Continuos Parameterization of the Median Surface of an Ellipsoidal Shell and Its Geometric Parameters

When analyzing the stress-strain state of thin-walled structural elements that have the shape of an ellipsoid, it becomes necessary to calculate the geometric characteristics of the ellipsoidal surface. When using the canonical ellipsoid equation, regions of uncertainty appear in the Cartesian coordinate system at the intersection points of the ellipsoid surface with the horizontal coordinate plane. To exclude these areas of uncertainty, we propose an expression of the radius vector of an ellipsoidal surface whose projections are functions of two parametric representations in mutually perpendicular planes. One of the planes is the vertical plane XOZ, and the other plane is the plane perpendicular to the axis O at the point with the x coordinate. The parameter T of the ellipse obtained from the intersection of the ellipsoid with the XOZ plane was chosen as the argument of the first parametric function. The argument of the second parametric function t is the parameter of an ellipse formed as a result of the intersection of an ellipsoidal surface with a plane perpendicular to the abscissa axis at a distance of x from the origin. The proposed representation of the ellipsoidal surface allowed us to exclude uncertainties at the intersection points of the ellipsoid with the HOWE coordinate plane. By differentiating the proposed radius-vector expression at an arbitrary point on an ellipsoidal surface, we obtain relations for the basis vectors of an arbitrary point and their derivatives represented by components in the same local basis. These relations are necessary for the development of algorithms for numerical analysis of deformation processes of engineering structures that have ellipsoidal surfaces.

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Study of modern software capabilities for complex shell analysis

Structural Mechanics of Engineering Constructions and Buildings ◽

10.22363/1815-5235-2020-16-1-45-53 ◽

2020 ◽

Vol 16 (1) ◽

pp. 45-53 ◽

Cited By ~ 1

Author(s):

Marina I. Rynkovskaya ◽

Timur Elberdov ◽

Enes Sert ◽

Andreas Öchsner

Keyword(s):

Fourier Series ◽

Series Expansion ◽

Complex Geometry ◽

Analytical Calculation ◽

Shell Structures ◽

Fourier Series Expansion ◽

Median Surface ◽

Stress Strain ◽

Engineering Structures ◽

Software Packages

Relevance. In design and calculation of civil engineering structures, several standard commercial software packages, which are successfully applied to solve everyday engineering problems, are traditionally used. However, when it is necessary to design the models of complex shape shell structures with defining surfaces based on parametric equations, such programs often have certain drawbacks. The aim of the work - analysis of existing types of commercial computational software packages in order to check which allow to design finite element models for shell structures with median surfaces of complex geometry given by parametric equations. Methods. The analysis of commercial computational software packages is carried out by studying the software manuals, and by building and calculating a model in the shape of a right helicoid as a test example. To evaluate the results of the stress-strain state of a shell with a middle surface in the form of a right helicoid, an analytical calculation method based on the Reissner’s equations and Fourier series expansion is used. Results. A review of modern commercial computational software packages as applied to models defined by parametric equations is carried out. A model for a shell structure with a median surface in the form of right helicoid is built. The numerical results of stress-strain behavior of the right helicoid are obtained and analyzed in comparison with the analytical solutions obtained using the Reissner’s equations with Fourier series expansion. The pros and cons of several popular means of software are presented.

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XXIII. Theorems on the attraction of ellipsoids for certain laws of force other than the inverse square

Philosophical Transactions of the Royal Society of London. (A.) ◽

10.1098/rsta.1895.0023 ◽

1895 ◽

Vol 186 ◽

pp. 897-950 ◽

Cited By ~ 9

Keyword(s):

Radius Vector ◽

Internal Point ◽

Ellipsoidal Shell ◽

Direction Cosines ◽

The Law

1. To find the potential at an internal point P of a thin ellipsoidal shell bounded by similar concentric ellipsoids, the law of force being the inverse k th power of the distance. Let ( x, y, z ) be the coordinates of P; ( l, m, n ) the direction cosines of a radius vector drawn from P to any point Q on the surface; and let PQ = r . Let ( a, b, c ) be the semi-axes of the ellipsoids, and ( α, β, γ ) their squared reciprocals. Then from the equation of the ellipsoid ( αl 2 + βm 2 + γn 2 ) r 2 + 2 ( αlx + βmy + γnz )r - (1 - αx 2 - βy 2 - γz 2 ) = 0.

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On One Property of a Circle on the Coordinate Plane

Geometry & Graphics ◽

10.12737/article_5953f2af770c35.65774157 ◽

2017 ◽

Vol 5 (2) ◽

pp. 13-24

Author(s):

Графский ◽

O. Grafskiy ◽

Пономарчук ◽

Yu. Ponomarchuk

Keyword(s):

Euclidean Geometry ◽

Coordinate Plane ◽

Non Euclidean Geometry ◽

Geometrical Form ◽

Constant Distance ◽

Intersection Points ◽

Definition Of ◽

Set Of Points ◽

First Time

Descartes’ and Fermat's method allowed to define many geometrical forms, including circles, on the coordinate plane by means of the arithmetic equations and to make necessary analytical operations in order to solve many problems of theoretical and applied research in various scientific areas, for example. However, the equations of a circle and other conics in the majority of research topics are used in the subsequent analysis of applied problems, or for analytical confirmation of constructive solutions in geometrical research, according to Russian geometrician G. Monge and others, including. It is natural to consider a circle as a locus of points, equidistant from a given point — a center of the circle, with a constant distance R. There is another definition of a circle: a set of points from which a given segment is visible under constant directed angle. Besides, a circle is accepted to model the Euclid plane in the known scheme of non-Euclidean geometry of Cayley-Klein, it is the absolute which was given by A. Cayley for the first time in his memoirs. It is possible to list various applications of this geometrical form, especially for harmonism definition of the corresponding points, where the diametral opposite points of a circle are accepted as basic, and also for construction of involutive compliances. The construction of tangents to a circle can be considered as a classical example. Their constructive definition is simple, but also constructions on the basis of known projective geometry postulates are possible (a hexagon when modeling a series of the second order, Pascal's lines). These postulates can be applied to construction of tangents to a circle (to an ellipse and hyperboles to determination of imaginary points of intersection of a circle and a line. This paper considers the construction of tangents to a circle without the use of arches of auxiliary circles, which was applied in order to determine the imaginary points of intersection of a circle and a line (an axis of coordinates). Besides, various dependences of parameter p2, which is equal to the product of the values of the intersection points’ coordinates of a circle and coordinate axes, are analytically determined.

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A High Performance Energy Analyzer for Use in Electron Scanning Microscopy

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100061975 ◽

1969 ◽

Vol 27 ◽

pp. 14-15

Author(s):

A. V. Crewe ◽

M. Isaacson ◽

D. Johnson

Keyword(s):

High Performance ◽

Vertical Plane ◽

Median Plane ◽

Electron Gun ◽

Uniform Field ◽

Loss Mechanism ◽

Electron Scanning Microscopy ◽

Sector Type ◽

Double Focusing ◽

Electron Energy Loss

A double focusing magnetic spectrometer has been constructed for use with a field emission electron gun scanning microscope in order to study the electron energy loss mechanism in thin specimens. It is of the uniform field sector type with curved pole pieces. The shape of the pole pieces is determined by requiring that all particles be focused to a point at the image slit (point 1). The resultant shape gives perfect focusing in the median plane (Fig. 1) and first order focusing in the vertical plane (Fig. 2).

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The time course of calcium deposition in shells of the barnacle (Balanus amphititre amphititre) during cyprid-juvenile metamorphosis

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100168621 ◽

1994 ◽

Vol 52 ◽

pp. 178-179

Author(s):

Nancy R. Wallace ◽

Craig C. Freudenrich ◽

Karl Wilbur ◽

Peter Ingram ◽

Ann LeFurgey

Keyword(s):

Electron Microscopy ◽

Scanning Electron Microscopy ◽

Time Course ◽

Vertical Plane ◽

Mantle Cavity ◽

Qualitative Assessment ◽

Calcium Deposition ◽

Free Swimming ◽

Freeze Dried ◽

Scanning Electron

The morphology of balanomorph barnacles during metamorphosis from the cyprid larval stage to the juvenile has been examined by light microscopy and scanning electron microscopy (SEM). The free-swimming cyprid attaches to a substrate, rotates 90° in the vertical plane, molts, and assumes the adult shape. The resulting metamorph is clad in soft cuticle and has an adult-like appearance with a mantle cavity, thorax with cirri, and incipient shell plates. At some time during the development from cyprid to juvenile, the barnacle begins to mineralize its shell, but it is not known whether calcification occurs before, during, or after ecdysis. To examine this issue, electron probe x-ray microanalysis (EPXMA) was used to detect calcium in cyprids and juveniles at various times during metamorphosis.Laboratory-raised, free-swimming cyprid larvae were allowed to settle on plastic coverslips in culture dishes of seawater. The cyprids were observed with a dissecting microscope, cryopreserved in liquid nitrogen-cooled liquid propane at various times (0-24 h) during metamorphosis, freeze dried, rotary carbon-coated, and examined with scanning electron microscopy (SEM). EPXMA dot maps were obtained in parallel for qualitative assessment of calcium and other elements in the carapace, wall, and opercular plates.

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Dynamic visual acuity when the required pursuit movement of the eye is in a vertical plane. (Joint Proj. Rep. No. 2).

PsycEXTRA Dataset ◽

10.1037/e419912004-001 ◽

1953 ◽

Cited By ~ 2

Author(s):

James W. Miller ◽

Elek Ludvigh

Keyword(s):

Visual Acuity ◽

Vertical Plane ◽

Dynamic Visual Acuity ◽

Pursuit Movement ◽

Plane Joint

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Lighting of Engineering Structures and Industrial Facilities New Aspects of the Topic

Light & Engineering ◽

10.33383/2017-071 ◽

2018 ◽

pp. 29-36

Author(s):

Nikolai I. Shepetkov ◽

George N. Cherkasov ◽

Vladimir A. Novikov

Keyword(s):

Fundamental Problem ◽

Lighting Design ◽

Aesthetic Value ◽

Engineering Structures ◽

Industrial Facilities ◽

Artificial Lighting ◽

Industrial Structures ◽

Design Solutions ◽

Architectural Lighting ◽

The Aesthetic

This paper considers the fundamental problem of artificial lighting in various types and scales of industrial facilities, focusing on exterior lighting design solutions. There is a lack of interest from investors, customers and society in highquality lighting design for industrial facilities in Russia, which in many cities are very imaginative structures, practically unused in the evening. Architectural lighting of various types of installations is illustrated with photographs. The purpose of the article is to draw attention to the aesthetic value of industrial structures, provided not only by the architectural, but also by a welldesigned lighting solution.

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Computing Model for Vibratory Digging-Out of Sugar Beet Roots

Agricultural science and practice ◽

10.15407/agrisp1.02.052 ◽

2014 ◽

Vol 1 (2) ◽

pp. 52-60

Author(s):

V. Bulgakov ◽

V. Adamchuk ◽

H. Kaletnyk

Keyword(s):

Differential Equations ◽

Sugar Beet ◽

Vertical Plane ◽

Forward Motion ◽

Admissible Velocity ◽

Beet Root ◽

Optimum Speed ◽

Analytical Research ◽

Sugar Beet Root ◽

The Given

The new design mathematical model of the sugar beet roots vibration digging-out process with the plowshare vibration digging working part has been created. In this case the sugar beet root is simulated as a solid body , while the plowshare vibration digging working part accomplishes fl uctuations in the longitudinal - vertical plane with the given amplitude and frequency in the process of work . The aim of the current research has been to determine the dependences between the design and kinematic parameters of the sugar beet roots vibra- tion digging-out technological process from soil , which provide the ir non-damage. Methods . For the aim ac- complishment, the methods of design mathematical models constructing based on the classical laws of me- chanics are applied. The solution of the obtained differential equations is accomplished with the PC involve- ment. Results . The differential equations of the sugar beet root’s motion in course of the vibration digging-out have been comprised . They allow to determine the admissible velocity of the vibration digging working part’s forward motion depending on the angular parameters of the latter. In the result of the computational simula- tion i.e., the solution of the obtained analytical dependence by PC, the graphic dependences of the admissible velocity of plowshare v ibration digging working part’s forward motion providing the extraction of the sugar beet root from soil without the breaking-off of its tail section have been determined. Conclusions . Due to the performed analytical research , it has been established that γ = 13 ... 16 ° , β = 20 ... 30 ° should be considered as the most reasonable values of γ and β angles of the vibration digging working part providing both its forward motion optimum speed and sugar beet root digging-out from the soil without damage . On the ground of the data obtained from the analytical rese arch, the new vibration digging working parts for the sugar beet roots have been designed; also the patents of Ukraine for the inventions have been obtained for them.

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