Page 374 - Cam Design Handbook

THB12 9/19/03 7:34 PM Page 362

362 CAM DESIGN HANDBOOK

Substituting z = x - y yields
+
+
˙
mz bz kz = - my ˙˙ (12.3)
˙˙
which places in evidence the prominent role of cam function acceleration y on the vibrat-
ing system, similar to a system response to a shock input at its foundation.
The general solution to Eq. (12.3) consists of the complementary or transient solution
plus the particular solution. The complementary solution is the solution to the homoge-
nous equation
˙ +
˙˙+
mz bz kz = 0
z ˙˙+ 2zw n z ˙ +w n 2 = 0
where w = ( km) 12
n
z = 1 2 b(1 km) 12
and is
( [
( [
12
12
z = Ae -zw t n sin 1 -z 2 ) w t]+ Be -zw t n cos 1 -z 2 ) w t] (12.4)
c n n
where A and B are determined in conjunction with the particular solution and the
initial conditions. One form of the particular solution is derivable from the impulse
response
1 2 12
z = ye ˙˙ -zw n ( t- ) t sin ( [ 1 -z 2 ) w n ( t - )] dt. (12.5)
t
p
12
(1 -z 2 ) w Ú 0
n
.
For finite y, the case here, z p (0) = z p (0) = 0, so that the general solution to Eq. (12.3) is
¨
È Ï ˙ 0 z()zw ˘ 12 12 ¸
z()+ 0
z() [
sin
z = z + Ì Í 12 n ˙ [ (1 -z 2 ) w n t]+ 0 cos (1 -z 2 ) w n t e ]˝ -zw t n (12.6)
p
Î Ó (1 -z 2 ) w n ˚ ˛
.
where the initial conditions z(0) and z(0) are subsumed in z c. Accordingly, for a system
initially at rest, the z p solution becomes the complete solution. However, the difficulty
with this solution is that y is periodic and integration would have to be performed
¨
over many cycles depending on the amount of damping present. Solutions would be
analyzed first to determine the input “transient” peak response, which is important in
cam-follower system performance. Incidentally, the “steady-state” response would be
reached when
znt = ( n + ) 1 T and znt) = ( ˙ zn + ) 1 T.
( ˙
()
Typical transient-response curves for one-degree-of-freedom systems are shown in
Fig. 12.2, where b is the cam angle for maximum displacement, radians. Hrones (1948)
mathematically analyzed and Mitchell (1940) tested the vibrations resulting from three
basic dwell-rise-dwell curves with a high-rigidity system (follower). They investigated the
transient dynamic magnification, providing the measure of the spring-dashpot coupling
effect on the follower. In Fig. 12.2, the dynamic magnification is equal to 2 at any dis-
continuity of the two curves, parabolic and simple harmonic. However, the cycloidal curve
has a smaller magnification, about 1.06 to 1. Thus a discontinuity means a sudden or tran-
sient application of the inertia load that produces a shock (twice the value of the inertia
load) in the cam-follower system. In design, this phenomenon is often called a “suddenly
applied load.”