Natural vibration of a non-uniform beam with multiple transverse cracks

For a class of non-uniform beams with derivable vibration mode function, a theoretical method is proposed to investigate the natural vibration of these beams with multiple transverse cracks. Assume that all transverse cracks are open cracks (cracks remain open during the vibration), and each crack can be equivalent to a massless bending spring. Based on the vibration mode function, the transfer matrix between the left and right cross sections of the cracked non-uniform beam can be easily derived, which contains parameters of cracks and non-uniform parameter. For the simply supported beam, the deflections and bending moments at the left and right ends are equal to zero, and the characteristic transfer matrix can be obtained. Assume that the determinant of this characteristic transfer matrix is zero, and each order natural frequency of the whole beam can be calculated, which are compared with those obtained by the finite element method. Results indicate that the theoretical method proposed has a high computational accuracy at lower-order natural frequencies, and cracks and non-uniform parameters have a great influence on natural frequencies of the non-uniform beam.