Longitudinal and transverse instabilities of moving nanoscale beam-like structures made of functionally graded materials

A mathematical model is proposed to explore vibrations and instabilities of moving functionally graded (FG) nanobeams. It is assumed that the FG nanobeam moves with a constant velocity and its material properties vary continuously across the thickness according to a power-law relation. The longitudinal and lateral equations of motion of the moving nanostructure are extracted by employing the nonlocal Rayleigh beam model. Using Galerkin approach and admissible mode shapes, the longitudinal and transverse frequencies are calculated. The effects of the power-law index, small-scale parameter, length of the FG nanobeam, and its velocity on the frequencies and stability of the moving nanostructure are comprehensively addressed. Both divergence and flutter instabilities of moving FG nanobeams are discussed, and the roles of influential factors on such phenomena are explained.