Page 353 - Cam Design Handbook

THB11 9/19/03 7:33 PM Page 341

CAM SYSTEM MODELING 341

1
PE = Kx 2
2
while the potential energy of the equivalent system is
1
PE = K x .
2
2 eq f
Since for small angles, the relationship between x and x f is
x x f
= .
l l
1 2
Using this expression for x above gives
2
l Ê ˆ
1
K = KÁ ˜ .
eq l Ë ¯
2
When l 1 > l 2, the spring appears stiffer than its physical stiffness, while for l 1 < l 2, the
equivalent stiffness is less than the physical stiffness of the spring because of the mechan-
ical advantage.
11.4.5 Massive Springs

Up to this point, the springs considered have been ideal springs and have had no mass
associated with them. In practice, it is often acceptable to model real springs as ideal
springs if the mass of other elements in the system dominates the mass of the springs. This
is not always the case, however, and real springs may need to be modeled as a combina-
tion of an ideal spring and a lumped mass. The equivalent mass of a massive spring
(Fig. 11.17) can be derived by applying the concepts used in the previous section.
Since the spring is a continuum, the kinetic energy is given by

1
()
2
KE = Ú v x dm. (11.55)
2
spring
If the mass is evenly distributed along the length of the spring,
V

V
l

X
L
FIGURE 11.17. Spring with mass.