Page 229 - Dynamics of Mechanical Systems

0593_C07_fm Page 210 Monday, May 6, 2002 2:42 PM

210 Dynamics of Mechanical Systems

or

Ia n = I a n (7.7.8)
ij j i aa i i
Equation (7.7.8) is equivalent to the three scalar equations:

ij (
Ia = I a or I − λδ ij) a = ( i = 1 ,, ) 3 (7.7.9)
2
0
ij j aa i i
where for simplicity in notation I is replaced by λ. When the index sums of these equations
aa
are written explicitly, they become:
11 (
I − ) + I a + I a = 0
λ
a
12 2
1
13 3
Ia +( I − ) λ a + I a = 0 (7.7.10)
21 1
22
23 3
2
Ia + I a +( I − ) λ a = 0
32 2
31 1
33
3
Equations (7.7.10) form a set of three linear equations for the three scalar components
a of n . If we can determine the a we have in effect determined and found n . However,
i
i
a
a
Eqs. (7.7.10) are homogeneous — that is, each term on the left sides contains one and only
one of the a , and the right sides are zero. This means that there is no solution (except the
i
trivial solution a = 0), unless the determinant of the coefficients of the a is zero. That is,
i
i
a nontrivial solution for the a exists only if:
i
11 (
I − ) λ I 12 I 13
22 (
I I − ) λ I = 0 (7.7.11)
21 23
33 (
I I I − ) λ
31 32
By expanding the determinant, we obtain:
2
3
λ − I λ + I λ − I = 0 (7.7.12)
I II III
where the coefficients I , I , and I are:
I
III
II
I = I + I + I (7.7.13)
I 11 22 33
I = I I − I I + I I − I I + I I − I I (7.7.14)
II 22 33 32 23 11 33 31 13 11 22 12 21
I = I I I − I I I + I II − I I I + I I I − II I (7.7.15)
III 11 22 33 11 32 23 12 31 23 12 21 33 21 32 13 31 13 22
These coefficients are seen to be directly related to the elements I of the inertia matrix.
ij
Indeed, they may be identified as:
I = sum of diagonal elements of I ij
I
I = sum of diagonal elements of the cofactor matrix of I ij
II
I = determinant of I ij
III