Suppose the lathe cuts 6 threads per inch when gears of equal size are used, and that the number of teeth in the gears available are 30, 35, 40 and so on, increasing by 5 up to 100. If 24 threads per inch are to be cut, the screw constant 6 is placed in the numerator and 24 in the denominator. The numerator and denominator are then divided into factors and each pair of factors is multiplied by the same number to find the gears, thus:
6 2 × 3 (2 × 20) × (3 × 10) 40 × 30 -- = ----- = ------------------- = ------- 24 4 × 6 (4 × 20) × (6 × 10) 80 × 60
The last four numbers indicate the gears which should be used. The upper two having 40 and 30 teeth are the _driving_ gears and the lower two having 80 and 60 teeth are the _driven_ gears. The driving gears are gear _a_ on the spindle stud and gear _c_ on the intermediate stud, meshing with the lead-screw gear, and the driven gears are gears _b_ and _d_. It makes no difference which of the driving gears is placed on the spindle stud, or which of the driven is placed on the lead-screw.
=Fractional Threads.=--Sometimes the lead of a thread is given as a fraction of an inch instead of stating the number of threads per inch. For example, a thread may be required to be cut, having 3/8-inch lead. The expression "3/8-inch lead" should first be transformed to "number of threads per inch." The number of threads per inch (the thread being single) equals:
To find the change gears to cut 2-2/3 threads per inch in a lathe having a screw constant of 8 and change gears varying from 24 to 100 teeth, increasing by 4, proceed as follows:
8 2 × 4 (2 × 36) × (4 × 24) 72 × 96 ----- = --------- = ----------------------- = ------- 2-2/3 1 × 2-2/3 (1 × 36) × (2-2/3 × 24) 36 × 64
As another illustration, suppose we are to cut 1-3/4 thread per inch on a lathe having a screw constant of 8, and that the gears have 24, 28, 32, 36, 40 teeth, etc., increasing by four up to one hundred. Following the rule:
8 2 × 4 (2 × 36) × (4 × 16) 72 × 64 ----- = --------- = ----------------------- = ------- 1-3/4 1 × 1-3/4 (1 × 36) × (1-3/4 × 16) 36 × 28
The gears having 72 and 64 teeth are the _driving_ gears, and those with 36 and 28 teeth are the _driven_ gears.
=Change Gears for Metric Pitches.=--When screws are cut in accordance with the metric system, it is the usual practice to give the lead of the thread in millimeters, instead of the number of threads per unit of measurement. To find the change gears for cutting metric threads, when using a lathe having an English lead-screw, first determine the number of threads per inch corresponding to the given lead in millimeters. Suppose a thread of 3 millimeters lead is to be cut in a lathe having an English lead-screw and a screw constant of 6. As there are 25.4 millimeters per inch, the number of threads per inch will equal 25.4 ÷ 3. Place the screw constant as the numerator, and the number of threads per inch to be cut as the denominator: