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4.4 FORCE AND TORQUE ANALYSIS
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A typical approach to dynamic analysis of a rigid cam system can be illustrated by an example of a mechanism with a reciprocating roller follower. A schematic drawing of such a mechanism is depicted in Fig. 4.19a. For the upward motion of the follower,
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Suggestion of Professor Charles R. Mischke, Iowa State University.
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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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CAM MECHANISMS 4.23
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CAM MECHANISMS
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FIGURE 4.18 Acceleration diagrams obtained in the static way. Curve 1 is from Eq. (4.28), curve 2 from Eq. (4.29).
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we assume that the follower s stem 4 contacts its guideway at points B and C. As a result of its upward motion, the Coulomb friction at B and C is fully developed and tan = . The free-body diagram of links 3 and 4 is shown in Fig. 4.19b. The cam force F23 can be resolved into two components: Pcr in the critical-angle ( cr) direction to sustain motion against friction, and Py in the y direction to produce accelerated motion or to oppose other forces. It can be found from the geometry of the follower that cr = 2s 2a d tan 1 1 l l 2 (4.30)
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where a = lB R0 Rr. For > cr, the cam-follower system is self-locking, and motion is impossible. From the force triangle in Fig. 4.19b and the rule of sines,
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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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CAM MECHANISMS 4.24
MACHINE ELEMENTS IN MOTION
FIGURE 4.19 Force analysis of reciprocating roller-follower cam system.
Py = F23
sin( cr ) sin cr
(4.31)
After finding the vertical component Py for constant 21 from the force-equilibrium equation, substituting into Eq. (4.31), and solving for F23, we have F23 = where sin cr(m 2 s + ks + P4 ) 21 sin ( cr ) (4.32)
m = mass of follower k = spring rate of retaining spring P4 = P4 + k = preset of spring k = P0; this force is called preload of spring
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.
CAM MECHANISMS 4.25
CAM MECHANISMS
For F23 = 0, roller and cam lose their contact. The result is called jump ([4.7], [4.8]). Assuming F23 = 0, we can find the jump speed of the cam from Eq. (4.31). The jump occurs for the upward movement of the follower at 21 ks P4 ms (4.33)
Since s is always positive, jump may occur only for negative values of s . To prevent jump, we increase preload P0 or the spring rate or both. The driving torque is T12 = sin cr sin (R0 + Rr + s)(m 2 s + ks + P 4 ) 21 sin ( cr ) (4.34)
We recall that according to Eq. (4.25), = tan 1 s s + R0 + Rr (4.35)
When motion is downward, the contact point of mating surfaces goes to the right side of the roller, cam force F23 changes inclination, and new contact points D and E in the follower s guideway replace old ones (B and C, respectively). The new point of concurrency is now at F . Since in most practical cases points F and F almost coincide, we can assume that both the point of concurrency and the line of action of force Pcr are unchanged. A new vector Pcr (broken line) is rotated by 180 with respect to the old one. It is easy to see in Fig. 4.19b that F23 for downward motion, when and Py equal those for the upward motion, is always smaller than F23 for upward motion. 4.4.1 Springs In cam-follower systems, the follower must contact the cam at all times. This is accomplished by a positive drive or a retaining spring. Spring forces should always prevent the previously described jump of the follower for all the operating speeds of the cam. Thus the necessary preload P0 of the spring and its spring rate k should be chosen for the highest possible velocity of the cam. By plotting inertial and spring forces, we can find values of preload P0 and spring rate k that will ensure sufficient load margin for the total range of the follower displacement. We use here only the negative portion of the acceleration curve. Since the follower must be held in contact with the cam, even while operating the system with temporary absence of applied forces, that part of the cam-system synthesis may be accomplished without applied forces. At the critical location, where both curves are in closest proximity, the spring force should exceed the inertial force with friction corrections included by not less than 25 to 50 percent.
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