Dynamic response and stability analysis of an unbalanced flexible rotating shaft equipped with n automatic ball-balancers

The paper presents analytical and numerical investigations of a system of unbalanced flexible rotating shaft equipped with n automatic ball-balancers, where the unbalanced masses are distributed in the length of the shaft. It includes the derivation of the equations of motion, the stability analysis on the basis of linearized equations of motion around the equilibrium position, and the results of the time responses of the system. The Stodola–Green rotor model, of which the shaft is assumed flexible, is proposed for the analysis step. The rotor model includes the influence of rigid-body rotations, due to the shaft flexibility. Utilizing Lagrange's method, the nonlinear equations of motion are derived. The study shows that for the angular velocities more than the first natural frequency and selecting the suitable values for the parameters of the automatic ball-balancers, which are in the stability region, the auto ball-balancers tend to improve the vibration behavior of the system, i.e., the partial balancing, but the complete balancing was achieved in a special case, where the imbalances are in the planes of the auto ball-balancers. Furthermore, it is shown that if the auto ball-balancers are closer to the unbalanced masses, a better vibration reduction is achieved.