Page 273 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

260 Computational Methods Chap. 8

[i.e., for the first mode, (0.63245)" + (0.70710)^ + (0.31623)2 = 1.0], whereas the
eigenvectors of the actual problem are M-orthogonal [i.e., (0.158)2 X 4 + (0.500)2 ^ ^
+ (0.632)2 y I ^ I 0j gy dividing each column by the last figure of each eigenvector,
the eigenvector normalized to 1.0 is obtained.
With the eigenvalues equal to A = co ^ m /k , the three natural frequencies are

found from
\f^,

2094— = 0.45761/ —
m \ m

= l.Ol/ —
^ \ m m
7905— = 1.3381-i/ —
m \ m

8.11 COMPUTER PROGRAM NOTES

In the computer program for Jacobi diagonalization, the term tan 20 = 2« ,-/
- Gjj) is first changed in terms of tan 0 by the identity
2 ta n 0 2fl,y 2W
tan 26
1 -- tan2 9 ciu DF
Multiplying out,
DF • 2 tan 6 = 2W - 2W 6
DF
tan2 6 + tan 0 - 1 = 0
KK
DF ,
tan0 = - 4 ^ ^ + 1/ 1 ^ 1
21V - i [ l iv j
-DF ± ^J{DFŸ + 41^^ HW
1
COS 0 =
VTl- tan2 0
sin 0 = cos 0 • tan 0
The computer programs on the disk are written in Fortran language. They
are more sophisticated than the basic discussion presented in the text. For
example, in the Jacobi diagonalization, the program searches for the largest
off-diagonal term in each iteration. For all the eigenvalue-eigenvector calculations,
the standard form is first developed. One can either decompose the mass matrix, in
which case, the eigenvalues are A a a)2, or if the stiffness matrix is decomposed,
the eigenvalues are  a 1 /co^.