Dynamic response of nanobeams with randomly distributed multiple vertical cracks

This paper aims to systematically study the steady-state forced vibrations of nanobeams weakened by multiple vertical cracks, which may conform to random or determined distributions. The variable separation and the Laplace transform methods are employed to obtain the Green’s function of beams with one crack for three typical kinds of beams, namely Hinged–Hinged, Clamped–Clamped, and Clamped–Free. Green’s functions for the nanobeams with an arbitrary number of cracks are figured out from the counterparts of beams with one crack by virtue of the massless elastic spring model and the transfer matrix method. The validity of the present solutions is discussed in comparison to those in literature in degenerated situations. The influence of statistical parameters of the random cracks on the dynamic responses of the weakened beams is studied in a systematic fashion. The present work may be beneficial to future structural designs on the nano-scales.