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CAM MECHANISMS 4.6
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MACHINE ELEMENTS IN MOTION
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4.2 BASIC CAM MOTIONS
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Basic cam motions consist of two families: the trigonometric and the polynomial.
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4.2.1 Trigonometric Family This family is of the form s = C0 + C1 sin a + C2 cos b (4.9)
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where C0, C1, a, and b are constants. For the low-speed systems where d < 10 4, we can construct all the necessary diagrams, symmetric and unsymmetric, from just two curves: a sine curve and a cosine curve. Assuming that the total rise or return motion s0 occurs for an angular displacement of the cam = 0, we can partition acceleration curves into i separate segments, where i = 1, 2, 3, . . . with subtended angles 1, 2, 3, . . . so that 1 + 2 + 3 + = 0. The sum of partial lifts s1, s2, s3, . . . in the separate segments should be equal to the total rise or return s0: s1 + s2 + s3 + = s0. If a dimensionless description / of cam rotation is introduced into a segment, we will have the value of ratio / equal to zero at the beginning of each segment and equal to unity at the end of each segment. All the separate segments of the acceleration curves can be described by equations of the kind s = A sin or s = A cos 2 (4.11) n n= 1 , 1, 2 2 (4.10)
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where A is the maximum or minimum value of the acceleration in the individual segment. The simplest case is when we have a positive drive with a symmetric acceleration curve (Fig. 4.7d). The complete rise motion can be described by a set of equations s = s0 2 1 sin 2 s = 2 s0 2 sin 2 (4.12) s0 2 s = 1 cos 4 2s0 2 s = cos 3
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The last term is called geometric jerk (s = 3s ). Traditionally, this motion is called cycloidal. The same equations can be used for the return motion of the follower. It is easy to prove that sreturn = s0 srise s return = s rise s return = s rise s return = s rise (4.13)
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CAM MECHANISMS 4.7
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CAM MECHANISMS
FIGURE 4.8 Trigonometric standard follower motions (according to the equation of Table 4.1, for c = d = 0).
All the other acceleration curves, symmetric and unsymmetric, can be constructed from just four trigonometric standard follower motions. They are denoted further by the numbers 1 through 4 (Fig. 4.8). These are displayed in Table 4.1. Equations in Table 4.1 can be used to represent the different segments of a follower s displacement diagram. Derivatives of displacement diagrams for the adjacent segments should match each other; thus several requirements must be met in order to splice them together to form the motion specification for a complete cam. Motions 1 through 4 have the following applications: Motion 1 is for the initial part of a rise motion. Motion 2 is for the end and/or the middle part of a rise motion and the initial part of a return motion. The value c is a constant, equal to zero only in application to the end part of a rise motion. Motion 3 is for the end part of a rise motion and/or the initial or middle part of a return motion. The value d is a constant, equal to zero only in application to the initial part of a return motion. Motion 4 is for the end part of a return motion. The procedure of matching the adjacent segments is best understood through examples. Example 1. This is an extended version of Example 5-2 from Shigley and Uicker [4.8], p. 229. Determine the motion specifications of a plate cam with reciprocating fol-