Page 419 - Cam Design Handbook

THB13 9/19/03 7:56 PM Page 407

CAM SYSTEM DYNAMICS—RESPONSE 407

For many years much research has been directed toward identifying and tabulating suit-
able functions to define cam motion segments which produce good dynamic response.
Neklutin (1952) contributed much in this area.
Freudenstein (1960) observed that cam dynamics could be improved by minimizing
the harmonic content of the motion and developed a family of low harmonic curves that
minimized the peak acceleration.
The next four sections of this chapter each explore a method for determining optimal
or at least near-optimal cam curves for any particular set of requirements (mainly lift, oper-
ating speed, system natural frequency, and duration). These methods allow a designer to
systematically obtain a near-optimal cam curve directly for whatever system is being
developed. The designer does not have to choose a curve which is known to have good
vibration characteristics and hope for the best nor must a variety of curves be selected to
find the best one through analysis.
Wiederrich (1981) showed, by applying modal analysis methods, that once the cam
curve has been designed using a single-DOF model to minimize vibration over the entire
speed range, from zero to maximum, then the actual system, with its many degrees of
freedom, has also been so optimized (i.e., the vibration of each of the modes has been
minimized). However, if the one-DOF model is used to optimize only at a single speed or
over a limited range of speeds, the vibration from the higher modes cannot be assumed to
be optimized and their combined vibration response could be significant compared to that
predicted by the single DOF analyzed. Therefore, it is generally recommended that opti-
mization using a single-DOF model be applied over the entire speed range. Of course, the
single-DOF model must also satisfy the conditions stated in Sec. 13.1 before any conclu-
sions can be made as to the adequacy of the design.
The methods presented are all relatively complex mathematically and thus require solu-
tion by computer using a program written for that purpose. Once the computer program
is available, application is relatively easy.

13.4 DYNAMIC SYNTHESIS OF CAMS USING
FINITE TRIGONOMETRIC SERIES

13.4.1 Introduction

In this section, we present cam design methods originated by Wiederrich and Roth (1978).
In these methods, follower or cam displacement is defined as a finite trigonometric series.
One method generates a “tuned” cam design, in which the cam profile is designed
to provide exactly the follower lift as a given finite trigonometric series function at one
particular cam speed. With this method, the performance at other speeds can be very
poor. Another method uses a finite trigonometric series function for cam lift and a mean
square error minimization technique to optimize the cam performance over a range of cam
speeds.
The methods employ the usual linear single degree-of-freedom mathematical system
model. The original 1975 paper provides a mathematical method to estimate the limita-
tions of such a model for any particular case. Basically, the method shows the single
degree-of-freedom model to be valid when the highest significant frequency of a Fourier
representation of the lift event is much lower than the second mode natural frequency of
the system. This is almost always true of a good design, where the highest significant fre-
quencies are significantly below the first mode natural frequency and therefore far below
that of the second mode.