A modeling method for vibration analysis of cracked beam with arbitrary boundary condition

Abstract This paper establishes a cracked Timoshenko beams model to investigate the vibration behavior based on the ultraspherical polynomials. Timoshenko beam theory is applied to model the free vibration analysis of the cracked beam and the numerical results are obtained by using ultraspherical orthogonal polynomials. The boundary conditions of both ends of the cracked beam are modeled as the elastic spring and the beam is divided into two parts by the crack section, and continuous conditions at the connecting face are modeled by the inverse of the flexibility coefficients of fracture mechanics theory. Ignoring the influence of boundary conditions, displacements admissible functions of cracked Timoshenko beam can be set up as ultraspherical orthogonal polynomials. The accuracy and robustness of the present method are evidenced through comparison with previous literature and the results achieved by the finite element method (FEM). In addition, the effects of flexibility coefficient on the natural frequencies are also investigated by using the proposed method. Numerical examples are given for free vibration analysis of cracked beams with various boundary conditions, which may be provided as reference data for future study.