Suppose, for example, that [Fig. 1235] represents the screw cutting gear or change wheels of a lathe, wheel d being the driver, i an intermediate wheel for transmitting motion from the driver d to the lead-screw wheel s. Suppose, also, that d has 32, i 80, and s 32 teeth, and we have a simple or single-geared lathe. In this case it may first be proved that we need not concern ourselves with the number of teeth in the intermediate i, because its number of teeth is of no consequence. For example, the 32 teeth in d will in a revolution move 32 of the teeth in i past the line of centres, and it is obvious that i will move the 32 teeth in s past the line of centres, causing it to make one revolution the same as d. If any other size of wheel be used for an intermediate, the effect will be precisely the same, the revolutions of d and of s remaining equal. Under these conditions the lathe would cut a thread whose pitch would be the same as that of the thread on the lead screw.
Fig. 1236.
Now let us turn to [Fig. 1236], representing an arrangement of gearing common in American practice, and we have within the lathe-head three gears, a, b, and c, which cannot be changed. Of these, b and c are simply intermediate wheels, the respective diameters of which have no effect upon the revolutions of the lead screw, except that they convey the motion to d. To demonstrate this, suppose the wheels to have the number of teeth marked respectively against them in the end view of the figure, c and d having each 20 teeth, and the one revolution of the live spindle wheel a will cause the lead-screw wheel to make one revolution, because a and s contain the same number of teeth. This may be made plain as follows: The 20 teeth in a will in one revolution cause b to make two revolutions, because b has but half as many teeth as a. The two revolutions of b will cause c to make but one revolution, because c has twice as many teeth as b has. Now, c and d are fast on the same shaft r; hence they revolve together, the one revolution of c simply being conveyed by the shaft r to d, and it is clear that the one revolution of a has been conveyed without change to d, and that, therefore, d may be considered to have simply taken the place of a, unaffected by the wheels b, c. Wheel i is again an intermediate, so that, whatever its diameter or number of teeth, one revolution of d will cause one revolution of s. Thus in this arrangement the lead screw will again revolve at the same speed as the live spindle, and the thread cut will be of the same pitch as the pitch of the lead screw. Practically, then, all the wheels between a and s, as thus arranged, act as simple intermediates, the same as though it were a single-geared lathe, which occurs because c and d have the same number of teeth, and we have, therefore, made no use of the shaft r to compound the gearing.
Fig. 1237.
The term “compounded” as applied to the change gears of a lathe, means that there exists in it a shaft or some equivalent means by which the velocity of the wheels may be changed. Such a shaft is shown at r in [Fig. 1236], and it affords a means of compounding by placing on its outer end, as at d, a wheel that has a different number of teeth to that in wheel c. In [Fig. 1237] this change is made, wheel d having 40 teeth instead of the 20 it had before. As in the former case, however, it will make one revolution to one of c or one of a, but having 40 teeth it will move 40 of the teeth in i past the line of centres, and this will cause the lead screw wheel s to make two revolutions, because it has 20 teeth only. Thus, the compounding of c and d on shaft r has caused s to make two revolutions to one of a, or, what is the same thing, one revolution of a will in this case cause s to make two revolutions, and the thread cut would be twice as coarse as the lead-screw thread. In the case of a lathe geared as in either [Fig. 1235] or [1236], all the wheels that we require to consider in calculating the change wheels are d and s. Now, the shaft r is called the “mandrel,” the “stud,” or the “spindle,” all three terms being used, and the wheel d is the wheel on the stud, mandrel, or spindle, while in every case s is that on the lead screw, and the revolutions of this wheel d and those of the lead screw will be in the same proportion as exists between their numbers of teeth. In considering their revolutions it is to be borne in mind that when d has more teeth than s the speed of the lead screw is increased, and the lathe will cut a thread coarser than that of its lead screw, or when d has less teeth than s the speed of the lead screw is diminished, and the pitch of thread cut will be finer than that of the lead screw.