Equation 9.12 (p. 241) gives the applied torque
*
Tc
*
in a camshaft. Equation 9.10b (p. 233) gives
the follower force for a force-closed system. Combining them gives

*
*

The
compensating cam needs to provide an equal and opposite torque
*
Tb
*
to
cancel the applied torque

*
*

where
the displacement of the compensating dummy follower is designated as
*
y
*
to
distinguish it from the primary follower’s displacement
*
x
*
.

Equating
the two torques gives

*
*

The
terms on the left-hand side (LHS) of equation 9.21 are all known. The factors
*
mb
*
,
*
cb
*
,
*
kb
*
, and the initial displacement
*
y
*
0
on the right-hand side (RHS) are all under the control of the designer. The
goal is to solve for the displacement function
*
y
*
of
the compensating cam. To do so requires iteration. Aviza[3] found that for
realistic values of driving cam parameters, if the acceleration of the
compensating cam follower train were included in equation 9.21, the numerical
method failed to converge. The equation was successfully solved by assuming a
zero mass for the compensating follower, thus eliminating the acceleration term
on the RHS.

Figure
9-28 shows the
*
s
*
,
*
v
*
, and
*
a
*
diagrams of a test cam
designed and built for this investigation. It is a simple, double-dwell design
with a 1-in modified sine rise over 90
°
, dwell for 90
°
, 1-in modified sine fall
over 90
°
, and dwell for 90
°
. Figure 9-29 shows the camshaft torque needed
to drive a follower mass of 0.004 bl against a follower spring of 30.23 lb/in
with a preload of 17.4 lb and 5% of critical damping.

Figure 9-30 shows the resulting
*
s
*
,
*
v
*
, and
*
a
*
diagrams of the required
compensating cam to cancel the torque assuming zero follower mass. Figure 9-31a
shows the original driving torque superposed on the countertorque from the
compensating cam. Figure 9-31b shows their difference—the residual torque in
the shaft—which is essentially zero.

However,
this computation results from the unrealistic assumption of a zero-mass
compensating cam follower train. When a realistic follower train mass for this
system of 0.003 bl is applied to the compensating cam train, the resulting
torque is as shown in Figure 9-32a, superposed on the original driving cam
torque. Figure 9-32b shows their difference or residual torque in the shaft,
now about 10% of the original, uncompensated torque. This technique, though
approximate, has resulted in a 90% reduction of peak torque in the camshaft. These
results are from a dynamic model of the system and were demonstrated
experimentally using the apparatus shown in Figure 9-27.[3]

Note
that it is possible in some cases for the contour of the calculated
compensating cam to have unacceptable pressure angles and/or radii of curvature
and thus be impractical to implement. In this study, the driving cam’s radii of
curvature and pressure angles were of reasonable values and so resulted in a
manufacturable compensating cam
.

Copyright 2004, Industrial
Press, Inc., New York, NY