Effect of axially functionally graded material on whirling frequencies and critical speeds of a spinning Timoshenko beam

This paper studies the free vibration of an axially functionally graded beam spinning with constant angular speed about its longitudinal axis. Material properties of the beam are assumed to be varied continuously in the axial direction according to a power law. Its whirling frequencies, critical speeds, and mode shapes are obtained in order to investigate the influences of axially functionally graded material (FGM). Spinning Timoshenko beam model is employed to incorporate the effects of transverse shear, rotational inertia, and gyroscopic motion. A spectral-Tchebychev method is employed to derive the discrete governing equation and to solve the dynamic characteristics of the beam. Results show that the axially FGM has significant influences on the whirling frequencies, critical speeds, and mode shapes. But these influences depend on boundary condition, whirling mode, and slenderness ratio. For cantilever beams, by choosing appropriate material gradient index, higher whirling frequencies and critical speeds, compared to those of homogeneous beams, can be obtained. However, the superiority of axially FGM can't be observed in beams with clamped-clamped, clamped-pinned, and pinned-pinned boundary conditions. Their critical speeds are smaller than those of one corresponding homogeneous beam.