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THB12 9/19/03 7:34 PM Page 389

CAM SYSTEM DYNAMICS—ANALYSIS 389

tX
t =
T 1
s sT ˙
G = G ˙ = 1
h h
x xT ˙ xT ˙˙ 2
˙˙
X = X ˙ = 1 X = 1 ,
h h h
2 p m
T =
T 1 k c
cT
x =
4 p
k Ê h ˆ 2
F = c Á ˜ b = wT ,
1
k t Ë R m b ¯
where h is the cam lift in a dwell-rise-dwell motion input and T 1 is the period of such
input. Substituting these into Eq. (12.26) yields
2 ˙
( 1+ G FX 2 1+ G Fx 2 ˙ ) Ê 2p ˆ ˙ + Ê 2p ˆ 2 X = ( 2 ˙ ) Ê 2p ˆ ˙ G + Ê 2p ˆ 2 . G (12.27)
˙˙
) + (
21+ G Fx
X
Ë T ¯ Ë T ¯ Ë T ¯ Ë T ¯
This is a second-order linear differential equation with variable coefficients.
Figure 12.23 shows the acceleration response of a cycloidal cam with T = 0.15 when
damping is not taken into account. Note that when F = 0 (corresponding to an infinitely
rigid shaft), the curve resembles that of nominal acceleration. With F increasing, the ampli-
tudes of the residual vibration increase markedly. The reason for this phenomenon is
that at big values of F, i.e., with a relatively flexible shaft, a large amount of energy is
stored in windup and bending during the first half of the cam lift. During the second half,
with the decreasing slope of the cam, the shaft relaxes and acts as a catapult. At the
increased angular velocity of the cam, the deceleration of the follower may appreciably
exceed the nominal value, so that heavy vibrations will persist after the end of the cam
rise. Figure 12.24 shows a plot of the residual amplitude of acceleration versus T, with F
as a varying parameter for an undamped cycloidal cam. Damping would change the effect
very little.

12.4.5 Multi-Degree-of-Freedom System

A complex dynamic model with eleven degrees of freedom was studied by Kim and New-
combe (1981). For more on multi-degree-of-freedom systems, see Pisano and Freuden-
stein (1983). Kim and Newcombe (1981) also presented a study to take into account
the random nature of flexibility and fabrication errors. Figure 12.25 shows the schematic
model consisting of driving torsional and driven translational subsystems. Figure 12.26
shows the three-degree-of-freedom torsional subsystem model. Figure 12.27 displays
the eight-degree-of-freedom translational subsystem model. The latter is divided into two
subsystems according to the vertical motion (y axis) and the horizontal motion (x axis) of
the parts.
Using Newton’s second law, we can write the following sets of equations of
motion.
For translational movement of the follower (see Fig. 12.27) along the y axis
direction