Dynamic analysis of a spinning Rayleigh beam

The whirl speed, critical speed and mode shape of a spinning beam in six general boundary conditions are investigated analytically in this paper. The beam is in Rayleigh model with rotatory inertia and gyroscopic effects. It is shown that the whirl speeds and critical speeds can be expressed analytically by a function of the slenderness ratio (l) defined by the beam length over its radius. Contrary to common belief, only finite number of critical speeds can be found in all speed ranges. The number is functional of l, but independent of the boundary conditions. The system's unbalanced response can therefore be expressed analytically by these finite precessional modes and the corresponding generalized coordinates.