Green׳s functions for the forced vibrations of cracked Euler-Bernoulli beams

In this paper, explicit expressions of the steady-state responses of a cracked Euler-Bernoulli beam submitted to a harmonic force are presented. The mechanical properties of cracked sections of the beam are characterized by five local stiffness models available in literature. Fundamental dynamic response of a beam with one crack is obtained by means of Green׳s function method. For a multi-cracked beam, the transfer matrix method is employed to derive the steady-state response, which can be readily reduced to those for a single-cracked beam. Numerical calculations are performed to validate the present solutions, to compare the dynamical behaviors of the beam corresponding to various classical local compliance models and to study the influences of crack geometry (depth and location) on the mechanical behavior of beam. Furthermore, the interactions of two cracks in the beam are particularly studied. The present analytical results can serve as a valuable benchmark to the future numerical simulations and experimental studies.