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Arc lead in thread milling.

Pretty, pretty, pretty sure the last reference is incorrect. And you are correct also about the lead in angle of 90* from the arc center. I'm glad this makes no sense to you because it doesn't make sense. The lead-in z motion must be calculated from the thread center, not the lead-in arc center.
Oh how great it feels to read that. Thank you. I thought I'd go crazy thinking about what I didn't understand about it.

Interesting how such a popular book could have such an error.
 
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I also suspect an error in the book (Z in the lead-in/out being 2X what it should be) but would need to see the full example to be certain. There's also something really clever in there.

For the sake of argument, let's work in polar coordinates centered on the hole (theta, R, Z). Thread milling has R constant and theta and Z in a linear relationship. Then a lead-in/lead-out is simply a reduction in R while theta and Z do their thing, and the metal that is supposed to remain there is guaranteed safe.

The choice of a lead that is half the radius of the cut has an interesting property-- 90 degrees of lead is 45 from the center, 45 is 22.5 from center and so on. That relationship too is linear. So you end up at the right Z height for a given theta with R decreasing. It's a perfect lead-in/out! And realistically, R only has to decrease a little bit before you're out of the cut.

This linear relationship between the angles breaks down for other ratios of cut to lead radii. Even with linearity out the window, it is possible to calculate good enough values for the change in Z during the leads. All you need to do is maintain linearity between theta (from hole center) and Z for that short time during the transition from cutting to leading.

Using the theta from the center of the hole to endpoint of the lead as discussed in previous posts is really pretty good for certain cases but not exact. The approximation gets worse for small lead radii, and if you had a 180 degree lead out, it wouldn't work at all since that would land you on the X axis.

As far as I can tell this modification from what's written in the book works every time:

L_t=A_lead*P/360*R_lead/R_thread

You can see that if R_lead=R_thread, you have normal thread cutting with L_t being the pitch. As R_lead gets smaller, so does the amount of change in Z, as it should. A small radius 180 degree lead would work fine if you're so inclined. In any case I am a believer in making the leads pretty short, because the lead at the bottom of the hole (whether it's lead-in or lead-out) isn't making usable thread.
 
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Please look at my crummy sketch.

The red dots might be called the "apparent cutting point" and the little hollow circles the "actual cutting point". Any reference to the helix angle of the lead-in arc seems to assume the cutting point is the apparent point, which is demonstrably wrong. The final cut takes place at the actual cutting point. The only place the apparent and actual cutting point are coincident is point 5.

One side of the actual cutting point will be above the lead and one side below, but if the cutter is sufficiently small with respect to the major diameter, and the thread angle sufficiently large, like 60 degrees, these areas are machined away by the actual cutting point, which is a function of only the angle from thread start and the lead.

This makes it easier to understand while low thread angles, like acme, might have some form error, and square threads will always show error. Even very large square threads with very small lead cut by thread milling will have form error, however slight.
20230903_200143.jpg
 
If you take lead in radius and calculate the arc length of that lead in, then calculate what this arc length corresponds to in degrees from hole center if you were to put the arc segment on the thread diameter instead, you can calculate the Z value from that.

Effectively the same thing but easier to do it like that if the start of your lead in is below centerline of the hole.

If your threadmill is relatively large, I think you would want a 180° lead in arc. Say starting point 0.5mm from thread minor diameter. Then the above method works well. Also this would result in a bigger arc lead in radius which is closer to the end result.
 
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