Page 265 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

252 Computational Methods Chap. 8

Solution: The transformation matrix is found from
U'^'U = K
Wi, 0 0 " «11 «12 «13 4 -1 o '
^12 U22 0 0 « 22 « 23 -1 2 -1
^13 «23 « 3 3 0 0 «33 0 -1 1
,
wfl W jW 12 u 11«13 4 -1 o '
^1,^12 («11 + ( u 12«13 + « 2 2 «23) = -1 2 -1
Mi iW,3 (^12^13 + W23l’^22 ) ( «?3 + «ii + «l3)_ 0 -1 1
By equating the eorresponding terms on each side, U is found:
2 -0.50 0
U =■■ 0 1.3228 -0.7559
0 0 0.6547

For the inverse of U, we let U *= [b^-] and solve the equation
i/f/ ‘ = /
2 -0.50 0 6,, 6 j2 6 i3 "1 0 o'
0 1.3228 -0.7559 0 ¿?22 ¿>23 = 0 1 0
0 0 0.6547 0 0 b.. 0 0 1

Again, equating the terms of the two sides, we obtain
0.50 0.1889 0.2182
= \b.^ = 0 0.7559 0.8726
0 0 1.5275
The dynamic matrix A using the decomposed stiffness matrix is

A =
0.50 0 0 ■4 0 O' 0.50 0.1889 0.2182
= 0.1889 0.7559 0 0 2 0 0 0.7559 0.8726
0.2182 0.8726 1.5275 0 0 1 0 0 1.5275
1.00 0.3779 0.4364'
= 0.3779 1.2857 1.4846
0.4363 1.4846 4.0476

The standard form is now
[-A 7 + ^ ']y = 0
where A = k/co^m and X = U~^Y.