One-sheet Hyperboloid Research Articles

Description of electromagnetic fields in uniformly accelerated frame. Revision of the radiation problem

It is shown that Maxwell equations for electromagnetic fields generated by a uniformly accelerated charge could be reduced to the Laplace equation in a co-moving frame (represented by the Łobaczewski geometry of the one-sheeted hyperboloid) for a single scalar potential. A full solution of this equation is derived. Then, the famous problem of radiation of a uniformly accelerated particle is revised. Finally, a description of the electromagnetic field on the scri is presented. Both of those approaches produce the same result, which, surprisingly, is slightly different to the well-established Larmor formula for radiation.

Laser Scanner-Based Hyperboloid Cooling Tower Geometry Inspection: Thickness and Deformation Mapping.

Hyperboloid cooling towers are counted among the largest cast-in-place industrial structures. They are an essential element of cooling systems used in many power plants in service today. Their main structural component, a reinforced-concrete shell in the form of a one-sheet hyperboloid with bidirectional curvature continuity, makes them stand out against other towers and poses very high construction and service requirements. The safe service and adequate durability of the hyperboloid structure are guaranteed by the proper geometric parameters of the reinforced-concrete shell and monitoring of their condition over time. This article presents an original concept for employing terrestrial laser scanning to conduct an end-to-end assessment of the geometric condition of a hyperboloid cooling tower as required by industry standards. The novelty of the proposed solution lies in the use of measurements of the interior of the structure to determine the actual thickness of the hyperboloid shell, which is generally disregarded in geometric measurements of such objects. The proposal involves several strategies and procedures for a reliable verification of the structure's verticality, the detection of signs of ovalisation of the shell, the estimation of the parameters of the structure's theoretical model, and the analysis of the distribution of the thickness and geometric imperfections of the reinforced-concrete shell. The idea behind the method for determining the actual thickness of the shell (including its variation due to repairs and reinforcement operations), which is generally disregarded when measuring the geometry of such structures, is to estimate the distance between point clouds of the internal and external surfaces of the structure using the M3C2 algorithm principle. As a particularly dangerous geometric anomaly of hyperboloid cooling towers, shell ovalisation is detected with an innovative analysis of the bimodality of the frequency distribution of radial deviations in horizontal cross-sections. The concept of a complete assessment of the geometry of a hyperboloid cooling tower was devised and validated using three measurement series of a structure that has been continuously in service for fifty years. The results are consistent with data found in design and service documents. We identified a permanent tilt of the structure's axis to the northeast and geometric imperfections of the hyperboloid shell from -0.125 m to +0.136 m. The results also demonstrated no advancing deformation of the hyperboloid shell over a two-year research period, which is vital for its further use.

Existence of Extremals for a Fourier Restriction Inequality on the One-Sheeted Hyperboloid

Existence of Extremals for a Fourier Restriction Inequality on the One-Sheeted Hyperboloid

Scheme Misalignment of Axes of Work Rolls with Support on Rolls with Barrel Surface in the Form of a One-Sheet Hyperboloid

Scheme Misalignment of Axes of Work Rolls with Support on Rolls with Barrel Surface in the Form of a One-Sheet Hyperboloid

Gaussian Curvature Effects on Graphene Quantum Dots

In the last few years, much attention has been paid to the exotic properties that graphene nanostructures exhibit, especially those emerging upon deforming the material. Here we present a study of the mechanical and electronic properties of bent hexagonal graphene quantum dots employing density functional theory. We explore three different kinds of surfaces with Gaussian curvature exhibiting different shapes-spherical, cylindrical, and one-sheet hyperboloid-used to bend the material, and several boundary conditions regarding what atoms are forced to lay on the chosen surface. In each case, we study the curvature energy and two quantum regeneration times (classic and revival) for different values of the curvature radius. A strong correlation between Gaussian curvature and these regeneration times is found, and a special divergence is observed for the revival time for the hyperboloid case, probably related to the pseudo-magnetic field generated by this curvature being capable of causing a phase transition.

The influence of shaping on the behavior of a mesh shell shaped as a one-sheeted hyperboloid

Introduction. The stress-strain state (SSS) of mesh shells that have the shape of a one-sheeted hyperboloid, consisting of one, two and three sections and subjected to vertical and wind loads coupled with their own weight is studied. The feature of these shells is the use of the Fibonacci number in their geometry, or in the ratio of their diameters and heights.
 Materials and methods. The position of rectilinear generatrices of shell depends on the angle of inclination of generatrices, which is determined by the mutual rotation of the upper and lower bases, and it determines various shapes of the structure. The stress-stain state of shells is found numerically using the LIRA–SAPR software package. The largest forces arising in the generatrices and the largest horizontal displacements of the upper edge of the shell, caused by wind loading, are compared.
 Results. Equations are obtained to identify the position of the generatrix and the size of the real semi-axis of the hyperboloid. Analytical models of shells with different inclinations of generatrices are provided. As a result of the data analysis, the shapes of shells, having different heights and numbers of storeys, are proposed to ensure their optimal strength and stiffness. If the structure is subjected to vertical loads and self-weight, the mutual rotation of upper and lower bases does not affect the SSS of e shells. When the wind load is applied, the situation is completely different. Hence, for single-section shells of medium height and high shells with optimal parameters of internal forces and displacements, the size of the real semi-axis coincides with the size corresponding to the “golden section”. At the same time, shells with a small angle of mutual rotation of the bases barely resist lateral loading. In two- and three-sectional shells, the value of the real half-axis is slightly greater.
 Conclusions. The use of mesh shells, having the shape of a one-sheeted hyperboloid that has the “golden section” ratio, allows obtaining more expressive structural solutions. Analytical models of such structures are presented.

Using the Theory of Functional Connections to Solve Boundary Value Geodesic Problems

This study provides a least-squares-based numerical approach to estimate the boundary value geodesic trajectory and associated parametric velocity on curved surfaces. The approach is based on the Theory of Functional Connections, an analytical framework to perform functional interpolation. Numerical examples are provided for a set of two-dimensional quadrics, including ellipsoid, elliptic hyperboloid, elliptic paraboloid, hyperbolic paraboloid, torus, one-sheeted hyperboloid, Moëbius strips, as well as on a generic surface. The estimated geodesic solutions for the tested surfaces are obtained with residuals at the machine-error level. In principle, the proposed approach can be applied to solve boundary value problems in more complex scenarios, such as on Riemannian manifolds.

Obtaining one-sheeted revolution hyperboloid surface by cutting

There are situations when obtaining surfaces that correspond to one-sheeted revolution hyperboloid surface is formulated in industrial practice. Solving such a problem may require the use of expensive procedures and equipment. The analysis of some solutions from the specialized literature led to the acceptance of some possibilities to obtain certain surfaces specific to the one- sheeted revolution hyperboloids by turning and milling. A technological solution that involves using a milling device adaptable to a universal lathe has been proposed.

Точное двумерное решение для токового сжатия тонкой осесимметричной оболочки и формирование перетяжки в X-пинче

Exact two-dimensional solutions are constructed that describe the dynamics of magnetic compression of a hollow shell, which is a one-sheeted hyperboloid of revolution. The solutions are applicable for interpreting the results of experiments on the formation of necks in X-pinches. In particular, they make it possible to relate the main parameters of the problem: the axial scale of the neck, the time of its formation, and the geometric characteristics of the system.

Exact two-dimensional solution for magnetic compression of a thin axisymmetric shell and neck formation in an X-pinch

Exact two-dimensional solutions are constructed that describe the dynamics of magnetic compression of a hollow shell, which is a one-sheeted hyperboloid of revolution. The solutions are applicable for interpreting the results of experiments on the formation of necks in X-pinches. In particular, they make it possible to relate the main parameters of the problem: the axial scale of the neck, the time of its formation, and the geometric characteristics of the system. Keywords: X-pinch, neck formation, thin shell model.

Controlled Change in the Dimensions of an Axisymmetric Spacecraft Descending in the Atmosphere of Mars

In the presented work, a controlled change by dimensions of a spacecraft descending in the atmosphere of Mars is considered. The aim of the work is to obtain a method for calculating the mass and mass-geometric characteristics of a spacecraft when changing its dimensions, which provides angular velocity passive control during the descent of this spacecraft in a low-density atmosphere. In the process of solving this problem, the geometric and mass-geometric characteristics of the descent spacecraft (volume, cross-sectional area, moments of inertia) were calculated. It is assumed that the outer shape of the spacecraft posterior to the incoming flow is a one-sheet rotational hyperboloid, which changes its dimensions during the spacecraft descent in the low-density atmosphere of Mars. As a result of solving the nonlinear programming problem, the minimum and maximum values of the main axial moments of inertia are obtained, which able to spin the spacecraft relative to the longitudinal axis of symmetry. The initial data for solving the nonlinear programming problem are the minimum volume and the maximum cross-sectional area of the hyperboloid, calculated according to the specified intervals of the variable controlling the dimensions of this surface. The method for calculating the mass and mass-geometric characteristics of a spacecraft when changing its dimensions ispresented, which makes it possible to control the magnitude of the angular velocity of a symmetric spacecraft in the low-density atmosphere of Mars without the use of onboard jet engines. In particular, it is shown in the work that as the height of the hyperboloid increases, the moment of inertia about the spacecraft longitudinal axis of symmetry decreases, accompanied by an increase in the moments of inertia about the transverse axes of symmetry. It can be shown that in this case there is an increase in the angular velocity of the spacecraft about the longitudinal axis, which makes it possible to achieve a stable orientation of the spacecraft upon entering the atmosphere. However, a more detailed study of the dynamics ofthe spacecraft relative motionwith a changeable shape in the atmosphere is beyond the scope of this work, but it can be presented in further publications.

Using the Properties of a One-Sheet Hyperboloid in Bearings with Cylindrical Rollers

The development of a new bearing with cylindrical rollers and raceways corresponding to the surface of a one-sheet hyperboloid is considered. The bearing is designed to withstand significant radial and axial loads and is intended to support the rollers in strip-rolling mills.

Chasles’ mystic tetrahedron

The results of the experimental study and testing of the theorem by M. Chasles are given. The theorem shows the peculiarities of the intersection of an arbitrary tetrahedron with an arbitrary quadratic surface (second-order surface). The multiple variants of the intersection stipulated in the theorem have been considered. It has been experimentally proven that for the majority of these variants the four intersecting lines constructed according to the algorithm of this theorem belong to the surface of a single one-sheeted hyperboloid. For two of the variants, the results that differ from the theorem have been received. The experiments implied building and studying 3D computer models obtained using the AutoCAD and SolidWorks packages. All the variants of the quadratic surface at different mutual position with respect to the tetrahedron have been considered. The methods of holding the experiments have been considered in detail. The proof of the theorem for one of its variants has been obtained and presented.

Roll Assembly of a Sheet-Rolling Stand Based on the Use of the Surface Properties of a One-Sheet Hyperboloid

We summarize the results of our research of a fundamentally new layout of rolls of a four-roll sheet-rolling stand. As the supports of rolls in this stand, we use new rolling bearings with cylindrical rollers. The analysis of the operation of this roll assembly is based on the properties of the surface of a one-sheet hyperboloid in the roll assemblies of four-roll sheet-rolling stands. We essentially use one property of the surface of one-sheet hyperboloid, namely, the presence of two families of rectilinear generatrices.

Refutation of a claim made by Fejes Tóth on the accuracy of surface meshes

AbstractFejes Tóth [3] studied approximations of smooth surfaces in three-space by piecewise flat triangular meshes with a given number of vertices on the surface that are optimal with respect to Hausdorff distance. He proves that this Hausdorff distance decreases inversely proportional with the number of vertices of the approximating mesh if the surface is convex. He also claims that this Hausdorff distance is inversely proportional to the square of the number of vertices for a specific non-convex surface, namely a one-sheeted hyperboloid of revolution bounded by two congruent circles. We refute this claim, and show that the asymptotic behavior of the Hausdorff distance is linear, that is the same as for convex surfaces.

MATHEMATIC MODEL OF AND METHOD FOR SOLVING THE DIRICHLET HEAT-EXCHANGE PROBLEM FOR ONE-SHEET ROTARY HYPERBOLOID

It is the first generalized 3D mathematical model developed for calculating temperature fields in the thin-wall one-sheet rotary hyperboloid, which rotates with constant angular velocity around the axis OZ, ; the model was created with the help of known equations of generating lines in cylindrical coordinate system with taking into account finite velocity of heat conductivity and in the form of the Dirichlet boundary problem for the hyperbolic equation of heat conduction under condition that heat-conduction properties of the body were constant, and no internal sources of the heat were available. At initial moment of time, the body temperature was constant; values of temperature on outside surfaces of the body were known and presented continuous function of coordinate.The hyperbolic heat-conductivity equation was derived from the generalized energy transfer equation for the moving element of continuous medium with taking into account finiteness of the heat conductivity velocity.In order to solve the boundary problem, the desired temperature field was represented as a complex Fourier series. The obtained boundary problems for the Fourier coefficients were found with the help of Laplace integral transformations and the new integral transformation for two-dimensional finite space. Intrinsic values and intrinsic functions for the integral transformation kernel were found by method of finite element and the Galerkin methods. Besides, the domain was divided into simplex element.As a result, the temperature field in the thin-wall one-sheet rotary hyperboloid was found in the form of convergent series in Fourier functions.

Structures and mechanical behaviors of soft nanotubes confining adhesive single or multiple elastic nanoparticles

The mechanical interaction between soft membrane nanotubes and confined nanoparticles plays essential roles in numerous cellular activities such as inter- and intracellular transport and signaling, membrane tubulation and pearling transformation driven by adsorbed nanocomponents, and cooperative cell uptake. The aim of this work is to theoretically analyze how structures and mechanical behaviors of particle-in-tube complexes are regulated by the particle elasticity, covering the cases of a single particle and multiple particles of spatial periodicity. Depending on the interplay of membrane tension, adhesion energy, nanoparticle size, and the rigidity ratio between the nanoparticles and soft tubes, characteristic interaction states in terms of wrapping degrees of the confined particles have been identified, and the corresponding wrapping phase diagrams are determined. As the particles become softer, the adhesion energy required for the partial-wrapping state becomes lower but that for full-wrapping is larger, and a rich variety of characteristic system configurations are observed including the bulged or undulated cylinders, necklaces of spheres or prolates, and sausage-string shapes. Moreover, two fundamental modes of interaction between multiple nanoparticles and the soft nanotube are revealed, the cooperative wrapping and individual wrapping. Based on perturbation analysis at small tube deformation and configurational assumptions of catenoid, torus, and one-sheeted hyperboloid at more general tube deformation, we have also obtained analytical solutions on the wrapping of rigid spherical nanoparticles in soft tubes. Comparison between wrapping of elastic particles by a flat membrane and by a soft membrane tube is discussed. This study enriches our knowledge on the diversity of membrane tubules in morphologies and structures.

Geometric modelling and materially nonlinear numerical analysis of shells in the shape of one-sheet hyperboloid of revolution

Aims of research. A surface of revolution is generated by rotation of a plane curve z = f(x) about an axis Oz called the axis of rotation. This paper provides information on hyperboloids of revolution surfaces and their classification. Their geometric modeling, linear and materially nonlinear analysis are worked out. Methods. Hyperboloids of revolution middle surface is plotted using the software MathCAD. The linear and materially nonlinear numerical analyses of thin shells of the shape of an hyperboloid of revolution surfaces on stress-strain state is given in this paper, using the finite elements method in a computer software R-FEM, the material which we use in our model is concrete with isotopic nonlinear 2D/3D stress-strain curve for materially nonlinear analysis and linear stress-strain curve for linear analyses. Comparison is done with the result of the finite elements linear analysis of their strain-stress results. Results. That displacements in the investigated shells subject to self-weight, wind load with materially nonlinear analysis are bigger than which done by linear analysis, in the other side the displacements is similarity subjected to free vibration load case. Based on these results, conclusions are made for the whole paper.

Hyperboloid offset surface in the architecture and construction industry

Abstract In this paper the issue of approximation of the hyperboloid offset surface off(S(t, v); d) at distance d by the hyperboloid surface S 1(ϕ, v) is considered. The problem of determining various surfaces approximating the hyperboloid offset surface off (S(t, v); d) is important due to the applications of the hyperboloid as a mathematical model for miscellaneous objects in the architecture and construction industry. The paper presents the method of determining the angles and coordinates of points of various surfaces approximating the hyperboloid of revolution. A two-sheet hyperboloid offset surface can be used for modelling double-layer domes. A one-sheet hyperboloid offset surface was used to model the reinforced structure of the cooling tower.

Frequency analysis of the response of a one-sheet hyperboloid to periodic longitudinal loading

The frequency analysis of the steady state forced vibrations of a thin shell of negative Gaussian curvature is performed. The one-sheet hyperboloid shell is subjected to the action of the periodic longitudinal load. The finite element model for the shell is built using flat quadrilateral shell elements with six degrees of freedom per node. The modal analysis of the shell is performed by the Lanczos method using linear formulation of the problems. Eigenfrequencies and eigenmodes of the shell are also defined using nonlinear formulation of the problems that takes into account the effect of a longitudinal under critical static force preliminary applied to the top edge of the shell. For this purpose the nonlinear statics problems and the natural vibration problems are sequentially solved by the modified Newton-Raphson method and Lanczos method respectively. A modal analysis showes that dependence of the shell eigenfrequencies on the static longitudinal loading is nonlinear. The steady state response of the shell to the periodic longitudinal loading is obtained both by means of direct and modal frequency analysis. It turned out that sensitivity of the hyperboloid of one sheet to the periodic longitudinal loading considerably increases when the static under critical longitudinal loading is taken into account. The responses of the shell are presented in the form of amplitude frequency dependences of the finite element model nodes.