Exact closed-form solution for the vibration modes of the Euler–Bernoulli beam with multiple open cracks

In this study, exact closed-form expressions for the vibration modes of the Euler–Bernoulli beam in the presence of multiple concentrated cracks are presented. The proposed expressions are provided explicitly as functions of four integration constants only, to be determined by the standard boundary conditions. The enforcement of the boundary conditions leads to explicit expressions of the natural frequency equations. Besides the evaluation of the natural frequencies, neither computational work nor recurrence expressions for the vibration modes are required. The cracks, that are not subjected to the closing phenomenon, are modelled as a sequence of Dirac's delta generalised functions in the flexural stiffness. The Eigen-mode governing equation is formulated over the entire domain of the beam without enforcement of any continuity conditions, which are already accounted for in the adopted flexural stiffness model. The vibration modes of beams with different numbers of cracks under different boundary conditions have been analysed by means of the proposed closed-form expressions in order to show their efficiency.