Rapid analysis of the effects of structural modifications on inconveniently situated eigenvalues

Abstract

The eigenvalues of a structure are its natural frequencies (vibration problems) or its critical load factors (buckling problems). It is often desirable that no eigenvalues should lie in a prescribed ‘prohibited range’, i.e. within a specified range of values of F, where F is the frequency (vibration problems) or load factor (buckling problems). Three alternative procedures, which are all based on a recent algorithm [1, 2], are presented. They enable the presence of eigenvalues within the prohibited range to be detected, and permit rapid determination of the effectiveness of structural modifications aimed at making this range free of eigenvalues. These procedures may involve considerably less computation than finding a single eigenvalue of the unmodified structure. The procedures are exact within their context, which is the stiffness matrix approach. They allow the use of sub-structures. Furthermore, they apply when the mathematical model used to represent the structure possesses a finite number of degrees-of-freedom (e.g. finite element representations) and when the model has an infinite number of degrees-of-freedom (e.g. models assembled from beams treated ‘exactly’ [7–9]).