Applying this to the example given, we have 6÷9 = 24÷36. The values of 36 and 24 are obtained by multiplying 6 and 9, respectively, by 4, which, of course, does not change the proportion. Any other number could be used as a multiplier, and if gears having 24 and 36 teeth were not available, this might be necessary. For example, if there were no gears of this size, some other multiplier as 5 or 6 might be used.
Suppose the number of teeth in the change gears supplied with the lathe are 24, 28, 32, 36, etc., increasing by four teeth up to 100, and assume that the lead-screw has 6 threads per inch and that 10 threads per inch are to be cut. Then,
| 6 | 6 × 4 | 24 | ||
| — | = | ——— | = | — |
| 10 | 10 × 4 | 40 |
By multiplying both numerator and denominator by 4, we obtain two available gears having 24 and 40 teeth, respectively. The 24-tooth gear goes on the spindle stud and, the 40-tooth gear on the lead-screw. The number of teeth in the intermediate or “idler” gear b, which connects the stud and lead-screw gears, is not considered as it does not affect the ratios between gears a and c, but is used simply to transmit motion from one gear to the other.
We have assumed in the foregoing that the spindle stud (on which gear a is mounted) and the main spindle of the lathe are geared in the ratio of one to one and make the same number of revolutions. In some lathes, however, these two members do not rotate at the same speed, so that if equal gears were placed on the lead-screw and spindle stud, the spindle would not make the same number of revolutions as the lead-screw. In that case if the actual number of threads per inch in the lead-screw were used when calculating the change gears, the result would be incorrect; hence, to avoid mistakes, the following general rule should be used as it gives the correct result, regardless of the ratios of the gears which connect the spindle and spindle stud:
Rule.—First find the number of threads per inch that is cut when gears of the same size are placed on the lead-screw and spindle, either by actual trial or by referring to the index plate. Then place this number as the numerator of a fraction and the number of threads per inch to be cut, as the denominator; multiply both numerator and denominator by some trial number, until numbers are obtained which correspond to numbers of teeth in gears that are available. The product of the trial number and the numerator (or “lathe screw constant”) represents the gear a for the spindle stud, and the product of the trial number and the denominator, the gear for the lead-screw.
Lathes with Compound Gearing.—When gearing is arranged as shown at A, [Fig. 29], it is referred to as simple gearing, but sometimes it is necessary to introduce two gears between the stud and screw as at B, which is termed compound gearing. The method of figuring compound gearing is practically the same as that for simple gearing. To find the change gears used in compound gearing, place the “screw constant” obtained by the foregoing rule, as the numerator, and the number of threads per inch to be cut as the denominator of a fraction; resolve both numerator and denominator into two factors each, and multiply each “pair” of factors by the same number, until values are obtained representing numbers of teeth in available change gears. (One factor in the numerator and one in the denominator make a “pair” of factors.)
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 |