Page 251 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

238 Computational Methods Chap. 8

by the computer is
x-’ - 5A^ + 4.5A - 1 = 0
which results in the reciprocal relation compared to

A^ - 4.5A^ + 5A - 1 = 0
-
(see Prob. 8-1). The solution to the equation in terms of A = (o^m/k gives

A, = 0.25536 w, = 0.50533^A;/m
A2 = 1.3554 «2 = 1.16422VVm
A3 = 2.8892 ÎO3 = \.69976^Jk/m
Note that the sum of the eigenvalues is 4.50.

8.3 MATRIX ITERATION

With knowledge of orthogonality and the expansion theorem, we are in a position
to discuss the somewhat different approach for finding the eigenvalues and
eigenvectors of a multi-DOF system by the matrix iteration procedure. Although
the method is applicable to the equations of motion formulated by either the
flexibility or the stiffness matrices, we use the flexibility matrix for demonstration.
The dynamic matrix A for this method need not be symmetric.
In terms of the flexibility matrix [a] = K \ the equation for the normal
mode vibration is
A X = X X (8.3-1)
where
A = [a][m] = K-^M

\ = 1/ (O^
The iteration is started by assuming a set of amplitudes for the left column of
Eq. (8.3-1) and performing the indicated operation, which results in a column of
numbers. This is then normalized by making one of the amplitudes equal to unity
and dividing each term of the column by the particular amplitude that was
normalized. The procedure is then repeated with the normalized column until the
amplitudes stabilize to a definite pattern. When the normalized column no longer
differs from that of the previous iteration, it has converged to the eigenvector
corresponding to the largest eigenvalue, which in this case is that of the smallest
natural frequency
Example 8.3-1
For the system shown in Fig. 8.3-1, write the matrix equation based on flexibility and
determine the lowest natural frequency by iteration.