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416 CAM DESIGN HANDBOOK

the usual rule of thumb that motions with discontinuous accelerations or high values of
jerk always have bad vibration characteristics.
2
From Fig. 13.10, we see that p X≤ max ≥ 6, but if desired, further reduction is possible
by increasing the ratios of A 3 /A 1 and/or A 3 /A 2 in Eq. (13.22).
13.4.5 Conclusion
The design of high-speed motions has been posed as an extremum problem. The solutions
thus obtained show that a trade-off exists between minimizing the imposed inertial force
and reducing the additional force and loss of control associated with large vibrations. In
addition, examples were shown in which motions with discontinuous acceleration were
found which have low vibration characteristics over a wide range of operating speeds, a
significant contradiction to the generalized statements concerning the bad vibration
characteristics of motions with high values of jerk. In view of this, it is recommended that
high-speed cam designs not be based upon rules of thumb, which rather arbitrarily specify
peak values of acceleration and jerk. Instead, at high speeds, each design should be eval-
uated separately based upon its particular requirements. The methods presented here are
well-suited for high-speed cam design.
All the important system parameters needed for these analyses are rarely known in
advance with extreme accuracy. In addition, damping will never be zero, as assumed in
the calculations for the response to one cycle. Therefore, the tuned cam design method of
Sec. 13.4.3 could well result in poor vibration characteristics at the design speed. Thus,
this method should rarely be used in practice; rather, the mean squared error minimiza-
tion method of Sec. 13.4.4 should be used. For a constant speed system this method should
be applied over a range of speeds bracketing the design speed and large enough to accom-
modate possible errors in the assumed parameters. Similarly, in a variable speed system
the desired speed range should be increased in the design process to account for errors.
Wiederrich (1981) showed that optimizing a design to minimize the residual vibrations
over the range of speeds from zero to design speed, using the single degree of freedom
model, also minimizes the response for the higher modes. This procedure minimizes the
deficiencies caused by the absence of higher modes in the one-degree-of-freedom model
to obtain a near-optimal solution.

13.5 APPLICATION OF OPTIMAL
CONTROL THEORY TO THE SYNTHESIS
OF CAM-FOLLOWER SYSTEMS

13.5.1 Introduction
Any cam design requires trade-offs between characteristics at the cam (such as contact
stresses, lateral forces, etc.) and at the output (such as vibrations, forces, etc.). Any pro-
cedure that optimizes only one of these characteristics penalizes the others. One method
that provides an approach to optimizing several of these characteristics simultaneously is
based on optimal control theory (Chew et al., Parts 1 and 2, 1983). Here, we define suit-
able optimality criteria and show how to use them to design systems to optimize several
parameters simultaneously.
13.5.2 Symbols

Dimensional parameters
f c = contact force between cam and follower
f ce = effective contact force between cam and follower