# Face spiral (scroll) on a disc turning a wheel one tooth per turn (24 : 1)

> How does a face spiral drive a toothed wheel?

- Category: Mechanisms › Gears
- Kind: Hardware
- Basis: Xem Studio design following Brown No. 29 (1868)
- Page: https://xemstudio.com/objects/face-spiral-disc-and-wheel-mechanism/ · JSON: https://xemstudio.com/api/geometry?object=face-spiral-disc-and-wheel-mechanism · Plain text: https://xemstudio.com/objects/face-spiral-disc-and-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: wheel axis; stays the same: disc, wheel, ratio; step to the nearest standard size (and redraw the hole or pocket it fits): pitch.
- 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.
- A single raised spiral on the face of a disc engages the teeth of a wheel whose axis lies across the face: as the disc turns, the rib at the wheel moves outward one pitch per turn and pushes the wheel on by one tooth — a worm wound flat on a disc. Ratio 24 : 1 for a 24-tooth wheel; the shafts are at right angles. Brown's No. 29; he notes the worm and worm wheel (No. 31) does the same more simply.
- The rib's radial section is the module 1.5 basic rack (pitch 4.71 mm), so in the plane through the disc's centre the wheel sees an ordinary rack. Across the wheel's 8 mm face the rib curves away and its section widens slightly, so the wheel's teeth were generated as the envelope of the rib's motion in seven planes across the face and lofted: an exact mate. Checked at eight positions over two turns of the disc: no overlap beyond contact.
- Drawn with the wheel's axis over radius 39 of the disc, 19.88 mm above the face, and the disc turned 60°.
- Parts (each has its own page): face-spiral-disc-m1-5-r30-4turns-d104, face-spiral-wheel-m1-5-24t-for-scroll-r39, linear-shafts-h6.
- Brown's movement No. 29 (1868) shows it; this model is a Xem Studio design following his figure, every part at true size.

## Sizes

| parameter | value |
|---|---|
| disc | Ø104 × 6, spiral r 30 → 48.85 (4 turns) |
| pitch | 4.71 mm per turn |
| wheel | 24 T, module 1.5, generated |
| ratio | 24 : 1 |
| wheel axis | 19.88 mm above the face, over r = 39 |

## 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. 29: figure, his text and an animation](https://507movements.com/mm_029.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/face-spiral-disc-and-wheel-mechanism/index.json · license: https://xemstudio.com/files/hardware/face-spiral-disc-and-wheel-mechanism/LICENSE.md

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

## Questions

**How does a face spiral drive a toothed wheel?**

A single raised spiral on the face of a disc engages the teeth of a wheel whose axis lies across the face: as the disc turns, the rib at the wheel moves outward one pitch per turn and pushes the wheel on by one tooth — a worm wound flat on a disc. Ratio 24 : 1 for a 24-tooth wheel; the shafts are at right angles. Brown's No. 29; he notes the worm and worm wheel (No. 31) does the same more simply.

**Is a face spiral the same as a worm?**

Kinematically yes: both advance the wheel one tooth per turn. The worm wraps the thread round a cylinder and keeps the same geometry along its length; the face spiral's rib changes radius every turn, so the wheel can only run over a limited band of the disc.

## Fits with

- [Face spiral (scroll) disc, pitch 4.71 (module 1.5), r 30 → 48.85, Ø104 × 6](https://xemstudio.com/objects/face-spiral-disc-m1-5-r30-4turns-d104/)
- [Wheel for a face spiral, module 1.5, 24 teeth, generated by the scroll at radius 39](https://xemstudio.com/objects/face-spiral-wheel-m1-5-24t-for-scroll-r39/)
- [Linear shafts Ø3 to Ø25 mm (h6), 100 to 1000 mm long](https://xemstudio.com/objects/linear-shafts-h6/)