Efficient eigenpair sensitivity analysis for symmetric damped structural systems with distinct and repeated eigenvalues

Abstract

The dynamic behavior of damped structural systems is often analyzed through quadratic eigenvalue problems. Performing sensitivity analysis on the associated eigenpairs is particularly complex in the presence of repeated eigenvalues. This paper is concerned with the development of an effective strategy for computing eigenpair sensitivities in symmetric damped systems, specifically addressing cases involving repeated eigenvalues. First, a method is developed to formulate non-singular coefficient matrices within the original space, allowing for the direct calculation of eigenvector derivatives in cases of distinct and repeated eigenvalues. Second, an iterative approach with the preconditioned conjugate orthogonal conjugate gradient (PCOCG) method is employed to solve the governing equations. The proposed method performs all computations in the original space, eliminating the need for matrix dimensionality expansion or edge addition, thereby reducing the “fill-in” operations typically associated with sparse matrices. Furthermore, this approach requires solely the modal information of interest, avoiding the need to compute particular solutions for eigenvector derivatives. The present method is confirmed through three engineering examples, i.e., a mass-spring-damper system, a square planar truss structure, and a water turbine generator.