Perturbed eigenvalue problem with the Davidon-Fletcher-Powell quasi-Newton approach for damage detection of fixed-fixed beams

Abstract

A damage detection method is formulated to estimate damage location and extent from non-kth perturbation terms of a specific set of eigenvectors and eigenvalues. The perturbed eigenvalue problem is established from the perturbations of stiffness matrix, eigenvector, and eigenvalue. Then stiffness parameters are estimated from this equation using the Davidon—Fletcher—Powell quasi-Newton approach. The optimization algorithm is iterative, and its process is monitored by d-norm and t-norm indicators. A fixed—fixed beam with an odd number of elements is used as a test structure to investigate the applicability of the method. In a five elements beam, t-norm convergences of the second-order algorithm are more effective for small and large-percentage damages. In medium-percentage damages, convergences of the first-order algorithm are faster for both indicators. Convergences of the general-order perturbation method are more effective for small and medium-percentage damages. Meanwhile, convergences of this method are slightly more effective in large-percentage damages. For seven elements medium-percentage damage and nine elements small-percentage damage, the second-order algorithm converges faster to the t-norm indicator. It is proven that convergence rate increases with the order of the algorithm.