Improved first-order approximation of eigenvalues and eigenvectors

A method based on reduced basis approximation concepts is presented for improved first-order approximation of eigenvalues and eigenvectors of modified structural dynamic systems. The terms of a local approximation based on Taylor or matrix power series are used as basis vectors for approximating the perturbed eigenparameters.For each eigenmode, a reduced eigensystem is generated by using the baseline eigenvector and the first-order approximation term as Ritz vectors. The solution of the reduced eigensystem leads to two possible estimates of each perturbed eigenvalue and eigenvector. Criteria for selection of the best approximation are presented. The zero- and first-order Rayleigh quotient approximation can be directly recovered as special cases of the present method. Results are presented for approximate dynamic reanalysis of a 25-bar planar truss structure. It is shown that high-quality approximation of the perturbed eigenparameters can be obtained for moderate perturbations in the stiffness and mass matrices. For very large local perturbations of the structure, including deletion of some structural members, it is shown that the present method yields reasonable-quality approximations