Using a continuous mathematical model to investigate the effect of a symmetric circumferential crack on the free vibrations of a simply supported Euler–Bernoulli beam

Abstract

This paper investigates the influence of a circumferential crack on a simply supported beam’s free vibration behavior analytically and numerically. The beam is modeled by Euler–Bernoulli beam theory with a circular cross-section. The circumferential crack is modeled by considering a continuous effect in the displacement field. An exponential function is considered in the displacement field with an exponential decay rate parameter which characterizes the influence of crack on its surroundings. The exponential function of crack depth ratio was proposed for the exponential decay rate of symmetric circumferential open cracks using a comparative method based on finite element simulation for the first time. The equation of motion is derived using Hamilton’s principle and solved by a Galerkin method, and natural frequencies are obtained. The crack depth and position influence on the natural frequency are analytically clarified and compared with a symmetric and single-edge crack. The possibility of identifying the location and relative depth of the crack has been investigated. Whatever the crack is, closer to the maximum amplitude vibration in each mode, the frequency ratio is more dropped, making it more detectible. Also, the circumferential crack is more affected on vibration behavior than the symmetric and single-edge crack. The analytical results have excellent conformity with the finite element results obtained by the ABAQUS 3-dimensional model. Also, some experimental results have been obtained to validate the finite element model. The finite element model has an excellent agreement with experimental results. The results can be used to identify circumferential cracks.