TABLE OF CONTENTS Some commonly occurring eigenvalue problems encountered in practice are as follows:
Free vibration of nonspinning structures.
Free vibration of spinning structures.
Free vibration formulation based on dynamic element formulation.
Stability of structural systems.
Their solution methods are described here in detail. At least three different procedures are relevant to their solution and these are classified as 1) single vector iteration, 2) progressive simultaneous iteration, and 3) the Lanczos method. Each technique has relative advantages over the others. The matrix formulation pertaining to the various eigenvalue problems may be conveniently derived from their respective dynamic equations. Each problem and its solution is discussed in the following sections.
The matrix equation of motion for a typical problem is given by a set of linear second-order differential equations
whose solution has the form
in which ? is the amplitude of the displacements q at time zero, ? is the frequency of oscillations, and t is the time variable, as usual. Substituting Eq. (9.2) in Eq. (9.1), and canceling the common factor e i ? t, yields the standard structural eigenvalue problem,
or alternatively,
where ? is an eigenvalue and ? the corresponding eigenvector. Solution of a wide range of eigenspectrum involving the number of eigenvalues p can be written in the form as
In Eq. (9.5) ? is an ( n p) matrix containing columns of p eigenvectors and ?
