On the unstable vibrations of a shaft having asymmetrical stiffness and/or asymmetrical rotor

In a rotating asymmetrical shaft having a keyway or a rectangular cross section, or in a rotating shaft with an asymmetrical rotor such as a two-pole generator or two-blade propeller, there occur two types of unstable vibrations. When bearing pedestals supporting a directional inequality in stiffness, each unstable region splits up into several regions. The position, width and number of the unstable regions and a dynamic behavior of the shaft are analytically obtained by approximation both for a rotating asymmetrical shaft and an asymmetrical rotor. The analytical results show a good coincidence with those obtained by an analog computer. In order to understand the mechanism for the occurrence of these two types of unstable vibrations, the authors clarify the conditions under which the time average of a torque applied to the shaft end is positive, so that the whirling amplitudes of the shaft increase and unstable vibrations occur. Vibratory solutions in the unstable region obtained by an analog computer are found to satisfy this instability condition.