Calculating the damping
*
c
*
based
on an assumed value of
requires specifying a value for the overall
system *k *and for its effective mass. The choice of *k *will
affect both the natural frequency of the system for a given mass and the
available force to keep the joint closed. Some iteration will probably be
needed to find a good compromise. A selection of data for commercially
available helical coil springs is provided in Appendix D. Note in equations
9.10 that the terms involving acceleration and velocity can be either positive
or negative. The terms involving the spring parameters *k *and
*Fpl *are the only ones that are always positive. To
keep the overall function always positive requires that the spring force terms
be large enough to counteract any negative values in the other terms.
Typically, the acceleration is larger numerically than the velocity, so the
negative acceleration usually is the principal cause of a negative force *Fc*.

The
principal concern in this analysis is to keep the cam force always positive in
sign as its direction is defined in Figure 9-1. The cam force is shown as
positive in that figure. In a force-closed system the cam can only push on the
follower. It cannot pull. The follower spring is responsible for providing the
force needed to keep the joint closed during the negative acceleration portions
of the follower motion. The damping force has an effect, but the spring must
supply the bulk of the force to maintain contact between the cam and follower.
If the force
*
Fc
*
goes negative at any time in the cycle, the
follower and cam will part company, a condition called
**
follower jump
**
. When they meet again, it will be with large and potentially
damaging impact forces. The follower jump, if any, will usually occur near the
point of maximum negative acceleration. Thus, we must select the spring
constant and preload to guarantee a positive force at all points in the cycle.
In automotive engine valve cam applications, follower jump is also called
*
valve float
*
, because the valve (follower) “floats” above the cam, also
periodically impacting the cam surface (sometimes called
*
valve crash
*
). This will occur if the cam rpm is increased to the point that
the larger negative acceleration makes the follower force negative. The
“redline” maximum engine rpm often indicated on its tachometer is to warn of
impending valve float above that speed that will damage the cam and follower.

Program
DYNACAM allows the iteration of equation 9.10 to be done quickly for any cam
whose kinematics have been defined in that program. The program's
*
Dynamics
*
button will solve equation 9.10 for all values of camshaft angle,
using the displacement, velocity, and acceleration functions previously
calculated for that cam design in the program. The program requires values for
the effective system mass
*
m
*
, effective spring constant
*
k
*
,
preload
*
Fpl
*
, and the assumed value of the damping ratio
. These values need to be determined for the
model by the designer using the methods described in Sections 8.9 and 8.10 (pp.
185-197). The calculated force at the cam-follower interface can then be
plotted or its values printed in tabular form. The system’s first natural
frequency, based on the model of Figure 9-1, is also reported when the tabular
force data are printed.

**
**

**
EXAMPLE 9-2
**

Kinetostatic
Force Analysis of a Force-Closed (Spring-Loaded) Cam-Follower

System.

**
**

**
Given:
**
**
**
A
translating roller follower as shown in Figure 9-1 is driven by a forceclosed

radial
plate cam which has the following program:

Segment
1: Rise 1 inch in 50
°
with modified sine acceleration

Segment
2: Dwell for 40
°

Segment
3: Fall 1 inch in 50
°
with cycloidal displacement

Segment
4: Dwell for 40
°

Segment
5: Rise 1 inch in 50
°
with 3-4-5 polynomial displacement

Segment
6: Dwell for 40
°

Segment
7: Fall 1 inch in 50
°
with 4-5-6-7 polynomial displacement

Segment
8: Dwell for 40
°

Camshaft
angular velocity is 18.85 rad/sec

Follower
effective mass is 0.0738 lb-sec2/in (blobs)

Damping
is 10% of critical (
=
0.10)

**
**

**
Problem:
**
**
**
Compute the
necessary spring constant and spring preload to maintain contact between cam
and follower, and calculate the dynamic force function for the cam. Calculate
the system natural frequency with the selected spring. Keep the pressure angle
under 30
°
.

**
**