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12.1 INTRODUCTION
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Worm gears are used for large speed reduction with concomitant increase in torque. They are limiting cases of helical gears, treated in Chap. 10. The shafts are normally perpendicular, though it is possible to accommodate other angles. Consider the helical-gear pair in Fig. 12.1a with shafts at 90 . The lead angles of the two gears are described by (lead angle is 90 less the helix angle). Since the shafts are perpendicular, 1 + 2 = 90 . If the lead angle of gear 1 is made small enough, the teeth eventually wrap completely around it, giving the appearance of a screw, as seen in Fig. 12.1b. Evidently this was at some stage taken to resemble a worm, and the term has remained. The mating member is called simply the gear, sometimes the wheel. The helix angle of the gear is equal to the lead angle of the worm (for shafts at 90 ). The worm is always the driver in speed reducers, but occasionally the units are used in reverse fashion for speed increasing. Worm-gear sets are self-locking when the gear cannot drive the worm.This occurs when the tangent of the lead angle is less than the coefficient of friction. The use of this feature in lieu of a brake is not rec-
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FIGURE 12.1 (a) Helical gear pair; (b) a small lead angle causes gear one to become a worm.
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Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
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WORM GEARING 12.3
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WORM GEARING
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FIGURE 12.2 Photograph of a worm-gear speed reducer. Notice that the gear partially wraps, or envelopes, the worm. (Cleveland Worm and Gear Company.)
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ommended, since under running conditions a gear set may not be self-locking at lead angles as small as 2 . There is only point contact between helical gears as described above. Line contact is obtained in worm gearing by making the gear envelop the worm as in Fig. 12.2; this is termed a singleenveloping gear set, and the worm is cylindrical. If the worm and gear envelop each other, the line contact increases as well as the torque that can be transmitted. The result is termed a double-enveloping gear set. The minimum number of teeth in the gear and the reduction ratio determine the number of threads (teeth) for the worm. Generally, 1 to 10 threads are used. In special cases a larger number may be required.
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12.2 KINEMATICS
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In specifying the pitch of worm-gear sets, it is customary to state the axial pitch px of the worm. For 90 shafts this is equal to the transverse circular pitch of the gear. The advance per revolution of the worm, termed the lead L, is L = pxNW This and other useful relations result from consideration of the developed pitch cylinder of the worm, seen in Fig. 12.3. From the geometry, the following relations can be found: d= NWpn sin (12.1)
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FIGURE 12.3 Developed pitch cylinder of worm.
Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
WORM GEARING 12.4
GEARING
d= tan = px = D= From Eqs. (12.1) and (12.5), we find
NWpx tan L NWpx = d d pn cos NGpn pxNG = cos
(12.2)
(12.3)
(12.4)
(12.5)
tan =
1 D NWD = NG d mG d
(12.6)
The center distance C can be derived from the diameters C= 1 pn NW mG + 2 cos sin (12.7)
which is sometimes more useful in the form 2 C pn NW mG 1 + = cos sin 2C mo NW cos 2C d sin U.S. customary units SI units either (12.8)
For use in the International System (SI), recognize that Diameter = Nmo = so that the substitution px = mo will convert any of the equations above to SI units. The pitch diameter of the gear is measured in the plane containing the worm axis and is, as for spur gears, D= NGpx (12.9) Npx
The worm pitch diameter is unrelated to the number of teeth. It should, however, be the same as that of the hob used to cut the worm-gear tooth.
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