Page 246 - Dynamics of Mechanical Systems

0593_C07_fm Page 227 Monday, May 6, 2002 2:42 PM

Inertia, Second Moment Vectors, Moments and Products of Inertia, Inertia Dyadics 227

By adding and subtracting these equations we have:

kj(
j)
Ia + b = (2λν a )( k + b ) (7.10.23)
−
k
j
and
kj(
j)
Ia − b =−(2λν a )( k − b ) (7.10.24)
+
j
k
These equations in turn may be expressed as:
⋅(
In + ) = (2λν n )( + n ) (7.10.25)
−
n
a b a b
and

⋅(
+
In − ) =−(2λν n )( − n ) (7.10.26)
n
a b a b
Thus, (n + n ) and (n – n ) are principal vectors of I corresponding to the principal values
b
b
a
a
(2λ – ν) and –(2λ + ν). From our previous discussions, however, we know that for the
general three-dimensional case with three distinct principal values, the principal values
contain the maximum, the minimum, and intermediate values of the moments of inertia.
For our discussion here, let these principal moments of inertia be designated as I , I ,
αα
ββ
and I , with principal unit vectors n , n , and n . That is,
α
β
γγ
γ
I⋅n α = I n , I⋅n β = I n β , I⋅n γ = I n γ ( no sum ) (7.10.27)
ββ
γγ
α
αα
Then, comparing Eqs. (7.10.25) and (7.10.26) with Eq. (7.10.27) we see that (2λ – ν) and
–(2λ + ν) are to be identified with I , I , and I and (n + n ) and (n – n ) are in the
αα
ββ
γγ
b
a
b
a
direction of n , n , and n . Specifically, if (2λ – ν) and –(2λ + ν) are identified with I and
αα
β
γ
α
I , we have:
ββ
2λν−= I and 2λν+= −I ββ (7.10.28)
ββ
Then, by adding these equations we have:
2λ = ( I αα − I ββ) 2 (7.10.29)
However, from Eq. (7.10.16) we see that:
⋅ ⋅
2λ= aI b = nI n = I (7.10.30)
kkj j a b ab
Therefore, the maximum absolute values of the products of inertia are the half differences
of the principal moments of inertia, and the directions where they occur are at 45° to the
principal directions of the moments of inertia.