The rolling of a flat figure in the form of an equilateral polygon on a curvilinear profile is considered. The profile is periodic. It is formed by a series connection of an arc of a symmetrical curve. The ends of the arc rely on a circle of a given radius. The equation of the curve, from which the curvilinear profile is constructed, is found. This is done provided that the centre of the polygon, when it rolls in profile, must also move in a circle. Rolling occurs in the absence of sliding. Therefore, the length of the arc of the curve is equal to the length of the side of the polygon. To find the equations of the curve of the profile, a first-order differential equation is constructed. Its analytical solution is obtained. The parametric equations of the curve are obtained in the polar coordinate system. The limits of the change of an angular parameter for the construction of a profile element are found. It is a part of the arc of the curve. According to the obtained equations, curvilinear profiles with different numbers of their elements are constructed.

Conveyance of a particle by a vertical screw, which is limited by a coaxial fixed cylinder Differential equations of relative particle motion along the periphery of a vertical screw have been deduced which rotates about its axis and is limited by a coaxial fixed cylinder. The equations have been solved by applying numerical methods and the trajectories of relative particle helical motion, that is to say, movement along the edge of a screw, which is common for a screw surface and for a limiting cylinder, have been built. The force of particle friction on the screw surface and on the surface of a cylindrical cover has been taken into consideration. The cases, when a helix angle, that is to say, the angle of screw edge inclination, is less than the angle of particle friction on the surface of a screw, when it is equal to a friction angle and when it is higher than a friction angle, have been considered. In addition, a specific case, when a screw is fixed and a particle moves downwards by gravity, which takes place in spiral chutes, has been analyzed. Kinematic characteristics-time curves have been presented.

The design of a helical surface tillage tool made of a developable helicoid section has been calculated using the methods of analytical and differential geometry, the theory of surfaces, the systems of computer-generated graphics and mathematics. A helical tool in the form of a skeleton cylinder made of bars, where there is a helical surface made of arranged sheet metal, has been developed. Such a design prevents a helical surface from becoming clogged with soil. The suggested tool operates as a roller and the soil can easily pass through the gaps between the bars. New helical harrow tools have been designed and made for this experiment. An experimental harrow equipped with helical tools has been made and used to conduct a field research. The research was aimed at determining the optimal kinematic parameters and the pattern of field movement, which allow performing the technological process of soil tillage with the performance indices that meet the agricultural requirements.

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The article deals with the interaction of a screw cultivator with soil particles. Due to the very wide application in technology, the term "helical surface" is usually understood as the surface of a helical conoid or auger. In this paper, we consider the surface of a deployable helicoid, also linear, but significantly different from the screw surface. The difference lies not only in the geometric shape, but also in the manufacturing technology. If the screw is made by punching or strip rolling with significant deformation of the billet, the unfolded helicoid can be made by simple bending with a minimum of plastic deformation. In terms of theory, if the thickness of the workpiece were zero, there would be no plastic deformation at all when bending it. The working body for soil cultivation consists of a strip of unfolded helical surface, the outer edge of which is sharpened and acts as a blade, and the inner one is rigidly attached to a lattice cylinder. The difference between the radius of the helical line of the blade and the cylinder determines the working depth. The lattice cylinder prevents clogging of the inter-screw space and at the same time performs the additional function of a roller. The body works like a disc tool, that is, the profile of the processed field has protrusions and depressions. At the moment when the moldboard touches the surface of the field, angles of attack and roll arise, similar to the angles of attack and roll of disc guns. The design parameters that provide these angles can be calculated from an analytical description of the surface. The section, that is, the drum with the rotating working surface of the auger, is located so that its axis makes a certain angle with the direction of motion of the unit. This causes an angle of attack and reaction forces that cause the drum to rotate with the surface. From the speed of the aggregate and considering the angle of attack, the angular velocity of the section can be found. The differential equation of motion of the particle after it hits the rotating surface is then generated. The differential equation is drawn in projections on the three axes of the stationary coordinate system. It includes three unknown dependencies: two variables describing the trajectory of the particle sliding on the surface, and the reaction force of the surface. The system is solved numerically. Trajectories of relative and absolute motion of the particle and graphs of changes in its relative and absolute velocities are plotted.