Particle motion over a plane, which rotates about a horizontal axis and makes a certain angle with it

Differential equations of relative material particle motion over a plane, which rotates about a horizontal axis, have been set up. Plane location relative to a rotation axis is set by a certain angle, which value can range from zero to ninety degrees. If angle value is equal to zero, a plane passes through a rotation axis; if angle value equals to 90 degrees, it is perpendicular to a rotation axis. The equations have been solved using numerical methods. In case of end positions of an angle, analytical solution has been found

EXTERNAL ROLLING OF A POLYGON ON CLOSED CURVILINEAR PROFILE

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

Investigation of deformation of the spring tooth of agricultural implements from the action of the force applied to it

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 soi

Transportation of a material particle by the operating mechanism of an agricultural machine in the form of a vertical screw confined by a coaxial stationary cylinder

The use of vertical augers in screw conveyors significantly expands the operational capabilities of such mechanisation equipment for performing loading and unloading technological operations. However, the existing designs of vertical screws do not fully meet the operational requirements. Their main disadvantages are increased energy consumption, especially with their significant overall dimensions, which depend on the influence of friction of the process material on the conveyor surfaces. Therefore, the study considers the movement of a material particle by the operating mechanism of an agricultural machine in the form of a vertical screw confined by a coaxial stationary cylinder and establishes the dependence of the speed of transportation of a material particle on the influence of the particle friction coefficient on the surface of the screw and on the surface of the limiting cylinder. The purpose of the study is to determine the parameters of the movement of agricultural material particles when they interact with the helical surface of the operating mechanism of a vertical conveyor in the form of a screw that rotates around the axis and the surface of a coaxial fixed cylindrical casing. Mathematical modelling of the processes of movement of soil particles along the helical surface of the operating mechanism is described on the basis of general laws and principles of analytical and differential mathematics, theoretical and analytical mechanics. As a result of the study, the differential equations of movement of a particle of agricultural material on the helical surface of the screw, which rotates around its axis in a fixed cylinder, were compiled. The equation is solved using numerical methods and the trajectories of the relative motion of the particle along the helical line – the edge of the screw, which is common to the surface of the screw and the limiting cylinder are constructed. The friction forces of the material particle on the surfaces of the screw and casing are also considered. The limit value of the helical line lifting angle is found when the particle lifting becomes impossible at a given angular velocity of screw rotation. The influence of the friction angles and radius of the limiting cylinder on the particle lifting speed is estimated. Graphs of kinematic characteristics as a function of time are given. The materials of the study can be used by researchers for further investigation and by practitioners in the selection of conveyors for transporting agricultural material

Form of axis of flexible incompressible bar at its pushing on rough ramp with permanent speed

Worked out differential equations of the motion of flexible incompressible bar on a rough ramp at its pushing with permanent speed. The special cases of the motion are considered without friction, at the different angles of slope of plane, at the different corners of entry of bar on a plane. The form of axis of bar for the considered cases was found by the numeral methods of integration

Study of the movement of soil particles on the surface of a screw tillage working body

This study considers the interaction of soil particles with a screw tillage body, the working surface of which is made in the form of a deployed helicoid and substantially differs from the screw. Furthermore, the unfolded surface is characterised by manufacturability of the working body, since it is possible to calculate the geometric dimensions of a flat workpiece, which is formed into a finished product by simple bending with minimal plastic deformations. The purpose of this paper was to conduct theoretical and experimental studies of the movement of soil particles along the helical surface of the tillage working body, which is made in the form of a deployed helicoid. Mathematical modelling of the processes of movement of soil particles along the helical surface of a tillage working body was described based on general laws and principles of analytical and differential mathematics, theoretical and analytical mechanics. Experimental studies were carried out using a methodology with elements of mathematical statistics, mathematical methods of optimal planning of a multifactor experiment. The analysis of the results of experimental and theoretical studies was performed using applied computer programmes and systems. Experimental studies were conducted according to standard and independently developed methods. Results: a mathematical model of the movement of soil particles along the helical surface of the tillage body was developed, and it was established that when the angle of attack of the helical tillage working body increases, the lifting value of soil particles increases and the area of their dispersion by the helical surface increases. The value of the coefficient of friction does not substantially affect the shape of the trajectory of the soil particle. The quality of loosening the soil most depends on the angle of attack of the screw surface and the angular speed of rotation of the working body. The materials of this paper can be used by scientists for further research and practices in the selection of tillage units

Constructing of flat curves in polar system of coordinates according to set properties at rotation around them pole

Приведено аналитическое описание кривых в полярной системе координат, отвечающих заданным свойствам механического (постоянство крутящего момента, создаваемого силой постоянной величины нажатия стержня вдоль нормали на кулачок, очерченный искомой кривой) и геометрического характера (постоянство угла пересечения заданной кривой с искомой при вращении последней вокруг полюса).Analytical description of curves in polar coordinates according to set properties of mechanical (constancy of twisting moment, created by permanent force of pressure of bar along normal on fist, outlined by found curve) and geometrical character (constancy of corner of crossing of set curve with founded after rotation last round pole) is resulted

A theoretical study of the limit path of the movement of a layer of soil along the plough mouldboard

There have been studied regularities affecting the magnitude of the dynamic component of the plough resistance for cylindrical mouldboards, and its peculiarities for cylindroid mouldboards depending on the direction of arrival of the soil layer on the ploughshare. The dynamic component of the draught resistance of the plough, which is directly proportional to the square of the ploughing velocity, can be estimated by the value of a dimensionless coefficient, conditionally named the "wrapping angle of the layer". This coefficient can be found for the mouldboard surface if the path of the movement of the mid-point of the section of the soil layer is known. By example of a cylindroid surface of the mouldboard it is shown how the dynamic component of the resistance varies depending on the design parameters of the mouldboard and the direction of arrival of the soil layer onto the ploughshare. As a result of the experimental study, a reduction of the longitudinal component of the thrust on the plough body, equipped with a mouldboard with an improved design made according to the results of the theoretical study, was achieved. This reduction was obtained by reducing the friction force due to the dynamic component of the resistance forces

Calculation of the bending parameters of a flat workpiece into a twist of a helicoid torso

Helical surfaces are deployable and non-deployable. The first is the surface of a helical conoid widely used in technology, known as the screw. The second one is the surface of an unfolding helicoid or torso-helicoid. In both cases, the surface is formed from a blank - a sweep in the form of a flat ring. If in the first case the sweep can be calculated approximately, since the exact one does not exist due to the properties of non-expanded surfaces, then in the second case it is calculated exactly. However, this does not mean that it is just as possible to form a ready-made product from it - a twist of a helicoid torso. In contrast to non-expanded surfaces, during the manufacture of which the workpiece is subjected to complex stretching and compression deformations, an expanded surface can be obtained by bending with minimal plastic deformations, the magnitude of which depends on the thickness of the sheet. Bending occurs along rectilinear generators, which are theoretically located on the workpiece. In the process of bending, the location of these generators should not change. In the theory of differential geometry, such a process can be described analytically and is called continuous bending. With regard to the manufacture of the twist of the torso-helicoid, this means a gradual increase in the step to the desired value. The work contains parametric equations that describe this method of bending. At the same time, not only stretching of the workpiece along the axis of the coil occurs, but also its twisting around the axis. A comparison of these two movements was made and it was found that the relationship between them is not linear. The work gives a formula describing this dependence. According to it, when the workpiece is uniformly stretched along the axis, the angle of its twist around the axis increases according to a dependence close to quadratic. In the work, examples of the use of the torso-helicoid are given, the surface is visualized, and graphs are constructed. Confirmation of the reliability of the obtained results is the first found quadratic form of the torso-helicoid, which does not change when the surface is ben

Construction of conical axoids on the basis of congruent spherical ellipses

Purpose: To carry out the transition from a cylindrical gear in which the centroids are congruent ellipses with centres of rotation in the foci, to a bevel gear on the basic of congruent spherical ellipses. Design/methodology/approach: Congruent ellipses with centres of rotation in the foci serve as centroids for the design of cylindrical gears with non-circular wheels. The article analytically shows that the analogues of ellipses on the plane - congruent spherical ellipses are the basis for the construction of the axoids of the corresponding bevel gears. An analogue of the centre-to-centre distance for ellipses in the plane is the angle between the axes of rotation of conical axoids. Findings: Based on the equality of the arcs of ellipses, the dependence of the angle of rotation of one axoid on the angle of rotation of another is found. Graphs of this dependence for separate cases are given. It is shown under what conditions the axes of axoids intersect at right angle. The parametric equations of spherical ellipses and corresponding axoids are given. They were used to construct spherical ellipses and corresponding conical axoids for different cases. For gears with right angle between the axes, separate positions of the axoids with different angles of their rotation around their axes are constructed. Practical implications: Spherical ellipses are directing curves for the construction of the corresponding conical axoids. Originality/value: The paper shows that congruent spherical ellipses act as centroids for the design of axoids of bevel gears. They roll one by one without sliding, rotating around axes that intersect in the centre of the sphere. To design such gears, it is important to know the interdependence between the geometric parameters, especially for common gears with a right angle between the axes