Page 244 - Dynamics of Mechanical Systems

0593_C07_fm Page 225 Monday, May 6, 2002 2:42 PM

Inertia, Second Moment Vectors, Moments and Products of Inertia, Inertia Dyadics 225

directed unit vector, the moment of inertia relative to the n direction may be expressed
a
as (see Eq. (7.5.5)):

I = a a I (7.10.5)
aa i j ij
where the a are the n components of n and where we are again employing the summation
a
i
i
convention. Because n is a unit vector, the a must satisfy the relation:
a
i
aa = 1 or 1 − aa = 0 (7.10.6)
ii ii
This latter expression is a constraint equation analogous to Eq. (7.10.2). Thus, from Eq.
(7.10.3), if we wish to find the direction n such that I is a maximum or minimum, then
a
aa
we can form the function h(a ,λ) as:
i
i (
D
ha ,λ) = a a I + ( 1 a a ) (7.10.7)
λ −
ij ij
i i
By setting the derivation of h with respect to the a equal to zero, as in Eq. (7.10.4), we have:
k
∂∂ = ∂ ( a ∂ )a I + a ∂a ∂a I
ha
a
k i k jij ( i j k ) ij
∂
− 2λ aa ∂a
i i k
= δ aI + a δ I − 2λ δ
a
ik iij i jk ij i ik
= Ia − 2λ a = 0
2
kj j k
or
Ia = λ a ( k = 12 3 ) (7.10.8)
,,
kj j k
where we have used the symmetry of the inertia dyadic components. By setting the
derivative of h with respect to λ equal to zero, we obtain Eq. (7.10.6):

aa = I (7.10.9)
kk

Equations (7.10.8) and (7.10.9) are identical to Eqs. (7.7.9) and (7.7.16), which determine
the principal moments of inertia and their directions. Therefore, the directions producing
the maximum and minimum central moments of inertia are the principal directions, and
the maximum and minimum moments of inertia are to be found among the principal
moments of inertia. Finally, the Lagrange multiplier λ is seen to be a principal moment
of inertia.
We can use the same procedure to obtain the direction for the maximum products of
inertia. From Eq. (7.5.5), the product of inertia relative to the directions of unit vectors n a
and n is:
b
I = a b I (7.10.10)
ab ij ij