Page 432 - Cam Design Handbook

THB13 9/19/03 7:56 PM Page 420

420 CAM DESIGN HANDBOOK

A second factor, C F , allows for cam and roller radii and width to relate the cam contact
force f c to the cam contact stress. This factor is obtained by a Hertzian stress formula and
is defined by
s = FC . (13.37)
2
c c F
To minimize the Hertzian contact stress at the cam, the quadratic formulation is used.
From Eqs. (13.36) and (13.37), the cost functional then becomes:
1
J = F N dt (13.38)
c Ú 0 f F
where

N = S C . (13.39)
F
F
F
The discussion after Eq. (13.31) explained that use of F f in the minimization criterion
caused reversal at the start of the motion. Similarly, the use of F f here has been shown to
do the same. Hence, the following formulation for minimizing the Hertzian cam contact
stress may be used without causing reversal:
1
J = F N dt (13.40)
2
2
c Ú 0 fe F
where
s = FN . (13.41)
2
ce fe F
The stress parameter, s ce , is that fraction of the Hertzian cam contact stress
resulting from the effective follower-spring force F fe , i.e., the preload effect has been
eliminated as was the case in the minimization of the follower-spring force. The
quadratic formulation in F fe is used to ensure that a negative value for force F fe is
undesirable.

13.5.6 Formulation of Optimization Problem
At this stage it is possible to define the problem of the optimization of a high-speed
cam-follower system using Eqs. (13.35) and (13.40), with weight factors W i (i = 1, 2, 3)
as follows:
Minimize
1
2 ˙˙
J = ( W F + W F + W F N d ) t (13.42)
2 ˙˙˙
2
2
Ú 0 1 fe 2 fe 3 fe F
subject to the system Eq. (13.30) and boundary conditions Eq. (13.32).
To design for desirable output characteristics, weight factor W 3 may be set to zero. To
design for favorable cam characteristics, weight factor W 3 is chosen as large as necessary
relative to weight factors W 1 and W 2 , with W 2 nonzero. Weight factor W 2 must be nonzero
to provide sufficient boundary conditions to ensure continuity of the cam acceleration
function.
When weight factor W 3 does not vanish, the problem cannot be solved by linear
methods due to the nonlinearity of parameter N F . It can, however, be solved using optimal
control theory.