Development of a Technique for the Geometrical Modeling of Conjugated Surfaces When Determining the Geometrical Parameters of an Engagement Surface Contact in Kinematic Pairs

The presented results of scientific research are aimed at increasing the efficiency of trenching for laying the utility lines using new less energy-consuming technologies of excavating the soil with the working equipment of multi-scraper excavators.<br><br>The proposed method of determining the efficient operating modes for excavators when digging a trench is based on the idea of cutting the soil with the blades at a critical depth level, which guarantees consumption of minimum specific energy and maximum efficiency of the machine. This becomes possible if the operation of such blades is provided with the absolute values and the ratio of the speeds of cutting and submitting the working body into the face.<br><br>To determine the efficient modes of multi-scraper excavators and the size of their edge side blades, the conditions of their effective unloading were identified and the patterns of changing the soil movement over the surface of unloading scrapers depending on the time of unloading were determined. For the same purpose, the dependences of the blocked cutting speed on the trench width were determined and the technical performance of the excavator was specified on the basis of determining the soil bearing capacity for one group of blades. It is found that the time of unloading the soil from the scrapers very slightly depends on their angular velocity within its change in the unloading zone. On this basis, the maximum angular speed of the scrapers is determined. The identified indicators are related to the width of the edge side blades performing asymmetric locked cutting, whose dimensions were determined by calculations.<br><br>The obtained efficient operating modes of scraper trench excavators and the size of their edge side blades allow developing practical recommendations for improving the working equipment of excavators of this type.


Introduction
In the modern stage of building machines with complex structural elements, the methods of descriptive geometry are widely used in design. One of the common methods of geometric object formation is geometric modeling, which makes it possible to obtain the desired geometry of an article and the required conjugation of surfaces at the design stage.
The emergence of new machine-building technologies for machining the articles in the automated industries poses new challenges for manufacturers. In turn, the accuracy of production depends significantly on the accuracy of the design, which led to the need to devise a fundamentally new designing technique, based on the use of the kinematic method.
Therefore, those studies are relevant that address the practical application of the theory for determining a contact surface in conjugated pairs, to improve the efficiency of design and fabrication operations. In particular, it is interesting to identify the prospects for further possibilities of this theory, combined with a variety of software developed to solve complex industrial tasks. This necessitates undertaking a deeper study into the multifaceted properties of the contact surface determination method as a unique tool for solving present and future technical tasks related to toothing and cutting tools.

Literature review and problem statement
The basic provisions and requirements for spatial toothing are stated in work [1]. It is noted that for toothing to operate normally it must have a common tangent plane and a common normal. It is difficult to achieve practical implementation of these requirements during design due to the need to build a large number of axoids. Paper [2] explores the tasks of forming a discretely-represented engagement curve on special points within the engagement area; however, issues related to the continuity of contact between the conjugated surfaces remain to be fully resolved.
Currently, the kinematic method in combination with computer technology is successfully used in the design of conjugated surfaces, reported in a series of studies. This approach is applied in work [3] but the quasi-statistical design error at high-speed processing becomes the main reason affecting the machining accuracy of articles with complex surfaces, which is a significant factor in the operation of kinematic pairs. An option to overcome the specified problem is proposed in studies [4,5]. The same approach was employed in work [6]. Reducing, and ideally eliminating, vibration in the machining process avoids the subsequent unstable working conditions. However, these studies are ineffective at the «manual» technique of achieving geometric parameters.
Paper [7] explores those issues that arise in the shape formation of complex conical screw surfaces of machines' work ele ments. The paper proposes an automated design system that implements a formalized calculation of the tool's trajectory in accordance with the suggested special strategy for screw surface machining. As the raw data, the system needs information about the geometry of the instrument and the screw surface in a parametric form to calculate the tool trajectory. Issues related to the definition of characteristics in the construction of a three-dimensional model of surfaces remained unresolved. A different approach to developing conservative intersections of curves with the boundaries of conjugated surfaces was used in work [8], which is more acceptable. A computer program to determine the curved surface of contacting elements with a fixed point was reviewed and described. Although it is possible to accidentally receive false triggering during the tests on the intersection between normals, the author was able to determine the actual intersection of spatial curves. The method developed and the geometric and computer modeling algorithms could be an option to overcome the emerging difficulties for the formation of a contact surface on CNC machines [9]. Researchers have identified intermediate machine adjustments that ensure milling stability, depending on the conditions and parameters of the process. This approach was applied in works [10,11]; but the use of a deliberate modification of contact surfaces reduces the sensitivity of machined articles, specifically, worm gears, and leads to errors in the manufacture of their elements. This fact is highlighted in paper [12].
The issues of determining the required parameters of the geometric shape of the contact spatial surface in conjugated kinematic pairs remain unresolved. All this suggests that research into a new technique for the geometrical determination of the shape of a contact surface in toothing involving computer technology is appropriate.

The aim and objectives of the study
The aim of this study is to devise a new technique for the geometric modeling of conjugated surfaces, which would make it possible to determine at the design stage the shape of the spatial surface of an engagement contact.
To accomplish the aim, the following tasks have been set: -to determine the geometric shape of a spatial contact screw surface; -to investigate the margin of error in the construction of a contact screw surface.

Materials and methods to determine the geometric shape of a spatial contact screw surface in the conjugated kinematic pairs
When designing the shapes of conjugated surfaces, we consider geometric modeling as a base for creating computer simulation techniques.
The type of contact surface interacting with two preset conjugated surfaces could predict manufacturing defects at the design stage. We suggest a technique to design the kinematic pairs of the required shape and precision of manufacture using a spatial contact surface. The geometric determination of the shape of the spatial contact surface with a curved transformation includes the circular and screw methods of converting the original curved surface along the curved axis.

The essence of the proposed technique for geometric modeling of conjugated surfaces
Let us determine the geometric shape of the contact screw surface in two conjugated curved surfaces. The original curved surface is set by the system of equations (1) and rotated at 30° around its axis (Fig. 1 To determine the line of intersection of two curved conjuga ted surfaces, let us assume that curves in space are set by a parametric equation whose solution yields a set of coordinates X, Y, and Z. We shall define the structure of the Х, Y, Z coordinates using an example of a curved line set by parametric expression (2).
where 0 ≤ t ≤ 2π. Each coordinate X, Y, Z is a one-dimensional array of values; we determine, from the arrays of coordinates of the original and rotated curved surfaces, the coordinates of the surface, which are equal to the coordinates of the original and rotated surfaces.
When rotated around a straight line (the coordinate axis, the surface's rotation axis), the ϑ parameter is constant for any value of ω. The ϑ parameter value is to be determined from the equality of the values of the X coordinates of the original and rotated surfaces. No. 1. The region is assigned for determining the ϑ parameter -the original surface within 0 ≤ ϑ ≤ π, and the contact line built does not match the actual line of intersection of surfaces (Fig. 2).
No. 2. The ϑ parameter of the original surface is within 0 ≤ ϑ ≤ π, the line built does not match the actual line of contact of the surfaces, but when the X parameter is mirrored relative to the XZ-plane passing through the surface rotation axis, the contact line coincides with the actual line of intersection of surfaces (Fig. 3).
No. 3. Based on calculation No. 2, but the X value calculation formula is changed (in the formula, sign «+» changed to sign «-»). The contact line coincided with the actual line of intersection of surfaces (Fig. 4). Taking into consideration the above results, we shall examine the permissible error in the construction of the geometric shape of the contact engagement surface in the curved conjugated pairs; we shall use the geometric methods of circular and screw transformation. The original surface of the paraboloid is set (Fig. 5) with parametric expression (3).  .
x x x y y y Shift the turning point C(x 0 , y 0 , z 0 ) relative to the origin of the coordinates and combine the systems of equations (6) to (8) in order to derive a general expression (9) for the rotation around the vertical axis passing through the set point: By fitting the parametric values from the system of equations (3) to expression (9) at a turning angle of 30°, we obtain expression (10) (Fig. 7).  When rotated around the vertical axis, the original surface passes through the contact screw engagement surface. This phenomenon leads to errors in the design of the kinematic pairs.
Consider another case implying the rotation of the curved surface around the curved axis at 30° (Fig. 8). In this case, there are errors in the contact of the conjugated surfaces.
Suppose the curved surface is located in an arbitrary place in the spatial domain and is set by the system of equations (11); rotating around the Z-axis yields expression (12) (Fig. 9).
When studying and modeling the geometric shape of an engagement surface contact, it was found that the shifted coordinates of the conjugated curved surfaces indicate an uneven change in its geometric parameters. After studying the geometric shape of the spatial contact surface, we revealed heterogeneity in the distribution of the vector beam cross-section field.

Discussion of results of studying the shape of a contact screw engagement surface
Our study of the geometric shape of the spatial surface of an engagement contact has shown that the contact surface is determined by two equalities: X and Y. The choice of one of these equalities depends on the region for determining the ϑ parameter of the original surface. By expressing the ϑ parameter through the equality X, we obtain the angle arc tangent within − ( ) π π 2 2 ; , and by expressing the ϑ parameter through the Y equality, we derive the angle arc tangent in the interval (0; π). Thus, the expression of the ϑ parameter through the equality X defines the region of the intersection line in the vicinity of the x coordinate axis, while the expression of the ϑ parameter through the Y equality -in the vicinity of the y coordinate axis.
The study reported here has made it possible to determine the surface of the contact based on the coordinates of the original surface. At the same time, the required values of the ϑ parameter correspond to the generatrix of the original surface, coinciding or parallel to the coordinate planes XZ or YZ.
The limitations of the geometric technique to determine the contact shape are as follows: -for each line of intersection, it is necessary to calculate the coordinates according to the formulae specific to them; -it is required to construct an additional algorithm to determine a formula to use when defining the line of intersection.
The proposed technique could be useful in machine building where it is necessary to determine contact engagement in kinematic pairs for effective geometric design of toothing and cutting tools.

Conclusions
1. This paper reports a practical technique for constructing a geometric shape of the contact between conjugated parts providing for the required technical parameters at the design stage. A special feature of the suggested procedure is the possibility to adapt it for use in software packages.
2. The procedure makes it possible to determine the permissible error in the geometric parameters for the shape of contact between conjugated pairs, which is set by the precision quality in the manufacture of toothing and cutting tools.