Page 477 - Dynamics of Mechanical Systems

0593_C13_fm Page 458 Monday, May 6, 2002 3:21 PM

458 Dynamics of Mechanical Systems

˙
ξ =−A ω sinω +t B ω cosω t (13.8.19)
3 3 3 3 3 3 3
Next, suppose that at time t = 0, the values of x , x , and x and their derivatives are x ,
1
10
2
3
x , x , ˙ x 10 , ˙ x 20 , ˙ x 30 , respectively. Then, from Eqs. (13.8.4), (13.8.5), and (13.8.6), the initial
20
30
values of the ξ , ξ , and ξ and of the ξ ˙ 1 , ξ ˙ 2 , and ξ ˙ 3 are determined. This in turn enables
1
3
2
us to use Eqs. (13.8.14) through (13.8.19) to determine the constants A and B (i = 1, 2, 3).
i
i
Finally, if we know the ξ , ξ , and ξ we can readily find the x , x , and x through Eqs.
3
2
2
1
1
3
(13.8.4), (13.8.5), and (13.8.6) in the matrix form:
ξ    12 2 2 12   
x
1
1
     
 ξ 2  =  22 0 − 2 2   x 2  (13.8.20)
 
x
3
3
 ξ     − 12 2 2 − 12   
or as:
ξ= Sx (13.8.21)
where S is regarded as a transformation matrix. An inspection of S shows that it is
orthogonal. That is, the inverse is equal to the transpose. Hence, we have:
x = S ξ = S ξ (13.8.22)
T
−1
or
ξ
x    12 2 2 − 12   
1
1
     
 x 2  =  22 0 2 2   ξ 2  (13.8.23)

ξ
3
 x     12 − 2 2 − 12  
3
 
or
x = ( 12)ξ 1 ( 2 2) ξ −( 12)ξ
+
1 2 3 (13.8.24)
x = ( 22) ξ + 0ξ 2 ( 2 2) ξ
+
2 1 3 (13.8.25)
x = ( 12)ξ 1 ( 2 2) ξ −( 12)ξ
−
3 2 3 (13.8.26)
13.9 Nonlinear Vibrations
The foregoing analyses are all based upon the solutions of linear differential equations.
A convenient property of linear equations is that we can superpose or add solutions.