Page 339 - Cam Design Handbook

THB11 9/19/03 7:33 PM Page 327

CAM SYSTEM MODELING 327

1
KE = Ú (v cg + w ¥ r cg ) ◊(v cg + ◊w ¥ r cg ) dm
2
rigid body
1
◊
= Ú ( [ v cg ◊ v cg )+◊ ( w ¥ r cg ) ◊(w ¥ r cg )+ ( 2 v cg (w ¥ r cg ))] dm
2
rigid body
1 1
= mv + J w 2 + Ú v cg (w ¥ r cg ) dm
◊
2
cg
cg
2 2
rigid body
1 1
= mv + J w 2 + 0. (11.27)
2
2 cg 2 cg
The first term is simply the translational kinetic energy of the body, the second term is the
rotational kinetic energy of the body and the third term goes to zero because of the property
of the center of mass given in Eq. 11.26. As a result, the kinetic energy of a body can be
described in terms of the mass of the body, the velocity at the center of mass, the moment of
inertia about the center of mass and the angular velocity. In modeling, this result provides
justification for describing a rigid body or a collection of rigidly connected rigid bodies by
a single point mass at the center of mass and a moment of inertia about the center of mass.
In terms of Newtonian mechanics, mass and inertia play integral roles in the forma-
tion of equations of motion. For translation, mass gives the relationship
F = m a = m˙˙ r (11.28)
between the force vector F applied to the rigid body and the acceleration vector a located
at the body’s center of gravity. The acceleration can also be expressed in terms of the
second derivative of the vector r, which represents the vector from any fixed inertial frame
of reference to the center of mass of the body. For rotation in the plane, the equivalent
expression is derived from the rate of change of angular momentum, giving
T = J w (11.29)
cg
cg
where T cg is the net torque about the center of mass and w is the angular velocity. Since
motion is restricted to the plane, rotation can occur only about one axis, so this is a scalar
equation in contrast to Eq. 11.28. When motion of the body does not lie in a single plane,
the situation becomes a bit more complicated, since the proper representation of inertia is
then a tensor, not a scalar.

11.3.1 Finding Mass and Moment of Inertia
Finding the mass of a component is straightforward. The element may either be weighed
directly on a scale or the mass can be determined from knowledge of the element’s density
and geometry. In this latter case, the mass is given simply by integrating the density over
the volume of the body:
m = Ú r dV. (11.30)
rigid body
Similarly, Eq. 11.21 can be directly used to obtain the moment of inertia of a body about
a given point. Moments of inertia of simple geometries can often be found in tables, Popov
(1976), Fenster and Gould (1985). As a first approximation, complex objects can often be
broken into one or more simple shapes and the moments of inertia (about the center of mass
of the combination of shapes) summed. Moments of inertia can also be obtained experi-