It’s all relative

I realise I threw around one particular gear ratio without any explanation.

It is usually considered important to maximise the potential life of 3D-printed gears. If (for example) one tooth happens to be misshapen, then it will apply slightly different pressure to each tooth that it meshes with than will the other teeth on that same gear. If the misshapen tooth meshes with only a subset of the teeth on the other gear then over time those teeth will accumulate a wear pattern different from the others and the motion of the driven gear will become irregular.

To ensure that wear is distributed evenly, we arrange for each tooth on the driving gear to mesh with every tooth on the driven gear once before meshing with any of them a second time. This is not as difficult as it might seem – the numbers of teeth on the two gears simply need to be relatively prime. Neither number need actually be prime, so 10 and 51 (both composite) having no common factors larger than 1 means they qualify, and they are slightly closer to 5:1 than 10 and 49.

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