РУХ ЧАСТИНКИ ПО СФЕРИЧНОМУ СЕГМЕНТУ З ВЕРТИКАЛЬНИМИ РАДІАЛЬНО ВСТАНОВЛЕНИМИ ЛОПАТКАМИ

Abstract

The relative motion of a particle on the inner surface of a horizontal spherical disk along a vertical blade mounted in the radial direction is considered in the article. The disk rotates around a vertical axis with a given angular velocity. A system of differential equations of motion of a particle is compiled and solved by numerical methods. The kinematic characteristics of the motion are found, the regularities of the relative motion of the particle on the surface of the cylinder are clarified. Graphs characterizing the motion of a particle at certain given parameters are constructed, namely: graph of angle change, which sets the position of the particle on the surface of the sphere in the direction of the meridian, graphs of absolute and relative velocities, graphs of change of forces of the reaction of the spherical disk and blade. Numerical integration of the obtained differential equation showed that in half a second the particle rises to the height of the hemisphere, and then begins to fall. In this case, the descent alternates with the rise to a complete stop of the particle at a certain height, i.e. the particle “sticks” and then rotates with the hemisphere. The angle of “sticking” can be found analytically. In addition, numerical calculation methods have shown that at zero value of the friction coefficient of the particle on the disk surface, i.e. at its absolutely smooth surface, and at the non-zero value of the friction coefficient of the blade surface, and at an unlimited increase of the disk angular velocity the particle “sticks” at the height of the center of the sphere. If both surfaces are absolutely smooth, then the damping oscillations of the angle that determines the position of the particle on the surface of the sphere in the direction of the meridian, occur indefinitely. The working surface of the disk of the centrifugal apparatus, which is made in the form of a spherical segment, provides the beginning of the flight of the particle at the time of ascent from the disk at a given angle to the horizontal plane, increasing the scattering area of the technological material. The analytical description of the particle motion obtained in the article makes it possible to investigate its acceleration along with the blades of the disk and to find the relative and absolute velocities at the moment of particle ascent from the disk. The found analytical dependencies allow determining the influence of constructive and technological parameters on the process of particle acceleration.