# Bevel gears, module 1.5, 20 : 40, shafts at 90°

> How do bevel gears turn motion through an angle?

- Category: Mechanisms › Gears
- Kind: Hardware
- Basis: Xem Studio design following Brown No. 25 (1868)
- Page: https://xemstudio.com/objects/bevel-gears-mechanism/ · JSON: https://xemstudio.com/api/geometry?object=bevel-gears-mechanism · Plain text: https://xemstudio.com/objects/bevel-gears-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 cone-shaped gears whose pitch cones share an apex roll on each other like friction cones: the shafts meet at the apex at any angle (90° here) and the ratio is the tooth ratio, 40 : 20 = 2 : 1. Equal gears at 90° are mitre gears.
- Pitch cone angles from tan δ₁ = sin Σ / (N₂/N₁ + cos Σ): 26.57° and 63.43°; cone distance 33.54 mm, face 11.2 mm. The teeth are exact spherical involutes, generated on a sphere about the apex, so the mesh is conjugate at every position; drawn turned 4°.
- Brown's No. 25 (bevel gears for shafts at an angle; mitre gears when the two are equal).
- Parts (each has its own page): bevel-gear-m1-5-20t-for-40t-90deg, bevel-gear-m1-5-40t-for-20t-90deg, linear-shafts-h6.
- Brown's movement No. 25 (1868) shows it; this model is a Xem Studio design following his figure, every part at true size.

## Sizes

| parameter | value |
|---|---|
| module (large end) | 1.5 |
| teeth | 20 / 40 |
| ratio | 2 : 1 |
| shaft angle | 90° |
| pitch cones | 26.57° / 63.43° |
| cone distance | 33.54 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. 25: figure, his text and an animation](https://507movements.com/mm_025.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/bevel-gears-mechanism/index.json · license: https://xemstudio.com/files/hardware/bevel-gears-mechanism/LICENSE.md

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

## Questions

**How do bevel gears turn motion through an angle?**

Two cone-shaped gears whose pitch cones share an apex roll on each other like friction cones: the shafts meet at the apex at any angle (90° here) and the ratio is the tooth ratio, 40 : 20 = 2 : 1. Equal gears at 90° are mitre gears.

**What is the pitch cone angle of a 20 and 40 tooth bevel pair at 90°?**

tan δ = 20/40: 26.57° for the pinion and 63.43° for the wheel.

**Why are bevel teeth spherical involutes?**

Every point of a bevel gear moves on a sphere about the apex; the tooth shape that rolls conjugately on that sphere is the spherical involute (Tredgold's back-cone method approximates it).

## Fits with

- [Bevel pinion, module 1.5, 20 teeth (for a 40-tooth wheel at 90°), spherical involute](https://xemstudio.com/objects/bevel-gear-m1-5-20t-for-40t-90deg/)
- [Bevel wheel, module 1.5, 40 teeth (for a 20-tooth pinion at 90°), spherical involute](https://xemstudio.com/objects/bevel-gear-m1-5-40t-for-20t-90deg/)
- [Linear shafts Ø3 to Ø25 mm (h6), 100 to 1000 mm long](https://xemstudio.com/objects/linear-shafts-h6/)