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.

Consequences:
Note that mapping a strait-flank trapezoidal rack to a cylindrical coordinate systems
maps the straight flanks to pieces of archimedian spiral (i.e. radius growing linear with angle).
And these are only a decent approximation for true evolventsfor rarge gears with many teeth.
The stiffer the material the less forgiving.

• Would be very bad for steel. Long range transmitted vibrations at turning frequency times tooth number + overtones.
• Is bad for many 3D printed plastics as most are quite stiff.
• May in some cases be acceptable for nanoscale atomically precise gears as
  atomic interactions are not so much like sharp surfaces.
  Especially since the atomistic granularity might cause bigger deviations from the ideal shape anyway.

(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
• Plücker coordinates

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 (broken?)
https://ik.imagekit.io/agmamedia/issues/0814/spiral-bevel-gears.pdf via internet archive

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