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.

Modeling of formation regularities of conical surfaces

The scheme of processing conical surfaces by grinding them to a flat tool is considered and a technical solution for the implementation of such processing is proposed. Using the created device allows implementing the group method of forming conical parts with a deviation of the generatrix of the cone from straightness of not more than ± 0.00012 mm. A mathematical model of the patterns of removal of stock from a conical part with a flat tool is developed. A formula is obtained for calculating the modulus of the sliding velocity at any point on the processed conical surface, which implements engineering methods for controlling the shaping of conical parts without conducting preliminary labor-intensive experimental studies. An optimization technique for the adjustment parameters of technological equipment was proposed. The most effective axicon processing modes were revealed at the stages of preliminary, medium and fine grinding, as well as at the polishing stage, depending on the technological heredity of the workpiece from the point of view of distribution of the stock to be removed over its surface. It has been established that changes in the eccentricity between the axes of rotation of the tool and the faceplate as well as the amplitudes of the reciprocating rotational movements of the latter practically do not affect both accuracy and processing productivity, therefore, in practice, these parameters can not be optimized, but their average values can be assigned. The operating modes of the basic lever grinding and polishing machine are established, at which the required accuracy of the working surface of the tool is provided, which directly affects the straightness of the generatrix of the cone. Studies of the regularities of the shaping of the side surface of a conical lens in the conditions of free grinding are carried out and the adjustment parameters of technological equipment that affect the quality and productivity of the processing process are determined.

Friction welding of carbon quality steel

The aim of the work is to study the possibility of using, instead of manual arc welding with a covered electrode, a method of friction welding of fittings and small-diameter branch pipes of hull power plants. The solution to this problem is possible on the basis of the application of the friction welding process of small diameter nipples and nipples. Given the insufficient theoretical and experimental study of the problem, it was necessary to solve the following scientific and technical problems: to develop the design of the models and the connected (conical) surfaces of the choke and branch pipe assemblies; substantiate the choice of optimal angles for preparing surfaces for welding; justify the choice and experimentally confirm the values of the welding mode parameters; to assess the quality of welded joints by non-destructive and destructive methods. The obtained positive results of the experimental study confirmed the technological feasibility and economic feasibility of using friction welding of carbon steel fittings and small-diameter nozzles.

The Hanging Drape: a Vertex Analogy

We present a novel, rigidly folding vertex inspired by the shape of the simplest hanging drape. Fold lines in the vertex correspond to pleats and ridges in the drape, and are symmetrically arranged to enable synchronised at folding of facet pairs. We calculate the folded rotation angles exactly using a spherical image specialised for inextensible vertex folding. We show that the vertex shape is bounded by a pair of conical surfaces whose apex semi-angles directly correspond with fold-line rotations, which expresses a geometrical equivalence between the external shape and internal folding motion of the vertex. We discuss how the vertex viz. drape perform as a novel type of conical defect based on its spherical image topography; and we highlight the meaning of bistable behaviour for the vertex, in analytical- and practical terms.

Equidistant, Standard and Secant Parallels

The paper describes a study of equidistant, standard and secant parallels in normal aspect cylindrical and conical projections. First, the explanation of cylindrical and conical projections as projections on cylindrical or conical surfaces is not recommended because it leads to misunderstanding projection properties. Furthermore, equidistant, standard and secant parallels are often assumed to be identical in references. After defining these three types of parallels, it is proved that it is necessary to differentiate them in the theory of map projections and teaching it.

Kinematic analysis of the method increasing the accuracy of treatment of conic surfaces

A method for processing high-precision conical surfaces under conditions of free grinding on a lever machine is proposed, which allows adjusting the process parameters taking into account the technological heredity of the workpiece from the point of view of the distribution pattern of the allowance to be removed from its surface. The scheme of implementation of the method in which the main links are a tool in the form of a faceplate and a separator for maintaining the required accuracy of its flatness. In order to develop a method of targeted control of the actuation of the working surface of the tool to maintain the required accuracy of its flatness, mathematical modeling of the motion patterns of the links of the actuator of the base machine, on which the proposed device for processing conical parts is installed, was performed. In the process of mathematical simulation structural analysis of the executive mechanism of the lever machine is carried out, connection between the generalized coordinate and provisions of links of the executive mechanism of the lever machine is established, position of the lever and the upper link in the vertical plane is probed, geometrical parameters of reflexive rotational motion of the upper link are considered, the ratio for calculation of length of a crank of the executive mechanism of the machine depending on amplitude of figurative movement of its output link is obtained. An analytical relationship has been established between the input and output links of the machine actuator, which makes it possible to calculate the length of the first of the marked links depending on the required amplitude of the oscillatory motion of the second of them and creates the prerequisites for the development of engineering methods for controlling the process of forming high-precision conical surfaces under conditions of free grinding.

An investigation on methods for axis detection of high-density generic axially symmetric mechanical surfaces for automatic geometric inspection

The detection of the symmetry axis from discrete axially symmetric surfaces is an interesting topic, which is transversal to various fields: from geometric inspection to reverse engineering, archeology, etc. In the literature, several approaches have been proposed for estimating the axis from high-density triangular models of surfaces acquired by three-dimensional (3D) scanning. The axis evaluation from discrete models is, in fact, a very complex task to accomplish, due to several factors that inevitably influence the quality of the estimation and the accuracy of the measurements and evaluations depending on it. The underlying principle of each one of these approaches takes advantage of a specific property of axially symmetric surfaces. No investigations, however, have been carried out so far in order to support in the selection of the most suitable algorithms for applications aimed at automatic geometric inspection. In this regard, ISO standards currently do not provide indications on how to perform the axis detection in the case of generic axially symmetric surfaces, limiting themselves to addressing the issue only in the case of cylindrical or conical surfaces. This paper first provides an overview of the approaches that can be used for geometric inspection purposes; then, it applies them to various case studies involving one or more generic axially symmetric surfaces, functionally important and for which the axis must be detected since necessary for geometric inspection. The aim is to compare, therefore, the performances of the various methodologies by trying to highlight the circumstances in which these ones may fail. Since this investigation requires a reference (i.e. the knowledge of the true axis), the methodologies have been applied to discrete models suitably extracted from CAD surfaces.