Page 344 - Cam Design Handbook

THB11 9/19/03 7:33 PM Page 332

332 CAM DESIGN HANDBOOK

1 1
KE = J w 2 + J ( + J )w 2
2 pinion in 2 gear load out
and the kinetic energy of the lumped parameter system is
1
KE = J w .
2
2 eq in
The equivalent inertia can be obtained by equating the two expressions for the kinetic
energy and incorporating the kinematics of the system. In this case, the kinematic rela-
tionship is simply the gear ratio or
Ê R pinion ˆ
w = Á ˜ w .
out in
Ë R ¯
gear
This gives
2
1 1 1 Ê R pinion ˆ
J w = J w + J ( + J )Á ˜ w 2
2
2
2 eq in 2 pinion in 2 gear load Ë R ¯ in
gear
so
2
Ê R pinion ˆ
J = J pinion + J ( gear + J )Á ˜ .
load
eq
Ë R ¯
gear
This would be the inertia felt by a motor or person turning the input shaft. Conversely, if
the motor were attached to the output shaft, the elements in Fig. 11.8 would have an equiv-
alent inertia of
2
Ê R gear ˆ
J = J + J + Á ˜ J .
eq gear load pinion
Ë R ¯
pinion
Because of the analogy to mechanical advantage, it is not surprising that the equivalent
inertia of a system depends upon the reference point chosen.
11.4 SPRINGS AND POTENTIAL ENERGY
In system modeling, the term “spring” refers not only to machine elements such as coil,
torsional, or leaf springs, but to any component of the system that deflects in response to
load. In other words, every part of the system that is not assumed to be perfectly rigid acts
as a spring for the purposes of modeling. Springs selected from a catalog and compliant
elements share several properties. Most fundamentally, they are described by some rela-
tionship between deflection and the force applied at either end of the spring. This is most
commonly modeled as a linear relationship
F = Kd (11.40)
keeping in mind that the force exerted by the spring always acts to oppose the compres-
sion or extension. The value K is known as the spring stiffness or spring rate and has units
of force/distance with lbf/in and N·m being common. In Eq. 11.40, the deflection d is
measured from the free or unstretched length, L f, of the spring and F is the force exerted
on the spring (Fig. 11.9).