Page 228 - Dynamics of Mechanical Systems

0593_C07_fm Page 209 Monday, May 6, 2002 2:42 PM

Inertia, Second Moment Vectors, Moments and Products of Inertia, Inertia Dyadics 209

second-moment vector is dependent upon the direction of the unit vector n , used in the
a
definition. That is, for a system of particles we have:

N
SO D ∑ p ×( n × )
I = m p (7.7.1)
a i i a i
i=1
Observe that for an arbitrary unit vector, it is unlikely that I SO will be parallel to n . If it
a
a
happens, however, that I SO is parallel to n , then n is said to be a principal unit vector or
a a a
an eigen unit vector.
Observe that if n is a principal unit vector and if n is perpendicular to n , then the
a
b
a
product of inertia I SO is zero. That is,
ab
I SO = I SO n ⋅ = 0 (7.7.2)
ab a b
Observe further that if n is a principal unit vector, the moment of inertia I SO is the
a
aa
magnitude of I S O . Then, we have:
a
I SO = I SO n ⋅ = I SO and I SO = I SO n (7.7.3)
aa a a a a aa a
The direction of a principal unit vector is called a principal direction. The corresponding
moment of inertia is called a principal moment of inertia or eigenvalue of inertia.
Consider again the inertia dyadic defined in Section 7.4. From Eq. (7.4.11) we have:

I SO = I SO n ⋅ (7.7.4)
a a
Hence, if n is a principal unit vector, we have:
a
I SO n ⋅ = I SO n (7.7.5)
a aa a
In view of these concepts and equations, the obvious questions that arise are do principal
unit vectors exist and, if they exist, how can they be found? Equation (7.7.5) is often used
as point of departure for an analysis to answer these questions and to discuss principal
unit vectors in general.
To begin the analysis, let Eq. (7.7.5) be rewritten without the superscripts:
In = I n (7.7.6)
⋅
a aa a
Next, let I and n be expressed in terms of mutually perpendicular unit vectors n (i = 1,
i
a
2, 3) as:
I = I n n and n = a n (7.7.7)
ij i j a k k
Then, Eq. (7.7.6) becomes:

⋅
⋅
⋅
In = I n n ⋅a n = I a n n n
a ij i j k k ij k i j k
= Ia n δ = Ia n
ij k i jk ij j i
= I n = I a n
aa a aa k k