Machinery's Handbook, 31st Edition
CHECKING GEAR SIZES
2315
The decimal part for 41.027 teeth is, by interpolation, .
42 40 41027 40 − − ^ ^ h h
× (0.2485 − 0.2473) +
0.2473 = 0.2479; (0.2479 + 2) ÷ 6 = 0.3747; and M = 0.3747 + 5.7939 = 6.1686. This measurement over wires or balls is based upon the use of 1.68/ P n wires or balls. If measurements over 1.728/ P n diameter wires or balls are preferred, use Table 1 to find the decimal part described above instead of Table 7. Measurement Over Two Wires or Two Balls for Odd Numbers of Teeth: The procedure is similar to that for two wire or two ball measurement for even tooth numbers except that a correction is made in the final M value to account for the wires or balls not being diametri cally opposite by one-half tooth interval. In addition, care must be taken to ensure that the balls or wires are kept in a plane of the gear’s rotation as described previously. A helical gear has 13 teeth of 8 normal diametral pitch, 14 1 ∕ 2 degree pressure angle, and 45 degree helix angle. Determine measurement M without allowance for backlash based upon the use of 1.728/ P n balls or wires. As before, D = 13 ∕ 8 × cos 45 ° = 2.2981; N e = 13/cos 3 45 ° = 36.770; and in the second column of Table 1 the decimal part of the measurement for 36 teeth is 0.4565 and that for 38 teeth is 0.4603. The decimal part for 36.770 teeth is, by interpolation, . 38 36 36770 36 − − ^ ^ h h × (0.4603 − 0.4565) + 0.4565 = 0.4580; (0.4580 + 2)/8 = 0.3073; and M = 0.3073 + 2.2981 = 2.6054. This measurement is correct for three-wire measurements but, for two balls or wires held in the plane of rotation of the gear, M must be corrected as follows: cos M M N 90 corrected BallDiam. BallDiam. # ° = − + ^ a h k . . . . cos 26054 8 1728 13 90 8 1728 25880 # ° = − + = a a k k Checking Spur Gear Size by Chordal Measurement Over Two or More Teeth.— Another method of checking gear sizes, that is generally available, is illustrated by the diagram accompanying Table 10. A vernier caliper is used to measure the distance M over two or more teeth. The diagram illustrates the measurement over two teeth (or with one inter vening tooth space), but three or more teeth might be included, depending upon the pitch. The jaws of the caliper are merely held in contact with the sides or profiles of the teeth and perpendicular to the axis of the gear. Measurement M for involute teeth of the correct size is determined as follows. General Formula for Checking External and Internal Spur Gears by Measurement Over Wires: The following formulas may be used for pressure angles or wire sizes not covered by the tables. In these formulas, M = measurement over wires for external gears or mea surement between wires for internal gears; D = pitch diameter; T = arc tooth thickness on pitch circle; W = wire diameter; N = number of gear teeth; A = pressure angle of gear; a = angle, the cosine of which is required in Formulas (2) and (3). First determine the involute function of angle a (inv a ); then the corresponding angle a is found by referring to the tables of involute functions beginning on page 112 , (1) (2) (3) Note: In Formulas (1), (2), and (3), use the upper sign for external and the lower sign for internal gears wherever a ± or 7 appears in the formulas. cos a A D T D A W N inv inv ! ! " π = cos cos M a D A W For even numbers of teeth, ! = , cos cos cos M a D A N W 90 For odd numbers of teeth ! ° = a a k k
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