Page 422 - Cam Design Handbook

THB13 9/19/03 7:56 PM Page 410

410 CAM DESIGN HANDBOOK

where the b n are subject to the constraints

 B n = H - ( V + V ) (13.12)
1
2
n–even n p 2
V
B ( V + )
 n = 2 1 (13.13)
n–odd n 2
 nB =0 (13.14)
n
n–even
 nB =0. (13.15)
n
n-odd
The associated cam profile tuned to operate at F 2 = F 2d is
H
y = T + Â C ( cos ( nT)+ D sin ( nT)) (13.16)
p n n n
where
-2 x
nB
C = 2 n (13.17)
n
[ F ( F + 4 x 2 2 )]
2
2
n
d
2
d
2
[ n -( F + 4 x 2 )] B
n
2
2
2
D = 2 d 2 n (13.18)
n
(
[ nF + 4 x 2 )]
2
n
2
2
2 d 2
and for a damped system Eq. (13.10) requires that
 C = 0 (13.19)
n
n–even
 C = 0. (13.20)
n
n–odd
As an example we consider the design of a unit rise dwell-rise-dwell motion subject to
the additional constraint that maximum value of X≤ be minimized. Three solutions to this
problem are tabulated in Table 13.1 and their acceleration profiles are shown in Fig. 13.6.
Solution number 3 is the preferred one since it provides the lowest maximum value of X≤.
Since we have assumed no damping in the development above, Eq. (13.4) reduces to
2
r ¢¢ + F r = ¢¢ y , r () = ¢() =r0 0 0. (13.21)
2
As shown by Eq. (13.6), R s is indicative of the tendency for a motion to excite large steady-
state vibrations.
Figures 13.7 and 13.8 illustrate typical residual vibration characteristics obtained using
solution numbers 1 to 3 tuned for F 2d =• and 10, respectively. For comparison we have
also shown the characteristics of parabolic and cycloidal motions. In plotting these curves
we have only indicated the peak values of R joined by dotted lines, except in the imme-
diate region of the designed operating point where we have plotted R exactly in order to
show the sensitivity of the performance to changes in the operating conditions.
From these figures the reader may verify several general observations. First, for large
values of F 2 (i.e., low speeds) the residual vibration of a family of cam profiles increases