Purpose. Creation of a mathematical model of conditions and factors of stability of the established motion of a machine unit whose working machine is a drum, permanently rotating around the horizontal axis, with a fluid fill of the chamber. Methodology. A filled drum is considered to be a system with permanent composition with variable inertial parameters whose variability is due to the redistribution at relative motion of masses of fluid fill of the chamber on the body of the drum. Based on the application of the principle of hardening of a mechanical system, the drum rotation equations take into account the whole mass of the fill, regardless of the nature of its interaction with the surface of the chamber. We employed methods of the mechanics of relative motion for the mathematical description of the undisturbed motion of the system. The principle of establishing a hierarchy of variables was applied. A hypothesis is accepted about proximity of the system's motion to the rotation with a slowly changing angular velocity. It is believed that dynamic and inertial pa- rameters of the filled drum are determined by the quasi-static dependences on the rotation velocity. We used a second-order Lagrange equation for a system with variable inertial parameters in order to determine the transitional motion. A direct Lyapunov method is applied to search for conditions of the motion stability. Findings. Dynamic and inertial parameters of the filled drum are formalized using differential equations of undisturbed steady and transitional rotation. We obtained a condition of asymptotic stability of the established motion of a tumbling mill machine unit. Instability factors of the system's motion are determined. Originality. Regularities of steady rotation of a drum mill are identified. It was established that factors of instability in the established, rotational around the horizontal axis, motion of the drum filled with a fluid medium, are the variations of variable inertial parameters - the axial moment of inertia and the moment of resistance of the fill to rotation. It is shown that the failure to comply with conditions of the motion stability could be caused by achieving the extreme negative values of derivatives from the inertial parameters of the inchamber fill by the angular velocity of drum rotation. Practical value. The developed mathematical model allows us to qualitatively determine conditions of steady rotational motion of the drum with a fluid fill of the chamber. Conditions for the occurrence of unstable motion have considerable practical significance because they cause self-excitation of auto-oscillations of the filled drum and determine improvement of effectiveness of grinding process in tumbling mills with traditional design solutions.

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