Page 412 - Cam Design Handbook

THB13 9/19/03 7:56 PM Page 400

400 CAM DESIGN HANDBOOK

13.1 INTRODUCTION TO RESPONSE

In Chap. 12 mathematical models of the differential equations were employed to establish
the performance of cam-follower systems. Modeling techniques were utilized to investi-
gate 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 fol-
lower 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 sufficiently 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 significant modes. Otherwise, the improvement by use of multi-
DOF 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 compari-
son to the natural frequency of the system. Therefore, vibrations excited by the accelera-
tion periods are not significantly 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 13.1 (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 influenced by the sudden application, reversal,
or removal of the excitation. The acceleration discontinuity in the harmonic cam profiles
leads to the high vibrations shown. As speed is increased, the cycloidal cam vibrations
will increase rapidly as the significant excitational frequencies approach the natural fre-
quency of the system. At sufficiently high speeds the cycloidal system vibrations will
approach those of harmonic profiles. As Sec. 13.4 will illustrate, we cannot generally
assume that profiles with acceleration discontinuities will always perform less well than
those without such discontinuities in high-speed systems. Sometimes the optimal solution
will have significant discontinuities in acceleration.