Machinery's Handbook, 31st Edition
2218 Spur Gearing Gears for Given Center Distance and Ratio.— When it is necessary to use a pair of gears of given ratio at a specified center distance C 1 , it may be found that no gears of standard diametral pitch will satisfy the center distance requirement. Gears of standard diametral pitch P may need to be redesigned to operate at other than their standard pitch diameter D and standard pressure angle φ . The diametral pitch P 1 at which these gears will operate is (1) where N p = number of teeth in pinion N G = number of teeth in gear and their operating pressure angle φ 1 is (2) Thus although the pair of gears are cut to a diametral pitch P and a pressure angle φ , they operate as standard gears of diametral pitch P 1 and pressure angle φ 1 . The pitch P and pressure angle φ should be chosen so that φ 1 lies between about 18 and 25 degrees. The operating pitch diameters of the pinion D p 1 and of the gear D G 1 are (3a) and (3b) The base diameters of the pinion D PB 1 and of the gear D GB 1 are (4a) and (4b) The basic tooth thickness, t 1 , at the operating pitch diameter for both pinion and gear is (5) The root diameters of the pinion D PR 1 and gear D GR 1 and the corresponding outside diameters D PO 1 and D GO 1 are not standard because each gear is to be cut with a cutter that is not standard for the operating pitch diameters D P 1 and D G 1 . The root diameters are (6a) and (6b) P C N N 2 1 + P G 1 = 1 φ = arccos cos φ P 1 P ( ) D P N P P 1 1 = D P N G G 1 1 = cos D D PB P 1 1 1 φ = cos D D GB G 1 1 1 φ = . P 15708 1 t 1 = D P N PR P b 2 P = − D P N GR G b 2 G = −
1
1
1
1
. tan
15708
t
−
P φ
f
p
(7a)
2
P
where
b b P c 1
= −
2
. tan
15708
t
−
P φ
and
(7b)
2
G
b b G c 1
= −
2
where b c is the hob or cutter addendum for the pinion and gear. The tooth thicknesses of the pinion t P 2 and the gear t G 2 are
N
. N 1 5708 inv inv 1 φ + −
=
P
t
(8a)
φ
c
m
P
2
P
P
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