A new crack detection method in a beam under geometrically nonlinear vibration

In this paper, by analyzing the geometrically nonlinear vibrations of a cracked beam under harmonic excitation, a new crack sensitive parameter is introduced to recognize the open edge crack in a beam. The beam is assumed to be clamped at both ends, so that, increasing the amplitude of vibration results in stretching of the beam longitudinally and causes the beam vibrations to be geometrically nonlinear. By extracting the nonlinear forced vibration equations and solving them using perturbation method, the steady state response of the beam to harmonic excitation with a frequency close to one of the three first natural frequencies is obtained analytically. These responses are then Hilbert transformed, and instantaneous amplitudes are obtained in each case. Based on these data, a new parameter is extracted which is shown to be dependent only on the modal parameters of the cracked beam. Then, this parameter which we call it “Crack Index” is used to detect the crack in a beam under the geometrically nonlinear vibration. It is shown that while the amount of data needed to obtain the crack indexes is the same as those needed to obtain the natural frequencies, these indexes are much more sensitive to crack than the modal parameters.