# Conoid (curved cone) pulleys: varied speed with a belt of constant length

> Why are the cones curved?

- Category: Mechanisms › Belt and friction drives
- Kind: Hardware
- Basis: Xem Studio design following Brown No. 10 (1868)
- Page: https://xemstudio.com/objects/conoid-pulleys-mechanism/ · JSON: https://xemstudio.com/api/geometry?object=conoid-pulleys-mechanism · Plain text: https://xemstudio.com/objects/conoid-pulleys-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

- Two straight cones head to tail keep r₁ + r₂ constant, which keeps a CROSSED belt the same length but not an open one: an open belt goes slack away from the middle. Curving the cones by the right amount makes the open-belt length exactly constant.
- Here Ø24–Ø56 × 64 conoids at 110 mm; at 20 mm along, Ø27.9 drives Ø51.9, ratio 0.54. The belt length 346.65 mm is the same at every position (straight cones of the same size lose -4.64 mm near the ends).
- Brown's No. 10 (cone pulleys of curved outline for gradually varying speed).
- Parts (each has its own page): conoid-pulley-24-56x64-bore8, flat-belt-10x1-5-conoid-pulleys-c110.
- Brown's movement No. 10 (1868) shows it; this model is a Xem Studio design following his figure, every part at true size.

## Sizes

| parameter | value |
|---|---|
| pulleys | 2 × Ø24–Ø56 × 64 conoids, head to tail |
| centres | 110 mm |
| ratio range | 0.43 – 2.33 (approx.) |
| drawn at 20 mm | ratio 0.54 |
| belt | 10 × 1.5, 346.65 mm |

## 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. 10: figure, his text and an animation](https://507movements.com/mm_010.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/conoid-pulleys-mechanism/index.json · license: https://xemstudio.com/files/hardware/conoid-pulleys-mechanism/LICENSE.md

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

## Questions

**Why are the cones curved?**

Two straight cones head to tail keep r₁ + r₂ constant, which keeps a CROSSED belt the same length but not an open one: an open belt goes slack away from the middle. Curving the cones by the right amount makes the open-belt length exactly constant.

## Fits with

- [Conoid (curved cone) pulley Ø24–Ø56 × 64, profile for constant open-belt length at 110 mm](https://xemstudio.com/objects/conoid-pulley-24-56x64-bore8/)
- [Flat belt 10 × 1.5 for two conoid pulleys at 110 mm (loop drawn at 20 mm along)](https://xemstudio.com/objects/flat-belt-10x1-5-conoid-pulleys-c110/)