Page:Practical Treatise on Milling and Milling Machines.djvu/67

Rh 72, 32, 48 and 40 teeth are selected. $${12 \over 10} = \left({72 \times 32 \over 48 \times 40}\right)$$ The first two are the driven, and the last two the drivers, the numerators of the fractions representing the driven gears. The 72 is the worm gear, 40 the first on stud, 32 the second on stud and 48 the screw gear. The two driving gears might be transposed, and the two driven gears might also be transposed without changing the spiral. That is, the 72 could be used as the second on stud and the 32 as the worm gear, if such an arrangement were more convenient. The following rules express in abridged form the methods of figuring change gears to cut given spirals, and of ascertaining what spirals can be cut with change gears.

Rules for Obtaining Ratio of the Gears Necessary to Cut a Given Spiral. Note the ratio of the required lead to 10. This ratio is the compound ratio of the driven to the driving gears. Example: If the lead of required spiral is 12 inches, 12 to 10 will be the ratio of the gears.

Or, divide the required lead by 10 and note the ratio between the quotient and 1. This ratio is usually the most simple form of the compound ratio of the driven to the driving gears. Example: If the required lead is 40 inches, the quotient is 40÷10 and the ratio 4 to 1.

Rule for Determining Number of Teeth of Gears Required to Cut a Given Spiral. Having obtained the ratio between the required lead and 10 by one of the preceding rules, express the ratio in the form of a fraction; resolve this fraction into two factors, raise these factors to higher terms that correspond with the teeth of gears that can be conveniently used. The numerators will represent the driven and the denominators the driving gears that produce the required spiral. For example: What gears shall be used to cut a lead of 27 inches?

$$\tfrac{27}{10}=\tfrac{3}{2}\times\tfrac{9}{5}=\left(\tfrac{3}{2}\times\tfrac{16}{16}\right)\times\left(\tfrac{9}{5}\times\tfrac{8}{8}\right)=\frac{48\times72}{32\times40}$$

From the fact that the product of the driven gears divided by the product of the drivers equals the lead divided by 10, or one-tenth of the lead, it is evident that ten times the product of the driven gears divided by the product of the drivers, will equal the lead of the spiral. Hence the rule:

Rule for Ascertaining what Spiral May be Cut by Any Given Change Gears. Divide ten times the product of the driven gears by the product of the drivers, and the quotient is the lead of the resulting spiral in inches to one turn. For example: What spiral