Page 431 - Cam Design Handbook

THB13 9/19/03 7:56 PM Page 419

CAM SYSTEM DYNAMICS—RESPONSE 419

Applying the optimality criterion discussed above requires reconciling the number of
desired boundary conditions with the number specifiable by the use of optimal-control
theory. To minimize the output residual vibrations, the ten boundary conditions given by
Eq. (13.32) are sufficient. These ensure zero initial and end cam velocities and accelera-
tions, thus maintaining continuity at the terminals.

13.5.4.2 Relative Vibration and Relative Vibrational Energy. With damping neglected
and using the definition of R v , Eq. (13.30) can be rewritten as
( K - ) 1 F ˙˙
˙˙
R + k (2pl ) 2 R = fe (13.33)
v v
K K
k k
where
F = K Y - Y. (13.34)
fe k c
After integrating Eq. (13.33) with respect to R v, the left-hand side becomes the rela-
tive vibrational energy of the output. It follows that this vibrational energy is controllable
¨
by the parameter, F fe. In the same way, control of the third derivative of the effective
...
follower-spring force, F fe, controls the time rate-of-change of relative vibrational energy.
Hence, the continuity of this time rate-of-change of the relative vibrational energy is
directly related to the continuity of the cam acceleration. The latter is specifiable through
the boundary conditions given by Eq. (13.32).
13.5.4.3 Problem Formulation Based on the Output Criterion. From the discussion
above, the problem formulation based on the output optimality criterion may be written
as follows:
Minimize
1
2 ˙˙
J = ( W F + W F d ) t (13.35)
2 ˙˙˙
Ú 0 1 fe 2 fe
such that system Eq. (13.30) and boundary conditions Eq. (13.32) are satisfied.

13.5.5 Development of an Optimality Criterion-2: Cam Criterion
13.5.5.1 General. The optimality criteria derived up to this point are for output char-
acteristics. It is also desirable to introduce criteria for cam contact stress to allow a com-
promise between output performance and cam contact stress.

13.5.5.2 Contact Stress at Cam-Follower Interface. The Hertzian contact stress at the
cam-follower interface is a nonlinear function of parameters such as pressure angles, cam
offset, roller-follower mass, preload, follower-guide dimensions, follower-guide friction,
and cam curvature. These parameters can be grouped into three nonlinear factors, each
highlighting certain aspects of the cam contact stress.
Factor S F allows for follower guide friction and follower pressure angle. It relates
the follower-spring force f f on the output mass to the cam contact force f c and is given in
normalized form as:
F = F S . (13.36)
F
c
f