Construction of a first lead screw?

[sfriedberg]

I still don't quite understand how an iterative process allows you to take a lead screw that's out by such and such tolerance and continue running it through a process until it reaches some arbitrary degree of accuracy,

So one technique that was used is error mapping, combined with a mechanical cam cut to compensate for the measured error at a particular position along the existing screw. This was not used on lathes used to make arbitrary parts, just on special lathes used to make precision lead screws. Basically, the feed nut was not held rigidly by the apron. It was pinned w.r.t left/right motion, but allowed to rotate on the existing screw. A cam follower was attached to the feed nut. A long linear cam was fastened to the bed. The cam follower (and thus feed nut rotation) was controlled by the linear cam. Because the amount of feed nut rotation was obviously limited, you could not take out large scale pitch errors this way. But you could readily correct for pitch variations by making an appropriate cam.
Another technique, which was used with less precise starting screws, was to have the carriage with two separated single-tooth followers, rather than a feed nut. This gives you an averaging behavior, and if you iterate the process, you will end up with a more uniform pitch along the final lead screw. What you won't have is good control over the final pitch, just that it's more uniform. Also you lose effective screw length on each generation, so you start with a substantially longer screw, or at some point in the process you reposition the carriage and pick up a thread on the screw you are cutting to extend it.
[Added in edit] Ah, I see Joe Gwinn described the first technique, and something closely related to the 2nd technique, back in post #5.

 

[rwa2004]

I always had an idea, but never looked into it. Couldn't you make a very accurate reliable first screw by wrapping a cylinder with a accurate shape such as a wire?

It is common to have very accurate wire diameter fabricated. If you take a rod and spiral wrap the wire around it, you will have an accurate screw with the pitch equal to the wide diameter.

 

[sfriedberg]

Couldn't you make a very accurate reliable first screw by wrapping a cylinder with a accurate shape such as a wire?

That is one way "first generation" screws were made. Both the wire and the mandrel it's wrapped around have to be accurate, and you have to compensate for the distortion of the wire cross-section as it's bent. You also cannot transmit much thrust with such a screw, but it can be enough for a specialized generating machine.

But we may have different criteria for "very accurate". For some people a "very accurate screw" means pitch accuracy of 1 or 2 millionths of an inch, over both short and long distances. For others it may mean pitch accuracy of merely 10 millionths of an inch. To put those numbers in context, Nook Industries offers ball screws in three accuracy categories: rolled 0.004"/ft, rolled 0.001"/ft, and ground 0.0005"/ft. I would guess that people in metrology and high-precision production would consider none of those to be "very accurate", but they are certainly acceptable for building machines with some other means of measuring axis travels, such as DRO scales. Typical commercial grade Acme screws are in the 0.004"/ft category, some a bit better, some a bit worse, and you can get ground Acme screws in the 0.0005"/ft category, too.

The wire wrapping technique is quite venerable. Blacksmiths made screws for vises and similar equipment by hand-forging and -welding square cross-section rod to a core, then finishing the result with files. But in that context, backlash of 0.025" between screw and nut would be quite "precise", and the fit was frequently much worse. In the context of precision lead screws, we want 10 times better than that for run-of-the-mill commercial lathes, and 100 times better than that for precision machines, plus pitch accuracy.

 

[john.k]

By the simple principle of high points wearing away first ,a screw in use ,as an olive or wine press ,will gradually become more uniform