Dynamic behavior of a spinning exponentially functionally graded shaft with unbalanced load

The dynamic behaviors of a pinned-pinned spinning exponentially functionally graded shaft with unbalanced loads are investigated. The shaft is simulated in the Rayleigh beam model considering rotary inertia and gyroscopic effects. The governing equation for the flexural vibration of the shaft is derived via the Hamilton principle. Based on the boundary conditions, both the exact and approximate whirl frequency equations of the system are obtained analytically. Also, the validity of the proposed model is confirmed by comparing it with the results reported in the literature. Finally, a numerical study on the basis of the analytical solutions is performed to evaluate the main parameters, including slenderness ratio (α), gradient index (β), the mass ratio (μ), and eccentric distance (γ) on the whirl frequency, critical spinning speed, mode shapes, and stability of the system. The results reveal that the vibration and instability of the spinning shaft are strongly dependent on the unbalanced load and material gradient.