Page 232 - Dynamics of Mechanical Systems

0593_C07_fm Page 213 Monday, May 6, 2002 2:42 PM

Inertia, Second Moment Vectors, Moments and Products of Inertia, Inertia Dyadics 213

Hence, the principal unit vector n corresponding to the principal value λ = 3 is:
a

n = 22 n + 2 4 n − 6 4 n (7.8.10)
a 1 2 3
Similarly, by taking the second principal value λ = 5, substituting into Eq. (7.8.2), and
using Eq. (7.8.8), we obtain the principal unit vector:

n = 32 n + 1 2 n (7.8.11)
b 2 3
Finally, for the third principal value λ = 6, the principal unit vector is:

n = 22 n − 2 4 n + 6 4 n (7.8.12)
c 1 2 3

Observe that the principal unit vectors n , n , and n are mutually perpendicular. This
b
c
a
is important because then the products of inertia relative to the directions of n , n , and
a
b
n are zero. To explore this further, recall from Eq. (7.7.6) that because n , n , and n are
b
c
a
c
principal unit vectors, we have:
⋅
⋅
In = λ n , I n = λ n , and In = λ n (7.8.13)
⋅
a 1 a b 2 b c 3 c
Hence, we have:
⋅
⋅
nI n = n I n = nI n = I = I = I = 0 (7.8.14)
⋅⋅
⋅⋅
b a c b a c ba cb ac
Equation (7.8.14) shows that the components of the dyadic I in the “mixed” directions
are zero; therefore, if we express I in terms of n , n , and n we have:
c
b
a
I = λ n n + λ n n + λ n n
1 a a 2 b b 3 c c
(7.8.15)
= 3 nn + 5 nn + 6 n n
a a b b c c
In view of Eqs. (7.8.1) and (7.8.15), we see that the matrices of I referred to n , n , and
2
1
n and to n , n , and n are vastly different. That is,
c
b
3
a
I  I I   92 −3 4 3 3  4
 11 12 13   
I = I I I = −3 4 39 8 3 8 (7.8.16)
ij  21 22 23   
I  I   
 31 I 32 33   33 4 3 8 37 8 
and
I  I I  3 0  0
 aa ab ac   
I αβ =  I ba I bb I bc = 0 5 0  (7.8.17)

 I  ca I cb I   0 0  6 
cc