Worm pairs

The Worm module rates cylindrical worm gear pairs with axes crossing at 90°: a hardened steel worm driving a bronze or cast iron wheel. It follows DIN 3996:2012-09, method C. Worm drives are usually limited by wear and temperature rather than by tooth breakage, so the module gives the efficiency and every loss, the oil sump temperature, the wear over the life, the pitting safety, the root safety of the worn tooth and the worm's deflection.

Inputs

Choose + Worm pair above the list (or Add worm pair in an empty module). The inspector groups:

Geometry

Label	Unit	Notes
Worm starts		z₁; more starts raise the efficiency.
Wheel teeth		z₂; the ratio is z₂/z₁.
Axial module m_x	mm
Diameter quotient q		The worm's mean diameter $d_{m 1} = q \cdot m_{x}$ .
Wheel profile shift x₂
Wheel face width b₂H	mm
Generating angle α₀	°	20 by default.
Flank form		A, N, K, I or C (DIN 3975); I by default.

Load and lubrication

Wheel torque (N·m): the nominal output torque T₂N; T₂ = K_A·T₂N.
Application factor KA, Worm speed (rpm), Life (h) and Starts per hour (0 in continuous duty).
Lubricant: Mineral oil (CLP), Polyalphaolefin or Polyglycol. The base oil changes the friction and the wear markedly.
Lubrication: Dip (oil sump) or Injection (then the injected oil flow is needed under Cooling).
Viscosity at 40 °C (220 mm²/s), Viscosity at 100 °C and Oil density at 15 °C (by lubricant when empty).
Friction μzm: empty takes the mean friction of DIN 3996, 7.4, from the reference gear's base value and the size, geometry, material and roughness factors.

Wheel material and requirements

Wheel material: CuSn12-C-GZ, CuSn12Ni2-C-GZ (the reference gear's), CuSn12Ni2-C-GC, CuAl10Fe5Ni5-C-GZ, EN-GJS-400-15 or EN-GJL-250. It supplies σHlim, τFlim, the modulus and the wear factors.
Accepted loss of wheel quality: None (reduced τFlim) or down to a grade from 8 to 12, which allows the full shear endurance limit.
Permissible wear: Backlash (k·m_x·cos γ) with Backlash factor k (0.3), Pointed tooth tip, Root safety 1.1 or Mass loss with a permissible mass loss in mg.
σHlim and τFlim: empty takes the material's values.
Min pitting safety (1.0), Min root safety (1.1), Min wear safety (1.1) and Min temperature safety (1.1), DIN 3996's minimums.

Details

Worm roughness Ra₁ (0.5 µm), wheel rim thickness and width, root diameters, Worm bearing span (2.5·d_m1 when empty) and First bearing to mesh for the deflection, Worm bearings (Adjusted pair or Locating / floating) and Shaft seals on the worm for their losses, Wheel dips into the oil, and the method B characteristic values pm, h and s* (empty: from the standard's equations for method C).

Cooling

Oil sump temperature: Heat balance (housing surface) with Housing surface and Heat transfer (10–15 still air, 20–30 fan) in W/(m²·K), or DIN 3996 method C (ribbed housing) with Fan on the worm shaft. Method C holds for a ribbed cast iron housing with centre distances of 63–400 mm, worm speeds of 60–3000 rpm and ratios of 10–40.
Ambient (20 °C) and Oil sump limit (90 °C mineral, 100 °C synthetic when empty).
With injection: Injected oil flow, Injection temperature and Oil temperature rise (3–5 K, cooler 10–20 K).

Results

The view card draws the wheel with the worm above it and shows the safeties: Pitting SH, Tooth root SF (worn), Wear SW, Temperature ST and Deflection Sδ. The key figures are the pitting safety, the wear safety with the wear life, the oil temperature and the efficiency (marked "self-locking" where that applies). Details lists the losses, sliding speed, flank pressure, wear against its limit, film thickness and wheel temperature, and notes where the pair leaves the standard's range.

In 3D the worm's starts are wound at their lead above the wheel at the centre distance, the wheel at its tooth count and face width; hovering turns the worm once and the wheel follows by starts over teeth. The 2D and 3D tabs above the drawing switch to a 3D model; the choice is kept for the module. In 3D, drag to orbit and scroll to zoom; the model stands still and plays its motion once when the pointer moves onto it or it gets the keyboard focus (never with reduced motion). Without WebGL the drawing stays, with a note.

The Worm drive template (two starts, 41 teeth, m_x 4 mm, q 10, b₂H 32 mm, 400 N·m output at 1450 rpm, CuSn12Ni2-C-GZ on polyglycol, dip lubrication, method C housing with a fan) gives:

Result	Value
Geometry	a 102 mm, i 20.5, d_m1 40 mm, d_m2 164 mm, γ_m 11.31°
Efficiency	82.59 % overall, mesh 89.35 % (wheel driving 88.19 %, starting 68.28 %)
Power	3.587 kW in, 2.963 kW out, 624.6 W lost (mesh 337.3 W, idling 149 W, bearings 83.68 W, seals 54.66 W)
Forces	wheel Ft 4878 N (worm axial), worm Ft 1181 N, radial 1968 N
Pitting	σHm 295.1 MPa of 452.9 MPa: SH 1.535
Wear	δWn 0.3391 mm of 1.177 mm (backlash): SW 3.47, wear life 63 096 h
Tooth root (worn)	τF 22.38 MPa of 90 MPa: SF 4.021
Worm deflection	3.382 µm against 80 µm
Oil sump	62.09 °C against 100 °C: ST 1.611

Reading the results

Efficiency: the mesh efficiency with the worm driving, the efficiency when the wheel drives (back-driving), and the starting efficiency with the friction at zero sliding speed. The overall efficiency adds idling, bearing and seal losses; with the wheel driving they come off what the mesh passes on to the worm, $(η_{z}^{'} \cdot P_{2} - losses) / P_{2}$ . A pair is reported self-locking when the wheel cannot drive the worm.
Wear: the wear depth δWn after the life from the wear intensity, which depends on the film thickness at the wheel's bulk temperature. SW is the permissible wear over the expected wear.
Pitting: the mean flank pressure σHm against the permissible value σHG with its life factors; the pitting life is the life at SH = 1.
Tooth root: the shear stress of the worn tooth against τFlim.
Temperature: the oil sump limit over the computed sump temperature.

The oil matters

The template on mineral oil instead of polyglycol: the efficiency falls to 79.57 %, the sump runs at 66.8 °C against a 90 °C limit (ST 1.347), and the wear grows to 1.648 mm, so the wear check fails with SW 0.7142 and a wear life of 12 986 h. A single-start worm on the same wheel (γ_m 5.711°) has an overall efficiency of 70.72 % and a starting efficiency of 52.87 %.

Fixing a failing check

Wear (SW): polyglycol or polyalphaolefin instead of mineral oil, a higher viscosity or a lower oil temperature (a thicker film), or a larger centre distance.
Temperature (ST): a fan or a larger housing surface, injection lubrication with a cooler, a synthetic oil (lower friction and a higher limit), more worm starts.
Pitting (SH): a larger centre distance or face width, a stronger bronze.
Tooth root (SF): a larger module or accept a loss of wheel quality where the application allows it.
Deflection: a shorter bearing span or a larger worm root diameter (higher q).

Standards and scope

DIN 3996:2012-09, method C: characteristic values pm, h and s* by the standard's equations (or given from a method B analysis), mean friction (7.4), losses and efficiency (7), wear (8), pitting (9), worm deflection (10), tooth root (11), oil sump temperature (12) and wheel bulk temperature (13). The rating reproduces the standard's three worked examples (axm verify).

Not covered: a scuffing check, the worm shaft's strength and its bearings' life (rate them in the shaft and bearings modules), and mounting the pair on the project's shafts.

From the command line

axm new worm.axm.json --template worm-drive
axm worm update worm.axm.json WG1 --patch '{"lubricant":"mineral"}'
axm solve worm.axm.json