Tatiana Volina’s research while affiliated with National University of Life and Environmental Sciences of Ukraine and other places

The interaction of the particles of the technological material with the working moving surfaces of the machines occurs during various technological processes. In the process of such interaction, the particles are forced to slide along the surface in a certain way in relative motion and describe a different trajectory in absolute motion. The absolute trajectory is the geometric sum of the relative sliding motion of the particle and the translational motion of the surface. To add these movements, it is convenient to use two coordinate systems: a moving one, in relation to which the relative movement of the particle is described, and a stationary one, in relation to which the translational movement of the surface and the absolute movement of the particle are described. A study of the relative motion of a particle in the tangent plane of the accompanying Frenet trihedron, which moves along a flat curve with variable curvature, was carried out. The relative motion of a particle in the tangent plane of the accompanying Frenet trihedron, which moves along a flat curve with variable curvature, is considered. Frenet's formulas were used to compile the system of differential equations of the relative motion of the particle. In this regard, unlike the traditional approach, the independent variable was not time, but the length of the arc of the guide curve along which the trihedron moves. The system of equations was composed in projections onto the vertices of a movable trihedron. It was solved by numerical methods. The proposed approach is considered on the example of the relative movement of a load in the car moving along a road with a curvilinear axis of variable curvature. The graphs of the relative sliding trajectory of the load and the relative speed for the given speed of the car were plotted.

Improvement of the screw working bodies for surface tillage is important for increasing the efficiency and quality of agricultural work. The purpose of the work was to calculate the design of the screw a working body for the surface cultivation of the soil from the compartment of the developable helicoid, which would perform the technological process with minimal resistance to plunging in the soil. The surface theory, analytical, and differential geometry were used to design the working body. It is shown in the paper that through a given spiral line, which is the cutting edge of the surface (blade), it is possible to draw developable helicoids with different inclinations of rectilinear generators. Designing the surface for the best plunging into the soil is important. It is established that the proposed working body is an alternative to the existing disc-type tools. It has been proven that for a long time, spherical discs for surface cultivation of the soil were fixed on a common shaft, which was installed on the unit with a certain angle between its axis and the direction of movement of the unit, which contributed to the plunging of the discs in the soil. It is confirmed that the disc-type tillage implements were improved, but this complicated the design of the unit since each disc received an individual axis of rotation. Studies have shown that this made it possible to additionally set the roll angle, that is, the deviation of the plane of the disc blade from the vertical direction, which improved the plunging of the disc in the soil and ensured more effective turning and mixing of plant residues. The proposed screw working body combines the simplicity of the design of the common shaft and the presence of the angle of attack and roll, which confirms its effectiveness in comparison with existing analogs. The necessary calculations were made, and the surfaces of the right and left courses were constructed with the designation of the necessary structural parameters. The application of the obtained results can simplify the design of the soil processing unit

The relevance of the study lies in the need to investigate the dependence of the force applied to the spring tooth on its parameters, which is an important task due to the widespread use of spring teeth in agricultural implements, such as balers, reapers, rakes, etc. The purpose of the study is to establish an analytical description of the spring tooth deformation depending on the amount of applied force. For this purpose, the theory of bending rods from the resistance of materials was applied, without simplifying it, as is common in construction, where the deflection of a beam is small compared to its length. The calculation is based on the well-known dependence of the curvature of the elastic axis of the beam (tooth) on the applied moment and the stiffness of its cross-section. The study considers a cantilevered tooth, which at the point of pinching is a spring with several turns, followed by a smooth transition to a rectilinear shape. The tooth is divided into two parts along its length: curvilinear and rectilinear. Calculation of the deformation, i.e., finding the shape of the elastic axis after the action of the applied force, is carried out for both parts separately. The need for this approach is dictated by the fact that the curvature of the elastic axis of the tooth in the free state changes abruptly from the stable value of the curvilinear part to a zero value of the straight part. The main result of the study is to find the shape of the elastic axis of individual parts of the tooth under the action of the applied force and combine them into one whole. This helps to determine the amount of movement of the free end of the tooth depending on the amount of force applied to it. The application of the obtained data can help in the development of more efficient and productive agricultural tools, and increase their durability and efficiency when interacting with the soil

Interpolation of a point series is a necessary step in solving such problems as building graphs de-scribing phenomena or processes, as well as modelling based on a set of reference points of the line frames defining the surface. To obtain an adequate model, the following conditions are imposed upon the interpolating curve: a minimum number of singular points (kinking points, inflection points or points of extreme curvature) and a regular curvature change along the curve. The aim of the work is to develop the algorithm for assigning characteristics (position of normals and curvature value) to the interpolating curve at reference points, at which the curve complies with the specified conditions. The characteristics of the curve are assigned within the area of their possible location. The possibilities of the proposed algorithm are investigated by interpolating the point series assigned to the branches of the parabola. In solving the test example, deviations of the normals and curvature radii from the corresponding characteristics of the original curve have been determined. The values obtained confirm the correctness of the solutions proposed in the paper.

Geodesic surface lines are analogous to straight lines on a plane. In addition to connecting two points of the surface by the shortest distance, they are the winding trajectories of the reinforcing threads in the strengthening of high-pressure cylinders. Just as a bundle of straight lines can be drawn from a given point on a plane in different directions, so there are geodesic lines on a surface that pass through a given point in different directions. Finding geodesic surface lines in the general case comes down to solving second-order differential equations. The purpose of the study is to investigate geodesic lines on the surface formed by the rotation of a given plane curve around a vertical axis and their transformation when this curve is shifted away from or towards the axis. For surfaces of revolution, the second-order differential equation can be reduced to the first order and even reduced to an integral based on the well-known Clerot formula. However, in this case, geodesic lines in all directions can be constructed only for a limited number of surfaces of rotation, and only limited fragments of geodesic lines can be constructed on the remaining surfaces. The article considers the construction of geodesic lines using the numerical solution of a second-order differential equation. The obtained results were visualized.

The common properties of images on a plane and a sphere are considered in the scientific works by scientists-designers of spherical mechanisms. This is due to the fact that the plane and the sphere share common geometric parameters. They include constancy at all points of the Gaussian curve, which has a zero value for a plane and a positive value for a sphere. Figures belonging to them can slide freely on both surfaces. With unlimited growth of the radius of the sphere, its limited section approaches the plane, and the spherical shape transforms into a plane. Thus, a loxodrome that crosses all meridians at a constant angle is transformed into a logarithmic spiral that intersects at a constant angle the radius vectors that come from the pole. The tooth profile of cylindrical gears is outlined by the involute of a circle. A spherical involute is used for the corresponding bevel gears. Other spherical curves are also known, which are analogs of flat ones. The formation of a cycloid and an involute of a circle are associated with the mutual rolling of a line segment with each of these figures. If the segment is fixed and the circle rolls along it, then the point of the circle describes the cycloid. In the case of a stationary circle along which a segment is rolled, the point of the segment will execute the involute. To move to the spherical analogs of these curves, it is necessary to replace the circle with a cone, and the straight line with a plane. The spherical prototype of the cycloid will be the trajectory of the point of the base of the cone, which rolls along the plane, that is, along the sweep of the cone. The sweep of a cone is a sector, the radius of the limiting circle of which is equal to the generating cone. If this sweep, like a section of a plane, is rolled around a fixed cone, when its top coincides with the center of the sector, then the point of the limiting radius of the sector will execute a spherical involute. This paper analytically implements these two motions and reports the parametric equations of the spherical analogs of the circle involute and the cycloid

This article deals with a spiral spring made of tape material with a rectilinear cross-section. A classic example of the use of this type of spring is mechanical type clockwork and starter mechanisms of internal combustion engines with manual start. If one end of a metal ruler is pinched, and a specific force is applied to the other end, it will bend under the action of the generated moment. The magnitude of the moment depends on the applied force and the ruler’s length. Therefore, the amount of deflection will increase with the increase of these parameters, and the shape of the ruler may take the form of a spiral. After the external load influence is terminated, the ruler will take its initial shape. That is, it will become straight. However, such spiral springs do not exist in practice. In its free state, the spring also has a spiral shape for compact sizes. This article uses the theory of elastic bending of rods for large deflections to determine this form. In addition, the initial curvature of its elastic axis in the free state must be considered for the spring. Simplifying linear bending formulas cannot be used to calculate the shape of the spring. The calculation of the shape of a spiral spring in a free state based on a given final shape with an applied moment is conducted in the article. The nonlinear bending theory is applied using the corresponding differential equations.KeywordsElastic AxisArc LengthForce MomentCurvatureDifferential EquationsProduct Innovation

The Frénet trihedron, known in differential geometry, is accompanying for a spatial and, as a special case, for a flat curve. Its three mutually perpendicular unit orts are defined uniquely for any point on the curve except for some special ones. Unlike the Frénet trihedron, the Darboux trihedron relates to the surface. Two of its unit orts are located in a plane tangent to the surface, and the third is directed normally to the surface. It can also be accompanying for the curve, which is located on the surface. To this end, one of the orts in the plane tangent to the surface must be tangent to the curve. Trihedra are movable and, with respect to a fixed coordinate system, change their position due to movement and rotation. The object of research is the process of formation of curves and surfaces, as a result of the geometric sum of the bulk motion of the Darboux trihedron and the relative motion of the point in its system under given conditions. In the study of the geometric characteristics of curves and surfaces, it is necessary to have formulas for the transition from the position of the elements of these objects in the system of a moving trihedron to the position in a fixed Cartesian coordinate system. This is exactly what needs to be solved. The results obtained are parametric equations of curves and surfaces that are tied to the initial surface. Nine guide cosines were found, three for each ort. A distinctive feature of this approach in comparison with the traditional one is the use of two systems: fixed and mobile, which is the Darboux trihedron. This approach allows us to consider in a new way the problem of the construction of curves and surfaces. The scope of practical application can be the construction of geometric shapes on a given surface. An example of such a construction is the laying of a pipeline along a given line on the surface. In addition, the sum of the relative motion of a point in a trihedron and the bulk motion of the trihedron itself over the surface gives an absolute trajectory of motion. Its sequential differentiation produces absolute speed and absolute acceleration without finding individual components, including the Coriolis acceleration. This could be used in point dynamics problems

After chopping the green mass with a cutting drum in forage harvesters, it must be loaded into a vehicle. A silo pipeline is used for this. It directs the movement of the crushed mass in the appropriate direction. The trajectory of particle movement is determined by the shape of a flat curve – the axis of the silo pipeline. The importance of the regularity of changes in the curvature of the axis of the silo pipeline or the trajectory of particle movement is shown analytically in the article. A differential equation of the movement of a single particle on the surface of the silo pipeline has been formulated. It is shown that the differential equation has an analytical solution for an axis in the form of a circle. For other cases, numerical methods should be used. A comparative analysis of the transportation process of crushed mass in silo pipelines with different axis shapes was made. Solutions of differential equations are accompanied by corresponding graphical illustrations.KeywordsProduct InnovationTrajectoryCurvilinear AxisVelocityParticle MovementDifferential EquationArc Length