Vibratory characteristics of cracked non-uniform beams with different boundary conditions

Non-uniform beams with bending moment of inertia and mass per unit length varying as I(x) = (1)(1+x)(+4) and m(x) = (2)(1+x) are widely used in various engineering fields, such as the civil and mechanical engineering etc. This paper presents an exact method to investigate the free vibration of cracked non-uniform beams with different conditions. Firstly, the closed form solution for the mode shape functions of the non-uniform beam is obtained based on the Euler-Bernoulli beam theory. Secondly, the beam is divided into several segments according to the different variable form, and each segment is further divided into many sub-segments by cracks. Four undetermined coefficients could represent the mode shape function of each sub-segment by simulating crack with the massless rotational spring. The undetermined transfer relationship in the same segment is obtained based on the principle of the transfer matrix method. The fourorder undetermined coefficient matrix is obtained by using continuity and equilibrium conditions between adjacent segments, and then the characteristic equation of the entire cracked beam is obtained after that. Finally, the results obtained from the finite element method and published papers are used to validate the correctness and reliability of the proposed method. The influences of crack depth, location and boundary conditions on natural frequencies of cracked non-uniform beams are discussed.