The Motion of a Particle on an Inclined Cylinder that Rotates Around its Own Axis

Abstract

It is well known fact, that parts of agricultural machinery often take a cylindrical shape. This form, for example, has a casing of hoisting and transport machines, in which the active working body rotates. In addition, drum dryers and triers contain an inclined cylinder that rotates around its own axis. The particles of the technological material interact with the rotating surface, which leads to their sliding, the character of which depends on the value of the angle of inclination of the cylinder. In this research by means of the methods of differential geometry, vector algebra, theoretical mechanics and numerical integration of differential equations the motion of a particle on the inner surface of an inclined cylinder rotating at a constant angular velocity around its own axis is considered. The axes of the fixed coordinate system were used to compile the differential equations of motion. It is established that the appropriate initial conditions when the particle is stationary at a certain distance from the lower generatrix of the cylinder in the direction of its rotation, can be determined analytically. In the case of motion on an inclined cylinder, the particle moves in the axial direction with a simultaneous decrease in the amplitude of oscillations. In addition, the angle of inclination of the cylinder has been found to play a significant role. If it is less than the angle of friction, then the oscillations stop, the motion of the particle is stabilized and it performs a rectilinear motion with a constant speed in the axial direction. If the angle of inclination of the cylinder is greater or equal to the angle of friction, the particle moves rapidly in the axial direction and its motion is not stabilized. The value of the angular velocity of rotation also plays a significant role. Its certain value provokes the "sticking" of the particle, which does not depend on the angle of the cylinder. The obtained analytical dependencies can be used in the design of cylindrical working bodies of agricultural machines.