Page 354 - Cam Design Handbook

THB11 9/19/03 7:33 PM Page 342

342 CAM DESIGN HANDBOOK

m
dm = s dx. (11.56)
L
Assuming that the spring compresses uniformly, the velocity profile along the spring
can be interpolated between the end points. At x = 0, the spring is fixed, so the velocity
is zero; at x = L, define the velocity as v L. The velocity at any point x between 0 and L is
therefore
x
v = v . (11.57)
L
L
Putting these together, the expression for kinetic energy gives
1 L x 2 m 1 mv 2 L 1 mv L 3 1 m
2
2
2
KE = Ú v 2 L s dx = s L Ú xdx = s L = s v . (11.58)
L
2 L 2 L 2 L 3 2 L 3 3 23
0 0
Equating this expression with the kinetic energy of a single equivalent mass attached at
the end of the spring (x = L), the equivalent mass is simply
m
m = s . (11.59)
eq
3
So a spring with mass can be represented by a combination of an ideal spring with the
same spring constant and an equivalent mass of one-third of the spring mass attached to
the moving end of the spring. Similar techniques can be used to obtain equivalent masses
for other distributed masses for which the rigid body assumption is not a good approxi-
mation. Such systems are better analyzed using other techniques, however (Ungar, 1985),
so this is not detailed here.

11.5 DAMPERS AND DISSIPATION

Unlike mass or inertia elements, which store kinetic energy, and springs, which store
potential energy, damping elements cannot store energy at all. Instead, they remove energy
from the system. Without dampers, systems modeled as combinations of masses and
dampers would always conserve energy. Since real systems can, at best, only approximate
conservative systems, models without dampers tend to underestimate the force or energy
needed to produce a certain system motion. The main reason for including dampers in a
system model is to capture this energy dissipation effect and more accurately describe the
magnitude of input necessary to produce a given output or the amplification at resonance.
As noted before, the addition of damping does not reduce the number of natural frequen-
cies or degrees of freedom of a system.
As with springs, damping elements describe both components intentionally designed
to remove energy from the system (such as dashpots and automotive shock absorbers) and
the inherent energy dissipation of other system elements. All elements contribute in some
way to energy dissipation, either through material damping in springs and structural
members or, more importantly, through friction in bearings and other surface contacts. In
contrast to mass and compliance (or spring) properties, it is generally extremely difficult
to obtain simple analytical expressions for the amount of damping in a system. Damping
in a gearbox, for instance, will depend upon a number of factors including the type and
condition of lubricant, the alignment of the shafts, and the condition and quality of the
gears. Unless data is available for the element from the manufacturer (as is the case with