Page 62 - Dynamics of Mechanical Systems

0593_C02_fm Page 43 Monday, May 6, 2002 1:46 PM

Review of Vector Algebra 43

ˆ
n n
3 3
ˆ
α n
2
α

n
2
FIGURE 2.11.2 ˆ
Unit vector sets n i and ˆ n i . n 1 , n 1
where s and c represent sinα and cosα. Hence, from Eq. (2.11.3), the matrix S has the
α
α
elements:

1 0 0 

S = S = 0 c α s − α   (2.11.14)
ij [] 

0 s α c  
α
Observe that SS and S S are:
T
T
1 0 0 1 0 0   1 0  0



T
SS = 0 c α s − α  0 c α s α  = 0 1 0   (2.11.15)



0 s α c 0 s − α c    0 0  1 

α
α
and
1 0 0 1 0 0   1 0  0



T
SS = 0 c α s α  0 c α s − α  = 0 1 0   (2.11.16)


0 s − α c 0 s α c    0 0   1


α
α
Observe in Figure 2.11.2 the rotation of the ˆ n i and ˆ n 1 through the angle α is a dextral
rotation. Imagine analogous dextral rotations of the about ˆ n 2 and ˆ n 3 through the angles
ˆ n
i
β and γ. Then, it is readily seen that the transformation matrices analogous to S of Eq.
(2.11.14) are:
 c 0 s  c s −  0
 β β   γ γ 
B =  0 1 0  and C =  γ c γ 0  (2.11.17)
s
 s −   0  1
 β 0 c β   0 
(In this context, the transformation matrix of Eq. (2.11.15) might be called A.)
Finally, let the ˆ n i have a general inclination relative to the n as shown in Figure 2.11.1.
i
The ˆ n i may be brought into this configuration by initially aligning them with the n and
i