Non Circular Gears

1
Non Circular Gears
Progress Report by Jeff Schöner for CS285 May 6,
2002
2
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

3
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.

4
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.

5
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.

6
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
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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.

8
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.

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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.

10
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.

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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.