Non Circular Gears

Progress Reportby Jeff Schöner

for CS285May 6, 2002

Review

Circular gears are well-understood. Non-circular gears are not, which makes them interesting.

I intended to produce 3 sets of NC gears.

Original purpose: art with industrial side-effects

Achievements

A general system that can generate elliptical (and perhaps other) gears.

Python program that produces SLF

Output passes the SIF test.

STL ready for first fabrication.

Problems

Ellipses do not have a closed form description of their perimeter or arc length.

Placing teeth is dramatically complicated. Approximations: How good do they need to be?

Not much literature on NC gears. Only one chapter in one book in the library.

Most gear texts discuss only how to make gears using existing machines.

Hard to find a mathematical description of involute curves.

More problems

Original naive algorithm did not work. Rolling distance must be taken into account as well as

angular rotation. Algorithm could be (and may still be) reworked. However, generating the shape description is not

nearly as difficult as creating an accuprate boundary representation.

Designing general software makes everything more complicated at first.

Ellipse Solutions

Representation Several parameters Two polar representations With one, placing hole is easier. With the other, computing curvature easier. Maxima makes computing nasty derivates

easier, although mistakes crop up in the data entry.

Images from http://mathworld.wolfram.com, Wolfram Research, makers of Mathematica

Ellipse Solutions: Placing Teeth

Perimeter and arc length contain elliptic integrals.

In math, just use E(t,k). In a computer, you need rational values.

Convert ellipse into a n-sided polygon.

Gears don't really have to be “curved”. In fact, must be a bunch of triangles in the end.

Ellipse Solutions: Placing Teeth

Algorithm Approximate the perimeter using a method like

Ramanujan's Divide by the number of teeth to get circular pitch. Set delta theta to something like 0.001 Walk in delta theta-sized steps along the perimeter,

marking section boundaries. Compute error. Refine value linearly. Repeat until no error or values cycle.

What remains to be done?

Fix some tooth orientation issues that don't occur with elliptical gears, but perhaps others.

Teeth need to be rotated away from the center of the gear.

Design 2 more sets of gears Ellipse driving an oval Oval driving an oval

FDM some real parts and make sure they work.

All original goals still seem do-able.

Conclusions: I've learned...

Gears may be well-understood, but textbooks are typically not very concerned with theory.

Current methods work, so new ones not in demand.

I don't know enough about mathematics as I'd like. I've forgotten a lot too.

A lot about ellipses, curvature, radii of curvature, involute curvers (circular and otherwise).

Where (and how) standard circular gear theory can be generalized and where it can't.

Conclusions: What would I do differently?

Structured my checkpoints differently. Learning theory of shapes and teeth proved to be not

as useful as I thought. Making the software took much more time than

expected. Coding approximations proved to be time consuming.

Focus more on boundary construction than on polar equation generation.