Analysis of the Nonlinear Behavior of the Free Vibration of a Cantilever Beam with a Fatigue Crack Using Lindstedt- Poincare's Method

Previous investigations indicate that using the open crack model for vibration analysis of cracked structures may lead to incorrect results. Such a simple model can only be used as a rough approximation for predicting the dynamic behavior of the cracked structures. Therefore, in order to predict the nonlinear dynamic behaviour of the structures with a fatigue crack more accurately, one has to consider the nonlinearity of the crack. In this paper, the nonlinear behavior of the free vibration of a cantilever beam with a Fatigue Crack is investigated. To this end, first, the lateral vibration of the cracked beam in its first mode is modeled as an SDOF system with an equivalent mass and stiffness. Then, a new model is introduced for the bilinear stiffness of the beam with a breathing crack. Using this model, the governing differential equation of motion is converted to the standard form that can be analyzed by Lindstedt- Poincare’s method. The results show that the response is composed of two parts. The main part is the response of a system with the mean equivalent stiffness of the systems corresponding to the closed crack and the open crack cases. The second part is composed of the first and second order correction terms, which reflects the effect of opening and closing of the crack on the vibration response. In fact, the correction terms consist of the higher harmonic components of the spectrum. The results have been validated by the experimental tests.