Vibration of Bi-Directional Functionally-Graded Material Beams with Spinning Motion

In this paper, a novel model of bi-directional functionally-graded beams with exponentially distributed material property parameters along the axial and radial directions is proposed. The whirl characteristics of the spinning bi-directional functionally-graded beam is investigated via the semi-analytical method. Considering the influence of rotary inertia, the governing equations of motion of the spinning beam are established based on the Rayleigh beam theory. The frequency equations for the beam with pinned–pinned boundary conditions are derived using the complex modal method. By comparing with the results from the literature, the accuracy of the proposed model is verified. Further, the whirl characteristics of the spinning beam system are analyzed. Also, the effects of gradient parameters, slenderness ratio, and dimensionless spinning speed on the whirl frequencies of the bi-directional functionally-graded beam are discussed in detail. The results show that for the spinning bi-directional functionally-graded beam, the natural frequency, critical speed, and vibration modes are mainly affected by the spinning speed, axial functional gradient parameter, and the slenderness ratio.