The tumbling mill rotation stability

Abstract

Геотехнічна і Гірнична механіка, машинобудування СтійкіСть обертання барабанного млина 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 parameters 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 in-chamber 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.

Establishing the effect of decreased power intensity of self-oscillatory grinding in a tumbling mill when the crushed material content in the intra-chamber fill is reduced

“…An analytical method of forecasting qualitative factors of auto-excitation of self-oscillations in a machine unit with a filled drum as its working machine was developed in [7]. Quantitative assessment of dynamic parameters of fill impact and energy and technological parameters of the innovative self-oscillatory grinding process was made in [8] for one discrete value of the degree of filling of the chamber with an intra-mill fill.…”

“…Characteristics of grinding processes in tumbling mills significantly depend on the content of particles of the crushed material in the intra-chamber grinding fill. However, the influence of variation of material content on the motion behavior of grinding bodies as an additional factor was not studied in [7][8][9]. A series of attempts were made for theoretical and experimental analysis of the influence of the processed material content on dynamic, energy and technological parameters of established conventional dry and wet grinding processes.…”

Revealing the effect of decreased energy intensity of grinding in a tumbling mill during self-excitation of auto-oscillations of the intrachamber fill

“…Paper [35] constructed an analytical model of quality conditions and stability factors for the steady motion of a machine unit whose operating machine is a permanently rotating drum with a granular filling. The proposed method makes it possible to predict the manifestation of factors related to the self-excitation of auto-oscillations in the considered dynamic system.…”

“…Under condition for the increased rotation speed of a tumbling mill, which is about 1…1.2 from critical (ψ ω ≈ 1…1.2), there occurs the self-excitation of self-oscillations of granular fill of the operating chamber without lifters [35]. In this case, the fill executes oscillatory flows under a two-phase mode with the formation in the chamber's cross section of a non-free fall zone and a pulsating zone ( Fig.…”

Establishing the effect of a decrease in power intensity of self-oscillating grinding in a tumbling mill with a reduction in an intrachamber fill

“…Impact action of the grinding bodies can be greatly increased during self-excitation of the pulsating behavior of the intrachamber fill. An analytical method for prediction of qualitative factors of excitation of self-oscillations of a unit in which the filled drum represents the working machine was developed in [8]. Quantitative estimation of dynamic parame ters of impact action of the and power and technological parameters of an innovative self-oscillatory grinding process was performed in [9].…”