# Worm and worm wheel, module 1.5, 1-start, 30 : 1

> How do a worm and worm wheel work?

- Category: Mechanisms › Gears
- Kind: Hardware
- Basis: Xem Studio design following Brown No. 31 (1868)
- Page: https://xemstudio.com/objects/worm-and-worm-wheel-mechanism/ · JSON: https://xemstudio.com/api/geometry?object=worm-and-worm-wheel-mechanism · Plain text: https://xemstudio.com/objects/worm-and-worm-wheel-mechanism/sheet.txt
- License: dimension sheet CC BY 4.0 (Xem Studio); files Xem Studio Design License (free to use and sell from, not to redistribute as files: https://xemstudio.com/license/)
- Use in every chat: add https://xemstudio.com/mcp as a connector (https://xemstudio.com/use-with-ai/)

**Exact dimensions.** True size, with no printing tolerance added.

## Notes

- Resize: multiply every length by one factor k and the motion stays identical: angles, ratios, timing and the formulas below do not change, because a mechanism's motion depends only on the ratios of its lengths. Here, scales with k: centre distance; stays the same: worm, wheel, ratio; step to the nearest standard size (and redraw the hole or pocket it fits): shafts.
- Resize: Gears scale by the module m (pitch Ø = m·z, centre distance = m·(z₁ + z₂)/2): keep the tooth counts so the ratio stays, and choose a standard module (0.5, 0.8, 1, 1.25, 1.5, 2, 2.5, 3).
- Resize: fit clearances stay in mm (a slide fit is the same gap at any size); keep every printed wall and pin at least two nozzle widths; mass grows with k³ but part strength only with k², so at k = 2 parts are about 8× heavier and 4× stronger: check pins and thin links when scaling up.
- An endless screw (the worm) on one shaft meshes with a toothed wheel on a shaft at right angles: each turn of the worm advances the wheel one tooth (one thread), so the ratio is the tooth count — 30 : 1 here — in a single, quiet stage. The same result as the face-spiral drive of Brown's No. 29, but simpler to make, so far more used.
- Worm: involute helicoid (ZI), axial module 1.5, single start, pitch Ø16, lead angle 5.36°. Wheel: 30-tooth involute helical gear with helix angle = lead angle, same hand — a non-throated worm wheel, so the pair is an exactly conjugate crossed-helical pair (point contact) at centre distance 30.5 mm. Checked at nine positions over 1.5 worm turns: no overlap beyond contact.
- With a lead angle of 5.36° (below ~6°) the drive is self-locking: the wheel cannot drive the worm. Brown's No. 31; drawn with the worm turned 30°.
- Parts (each has its own page): worm-zi-m1-5-1start-d16-l40-bore6, worm-wheel-helical-m1-5-30t-for-1start-d16, linear-shafts-h6.
- Brown's movement No. 31 (1868) shows it; this model is a Xem Studio design following his figure, every part at true size.

## Sizes

| parameter | value |
|---|---|
| worm | module 1.5, 1-start, pitch Ø16, lead angle 5.36° |
| wheel | 30 T helical, helix 5.36° |
| ratio | 30 : 1 |
| centre distance | 30.5 mm |
| shafts | 90°, non-intersecting |

## Where the files are

- Henry T. Brown (1868, public domain): [Five Hundred and Seven Mechanical Movements — scan on archive.org](https://archive.org/details/fivehundredseven00browiala) — web — Publisher's page; dimensions are facts
- 507movements.com: [Brown's movement No. 31: figure, his text and an animation](https://507movements.com/mm_031.html) — web — Publisher's page; dimensions are facts

## Free STEP/STL (1 parts, Xem Studio Design License, Xem Studio)

Index: https://xemstudio.com/files/hardware/worm-and-worm-wheel-mechanism/index.json · license: https://xemstudio.com/files/hardware/worm-and-worm-wheel-mechanism/LICENSE.md

- worm-and-worm-wheel-mechanism-assembly: [STEP](https://xemstudio.com/files/hardware/worm-and-worm-wheel-mechanism/worm-and-worm-wheel-mechanism-assembly.step) · [STL](https://xemstudio.com/files/hardware/worm-and-worm-wheel-mechanism/worm-and-worm-wheel-mechanism-assembly.stl) — Assembled mechanism, all parts placed at one pose (STEP keeps each part separate)

## Questions

**How do a worm and worm wheel work?**

An endless screw (the worm) on one shaft meshes with a toothed wheel on a shaft at right angles: each turn of the worm advances the wheel one tooth (one thread), so the ratio is the tooth count — 30 : 1 here — in a single, quiet stage. The same result as the face-spiral drive of Brown's No. 29, but simpler to make, so far more used.

**What is the ratio of a worm drive?**

Wheel teeth divided by worm starts: a single-start worm and a 30-tooth wheel give 30 : 1 in one stage.

**Why is the worm wheel a helical gear here?**

A worm is a helical gear with very few teeth; a helical gear of the same normal module and hand, with helix angle equal to the worm's lead angle, meshes with it conjugately on crossed axes. Throated (hobbed) wheels add line contact for power, at the cost of exact axial positioning.

## Fits with

- [Worm (involute helicoid, ZI), axial module 1.5, 1-start, pitch Ø16 × 40, Ø6 bore](https://xemstudio.com/objects/worm-zi-m1-5-1start-d16-l40-bore6/)
- [Worm wheel (helical, non-throated), module 1.5, 30 teeth, helix 5.36° RH, for the 1-start Ø16 worm](https://xemstudio.com/objects/worm-wheel-helical-m1-5-30t-for-1start-d16/)
- [Linear shafts Ø3 to Ø25 mm (h6), 100 to 1000 mm long](https://xemstudio.com/objects/linear-shafts-h6/)