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1234 Two-Degree-of-Freedom System Let us consider a two-degree-of-freedom linear, closed-track cam-follower system as shown in Fig 129 For other applications of two DOF systems, see Hanachi and Freudenstein (1986) and Eliss (1964) Let m1, m2 = respective mass: 1, 2, lb k1, k2 = respective spring, 1, 2, lb/in b1,b2 = respective damping coef cient, 1, 2, lb/(in/sec)

3 2 1 0 1 2 3

FIGURE 128 Cam example (chatter resulted from resonance with strong 18th harmonic number)

d (nl'

Based on the free-body diagrams, applying Newton s second law, we obtain m1 x1 = b2 z2 + k2 z2 - b1z1 - k1z1 m2 x2 = - b2 z2 - k2 z2 where z1 = x1 - y and z2 = x2 - x1 are the relative coordinates Eq (129) can be rearranged as m1z1 + b1z1 + k1z1 = b2 z2 + k2 z2 - m1 y m2 z2 + b2 z2 + k2 z2 = - m2 ( z1 + y) (1210) (129)

27 8 1 0 1 8 27

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Substituting the rst expression for 1 into the second Eq (1210) leads to the following z z1 + 2z 1w 1z1 + w 12 z1 = -w 12d 1 2 2 z2 + 2z 2w 2 z2 + w 2 z2 = -w 2 d 2 , where y b2 k - z - 2z w 12 k1 2 k1 2 w2 b b m d 2 = - 1 2 z1 + 2 2 z2 - 12 z1 + 2 z2 w2 m1w 2 m1w 2 m1 d1 = w 12 = k1 k 2 , w2 = 2 m1 m2

We plot these points on a single set of axes (Figure 121) Supposing that the curve we seek to draw is a smooth interpolation of these points (calculus will later show us that this supposition is correct), we find that our curve is as shown in Figure 122 This curve is called a cubic

(1211)

z1 =

= 4 x 2 and ,/l+($)

It is convenient to make a table of values:

1235 Some Cam-System Dynamic Phenomena 12351 Balancing In this section miscellaneous subjects of interest in the dynamics of cam-follower systems are covered, such as the balancing of rotating elements of the cam body, the crossover shock that occurs in closed-track cam-follower systems, the spring surge (resonance) that occurs with open-track cams, and harmonic analysis of cams The balance of the dynamic members of all complex machinery is a critical factor in the performance Thus it is with high-speed cam-follower systems The imbalance of the asymmetrical cam mass itself with respect to the center of rotation is the rst issue of concern Obviously, strong vibrations are induced in the system at the cam speed fundamental harmonic This vibration can be kept to a minimum, although not completely eliminated, by intelligent practical cam proportions, using a well-ribbed, low-mass structure with designed counterweights as part of the cam body Some of the other sources of vibration due to imbalance that may be investigated for cam-follower systems are:

4 3 2 1 0 1 2 3

.

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