency worms CAN back-drive!
Right-angle worm gearsets achieve compact 10:1 to 80:1 single-stage reductions, yet pure sliding contact along the worm helix creates a fundamental engineering trade-off governed by Worm Lead Angle tan(lambda) = Worm_Starts_Nw / Diametral_Quotient_q (where q = d_worm / module). With normal pressure angle phi_n and running friction coefficient mu_k (bronze gear on hardened ground steel worm), worm-driving efficiency is eta = ((COS(phi_n) - mu_k * TAN(lambda)) / (COS(phi_n) + mu_k / TAN(lambda))) * 100! Crucially, static self-locking against gravity load back-driving ONLY holds when TAN(lambda) < mu_static * COS(phi_n) (~8.1° for mu_static = 0.15)—meaning high-efficiency 2-start or 4-start worms WILL overhaul under hoist loads unless fitted with a spring-set brake!
Verify on a 2-Start Conveyor Worm Reducer—Worm_Starts N_w = 2 (B2), Gear_Teeth N_g = 50 (B3 -> Ratio = 25.00 in E4), Diametral_Quotient q = 10.0 (B4 -> tan(lambda) = 0.20, Lead Angle = 11.31 deg in E2), Normal_Pressure_Angle phi_n = 20.0 deg (B5), Running_Friction mu_k = 0.05 (B6), and Static_Friction mu_s = 0.15 (B7): Running Efficiency reaches 78.15% (E3), while Static Lock Check flags "BACK-DRIVABLE" (E5) because tan(11.31°) = 0.20 > 0.15 * cos(20°) = 0.1410!
| Worm Drive KPI (Nw=2, Ng=50, q=10, phi_n=20°, mu_k=0.05, mu_s=0.15) | Formula | Verified Output |
|---|---|---|
| Worm Lead Angle lambda (deg) | DEGREES(ATAN(Worm_Starts_Nw / Quotient_q)) | 11.31° (5.71° @ Nw = 1) |
| Worm-Driving Efficiency (%) | ((COS(phin) - muk*TAN(lam)) / (COS(phin) + muk/TAN(lam)))*100 | 78.15% (64.92% @ Nw = 1) |
| Gearset Speed Ratio (i) | Gear_Teeth_Ng / Worm_Starts_Nw | 25.00 (50.00 @ Nw = 1) |
| Static Back-Driving Status | TAN(lam) < mu_s * COS(phin) -> SELF-LOCKING | BACK-DRIVABLE (SELF-LOCKING @ Nw=1) |
Worm-Driving Sliding Efficiency & Static Self-Locking Formula
Validate integer N_w in [1, 8], N_g >= 15, Quotient q in [5, 25], phi_n in [14.5, 30], and 0 < mu_k <= mu_s <= 0.40!
=LET(nw_s, B2, ng_t, B3, q_dm, B4, ph_d, B5, mu_k, B6, mu_s, B7, IF(OR(nw_s < 1, nw_s > 8, INT(nw_s) <> nw_s, ng_t < 15, q_dm < 5, q_dm > 25, ph_d < 14.5, ph_d > 30, mu_k <= 0, mu_s < mu_k, mu_s > 0.4), "Invalid Worm Input", LET(tl_m, nw_s / q_dm, cp_h, COS(ph_d * PI() / 180), ROUND(((cp_h - mu_k * tl_m) / (cp_h + (mu_k / tl_m))) * 100, 2))))Single-Start Hoist Self-Locking vs Efficiency Tradeoff Check (Set Starts N_w = 1)
Switch Worm_Starts B2 from 2 down to a single-start worm (N_w = 1): Lead Angle E2 drops to 5.71 deg (tan = 0.10 < 0.1410), flipping Status E5 to "SELF-LOCKING" while Speed Ratio E4 doubles to 50.00 and Running Efficiency E3 drops to 64.92%!
=IF(ISTEXT(E3), E3, IF((B2 / B4) < B7 * COS(B5 * PI() / 180), "SELF-LOCKING", "BACK-DRIVABLE"))Try the example
Use the original values or change them to test your understanding.