Page 240 - Dynamics of Mechanical Systems

0593_C07_fm Page 221 Monday, May 6, 2002 2:42 PM

Inertia, Second Moment Vectors, Moments and Products of Inertia, Inertia Dyadics 221

From Eq. (7.7.5), if n is a principal unit vector, we have:
a
⋅
In = I n no sum on a (7.9.38)
a aa a
The associated scalar equations, Eq. (7.7.10), then become:
11 (
I − ) + I a = 0
a
I
1
12 2
aa
Ia +( I − ) a =
21 1 22 I aa 2 0 (7.9.39)
33 (
I − ) a = 0
I
3
aa
where, as before, a , a , and a are the components of n relative to n , n , and n . By
3
1
a
2
1
2
3
inspection, a solution to these equations is:
I = I , a = 1, a = a = 0 (7.9.40)
aa
3
2
1
33
If, however, I is not equal to I , then a is zero and the equations for a , a , and I reduce to:
3
2
aa
1
aa
33
11 (
I − ) + I a = 0
a
I
aa
12 2
1
Ia +( I − ) a =
21 1 22 I aa 2 0 (7.9.41)
a + a = 1
2
2
1 2
Because the first two of these equations are linear and homogeneous, the third equation
will be violated unless the determinant of the coefficients of the first two equations is zero.
That is,
11 (
I − ) I 12
I
aa
22 (
I I − ) = 0 (7.9.42)
I
21 aa
Expanding the determinant we obtain:
I − ( I + ) + I I − I =
2
2
aa I aa 11 I 22 11 22 12 0 (7.9.43)
Solving for I we find:
aa
I + I  I − I  2  12
I = 11 22 ±    11 22  + I  (7.9.44)
2
aa   12
2   2  
When I has the values as in Eq. (7.9.44), the first two equations of Eq. (7.9.41) become
aa
dependent. Hence, by taking the first and third of Eq. (7.9.41) we have a and a to be:
2
1
aa ]
a = I 12 [ I +( I − ) 2 12 (7.9.45)
2
I
1 12 11