The effects of damping ratio
can best be seen in Figure 9-6d, which shows a
3-D plot of forced vibration amplitude as a function of both frequency ratio
and damping ratio. The addition of damping reduces the amplitude of vibration
at the natural frequency, but very large damping ratios are needed to keep the
output amplitude less than or equal to the input amplitude. This is much more
damping than is found in cam-follower systems and most machinery. About 50 to
60% of critical damping will eliminate the resonance peak. Unfortunately,

most
cam follower systems have damping ratios of less than about 10% of critical. At
those damping levels, the response at resonance is about five times the static
response. This will create unsustainable stresses in most systems if allowed to
occur.

It
is obvious that we must avoid driving an underdamped system at or near its
natural frequency. One result of operation of an underdamped cam-follower
system near
*
n
*
can
be
**
follower jump
**
. The system of follower ass and spring can
oscillate violently at its natural frequency and leave contact with the cam.
When it does reestablish contact, it may do so with severe impact loads that can
quickly fail the materials.

The
designer has a degree of control over resonance in that the system’s mass
*
m
*
and
stiffness
*
k
*
can be tailored to move its natural frequency
away from any required operating frequencies. A common rule of thumb is to
design the system to have a fundamental natural frequency
*
n
*
at
least ten times the highest forcing frequency expected in service, thus keeping
all operation well below the resonance point. This is often difficult to
achieve in mechanical systems. One tries to achieve the largest ratio
*
n
*
/
*
f
*
possible nevertheless. As will be shown in
later chapters, the high harmonic content of cam functions can require that
even higher ratios than 10 are needed in some cases.

**
**

**
Follower Rise Time
**

For
cams with finite jerk having a dwell after the rise of the same order of
duration as the rise, Koster
[2]
defines a
dimensionless ratio,

where
*
T
*
*
n
*
is the period of the follower system’s natural
frequency and
*
T
*
*
r
*
is the time to complete the rise portion of the
follower motion. To minimize the effects of the transient vibrations on the
system, this ratio should be as small as possible and always less than 1. A
small value of
is equivalent to a small ratio of *f */
*n*. The error in acceleration of the follower due
to vibrations during the dwell period will be proportional to the first power
of , and the error in follower position will be
proportional to the third power of , for
values of < 0.5. If the dwell’s duration is about as
long or longer than that of the rise, these transient vibrations will tend to
die out by the end of the dwell.[2] The
next fall or rise will again provide an input to the system and cause a new
transient response.

Thus,
the response of a cam-follower system of this type will be dominated by the
recurring transient responses rather than by the forced, or steady-state,
response. It is important to adhere to the fundamental law of cam design and
use cam programs with finite jerk in order to minimize these residual
vibrations in the follower system. Koster
[2]
reports
that the cycloidal and 3-4-5 polynomial programs both gave low residual
vibrations in the double-dwell cam. The modified sine acceleration program will
also give good results. All these have finite jerk. Other functions that offer
even lower residual vibration levels will be introduced in later chapters.

Some
thought and observation of equation 9.1d (p. 216) will show that we would like
our system members to be both light (low
*
m
*
) and stiff
(high
*
k
*
) to get high values for
*
n
*
and
thus low values for
.
Unfortunately, the lightest materials are seldom also the stiffest. Aluminum is
one-third the weight of steel but is also about a third as stiff. Titanium is
about half the weight of steel but also about half as stiff. Some of the exotic
composite materials such as carbon fiber/epoxy offer better stiffness-to-weight
ratios, but their cost is high and processing is difficult.

Note
in Figure 9-6 (p. 225) that the amplitude of vibration at large frequency
ratios approaches zero for forced and one for self-excited systems. So, if the
system can be brought up to speed through the resonance point without damage
and then kept operating at a large frequency ratio, the vibration will be
minimal, especially if it is a forced system. An example of systems designed to
be run this way are large self-excited devices that must run at higher speed
such as electrical power generators. Their large mass creates a lower natural
frequency than their required operating speeds. They are “run up” as quickly as
possible through the resonance region to avoid damage from their vibrations and
“run down” quickly through resonance when stopping them. They also have the
advantage of long duty cycles of constant speed operation in the safe frequency
region between infrequent starts and stops.

**
**