Dynamic instability analysis of a cantilever beam with breathing crack

In this paper, dynamic characteristics of a beam with breathing crack is considered. Breathing crack is modeled as a bilinear oscillator. The stiffness of the cracked beam is estimated by using influence coefficients based on castigliano's theorem and strain energy release rate (SERR). The equation of motion of breathing cracked beam is formulated using finite element method using Hamilton's principle. The equation of motion of breathing cracked beam is converted into Mathieu - Hill type equation to obtain the regions of dynamic instability of beam using Harmonic balance method and it is further solved for Eigen frequency of the cracked beam. The increase in breathing crack depth increases the instability region and it is found that the effect of the crack location near to the fixed end is more for the cantilever beam, and this also increases the instability region. It is found that increase in dynamic instability index increases the instability regions of the cracked structure. In addition to that, the effect of static and dynamic loads are also investigated and discussed. The study has been conducted for the first two instability boundaries of the cracked structure only. It is hence seen that assuming the crack to remain open underestimates the stability boundaries of the system. Permitting the crack to open and close (breath) yields a stability boundaries in between the open and uncracked beam.