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106-DIMENSIONAL SPACE GEOMETRIC TOOLS APPLICATION IN ENGINEERING PRINCIPLES
Prudence Musoko
Far Eastern Federal University, Vladivostok, Russia
ABSTRACT
The 6-dimensional geometric space provides tools and methods that can help eliminate 2 main problems in the representation of geometric objects in the machine parts manufacturing industry. The problems are, firstly that modern geometry tools cannot operate with non-ideal shapes and configurations of material objects and secondly, there aren't any 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. This article will look at the application of the tools and methods for the 6-dimensional space in solving these 2 problems.
Keywords: Geometric configuration, Dimensional chain analysis, non-ideal geometric objects
Introduction
Throughout the use of geometry configurations in the representation of objects, attention has been given to the presentation of ideal geometric parts with little or no attention given to non-ideal parts. However, in real practical examples in the engineering field, errors are unavoidable resulting in non-ideal geometric objects [1, 2, 3]. To incorporate these objects, Lelyukhin V.E. and Kolesnikova O.V. propose a 6-dimensional space model to analyse non-ideal object (real parts) during modelling [3, 4]. In their model, they introduce some
concepts to use when handling engineering problems and this article serves to discuss these principles.
Definition of a 6-dimensional space
The 6-dimensional space is defined by 3 angular vectors and 3 linear vectors. These vectors make up 2 coordinate systems that are non-coplanar and orthogonal to each other. The angular vector is defined as the angular value (angle) between the two linear vectors lying in one plane with an indication of the start and end of rotation. Fig. 1 below shows the 6- dimensional space [5, 6].
Fig. 1. Forming the basis of six-dimensional space
Shown in Fig. 1 basis vectors called basis vectors, the ocx, oCy, ocz vectors are angular arranged in mutually perpendicular planes, which in turn are perpendicular to the respective ex, ey, ez linear vectors.
Presentation of coordinates
The vectors are represented in form of a table as shown in figure 2 below
Fig. 2. Presentation of coordinates in 6-dimensional space used is binary, 1 and 0
0- Represents the absence of restriction
1- Represents the lack of movement Formation of geometric configurations of parts Formation of geometric configurations of parts in
the 6-dimension depend on points, lines and surfaces as the elementary objects. These elementary objects are defined as:
Point - is the original " unit block" 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 (xh yh zt) [4].
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 and circles which correspond to angular vector [4, 5]. A surface is defined as and the elementary surfaces used to define the parts are cylinder, plane and sphere.
The plane is formed by a series of straight lines which are not limited in dimension or position. The lines can be of different size and can lie in any position, allowing the presentation of a non-ideal surface (plane).
From figure 2 it can be observed that the representation of the plane is flexible enough to represent the non-ideal geometric object by allowing deviation from the usual straight edges of the planes and the flatness of the plane.
Fig. 3. Non-ideal plane constructed from a series of straight lines
The lines that form the plane are not bound to be equal in length l hence possibility to present rough plane edges. These deviations are common during preparation of parts and by use of this method, they can be represented.
A cylinder can be represented by a series of circular lines (lc) or li [4]. The 2 options in which a cylinder can be created are represented in the images figure 4 below:
Fig. 4. Construction of cylindrical surface using lines lc and ll
The center line marks the center points of these lc constructing the cylindrical surface. However the radius of the circles can be different in the case of an irregular surface (in non-ideal objects). These surfaces are then used to formulate different machine part structures. A regular cube is constructed by a set of 6 planes, each plane representing a surface on the cube. This fea-
ture makes it possible to make clearly establish the generation of this cube. A cubic structure can be obtained from 6 planes attached to each other. Since these plane locations are fully defined by the 6-dimensional coordinate system, it can be possible to deduce how each surface (plane) will influence the whole structure. The location of a single plane can be represented as in a 6-dimensional space as shown in figure 5 below:
X y
0 0 1
1 1 0
aj
Fig. 5. Presentation of a plane on 6 geometric space
The plane in the figure above is defined based on its ability to freely move (degrees of freedom). The plan is free to move (extend) linearly in the x and y direc-
tions and free to rotate about the z axis therefore represented by 0. To look at how attaching another plane can be of influence, we will define 3 planes we will use as
shown in figure 6 below:
Fig. 5. Three planes in 6-dimensional space
Now using the definition of planes as shown in figure 6, we will look at how each plane can influence the other. Suppose we have the first plane alone ni, the freedom of movement is described above. Now attaching 112 will result in restriction of the linear movement
in x as well as the rotational movement in z which were initially free in ni before the attachment. This can be presented in by coordinates as follow:
The attachment of the plane n2 shows that the degrees of freedom have been reduced linearly in x direction and the rotation in the z axis as well is now restricted. Likewise, attaching n3 to ni will result in one free directional coordinate thus the unconfined y-axis linear movement. Attaching all the 3 planes will result in a complete restriction in all directions. This mechanism can be of used to define a geometric configuration based on the confinements required. Dimensional chain diagrams Another tool provided by the 6-dimensional space model is the graphical presentation of the dimensional chain analysis. In general, a linear dimensional chain is a set of independent parallel dimensions which continue each other to create a geometrically closed circuit [7]. Dimensional chain analysis is then defined as the analysis of these linear dimensional chains. There are different methods used for this analysis and a brief description of 3 of them is given below: Worst Case Method (WC). In worst case analysis, the tolerance is calculated by adding the component tolerances in linear form. The maximum and minimum limits of the tolerances are used to compute the resultant. This method does not address the possible other variations that can be available and the equation below is used
Where R- resultant, n- number of components, i-individual/ independent component
The resultant is calculated along a certain direction.^]
RSS Method.
Root Sum Square (RSS) method is a statistical tolerance analysis that considers each part's tolerance as a statistical distribution. The statistical model limits the possibility of the occurrence of defectives, hence becomes not convincing to the designers as it is important to analyze these possibilities before manufacture [8]. The equation used is as follows:
T.
final
¿=i
R
=
Where Tt is the standard deviation of the i'th part, And, n is the number of parts in the stack, And, Tfinal is the standard deviation of the stack. All the above methods, serve the purpose of finding, calculating the closing loop. The closing loop however can have different iterations and the number of iterations is very important. From a study [1], "The GD&T values in the dimension chains are assigned by designers on a trial and error basis, in the light of their previous experience (in design, manufacturing systems capabilities and assembly tolerances) and the tolerance stack up of the standard parts." [1]. This show that there is a possibility of assigning the dimensions to the closing loop and have a completely different closing loop,
and that is what Lelyukhin V.E. and Kolesnikova O.V. addressed by introducing a system of using graphs to analyze the dimension. This would help to reduce the deviation resulting from the whole structure and improve efficiency by selection of the most optimum option. To illustrate the principle of dimensional chain diagrams in 6-dimensional space, figure 6 below will be used as an example.
4+
From the figure 7 above, the dimensional graphs are drawn in the x, y and z direction. From the z-direc-tion graph, the vertices are only 2 with one edge. This scenario presents no other variations in terms of tolerance. The y-directional graph has 3 vertices and 2
The method utilizes the representation of a graph by a pair of sets (V, E). The set V = {v1,v2, ...,vn] which consists of a finite set of vertices will represent the surfaces of the geometric configuration, and the set E = {e1,e2, ...,em} which consists a finite set of edges which characterize the connection between the pairs of vertices, represent the dimensions between the surfaces [5, 6].
tree dia-
©
edges. The 3rd edge is the closing-loop dimension. This scenario can result in 3 different variations as in the figure 8 below:
Fig. 6. Illustration of dimensional chain (geometric configuration with numbered (labeled) surface
From figure 6, the surfaces are labeled in circles and from that diagram, the following dimensional grams are constructed as in figure 7 below:
Fig. 7. Dimensional tree diagrams obtained from the geometric configuration in figure 6
Fig. 8. Possible dimensional tree diagram variations along the y-axis
The designing engineer can then analyze these three variations and select the variation that has the least error. The using the 6-coordinate system, an algorithm can be used to decide also on the variation giving options for the processing sequence of the surfaces.
Now considering the x-direction, there are 4 vertices and 3 edges. The 4th edge, which is the closing edge can be calculated from the other 3 edges. The choice on which of the dimensions to assign and which to deduce from calculations can be altered by considering the different variations possible. For the 4 vertices, the 6-di-mensional model gives a formula that when applied gives a resultant of 16 different possible variations.
Fig. 9. Summary of the16possible variations along the x-axis.
Number of possible variations is calculated from the equation:
Vn = nn-2
, where Vn - total number of variations for n connected surfaces and n - number of connected surfaces. For n= 4,
V4 = 44-2 = 42 = 16 The 16 variations can be summarized by figure 9 below:
The 16-dimension tree diagrams can be drawn by considering any 4 edges from the ones given in figure 9 above, one of these 4 edges being the closing edge. In "Analysis and calculation of dimensional chains based on graphs of dimensional ties" [4], the 16 variations are characterised. The formula can be applied to any number of surfaces, and the more surfaces will result in more variations to consider.
Conclusion
This article analyzes the 2 features developed for handling non-ideal parts during modelling. These 2 features are:
1. The development of a geometric model using the 3 main element, the point, lines (lc and l¡) and surfaces. This feature clearly outlines the formation of a geometric configuration regarded much efficient for the presentation of non-ideal objects other than creation of solids through boundary definition.
2. Use of dimensional chain graphs to deduce the number of variations that can be obtained from a geometric configuration.
The first step in solving a problem is to identify it (the ability realizes that there is a problem) and completely understand its attributes. This applies to the effectiveness of these 2 features in the modelling of parts. In order to manufacture more accurate machine parts, its important to identify all possible variations of nonideal objects that can result. The ability to visualize the non-ideal objects and understanding all variations possible will improve the capability of producing much more efficient machine parts with precise components.
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