Free Vibration of the Cracked Non-uniform Beam with Cross Section Varying as Polynomial Functions

This paper presents an approach for free vibration analysis of the cracked non-uniform beam with general boundary conditions, whose mass per unit length and bending moment of inertia varying as polynomial functions. Firstly, the general expression of mode shape function of the non-uniform beam with four undetermined coefficients is solved by a generalized power series method based on the Euler-Bernoulli beam theory. Then, the transfer matrix method is combined into the mode shape function of the non-uniform beam to improve the computational efficiency owe to the fewer segments divided by the cracks and joints at the different forms of variable cross-section. The massless rotational springs are adopted to simulate the cracks to derive the transfer relationship of the undetermined coefficients in the same segment. The four undetermined coefficients matrix is obtained by using equilibrium and continuity conditions between two adjacent segments, and then the characteristic equation of the entire cracked beam is formed. Finally, the correctness and reliability of the proposed method in this paper is verified by the methods presented by other published papers and the Finite Element Method (FEM). In addition, the crack parameters on the vibratory characteristics are investigated through the numerical examples.