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The authors of the study work out the differential equations of motion of a vertical rotor model on an elastic-dissipative suspension, balanced by a ball-type automatic balancing device. Often, the cross-section of the cavity of the body of the automatic balancing device is rectangular and during rolling the balls have two points of contact, in one of which the balls slide along the surface of the cavity. To prevent the balls from sliding, the inner surface of the cavity of the automatic balancing device is made in the shape of a torus, which provides one point of contact. The forces of gravity and the forces of resistance to the movement of the correcting weights are taken into account, and the model is drawn up for both viscous and dry friction forces inside the body of the automatic balancing device. The obtained mathematical model of the rotor makes it possible to study the transient and steady-state modes of motion of the rotor system.