Page 421 - Cam Design Handbook

THB13 9/19/03 7:56 PM Page 409

CAM SYSTEM DYNAMICS—RESPONSE 409

˙
˙˙
X + 2xw X +w 2 X = w 2 y + 2x w ˙. y (13.2)
2 2 2 1 1 1
Before we can proceed further, we need to mathematically define a vibration criterion.
The quality of the dynamic behavior of any system is reflected by its displacement rela-
tive to its static equilibrium position. We define the relative vibration as
w 2
r = 1 y x. (13.3)
-
w 2
2
In terms of r, Eq. (13.2) becomes
w 2 w
˙˙ r + 2xw ˙ r +w 2 r = 1 ˙˙ y + ( 2 wx -x w ) 1 ˙. y (13.4)
2 2 2 12 1 2
w 2 w
2 2
Our primary goal is to minimize the vibration levels in the steady state operation of
the cam-follower system. This level, R s, is dependent on the magnitude R of r induced by
one cycle of y and the amount of damping present. The residual vibration (R) is defined
as
12
Ï r¢ () ¸
2
t
2
R = Ì r t ()+ ˝ (13.5)
Ó F 2 2 ˛
where t would normally be the time at the end of the cam lift event (at the start of the
dwell period that continues until the next lift event). Alternatively, t could be the time at
some other critical part of the cam rotation. Then, R s can be expressed as
Ï È -2px w ˘¸
/
R £ R Ì 1 - exp 2 2 ˝ . (13.6)
s Í ˙
Ó Î w ˚˛
Eq. (13.2) can be written in normalized form as
+
2
2
X ¢¢ + 2x F X ¢ + F X = F y 2x F ¢ y (13.7)
22 2 2 22
where we have used the definitions of F 2, y¯, and ( )¢ given in the nomenclature and assumed
that x 1 w 2 = x 2 w 1 .
The required motion is to occur over an angle of cam rotation q 0 , and the motion should
begin and end with R identically zero so that it does not contribute to R s. In addition we
¨
require that the motion begin and end with y equal to zero. Substituting these conditions
into Eq. (13.7) it follows that the required motion is subject to the constraints
0
p
X 0 () = ¢¢() = X¢¢¢() = X¢¢() = X¢¢¢() = 0
p
X 0
(13.8)
X() =p H, X 0 = V , X¢() =p V .
¢()
1 2
In addition, if x 1 = x 2 = 0 then
p
X 0 () = X () = 0 (13.9)
iv
iv
and if x 1 and x 2 are not zero we must specify that
y 0 () = 0, y() = H. (13.10)
p
These conditions are satisfied by a motion of the form
H B
X = T - Â n sin ( nT) (13.11)
p n n 2