One of the new methods for reducing the vibrations of rotors with variable imbalance is implementing automatic ball balancer (ABB). Although, the ABB has numerous advantages, it has one major deficiency; increasing the rotor vibration amplitude at transient state that limits the use of this type of balancers. In the previous studies for diminishing the mentioned deficiency, a new type of ball balancer which is called the ball-spring ABB, is introduced and the dynamic behavior of Jeffcott rotor equipped with the ball-spring ABB is investigated. In the Jeffcott rotor model the gyroscopic effect is not considered, however, in practice and in many applications, due to asymmetry which comes from the offset of the rotor from the shaft mid-span, the gyroscopic effect is generated. In such conditions, the results of Jeffcott model are not reliable and dynamic behavior of the ball-spring ABB should be investigated in the presence of gyroscopic effect. In this paper by considering the asymmetry in the rotor-shaft system and taking into account the gyroscopic effect, the equations of motion of a rotor equipped with the ball-spring ABB are derived. The time responses of the system are computed and based on the Lyapanov first method, the stable regions are extracted. The results show that not only the gyroscopic effect does not affect on the performance of the ball-spring ABB, but also the magnitude of the Eulerian angles of the rotor equipped with the ball-spring ABB is less those the rotor equipped with the traditional one.