Abstract. Generalized differential equations of relative particle motion over the internal surface of an inclined cylinder, which performs an oscillating motion, have been formulated. All the points of a cylinder trace ellipses in vertical planes. Such cases of cylinder oscillation, when either semi-axes or one of them is equal to zero, that is to say, a cylinder performs reciprocating motion, have been considered. The equations have been solved applying numerical methods and trajectories of relative particle motion over the surface of a cylinder have been built. Graphs of other kinematic characteristics as a timevarying function have been presented. Certain cases, when a cylinder axis is located horizontally or at an angle to a horizontal plane, have been considered.

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.

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.

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Relative particle motion on the internal rough surface of a concave soil-tilling disk, which rotates around horizontal axis under a soil reaction force, has been considered. A disk blade is positioned in a vertical plane, which makes an incidence angle with the direction of the machine movement. This angle has its acceptable limit and when it exceeds the limit, it causes disk dragging. In the paper, it has been assumed that dragging was non-existent, the rotational rate of a disk is stable and it depends on stabilized velocity of the machine movement and on an incidence angle. When a machine is operated, soil particles get onto an internal disk surface and perform a relative motion, which determines the slipping trajectory of a particle on a disk surface. The trajectory of the absolute particle motion relative to a fixed coordinate system allows tracing the rise height of a particle after its gets onto a disk surface. Differential equations of particle motion have been developed and solved using numerical methods. A particle gets onto a disk with specified initial criteria, namely the direction of its entering on a disk and its initial velocity. Two models of particle motion on a disk have been considered. According to the first model, we assume that further particle motion after its getting onto a disk surface is performed due to the kinetic energy of a particle at the moment of its getting onto a disk surface. According to the second model, the backup force of other particles, which breaks the force of particle friction on a surface, is taken into account. Differential equations of particle motion have been developed in projections onto a fixed solid system of axes. Thus, the system includes three differential equations in three unknown functions. One of the unknown functions is the surface reaction force and the dependences, which describe relative particle motion on a disk surface. For the purpose of the quality of soil loosening, we conducted multivariate experiment to determine the area of the soil dissipation. On the basis of field experiments, it has been determined that the production process quality of tillage machine meets agrotechnical requirements.