Comparison of analytical and numerical methods in computing the derivatives of vibration eigen-pairs

Abstract

The derivatives of eigenvalues and eigenvectors of structural vibrations are widely used in system optimization, finite element model updating, structural damage detection, and other fields. Traditional analytical methods, including the modal superposition method, Nelson's method, and their derivatives, have limitations in computational efficiency when dealing with large-scale structures possessing thousands or tens of thousands of degrees of freedom. To address this issue and also to elucidate the advantages of numerical algorithms to graduate students, this paper introduces a rapid numerical algorithm specifically designed for efficiently calculating the derivatives of vibration characteristic pairs in large-scale structures with numerous degrees of freedom. The technique comprises three primary steps: Firstly, the Neumann series expansion is leveraged to swiftly compute the inverse of the perturbed stiffness matrix resulting from minor structural modifications. Secondly, this inverse of the perturbed stiffness matrix is integrated into the subspace iteration method to accurately determine the perturbed eigenvalues and eigenvectors. Lastly, the forward difference algorithm is utilized to precisely calculate the derivatives of the eigenvalues and eigenvectors. Through four numerical examples, the efficiency of this rapid algorithm is evident, as it significantly reduces computational time while maintaining comparable accuracy compared to existing methods. This comparative study also helps graduate students deepen their understanding of the advantages of numerical methods and increase their interest in research on numerical methods.