Page 256 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 8.5 Convergence to Higher Modes 243

The constant then becomes

O', = — ------ (8.5-8)

which substituted into Eq. (8.5-7) gives

X 2 = X^ —a(f>i = A'l —</>!«
/
<>
_
./ a 1 'T ' 1
Thus,

1 -

is another expression for the sweeping matrix, which can be more easily pro­
grammed.
Example 8.5-1
Consider the same system of Example 8.3-1, in which the eigenvalue and eigenvector
for the first mode were found as
( 1) '0.250]
14.32 = i 0.790
1.000 I

To determine the second mode, we form the sweeping matrix given by Eq. (8.5-3):
1 / 1.00 r

0 2 [ 0.25 j 4 [ 0.25 j "0 - 1.58 - r
1 / 0.79 \
0 1 0 = 0 1 0
_0 0 1 _0 0 1 _
The new equation for the second mode iteration is found from Eq. (8.5-6):
[AS]X = X
\
-4 2 r ■0 -1.58 - r
4 8 4 0 1 0
4 8 7 0 0 1

■0 4.32 -3 .0 '
0 1.67 0 ■^2
0 1.67 3.0 1^3.
Knowing that the second mode would have a node, we might start the iteration with