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THB11 9/19/03 7:33 PM Page 329

CAM SYSTEM MODELING 329

11.3.2 Moving Moments of Inertia to Other Points

Sometimes the moment of inertia about the center of mass is known, but the moment of
inertia about another point on the body is required. While the moment of inertia about the
desired point could be calculated directly, it is often easier to simply move the moment of
inertia. If the distance between the center of mass and the point O where the moment of
inertia is desired is given by d, the two moments of inertia are related by the parallel-axis
theorem
2
J = J + md . (11.31)
o
cg
As with the previous results, planar motion is assumed (though an analogous result holds
for spatial motion with proper definition of the distance d and the axes of rotation). This
relationship can be proven easily by substituting the expression
x = x + x y = y + y (11.32)
o cg o cg
into Eq. 11.21 and realizing that
d = x + y . (11.33)
2
2
2
EXAMPLE Continuing with the example of the flat plate in Fig. 11.6, suppose that the
plate is pinned to rotate around point O. From the discussion above, the kinetic energy
can be described in terms of the angular velocity and the moment of inertia about point
O. To derive the moment of inertia about point O from the moment of inertia about the
center of mass, the parallel axis theorem can be used
Ê s ˆ 2 1
2
J = J + m Ë ¯ = J + ms .
cg
o
cg
2 2
As a check, the moment of inertia about O can be derived using the basic formula:
s s
2
2
2
J = Ú r dm = Ú Ú ( x + y ) r tdxdy
o
o
plate y=0 x=0
s
x 3 2r ts 4 2 1
= 2r ts = = ms = J + ms 2
2
3 x=0 3 3 cg 2
which gives the same result, albeit with a fair amount more work.
11.3.3 Equivalent Mass or Inertia
While the mass and inertia of each component in the system can be determined, the mass
of interest in dynamic modeling is generally not the mass of each individual component,
but rather the “equivalent mass” of the components connected as they are in the system.
This is particularly true when model reduction is used to reduce the number of degrees of
freedom and lump multiple masses and inertias together. The equivalent mass is, loosely
speaking, the mass “felt” by an observer at a particular reference point in the system. While
this may seem to be a somewhat abstract concept, it is nothing more than a consequence
of the familiar concept of mechanical advantage.
Figure 11.7 demonstrates this. Suppose that the mass m is rigidly attached to a mass-
less rod a distance l 1 from the pivot and a force F is applied to the rod a distance l 2 from