An analytical method for damped free vibration analysis of a cracked beam considering the coupled multimode equations

The multimodal free vibration of a beam with a breathing crack excited by arbitrary initial conditions is investigated. Taking the initial conditions to be arbitrary makes more than one mode of the beam to be excited simultaneously. By considering the bending moment at the crack position, a multi-harmonic function describing the instantaneous opening and closing of the crack is extracted. Since the modal stiffnesses of the beam are dependent on the crack parameters, the extracted crack breathing function will appear in the equations of motion and makes them to be coupled. These equations are solved using the perturbation method. Then, the free response of the beam is extracted under three cases of initial conditions: excitation of the first mode, simultaneous excitation of the first and second modes, and simultaneous excitation of the first three modes. The results show that by exciting the first mode solely, the harmonic components of the response offer very limited information about the crack, however, by exciting the first several modes simultaneously, many other harmonic components appears at the frequency response curves which are more sensitive to the crack and contain more comprehensive information about the crack parameters.