Direct and inverse methods on free vibration analysis of simply supported beams with a crack

An analytical transfer matrix method is used to solve the direct and inverse problems of simply supported beams with an open crack. The crack is modeled as a rotational spring with sectional flexibility. By using the Timoshenko beam theory on two separate beams respectively and applying the compatibility requirements of the crack, the characteristic equation for this cracked system can be obtained explicitly. This characteristic equation is a function of the eigenvalue (natural frequency), the location of the crack and its sectional flexibility. When any two natural frequencies in this cracked system are measured, the location and the sectional flexibility can be determined using the characteristic equation. The crack size is then computed by using the relationship between the sectional flexibility and the crack size. The theoretical results are also validated by a comparison with experimental measurements.