Analysis of Coupled Transverse and Axial Vibrations of Euler Bernoulli and Timoshenko Beams with Longitudinal Crack for Its Detection

Abstract

This paper presents a method to analyze the vibration of monolithic beams with longitudinal cracks for its detection. Both forward problem of determination of natural frequencies knowing the beam and crack geometry details as well as inverse problem of detection of crack with the knowledge of changes in the beam natural frequencies has been examined. Both long (Euler-Bernoulli) and short (Timoshenko) beams have been studied. For modeling a crack located at the free end of a cantilever, the beam is divided into three segments. For an internal crack located away from the free end of the beam, it is split into four segments. In both cases, two of the segments take care of beam portions above and below the crack. The cracked segments are constrained to have the same transverse displacements but different axial movements. The modeling shows good accuracy for both the forward and inverse problems. The formulation predicts the first five fundamental natural frequencies in the forward problems with a maximum difference of 5 % with reference to finite element solutions for short beams with edge or inner cracks. Further, in the case of inverse problems, edge crack with sizes varying 5 to 50 % of the beam length has been detected with errors less than 3 % in both short as well as long beams. In the case of inner crack located at the mid-span with sizes varying from 5 to 45 % of the beam length has been detected with errors less than 3 % in location and 6 % in size. The results thus show encouraging possibility of exploitation of the proposed method for crack detection in practice.