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6ИСПОЛЬЗОВАНИЕ 6-ТИ МЕРНОГО ПРОСТРАНСТВА ДЛЯ ОПИСАНИЯ ГЕОМЕТРИЧЕСКОЙ
КОНФИГУРАЦИИ ДЕТАЛЕЙ МАШИН
Пруденс Мусоко
Дальневосточный федеральный университет, Владивосток, Россия
USE OF 6-DIMENSIONAL SPACE FOR DESCRIPTION OF GEOMETRIC CONFIGURATION OF
MACHINE PARTS
Prudence Musoko
Far Eastern Federal University, Vladivostok, Russia
Аннотация
При изготовлении деталей результат изготовления не так идеален, как геометрически представлен во время моделирования. Встречаются ошибки формы поверхности и ошибки местоположения, что приводит к неидеальным объектам, которые, если их не учитывать при прохождении процедуры моделирования, приведут к большому количеству брака и, следовательно, к потерям для компании-производителя. Проводятся исследования различных геометрических пространств по представлению неидеальных геометрических частей и инструментов, предлагаемых для решения геометрических задач в 6-мерном пространстве.
Abstract
In the production of parts, the outcome of the manufacture is not as ideal as it is geometrically represented during its modelling. Surface shape errors and location errors errors are encountered and hence resulting in nonideal objects which if not considered whilst going through the modelling procedure will result in a lot of rejects being produced and hence losses to the manufacturing company. An investigation is carried out different geometric spaces on the presentation of non-ideal geometric parts, and the tools proposed for solving geometric problems in 6-dimensional space.
Ключевые слова: геометрическое пространство, геометрические объекты, неидеальные объекты, детали машин, технология машиностроения.
Keywords: geometric space, geometric objects, non-ideal objects, machine parts, engineering technology.
Introduction
Manufacturing technology, grouped into addictive, molding and subtractive, serves one common purpose in all these groups, thus ensuring an intended geometric configuration (within the permissible errors). [6]. A geometric configuration is regarded as a representation of a combinatorial configuration into a geometric space and also identified as a set of points in a geometric space that is used to represent a geometric object [1]. These geometric objects consist of a point, segment, line, curve, polygon, polyhedron of a given dimension [2]. The geometric objects play an important role in the modelling of machine parts using CAD system before the actual manufacturing process.[3] "Geometrical modelling is a general term applied to three-dimensional computer-aided design techniques. There are three main types of geometrical modelling used, namely: line or wireframe modelling, surface modelling and solid modelling. Each have their own applications in the design of engineering components which is dependent on the ability of the method to model certain geometric structures effectively and generate the correct data for analysis" [1]. This can only be achieved provided, the conditions of the geometric space support it. However, with the traditional geometric spaces, there are 2 unsolved problems. The first of these is that modern geometry tools cannot operate with non-ideal shapes and configurations of material objects. The second problem - is the absence of the methods and tools for describing generation circuits geometric objects, from the manufacturing lines and the ending of the
structure which characterizes the relative location of the surfaces
In this article, we will look through the traditional geometrics spaces that are in use to this day, which are the Euclidean 3D geometric space, Lobachevski geometric space and the Hilbert's geometric space, in-line with the problems afore mentioned. We will also look at the six-dimensional space proposed by Leluykhin and Kolesnikov, on how it addresses the 2 problems. Precise presentation of geometric objects helps to improve the efficiency in solving engineering problems, as well as in other fields.
Brief history on the development of different geometric spaces
The use of diagrams in geometry has a long history which dates to around 1700 B.C Babylonian clay tablets found by archeological investigations of ancient Mesopotamian city mounds at the end of the nineteenth century [4, 5]. One of the first geometric books Euclid's Elements written by Euclid around 300 B.C which incorporated most of the preexisting mathematics. Euclid set 5 axioms and gave explanations to prove geometrical facts. In this book, first notions of mathematical proof and the logical development of a subject encountered in Greek mathematics. There was a misunderstanding with the 5th axiom (known as parallel axiom). This led to the discovery of non-Euclidean geometry.
In the 1820s, Gauss, Lobachevsky, and Bolyai individually discovered the non-Euclidean geometries as they were unconvinced with Euclid's fifth postulate.
Bernard Riemann a student of Gauss who further studied Gauss's work on non-Euclidean geometries, also developed his own alternate to the Parallel Postulate. Riemann's Alternate developed the idea of non-exist-ents of parallel lines geometries. The non-Euclidean geometry developed by Riemann could be modeled on a sphere hence referred to as a spherical geometry or elliptical geometry.
David Hilbert's Foundations of Geometry, the first version of which was written in 1899 [8]. Hilbert developed axioms However, it turned out that the goal of finding a finite set of axioms from which all of mathematics could be derived was impossible to achieve. and showed that there was a unique geometry that satisfies his axioms, so that any fact that is true in that geometry is a logical consequence of his axioms. He then developed Hilbert space. The mathematicians such as Archimedes, Descartes, Fermat, Leibniz, Gauss, Loba-chevsky, and Boole played a role in the discovery of non-Euclidean geometry, the arithmetic mathematics, and the formalization of logic; the logic diagrams; up to the recent developments in the theory of reasoning with diagrams[7].
Euclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: The Elements. The parallel axiom occupies a special place in the axiomatics of Euclidean geometry as it is what distinguishes it from non-Euclidean geometry [4, 5].The space of Euclidean geometry is usually described as a set of objects of three kinds, called "points", "lines" and "planes"; the relations between them are incidence, order ("lying between"), congruence (or the concept of a motion), and continuity.
Euclid defines a solid as that which has length, breadth, and depth and the face of a solid is a surface in his first and second definitions. He further describes other geometric solids such as cube, sphere, cylinder, cone and many others [6].
The properties of Euclidean space are described by a finite-dimensional real vector space, which is an inner product space over the real numbers. Euclidean 3-di-mensional space is composed of 3 orthogonal planes known as x, y and z planes. A point P in the 3-dimen-sional space is represented by as (xp, yp, zp) where each of the x, y and z values represent the position of the point in each plane (Fig. 1).
Fig. 1. 3-dimensional Euclidean geometric space a) point P in Euclidean space b) co-planer planes in Euclidean space
In the Euclidean 3-dimensional space geometry for parts modelling in engineering, the CAD system uses various methods such as Exhaustive enumeration, cell decomposition, constructive solid geometry and boundary representation, among others.
Now considering these applications, it show that the Euclidean geometry cannot be used to illustrate the generating loops on how the parts can be produced as the elements applied are already finished parts for example, a sphere is just defined by the center and radius, without any attempt to does not just automatically appear but one has to carry out some processes to obtain it.
Non-Euclidean geometry
Euclidean geometry refers to "flat" geometry, which is also known as parabolic geometry, while non-
Euclidean geometries are referred to as hyperbolic geometry which is the geometry that was established by Lobachevsky-Bolyai-Gauss geometry) and elliptic geometry by Riemannian.
2.2. Lobachevski space
Nikolai Ivanovich Lobachevsky was a Russian mathematician and geometer, who invented hyperbolic geometry, otherwise known as Lobachevskian geometry. Lobachevskii space is a non-Euclidean hyperbolic space whose definition relates to concepts of the geometry of a pseudo-Euclidean space defined by the axioms of Lobachevskii geometry. In Lobachevsky's geometry, instead of the fifth Euclidean postulate, the following axiom is accepted: at least two straight lines lying on the same line and not intersecting it pass through a point not lying on a given line (Fig. 2).
Fig. 2. Lobachevski geometric space a) 5-axiom interpretation in Lobachevsky space b) Lobachevsky plane surface
Lobachevsky geometry can be distinguished from Euclidean geometry by considering the presentation of a triangle (Fig. 3).
EUCLIDEAN HYPEFBDLCC
Fig. 3. Difference in triangle presentation, in Euclidean space and in Lobachevski space
In Lobachevsky's geometric space, straight lines in a plane either intersect, are parallel, or are diverging. In Lobachevsky's geometry, all theorems that can be proved without using the parallelism axiom are preserved.
2.3. Hilbert geometric space
David Hilbert (1852-1943), a German mathematician and physicist initiated many of the most common infinite-dimensional spaces which are today classified as Hilbert spaces , which are a natural extension of the concept of a finite-dimensional vector space with a scalar product (i.e. a finite-dimensional Euclidean space or a finite-dimensional unitary space). In fact, if a scalar product is specified in a finite-dimensional vector space (over the field of real or complex numbers), then property 6), which is called the completeness of the Hilbert space, is automatically satisfied.
David Hilbert was motivated the need to understand sets of functions to create and study infinite-dimensional spaces which is known as functional analysis (study of abstract sets of functions). Hilbert refined axioms (1) and (5) as follows:
- For any two different points, (a) there exists a line containing these two points, and (b) this line is unique (1).
- For any line L and point p not on L, (a) there exists a line through p not meeting L, and (b) this line is unique (5).
Most of the basic properties of Hilbert spaces are relatively straightforward generalizations of the properties of the "flat," finite-dimensional spaces that we have already encountered. We can distinguish Hilbert space by considering the following:
- a point in n-dimensional space n-tuples, (xx, x2, x3, x4,..., x„), whereas a point in Hilbert spaces (infinite dimensional space) a point is represented by an infinite sequence of numbers, xx, x2, x3, x4,...).
Now having these points as shown above, to further describe Hilbert space, we will use the theory of calculation the distance of the points from the origin. Firstly, we consider finite-dimensional spaces:
For an n-dimensional space using the same principle, distance from the origin to point is
= V((*i - 0)2 + (x2 - 0)2 + (X3 - 0)2 + ... (x„ - 0)2)
/„ = V(x12 + X 2
2
2 + X32 + + )
For a point in an infinite-dimensional space the distance from the origin is
= Va*i - 0)2 + (X2 - 0)2 + (X3 - 0)2
+ (X4-0)2+ ...)
¿ra = V(xx2 + x22 + x32 + x42 + ...)
Now using the distance formula above from the origin, the sum underneath the square root sign can either
i. "diverges," that is, the sum becomes larger than any number that we can imagine provided we add enough terms in the series or
ii. "converge"; that means that no matter how many terms in the sum we add together, our answer remains smaller than some fixed number. Thus the "infinite sum" under the square root sign represents some number.
If the sum inside the square root sign converges, then the point (x1,x2,x3,x4,...). belongs to the Hilbert space. If, however, the sum diverges, then we conclude the corresponding infinite sequence of numbers does not represent a point in Hilbert space. For example, a point Pm =(2, 2, 2, 2 ...) does not correspond with Hilbert space
The geometric theories described above have helped in the manufacturing field to:
1. Create, draw parts in various shapes and forms
2. Visualize and adjusting the parts that are to be constructed
However as perfect objects, which does not apply in the actual manufacturing process. In summary, the 3 different approaches do not solve the problems afore mentioned so now we will look at the 6-dimensional space that was proposed
a)
Having defined the main assumptions used in the 6-dimensional space, we can then look at the coordinate
a)
6-dimensional space
The 6-dimensional space was proposed by 2 Russian professionals in the field of mathematics and engineering, Lelyuhin V.E. and Kolesnikova O.V. [3, 4].
The 6-dimensional geometric space is established based on the principle of the degrees of freedom used when machining parts. The degrees of freedom use the theory of transformation, 3 rotational (angular) and 3 translational/ (along the plane) giving a resultant of 6, hence 6 degrees of freedom. Just like the other geometric spaces discussed above, a set of postulates are set and in this case the following are the proposed:
- "Considering a particular finite- dimensional space as the point of origin and the existence of real geometric objects whose properties match the properties of the surrounding material objects in the aggregate state of the rigid body with a metastable structure."
- "To form objects, it is postulated that the geometric configuration of an object is formed by a finite number of complex conjugate or intersecting surfaces n (n> 1) that form a closed space filled with "material" substance with the specified properties." [3]
- Geometric space is defined by 2 types of vectors, linear and angular vectors as shown in the figure below, where the angular vector is the value (angle) between the two linear vectors lying in one plane indicating of the direction of rotation (Fig. 4) [3]
"A
system. The 6-dimensional space consist of 2 orthonormal coordination system with the same origin.
6)
Fig. 4. Two types of vectors: a) linear; b) angular
Fig. 5. Coordination space in 6-dimensional space a) coordination systems consisting of three linear vectors b) coordination system consisting of three angular vectors
The 2 coordination systems are then supper imposed by transferring the origins O' and O" about a point O. this is shown in the Fig. 6.
Fig. 6. Forming the basis of a six-dimensional space
As shown in Fig. 6, base vectors ex, ey, ez and «x, «y, «z known as basic vectors and the angular vectors «z are arranged in mutually perpendicular planes, which in turn are perpendicular to the respective linear vectors ex, ey, ez.
The elements that make up basic geometrical configurations in the 6-dimensional space are points, lines and surfaces and are defined as:
- Point - is the original "brick" to generate other more complex elements. This element has zero dimensions. The position in space of i-th point is determined by only three linear coordinates (Xj, yu Zj).
- Line - an infinite one-dimensional space represented by an inseparable set of points. Characteristically, in the vicinity of any point belonging to the line, are not more than two adjacent points that are not in contact with each other. It may be noted that each line is continuous, smooth and infinite. There are 2 basic
lines which are: straight lines - corresponds with linear vector. The first derivatives in all its points coincide (are constant) and circle (/c) - corresponds with angular vector. Circle is a line at which the first derivative of the increment between any two pairs of equally spaced dots are equal.
- Surface - an endless, smooth two-dimensional space formed by an inseparable plurality of lines in the vicinity of each of which there are not more than two adjacent, but not contiguous lines. Full set of surface S is limited by any various forms acceptable for the existence in the above six-dimensional space. Three surfaces are considered elementary surfaces: plane, cylinder and a sphere
Plane, cylinder, sphere is elementary (Fig. 7), since the fact that only the surface kinematically may be formed using two elementary lines producing - line and the circle.
Fig. 7. Example of the basic object configuration in 6-dimensional space using elementary lines
Conclusion
This article analysis 4 geometric space models mainly focusing on their application in the configuration of geometric objects and their relevancy in part modelling during machinery production. From the history given in the text, it is evident that a lot of attempts have been made in enhancing the geometric field over
the past. However, most of these contributions only address the general geometry without focusing on the application of these spaces in the mechanical engineering field especially on when modelling of parts. Good enough, Lelyuhin VE, Kolesnikova OV, propose a geometric configuration system that addresses 2 main
problems related to the configuration of geometric objects, applying some engineering principles. The author analyses 4 geometric spaces so as to in an attempt to resolve the above-mentioned problems. A conclusion was then made that applying the 6-dimensional geometric space model due to the following listed advantages
- unlike all the other 3 that only rely on linear properties to define an entity, the 6-dimensional geometric model gives a more specific position through the angular coordinates. This is essential in the modelling of the parts as it can assist in determining the fixation of a geometric object.
- The model clearly outlines how the solids are constructed using the points and lines, not as boundary determiners but as generated by the basic components. This makes it possible to present real objects other than the other systems that are only restricted to ideal objects which are not applicable in real engineering situations.
- The solid generation method proposed in allows for the presentation of non-ideal geometric objects as the lines are not bound to be of the same size. For example a cylinder generated from lines lc (circles) can be of different radius giving an imperfect profile thus non-ideal parts profile (real parts).
The work is done under the supervision of Candidate of Technical Sciences, Associate Professor Le-lyukhin Vladimir Egorovich at the FEFU Engineering School.
References
1. Dr. C. Erdem, Computer Aided Technical Drawing, Three Dimensional Design and Solid Modelling, 2002 https://transport.itu.edu.tr/docs/librariespro-vider99/dersnotlari/dersnotlarires112e/not/cadd-6.pdf?sfvrsn=4
2. Lelyukhin V.E., Kolesnikova O.V. Analysis and calculation of dimensional chains based on graphs
of dimensional ties / Bulletin of the Engineering School of the Far Eastern Federal University. 2015. No. 4. P. 3-12.
3. Lelyukhin V.E., Kolesnikova O.V. Geometry of real objects in shipbuilding and ship repair. Vestnik of Astrakhan State Technical University. Series: Marine Engineering and Technologies. 2020;1:31-44. (In Russ.) DOI: 10.24143/2073-1574-2020-1-31-44.
4. V.E. Lelyukhin, O.V. Kolesnikova and O.M. Ponkratova Formation Automation of Geometric Configuration of Real Machine Parts. A.A. Radionov et al. (eds.). Proceedings of the 5th International Conference on Industrial Engineering (ICIE 2019), Lecture Notes in Mechanical Engineering, pp. 111-119.
5. A HISTORICAL INTRODUCTION TO ELEMENTARY GEOMETRY http://mypages.iit.edu/~maslanka/119notes1.pdf
6. V.E. Lelyukhin, O.V. Kolesnikova and T.A. Kuzminova, "Classification of Methods for Forming Surfaces When Machining Parts on Machine Tools," 2019 International Science and Technology Conference "EastConf", Vladivostok, Russia, 2019, pp. 1-5. doi: 10.1109/EastConf.2019.8725366
7. Pisanski, Tomaz & Servatius, Brigitte. (2013). Geometric Configurations. 10.1007/978-0-8176-8364-1_6.
https://www.researchgate.net/publication/315543117_ Geometric_Configurations
8. Franz-Erich Wolter, Martin Reuter Geometric Modelling for Engineering Applications Book August 2009
9. Reverse Engineering: Algebraic Boundary Representations to Constructive Solid Geometry* by S.F. Buchele** and W.A. Ellingson https://www.ash-esi.org/wp-content/uploads/2016/03/Suzanne-Buchele-Reverse-Engineering-Algebraic-Boundary-Representations-to-Constructive-Solid-Geometry.pdf
РАЗРАБОТКА УСТРОЙСТВ ЗАЩИТЫ АВТОНОМНОГО АСИНХРОННОГО ГЕНЕРАТОРА ОТ
ПОВРЕЖДЕНИЙ В ОБМОТКЕ СТАТОРА
Соболь А.Н.
Кандидат технических наук., доцент ФГБОУ ВО Кубанский ГАУ, Краснодар, РФ
Андреева А.А. Студент факультета энергетики ФГБОУ ВО Кубанский ГАУ, Краснодар, РФ
DEVELOPMENT OF DEVICES FOR PROTECTING AN AUTONOMOUS ASYNCHRONOUS GENERATOR FROM DAMAGE IN THE STATOR WINDING
Sobol A.
Candidate of Technical Sciences., Associate Professor FSBEI HE Kuban SA U, Krasnodar, Russian Federation
Andreeva A. student of the Faculty of Energy FSBEI HE Kuban SA U, Krasnodar, Russian Federation
Аннотация
В настоящее время применение автономных асинхронных генераторов в качестве альтернативных источников электроснабжения предприятий, а также для личных нужд имеет достаточно большие перспективы. Их эксплуатация встречает затруднения из-за необходимости выявления повреждений в обмотках
