COMPARING THE PERTURBED EIGENSOLUTIONS OF A GENERALIZED AND A STANDARD EIGENVALUE PROBLEM

✍ Scribed by P.D. CHA; W. GU

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 132 KB
• Volume: 227
• Category: Article
• ISSN: 0022-460X
• DOI: 10.1006/jsvi.1999.2375

✦ Synopsis

A frequently encountered scenario in structural dynamics is determining the changes in the eigensolution of a system after certain modi"cations are introduced. Clearly, if these modi"cations are substantial, then a new analysis and computational cycle are necessary in order to compute the new eigendata. However, if the changes made are small, then the perturbation theory can be applied whereby the initial modal characteristics are used as a basis to extract the new eigensolution of the modi"ed system without performing a new and possibly costly analysis. Over the years the perturbation theory has been used in the solution of many di!erent problems, and hence only a few selected references are given [1}7].

In this technical note, the perturbation theory will be used to determine the "rst order eigensolutions of a slightly perturbed symmetric generalized eigenvalue problem and its corresponding standard eigenvalue problem. The "rst order perturbation results obtained in this technical note are well known and certainly not new. The objective of this technical note is not to show how the perturbation theory can be applied to extract the modes of vibration of slightly modi"ed structures. Instead, the goal is to highlight the similarities and di!erences in the "rst order perturbed eigensolutions for the same system obtained by solving a generalized eigenvalue problem and its corresponding standard eigenvalue problem. In particular, it will be shown that a certain coe$cient that is commonly assumed zero (in the standard eigenvalue problem formulation) cannot be neglected. Numerical examples and comparisons will be made to illustrate the importance of this term.