Page 268 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 8.10 Jacobi Diagonalization 255

¿73 5 in a 6 X 6 matrix, the rotation matrix is
1 0 0 0 0 0
0 1 0 0 0 0
0 0 cos 6 0 —sin d 0

R (8.10-7)
0 0 0 1 0 0
0 0 sin 6 0 cos d 0
0 0 0 0 0 1
and 6 is determsined from the same equation as before.

tan 20 = ( 8.10-8)
- a. a-. - a:
If = Ujj, 20 = ±90° and 6 = ±45°. Although 26 can also be taken in the left
half space, there is no loss of generality in restricting 0 to the range ± 45°. Due to
the symmetry of matrix A, this step reduces one pair of the off-diagonal terms to
zero, and must be repeated for every pair of the off-diagonal terms of matrix A.
However, in reducing the next pair to zero, it introduces a small nonzero term to
the previously zeroed element. So having zeroed all the off-diagonal elements,
another sweep of the process must be made until the size of all the off-diagonal
terms is reduced to the threshold of the specified value. Having reached this level
of accuracy, the resulting diagonal matrix becomes equal to the eigenvalue matrix
A, and the eigenvectors are given by the columns of the products of the rotation
matrices. In summary, letting subscript / stand for the last iteration.
Ai = R ■R lR ^ A Ri = A

lim R 1R 2 ■R, = P (8.10-9)
Although the proof of convergence of the Jacobi iteration is beyond the
scope of this text, experience has shown that rapid convergence is generally found,
and usually acceptable results are obtained in less than five sweeps, and often in
one or two sweeps when the off-diagonal elements in the original matrix are small
in comparison to the diagonal elements. The number of calculations is also quite
limited in that in spite of the size of the matrix, only two rows and two columns are
involved for each iteration.
Example 8.10-1
When the mass matrix is decomposed in Fig. 8.9-1, the standard form of the equation
of motion becomes
1.0 -0.3536 0 ]■

—\I + -0.3536 1.0 -0.7071
0 -0.7071 1.0 J
where A = co^m/k. By using the Jacobi method, diagonalize the dynamic matrix, and

determine the eigenvalues and the eigenvectors for the system.