Modern Machine-Shop Practice, Volumes I and II - Joshua Rose

When we have to deal with a lathe compounded as in [Fig. 1238], in which the combination can be altered in two places—that is, between a and c and between d and s—the wheel a remaining fixed, and the pitch of the lead screw is 2 per inch, and it is required to cut 8 per inch—this gives us the fraction ²⁄₈, which is at once the proportion that must exist between the revolutions of the wheel a and the wheel s. But in this case the fraction gives us the number of revolutions that wheel s must make while the wheel a is making two revolutions, and it is more convenient to obtain the number that s requires to make while a is making one revolution, which we may do by simply dividing the pitch required to be cut by the pitch of the lead screw, as follows: Pitch of thread required, 8; pitch of lead screw, 2; 8 ÷ 2 = 4 = the revolutions s must make while a makes one. We have then to reduce the revolutions four times, which we may do by putting on at c a wheel with twice as many teeth in it as there are in a, and as a has 32, therefore c must have 64 teeth. When we come to the second pair of wheels, d and s, we may put any wheel we like in place of d, providing we put on s a wheel having twice as many.

But suppose we require to cut a fractional pitch, as, say, 4¹⁄₈ per inch, the pitch of lead screw being 2, all we have to do is to put the pitch of the lead screw into eighths, and also put the number of teeth in a into eighths; thus: In two there are 16 eighths, and in the pitch required there are 33 eighths; hence for the pitch of the lead screw we use the 16, and for the thread required we use the 33, and proceed as before; thus:

Pitch of thread required.		Pitch of lead screw.
33	÷	16	=	2¹⁄₁₆	=	the revolution which a must make while wheelb makes one revolution.

The simplest method of doing this would be to put on at c a wheel having 2¹⁄₁₆ times as many teeth as there are in a. Suppose then that a has 32 teeth, and one sixteenth of 32 = 2, because 32 ÷ 16 = 2. Then twice 32 is 64, and if we add the 2 to this we get 66; hence, if we give wheel c 66 teeth, we have reduced the motion the 2¹⁄₁₆ times, and we may put on d and s wheels having an equal number of teeth. Or we may put on a wheel at c having the same number as a has, and then put on any two wheels at d and c, so long as that at s has 2¹⁄₁₆ times as many teeth as that at d.

Again, suppose that the pitch of a lead screw is 4 threads per inch, and that it be required to find what wheels to use to cut a thread of ¹¹⁄₁₆ inch pitch, that is to say, a thread that measures ¹¹⁄₁₆ inch from one thread to the other, and not a pitch of ¹¹⁄₁₆ threads per inch: First we must bring the pitch of the lead screw and the pitch to be cut to the same terms, and as the pitch to be cut is expressed in sixteenths we must bring the lead screw pitch to sixteenths also. Thus, in an inch of the length of the lead screw there are 16 sixteenths, and in this inch there are 4 threads; hence each thread is ⁴⁄₁₆ pitch, because 16 ÷ 4 = 4. Our pitch of lead screw expressed in sixteenths is, therefore, 4, and as the pitch to be cut is ¹¹⁄₁₆ it is expressed in sixteenths by 11; hence we have the fraction ⁴⁄₁₁, which is the proportion that must exist between the wheels, or in other words, while the lathe spindle (or what is the same thing, the work) makes 4 revolutions the lead screw must make 11.

Suppose the lathe to be single geared, and not compounded, and we multiply this fraction and get—

	4	×	4	=	16	=	wheel	to go on	lead screw.
	11	×	4	=	44	=	„	„	stud or mandrel.

Or,	4	×	5	=	20	=	wheel	to go on	lead screw.
Or,	11	×	5	=	55	=	„	„	stud or mandrel.

Or,	4	×	6	=	24	=	wheel	to go on	lead screw.
Or,	11	×	6	=	66	=	„	„	stud or mandrel.

But suppose the lathe to be compounded as in [Fig. 1235], and we may arrange the wheels in several ways, and in order to make the problem more practical, we may suppose the lathe to have wheels with the following numbers of teeth, 18, 24, 36, 36, 48, 60, 66, 72, 84, 90, 96, 102, 108, and 132.