DESIGNING OF DEVELOPED SURFACES OF COMPLEX PARTS

Dep. «Higher Mathematics», Dnipropetrovsk State Agrarian-Economic University, Voroshilov St., 25, Dnipro, Ukraine, 49600, tel. +38 (056) 713 51 86, e-mail mozganet@mail.ua, ORCID 0000-0003-4860-4818 Dep. «Grafics», Dnipropetrovsk National University of Railway Transport named after Academician V. Lazaryan, Lazaryan St., 2, Dnipro, Ukraine, 49010, tel. +38 (056) 373 15 38, e-mail krasnyuk@mail.diit.edu.ua, ORCID 0000-0002-1400-9992 Dep. «Grafics», Dnipropetrovsk National University of Railway Transport named after Academician V. Lazaryan, Lazaryan St., 2, Dnipro, Ukraine, 49010, tel. +38 (067) 724 47 22, e-mail ulchenkotv@ya.ru, ORCID 0000-0003-2354-7765 Dep. «Grafics», Dnipropetrovsk National University of Railway Transport named after Academician V. Lazaryan, Lazaryan St., 2, Dnipro, Ukraine, 49010, tel. +38 (056) 373 15 59, e-mail pro-f@ukr.net, ORCID 0000-0003-1340-0284


Introduction
For a multiple production the most important among the technological equipment is mechanical processing equipment characterized by the need of frequent change, optimal service life is 5-7 years. Analysis of design methods of technological and mechanical processing equipment showed that at the early design stages it is very difficult to assess and take into account the variety of factors affecting the quality of its work. There is a problem of automated selection of design and process parameters the mating surfaces to ensure operating pur-pose of the product. At the same time, there is no scientific methodological framework that allows taking into account the design and technological features of the designed equipment.
One of the main directions of descriptive geometry is designing surfaces, linear and twodimensional contours, which correspond to different positional, metric, differential conditions. Currently, these areas received significant results that meet the modern requirements of design planning.
The works [10,11] present the geometric model of ruled surfaces with multiparameter sets of lines and special lines of surfaces, but a large number of parameters and difficult determination of special lines, such as curvature, make the use of these models difficult in practice. The works [12][13][14][15] are devoted to the development of geometric models of tillage working element surfaces. The disadvantage of these models is that they are specified for the working element, which reduces the possibility of their use in the design of other tools. A common shortcoming of these models is that they are presented in general or for a specific working element. This leads to the fact that in each case it is necessary to develop the own algorithm for surface building.

Purpose
The main directions of the descriptive geometry of surfaces formulated by I. I. Kotov [7,8] and including development of methods for designing linear and two-dimensional contours, continuous frames in compliance with the set differentialisometric conditions, generation of contours and surfaces equations, are still up to date. Furthermore, increasing requirements for accuracy characteristics, increase in speed and the need to improve the dynamic characteristics of surface interactions with the environment made the surface geometry conditions more complicated. For example, the transition curves of railways need continuous fourth derivative at the junctions [4,18]; bidimensional contours must have degree of smoothness at the joint not lower than the second one [3,9] for the surfaces, operating in conditions of high density environment (working elements of tillage tools, machines and mechanisms in the carlocomotive facilities of railways, ship contours, etc.) or of high speeds.
The above generates an urgent need to develop new methods for continuous mathematical surface models (MSM), which correspond to a large number of differential geometric terms with a given degree of accuracy and solution of various applied problems on the received MSM.
An important sector of technology, which is connected with engineering progress, are those units that are engaged in designing and manufacturing of products, whose main functional element is the surface, working in conditions of high speeds or a large density, particularly the tillage elements and machines. Energy costs thus depend on the geometric properties of surfaces.
One of the most common technical surfaces are the ruled ones, among which a special position is occupied by developed surfaces thanks to their differential-parametric properties: -surface tangent plane is n contact along the rectilinear generator and does not change its position in space when changing the point of contact; -the surfaces can be made by bending sheet metal.
These provisions enable a product manufacturer to save significant material and energy means, therefore, the development of geometric models of such surfaces is an important task.
The work purpose is to ensure the rational choice of parameters of the mating surfaces of parts when designing process equipment based on the methods of artificial intelligence.
The study object is a combination of machining and mechanical processing equipment, tools and other machine parts, working in conditions of friction under boundary lubrication and made of general purpose structural materials.
The paper considers the geometric model of a ruled developed surface, the conditions of existence of such a surface and provides a generalized algorithm for surface plotting regardless of the type of the working element or the machinebuilding product [6].

Methodology
We know that a developable surface is given by two curves which have the property: if through any point of the first curve to draw the area tangent to both curves, this tangent property is kept in many geometric transformations.
Assume that we have two set curves: : which separate a congruence from a set of lines: Herewith the parameters , , , a b c d depend on u and v and are determined by the equations (1) and (2) To separate a developable surface, we introduce additional condition as compatibility of equations defining the framing of both curves with the standards of future surface [1, 2]: where primes indicate the rates of the functions that define curves by their parameters, and differential equation of ruled surface: which is equivalent to the equation: The equation (6) expresses the fact that the surface standards at the relevant points of the set curves (which belong to the same straight line generator) are parallel that is equivalent to existence of general tangent area to the surface in these points. Indeed, given (4) and (5), the equation (8) takes the form: i.e. vectors tangent to the curve and generators are coplanar. Equation (9) allows determining the relationship between the parameters u and v provided the surface developability: that together with (3), (4), (5) gives the desired surface equation.
If one of the parameters (for example u ) can be from (10) expressed clearly through the other parameter, then with (4), (5) and (11) surface equation will look like: To find the edge of regression l, we differentiate (12) by v : One of the equations (13) together with (12) will define l.
Obviously, not every two lines with their shape and position in space will enable to plot a developable surface. In solving the determinant (9) there may be the following cases (except the discussed above, which leads to (10)): -determinant identically equals to zero (curves are in the same area, which is the sought surface regardless of function  ); -as a result of the solution (9) becomes the equation of one parameter (there is no developable surface, there are separate generators, whose number equals to the number of roots of equations, where there is a common tangent area by given curves); -determinant (9) is not equal to zero (developable surface does not exist).
Thus, the desired result gives only the case that leads to (10).
We consider the equation (6) and (9). Minor determinants 2 2  of the first two rows of the determinant (9) are the coordinates of the surface Plotting of the surface is greatly simplified when the guide curves are contour lines. Algorithm for plotting the surface.
5. We write the surface equation (3). The given model is used for building the semidigger moldboard of the upper deck of a doubledeck plow PNY-4-40 ( Fig. 1), while the lower deck had digger moldboards. As the guide curves we selected boundary movement trajectories of soil beds [5,16,17].

Findings
The assessment of the operation quality of the double-deck plow with digger and semi-digger moldboards of the upper deck was conducted by the following indicators: -depth of plowing of plant residues; -percentage of plowing of plant residues Experimental studies have shown the application prospectivity of semi-digger moldboards on moldboard plows, particularly on the double-deck ones. Taking into account the operating speed of the plow 2.8 m/s, the plant residues plowing percentage for plow with semi-digger moldboards is 98.9%, and with the digger ones -96.1%, which is 4.3% higher, that is why semi-digger moldboards mounted on the upper deck of the double-deck plow outweigh the digger moldboards by agrotechnical parameters [19].

Originality and practical value
1. The approaches to solving the problem of recognition of wear conditions of the tested interface, depicted by its conceptual model, were elaborated; the corresponding algorithms of the computational procedures were built. group, each of which corresponds to an individual computational model of surface quality parameters normalization.

Conclusions
1. The analysed method of putting together the analytic formulas of specific transformations allows us to study the converted lines and surfaces using the methods of analytic geometry.
2. The designed model of developed surface is expedient to be used for designing various working elements, including tillage tools.
3. Using the system, we elaborated the algorithms and compiled the programs for analytical calculation of line, non-line and equidistant to them surfaces by given differential geometric terms, which have shown high efficiency of the system application in solving the above problems.