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A new perturbation method has been proposed to investigate the sensitivities of the eigen-parameters (frequencies and mode shapes) and the modal strain energy density of a damaged beam. The sensitivities were obtained via the Rayleigh quotient and Taylor series expansion. The damage was simulated by the reduction in the cross-sectional area of the beam. The theoretical relationships were established between the damage parameter and the variations of the frequency and modal strain energy. The analytical formulae explicitly reveal the underlying mechanism that, the frequency, which is closely related to the modal strain energy of the entire structure, is less sensitive to the local damage; the modal strain energy density, on the other hand, is able to reflect the local damage in a small region. The results of the perturbation analyses were validated with the finite element analyses and experimental tests of the beam models with 2 damage levels. The first resonant frequencies decrease slightly with the increases of the damage. When 30 % of the cross-sectional area is cut off in the damaged zone, the first resonant frequencies are still 95 % and 98 % those of the intact beam for the bending vibration around x-axis and around y-axis; meanwhile, the modal strain energy increases 94 % and 54 % in the damage zone, which closely correlates with the local damage parameters. The proposed perturbation method has the potential to assess the quality of damage indictors faster and more easily than FE and experimental methods.

Sensitivity analyses of resonant frequencies and modal strain energy of damaged beams by perturbation method