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Non Circular Gears

Non Circular Gears. Progress Report by Jeff Sch ö ner for CS285 May 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.

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Non Circular Gears

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  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

  7. 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.

  9. 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.

  11. 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.

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