### fixing a gear ratio problem

After various mechanical tweaks to allow too-tight gears to turn, the prototype is starting to come to life. But even just moving a number from one digit stack to another disclosed a bug: in the process it multiplies by 1.35!

Sort of. Here's the deal...

Changing a number wheel by one digit up or down means rotating it about the axis by 360/(3*10) = 12 degrees, because there are three repetitions of the digits 0 through 9 around the circumference. Since there are 30 teeth on a digit wheel, each full tooth movement represents a change of one digit.

When a chain of gears is linked together, a movement by exactly one tooth is propagated unchanged through all the gears -- regardless of the diameter and number of teeth of the intermediate gears -- as long as all the gears have the same tooth pitch. That is a parameter called the Diametral Pitch (DP), which is the number of teeth per inch of diameter of the contact circle of the gears.

Following Babbage's Plans from 13 through 28, I used a DP of 10 consistently. I naively assumed, therefore, that one tooth movement at the beginning of a chain would create one tooth movement at the end, regardless of the gear sequence.

Wrong! It works for any combination meshed gears on separate axes, but it fails when two gears are concentric on a single axis and are locked together. If those gears have different diameters but the same DP, they will have a different number of teeth. A movement by one tooth on the smaller gear will obviously produce a movement by more than one tooth on the larger gear.

If the larger gears on the concentric shaft only ever mate with gears of the same size, as that drawing shows, then everything works fine. But my design violated that rule for the short pinion that connects a long pinion to the number stack of the anticipating carriage. Just for convenience of the layout, it linked to the larger of the pinion's gears. I didn't realize that would cause a problem, but since the small gear has 20 teeth and the large gear has 27 teeth, transferring through that pinion make one tooth movement become 27/20 = 1.35 of a tooth movement. (Ok, that's not exactly "multiplication", but it's not good.)

I have now remade those pinion gears to be slightly larger so they can reach the inner small gear of the long pinions, which is what is linked to the digit wheels of the stack on the left. I also had to lower all the anticipating carriage axes by 1/2" so the meshes line up.

Tim Robinson and I had previously noted that Babbage violates the "all gears have DP 10" rule for the long pinions, as shown on the following snippet from drawing M/008 of 1837.

The larger gears have the same number of teeth as the smaller gears, and hence a smaller DP.

If larger gears only ever mesh with larger gears, that isn't technically necessary to keep the gear ratios correct. But as Tim points out, it __is__ necessary so that when the movable long pinions change meshing by being raised or lowered, the gear teeth are aligned and slip neatly into one another.

My remade long pinions now have DP 10 for the small 2" diameter gears, and DP 7.407 for the larger 2.7" diameter gears, so they both have 20 teeth. I can now transfer the number 123 and get 123, and the long pinions have hope of meshing correctly for shifts.

On to the next bug.

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