Machinery's Handbook, 31st Edition
HELICAL GEARING
2283
2) D = pitch diameter of gear = cos P N n α 3) d = pitch diameter of pinion = cos P n n β 4) O = outside diameter of gear = D + P 2 n 5) o = outside diameter of pinion = d + P 2 n
6) N ′ = number of teeth marked on formed cutter for gear (see below) 7) n ′ = number of teeth marked on formed cutter for pinion (see below)
8) L = lead of helix on gear = π D cot α 9) l = lead of helix on pinion = π d cot β
Selecting Cutter for Milling Helical Gears.— The proper milling cutter to use for spur gears depends on the pitch of the teeth and also upon the number of teeth as explained on page 2227 but a cutter for milling helical gears is not selected with reference to the actual number of teeth in the gear, as in spur gearing, but rather with reference to a calculated number N ′ that takes into account the effect on the tooth profile of lead angle, normal dia metral pitch, and cutter diameter. In the helical gearing examples starting on page 2276 the number of teeth N ′ on which to base the selection of the cutter has been determined using the approximate formula N ′ = N ÷ cos 3 α or N ′ = N sec 3 α , where N = the actual number of teeth in the helical gear and α = the helix angle. However, the use of this formula may, where a combination of high helix angle and low tooth number is involved, result in the selection of a higher number of cutter than should actually be used for greatest accuracy. This condition is most likely to occur when the aforementioned formula is used to calculate N ′ for gears of high helix angle and low number of teeth. To avoid the possibility of error in choice of cutter number, the following formula, which gives theoretically correct results for all combinations of helix angle and tooth numbers, is to be preferred: (1) where: N ′ = number of teeth on which to base selection of cutter number from table on page 2229 ; N = actual number of teeth in helical gear; α = helix angle; P n = normal diame tral pitch of gear and cutter; and D c = pitch diameter of cutter. To simplify calculations, Formula (1) may be written as follows: (2) In this formula, K , K ′ and Q are constants obtained from the tables on page 2284 . Example: Helix angle = 30 degrees; number of teeth in helical gear = 15; and normal diametral pitch = 20. From the tables on page 2284 K , K ′ , and Q are, respectively, 1.540, 0.333, and 37.80. . . . . . . , , N 15 1 540 37 80 0 333 23 10 12 60 3570 say36 # # = + = + = ′ ^ h ^ h Hence, from page 2229 select a number 3 cutter. Had the approximate formula been used, then a number 5 cutter would have been selected on the basis of N ′ = 23. N NK QK = + ′ ′ sec tan N N = ′ P D n c 3 2 α α +
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