Harmonic extension through conical surfaces

This paper establishes extension results for harmonic functions which vanish on a conical surface. These are based on a detailed analysis of expansions for the Green function of an infinite cone.

Study on the transient temperature distribution of new conical friction plate under sliding friction

Conical friction surface is a novel configuration for friction plate in transmission. Numerical FEA models for transient heat transfer and distribution of conically grooved friction plate have been established to investigate the thermal behavior of the conical surface with different configurations. The finite element method is used to obtain the numerical solution, the temperature test data of conical surface are obtained by the friction test rig. In order to study and compare the temperature behavior of conically grooved friction plate, several three-dimensional transient temperature models are established. The heat generated on the friction interface during the continuous sliding process is calculated. Two different pressure conditions were defined to evaluate the influence of different load conditions on temperature rise and the effects of conical configuration parameters on surface temperature distribution are investigated. The results show that the radial temperature gradient on conical friction surface is obvious. The uniform pressure condition could be used when evaluating the temperature rise of conically grooved friction plate. The increase of the cone height could improve the radial temperature gradient of the conically grooved friction plate.

Method of Designing a Friction-Based Wedge Anchorage System for High-Strength CFRP Plates

The cables of high-strength carbon fiber reinforced polymer (CFRP) plates are starting to be applied to large spatial structures. However, their main anchorage systems rely on the adhesive force, which entails risks to their integrity resulting from aging of the binding agent. In this study, a friction-based wedge anchorage system was designed for CFRP plates. The working mechanism of the proposed anchorage system was explored both theoretically and experimentally. The anti-slip mechanism and condition of CFRP plates were formulated so that the equivalent frictional angle of the contact surface between a CFRP plate and wedges must not be smaller than the sum of the dip angle of the wedge external conical surface and the frictional angle between the wedges and barrel. An analysis of the stress distribution in the anchorage zone of the CFRP plate was conducted using the Tsai-Wu failure criterion, which concluded that the compressive stresses should be reduced on the section closer to the load-bearing end of the anchorage system. Furthermore, the anchorage efficiency coefficient was proposed, which depends on stress concentration coefficients, plate thickness, length of anchorage zone, dip angle of wedge external conical surface, and its frictional angle. Then, it was determined that the minimum length of an anchorage zone for the CFRP plates with various specifications should be at least 49 times larger than the CFRP thickness. A finite element analysis and static tensile tests on six specimens were carried out. The experimental results revealed that the anchorage efficiency coefficient of the optimized anchor reached 97.9%.

COMBINED EXTRUSION OF GLASSES WITH A CONICAL BOTTOM. KINEMATIC AND STRESS STATES OF THE PLASTIC REGION LOCATED UNDER THE CONICAL SURFACE OF THE PUNCH

For extrusion of glasses with a conical bottom using the method of plastic flow by A. L. Vorontsov, the kinematic and stress states of the extruded metal in the area of the plastic deformation center located under the conical surface of the punch were determined. The resulting formulas will be used to determine the stress state in the region of the hearth located under the central flat part of the punch-son՚s working end. In the future, the results of this mechanical and mathematical analysis will also make it possible to investigate the question of the presence of the taper of the punch, which is optimal in terms of strength.

DETERMINATION OF THE MODE OF FUNCTIONING OF THE CENTRIFUGAL-GRAVITATIONAL SEPARATOR OF THE GRINDING DEVICE

The peculiarity of harvesting legumes is the need to process their heap on grater devices with subsequent separation. To improve the technical and economic performance of seed heap processing equipment, it is advisable to combine wiping and separation operations by combining a grater working body and a separating rotating sieve of cylindrical or conical shape. The analysis of influence of geometry of rotary sieve drums allows to define the rational form and parameters of work of the separating device which will provide the necessary quality of seed material. Analysis of the literature on the processes of sieve separation of grain and seed mixtures showed the advantages of centrifugal separators with a rotating sieve surface. Improving the efficiency of separation of mixtures by rotary sieves is achieved by the simultaneous use of centrifugal, Coriolis and gravitational forces. The aim of the study is to increase the efficiency of separation of seed heaps of legumes after treatment with a grater device by determining the trajectory of the seeds along the rotational surfaces. Earlier, the authors developed a grater-separating unit in which a cylindrical perforated drum is used to remove unworn seeds. This design does not use the surface of the sieve effectively enough. To increase the uniformity of the load on the steaming surface of the rotary drum, it is advisable to reduce its area in the direction of seed movement. To do this, use a conical separating surface to reduce the radius of the cone in the direction of movement of the material. When considering the movement of a particle, it is represented as a material point with mass m moving along a conical surface that rotates around a vertical axis. As a result of theoretical researches the dependence which defines time of stay of a particle on a sieve depending on a coordinate (length of a generating cone) is received. A graphical interpretation of this dependence for certain values of design parameters is also presented. As a result of research, it was found that increasing the residence time of the material on the sieve due to the use of a conical surface increases the yield of pure seeds, and also contributes to the uniform loading of the sieve surface, which improves the quality of the source material.

Teaching the Notion Conic Section with Computer

The article discusses the problem of introducing and constructing mathematical concepts using a computer. The Wolfram Mathematica 12 symbolic calculation system is used at each stage of the complex spiral process to form the notion of conic section and the related concepts of focus, directrix and eccentricity. The nature of these notions implies the use of appropriate animations, 3D graphics and symbolic calculations. Our vision of the process of formation of mathematical concepts is presented. The notions ellipse, parabola and hyperbola are defined as the intersection of a conical surface with a plane not containing the vertex of the conical surface. The conical section is represented as a geometric location of points on the plane for which the ratio of the distance to the focus to the distance to the directrix is a constant value. The lines of hyperbola and ellipse are determined by their foci. The equivalence of different definitions for conical sections is commented.

COMBINED EXTRUSION OF GLASSES WITH A CONICAL BOTTOM. KINEMATIC AND STRESS STATES OF THE PLASTIC REGION IN CONTACT WITH THE CONICAL SURFACE OF THE MATRIX

For the extrusion of glasses with a conical bottom using the method of plastic flow by A. L. Vorontsov, the kinematic and stress states of the extruded metal in the area of the plastic deformation zone in contact with the conical surface of the matrix were determined. The results are combined with the results obtained earlier for the cylindrical region located under the forming glass wall. The resulting formulas will be used to determine the stress state in the areas of the focus located under the end of the punch. The derived formula for the maximum pressure on the matrix wall is necessary for calculating the matrix strength and making an informed decision about the need for its banding.