In [Fig. 615], for example, we have a pitch of lead screw of three threads per inch, and the gears arranged to cut six threads per inch on the work. As the bottom wheel has twice as many teeth as the top one, it is clear that, while the top one makes one, the bottom one will make half revolution, and the lead screw will make half a turn for every turn the work makes. Now, suppose the tool point to stand opposite to space a, and the nut (supposing it to have but one thread only, which is all that is required for our purpose), stand opposite to space d. Suppose, further, that the lathe makes one revolution, and space b on the work will have moved to occupy the position occupied by space a, or, rather, there will still be a place at a fully in front of the tool, as should be the case, but the lead screw will have made half revolution, the top e of the thread coming opposite to the feed nut, as in the position of tool and nut shown in the figure at t and n; hence the nut would not engage, without moving the lathe carriage sideways, and thus throwing the tool to one side of the thread in the work. When, however, the work had made another revolution, both the feed screw and the work would again come into position for the tool and nut to engage properly, and it follows that in this case the tool will always fall into proper position for the nut to be locked.
It is obvious, however, that had the lead screw thread been a square one, and the nut thread to accurately fit to the lead screw thread, so as to completely fill it, then the nut could not engage with the lead screw until the lathe had made a complete revolution, at which time the work will have made two full or complete revolutions, and the tool would, therefore, fall into proper position to follow in the groove or part of a thread cut at the first tool traverse.
Fig. 616.
In [Fig. 616], we have the same lead screw geared to cut five or an odd number of threads per inch. The tool and the nut are shown in position to properly engage, but suppose, the nut being disengaged, that the work makes one revolution, and during this period the lead screw will have made 3⁄5ths of a revolution, hence the nut will not be in position to engage properly, because, although space b will have travelled forward so as to occupy the position of space a in the figure (that is, there will be a space fairly in front of the tool point), yet the nut will not engage properly, because the nut point will not be opposite to the bottom of the lead screw thread. When the work has made its second revolution, and space c moves to the position occupied by a, the lead screw will have made 6⁄5 or 11⁄5 revolutions, and the nut cannot engage properly; when the lathe has made its third revolution, the lead screw will have made 14⁄5 revolutions and the nut will still fall to one side of the thread space, and will not lock properly. The work having made its fourth turn, the lead screw will have made 22⁄5 turns, and the nut will not be in position to lock fairly. The work having made its fifth turn, however, the lead screw will have made three turns, and the threads will fall into the same position that they occupy in the figure, and both tool and feed nut will fall into their proper positions in their respective threads. It does not follow, however, that, the lead screw having a V-shaped thread, the nut cannot be forced to engage but once in every five turns of the lead screw, because, were this the case, it would be impossible to lock the nut in an improper position.
Fig. 617.
Suppose, for example, that we have in [Fig. 617], the same piece of work and lead screw as in [Fig. 616], and that a first groove, a, has been cut with the tool in the position shown, and the nut engaged in the position marked 1. Now, suppose the nut be disengaged and the work allowed to make one revolution, then the lead screw will, during this revolution, revolve 3⁄5 of a revolution, and the position of the nut point with relation to the lead screw will be as at position 2. If, then, the nut was forced into the lead screw thread, it would, acting on the wedge principle, move the carriage to the right sufficiently to permit the nut to engage fully in thread g, and the tool would then cut a second groove on thread b. If the nut then be withdrawn from thread g, and the work allowed to make another revolution, the nut will stand in a precisely similar position with relation to the lead screw thread as it did in position 2, and by forcing it down into thread h the carriage would be again forced to the right, causing a third thread, c, to be cut. By repeating the operation of withdrawing the nut, letting the work make another revolution and then engaging the nut again, it will seat in thread k, and a fourth thread d will be cut. On again repeating the operation, however, the nut will come into position 5, and, on being drawn home into thread, or, rather, into space l, the tool will fall into groove a again. Thus there will be four threads, each having a pitch equal to that of the lead screw. The second (b) of these four will fall to the left of thread a to an amount or distance equal to 2⁄5 of the pitch of the lead screw, because, in forcing the nut from position 2 down into the lead screw, the slide rest, and therefore the tool, will be moved to the right 2⁄5 of the pitch of the lead screw. The third thread c will fall to the left of thread b also to an amount equal to 2⁄5 of the pitch of the lead screw, because, in forcing the thread to seat itself into thread h from position 3, the slide rest was again moved (to that amount) to the right. The fourth thread d will fall to the left of thread c to the same amount and for the same reason.