On the hollowness ratio effect on the dynamics of a spinning Rayleigh beam

The analytical solutions of a spinning Rayleigh beam with rotatory moment inertia and gyroscopic effect are presented in this paper. The critical speeds can be written analytically in a function of the length-to-radius ratio (l) defined by the beam's length over its outer radius and the hollowness ratio (α) defined by the hollow area over the total area of the cross section. The sensitivity analyses show that the critical speed is decreasing with l, but increasing with α. Moreover, α is more sensitive to the critical speeds. The design of a spinning beam should therefore be emphasized more on the hollowness factor. Contrary to common belief, only finite critical speeds exist and the number is independent of the boundary conditions. It increases monotonically with l, but decreases with α. The steady state unbalanced response can therefore be expressed analytically by the finite precessional modes and the corresponding generalized coordinates.