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I'm in the works of writing my own gear program, but I've stumbled on something that textbooks and the internet don't seem to answer. A lot of the time, the dedendum of a gear cuts into the base circle. When this happens, the curve below the base circle cannot be involute. Here is an example:

enter image description here

I'm using a simple line at the moment, but the actual curve looks closer to a fillet/spline curve. Here is an example of what it should look like:

enter image description here

This image comes from Shigley's Machine Design. This phenomena is known as undercutting and I try to avoid it, but sometimes I am limited by size and have to design a gear with a little undercutting.

I was wondering if anyone had any knowledge of this before I just make up some spline relationship.

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Depending on your goal, a simple spline or circular arc should be sufficient. The actual geometry of the root below the involute flank depends on the manufacturing process. While you might be able to design an optimized geometry for a profile milled gear, for meshing processes like hobbing the geometry depends on the meshing and determination is not trivial. The actual shape of the root is often simplified as long as you don't have to look closely at the maximum of root stresses. I know of programms "just plotting" the edge of a gear rack meshing with the gear as a good approximation of a hobbed root fillet.

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  • $\begingroup$ I ended up using the same involute curve, but I reflected the curve about a horizontal axis going through the base circle. It seems to be working fine. Thanks for your input! $\endgroup$ May 11, 2023 at 19:43

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