Page 252 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 8.3 Matrix Iteration 239

Figure 8.3-1.
Solution: The mass and the flexibility matrices for the system are
■4 0 o' 1 1 r
m] = m 0 2 0 W - 5 f 1 4 4
0 0 1 1 4 7
and substituting into Eq. (8.3-1), we have
1 1 r ■4 0 0-
1 4 4 0 2 0
1 4 7 0 0 1
or

To start the iteration, we arbitrarily assume
/ 0.2
0.6
^1 = " 4 =
W \l.O
0.2 ) / 3.0 ^0.238 \

AX. = 0.6 = 9.6 12.6( 0.762
1.0 / 1 12.6 1.000 j
By using the new normalized column for X2, the second iteration yields
0.238] 3.476] (0.247]
0.762 11.048 = 14.048 0.786
1.000 j 14.048) \ 1.000)

In a similar manner, the third iteration gives
0.247 3.560] (0.249]
AX, = 0.786 ) = 11.276 = 14.276 0.790
1.000 14.276) ( 1.000)
By repeating this procedure a few more times, the iteration procedure converges to
'0.250^ M i 3k \ '0.250^
14.324( 0.790 } = A{ ^2 = — 0.790

1.000J h 1.000