Machinery's Handbook, 31st Edition
Spur Gearing 2233 Minimum Number of Teeth to Avoid Undercutting by Hob.— The data in the above tables give tooth proportions for low numbers of teeth to avoid interference between the gear tooth tip and the pinion tooth flank. Consideration must also be given to pos- sible undercutting of the pinion tooth flank by the hob used to cut the pinion. The mini- mum number of teeth N min of standard proportion that may be cut without undercut is: N min = 2 P csc 2 φ [ a H − r t (1 − sin φ )] where: a H = cutter addendum; r t = radius at cutter tip or corners; φ = cutter pressure angle; and P = diametral pitch. Gear to Mesh with Enlarged Pinion.— Data in the fifth column of Table 8 show minimum number of teeth in a mating gear which can be cut with hob or rack type cutter without undercut, when outside diameter of gear has been reduced an amount equal to the pinion enlargement to retain the standard center distance. To calculate N for the gear, insert ad- dendum a of enlarged mating pinion in the formula N = 2 a × csc 2 φ . Example: A gear is to mesh with a 24-tooth pinion of 1 diametral pitch which has been enlarged 0.4954 inch, as shown by the table. The pressure angle is 14 1 ∕ 2 degrees. Find min imum number of teeth N for reduced gear. . . . . .( ) N 1 0 4954 2 1 2477 2 12477 1595 398 40 Pinion addendum Hence, use ' # # = + = = = ^ h In the case of fine pitch gears with reduced outside diameters, the recommended mini mum numbers of teeth given in Table 9b, Table 9c, and Table 9d are somewhat more than the minimum numbers required to prevent undercutting and are based upon studies made by the American Gear Manufacturers Association . Standard Center-Distance System for Enlarged Pinions.— In this system, sometimes re- ferred to as “long and short addendums,” the center distance is made standard for the numbers of teeth in pinion and gear. The outside diameter of the gear is decreased by the same amount that the outside of the pinion is enlarged. The advantages of this system are: 1) No change in center distance or ratio is required; 2) The operating pressure angle remains standard; and 3) A slightly greater contact ratio is obtained than when the center distance is increased. The disadvantages are 1) The gears as well as the pinion must be changed from standard dimensions; 2) Pinions having fewer than the minimum number of teeth to avoid undercut cannot be satisfactorily meshed together; and 3) In most cases where gear trains include idler gears, the standard center-distance system cannot be used. Enlarged Center-Distance System for Enlarged Pinions.— If an enlarged pinion is meshed with another enlarged pinion or with a gear of standard outside diameter, the center dis- tance must be increased. For fine-pitch gears, it is usually satisfactory to increase the center distance by an amount equal to one-half of the enlargements (see eighth column of Table 9b). This is an approximation as theoretically there is a slight increase in backlash. The advantages of this system are: 1) Only the pinions need be changed from the stan dard dimensions; 2) Pinions having fewer than 18 teeth may engage other pinions in this range; 3) The pinion tooth, which is the weaker member, is made stronger by the enlarge ment; and 4) The tooth contact stress, which controls gear durability, is lowered by being moved away from the pinion base circle. The disadvantages are: 1) Center distances must be enlarged over the standard; 2) The operating pressure angle increases slightly with different combinations of pinions and gears, which is usually not important; and 3) The contact ratio is slightly smaller than that obtained with the standard center-distance system. This consideration is of minor importance as in the worst case the loss is approximately only 6 percent. Enlarged Pinions Meshing without Backlash: When two enlarged pinions are to mesh without backlash, their center distance will be greater than the standard and less than that for the enlarged center-distance system. This center distance may be calculated by the formulas given in the following section.
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