Free vibration analysis of non-uniform multi-cracked Euler–Bernoulli beam

This paper provides an exact approach for free transverse vibration analysis of a simply supported non-uniform Euler-Bernoulli beam with an arbitrary number of concentrated cracks. The equation of motion is normalized and written based on a set of dimensionless parameters. The general solution is obtained based on the Bessel functions. The differential equation is modified to include cracks. The cracks are modeled with a massless rotational springs. Determination of natural frequencies and mode shapes are simplified by expressing the general solution based on linear combination of the Bessel functions. The main advantage of the proposed method is elimination of numerical computation of the high order determinant. So, the eigenvalue equation of a non-uniform beam with any number of cracks can be determined from a second order determinant. Numerical computation is given to illustrate the proposed method and to investigate the effects of number, position and intensity of cracks on the characteristics vibration. Finally, the results of cracked beam are validated via comparison with those are computed from Differential Quadrature Element Method (DQEM). There is a good agreement between the two results obtained.