The Relations between Deflection, Stored Energy and Natural Frequencies, with Application in Damage Detection

摘要: The paper firstly introduces the mathematical relations between the deflection of a cantilever beam at the free end and its capacity to store energy. It is shown by means of the finite element method FEM that by the occurrence of a crack, the stored energy decreases proportionally with the deflection increase. This means that for beams with geometric discontinuities, an inverse formulation of Castgliano’s theorem is possible. Tests are performed for more applied forces and damage depths, for cracks located near the fixed end. It was demonstrated that the energy decrease is not influenced by the force amplitude if deformation remains in the linear domain. Afterward, it is proved that a crack of given depth produces similar dimensionless frequency decreases for all bending modes at the fixed end (ie location where the bending moment achieves maxima). Finally, the energy distribution for several bending modes is pointed out, and it is shown that the beam’s diminished capacity to store energy, subsequent the frequency decrease, is influenced by the local bending moment.