A New Nonlinear Model for Flexural Vibration Analysis of a Cracked Beam with a Fatigue Crack

In this paper, the continuous model for vibration analysis of a cracked beam developed by Shen and Pierre is modified. For this end, by some realistic assumptions, new functions for displacement and stress fields are proposed. Then, the equation of motion of the cracked beam with breathing crack is obtained via the Hu-Washizu variational principle. The new obtained equation of motion is self-adjoint. Moreovers, by employing the Galerkin method, the modes shape of beam with a breathing crack are obtained. Then, in order to obtain the time response of the cracked beam, a new bilinear model is introdused for the stiffness corresponding to each mode. Using this model, the governing equation of motion is converted into the standard form which can be analyzed by Lindstedt-Poincar’ method. The results show that response obtained throught the perturbation metod (Lindstedt-Poincar method) is composed of two parts. The main part is the response of a system with the equivalent stiffness, whith is equal to the main value of the stiffness corresponding to the fully open and fully close crack cases. The remaining part of the response consistence of the correction terms, which reflects the effect of opening and closing the crack during vibration. The results show that for a given crack parameters, redaction in natural frequencies for a fatigue-breathing crack are smaller than the ones caused by open cracks. Also, the results have been validated by the experimental and theoretical data reported in the literature. There is a good agreement between the results obtained through the proposed method and those obtained from the reported experimental data.