Page 151 - Cam Design Handbook

THB5 8/15/03 1:52 PM Page 139

CAM MOTION SYNTHESIS USING SPLINE FUNCTIONS 139

d
d ¢¢ + ¢(C + ) (Mw )+ ( d K + K ) (Mw 2 ) =-S h w 2 K
() 2
C
s f s f c c d f
2
K
() 1
(K f + ) wb 2 - S h w d (5.31)
c
s
c
(
[KC s + ) (K s + K f ) - ] (Mw 2 ) b.
C
C
f
f
f
Here initial conditions at the start of the rise are taken as d = 0 and d¢= 0.
If the cam-follower system is free from damping, then in terms of the synthesized
output motion Eq. (5.31) becomes:
4
d ¢¢ + ( d K + K ) (Mw 2 ) =-S () 4 (hM )(w b ) (K + K ) w 2
s f d s f
(5.32)
2
)
- S () 2 (h w 2 )(w b .
d
It is obvious from Eq. (5.31) that the vibrational response is governed by the cam veloc-
(1)
ity and acceleration if the parameters for mass, springs, and dampers are fixed. Here, S c
(2)
and S c are determined from the synthesized output motion based on the design speed w d
and cam angle b. The solution of Eq. (5.31) yields the primary vibration at any rotational
speed w. As for residual vibration (when t > 1), the right-hand side of Eq. (5.31) dis-
appears. Therefore, from the homogeneous solution of Eq. (5.31) the amplitude of the
residual vibration at any rotational speed w can be found as:
- ) }
2
]
w
d = { d ()+[ d¢() + d() 1 xw n (w n 1 x 2 2 1 2 (5.33)
1
2
1
where:
n (
w n = the natural frequency of the cam-follower system, w = ( K + K f ) ) and
M
s
x = the damping ratio, (x = 0.5(C s + C f)/M/w n).
When the cam is rotating at the frequency of w = w d/b the values of d(1) and d¢(1)
(t = 1) in Eq. (5.32) will vanish and the amplitude of the residual vibration becomes zero.
To assess vibrational responses, both primary and residual vibrations can be examined in
a form of response spectrum for each synthesized output motion (Chen, 1981 and 1982;
Mercer and Holowenko, 1958; Rees Jones and Reeves, 1978; Neklutin, 1954). The vibra-
tional characteristic for the camshaft at various rotational speeds can be observed from the
response spectrum even though the output motion is synthesized for the camshaft at a con-
stant speed, w d.
Example Application
Example 8: Synthesis of Cam Motion with Nonrigid Follower An example appli-
cation is provided here to illustrate the mechanics of applying the procedure described
earlier. A DRD motion is synthesized using the spline interpolation procedure. The
dynamic behavior of the follower system is then investigated and the cam motion is found
using the spline collocation procedure. In preparing this example a case was selected from
the literature for which a cam had been designed using an optimized polynomial (Peisakh,
1966; Chen, 1981). The solution obtained using the optimized polynomial gives a con-
venient basis for evaluating the performance of the methods described here, at least for
one case. The results shown were refined iteratively by trying a total of ten different com-
binations of spline parameters and sets of constraints. Space limitations prevent the pres-
entation of all the iterations; however, it should be clear that the procedures being used
allow great freedom in specifying motion constraints and can easily evaluate the effects
of changing these constraints and of altering the spline parameters.
In this example, each step of the process is illustrated even though some of the
operations can be easily reduced to an algorithm and eliminated as a concern to the