But in this case, as before, if the lead screw had a square thread and the nut threads completely filled the spaces between the lead screw threads, then the nut could not engage at the 2nd, 3rd, or 4th work revolution, hence the false threads b, c, and d, could not have been cut, even though the feed nut was disengaged and the lathe carriage was traversed back by hand.
Fig. 618.
Now, suppose that two threads on the work measure less than the amount the lead screw advances during the time that the work makes a revolution, and if the lead screw has a V-shaped thread, the case is altered. We have, for example, in [Fig. 618], a pitch of lead screw of 3 to cut 12 and 13 threads respectively. In the case of the 13 threads it will be seen that, supposing there to have been a first cut taken on the work, and the feed nut to be disengaged while the work makes a revolution, then the lead screw will revolve 3⁄13 revolution and the point a on the lead screw will have moved up to point b, and the nut point remaining at n, seating it in the thread, would cause it to engage with the same thread that it did before, and no second thread would be cut. If the nut be then released, the work allowed to make another revolution and the nut again closed, the operation would be the same as before, and no error would be induced, and so on. Suppose, further, that after the nut was disengaged the lathe was permitted to make two revolutions, and the lead screw would make 6⁄13, or less than half a turn, and closing it would still cause it to pass back into the same thread on the lead screw and produce correct work. But if after the nut was released the work made three turns, the lead screw would make 9⁄13 of a turn, and the nut would fall on the right-hand side of the lead screw thread, and in closing would move the lathe carriage to the right, causing the tool to cut a second thread. Now, the same operation that occurred with the first thread would during the next three trials occur with the second thread, and at the next or seventh trial a third thread would be cut, which would be again operated upon during the next succeeding three trials. At the eleventh trial a fourth thread would be cut, but on the next three trials the tool would again fall into the groove first cut and the work proceed correctly. In the case of the 12 threads, the thread cut at the first and second trials would be correct. At the third trial the nut would seat itself in the groove c of the lead screw, causing the carriage to move to the right to a distance equal to twice the pitch of thread being cut, but the tool would still fall into the same groove in the work, as it also would on the fourth. At the fifth trial the process would be repeated, and so on, so that no second thread would be cut.
Fig. 619.
It may now be noted that if we draw the lead screw and the thread to be cut as in the figure, and draw the dotted lines shown, then those that meet the bottom of the thread on the lead screw, and also meet the groove cut on the work, at the first trial, represent the cases in which the nut will fall naturally into its proper position for the tool to fall into the correct groove, while whenever the nut is being forced home it seats in a groove in the lead screw, the bottom of which groove meets a line drawn from the first thread cut; the results obtained will be made correct by reason of the movement given to the slide nut when artificially seating the nut. This is shown to be the case in [Fig. 619], which represents a lead screw having an even number of threads per inch, and from which it appears that in cutting 12 threads (an even number also) the nut cannot be engaged wrong, whereas in the case of 13 threads it can be engaged right three times in 13 trials, and 10 times wrong, the latter causing the tool to cut three wrong threads.
To prevent end motion of a lead screw it should have collars on both sides of one bearing, and not one at each bearing. By this means the screw will be permitted to expand and contract under variations of atmospheric temperature, without binding against the bearing faces.