Vibrations of axially moving flexible beams made of functionally graded materials

Problems related to the vibrations of axially moving flexible beams made of functionally graded materials are addressed. The problem of an axially moving beam may be interpreted as a telescopic system in which the mass is not constant, the mechanism of elastic deformation is transverse bending. A thin-walled beam with annular cross-section is analyzed, in which a continuously graded variation in the composition of ceramic and metal phases across the wall thickness with a simple power law is considered. In this paper a finite element scheme is employed to obtain numerical approximations to the variational equation of the problem. Normally, finite element approaches use fixed-size elements, however, for this kind of problems the increase of the number of elements, step by step as the mass enters, is a cumbersome task. For this reason an approach based on a beam-element of variable domain is adopted. The length of the element is a prescribed function of time. Results highlighting the effects of the beam flexibility, tip mass and material constituents on the dynamics of the axially moving beams are presented and the corresponding conclusions are given.