Page 237 - Dynamics of Mechanical Systems

0593_C07_fm Page 218 Monday, May 6, 2002 2:42 PM

218 Dynamics of Mechanical Systems

Because I is symmetric we have:

α = β (7.9.23)
b a

From Eq. (7.9.7), the invariants I , I , and I are:
II
I
III
I = 3 I = α + β + I
I a b
2
I = 3 I = β I + α I + α β − α 2 (7.9.24)
II b a a b b
2
3
I = I = I(αβ − α b)
III a b
where the right sides of these expressions are obtained from the description of I , I , and
I
II
I in Eqs. (7.7.13), (7.7.14), and (7.7.15) and from the matrix of I in Eq. (7.9.22). By solving
III
for α , α , and β , we obtain:
b
b
a
α = I, α = β = , β = I (7.9.25)
0
a b a b
Hence, αα αα and ββ ββ are:
αα = In and ββ = In (7.9.26)
a a b
Finally, I becomes (see Eq. (7.9.21)):

I = I n n + I n n + I n n (7.9.27)
a a b b c c

Then, if ηη ηη is any vector:
η
I⋅= ηη (7.9.28)
η
I
Hence, when there is a triple root of Eq. (7.7.12) all unit vectors are principal unit vectors.
Regarding the question of real roots of Eq. (7.7.12), suppose a root is not real. That is,
let a root λ have the complex form:

λ = α + i β (7.9.29)

where α and β are real and i is the imaginary −1. Let n be a principal unit vector
associated with the complex root λ. Then, from Eq. (7.7.5), we have:

In⋅= λ n (7.9.30)

By following the procedures of the previous sections we can obtain n by knowing λ.
Because λ is complex, the components of n will be complex. Then, n could be expressed
in the form:

n = n + i n β (7.9.31)
α