Page 223 - Dynamics of Mechanical Systems

0593_C07_fm Page 204 Monday, May 6, 2002 2:42 PM

204 Dynamics of Mechanical Systems

D = d n n + d n n + d n n
11 1 1 12 1 2 13 1 3
+ d nn + d nn + d nn (7.4.6)
21 2 1 22 2 2 23 2 3
+ d nn + d nn + d nn = d n n
31 3 1 22 3 2 33 3 3 ij i j

Observe that a dyadic may be thought of as being a vector whose components are vectors;
hence, dyadics are sometimes called vector–vectors. Observe further that the scalar com-
ponents of D (the d of Eq. (7.4.6)) can be considered as the elements of a 3 × 3 matrix and
ij
as the components of a second-order tensor (see References 7.5, 7.6, and 7.7).
In Eq. (7.3.12), we see that the moments and products of inertia may be assembled as
elements of a matrix. In Eq. (7.4.3) these elements are seen to be I (i, j = 1, 2, 3). If these
ij
matrix elements are identified with dyadic components, we obtain the inertia dyadic
defined as:

=
I PO D I n n + I n n + I n n
11 1 1 12 1 2 13 1 3
+ I nn + I nn + I nn (7.4.7)
21 2 1 22 2 2 23 2 3
+ I nn + I nn + I nn
31 3 1 32 3 2 33 3 3

or

I PO = I n n (7.4.8)
ij i j

By comparing Eqs. (7.4.2) and (7.4.7), we see that the inertia dyadic may also be expressed
in the form:

I PO = n I PO + n I PO + n I PO (7.4.9)
1 1 22 33
Equation (7.3.5) shows that the matrix of moments and products of inertia is symmetric
(that is, I = I ). Then, by rearranging the terms of Eq. (7.4.7), we see that I P/O may also be
ij
ji
expressed as:
I PO = I PO n + I PO n + I PO n (7.4.10)
1 1 2 2 3 3
The inertia dyadic may thus be interpreted as a vector whose components are second-
moment vectors.
A principal advantage of using the inertia dyadic is that it can be used to generate
second-moment vectors, moments of inertia, and products of inertia. Specifically, once the
inertia dyadic is known, these other quantities may be obtained simply by dot product
multiplication with unit vectors. That is,

⋅
I PO = I PO n ⋅ = n I PO i = ( 12 ) 3 (7.4.11)
,,
i i i
and

⋅
I PO = n I PO n ⋅ i = ( 12 ) 3 (7.4.12)
,,
ij i j