Page 381 - Cam Design Handbook

THB12 9/19/03 7:34 PM Page 369

CAM SYSTEM DYNAMICS—ANALYSIS 369

• F cos ( wt - )
a
R = kz = Â n n (12.8)
k
m b 12
n=0 È ( w - w )+ w 2 ˘
2
2
Î Í o n m 2 n ˚ ˙
where
a = phase angle
b = clamping factor
R = resultant dynamic force
F(t) = external driving force
F n = amplitude of nth harmonic of harmonic analysis of F(t)
k = system elasticity
m = effective mass of cam-follower system from equilibrium position
z = deflection of cam-follower system
w o = (k) 1/2 = natural frequency of system, rad/sec
w n = frequency of nth harmonic, rad/sec

Examination of the equation shows that resonance occurs at speeds that are multiples of
the natural frequency of the system. With small b, the value under the square-root sign
becomes very small when w o = w n; consequently, the contribution of that member which
is in resonance to the complete solution is by far the largest. With a given cam-follower
system, the magnitude of the response at resonance will depend primarily on the magni-
tude of the component F n. By performing a harmonic analysis of the forces acting on the
system, it is possible to predict the response with a given cam operating at a given speed.
The principal force acting on the system is the acceleration force. Therefore, a harmonic
analysis of the acceleration forces produced by a given cam design will give an indica-
tion of the performance to be expected.
In general, any periodic function, provided that it contains no more than a finite number
of discontinuities, has a convergent Fourier series that represents it. However, the number
and extent of the discontinuities have an important bearing on the rapidity of the conver-
gence, i.e., the rate at which the harmonics approach zero as n becomes large. If a func-
tion itself has discontinuities, the coefficients will decrease in general as 1/n, whereas if
discontinuities do not exist in the function itself but only in its first derivative, the coeffi-
2
cients will fall off approximately as 1/n .
Examination of the three acceleration forms considered, namely constant acceleration,
simple harmonic, and cycloidal, shows that only the cycloidal is completely continuous.
It was stated in Mitchell (1950) that the cycloidal curve cam-follower system used has a
natural frequency of 162.5 cycles per sec. At a cam speed of 140rpm, resonance occurs
with the harmonic 9750/140 = 70th harmonic. Therefore, we would anticipate that the res-
onant response of the system to the cycloidal cam profile would be on the order of 1/70
of the response to the other two profiles at this speed.

EXAMPLE A closed-track radial cam turning once every 4sec operates a roller follower
on a push rod. This rod moves a rocking lever and, by means of a connecting link, a sliding
table weighing 185lb. Some basic dimensions of the roller follower displacement from
dwell to dwell are shown in Table 12.1. The strength of the connecting members are: under
100lb of compressive load, the push rod deflects 0.0012in, the connecting link 0.0165in,
and the rocking lever bends 0.237in. The masses of these members are small in compar-
ison with the table. While the cam is operating, a troublesome chatter develops in the
table. Find the cause. Note: it is recognized that the machine is very old and the cam
surface is worn and has serious surface imperfections.