Page 235 - Dynamics of Mechanical Systems

0593_C07_fm Page 216 Monday, May 6, 2002 2:42 PM

216 Dynamics of Mechanical Systems

By expanding Eq. (7.9.6) and by comparing the coefficients with those of Eq. (7.9.5), we
discover that the roots are related to the coefficients (the invariants) by the expressions:

I = I + I + I
I aa bb cc
I = I I + I I + I I (7.9.7)
II aa bb bb cc cc aa
I = I I I
III aa bb cc
Now, if I is distinct from I (= I ), then by the reasoning of the foregoing paragraph the
aa
bb
cc
two corresponding principal unit vectors n and n will be distinct and perpendicular.
a
c
That is,
⋅
⋅
⋅
In = I n , I n = I n , and n n = 0 (7.9.8)
a aa a c cc c a c
Let n be a unit vector perpendicular to n and n . Then, I • n will be some vector, say
a
b
c
b
β. However, because n , n , and n are mutually perpendicular, I can be expressed in the
b
c
a
form:
I = ( I n n ) +( I n n ) +( I n n )
⋅
⋅
⋅
a a b b c c (7.9.9)
Then, in view of Eq. (7.9.8) and if I · n is ββ ββ, we have:
b
I = I n n + ββ n + I n n (7.9.10)
aa a a b cc c c
Because I is also symmetric, we have:
ββn = n ββ (7.9.11)
b b
Therefore, ββ ββ must be parallel to n . That is,
b
=
ββββn (7.9.12)
b
where β is the magnitude of ββ ββ. Hence, I takes the form:

I = I n n + β n n + I n n (7.9.13)
aa a a b b cc c c
Then, because I is the sum of the diagonal elements of the matrix of I (independently of
I
the unit vectors in which I is expressed) and in view of the first of Eq. (7.9.7), we have:

I = I + + I = I + I + I (7.9.14)
β
I aa cc aa bb cc
or

β= I (7.9.15)
bb