**
9.4 KINETOSTATICFORCE ANALYSIS OF THE
FORCE-CLOSED CAMFOLLOWER
**

The
previous sections introduced forward dynamic analysis and the solution to the system
differential equation of motion (9.2b). The applied force
*
Fc
*
(
*
t
*
)
is presumed to be known, and the system equation is solved for the resulting
displacement
*
x
*
from which its derivatives can also be
determined. The
**
inverse dynamics
**
, or
**
kinetostatics
**
, approach provides a quick way to determine
how much spring force is needed to keep the follower in contact with the cam at
a chosen design speed. The displacement and its derivatives are defined from
the kinematic design of the cam based on an assumed constant angular velocity
ù
of
the cam. Equation 9.2b can be solved algebraically for the force
*
Fc
*
(
*
t
*
)
provided that values for mass
*
m
*
, spring constant
*
k,
*
preload
*
Fpl
*
, and damping factor
*
c
*
are
known in addition to the kinematic displacement, velocity, and acceleration
functions.

Figure
9-1a (p. 214) shows a simple plate or disk cam driving a spring-loaded, roller
follower. This is a force-closed system which depends on the spring force to
keep the cam and follower in contact at all times. Figure 9-1b shows a lumped
parameter model of this system in which all the
**
mass
**
that
moves with the follower train is lumped together as
*
m
*
,
all the springiness in the system is lumped within the
**
spring constant
**
*
k
*
,
and all the
**
damping
**
or resistance to movement is lumped together as
a damper with coefficient
*
c
*
.

The
designer has a large degree of control over the system spring constant
*
keff
*
as
it tends to be dominated by the
*
ks
*
of the physical return spring
in this model. The elasticities of the follower parts also contribute to the
overall system
*
keff
*
but are usually much stiffer than the physical
spring. If the follower stiffness is in series with the return spring, as it
often is, equation 8.15c (p. 191) shows that the softest spring in series will
dominate the effective spring constant. Thus, the return spring will virtually
determine the overall
*
k
*
unless some parts of the follower train have
similarly low stiffness.

The
designer will choose or design the return spring and thus can specify both its
*
k
*
and
the amount of preload to be introduced at assembly. Preload of a spring occurs
when it is compressed (or extended if an extension spring) from its
*
free length
*
to its initial assembled length. This is a necessary and desirable
situation as we want some residual force on the follower even when the cam is
at its lowest displacement. This will help maintain good contact between the
cam and follower at all times. This spring preload
*
Fpl
*
adds
a constant term to equation 9.2b which becomes:

or
:

The
value of
*
m
*
is determined from the effective mass of the
system as lumped in the single-
*
DOF
*
model of Figure 9-1. The value
of
*
c
*
for most cam-follower systems can be estimated
for a first approximation to be about 0.06 of the critical damping
*
cc
*
as
defined in equation 9.2i (p. 218).