Page 247 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

8

Computational Methods

In the previous chapters, we have discussed the basic procedure for finding the
eigenvalues and eigenvectors of a system. In this basic method, the eigenvalues of
the system are found from the roots of the polynomial equation obtained from the
characteristic determinant. Each of the roots (or eigenvalues) was then substituted,
one at a time, into the equations of motion to determine the mode shape (or
eigenvectors) of the system.
Although this method is applicable to any N-DOF system, for systems with
DOF greater than 2, the characteristic equation results in an algebraic equation of
degree 3 or higher and the digital computer is essential for the numerical work.
As an alternative to this procedure, there is an implicit method of transfor­
mation of coordinates coupled with an iteration procedure that results in all the
eigenvalues and eigenvectors simultaneously. In this method, the equation of
motion

[ - AM + K ]X = 0 (a)

must first be converted to the standard eigenvalue form utilized in most of the
computer programs. This standard form is

[A - A/]y = 0 (b)

where ^4 is a square symmetric matrix, and Y is a new displacement vector
transformed from X. Because these methods all involve the iteration procedure,
we precede the transformation method with the computer application to the basic
method and the method of matrix iteration.

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