Page 335 - Cam Design Handbook

THB11 9/19/03 7:33 PM Page 323

CAM SYSTEM MODELING 323

KK
K = s t . (11.13)
eq
K + K
s t
The resulting natural frequency is
K
w = eq = 6 22 rad s = 0990 Hz.
.
.
n
M
Approach 3: Lump Mass and Stiffness Independently. Assume that a single mass
and spring are required and independently combine masses and springs into a single equiv-
alent mass and a single equivalent spring using techniques discussed in later sections. This
is the approach suggested by Chen (1982) for reducing models of cam-follower systems
and is illustrated schematically in Fig. 11.3d. The equivalent stiffness is again given by
Eq. (11.13) while the equivalent mass is simply the sum of the masses
M = M m. (11.14)
+
eq
The natural frequency of the simplified system is
K
.
.
w = eq = 5 09 rad s = 0 940 Hz.
n
M eq
Relative to the exact model, the error in natural frequency prediction for the three
approaches is about 4.2 percent, 0.03 percent, and 5.1 percent, respectively. While none
of these simplifications give exactly the same result as the exact model, all are close and
the error involved in approach 2 is below the error introduced by any reasonable meas-
urement system. Strictly speaking, all of these simplifications are wrong in that they do
not precisely predict the natural frequency. Nevertheless, they are useful since they reduce
complexity, produce reasonable approximations, and demonstrate which physical param-
eters are significant in determining the natural frequency of the body mode. For instance,
it is much clearer from these simple models that the sprung mass and the suspension spring
stiffness are the major determinants of the body mode natural frequency. While it would
be possible to choose the equivalent mass above and then choose an equivalent stiffness
that produced a value numerically equal to the exact natural frequency of the body mode,
this stiffness would lose its connection to physical parameters. Hence its usefulness as a
design tool would be reduced and the model would be valid only for the one specific
system for which it was developed.
The fact that the three approaches above all produce similar results provides
some measure of confidence in removing the tire degree of freedom. In this case, it follows
from the fact that the tire mass is an order of magnitude smaller than that of the body,
while the tire stiffness is about an order of magnitude larger. Hence, assuming that the
mass is zero or the stiffness is infinite are both reasonable assumptions to make. In other
systems, only one of these assumptions may be appropriate, based upon the physical
parameters.
A reasonable approach to producing dynamic models of cam-driven systems, therefore,
is to start with some understanding of the mass and stiffness of each component in the
system. With this information, the model can be reduced such that the most compliant
(least stiff) or most massive components are retained and the models ignore natural fre-
quencies above the frequency range of interest. This ensures that the model remains as
simple as possible. Of course, it is then necessary to verify (by verifying that the natural
frequencies associated with the components assumed to be rigid are outside the region of
interest or, preferably, though experimental validation) that the model is not too simple.
Models can be scaled up in complexity by replacing some of the stiffnesses or masses