codeproject vb.net barcode generator Data of First Half of Rise Motion Used for Calculations of Second Half in Software

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TABLE 4.3 Data of First Half of Rise Motion Used for Calculations of Second Half
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All the trigonometric curves of this section were calculated with finite values of jerk, which is of great importance for the dynamic behavior of the cam mechanism. An example of the jerk diagram is given in Fig. 4.10. The jerk curve j was plotted by using the dimensionless expression j= s s0/ 3 0
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(4.17)
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This form of the jerk description can also be used to compare properties of different acceleration diagrams.
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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 4.14
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MACHINE ELEMENTS IN MOTION
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4.2.2 Polynomial Family The basic polynomial equation is s = C0 + C1 + C2 0 0
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+
(4.18)
with constants Ci depending on assumed initial and final conditions. This family is especially useful in the design of flexible cam systems, where values of the dynamic factor are d 10 2. Dudley (1947) first used polynomials for the synthesis of flexible systems, and his ideal later was improved by Stoddart [4.9] in application to automotive cam gears. The shape factor s of the cam profile can be found by this method after a priori decisions are made about the motion y of the last link in the kinematic chain. Cams of that kind are called polydyne cams. When flexibility of the system can be neglected, the initial and final conditions ([4.3], [4.4], and [4.8]) might be as follows (positive drive): 1. Initial conditions for full-rise motion are =0 0 2. Final (end) conditions are =1 0 s = s0 s = 0 s = 0 s=0 s = 0 s = 0
The first and second derivatives of Eq. (4.18) are s = C1 + 2C2 + 3C3 0 0
+ 4C4
0
+ (4.19)
s = 2C2 + 6C3 + 12C4 0 0
+
Substituting six initial and final conditions into Eqs. (4.18) and (4.19) and solving them simultaneously for unknowns C0, C1, C2, C3, C4, and C5, we have s = 10s0 s0 0 0
1.5
0 0
+ 0.6 0
0
s = 30
0
+ 0
(4.20)
s = 60 and for a jerk s = ds /d , or
s0 3 2 0 0 0
s = 60
s0 1 6 +6 3 0 0 0
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CAM MECHANISMS 4.15
CAM MECHANISMS
This is called the polynomial 3-4-5, since powers 3, 4, and 5 remain in the displacement equation. It provides a fairly good diagram for the positive drives. Equations for the full-return polynomial are s(return) = s(rise) + s0 s (return) = s (rise) s (return) = s (rise) s (return) = s (rise) (4.21)
All the characteristic curves of the full-rise 3-4-5 polynomial are shown in Fig. 4.11. They were generated by the computer for s0 = 1 displacement unit (inches or centimeters) and 0 = 1 rad.
FIGURE 4.11 Full-rise 3-4-5 polynomial motion.
After a proper set of initial and final conditions is established, the basic equation [Eq. (4.18)] can be used for describing any kind of follower motion with an unsymmetric acceleration diagram. Details concerning the necessary procedure can be found in Rothbart [4.7].
4.2.3 Other Cam Motions The basic cam motions described in the previous sections cover most of the routine needs of the contemporary cam designer. However, sometimes the cost of manufacturing the cam profile may be too high and the dynamic properties of the cam motion may not be severe. This is the case of cams used for generating functions. There is a very effective approach, described by Mischke [4.2], concerning an optimum design of simple eccentric cams. They are very inexpensive, yet can be used even for generating very complicated functions.
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 4.16
MACHINE ELEMENTS IN MOTION
The other approach, when we are interested in inexpensive cams, is to use circulararc cams or tangent cams. They are still used in low-speed diesel IC engines since the cost of their manufacture is low (compare with Fig. 4.15). An extensive review of these cams can be found in Rothbart [4.7]. They were used quite frequently in the past when the speed of machines was low, but today they are not often recommended because their dynamic characteristics are poor. The only exception can be made for fine- or light-duty mechanisms, such as those of 8- and 16-mm film projectors, where circular-arc cams are still widely used. Those cams are usually of the positive drive kind, where the breadth of the cam is constant. The cam drives a reciprocating follower with two flat working surfaces a fixed distance apart, which contact opposite sides of the cam. The constant-breadth cam is depicted in Fig. 4.12. For given values of radius , total angle of cam rotation 0, and total lift of the follower s0, the basic dimensions of the cam can be found from the relations ([4.11]) R1 = b(s0 + ) 1 b r = R1 s0 (4.22)
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