Symmetry preserving eigenvalue embedding in finite-element model updating of vibrating structures

The eigenvalue embedding problem addressed in this paper is the one of reassigning a few troublesome eigenvalues of a symmetric finite-element model to some suitable chosen ones, in such a way that the updated model remains symmetric and the remaining large number of eigenvalues and eigenvectors of the original model is to remain unchanged. The problem naturally arises in stabilizing a large-scale system or combating dangerous vibrations, which can be responsible for undesired phenomena such as resonance, in large vibrating structures. A new computationally efficient and symmetry preserving method and associated theories are presented in this paper. The model is updated using low-rank symmetric updates and other computational requirements of the method include only simple operations such as matrix multiplications and solutions of low-order algebraic linear systems. These features make the method practical for large-scale applications. The results of numerical experiments on the simulated data obtained from the Boeing company and on some benchmark examples are presented to show the accuracy of the method. Computable error bounds for the updated matrices are also given by means of rigorous mathematical analysis.