Machinery's Handbook, 31st Edition
2186
Thread Cutting
4 2306140
L C
.
=
=
. 17345
may be used. The method of conjugate fractions shown on page 5 will find the ratio, 113 ∕ 49 = 2.306122, which is closer than any other having suitable factors. This ratio is in error by only 2.306140 − 2.306122 = 0.000018. Therefore, the driver should have 113 teeth and the driven gear 49 teeth. Relieving Helical-Fluted Hobs.— Relieving hobs that have been fluted at right angles to the thread is another example of approximating a required change-gear ratio. The usual method is to change the angle of the helical flutes to agree with previously calculated change-gears. The ratio between the hob and the relieving attachment is expressed in the formula:
cos C N #
driven driver gears
=
2
h
^
α
and
P
tan
α=
H
c
in which: N = number of flutes in hob; α = helix angle of thread from plane perpendicular to axis; C = constant of relieving attachment; P = axial lead of hob; and H c = hob pitch cir cumference, = 3.1416 times pitch diameter. The constant of the relieving attachment is found on its index plate and is determined by the number of flutes that require equal gears on the change-gear studs. These values will vary with different makes of lathes. For example, what four change-gears can be used to relieve a helical-fluted worm-gear hob, of 24 diametral pitch, six starts, 13 degrees, 41 minutes helix angle of thread, with eleven helical flutes, assuming a relieving attachment having a constant of 4 is to be used? h Using the conjugate fractions method discussed on page 5 , the following ratios are found to provide factors that are successively closer approximations to the required change-gear ratio 2.913136. Numerator / Denominator Ratio Error 67 × 78/(39 × 46) 2.913043 − 0.000093 30 × 47/(22 × 22) 2.913223 +0.000087 80 × 26/(21 × 34) 2.913165 +0.000029 27 × 82/(20 × 38) 2.913158 +0.000021 55 × 75/(24 × 59) 2.913136 +0.0000004 74 × 92/(57 × 41) 2.913136 +0.00000005 . 4 0944045 11 # . 2913136 cos cos C N 4 13 41 ° ′ 11 2 2 # # α = = = ^ ^ ^ h h
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