Page 264 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 8.9 Cholesky Decomposition 251

Step 5: For this simple problem, the eigenvalues and eigenvectors in y coordi­
nates are found from the usual procedure:
(0.6668 - A) 0.4715
0
0.4715 (1.8338 -A )
A^ - 2.50A + 1.0 = 0
yi ( 1) 0.3537 \
A, = 2.0 1.000 /
>^2
( 2)
- 2.8267 \
A, = 0.50 1.000 /
i:)
Step 6: Eigenvalues are not changed by the transformation of coordinates.
The eigenvectors in the original x coordinates are found from the transformation
equation.
= U'Y

"0.5774 0.4083" "0.3537 - 2.8267"
<f>(x) 0 1.2249 1.000 1.000
"0.6125 -1.22381 r0.50 -1.00
— 1.2249 1.2249 = 1.00 1.00

( 1) 0.50
1.00
(2)
- 1.00
4>2(x) 1.00

Example 8.9-2
Figure 8.9-1 shows a 3-DOF model of a building for which the equation of motion is

4 0 O’ 4 -1 0 ]' '0
0 2 0 + -1 2 -1 = { 0

- ( t )
0 0 1 . 0 1J l i l M o ,
Reduce the equation to the standard form by decomposing the stiffness matrix.
m

Zm

4m
3A
Figure 8.9-1.