Page 201 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

188 Properties of Vibrating Systems Chap. 6

where A is the diagonal matrix of the eigenvalues.
A, 0 0
A 0 A2 0 (6.7-7)
0 0 A3
Example 6.7-1
Verify the results of the system considered in Example 5.1-1 (see Fig. 6.7-1) by
substituting them into the equations of Sec. 6.7.

I
|--W A r- m —W W — ¿Im^VWV-p
Figure 6.7-1.
Solution: The mass and stiffness matrices are
1 0 2 - 1
M = m 0 2 K = k -1 2
eigenvectors for Example 5.1-1 are
(o'm 10 '

A, = —r - = 0.634 ^1 =
* k 1.000 /

(oim 1- 2.73
A2 = —^— = 2.366 4>2 = 1.00
Forming the modal matrix F, we have
’0.731 -2.73
P = 1.00 1.00

0.731 1.0 o ’ 0.731 -2.73
P^MP =
-2.73 1.0 2 1
2.53 0 ^11 0
0 9.45 0 M22
Thus, the generalized mass are 2.53 and 9.45.
If instead of P we use the orthonormal modes, we obtain
1 f 0.731' \ / - 2 . 7 3 \ ' 0.459 - 0 . 8 8 8
P = 1
i / l 5 3 ' \1.00 , / \ 1 .0 0 / 0.628 0.3253
0.459 0.628 1 0 0.459 - 0.888' ■1.00 0

- 0.888 0.325 0 2 0.628 0.325 0 1.00
0.459 0.628 2 - 1 0.459 - 0.888’
P^KP
- 0.888 0.325 - 1 2 _ 0.628 0.325 _
0
0.635 0
0 2.365 0 A.
Thus, the diagonal terms agree with the eigenvalues of Example 5.1-1.