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In Chap 12 mathematical models of the differential equations were employed to establish the performance of cam-follower systems Modeling techniques were utilized to investigate the vibratory response of the follower in the time domain In this chapter much of the dynamic study is done as an input of time transients to the cam-follower system Also, Wiederrich (2001) has contributed in the development of this chapter In the beginning, we will be operating in the frequency domain The dynamic response of a cam-follower system has the following three considerations: The driving motion produced by the cam, called the base excitation Note that other external disturbing forces may also act on the follower at the same time The mass, elasticity, and damping of the system between the base excitation and the follower end point The behavior of the follower caused by the excitation, which is called the response The studies presented in this chapter all use the single degree-of-freedom (DOF) model As stated in Chap 12, one DOF is suf ciently accurate to model most cam-follower systems This DOF, the fundamental mode of the system, usually represents the great majority of the dynamic deformation of the system In systems in which one DOF is not clearly dominant, the error in a one-DOF model may be too great When this occurs, either a multi-DOF system must be used or the system structurally redesigned to mitigate the adverse effect of the other signi cant modes Otherwise, the improvement by use of multiDOF does not justify the additional analytical and modeling complexity or the additional data required Multi-DOF system responses are occasionally referred to but will not be treated in this chapter The response of a typical dynamic system consists of both steady state and transient responses For cam-follower systems it is the transient response that is pertinent Steady state vibration is usually not a concern since the cam angular velocity is low in comparison to the natural frequency of the system Therefore, vibrations excited by the acceleration periods are not signi cantly reduced or reinforced by succeeding cam cycles For ease of analysis and simplicity to compare the different cam curve responses we usually assume that vibration damps out during the dwell period and does not carry over to the next cycle The designer will consider The primary response produced during the application of the base excitation or stroke The residual response that remains at the start of the dwell after the removal of the excitation Figure 131 (Hrones, 1948 and Mitchell, 1950) shows the primary and residual responses of a relatively low speed cam-follower system Note that the vibrations occur at the natural frequency of the system Vibrations take place during the stroke and the dwell periods, with their peak magnitudes in uenced by the sudden application, reversal, or removal of the excitation The acceleration discontinuity in the harmonic cam pro les leads to the high vibrations shown As speed is increased, the cycloidal cam vibrations will increase rapidly as the signi cant excitational frequencies approach the natural frequency of the system At suf ciently high speeds the cycloidal system vibrations will approach those of harmonic pro les As Sec 134 will illustrate, we cannot generally assume that pro les with acceleration discontinuities will always perform less well than those without such discontinuities in high-speed systems Sometimes the optimal solution will have signi cant discontinuities in acceleration.

Nominal (Theoretical) curve:

78 Calculate (a) (b) (c) (d) (e)

Second Solution: The general rule for differentiating a fraction is:

Actual curve:

79 Calculate 3sin x cos x dx (a) 3sin x cos x ln 3 (b) 3cos x ln 3 (c) 3sin x ln 3 (d) 3sin x / ln 3 (e) 3cos x / ln 3 80 The derivative of g (x) = x sin x is (a) (b) (c) (d) (sin x)(cos x)x sin x (sin x) x cos x (cos x) x sin x (sin x) x sin x 1 sin x x sin x (e) cos x ln x + x

Tolerance:

FIGURE 106 Precision measuring machine example Leitz PMM, Browne Sharpe Crop (Courtesy CAMCO Corporation, Wheeling, Ill)

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81 A population of bacteria tends to double every three hours If there are 10,000 bacteria at 10:00 am, then how many will there be at noon (a) 14,444 (b) 16,355 (c) 12,991 (d) 15,874 (e) 13,565

acceptable accuracy of fabrication is determined by the experience of the shop artisans and the availability of effective machine tool equipment The subject of proper cam production is most signi cant to the ultimate performance and acceptability by the customer Errors in any machining operation are statistical phenomena and the probabilistic approach to their study will produce meaningful results Kim and Newcombe (1978, 1982) applied probabilistic techniques in the study of the effect of manufacturing tolerance in high-speed cams Grewal and Newcombe (1988) and Newcombe and Kim (1983) applied the same techniques, separating the size and waviness components of the errors The actual manufactured cam-follower system to perform its theoretical design function is critically desired The deviation of the actual follower s action from the theoretical can be listed in four categories of manufacturing in which in each mechanism can be the source of more than one type of error: structural errors xed backlash errors

By applying this rule the problem becomes much easier:

82 Calculate (a) (b) (c) (d) (e)

1041 Structural Errors Structural errors are mistakes in the dimension of the part or the utilization of wrong parts in building the machine Another error of this type is one in the computer programming data, for example, in the case of a cam driving an oscillating follower If the lever were the wrong length it would give the follower the wrong information because it would rest on the improper part of the cam curve

1042 Fixed Backlash Errors Most mechanisms have clearances necessary for assembly and operation, ie, clearance between gear teeth and clearance between the roller follower and the cam groove The smallest necessary clearance will result in a minimum backlash error For cam-follower systems, backlash is a major source of error and the designs should be established to reduce it as much as possible

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