Machinery's Handbook, 31st Edition
2328 Gear Teeth epicycloidal for tooth outlines. This was about 1695 and not long after Roemer had first demonstrated the epicycloidal form. The applicability of the involute had been further elucidated by Leonard Euler, a Swiss mathematician, born at Basel, 1707, who is credited by Willis with being the first to suggest it. Willis devised the Willis odontograph for laying out involute teeth. A pressure angle of 14 1 ∕ 2 degrees was selected for three different reasons. First, because the sine of 14 1 ∕ 2 degrees is nearly 1 ∕ 4 , making it convenient in calculation; second, because this angle coincided closely with the pressure angle resulting from the usual construction of epicycloidal gear teeth; third, because the angle of the straight-sided involute rack is the same as the 29-degree worm thread. Calculating Replacement-Gear Dimensions from Simple Measurements.— The following Table 1a, Table 1b, and Table 1c provide formulas with which to calculate the dimensions needed to produce replacement spur, bevel, and helical gears when only the number of teeth, the outside diameter, and the tooth depth of the gear to be replaced are known. For helical gears, exact helix angles can be obtained by the following procedure. 1) Using a common protractor, measure the approximate helix angle A at the approxi mate pitch line. 2) Place sample or its mating gear on the arbor of a gear hobbing machine. 3) Calculate the index and lead gears differentially for the angle obtained by the measurements, and set up the machine as though a gear is to be cut. 4) Attach a dial indicator on an adjustable arm to the vertical swivel head, with the indi cator plunger in a plane perpendicular to the gear axis and in contact with the tooth face. Contact may be anywhere between the top and the root of the tooth. 5) With the power shut off, engage the starting lever and traverse the indicator plunger axially by means of the handwheel. 6) If angle A is correct, the indicator plunger will not move as it traverses the face width of the gear. If it does move from 0, note the amount. Divide the amount of movement by the width of the gear to obtain the tangent of the angle by which to correct angle A , plus or minus, depending on the direction of indicator movement. Table 1a. Formulas for Calculating Spur Gear Dimensions
Tooth Form and Pressure Angle
American Standard 14 1 ∕ 2 -and 20-degree full depth American Standard 20-degree stub Fellows 20-degree stub
O N + 2
. P 31416 P 31416 . . P 31416 N
P 1157 .
P 2157 .
P 0157 . . P 02 .
P 15708 P 15708 . P 15708 N
P N + 2
P 1 .
P N P N
. O N + 16
. P N + 16 . P 08
. P 18
P 1
P 2 N D +
1
. P 125 D
. P 225 D
. P 025 D
a P N N
N
See Note
P
P
D
a In the Fellows stub-tooth system, P N = diametral pitch in numerator of stub-tooth designation and is used to determine circular pitch and number of teeth, and P D = diametral pitch in the denominator of stub-tooth designation and is used to determine tooth depth. N = number of teeth.
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