Analysis of the Forced Nonlinear Vibration Behavior of a Cantilever Beam with a Fatigue Crack Using the Lindstedt- Poincarè Method

In this paper, the nonlinear vibration behavior of a cantilever beam with a Fatigue Crack under harmonic excitation is investigated from an analytical point of view using one of the Perturbation Method’s named Lindstedt Poincarè Method. To this end, first the lateral vibration of the cracked beam in its first vibration mode is modeled as a SDOF system with an equivalent mass and stiffness. Then by introducing a new model for the bilinear stiffness of the beam with a breathing crack, the governing differential equation of motion is converted to the standard form that can be analyzed by Lindstedt- Poincarè 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 which is referred as to corresponding linear system. The second part is composed of the first and the second order correction terms, which reflects the effect of system nonlinearity due to opening and closing the crack on the vibration response. In fact, the correction terms consist of the super harmonic components of the response. Thus in this work an analytical relation is established between super harmonic components of the dynamic response and the system characteristics and the crack parameters. Results have been validated by a numerical method and experimental results of literature.

کلیدواژه ها:

Cracked beam ، breathing crack ، super harmonic components ، lindstedt poincarè method