Analytical solution of whirl speed and mode shape for rotating shafts

The analytical solution of whirl speed and mode shape of a rotating shaft in six boundary conditions is presented in this paper. The shaft is modelled by a Rayleigh beam with rotatory inertia and gyroscopic effects, and the boundary conditions are (1) short-short, (2) long-long, (3) long-free, (4) free-free, (5) long-short, and (6) short-free bearings. It is shown that the whirl speed can be written analytically by a function of the whirl ratio (λ) defined by the rotating speed over the whirl speed and the slenderness ratio (l) defined by the length of the shaft over its radius. The number of whirl speeds, contrary to common belief, is finite when λ > 1/2 . For the first time, the rotating system's unbalanced response can be written analytically in an exact form by a finite number of vibration modes with the corresponding generalized coordinates.