Page 231 - Dynamics of Mechanical Systems

0593_C07_fm Page 212 Monday, May 6, 2002 2:42 PM

212 Dynamics of Mechanical Systems

 ( 92 − ) λ − ( 3 4) ( 3 3 4) 
 
 − ( 3 4) ( 39 8 − ) λ ( 3 8)  = 0 (7.8.3)
 33 4) ( 3 8) ( 
λ
 (  37 8 − ) 

Expanding the determinant, we obtain:

−
λ − 14 λ + 63 λ 90 = 0 (7.8.4)
2
3
The coefficients of Eq. (7.8.4) are seen to be the parameters I , I , and I of Eqs. (7.7.13),
II
I
III
(7.7.14), and (7.7.15).
By solving Eq. (7.8.4) we obtain:
λ = λ = 3, λ = λ = 5, λ = λ = 6 (7.8.5)
1 2 3
These are the principal moments of inertia. By substituting the first of these into Eq. (7.8.2),
we obtain:

( 92 3) +− ( 3 4) ( 3 3 4) a 3 = 0
−
+
a
a
1
2
− ( 3 4) +( 39 8 3) ( 3 8) a = 0 (7.8.6)
−
+
a
a
1 2 3
( 33 4) +( 3 8) a 2 ( 37 8 3)a 3 = 0
−
+
a
1
By eliminating fractions we have:
2a − a + 3a = 0
1 2 3
− 6a + 15a + 3a = 0 (7.8.7)
1 2 3
18a + 3a + 13 3a = 0
1 2 3
These equations are seen to be dependent (multiplying the first by 12 and adding the
result to the second produces the third); hence, only two of the equations are independent.
A third independent equation may be obtained from Eq. (7.7.16) by requiring that n be
a
a unit vector:
a + a + a = 1 (7.8.8)
2
2
2
1 2 3
Thus, by selecting any two of Eq. (7.8.7) and appending them to Eq. (7.8.8) we can
determine a , a , and a . The result is:
1
3
2
a = 22, a = 2 4, a = − 6 4 (7.8.9)
1 2 3