The main parameter that determines the quality of rolled products is the accuracy of geometric dimensions. The accuracy of the pipe wall thickness is characterized by the nominal value and the range of permissible values, i.e., the tolerance range. To obtain the nominal values of the pipe wall thickness in the rolling mill, the same roll profile (groove system) and a limited pool of diameters of cylindrical mandrels are used. Based on the diameter of rolling mandrels selected from the existing database and the need to obtain the nominal thickness of the pipe wall, the position of the rolls in the stands of the finishing group is determined, which determines the actual accuracy of wall thickness. The article presents an algorithm for calculating the rational diameters of cylindrical mandrels and provides some results of pilot testing of this technical solution.

For a function f(z) analytic in a disk, a method of approximate reconstruction from known (or arbitrarily specified) boundary values of its real part (under the condition of its continuity) using interpolation wavelets is proposed; the method is easy to implement numerically. Despite the fact that there are known exact analytical formulas for solving this problem, the explicit formulas for approximating the function f(z) proposed here are much easier to apply in practice, since the previously known exact formulas lead to the necessity to apply numerical integration methods when calculating convolutions of functions with Poisson or Schwartz kernels. For the approximations used in this paper, effective upper bounds are obtained for the error of approximation of functions analytic in the disk by interpolation wavelets in the spaces L_p(0,2π) for any p≥ 2 . These estimates can be used to find the parameters of the wavelets from a desired accuracy of recovering the function f(z) . Note that if the real part of f(z) is continuous on the boundary of the disk, the continuity of f(z) in the closure of the disk cannot be guaranteed; that is why it is impossible to estimate the approximation error for f(z) in the uniform metric (for p=∞ ) in the general case.

This article describes experimental studies of reducing during one pass rolling of pipes from ShKh-15 steel at laboratory rolling mill. Designing and processing of experimental data have been performed by means of complete factorial experiment. According to the results, the changes in the wall thickness depend on partial deformation of external diameter. The wall thickening increases with partial deformation. Analysis has demonstrated that the curvature of the end area decrease with decrease in temperature, the degree of wall thinness of workpieces, and the partial deformation in reducing mill stands. The temperature has exposed the minimum influence of the mentioned parameters. As a consequence of the work, the experimental dependence has been developed describing the influence of the deformation extent on the changes in wall thickness, the critical degree of partial deformation has been specified. It has been concluded that the wall thickening and probability of defect occurrence can be reduced due to decrease in the workpiece diameter before the reducing mill.

DOI: 10.21538/0134-4889-2022-28-4-9-16 Поступила 9.09.2022 Принята к публикации 9.09.2022 Акопян Роман Размикович д-р. физ.-мат. наук, доцент, зав. отд. Институт математики и механики имени Н.Н. Красовского УрО РАН г. Екатеринбург e-mail: RRAkopyan@mephi.ru Антонов Николай Юрьевич д-р физ.-мат. наук, зам. директора Институт математики и механики имени Н.Н. Красовского УрО РАН

Existence of defects on pipe surface deteriorates the product quality. This article presents the results of integrated studies of surface defects such as: dent, skin, and lap. Regarding the TPA-140 rolling mill, the locations, reasons of occurrence, and remedial methods are specified.

The main trend in the world's pipe rolling production is the use of a continuous longitudinal rolling mill in a pipe-rolling line. In such mills, shells are rolled into pipes using long cylindrical mandrels, which are subject to severe temperature and force conditions. In addition, mandrels are the most expensive tool of continuous mills. The wear of mandrels during rolling, producing different wall thickness with a mandrel of the same diameter, and producing the same wall thickness with mandrels of different diameters necessitate adjusting the position of the rolls in the mill stands. The adjustment of the position of the rolls in the stands of a mill is known to lead to changes in the wall thickness and the formation of defects on the surface of the mother pipe, which affects other parameters of the process. This necessitates experimental study into the process of longitudinal rolling of pipes in order to evaluate the effect of the mandrel diameter on the accuracy of pipes, the angle of contact between the mandrel and the metal, and the overfill of the groove. The defects caused by the irrational choice of the diameter of a cylindrical mandrel are characterized. Lead was used as a model material for making samples during the study.

Production of seamless pipes made from 08-12Cr18Ni10Ti austenitic stainless steel is developed in cooperation of JSC VTZ and JSC RusNITI involved in the PJSC TMK. Development was preceded by the complex of investigations on the technology base of PJSC TMK in JSC RusNITI. This paper shows the study results of longitudinal rolling of pipes using physical and computer modeling. Adaptation of the study results and test pipe rolling were realized at the manufacturing area of JSC VTZ in a PRM 159-426 line of pipe-rolling plant possessing an MPM mandrel mill with retained mandrel and two-roll mill housing.

Stainless steels of austenite class 08–12Х18Н10Т have a high corrosion resistance, which stipulates for their wide application in various areas of industry. Technology of pipes production of the steels is rather specific and requires observation of some conditions. It was shown that temperature of a work-piece heating before deformation is an important parameter of the technology. It was noted that for the piercing of a steel work-piece with various chrome content, there is a rational temperature interval. Nonobservation of the temperature can lead to defects formation on internal pipe surface because of earlier destruction and opening of metal cavity during piercing. The choice of the rolling-out scheme has a direct effect on the work-piece forming in cross-sections. Results of hot rolling of Ø37×2,5 tube samples, manufactured of 08–12Х18Н10Т steel and carbon steel of grade 40 presented. The rolling was done at a laboratory mill. As a result of the experiment the lower limit of ovality of rolls grooves was specified for conditions of rolling of pipes from 08–12Х18Н10Т steels by 2-roll scheme. At the rolling with ovality B / H ≤1,07, defects appeared on the internal tube surface in the form of scratches caused by the mandrel. The rational range of ovality of grooves at multi-stand rolling can be from 1.08 to 1.15. According to criterion of groove overfilling by metal for steels 08–12Х18Н10Т, requirements were formed towards the groove width of the first stand of longitudinal rolling mill. The groove width must be larger than the sleeve diameter: for 3-roll scheme – at least by 2–3%, for 2-roll scheme – at least by 7.0–7.5%. Potential advantages of 3-roll scheme comparing with 2-roll scheme for rolling of 08–12Х18Н10Т steels were established as follows: lower probability of grooves overfilling by metal of the work-piece, absence of defects (scratches caused by mandrel), on the tube internal surface at minimal ovality level of 1.07, lower level of transverse pipe wall non-uniform thickness.

The author's scheme for constructing a multiresolution analysis on a sphere in R-3 with respect to the spherical coordinates, which was published in 2019, is extended to spheres in R-n (n >= 3). In contrast to other papers, only periodic wavelets on the axis and their tensor products are used. Approximation properties are studied only for the wavelets based on the simplest scalar wavelets of Kotel'nikov-Meyer type with the compact support of their Fourier transforms. The implementation of the idea of a smooth continuation of functions from a sphere to 2 pi-periodic functions in the polar coordinates analytically (without the complicated geometric interpretation made by the author earlier in R-3) turned out to be very simple.

Complementing the authors' earlier joint papers on the application of orthogonal wavelets to represent solutions of Dirichlet problems with the Laplace operator and its powers in a disk and a ring and of interpolating wavelets for the same problem in a disk, we develop a technique of applying periodic interpolating wavelets in a ring for the Dirichlet boundary value problem. The emphasis is not on the exact representation of the solution in the form of (double) series in a wavelet system but on the approximation of solutions with any given accuracy by finite linear combinations of dyadic rational translations of special harmonic polynomials; these combinations are constructed with the use of interpolating wavelets. The obtained approximation formulas are simply calculated, especially if the squared Fourier transform of the Meyer scaling function with the properties described in the paper is explicitly defined in terms of the corresponding elementary functions.

Under contemporary manufacturing conditions, the problem of improvement of the quality of pipes (used as components of the equipment installed at various plants) that can be realized without significant additional investments proves to be quite urgent. A significant influence on the quality of seamless pipes in the process of hot rolling is exerted by the accuracy of adjustment of the technological axis of a continuous rolling mill. The application of contemporary methods of monitoring of the geometric parameters of equipment based on the use of contactless measuring 3D systems under the conditions of pipe-rolling mills (PRM) equipped with fine-quality mills opens new possibilities for the improvement of the quality of pipes. We also present the data on the introduction of measuring 3D systems, which realize an absolutely new approach to the determination and fine adjustment of the technological axes of continuous pipe-rolling mills, in the process of manufacturing.

There are several works where bases of wavelets on the sphere (mainly orthogonal and wavelet-like bases) were constructed. In all such constructions, the authors seek to preserve the most important properties of classical wavelets including constructions on the basis of the lifting-scheme. In the present paper, we propose one more construction of wavelets on the sphere. Although two of three systems of wavelets constructed in this paper are orthogonal, we are more interested in their interpolation properties. Our main idea consists in a special double expansion of the unit sphere in \(\mathbb{R}^3\) such that any continuous function on this sphere defined in spherical coordinates is easily mapped into a \(2\pi\)-periodic function on the plane. After that everything becomes simple, since the classical scheme of the tensor product of one-dimensional bases of functional spaces works to construct bases of spaces of functions of several variables.

July 16, 2018, was the 75th birthday of famous Russian scientist, prominent mathematician, Doctor of Physics and Mathematics, Professor Vitalii Vladimirovich Arestov.

We propose and validate a simple numerical method that finds an approximate solution with any given accuracy to the Dirichlet boundary value problem in a disk for a homogeneous equation with the Laplace operator. There are many known numerical methods that solve this problem, starting with the approximate calculation of the Poisson integral, which gives an exact representation of the solution inside the disk in terms of the given boundary values of the required functions. We employ the idea of approximating a given 2π-periodic boundary function by trigonometric polynomials, since it is easy to extend them to harmonic polynomials inside the disk so that the deviation from the required harmonic function does not exceed the error of approximation of the boundary function. The approximating trigonometric polynomials are constructed by means of an interpolation projection to subspaces of a multiresolution analysis (approximation) with basis 2π-periodic scaling functions (more exactly, their binary rational compressions and shifts). Such functions were constructed by the authors earlier on the basis of Meyer-type wavelets; they are either orthogonal and at the same time interpolating on uniform grids of the corresponding scale or only interpolating. The bounds on the rate of approximation of the solution to the boundary value problem are based on the property ofMeyer wavelets to preserve trigonometric polynomials of certain (large) orders; this property was used for other purposes in the first two papers listed in the references. Since a numerical bound of the approximation error is very important for the practical application of the method, a considerable portion of the paper is devoted to this issue, more exactly, to the explicit calculation of the constants in the order bounds of the error known earlier.

The most productive and modern method is the production of pipes on pipe-rolling machines using continuous rolling mills with a retained mandrel. The production of seamless hot rolled tubes on such aggregates can be accompanied by the appearance of defects on the surface of the pipes, which leads to an increase in the amount of defective products. Taking into account the current state of the art, the process of defect formation in the production of pipes has not been studied sufficiently. Defects occur on the surface of pipes when rolling in the mills of the line of the pipe-rolling unit. Typical types of surface defects are surface cavity, rolling skin on the outer and inner surfaces of the pipes, pigeon hole. In this article, we consider the quality of pipes with an outer diameter in the range of 219÷325 mm with a wall thickness in the range of 8÷24 mm obtained by rolling on aggregates that have continuous rolling mills with 2- and 3-roll stands in the line. The quality is considered from the point of view of the influence of technological factors of rolling (the drawing coefficient, the reduction in the external diameter, the wall thickness, etc.) on the amount of reject for certain types of surface defects. As a universal, previously little studied, the size of which can be judged on the level of defectiveness of technology, we consider the factor of the ratio of deformations in diameter to deformation along the wall thickness in the rolling mill. A number of models have been obtained by statistical methods, guided by which it is possible to predict the expected level of defects in surface defects in the production of seamless pipes, and also to select deformation conditions that minimize the amount of defect.

In this paper a simple iterative synthesis method is presented for the formation of the shape of the reflector surface with a single feed element to produce the desired contour beam. This is the method of the optimal phase synthesis of the appropriate field in the reflector aperture similar to other works. But unlike them, we solve the problem in a very simple way using the properties of complex-valued functions and Fourier transforms and not applying complicated methods of numerical minimization theory.