Perturbation analysis of natural oscillations of elastic structures with damage

Abstract

A mathematical justification is given for the energy based approach to estimating the changes in the natural frequencies of an elastic structure when a small flexibility or mass is added. In particular, we consider the situation where the vibration modes of the perturbed structure are not kinematically admissible for the original one. Local damage (cracks), imperfect boundary conditions (elastic supports instead of ideal ones), shear compliance of a beam or of a plate are examples of such structural changes. The rigorous asymptotic procedure establishes a relationship between the frequency shift and the complementary energy of the added compliance calculated for the vibration mode of the original structure, whatever its type, be it a rod, a plate or a shell. Special treatment is required in case of repeated (multiple) natural frequencies. Examples ranging from simple toy models to a practically relevant model of a plate with a crack are demonstrated. The approach is particularly efficient when the effort for the vibration analysis of the original structure is essentially lower than that of the perturbed one. Parametric studies with varying damage severity and configuration result as a post-processing of a single solution for the unperturbed structure.