A Theoretical and Experimental Investigation of Free Vibration Behavior of a Cantilever Beam with a Breathing Crack

In this paper, free vibration behavior of a cracked cantilever beam is investigated. The lateral vibration of the cracked beam in its first mode is modeled as a SDOF system with an equivalent mass, stiffness, and damping. In this model, it is assumed that the equivalent stiffness of the system is varying as a harmonic function due to the opening and closing of the crack during vibration. The main purpose of this investigation is to derive an analytical relation to describe the effects of the system characteristics and the crack parameters on superharmonic components of the response spectrum. To this end, the method of multiple scales is employed to solve the governing differential equation of motion. The results show that the response is composed of two parts, one of which (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 and the second part is the first order correction term, which reflects the effect of opening and closing the crack on the vibration response. In fact, the correction term consists of the superharmonic components of the spectrum. Results have been validated by numerical methods and experimental tests.