A transfer matrix method for free vibration analysis and crack identification of stepped beams with multiple edge cracks and different boundary conditions

This paper illustrates an analytical approach to investigating natural frequencies and mode shapes of a stepped beam with an arbitrary number of transverse cracks and general form of boundary conditions. A new method to solve the inverse problem of determining the location and depth of multiple cracks is also presented. Based on the Euler–Bernoulli beam theory, the stepped cracked beam is modeled as an assembly of uniform sub-segments connected by massless rotational springs representing local flexibility induced by the non-propagating edge cracks. A simple transfer matrix method is utilized to obtain the general form of characteristic equation for the cracked beam, which is a function of frequency, the locations and sizes of the cracks, boundary conditions, geometrical and physical parameters of the beam. The proposed method is then used to form a system of 2N equations in order to identify N cracks exploiting 2N measured natural frequencies of the damaged beam. Various numerical examples for both direct and inverse problem are provided to validate the present approach. The results are in good agreement with those obtained by finite element and experimental methods.