Machinery's Handbook, 31st Edition
2234 Spur Gearing Center Distance at Which Modified Mating Spur Gears Will Mesh with No Backlash.— When the tooth thickness of one or both of a pair of mating spur gears has been increased or decreased from the standard value ( π ÷ 2 P ), the center distance at which they will mesh tightly (without backlash) may be calculated from the following formulas:
cos
n N = + + + − P t T ^ φ h
C P n N C = +
φ
π
inv inv 1 φ
=
C
2
#
cos
1
φ
1
In these formulas, P = diametral pitch; n = number of teeth in pinion; N = number of teeth in gear; t and T are the actual tooth thicknesses of the pinion and gear, respectively, on their standard pitch circles; inv φ = involute function of standard pressure angle of gears; C = standard center distance for the gears; C 1 = center distance at which the gears mesh without backlash; and inv φ 1 = involute function of operating pressure angle when gears are meshed tightly at center distance C 1 . Example: Calculate the center distance for no backlash when an enlarged 10-tooth pinion of 100 diametral pitch and 20-degree pressure angle is meshed with a standard 30-tooth gear, the circular tooth thickness of the pinion and gear, respectively, being 0.01873 and 0.015708 inch. From the table of involute functions, page 112 , inv 20-degrees = 0.014904. Therefore, . . . . . . . . . . 092175 0 93969 0 2000 0 2039 . . cos cos C P n N C 20 10 30 100 0 01873 0 015708 + + ^ ^ h 0014904 4 0 34438 0 31416 − 0022459 22 49 2 = + = + = = = 2 100 10 30 02000 # 22 49 20 02000 inv inv inv 1 φ φ = = then, from the involute table inch inch 1 1 1 # # ° ° ′ ° ′ ° π − = φ + + = = h Contact Diameter.— For two meshing gears it is important to know the contact diameter of each. A first gear with number of teeth, n , and outside diameter, d 0 , meshes at a standard center distance with a second gear with number of teeth, N , and outside diameter, D 0 ; both gears have a diametral pitch, P , and pressure angle, φ , a , A , b , and B are unnamed angles used only in the calculations. The contact diameter, d c , is found by a three-step calcula- tion that can be done by hand using a trigonometric table and a logarithmic table or a desk calculator. Slide rule calculation is not recommended because it is not accurate enough to give good results. The three-step formulas to find the contact diameter, d c , of the first gear are: (1) (2) (3) Similarly the three-step formulas to find the contact diameter, D c , of the second gear are: (4) cos cos A D P N 0 # φ = tan tan tan tan b n N A φ φ = − − ^ h cos cos d P b n c φ = cos cos a d P n o # φ =
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