Determination of crack location in beams using natural frequencies

In this paper, we describe a numerical method for determining the location of a crack in a beam of varying depth when the lowest three natural frequencies of the cracked beam are known. The crack is modelled as a rotational spring and graphs of spring stiffness versus crack location are plotted for each natural frequency. The point of intersection of the three curves gives the location of the crack. Earlier work in this area involved the use of the Frobenius technique for solving the governing differential equation analytically and then using a serai-numerical approach to obtain the crack location. In this work, we use the finite element approach to solve the same problem. The beam is modelled using beam elements and the inverse problem of finding the spring stiffness, given the natural frequency, is shown to be related to the problem of a rank-one modification of an eigenvalue problem. Examples outlining the accuracy and ease of using this method are shown. The results are compared with those from semi-analytical approaches. The biggest advantage of this method is the generality in the approach; different boundary conditions and variations in the depth of the beam can be easily modelled. (C) 2001 Academic Press.