A mathematical model of particles movement of separated material in a friction separator is proposed. It includes equations of their movement at each stage of separation (along a rough inclined plane, on a curvilinear part of a springboard, impact of a particle on the surface of a drum, free motion before leaving the separation zone), and differential equations of particle movement in circulating air flow created by a rotating drum; Each of these phases of motion is described by a system of equations obtained on the basis of basic laws of mechanics. An analysis of a particle motion along aslope plane was carried out on the basis of the law on energy changes. Depending on the ratios of the recovery factor and instantaneous friction, being random variables and also the coordinates of a point impact the particle is possible to move along different trajectories with different initial conditions. The mathematical model of the process of separation of loose multicomponent materials on a friction drum-shelf separator makes it possible to investigate comprehensively the process of separation of particles on friction and elastic properties and to optimize the operating modes of the apparatus at relatively low costs without using macaques and prototypes and investigate them experimentally. On the basis of analysis of the proposed theories of particles motion, the main ways are formulated of further improvement of methodology for designing and creating friction devices.

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