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It is well known that parts of agricultural machinery often have a cylindrical shape. This shape, for example, can be observed in the casing of lifting and transport machines, where the active working body rotates. Furthermore, drum grain dryers and triers use an inclined cylinder that rotates around its axis. In this case, the particles of the technological material interact with the rotating surface, which leads to their sliding, the nature of which depends on the value of the angle of inclination of the cylinder. In this study, the methods of differential geometry, vector algebra, theoretical mechanics, and numerical integration of differential equations consider the motion of a particle along the inner surface of an inclined cylinder rotating at a constant angular velocity around its axis. The axes of a fixed coordinate system are used to compose differential equations of motion. It was established that the proper initial conditions under which the particle would be stationary at a certain distance from the lower forming cylinder towards its rotation can be determined analytically. In case of movement along an inclined cylinder, the particle moves, among other things, in the axial direction, while reducing the amplitude of vibrations. Furthermore, it was found that the angle of inclination of the cylinder plays a significant role. If the latter is less than the angle of friction, then the vibrations stop, the movement of the particle stabilises, and it performs a rectilinear movement at a constant speed in the axial direction. If the angle of inclination of the cylinder is greater than or equal to the angle of friction, then the particle moves rapidly in the axial direction and its movement does not stabilise. The value of the angular velocity of rotation also plays a significant role. A certain amount of it provokes “sticking” of the particle, which does not depend on the inclination angle of the cylinder. The obtained analytical dependences can be used in the design of cylindrical working bodies of agricultural machines.

Currently, studies of the movement of agricultural materials on gravity surfaces have been carried out. These works also indicate the possibility of solving the inverse problem - the construction of such a surface that would ensure the given trajectory of the particle's movement. In the article, we will consider the construction of a linear surface, which ensures the movement of a particle along a slant line. The property of such lines is a constant angle between the tangent line drawn to the curve at any point and the horizontal plane (the angle of rise of the curve), as well as the parallelism of the main normal of the curve to the horizontal plane The purpose of the research is to construct a linear surface according to the calculated trajectory of the movement of a material particle along it. The construction of a linear surface according to a given trajectory of the movement of a material particle under the action of the force of its own weight is considered. A system of equations was obtained that describes the movement of a material particle along a linear gravitational surface. The considered example is the construction of a linear surface, which, with a known coefficient of friction, would ensure the movement of a particle along a helical line given by the angle of elevation and a constant curvature. An example of the construction of a linear surface is also given, which, with a known coefficient of friction, would ensure the accelerated movement of a particle along the surface with a constant angle. The results of the conducted research make it possible to construct a linear surface according to a given trajectory of the movement of a material particle under the action of its own weight.

The purpose of the study is the development of Maple-model of motion of a particle on a sloping plane, which performs reciprocating oscillations. A computational experiment was conducted to study the trajectory-kinematic properties of the motion of a particle on a sloping plane, which performs reciprocating oscillations. It is shown that for a horizontal rough plane, which performs reciprocating oscillation displacements, the absolute and relative trajectories of the particle are most rapidly stabilized by the largest coefficient of friction. Particles with a lower coefficient of friction have a smaller amplitude of absolute trajectory. The initial velocity and direction of throwing influence the time and place of the stabilization of the movement.

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