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GEAR TRAINS 5.2
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MACHINE ELEMENTS IN MOTION
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FIGURE 5.1 Ordinary gear trains. (a) Simple; (b) compound.
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n5 =
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N2 n2 N5
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(5.3)
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where the minus sign is now introduced to indicate contrarotation of the two gears. The compound train in Fig. 5.1b has the following velocity ratios for the pairs of driver and driven gears: n3 = N2 n2 N3 and n5 = N4 n4 N5 (5.4)
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and, of course, n4 = n3. Combining the equations yields
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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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GEAR TRAINS 5.3
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GEAR TRAINS
FIGURE 5.2 Gears 2 and 5 are required if idler gears are not used.
n5 =
N2N4 n2 N3N5
(5.5)
and the thing worthy of note here is that the numbers of teeth of all gears constituting a mesh with a compounded pair are required to determine the velocity ratio through the system. Compound gear trains have an advantage over simple gear trains whenever the speed change is large. For example, if a reduction of 12/1 is required, the final gear in a simple train will have a diameter 12 times that of the first gear.
5.2 GEAR TYPE SELECTION
The disposition of the axes to be joined by the gear train often suggests the type of gear to choose. If the axes are parallel, the choices can be spur gears or helical gears. If the axes intersect, bevel gears can be used. If the axes are nonparallel and nonintersecting, then crossed helicals, worm and gear, or hypoid gears will work. In Fig. 5.3, a train having various types of gears is shown. Gears 2 and 3, parallel helical gears, have a speed ratio n3 = N2 n2 N3 (5.6)
Gears 4 and 5, bevel gears, have a speed ratio n5 = N4 n4 N5 (5.7)
Gears 6 and 7, worm and gear, are considered in a slightly different manner. A worm is generally spoken of as having threads, one, two, three or more (see Chap. 12). A
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GEAR TRAINS 5.4
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FIGURE 5.3 Various gears used in a train.
worm with one thread would have a lead equal to the pitch of the thread. A worm with two threads would have a lead equal to twice the pitch of the thread. Thus n7 = number of threads on 6 n6 N7 (5.8)
Joining Eqs. (5.6), (5.7), and (5.8), we find n7 = N6 N4 N2 n2 N7 N5 N3 (5.9)
where N6 represents the number of threads of the worm gear. To determine the direction of rotation of gear 7, an inversion technique can be used. Fix gear 7 and allow the worm to translate along its axis as it rotates. Here it is necessary to note the hand of the worm, which can be either right or left. In the figure, gear 6 rotates in the same direction as gear 5 and, having a right-hand thread, will move downward (in the drawing). Now, inverting back to the original mechanism, the worm is moved in translation to its proper position, and by doing so, gear 7 is seen to rotate clockwise.
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.
GEAR TRAINS 5.5
GEAR TRAINS
5.3 PLANETARY GEAR TRAINS
Planetary gear trains, also referred to as epicyclic gear trains, are those in which one or more gears orbit about the central axis of the train. Thus, they differ from an ordinary train by having a moving axis or axes. Figure 5.4 shows a basic arrangement that is functional by itself or when used as a part of some more complex system. Gear 2 is called a sun gear, gear 4 is a planet, link 3 is an arm, or planet carrier, and gear 5 an internal-toothed ring gear. Planetary gear trains are, fundamentally, two-degree-of-freedom systems. Therefore, two inputs are required before they can be uniquely analyzed. Quite frequently a fixed gear is included in the train. Its velocity is zero, but this zero velocity constitutes one of the input values. Any link in the train shown except the planet can serve as an input or an output link. If, for example, the rotations of link 2 and link 5 were the input values, the rotation of the arm would be the output. The term link refers to the individual machine elements comprising a mechanism or linkage, and gear trains are included in this broad array of systems. Each link is paired, or joined, with at least two other links by some form of connection, such as pin points, sliding joints, or
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