MATHEMATICAL MODELS FOR THE ANALYSIS OF THE PARAMETERS OF CHANNELS IN THE PLANNING OF MECHANICAL PROCESSING AND WELDING OPERATIONS Volodymyr Shanaida 1, Ruslan Skliarov 1 and Valeriy Lazaryuk 1 1 Ternopil Ivan Puluj National Technical University, Ruska str., 56, Ternopil, 46001, Ukraine Abstract The introduction and use of information technologies is an integral part of the successful functioning of modern production. The analysis of the production processes of individual enterprises made it possible to determine specific requirements for planning their production activities. In many cases, planning departments create their own intellectual and informational systems for comprehensive planning of the production process even when accepting a production order. We have proposed a series of mathematical models for describing the geometric parameters of the part, which have a significant impact on the indicators of the energy consumption of the production process and the costs of performing assembly operations. Mathematical models are obtained by implementing a non-linear regression algorithm of a general type. The adequacy of mathematical models was checked by the value of the coefficients of determination R2 for the proposed approximating functions and input sets of discrete data. Keywords 1 band saw technologies, rolled section, mathematical modeling, information systems, welding operation 1. Introduction The use of profile blanks for the manufacture of body and frame structures involves the analysis of several technical and economic indicators. Among the technical indicators, it is worth noting such as the cross-sectional area of the profile, and among the technical and economic ones, the indicator for accounting for the length of the weld seam. The first indicator has the significance of the choice of equipment to ensure the mechanical processing of the used profile, and, accordingly, the power consumed per unit of time. The second indicator indicates the actual costs of consumables and the time required to perform a welding operation by an employee of a particular qualification. These indicators have a direct impact on the employee's salary. In modern blank production, about 80% of blanks are cut using band saw technologies. These technologies are high-tech, high-performance energy and resource-saving processes. Band sawing technologies cover a wide range of workpiece cross-sections - from sheets with a thickness of 0.5 mm to rolled products of 1.5 m. Band saws process steel blocks, long products, hard-to-cut steels, nickel- based and titanium-based alloys, non-ferrous metals and their alloys, granite, concrete, and other materials of various shapes and sizes. 2. Related works A lot of researchers have been studying how band saws perform when used for machining. They've been focusing on a few key areas, including analyzing the temperature in the cutting zone [1], looking Proceedings ITTAP’2023: 3rd International Workshop on Information Technologies: Theoretical and Applied Problems, November 22–24, 2023, Ternopil, Ukraine, Opole, Poland EMAIL: Shanayda-vv@ukr.net (A. 1); kalibr2011@gmail.com (A. 2); lazaryuk@gmail.com (A. 3) ORCID: 0000-0002-9743-9110 (A. 1); 0000-0001-6112-964X (A. 2); 0000-0003-3731-2828 (A. 3) ©️ 2020 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). CEUR Workshop Proceedings (CEUR-WS.org) CEUR ceur-ws.org Workshop ISSN 1613-0073 Proceedings at different types of dynamic loads and how they affect the process [2], and examining how different geometric parameters and the quality of the metal being cut impact energy costs [3, 4]. Another important factor in this type of machining is the shape of the chip created by the cut layer, which can have a big impact when working in tight spaces [5]. This issue has been studied in various forms regarding the impact of geometric parameters of the sheared metal layer on the power consumed during mechanical processing [6]. Numerous articles have identified cutting power as the primary indicator of energy consumption during processing procedures [7-9]. Band saws typically indicate the recommended thickness and height, among other technical characteristics [10,11]. Altering the cross- sectional area for specified saw blade parameters will significantly affect the consumed cutting power [12]. When planning a welding operation, it's crucial to consider the cross-sectional area and perimeter of the channel. These parameters are necessary for calculating welding modes and working time. The goal is to ensure that the welded structure has the same strength as the original material. To achieve this, it's important to analyze the softening heat-affected zone parameters in the welded joint. The geometric parameters of this zone are determined by the cross-sectional area and perimeter of the rolling products section [13, 14]. In order to determine how long it will take to weld each piece, we must combine the main arc burning time and auxiliary time. The main arcing time is proportional by the size of the weld's cross-section and inversely proportional by the arc current. The auxiliary time considers the length of the weld and the number of passes required, which are determined by the cross-section size. If the channel's cross- sectional area changes, the welding time will also change. It's important to note that the welding speed is inversely affected by the cross-sectional area of the seam [15]. Welded joints of channels often use butt seams. This type of connection is practical, straightforward, and cost-effective. Welding is typically done from both sides to ensure adequate depth of penetration. However, creating a proper edge preparation can be challenging for butt joint profiles, as incomplete penetration can occur at the entrance corners. For low-stress structures with shaped profiles, overlapping strapped butt joints are preferred. It's important to note that welding the strapping causes a significant stress concentration due to the sudden change in the joint's cross-section [14]. In certain situations, the structure can be put under too much stress due to the active loads, causing the weld's tensile strength calculation to be exceeded. To address this task, welded beams are assembled for stretched belts of structures to make assembly joints. An oblique butt joint is created during this welding operation, which is just as strong as the main section of the beam. To ensure it's strong enough, you can use information about the cross-sectional area of the channel in the oblique joint to select the optimal angle of inclination [16]. The advancement of production processes through automation and the creation of automated production preparation systems [17] requires the implementation of mathematical models to formalize technology and management tasks. Additionally, intelligent information systems and technologies rely on mathematical models of varying degrees of difficulty, making the development and verification of such models a pressing matter. 3. Proposed methodology During the research, non-linear regression of the general type, with 3D modeling and discrete set analysis algorithms was used. 4. Results According to DSTU 3436-96 "Hot-rolled steel channels (Rolling products)" the channel's geometric profile is determined by its dimensional characteristics (Fig. 1, a) and mass-geometric indicators (Fig. 1, b). а) embodiment 1 embodiment 2 b) Figure 1: The part configuration and mass-geometrical adjectives of the channel In this article, we consider two main parameters: b, which represents the width of the channel shelf, and h, which represents the height of the channel. When constructing frame structures and trusses, there are two options for cutting the channel profile. The first option involves cutting along the channel shelf, and the height of the channel remains unchanged. The second option involves cutting along the profile height, and the width of the shelf remains unchanged. In both cases, depending on the size of the cutting angle (displacement of the cutting blade of the saw along one of the geometric parameters), we get the values of the areas and perimeters, which will not be proportional to the values of the areas and perimeters in the normal section according to the right triangle rule. The angle value affects the resulting area and perimeter measurements non-proportional. Therefore, it is advisable to perform a study of changes in areas and perimeters for both cases regarding the most frequently used channel numbers both in general mechanical engineering and in other branches of economic activity. 4.1. The research and analysis alterations in the cross-sectional area and perimeter while displacement along the channel shelf When moving the saw along the shelf of channel number 5U, which is manufactured according to DSTU 3436-96 " Hot-rolled steel channels (Rolling products)" (Fig. 2) Figure 2: The scheme of the displacement of the saw during an oblique cut along the width of the channel shelf the value of the cross-sectional area depending on the amount of displacement along the channel shelf can be described by the equation: 𝑃𝑙(𝑥) = 1.559 ∙ 𝑥 1.559 − 2.684 ∙ 𝑥 + 626.714 where 𝑥 - displacement, mm. After estimating the values of the studied parameter according to the proposed dependence, a comparison of the areas of the sheared layer with the actual indicators was carried out (see Table 1). Table 1 Comparative data of cross-sectional areas for straight and oblique cuts Displacement, mm 0 10 15 20 25 32 Cross-sectional area: Actual, mm2 626.689 656.576 692.123 739.022 795.266 886.272 2 Estimated, mm 626.714 656.271 692.552 739.158 794.833 886.419 Relative error, % 4.008e-3 0.046 0.062 0.018 0.054 0.017 For the same channel, the equation of describing the perimeter of a channel's cross-section varies based on the displacement along the shelf of the channel too: 𝑃𝑒𝑟(𝑥) = 0.105 ∙ 𝑥 1.608 − 0.201 ∙ 𝑥 + 149.708 After estimating the values of the studied parameter according to the proposed dependence, a comparison of the perimeters of the sheared layer with the actual indicators was carried out (Table 2). Table 2 Comparative data of the cross-sectional perimeter for straight and oblique cuts Displacement, mm 0 10 15 20 25 32 Perimeter: Actual, mm 208.918 214.427 220.989 229.661 240.081 256.982 Estimated, mm 208.923 214.372 221.066 229.686 240.003 257.00 Relative error, % 2.199e-3 0.026 0.035 0.011 0.032 0.01 The coefficients of determination R2 were calculated for the proposed approximating functions using a discrete set of input data: when using mathematical dependence for: cross-sectional area cross-section perimeter R2 0.997 0.997 Other standard channel sizes were also studied: - Channel №6.5U A proposed function for approximating the cross-sectional area is presented: 𝑃𝑙(𝑥) = 1.478 ∙ 𝑥 1.572 − 2.686 ∙ 𝑥 + 762.796 The outcomes of measurements and computations have been condensed into a table 3. Table 3 Comparative data of cross-sectional areas for straight and oblique cuts Displacement, mm 0 10 15 20 25 30 36 Cross-sectional area: Actual, mm2 762.737 791.617 826.298 872.540 928.616 992.861 1078.673 Estimated, mm2 762.796 791.077 826.797 872.993 928.426 992.243 1.079e3 Relative error, % 7.70e-3 0.068 0.06 0.052 0021 0.062 0.031 A proposed function for approximating the cross-sectional perimeter is presented: 𝑃𝑒𝑟(𝑥) = 0.244 ∙ 𝑥 1.582 − 0.444 ∙ 𝑥 + 254.314 The outcomes of measurements and computations have been condensed into a table 4. Table 4 Comparative data of the cross-sectional perimeter for straight and oblique cuts Displacement, mm 0 10 15 20 25 30 36 Perimeter: Actual, mm 254.304 259.282 265.266 273.255 282.958 294.091 308.988 Estimated, mm 254.314 259.19 265.35 273.332 282.926 293.987 309.045 Relative error, % 3.957e-3 0.035 0.032 0.028 0.011 0.035 0.018 The coefficients of determination R2 were calculated for the proposed approximating functions using a discrete set of input data: when using mathematical dependence for: cross-sectional area cross-section perimeter 2 R 0.996 0.996 - Channel №8U A proposed function for approximating the cross-sectional area is presented: 𝑃𝑙(𝑥) = 1.406 ∙ 𝑥 1.584 − 2.684 ∙ 𝑥 + 911.083 The outcomes of measurements and computations have been condensed into a table 5. Table 5 Comparative data of cross-sectional areas for straight and oblique cuts Displacement, 0 10 15 20 25 30 35 40 mm Cross-sectional area: Actual, mm2 910.975 939.012 972.922 1018.501 174.265 1138.719 1210.475 1288.314 Estimated, 911.083 938.212 973.41 1.019e3 1.074e3 1.138 e3 1.21 e3 1.289 e3 mm2 Relative error, 0.012 0.085 0.05 0.07 0.014 0.05 0.055 0.045 % A proposed function for approximating the cross-sectional perimeter is presented: 𝑃𝑒𝑟(𝑥) = 0.218 ∙ 𝑥 1.594 − 0.416 ∙ 𝑥 + 299.166 The outcomes of measurements and computations have been condensed into a table 6. Table 6 Comparative data of the cross-sectional perimeter for straight and oblique cuts Displacement, 0 10 15 20 25 30 35 40 mm Perimeter: Actual, mm 299.149 303.674 309.152 316.522 325.549 335.997 347.645 360.297 Estimated, 299.166 303.547 309.229 316.635 325.573 335.908 347.539 360.388 mm Relative error, 5.797e-3 0.042 0.025 0.036 7.298e-3 0.027 0.03 0.025 % The coefficients of determination R2 were calculated for the proposed approximating functions using a discrete set of input data: when using mathematical dependence for: cross-sectional area cross-section perimeter R2 0.996 0.996 - Channel №10U A proposed function for approximating the cross-sectional area is presented: 𝑃𝑙(𝑥) = 1.287 ∙ 𝑥 1.597 − 2.63 ∙ 𝑥 + 1.11 ∙ 103 The outcomes of measurements and computations have been condensed into a table 7. Table 7 Comparative data of cross-sectional areas for straight and oblique cuts Displacement, mm 0 10 15 20 25 Cross-sectional area: Actual, mm2 1109.957 1135.882 1167.479 1210.329 1263.289 Estimated, mm2 1.11 e3 1.135 e3 1.168 e3 1.211 e3 1.264 e3 Relative error, % 0.019 0.106 0.029 0.08 0.054 Continua of table 7 Displacement, mm 30 35 40 46 Cross-sectional area: Actual, mm2 1325.147 1394.718 1470.910 1569.716 Estimated, mm2 1.325 e3 1.394 e3 1.47 e3 1.571 e3 Relative error, % 9.1e-3 0.061 0.059 0.054 A proposed function for approximating the cross-sectional perimeter is presented: 𝑃𝑒𝑟(𝑥) = 0.19 ∙ 𝑥 1.606 − 0.391 ∙ 𝑥 + 361.55 The outcomes of measurements and computations have been condensed into a table 8. Table 8 Comparative data of the cross-sectional perimeter for straight and oblique cuts Displacement, mm 0 10 15 20 25 Perimeter: Actual, mm 361.517 365.5 370.358 376.952 385.109 Estimated, mm 361.55 365.318 370.408 377.098 385.212 Relative error, % 9.004e-3 0.05 0.014 0.039 0.027 Continua of table 8 Displacement, mm 30 35 40 46 Perimeter: Actual, mm 394.646 405.385 417.159 432.446 Estimated, mm 394.629 405.258 417.027 432.573 Relative error, % 4.35e-3 0.031 0.032 0.029 The coefficients of determination R2 were calculated for the proposed approximating functions using a discrete set of input data: when using mathematical dependence for: cross-sectional area cross-section perimeter R2 0.995 0.995 - Channel №12U A proposed function for approximating the cross-sectional area is presented: 𝑃𝑙(𝑥) = 1.157 ∙ 𝑥 1.616 − 2.457 ∙ 𝑥 + 1.347 ∙ 103 The outcomes of measurements and computations have been condensed into a table 9. Table 9 Comparative data of cross-sectional areas for straight and oblique cuts Displacement, mm 0 10 20 30 40 52 Cross-sectional area: Actual, mm2 1347.239 1371.924 1443.451 1555.369 1699.72 1905.283 2 Estimated, mm 1.347 e3 1.371 e3 1.445 e3 1.556 e3 1.698 e3 1.906 e3 Relative error, % 0.019 0.088 0.096 0.031 0.088 0.029 A proposed function for approximating the cross-sectional perimeter is presented: 𝑃𝑒𝑟(𝑥) = 0.161 ∙ 𝑥 1.624 − 0.342 ∙ 𝑥 + 423.723 The outcomes of measurements and computations have been condensed into a table 10. Table 10 Comparative data of the cross-sectional perimeter for straight and oblique cuts Displacement, mm 0 10 20 30 40 52 Perimeter: Actual, mm 423.686 427.235 437.527 453.654 474.49 504.223 Estimated, mm 423.723 427.064 437.723 453.724 474.28 504.302 Relative error, % 8.636e-3 0.04 0.045 0.015 0.044 0.016 The coefficients of determination R2 were calculated for the proposed approximating functions using a discrete set of input data: when using mathematical dependence for: cross-sectional area cross-section perimeter R2 0.995 0.995 - Channel №14U A proposed function for approximating the cross-sectional area is presented: 𝑃𝑙(𝑥) = 1.121 ∙ 𝑥 1.62 − 2.549 ∙ 𝑥 + 1.587 ∙ 103 The outcomes of measurements and computations have been condensed into a table 11. Table 11 Comparative data of cross-sectional areas for straight and oblique cuts Displacement, 0 10 20 30 40 50 58 mm Cross-sectional area: Actual, mm2 1586.583 1609.993 1678.262 1786.255 1927.307 2094.749 2243.768 Estimated, mm2 1.587 e3 1.608 e3 1.680 e3 1.788 e3 1.926 e3 2.093 e3 2.245 e3 Relative error, % 0.027 0.108 0.082 0.07 0.045 0.077 0.052 A proposed function for approximating the cross-sectional perimeter is presented: 𝑃𝑒𝑟(𝑥) = 0.149 ∙ 𝑥 1.627 − 0.339 ∙ 𝑥 + 486.283 The outcomes of measurements and computations have been condensed into a table 12. Table 12 Comparative data of the cross-sectional perimeter for straight and oblique cuts Displacement, mm 0 10 20 30 40 50 58 Perimeter: Actual, mm 486.224 489.429 498.783 513.596 532.970 556.005 576.534 Estimated, mm 486.283 489.194 498.967 513.765 532.853 555.787 576.692 Relative error, % 0.012 0.048 0.037 0.033 0.022 0.039 0.027 The coefficients of determination R2 were calculated for the proposed approximating functions using a discrete set of input data: when using mathematical dependence for: cross-sectional area cross-section perimeter R2 0.995 0.995 - Channel №16U A proposed function for approximating the cross-sectional area is presented: 𝑃𝑙(𝑥) = 1.085 ∙ 𝑥 1.623 − 2.61 ∙ 𝑥 + 1.837 ∙ 103 The outcomes of measurements and computations have been condensed into a table 13. Table 13 Comparative data of cross-sectional areas for straight and oblique cuts Displacement, 0 10 20 30 40 50 64 mm Cross-sectional area: Actual, mm2 1836.125 1858.404 1923.692 2027.840 2165.247 2330.036 2596.673 Estimated, mm2 1.837 e3 1.856 e3 1.925 e3 2.03 e3 2.165 e3 2.328 e3 2.598 e3 Relative error, % 0.033 0.118 0.066 0.089 0.011 0.096 0.038 A proposed function for approximating the cross-sectional perimeter is presented: 𝑃𝑒𝑟(𝑥) = 0.138 ∙ 𝑥 1.631 − 0.332 ∙ 𝑥 + 548.485 The outcomes of measurements and computations have been condensed into a table 14. Table 14 Comparative data of the cross-sectional perimeter for straight and oblique cuts Displacement, mm 0 10 20 30 40 50 64 Perimeter: Actual, mm 548.407 551.326 559.885 573.550 591.602 613.281 648.417 Estimated, mm 548.485 551.042 560.046 573.784 591.572 612.992 648.546 Relative error, % 0.014 0.051 0.029 0.041 0.00512 0.047 0.02 The coefficients of determination R2 were calculated for the proposed approximating functions using a discrete set of input data: when using mathematical dependence for: cross-sectional area cross-section perimeter R2 0.994 0.994 - Channel №18U A proposed function for approximating the cross-sectional area is presented: 𝑃𝑙(𝑥) = 1.03 ∙ 𝑥 1.631 − 2.614 ∙ 𝑥 + 2.099 ∙ 103 The outcomes of measurements and computations have been condensed into a table 15. Table 15 Comparative data of cross-sectional areas for straight and oblique cuts Displacement, 0 10 20 30 40 55 70 mm Cross-sectional area: Actual, mm2 2098.175 2119.476 2182.134 2282.746 2416.575 2668.353 2967.267 Estimated, mm2 2.099 e3 2.117 e3 2.183 e3 2.285 e3 2.417 e3 2.665 e3 2.969 e3 Relative error, % 0.038 0.123 0.045 0.092 0.015 0.108 0.043 A proposed function for approximating the cross-sectional perimeter is presented: 𝑃𝑒𝑟(𝑥) = 0.126 ∙ 𝑥 1.638 − 0.32 ∙ 𝑥 + 611.045 The outcomes of measurements and computations have been condensed into a table 16. Table 16 Comparative data of the cross-sectional perimeter for straight and oblique cuts Displacement, mm 0 10 20 30 40 55 70 Perimeter: Actual, mm 610.945 613.627 621.518 634.198 651.081 682.893 720.728 Estimated, mm 611.045 613.305 621.637 634.457 651.127 682.534 720.885 Relative error, % 0.016 0.052 0.019 0.041 0.007 0.053 0.022 The coefficients of determination R2 were calculated for the proposed approximating functions using a discrete set of input data: when using mathematical dependence for: cross-sectional area cross-section perimeter 2 R 0.994 0.994 4.2. The research and analysis alterations in the cross-sectional area and perimeter while displacement along the height of the channel When shifting the saw along the wall (leg) of channel number 5U with height h, which is manufactured according to DSTU 3436-96 " Hot-rolled steel channels (Rolling products)" (Fig. 3) the Figure 3: The scheme of the displacement of the saw during an oblique cut along the height of the channel value of the cross-sectional area depending on the amount of displacement along the channel shelf can be described by the equation: 𝑃𝑙(𝑥) = 0.614 ∙ 𝑥 1.604 − 1.314 ∙ 𝑥 + 626.844 After estimating the values of the studied parameter according to the proposed dependence, a comparison of the areas of the sheared layer with the actual indicators was carried out (see Table 17). Table 17 Comparative data of cross-sectional areas for straight and oblique cuts Displacement, mm 0 10 15 20 25 Cross-sectional area: Actual, mm2 626.689 639.100 654.283 674.965 700.660 2 Estimated, mm 626.844 638.356 654.368 675.488 701.155 Relative error, % 0.025 0.116 0.013 0.078 0.071 Continua of table 17 Displacement, mm 30 35 40 45 50 Cross-sectional area: Actual, mm2 730.839 764.971 802.554 843.124 886.272 Estimated, mm2 730.98 764.67 801.996 842.769 886.831 Relative error, % 0.019 0.039 0.07 0.042 0.063 A proposed function for approximating the cross-sectional perimeter is presented: 𝑃𝑒𝑟(𝑥) = 0.086 ∙ 𝑥 1.618 − 0.186 ∙ 𝑥 + 208.941. The outcomes of measurements and computations have been condensed into a table 18. Table 18 Comparative data of the cross-sectional perimeter for straight and oblique cuts Displacement, mm 0 10 15 20 25 Perimeter: Actual, mm 208.918 210.770 213.038 216.130 219.977 Estimated, mm 208.941 210.662 213.049 216.206 220.049 Relative error, % 0.011 0.051 5.38e-3 0.035 0.033 Continua of table 18 Displacement, mm 30 35 40 45 50 Perimeter: Actual, mm 224.501 229.627 235.279 241.391 247.902 Estimated, mm 224.522 229.584 235.198 241.339 247.983 Relative error, % 9.539e-3 0.019 0.034 0.021 0.033 The coefficients of determination R2 were calculated for the proposed approximating functions using a discrete set of input data: when using mathematical dependence for: cross-sectional area cross-section perimeter R2 0.995 0.995 - Channel №6.5U A proposed function for approximating the cross-sectional area is presented: 𝑃𝑙(𝑥) = 0.466 ∙ 𝑥 1.613 − 1.148 ∙ 𝑥 + 763.042. The outcomes of measurements and computations have been condensed into a table 19. Table 19 Comparative data of cross-sectional areas for straight and oblique cuts Displacement, mm 0 10 20 30 40 Cross-sectional area: Actual, mm2 762.737 771.711 798.027 840.056 895.590 Estimated, mm2 763.042 770.660 798.499 840.951 895.807 Relative error, % 0.04 0.136 0.059 0.107 0.024 Continua of table 19 Displacement, mm 45 50 55 60 65 Cross-sectional area: Actual, mm2 927.687 962.294 999.150 1038.015 1078.673 Estimated, mm2 927.452 961.736 998.553 1.038e3 1.079e3 Relative error, % 0.025 0.058 0.06 0.02 0.07 A proposed function for approximating the cross-sectional perimeter is presented: 𝑃𝑒𝑟(𝑥) = 0.072 ∙ 𝑥 1.623 − 0.178 ∙ 𝑥 + 254.352 The outcomes of measurements and computations have been condensed into a table 20. Table 20 Comparative data of the cross-sectional perimeter for straight and oblique cuts Displacement, mm 0 10 20 30 40 Perimeter: Actual, mm 254.304 255.750 259.994 266.780 275.762 Estimated, mm 254.352 255.585 260.067 266.921 275.797 Relative error, % 0.019 0.065 0.028 0.053 0.013 Continua of table 20 Displacement, mm 45 50 55 60 65 Perimeter: Actual, mm 280.960 286.570 292.551 298.864 305.475 Estimated, mm 280.924 286.483 292.457 298.831 305.594 Relative error, % 0.013 0.03 0.032 0.011 0.039 The coefficients of determination R2 were calculated for the proposed approximating functions using a discrete set of input data: when using mathematical dependence for: cross-sectional area cross-section perimeter 2 R 0.994 0.995 - Channel №8U A proposed function for approximating the cross-sectional area is presented: 𝑃𝑙(𝑥) = 0.37 ∙ 𝑥 1.626 − 1.035 ∙ 𝑥 + 911.461 The outcomes of measurements and computations have been condensed into a table 21. Table 21 Comparative data of cross-sectional areas for straight and oblique cuts Displacement, mm 0 10 20 30 40 Cross-sectional area: Actual, mm2 910.975 918.065 939.012 972.922 1018.501 Estimated, mm2 911.461 916.768 939.090 973.857 1.019e3 Relative error, % 0.053 0.141 8.329e-3 0.096 0.073 Continua of table 21 Displacement, mm 50 60 70 80 Cross-sectional area: Actual, mm2 1074.265 1138.719 1210.475 1288.314 Estimated, mm2 1.074 e3 1.138 e3 1.21 e3 1.289 e3 Relative error, % 9.835e-3 0.079 0.068 0.068 A proposed function for approximating the cross-sectional perimeter is presented: 𝑃𝑒𝑟(𝑥) = 0.059 ∙ 𝑥 1.635 − 0.167 ∙ 𝑥 + 299.229 The outcomes of measurements and computations have been condensed into a table 22. Table 22 Comparative data of the cross-sectional perimeter for straight and oblique cuts Displacement, mm 0 10 20 30 40 Perimeter: Actual, mm 299.149 300.334 303.835 309.506 317.138 Estimated, mm 299.229 300.122 303.846 309.659 317.260 Relative error, % 0.027 0.071 3.777e-3 0.049 0.039 Continua of table 22 Displacement, mm 50 60 60 65 Perimeter: Actual, mm 326.485 337.304 349.365 362.467 Estimated, mm 326.469 337.157 349.23 362.61 Relative error, % 4.953e-3 0.043 0.039 0.04 The coefficients of determination R2 were calculated for the proposed approximating functions using a discrete set of input data: when using mathematical dependence for: cross-sectional area cross-section perimeter 2 R 0.994 0.994 - Channel №10U A proposed function for approximating the cross-sectional area is presented: 𝑃𝑙(𝑥) = 0.313 ∙ 𝑥 1.627 − 1.016 ∙ 𝑥 + 1.111 ∙ 103 The outcomes of measurements and computations have been condensed into a table 23. Table 23 Comparative data of cross-sectional areas for straight and oblique cuts Displacement, mm 0 10 20 30 40 50 Cross-sectional area: Actual, mm2 1109.957 1115.493 1131.938 1158.829 1195.460 1240.970 Estimated, mm2 1.111e3 1.114e3 1.131e3 1.159e3 1.197e3 1.242e3 Relative error, % 0.074 0.145 0.048 0.057 0.097 0.069 Continua of table 23 Displacement, mm 60 70 80 90 100 Cross-sectional area: Actual, mm2 1294.421 1354.875 1421.438 1493.294 1569.716 Estimated, mm2 1.294e3 1.354e3 1.42e3 1.493e3 1.571e3 Relative error, % 4.579e-3 0.059 0.085 0.045 0.081 A proposed function for approximating the cross-sectional perimeter is presented: 𝑃𝑒𝑟(𝑥) = 0.052 ∙ 𝑥 1.635 − 0.17 ∙ 𝑥 + 361.657 The outcomes of measurements and computations have been condensed into a table 24. Table 24 Comparative data of the cross-sectional perimeter for straight and oblique cuts Displacement, mm 0 10 20 30 40 50 Perimeter: Actual, mm 361.517 362.469 365.300 369.931 376.243 384.091 Estimated, mm 361.657 362.196 365.208 370.042 376.438 384.237 Relative error, % 0.039 0.075 0.025 0.03 0.052 0.038 Continua of table 24 Displacement, mm 60 70 80 90 100 Perimeter: Actual, mm 393.317 403.761 415.271 427.709 440.951 Estimated, mm 393.328 403.627 415.068 427.596 441.165 Relative error, % 2.765e-3 0.033 0.049 0.027 0.048 The coefficients of determination R2 were calculated for the proposed approximating functions using a discrete set of input data: when using mathematical dependence for: cross-sectional area cross-section perimeter R2 0.994 0.994 - Channel №12U A proposed function for approximating the cross-sectional area is presented: 𝑃𝑙(𝑥) = 0.293 ∙ 𝑥 1.62 − 1.044 ∙ 𝑥 + 1.348 ∙ 103 The outcomes of measurements and computations have been condensed into a table 25. Table 25 Comparative data of cross-sectional areas for straight and oblique cuts Displacement, mm 0 20 40 60 80 100 120 Cross-sectional area: Actual, mm2 1347.239 1365.822 1420.114 1506.258 1619.179 1753.712 1905.283 Estimated, mm2 1.348e3 1.364e3 1.421e3 1.507e3 1.619e3 1.752e3 1.906e3 Relative error, % 0.03 0.113 0.077 0.079 0.033 0.087 0.049 A proposed function for approximating the cross-sectional perimeter is presented: 𝑃𝑒𝑟(𝑥) = 0.048 ∙ 𝑥 1.627 − 0.173 ∙ 𝑥 + 423.754 The outcomes of measurements and computations have been condensed into a table 26. Table 26 Comparative data of the cross-sectional perimeter for straight and oblique cuts Displacement, mm 0 20 40 60 80 100 120 Perimeter: Actual, mm 423.686 426.859 436.135 450.867 470.203 493.272 519.301 Estimated, mm 423.754 426.598 436.317 451.068 470.115 493.015 519.456 Relative error, % 0.016 0.061 0.042 0.044 0.019 0.052 0.03 2 The coefficients of determination R were calculated for the proposed approximating functions using a discrete set of input data: when using mathematical dependence for: cross-sectional area cross-section perimeter R2 0.994 0.994 - Channel №14U A proposed function for approximating the cross-sectional area is presented: 𝑃𝑙(𝑥) = 0.158 ∙ 𝑥 1.705 − 0.498 ∙ 𝑥 + 1.586 ∙ 103 The outcomes of measurements and computations have been condensed into a table 27. Table 27 Comparative data of cross-sectional areas for straight and oblique cuts Displacement, mm 0 20 40 60 Cross-sectional area: Actual, mm2 1586.583 1602.691 1650.072 1726.151 2 Estimated, mm 1.586e3 1.603e3 1.652e3 1.727e3 Relative error, % 0.017 9.611e-3 0.103 0.043 Continua of table 27 Displacement, mm 80 100 120 140 Cross-sectional area: Actual, mm2 1827.349 1949.758 2071.504 2243.768 Estimated, mm2 1.825e3 1.944e3 2.082e3 2.239e3 Relative error, % 0.136 0.307 0.522 0.194 A proposed function for approximating the cross-sectional perimeter is presented: 𝑃𝑒𝑟(𝑥) = 0.026 ∙ 𝑥 1.712 − 0.081 ∙ 𝑥 + 486.174 The outcomes of measurements and computations have been condensed into a table 28. Table 28 Comparative data of the cross-sectional perimeter for straight and oblique cuts Displacement, mm 0 20 40 60 Perimeter: Actual, mm 486.224 488.955 496.993 509.907 Estimated, mm 486.174 488.937 497.281 510.028 Relative error, % 0.01 3.78e-3 0.058 0.024 Continua of table 28 Displacement, mm 80 100 120 140 Perimeter: Actual, mm 527.100 547.919 568.646 598.008 Estimated, mm 526.674 546.904 570.491 597.263 Relative error, % 0.081 0.185 0.324 0.125 The coefficients of determination R2 were calculated for the proposed approximating functions using a discrete set of input data: when using mathematical dependence for: cross-sectional area cross-section perimeter R2 0.979 0.979 - Channel №16U A proposed function for approximating the cross-sectional area is presented: 𝑃𝑙(𝑥) = 0.244 ∙ 𝑥 1.624 − 1.043 ∙ 𝑥 + 1.837 ∙ 103 The outcomes of measurements and computations have been condensed into a table 29. Table 29 Comparative data of cross-sectional areas for straight and oblique cuts Displacement, mm 0 25 50 75 Cross-sectional area: Actual, mm2 1836.125 1858.404 1923.692 2027.840 2 Estimated, mm 1.837e3 1.856e3 1.925e3 2.03e3 Relative error, % 0.034 0.119 0.064 0.089 Continua of table 29 Displacement, mm 100 125 150 160 Cross-sectional area: Actual, mm2 2165.247 2330.036 2516.836 2596.673 Estimated, mm2 2.165e3 2.328e3 2.516e3 2.598e3 Relative error, % 8.072e-3 0.088 0.027 0.056 A proposed function for approximating the cross-sectional perimeter is presented: 𝑃𝑒𝑟(𝑥) = 0.04 ∙ 𝑥 1.63 − 0.171 ∙ 𝑥 + 548.51 The outcomes of measurements and computations have been condensed into a table 30. Table 30 Comparative data of the cross-sectional perimeter for straight and oblique cuts Displacement, mm 0 25 50 75 Perimeter: Actual, mm 548.407 552.143 563.095 580.579 Estimated, mm 548.51 551.776 563.298 580.878 Relative error, % 0.019 0.066 0.036 0.052 Continua of table 30 Displacement, mm 100 125 150 160 Perimeter: Actual, mm 603.669 631.392 662.856 676.314 Estimated, mm 603.641 631.054 662.745 676.553 Relative error, % 4.578e-3 0.053 0.017 0.035 The coefficients of determination R2 were calculated for the proposed approximating functions using a discrete set of input data: when using mathematical dependence for: cross-sectional area cross-section perimeter R2 0.995 0.995 - Channel №18U A proposed function for approximating the cross-sectional area is presented: 𝑃𝑙(𝑥) = 0.231 ∙ 𝑥 1.624 − 1.064 ∙ 𝑥 + 2.099 ∙ 103 The outcomes of measurements and computations have been condensed into a table 31. Table 31 Comparative data of cross-sectional areas for straight and oblique cuts Displacement, mm 0 25 50 75 Cross-sectional area: Actual, mm2 2098.175 2118.315 2177.619 2273.022 Estimated, mm2 2.099e3 2.115e3 2.178e3 2.275e3 Relative error, % 0.044 0.134 0.036 0.101 Continua of table 31 Displacement, mm 100 125 150 180 Cross-sectional area: Actual, mm2 2400.225 2554.482 2731.211 2967.267 Estimated, mm2 2.401e3 2.553e3 2.729e3 2.969e3 Relative error, % 0.041 0.055 0.088 0.056 A proposed function for approximating the cross-sectional perimeter is presented: 𝑃𝑒𝑟(𝑥) = 0.037 ∙ 𝑥 1.63 − 0.172 ∙ 𝑥 + 611.096 The outcomes of measurements and computations have been condensed into a table 32. Table 32 Comparative data of the cross-sectional perimeter for straight and oblique cuts Displacement, mm 0 25 50 75 Perimeter: Actual, mm 610.945 614.275 624.082 639.868 Estimated, mm 611.096 613.812 624.209 640.242 Relative error, % 0.025 0.075 0.02 0.059 Continua of table 32 Displacement, mm 100 125 150 180 Perimeter: Actual, mm 660.931 686.498 715.817 755.021 Estimated, mm 661.095 686.271 715.422 755.290 Relative error, % 0.025 0.033 0.055 0.036 The coefficients of determination R2 were calculated for the proposed approximating functions using a discrete set of input data: when using mathematical dependence for: cross-sectional area cross-section perimeter R2 0.994 0.994 5. Conclusions In the process of cutting a part of a complex geometric profile (channel) at different angles, it has been observed that the perimeter and cross-sectional area do not change proportionally to the angle of the cut. This applies to both angular cuts made along the height and width of the profile. For each standard size of the channel, we defined mathematical construction that explains how the perimeter and cross-sectional area of the profile change based on the displacement of the metal-cutting tool relative to the base points in the normal section. Studies have shown that the best way to present these mathematical models is through a non-linear regression of the general type. The accuracy of the mathematical models developed was confirmed by calculating the coefficient of determination R2. The values obtained ranged from 0.994 to 0.997, with only one case showing 0.979. These results indicate that the proposed mathematical models accurately depict the measurement results obtained from both mathematics analysis and 3D modeling. The proposed mathematical models provide effective design of welded joints of metal structures. 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