Page 236 - Dynamics of Mechanical Systems

0593_C07_fm Page 217 Monday, May 6, 2002 2:42 PM

Inertia, Second Moment Vectors, Moments and Products of Inertia, Inertia Dyadics 217

Hence, by assuming that I and I are equal, Eq. (7.9.13) produces an expression for I in
aa
bb
the form:
I = I n n + I n n + I n n
aa a a aa b b cc c c
= ( nn + nn ) + I n n (7.9.16)
I
aa a a b b cc c c
Then, if ηη ηη is any unit vector parallel to the plane of n and n (that is, perpendicular to
a
b
n ), we have:
c
I⋅= ( n n + n n ) ⋅+ I n n ⋅ηη
η
η
ηη I
aa a a b b cc c c
(7.9.17)
= I ηη + 0
aa
Therefore, ηη ηη is a principal unit vector (see Eq. (7.7.5)). This means that all unit vectors
parallel to the plane of n and n are principal unit vectors. Thus, if two of the principal
a
b
moments of inertia are equal, an infinite number of principal unit vectors are parallel to
the plane that is normal to the principal unit vector associated with the distinct root of
Eq. (7.7.12).
We can consider the case when all three of the roots of Eq. (7.7.12) are equal by similar
reasoning. In this case, only one principal unit vector is found by the procedure of the
previous sections. Let this vector be n and let n and n be unit vectors perpendicular to
b
a
c
n such that n , n , and n form a mutually perpendicular set. Then, I has the form:
a
c
c
b
I = αα n + ββ n + I n n (7.9.18)
a b c c
where αα αα and ββ ββ are vectors to be determined and I is the triple root of Eq. (7.7.12). Because
I is symmetric, we have:
+
ααn + ββn = n αα n ββ (7.9.19)
a b a b
or, equivalently, αα αα and ββ ββ must be parallel to a plane normal to n . Thus, αα αα and ββ ββ have the
c
forms:

αα = α n + α n and ββ = β n + β n (7.9.20)
a a b b a a b b
Hence, I becomes:

I = α n n + β n n + α n n + β n n + I n n (7.9.21)
a a a a a b b b a b b b c c
The matrix of I relative to n , n , and n is:
b
a
c
 α a β a 0
 
 α b β b 0  (7.9.22)
  0 0 1 