Page 263 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

250 Computational Methods Chap. 8

both U and U ' have been evaluated and expressed in simple equations for the
computer subroutine in Appendix C.5.
Example 8.9-1
Solve Example 8.8-1 by deeomposing the stiffness matrix. The two matriees for the
problem are
‘ 2 0 3 -1
M = m K = k
0 1 - 1 1
Solution: For the 2 x 2 matrix, the algebraic work for the decomposition is small and we
carry out all the steps.
Step 1:
U^U = K
2
,
n,! 0 ' w,, W2 ^ll ^11^12 3 -1

^12 W22 0 H22 ^11^12 i^^\2 ^22} - 1 1
W],
W] 1^12 ^ -1 - 1/1.732 = -0.5774
,
uli = I - w?2 ^22 = \/l - (-0.5774)^ 0.8164
1.732 -0.5774
U =
0 0.8162
Check by substituting back into U^U = K.
Step 2: Find the inverse of U from UU
’ 1.732 -0 .5 7 7 4 ' 'bu '>12' '1.732/7,, (1.732/7,2 - 0.5774/722)' ’ 1 o '
0 0.8162 0 '>22 0 0.8164/722 0 1
, ^ 1 = 0.5774 A = ^ = 1.2249
,

1.732 ^22 0.8164
1
60 = (0.5774 X 1.2249) = 0.4083
0.5774 0.4083
=
0 1.2249

Check by substituting back into UU ^ = /.
Step 3:
0.5774 0 2 0 0.5774 0.4083'
A = U-^MU~^ =
0.4083 1.2249 .0 1. 0 1.2249
0.6668 0.4715
0.4715 1.8338
Note that A is symmetric.
Step 4: The equation of motion is now in standard form, but in y coordinates:
_
■f 0.6668 0.4715 1 0]'

[0.4715 1.8338 - A 0 1J A =