Crack identification in a stepped beam carrying a rigid disk

In this study, a simple algorithm based on a mathematical model is proposed to identify crack location and depth in a stepped cantilever Euler–Bernoulli beam carrying a rigid disk at its tip. The mathematical model that describes the lateral vibration of the beam is derived using the assumed mode method that coalesces with the Lagrange's equation. A massless torsional spring whose stiffness depend on the severity of the crack is used as a crack model. Using this crack modeling method combined with the assumed mode method, the crack effect is introduced to the system flexibility as global additional structural flexibility. For the assumed mode method, the mode shapes for two uniform beams connected by a massless torsional spring (simulating the cracked beam) are adopted as trial functions. The proposed identification algorithm utilizes the first three natural frequencies shift of the beam caused by a crack to estimate its location and depth. In addition, the proposed mathematical model is used to illustrate the effect of the crack depth and its location on the dynamic characteristics of the system. Using the commercial finite element (FE) software (ANSYS 8.0), three-dimensional finite element analysis (FEA) is carried out to show the accuracy of the derived mathematical model and to demonstrate the reliability of the proposed crack identification algorithm. The analysis showed consistency with the assumed mode results. It showed that the error in concurrent prediction of crack depth and its location using the proposed algorithm is about 10%.