# Tusi couple: straight line from a gear rolling inside a ring twice its size

> How does a Tusi couple make exact straight-line motion?

- Category: Mechanisms › Gears
- Kind: Hardware
- Basis: Xem Studio design following Brown No. 329 (1868)
- Page: https://xemstudio.com/objects/tusi-couple-straight-line-mechanism/ · JSON: https://xemstudio.com/api/geometry?object=tusi-couple-straight-line-mechanism · Plain text: https://xemstudio.com/objects/tusi-couple-straight-line-mechanism/sheet.txt
- License: dimension sheet CC BY 4.0 (Xem Studio); linked files under their publishers' licenses
- 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

- A planet of half the ring's pitch diameter rolls inside it (30 T in 60 T, carrier 15 mm): any point on the planet's pitch circle moves on a straight line through the ring's centre, 4 × 15 = 60 mm long — exact, not approximate.
- At the drawn carrier angle 40° the planet has turned −40° and the tracer pin sits at x = 22.98, y = 0; x = 60/2 · cos θ.
- Brown's No. 329 uses it to guide a piston rod without slides (cog-wheel inside a fixed internal gear of twice its diameter).
- Parts (each has its own page): internal-ring-gear-m1-60t, spur-gears-module-1, tusi-carrier-15, tusi-tracer-arm-15, dowel-pins-metric.
- Brown's movement No. 329 (1868) shows it; this model is a Xem Studio design following his figure, every part at true size.

## Sizes

| parameter | value |
|---|---|
| ring | 60 T internal |
| planet | 30 T |
| carrier | 15 mm |
| stroke | 60 mm, straight |

## 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. 329: figure, his text and an animation](https://507movements.com/mm_329.html) — web — Publisher's page; dimensions are facts

## Free STEP/STL (1 parts, CC BY 4.0, Xem Studio)

Index: https://xemstudio.com/files/hardware/tusi-couple-straight-line-mechanism/index.json

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

## Questions

**How does a Tusi couple make exact straight-line motion?**

A planet of half the ring's pitch diameter rolls inside it (30 T in 60 T, carrier 15 mm): any point on the planet's pitch circle moves on a straight line through the ring's centre, 4 × 15 = 60 mm long — exact, not approximate.

## Fits with

- [Internal (ring) gear, module 1, 60 teeth, Ø80](https://xemstudio.com/objects/internal-ring-gear-m1-60t/)
- [Spur gears, module 1: 10 to 80 teeth (20° involute)](https://xemstudio.com/objects/spur-gears-module-1/)
- [Carrier arm, 15 mm centres, Ø10 holes](https://xemstudio.com/objects/tusi-carrier-15/)
- [Tracer arm, Ø10 hub, Ø3 pin at 15 mm](https://xemstudio.com/objects/tusi-tracer-arm-15/)
- [Dowel pins, 2–6 mm (ISO 2338 / DIN 6325)](https://xemstudio.com/objects/dowel-pins-metric/)