Page 233 - Dynamics of Mechanical Systems

0593_C07_fm Page 214 Monday, May 6, 2002 2:42 PM

214 Dynamics of Mechanical Systems

where the Greek subscripts α and β refer to the indices a, b, and c. We can obtain a direct
relationship between these matrices by using the transformation matrices introduced in
Chapter 2. Specifically, from Eq. (2.11.3) the elements of the transformation matrix between
the n (i = 1, 2, 3) and the n (α = 1, b, c) are:
α
i
⋅
S = nn and n α = S n i (7.8.18)
iα
iα
α
i
Let I be expressed in the forms:
I = I n n = I n n (7.8.19)
ij i j αβ α β
Then, the I and the I are related by the expressions:
αβ
ij
I = S SI and I = S SI (7.8.20)
ij iα jβαβ αβ iα jβ ij

From Eqs. (7.8.10), (7.8.11), (7.8.12), and (7.8.18), we see that for our example the S are:
iα
 22 0 2 2 
S =   24 3 2 − 24   (7.8.21)
iα
 
 − 64 1 2 64 

Then, in matrix form, Eq. (7.8.20) becomes:

 92 − 3 4 3 3 4  22 0 2 2   3 0 0
     
 − 3 4 39 8 3 8  =  24 3 2 − 24 0 5 0 
 

   − 64  6
 33 4 3 8 37 8   64 1 2   0 0 
(7.8.22)
 2 2 24 − 24
 
 0 3 2 1 2 
 − 
 22 2 4 6 4 
and

3  0 0  22 2 4 − 6 4  92 − 3 4 3 3 4
     

 0 5 0 =  0 3 2 1 2   − 3 4 39 8 3 8 
 0  0 6    22 − 2 4 6 4   33 4 3 8 37 8  
 
(7.8.23)
 22 0 2 2 
 
 24 3 2 − 24 
 − 
 64 1 2 64 

Finally, observe that the columns of the transformation matrix [S ] are the components
iα
of the principal unit vectors n , n , and n .
c
a
b