Page 218 - Dynamics of Mechanical Systems

0593_C07_fm Page 199 Monday, May 6, 2002 2:42 PM

7

Inertia, Second Moment Vectors, Moments

and Products of Inertia, Inertia Dyadics

7.1 Introduction
In this chapter we review various topics and concepts about inertia. Many readers will
be familiar with a majority of these topics; however, some topics, particularly those
concerned with three-dimensional aspects of inertia, may not be as well understood, yet
these topics will be of most use to us in our continuing discussion of mechanical system
dynamics. In a sense, we have already begun our review with our discussion of mass
centers in the previous chapter. At the end of the chapter, however, we discovered that
we need additional information to adequately describe the inertia torque of Eq. (6.9.11),
shown again here:

 N   N 
T =− ∑ m r ×(αα × r ) − ωω ×  ∑ m r ×(ωω × r ) (7.1.1)
*
 ii i   ii i 
 = i 1   = i 1 
Indeed, the principal motivation for our review of inertia is to obtain simplified expres-
sions for this torque. Our review will parallel the development in Reference 7.4 with a
basis found in References 7.1 to 7.3. We begin with a discussion about second-moment
vectors — a topic that will probably be unfamiliar to most readers. As we shall see, though,
second-moment vectors provide a basis for the development of the more familiar topics,
particularly moments and products of inertia.

7.2 Second-Moment Vectors
Consider a particle P with mass m and an arbitrary reference point O. Consider also an
arbitrarily directed unit vector n as in Figure 7.2.1. Let p be a position vector locating P
a
relative to O. The second moment of P relative to O for the direction n is defined as:
a
PO D p ×( p)
I = m n × (7.2.1)
a a

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