PARTICLE MOTION ON THE SPHERICAL SEGMENT WITH VERTICALS RADIALLY INSTALLED BLADES

The relative motion of a particle on the inner surface of a horizontal spherical disk along a vertical blade mounted in the radial direction is considered in the article. The disk rotates around a vertical axis with a given angular velocity. A system of differential equations of motion of a particle is compiled and solved by numerical methods. The kinematic characteristics of the motion are found, the regularities of the relative motion of the particle on the surface of the cylinder are clarified. Graphs characterizing the motion of a particle at certain given parameters are constructed, namely: graph of angle change, which sets the position of the particle on the surface of the sphere in the direction of the meridian, graphs of absolute and relative velocities, graphs of change of forces of the reaction of the spherical disk and blade. Numerical integration of the obtained differential equation showed that in half a second the particle rises to the height of the hemisphere, and then begins to fall. In this case, the descent alternates with the rise to a complete stop of the particle at a certain height, i.e. the particle “sticks” and then rotates with the hemisphere. The angle of “sticking” can be found analytically. In addition, numerical calculation methods have shown that at zero value of the friction coefficient of the particle on the disk surface, i.e. at its absolutely smooth surface, and at the non-zero value of the friction coefficient of the blade surface, and at an unlimited increase of the disk angular velocity the particle “sticks” at the height of the center of the sphere. If both surfaces are absolutely smooth, then the damping oscillations of the angle that determines the position of the particle on the surface of the sphere in the direction of the meridian, occur indefinitely. The working surface of the disk of the centrifugal apparatus, which is made in the form of a spherical segment, provides the beginning of the flight of the particle at the time of ascent from the disk at a given angle to the horizontal plane, increasing the scattering area of the technological material. The analytical description of the particle motion obtained in the article makes it possible to investigate its acceleration along with the blades of the disk and to find the relative and absolute velocities at the moment of particle ascent from the disk. The found analytical dependencies allow determining the influence of constructive and technological parameters on the process of particle acceleration.

Modeling of Natural Lighting Parameters in the Open Air with Intermeradiandiate Luminance Distribution

When simulating outdoor natural lighting, a spherical sky model is used. This is true in a clear sky, but in the presence of clouds, this model does not correspond to reality. This paper presents the substantiation of the sky model in the form of a spherical segment with a standard distribution of the luminance of the semi-clear (intermediate) sky. Moreover, instead of the ratio of illumination, under given cloudiness conditions to illumination with ideal transparency of the atmosphere, which are usually used in European standards, the direct value of cloudiness is used here, taken from the results of long-term observations at meteorological stations. To modeling the parameters of outdoor natural lighting, a more effective and simple mathematical apparatus of point calculus is used, with the help of which a point set of a spherical segment is formed. On the basis of this set, a field of elementary pyramids is created. For each pyramid, using well-known formulas, elementary values of the parameters of natural lighting are determined.

The justification of interaction of fruits with the rolling-in device of a combine-harvester for harvesting cucurbit crops

The role of cucurbit crops in the life of man has been studied and presented. The process of interaction between the rolling-in device of a combine-harvester and a fruit, as well as the process of hitting water-melons into different surfaces, have been considered. The diagram of the distribution of velocities when the rolling-in device interacts with a fruit, as well as the area of expansion of plastic deformation as a result of the impact of a fruit, have been presented. The equation of work, which arises when a fruit hits into the rolling-in device, has been considered, the work being decomposed into plastic work of deformation and elastic work of deformation. The work of plastic deformation, which arises when fruits come into collision with a surface, is considered in more detail, the work being decomposed into plastic work of deformation in the volume, which is limited by a spherical segment and a cone. The absolute and allowable velocities after the impact have been theoretically determined and substantiated. The parameters of allowable velocities for the varieties "Crimson sweet" and "Kholodok" have been theoretically found and the confirmation of the theoretical analysis was carried out in experimental research, which was conducted on a laboratory installation for determining the critical speed of impact on fruits of cucurbit crops. The comparison of the theoretical and experimental values of the terminal velocities confirms the credibility of the research.

MOVEMENT OF THE PARTICLE ON THE INTERNAL SURFACE OF THE SPHERICAL SEGMENT ROTATING ABOUT A VERTICAL AXIS

The particle relative motion on a spherical segment rotating about a vertical axis was considered in the article. The differential equations of the relative displacement of a particle were completed and solved by numerical methods. The relative and absolute trajectories of particle motion and graphs of relative and absolute velocity changes were constructed. The regularity of particle motion as it is lifted over the surface was found out. The conducted experimental research has confirmed the received theoretical results.

Determining the exponent of the unloading curve when flattening a sphere

To study the flattening of the sphere, it is proposed to use the kinetic indentation diagram by the plane. Given the known values of the reduced elastic modulus, applied force, maximum and residual deformation, it is possible to determine the contact area. It is indicated that in this regard, the exponent of the unloading curve of a pre-loaded sphere with a flat rigid surface plays an important role. The analysis of methods for determining the unloading curves of unloading for the finite element models, taking into account strain hardening, is carried out. It is shown that dependences of the unloading curves during flattening on the relative indentation in the form and the range of values differ from the similar ones during indentations of the sphere. The dependence between the exponents of the unloading curves for the force and for the area is determined. The range of correct use of the results of the finite element analysis of a hemisphere for rough surfaces is indicated. The exponent of the unloading curve after flattening the spherical segment from the half-space property is determined.

Analysis of the Sheet Shell's Curvature with Lame's Superellipse Method during Superplastic Forming

The prospects and novelty of using the expression of Lame's superellipse for approximating the curvature of shells in superplastic forming (SPF) and for predicting the geometry of a product are shown. Different versions of the SPF facilitate the realization of different radii of curvature of the shell contours, which differ significantly from the radius of the spherical segment. The regularities of the change in the radius of conjugation of the bottom and the wall of the spherical shell for various SPF variants are established.