Improvement of Normalization Methods for Eigenvector Derivatives

A theoretical method is developed for improving certain calculations for eigenvector derivatives in linear systems. The subject algorithms are all forward analysis, i.e., the eigenvector changes are being driven by design parameter perturbations. The new method emphasizes proper mass normalization and is most needed when iteratively computing eigenvector perturbations within a reduced-dimension space. Previous schemes handled the normalization task adequately only for the case of small mode changes because these schemes drop certain terms from the governing normalization equation. When moderate (± 10%) mode changes are evident, as in iterative convergence difficulties or cases of moderate design variable changes, it is necessary to implement a normalization scheme that considers higher-order terms. Two roots to the governing normalization equation exist, only one of which can be used. Criteria for choice of the proper root are developed and a benchmark problem is analyzed employing the new technology. Use of this normalization is required for moderate-change algorithms and can improve existing small-change algorithms.