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0593_C07_fm Page 203 Monday, May 6, 2002 2:42 PM

Inertia, Second Moment Vectors, Moments and Products of Inertia, Inertia Dyadics 203

7.4 Inertia Dyadics
Comparing Eqs. (7.3.6), (7.3.7), and (7.3.8) with (7.3.9) we see that the second moment
vectors may be expressed as:

I PO = I n + I n + I n
x xx x xy y xz z
I PO = I n + I n + I n (7.4.1)
y yx x yy y yz z
I PO = I n + I n + I n
z zx x zy y zz z

where the superscripts on the moments and products of inertia have been deleted. We
can simplify these expressions further by using the index notation introduced and devel-
oped in Chapter 2. Specifically, if the subscripts x, y, and z are replaced by the integers 1,
2, and 3, we have:

n +
I PO = I 11 1 I 12 n + I 13 n = I 1j n j
1
3
2
n +
I PO = I 21 1 I 22 n + I 23 n = I 2 j n j (7.4.2)
3
2
2
n +
I PO = I 31 1 I 32 n + I 33 n = I 3j n j
3
3
2
or
,,
I PO = I n ( i = 12 3 ) (7.4.3)
i ij i
where the repeated index denotes a sum over the range of the index.
These expressions can be simplified even further by using the concept of a dyadic. A
dyadic is the result of a product of vectors employing the usual rules of elementary algebra,
except that the pre- and post-positions of the vectors are maintained. That is, if a and b
are vectors expressed as:

a = a n + a n + a n and b = b n + b n + b n (7.4.4)
1 1 2 2 3 3 1 1 2 2 3 3
then the dyadic product of a and b is defined through the operations:

D n )
ab = (a n + a n + a n )(b n + b n + b
1 1 2 2 3 3 1 1 2 2 3 3
= ab nn + ab nn + ab nn
1 111 1 2 1 2 1 3 1 3
(7.4.5)
+ ab nn + ab nn + ab nn
2 1 2 1 2 2 2 2 2 3 2 3
+ ab nn + ab nn + ab nn
3 1 3 1 3 2 2 2 3 3 3 3
where the unit vector products are called dyads. The nine dyads form the basis for a general
dyadic (say, D), expressed as: