Direct Multiscale Analysis of Stability of an Axially Moving Functionally Graded Beam with Time-Dependent Velocity

In this study, the transverse vibration of a traveling beam made of functionally graded material was analyzed. The material gradation was assumed to vary continuously along the thickness direction of the beam in the form of power law exponent. The effect of the longitudinally varying tension due to axial acceleration was highlighted, and the dependence of the tension on the finite support rigidity was also considered. A complex governing equation of the functionally graded beam was derived by the Hamilton principle, in which the geometric nonlinearity, material properties and axial load were incorporated. The direct multiscale method was applied to the analysis process of an axially moving functionally graded beam with time-dependent velocity, and the natural frequency and solvability conditions were obtained. Based on the conditions, the stability boundaries of subharmonic resonance and combination resonance were obtained. It was found that the dynamic behavior of axial moving beams could be tuned by using the distribution law of the functional gradient parameters.