Page 425 - Cam Design Handbook

THB13 9/19/03 7:56 PM Page 413

CAM SYSTEM DYNAMICS—RESPONSE 413

as the peak acceleration is decreased. Second, tuning a cam for higher speeds increases
the tendency for the cam to excite vibrations at lower speeds. Finally, we can verify that
the vibration behavior of these tuned cams is better than that of a comparable cycloidal
cam for speeds below and slightly above the designed speed if the designed operating
point, F 2d , is greater than the harmonic content of the tuned cam.
Tuned cams are appropriate for use in those systems that are intended to operate at a
constant or nearly constant speed. Where applicable, the method of tuned cam design may
be the most powerful method available in that it allows the designer to specify entirely
the output motion at the design speed. However, the performance of the system away from
the design speed may deteriorate depending on the designer’s choice of optimization
criteria and constraints.
In many applications, cam systems are required to operate over a wide range of oper-
ating speeds. In the next section we develop methods for designing cams to have low
vibration characteristics over a range of operating speeds. A primary objective of these
methods is the elimination of all unnecessary constraints to obtain the largest possible
family of solutions.

13.4.4 Cam Design Based on Mean Squared Error Minimization
Cams are often designed to operate over a wide range of speeds. Since it is not possible
to design a cam to be optimal at more than one speed at a time, the question becomes how
best to design the cam for the best overall performance. This section presents a method
of designing cams for optimized performance over a range of operating speeds.
The cam performance index, which reflects the deviation of the system from its desired
operating conditions at any one speed, is defined as
( XT - X¢) 2 p
¢()
2
P = Â ( X T () - X ) + Â j j + A X dT. (13.22)
¢¢
A
A
2
1 i i 2 F 2 3 Ú 0
i j 2
Minimizing P is an indication of optimal cam design, since this is a measure of the
deviation in follower position, velocity, and acceleration.
In addition, it may also be desired to impose constraints on the cam profile of the form
¢()
,
l
1
yT () = y , y T = y¢, k = ◊◊◊, . (13.23)
k
k
k
k
We want to optimize the performance over the operating range of speeds
1 1 1
£ £ . (13.24)
F 21 F 2 F 2h
To optimize the performance, we want to minimize the average value of the mean squared
¯
error P over this speed range. Thus, we want to minimize
1 F h Ê 1 ˆ
2
P = PdÁ ˜. (13.25)
F Ú 1 Ë F ¯
21
2
Assuming that
•
y = Â (a cos ( nT)+ sin ( nT)) (13.26)
b
n n
n=0
then the forced response of the follower for the worst case of an undamped system
(x 1 = x 2 = 0) is, from Eq. (13.7),