An analysis of the motion of a rotor on a massless shaft having unequal cross-sectional stiffnesses

The equations of motion of a heavy rotor attached to a massless horizontal shaft with unequal cross-sectional moments of inertia have been developed and solved, including the effects of mass eccentricity and gravity on the motion. The effects of both internal and external damping have been included by linearizing the damping forces in terms of equivalent viscous damping. The existence of several areas of stability and instability have been shown, based on the initial parameters of the problem. In particular, the existence of an absolute lower bound and an absolute upper bound on stability have been shown to exist, together with an interval of absolute instability which is independent of the upper and lower bounds. Plots of the motion and orientation of the rotor have been made, showing the effects of different values of damping and stiffnesses on the curve traced by the center of the rotor. This curve is shown to consist of three parts—a constant deflection and a whirl at angular velocity 2 [angular velocity] due to the effect of gravity, and a whirl at angular velocity [angular velocity] due to the effects of mass eccentricity, where cu is the angular velocity of the shaft.