rethinking gear teeth

The prototype mechanism has been reinstalled using the new framing plates made on an NC milling machine, and there is clearly an improvement. The gears move more smoothly and, unlike before, perform the same on top as on bottom, thanks to the now precise and vertical axle positioning.

But the crippling backlash problem remains, despite the fact that the center-to-center spacing is now nominal for the gear diameters, not expanded. The problem, I've concluded, is in the gear tooth design.

I've been naively using the Solidworks "Toolkbox" to design what I expected to be gears with standard 20-degree pressure angle involute teeth. Now I discover this disclaimer in the fine print of their documentation: "Toolbox gears are representations for machine design purposes. They are not true involute gears that you can use for manufacturing."  Nice.

What's more, there doesn't seem to be a backlash adjustment in their complicated set of equations, which are anyway impossible to modify. So it's time to abandon the Toolbox and design gear teeth from scratch.

Luckily there are many helpful resources on the web. I adopted an 8-year-old scheme by Vinod Rex (https://grabcad.com/library/parametric-involute-gear-1) that uses the standard parametric equations for involute curves, and also has easily modifiable parameters for backlash, dedendum expansion, and more. I set about experimenting, using a test jig to measure the backlash in a 5-gear train.

The Toolbox-designed gear chain has a cumulative backlash of about 3/4 of a tooth. The best of the four alternate true involute gear designs I tried, which uses a 14.5-degree pressure angle, 0 backlash, and 15% dedendum expansion, has cumulative backlash of about 1/3 of a tooth, which is a considerable improvement. 

The various designs trade off backlash for smoothness of motion. There may be better compromises, but I'm  going to stop experimenting for now and try the new tooth design in a complete set of nine gears for one decimal position of the actual prototype: two digit wheels, two long pinion connectors, fixed and movable long pinions,  a carriage connector pinion, a carriage digit wheel, and a carry sector. Then I can see how it compares to the existing Toolbox-designed chain in the other decimal positions.

Here's a comparison of the old (left) and new (right) gear teeth:

The irony is that the new teeth look much more like those in some of Babbage's drawings!