FIGURE 5.4 A basic planetary train. in Software

Encoder EAN13 in Software FIGURE 5.4 A basic planetary train.

FIGURE 5.4 A basic planetary train.
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GEAR TRAINS 5.6
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MACHINE ELEMENTS IN MOTION
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direct contact, a pairing that is prevalent in cam-and-gear systems. An explanation and an illustration of the joint types are found in Refs. [5.1] and [5.2] as well as others (see Chap. 3). There are several methods for analyzing planetary trains. Among these are instant-centers, formula, and tabular methods. By instant centers, as in Ref. [5.3] and on a face view of the train, draw vectors representing the velocities of the instant centers for which input information is known. Then, by simple graphical construction, the velocity of another center can be found and converted to a rotational speed. Figure 5.5 illustrates this technique.
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FIGURE 5.5 Instant-centers method of velocity analysis.
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Calculate VIC24 and VIC45 from V = r (5.10)
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where r = radius dimension and = angular velocity in radians per second (rad/s). Draw these vectors to scale in the face view of the train. Then VIC24 and VIC45 will emanate from their instant-center positions. Now draw a straight line through the
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GEAR TRAINS
termini of the velocity vectors. The velocity of IC34 will be a vector perpendicular to the line of centers and having its terminus on the velocity gradient. Determine of link 3 by using Eq. (5.10). Thus, VIC24 = r2 2 and VIC45 = r5 5
Choose a scale and construct the two vectors. Next, draw the gradient line and construct VIC34. Scale its magnitude and determine n3 according to n3 = VIC34 60 2 r3 (5.11)
where r3 = radius of the arm and n3 is in revolutions per minute. If gear 5 is fixed, then VIC45 = 0; using VIC24, connect the terminus of VIC24 and IC45 with a straight line, and find VIC34 as before. See Fig. 5.6.
This line can be called a velocity gradient for link 4.
FIGURE 5.6 Gear 5 is fixed.
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GEAR TRAINS 5.8
MACHINE ELEMENTS IN MOTION
By formula, the relative-motion equation will establish the velocity of the gears relative to the arm; that is, n23 = n2 n3 n53 = n5 n3 Then, dividing (5.13) by (5.12), we see that n53 n5 n3 = n23 n2 n3 (5.14) (5.12) (5.13)
which represents the ratio of the relative velocity of gear 5 to that of gear 2 with both velocities related to the arm. The right-hand side of the equation is called the train value. If the arm should be held fixed, then the ratio of output to input speeds for an ordinary train is obtained. The equation for train value, which is seen in most references, can be written e= where nF = speed of first gear in train nL = speed of last gear in train nA = speed of arm nL nA nF nA (5.15)
The following example will illustrate the use of Eq. (5.15). Example 1. Refer to the planetary train of Fig. 5.4. The tooth numbers are N2 = 104, N4 = 32, and N5 = 168. Gear 2 is driven at 250 r/min in a clockwise negative direction, and gear 5 is driven at 80 r/min in a counterclockwise positive direction. Find the speed and direction of rotation of the arm. Solution. nF = n2 = 250 r/min nL = n5 = +80 r/min e= In Eq. (5.15), 80 n3 13 = 21 250 n3 n3 = 46.2 r/min N2 N4 N4 104 = N5 32 13 32 = 21 168
By tabular method, a table is first formed according to the following: 1. Include a column for any gear centered on the planetary axis. 2. Do not include a column for any gear whose axis of rotation is fixed and different from the planetary axis. 3. A column for the arm is not necessary. 4. The planet, or planets, may be included in a column or not, as preferred. Gears which fit rule 2 are treated as ordinary gear train elements. They are used as input motions to the planetary system, or they may function as output motions. The table contains three rows arranged so that each entry in a column will constitute one term of the relative-motion equation
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