Investigation of nonlinear vibrations of a cantilever beam with a transverse fatigue crack

Nonlinear phenomena widely exist in engineering applications. A common example of this phenomenon in aerospace structures is the creation of fatigue cracks that open and close continuously under dynamic loads during the vibration cycle, causing a bilinear behavior in the structure. Late detection of such cracks can lead to a catastrophic failure. Therefore, identifying the nonlinear behavior of the cracked structure is very important for the prevention of structural failures. In this study, the nonlinear vibration behavior of a cantilever beam with a transverse breathing crack with bilinear behavior has been studied. For this purpose, we first estimate the polynomial function equivalent to the bilinear function of the beam stiffness. Then, using the well-known perturbation method, the method of multiple scales, the nonlinear equation of the beam is solved, and the frequency response equations for both harmonic and superharmonic resonances are extracted. Then, the sensitivity of the responses to the crack parameters such as depth, location, excitation force, and damping coefficient are investigated. The beam frequency response curves in the main resonance have become highly nonlinear due to the increase of the crack parameters and causes softening of the curves. Also, according to the results of the beam vibrations in superharmonic resonance, it was observed that the behavior of the beam in superharmonic resonance has a higher sensitivity to the existence of the breathing crack.