Generalized gears

There is a completely general way
to generate gear pairs for any combination of axes.

Axes can be …

• parallel (spur gear pairs)
• antiparallel (spur gear in ring gear)
• angleled and crossing (conical bevel gears)
• angled and not crossing (hyperboloid gears)

This is stated to only be …

• only solved for cycloid profiles though and
• not for evolvent or other profiles.

(wiki-TODO: Implementation details so far understood.)

Imprecise naive approach

Factoring the tooth profiles apart from the rolling surface.

First: Modelling the teeth profiles as a rack (cycloidal or trapezoidal)
modeling it highly subdivided even for flat surfaces. Second: Doing for all the vertices simply a coordinate transformation to the rolling surfaces given as a coordinate system with x & y on the spanning the rolling surface z normal to it for the local tooth profile height.

(wiki-TODO: Publish cleaned up existing OpenSCAD implementation and link to it from here.) (wiki-TODO: Add illustrative demo image of results.)

Related

• Atomically precise gears
• 3D modeling

External links

• Plücker coordinates
• Screw theory
• Dual numbers

References

Winfough2014

W. R. Winfough and D. B. Dooner, "Testing of Alternative Spiral Bevel and Hypoid Gear Theory", Power Transmission Engineering, August 2014, pp. 30-36. https://www.powertransmission.com/ext/resources/issues/0814/spiral-bevel-gears.pdf

Figliolini2013

Giorgio Figliolini, Hellmuth Stachel, Jorge Angeles, "On the Synthesis of Spatial Cycloidal Gears", Meccanica, Vol. 48, pp. 1239–1249, 2013. https://doi.org/10.1007/s11012-012-9664-9

Figliolini2006

Giorgio Figliolini, Jorge Angeles, "The Synthesis of the Pitch Surfaces of Internal and External Skew-Gears and Their Racks", Journal of Mechanical Design, Vol. 128, No. 4, pp. 794–802, July 2006. https://doi.org/10.1115/1.2202875

People & institutions

Hellmuth Stachel
https://en.wikipedia.org/wiki/Hellmuth_Stachel
https://www.geometrie.tuwien.ac.at/stachel/
https://www.geometrie.tuwien.ac.at/

Giorgio Figliolini
https://dblp.org/pid/39/1517.html
https://www.researchgate.net/profile/Giorgio-Figliolini https://ieeexplore.ieee.org/author/37727132100

Jorge Angeles
https://www.mcgill.ca/mecheng/people/faculty/staff/jorgeangeles

W. R. Winfough
https://www.scopus.com/authid/detail.uri?authorId=6507515209

D. B. Dooner
https://www.researchgate.net/scientific-contributions/David-B-Dooner-82040915
https://www.uprm.edu/inme/people/faculty/dooner-david-b/ via internetarchive