In this paper, a single degree of freedom system was used to model a beam with a breathing crack. Then analytical approximate solution approach was employed to solve this nonlinear vibration equation based on the Homotopy Perturbation Method (HPM). Nonlinear free vibration frequencies under different crack depth to thickness ratios were calculated and compared to the results that were predicted by the bilinear oscillator model. Both predictions were validated experimental data. It has been clearly shown that the fundamental frequency of a beam with a breathing crack is lower than the case in which the crack is assumed to be open and remains open. Nonlinear forced vibration responses containing all harmonics were determined analytically. Numerical simulation results were used to validate our analytical approximation solutions. A damage detection scheme was proposed to relate the crack depth to thickness ratio with a damage index, which is derived from the nonlinear forced responses. This damage index is able to accurately assess the breathing crack condition even for a very small crack depth.

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