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

How does a Tusi couple make exact straight-line motion? · Hardware · basis: Xem Studio design following Brown No. 329 (1868)

Also called: tusi couple, cardan straight line, hypocycloid straight line, cardan gear, par de Tusi, Kardankreis (Tusi-Paar)

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

sizes
4 in table

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.

Tusi couple: straight line from a gear rolling inside a ring twice… size chart

parametervalue
ring60 T internal
planet30 T
carrier15 mm
stroke60 mm, straight

Where the files are

Other publishers' files are linked, not hosted. "verify" = check the license at the link before commercial use.

Henry T. Brown (1868, public domain)Five Hundred and Seven Mechanical Movements — scan on archive.orgwebPublisher's page; dimensions are facts
507movements.comBrown's movement No. 329: figure, his text and an animationwebPublisher's page; dimensions are facts

Free files

★ Free files 1 parts. Generated by Xem Studio from the standard; threads are plain cylinders at major diameter. CC BY 4.0. index.json

tusi-couple-straight-line-mechanism-assemblyAssembled mechanism, all parts placed at one pose (STEP keeps each part separate)STEP · STL
Let your AI pull this. Add https://xemstudio.com/mcp to Claude, ChatGPT, Cursor or any MCP client (free, no account, one minute) and it can look up exact sizes, mating fits and free STEP/STL from every chat.
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Plain text: sheet.txt · JSON: /api/geometry?object=tusi-couple-straight-line-mechanism

Tusi couple: straight line from a gear rolling inside a ring twice its size, isometric render of a generated part
Render of tusi-couple-straight-line-mechanism-assembly from the free STEP; not to scale.

Fits with

Internal (ring) gear, module 1, 60 teeth, Ø80, isometric render of a generated part
Internal (ring) gear, module 1, 60 teeth, Ø8080.0×80.0×8.0 mm★ 1 file
Spur gears, module 1: 10 to 80 teeth (20° involute), isometric render of a generated part
Spur gears, module 1: 10 to 80 teeth (20° involute)26 sizes★ 26 files
Carrier arm, 15 mm centres, Ø10 holes, isometric render of a generated part
Carrier arm, 15 mm centres, Ø10 holes33.0×18.0×5.0 mm★ 1 file
Tracer arm, Ø10 hub, Ø3 pin at 15 mm, isometric render of a generated part
Tracer arm, Ø10 hub, Ø3 pin at 15 mm33.0×18.0×5.0 mm★ 1 file
Dowel pins, 2–6 mm (ISO 2338 / DIN 6325), isometric render of a generated part
Dowel pins, 2–6 mm (ISO 2338 / DIN 6325)5 sizes★ 23 files

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.