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324 CAM DESIGN HANDBOOK

removed by model reduction, thus generating additional degrees of freedom and additional
natural frequencies at successively higher frequencies.

11.3 MASS, INERTIA, AND KINETIC ENERGY

The ultimate goal of modeling a dynamic system is to produce equations of motion that
may be used for simulation and analysis. These equations can be produced by relying on
either Newtonian dynamics or analytical dynamics such as Lagrange’s equations of motion
Layton (1998). In the case of the former, the basic approach is to use Newton’s second
law to relate force and acceleration; in the latter, the basic equations of motion are derived
from expressions of the kinetic and potential energy of the system. In either case, the con-
cepts of mass and inertia are fundamental to the derivation of equations of motion and,
therefore, to dynamic systems modeling. From the Newtonian perspective, mass and
inertia provide the link between the forces acting on the system (kinetics) and the accel-
eration (kinematics) experienced by the system (or more rigorously, between the forces
and the change of momentum). From the perspective of analytical dynamics, mass and
inertia are essential in forming the kinetic energy of the system. The following section
develops relationships among mass, inertia, and kinetic energy for a variety of cases
involving rigid body motion.
The kinetic energy of a single particle of mass m (with units of lbm or kg) is given by
1 1
vv
KE = m ◊ = mv 2 (11.15)
2 2
where v is the velocity vector of the particle in a three-dimensional space and v is the mag-
nitude of the velocity (a scalar). Clearly, the kinetic energy depends only upon the mag-
nitude of velocity and not on the direction. The kinetic energy of a system of particles or
a rigid body can be derived by integrating over all of the particles that comprise the body,
giving
1
KE = Ú ◊ vv dm. (11.16)
2
rigid body
This equation is rather cumbersome, but a number of simplified forms exist for the
common cases of motion needed in modeling cam systems. For a rigid body that trans-
lates without rotating, the angular velocity is zero and the linear velocity at every point
on the body is the same since the distance between points on a rigid body cannot change.
In this case, if v represents the velocity vector at any point on the body, v is its magnitude
and m is the total mass of the body, the kinetic energy takes the familiar form
1 1 1
2
◊
KE = Ú ◊ vv dm = vv Ú dm = mv . (11.17)
2 2 2
rigid body rigid body
Analysis of the kinetic energy is simplified greatly if the rigid body is restricted
to move in a single plane of motion. Under this assumption, the motion can be generally
described by the linear velocity at one point and the angular or rotational velocity of
the body. While the linear velocity can, in general, vary from point to point in the body,
the angular velocity is the same anywhere on the body and should always be described
in units of radians/s in both SI and English systems of units. A further simplification
may be made if the body rotates in the plane about some point on the body. A body
rotates about a point if that point on the body remains stationary with respect to an iner-