Page 345 - Cam Design Handbook

THB11 9/19/03 7:33 PM Page 333

CAM SYSTEM MODELING 333

(a) Free.

L f
d

(b) Deflected.
F

FIGURE 11.9. Spring deflection.

There are several reasons why a linear spring model is a useful approximation. First,
systems described by combinations of masses, linear springs, and linear dampers possess
linear equations of motion. These are more readily analyzed than general nonlinear forms.
The linear description is also compact, requiring only a single parameter (the rate of stiff-
ness) to describe the spring. From a practical standpoint, linearity proves to be a reason-
able assumption for most physical springs, at least within a small range of deflection. If
more accuracy is desired, however, springs can instead be represented as a general non-
linear function such as:
F = () d . (11.41)
f
s
Such expressions can be useful in describing the bottoming out behavior of coil springs,
for instance. The only constraint is that the function must be a nondecreasing function of
displacement in order to capture the desired physical characteristics of a spring.
A second property that mechanical springs and compliant elements share is the storage
of energy. It requires work to displace a spring from its free length and, if the spring is
assumed ideal, all of that work is stored in the spring. As a result, the amount stored (the
spring’s potential energy) can be found by integrating the work done on the spring. For
the general nonlinear spring form given above,

d
s Ú
PE d () = f h () dh. (11.42)
0
For the linear spring in Eq. 11.41, displaced a distance x from its unstretched length, this
translates to the familiar expression
d 1
2
PE = Ú K d = Kd . (11.43)
hh
0 2
As with the force, the only spring properties needed to determine the amount of energy
stored are the spring rate and the unstretched length. Spring rates can be determined
experimentally by applying a known force and measuring deflection (or deflecting a