A CONTRIBUTION TO DIGITIZING THE DEFORMATION ZONE OF COMPLEX GEOMETRY Текст научной статьи по специальности «Компьютерные и информационные науки»

S.Spuzic

The University of South Australia

D.I.Kinzin

Nosov Magnitogorsk State Technical University

A CONTRIBUTION TO DIGITIZING THE DEFORMATION ZONE OF COMPLEX GEOMETRY

Abstract. The efforts for actualizing a breakthrough in technology and rising the level of the sustainability of manufacturing systems via implementation of I4 and I5 have given significant impetus to the concept of the Digital Twins. In the context of the rolling industry, a propitious version of a Digital Twin is a replica of the calibration (roll pass design). The calibration is an embodiment of the instructions for the realisation of the rolling process, including design, monitoring, control and maintenance. An aspect of fundamental importance in the calibration and in the design of digital twins is the mathematical definition of the geometry of the deformation zone and, in particular, digitizing the incoming and exiting contours of the rolled solids. The proposed method for digitizing the pass morphometry opens the gate for mobilizing the existing computing potentials and analytic theories for improving the sustainability of rolling systems.

Keywords: calibration (roll pass design), calibre, groove, deformation zone, digital twin, Chebyshev polynomials.

Introduction

The calibration (roll pass design) is an essential part of the technology of rolling long products. One of the central issues in calibration is the mathematical description of the geometry of the deformation zone and, in particular, digitizing the incoming and exiting contours of the rolled solids.

The difficulty of choosing an appropriate mathematical description is due to the great variety and complexity of the calibre forms, and it is inevitable that different approaches to this particular task are required to solve different calibration tasks.

The earliest and simplest approaches in this matter were to reduce the complex shape of the deformation zone to the simplest case of rolling a solid of a rectangular cross-section in the groove-less rolls. This allowed for greater simplicity of performing calculations and analyzing the results obtained in comparison with rolling in calibers, but also for the robust methodological approach, according to which the calculations are reduced to the deformation of a conventional rectangular solid rolled in grooveless ("flat") rolls. This includes a widely known method of using the corresponding rectangles (e.g. the rectangles derived by maintaining the actual cross-section width while calculating the average height as the ratio of the cross-section area and the width).

The next approximation was to develop different calibration methods for specific calibre "systems" characterized by geometrical similarity of the contours occurring in a sequence of the rolling passes. This approach is still the most common for the calibration of the both simple and the complex geometries. Consideration of different calibre systems of a given shape is a big breakthrough in relation to the above method of the corresponding rectangular sections. Nevertheless, this method is very selective i.e. limited to analysing different geometries separately, without the possibility of extracting the analogous inferences). This limitation does not correspond to the current level of development in the sciences and engineering.

The isolation of individual calibration systems from an infinite variety of possible options deprives us of the ability to find optimal solutions that can lie beyond the apparent boundaries. In other words, a significant disadvantage of analysing the isolated groups is that the relations defined exclusively for a specific calibre system cannot be applied to other systems. Moreover, the precise details and boundaries of the calibre grooves are blurred thus allowing only for designing rolling passes that are practically possible, but not for designing the calibres and passes that are optimised.

For example, in the context of the calibre generalization, the rhombic calibre in Fig. 1 can be transformed into a round calibre by increasing the fillet radius. Ignoring the reality that both shapes are topologically equivalent prevents extrapolating and interpolating the interrelations obtained experimentally and/or theoretically from the broader family of the topologically equivalent geometries [1, 2].

Fig. 1. Sample for discussing the calibre topology

Further development, prompted by extensive use of personal computers, led to defining the calibres by means of analytical functions. For example, V.S.Berkowski suggested using the Lame curve, Eqn (1), to describe the calibres of a simple form.

X a Y

+ —

h b

P

= 1.

(1)

This approach allows for defining a wide variety of geometries with sufficient accuracy (Fig. 2).

Fig. 2. Analytic description of simple geometries

Despite the apparent simplicity and elegance, this solution is associated with significant difficulties in calculating various calibration parameters. For example, calculating the calibre area involves a gamma function, which requires decomposing the function into polynomials. In addition, there are subtle deviations from the ideal forms defined by Eqn (1) which in practice result in significant effects such as preventing the calibre overfill, mitigating the extreme wear, etc.

For that reason, various authors suggested using partial functions to describe the calibre segments for both the simple and the complex geometries (Fig. 3).

The disadvantages of such approach are not only the complexity, but also the inability to embrace in the common observation space the calibres that otherwise belong to the same system.

In attempt to overcome these problems, a method was proposed for describing the calibres using vectors (Fig. 4).

This method allows for considerable accuracy with a large number of vectors and is well suited for computerised analyses of the broad variety of simple geometries. This straightforward method allows for adjusting individual vectors, which allows accounting for local changes in the shape and the wear of the groove contours [4].

Fig. 4. Vector description of the calibres [4]

In general, all above presented approaches imply that different methods of description are required to solve different tasks. Furthermore, it is possible to combine these different methods of description and integrate them in a hybrid models useful for solving a variety of the optimization problems.

More recently, it has been proposed that the category of simple symmetrical grooves can be efficiently defined using Chebyshev polynomials yielding the same advantages as the vector method [5]. In addition to enabling the convenient calculations of the calibre parameters, this approach is applicable also to complex symmetrical geometries via convenient mathematical manipulations [5].

In the pursuit for devising a generic transformation for the broader variety of the calibres including the asymmetric cases, the method embracing most if not all geometries of a practical significance is proposed below.

Rationale

The contemporary science and engineering are characterised by an impressive development of the information processors propelled by the so called Information Technology and the Computer Science, which together resulted in the accelerated accumulation of industrial records collected from the actual manufacturing processes.

The efforts to improve the industrial systems led to evolving the concept of digital twin-based intelligent manufacturing through the integration of artificial intelligence, Internet of things, big data, cloud computing, information communication, to finally realize the online machine learning applications within the I4 and I5 systems [6-8].

In the case of the rolling technology, this means that the possibility of extracting the new useful knowledge by analysing the immense repositories of industrial records has become a certainty. The suitable route for applying big data analytics and constructing the digital twins includes digitizing the industrial records.

A significant number of variables defining a rolling mill, the operation system and the rolling process itself (such as the tool dimensions, the distances along the rolling line, the number of the available passes, the temperatures, velocity, roll separating force and torque, the yield, productivity, the duration of the delays, the resource consumption data, etc.) are already recorded in digital format. However, the present format of the digital data defining the deformation zone parameters, in particular the morphometry of the entry and the exit cross-sections of the rolled solid needs to be translated into a more generic format. This is because the existing dimensioning conventions, and indeed the geometries involved in manufacturing the similar products, differ to an extent that makes the relevant industrial records mutually incompatible from the point of view of constructing a database suitable for the statistical analyses.

Levandovskij [9] and Mikhailenko et al. [10] discussed the issues that emerge when the pass geometry is described using mathematical terms. While the former has addressed the degree of discretization needed for sufficient representing the groove features, the later proposed fractioning the pass contour in the three-dimensional space.

In fact, there is no need for introducing the 3-dimensional space since the rolling pass is, for the most of analytic purposes, defined sufficiently by the entry and the exit cross-sections and by the roll diameter. However, the simple analytic functions (a line, an arc, a parabola) are not sufficient for identifying possible optimised details of the pass contour.

An additional issue of significance for controlling the deformation zone is the gradual change in the groove morphometry due to continuous wear across the interface of the tool - the roll - and the rolled material. The curves defining the profiles of the new and the worn surface need to be defined using the same mathematical form. In this way, the tribological variables can be correlated with the mathematical functions describing the change in the groove contour. From a purely mor-phometric viewpoint, given other the same, the change in the groove contour due to the wear can be correlated to the initial definition of the deformation zone, i.e. the definition of the unworn pass.

With having in mind the above, a viable solution for double symmetrical shapes and for moderately complex cases with one axis of symmetry have been presented in [5]. In the following section one solution for the more complex asymmetrical geometries is discussed.

A uniform procedure for defining the deformation zone morphometry

As outlined in [6] the deformation zone is sufficiently defined by the entry and the exit cross-sections of the rolled solid and by the relevant measure of the roll diameters. While for the simple cases discussed in [6] the orientation of the roll diameter(s) relative to the cross-section geometry (or vice-versa) is straightforward, this orientation requires an elaboration for the asymmetrical cas-

es. Figs. 5 to 9 provide illustrations for the proposed new method of solving this problem. For the simplicity, the two-roll configuration is used, nevertheless, in the case of a three or four roll configurations the procedure for defining the diameter orientation is analogous to the case of two-rolls.

a b

Fig. 5. The initial double symmetrical calibre (a), and the cross-section contour after rotating it for 90° (b)

The cross-section contour of the solid produced during the 1st pass in the double symmetrical «box» calibre (Fig. 5) is entered during the 2nd pass into the asymmetrical calibre shown in Fig. 6.

NEUTRAL LINE

Fig. 6. The calibre where the exit cross-section is formed during the second pass

The cross-section contour entering the 2nd pass is defined by the contour positioned in the 1st quadrant. The contour exiting the 2nd pass needs to be positioned in the coordinate system in order to define its partition relative to the initial coordinate system. The horizontal axis could be selected to coincide with the so-called «calibre neutral line» (N.L.) yet, as shown in Fig. 7, there is an orientation discrepancy between the neutral line and the initial coordinates. The two-dimensional scheme in Fig. 7 shows an overlay in which the entering contour is superimposed on the cross section of the calibre grooves at the instant of the first contact.

It should be noted that the locations of the initial contact points and the orientation of the entering contour are based on a compromise, and in reality, do not necessarily coincide perfectly with their actual configuration in the deformation zone. The disposition presented in Fig. 7 is based on mathematical idealisation of gripping a rigid two-dimensional groove contours shifted to the equal distances from the calibre neutral line. The entering «rigid» cross-section assumes the position that allows for minimising the calibre gap under the condition that no part of the involved contours is deformed.

To eliminate this misalignment, the complete Fig. 7 is rotated counter-clockwise through the angle defined by the vertex P and the conjoint axes marked in blue colour. The rotation centre is at the initial origin defined in Fig. 5 by the intercept of the axes of symmetry.

This allows uncovering a feasible position for the exit cross-section in the initial coordinate system. This is shown in Fig. 8 and 9, where the box cross-section assumes its original position. The pass coordinate system then allows for dividing the exit cross-section contour according to the procedure defined in [5]. The orientation (angle) for the groove roll diameter is defined relative to the vertical coordinate.

Fig. 7. First contact at the entry of the deformation zone in the 2nd pass

Fig. 8. First contact at the entry of the deformation zone and the groove inclination to fit the position of the entry contour in the coordinate system

The following asymmetric passes downstream the rolling line are positioned analogously by holding onto the axes of the initial box groove (the starting double symmetrical case) while rotating the entry & exit cross-sections pair locked into their grip-contact points.

During the process of finding the contact points between the entering cross-section and the calibre, the entering contour is been rotated until maximum four and minimum two contact points are established. In the next step, both contours are rotated backwards about the origin (0;0) of the initial coordinates (defined in the antecedent contour) until they return to the initial position. The procedure for positioning the next following pass is illustrated in the Appendix.

Fig. 9. The exit cross-section (coloured in red) positioned consistently with the coordinates of the entering box cross-section

The described manipulation allows for portioning both entry and the exit cross section in the four quadrants, as well as for defining the orientation angles for the roll diameters for each of the passes.

The digitizing is then finalised by applying the procedure explained in [5] onto each of the four portions of the contour separately. Each portion starts at its intercept with the _y-axis and ends as explained in [5].

After including other variables (chemical composition, temperatures, velocities, rolling loads, the tool-roll-characteristics, etc.) the order of the vector dimensions reaches the level of 102.

The basic principles of calibration in rolling technology allow for routinely deriving the details of the groove contour transition to the roll collars - the light-blue coloured roll gap (roll clearance) in Fig. 9 - while maintaining the optimised pass morphometry inferred by analysing the digitized database.

Conclusions

The efforts for actualizing a breakthrough in technology and rising the level of the sustainabil-ity of manufacturing systems via implementation of I4 and I5 have given significant impetus to the concept of the Digital Twins. In the context of the rolling industry, a propitious version of Digital Twin is a replica of calibration which is the embodiment of the instructions for the realisation of the rolling process, including the design, monitoring, control and maintenance.

While attracting strong interest from the industry (the market is forecasted to reach over $15 billion by 2023) Digital Twin related engineering research is in its infancy [6], and hence the need for the contributions such as presented in this manuscript. An aspect of fundamental importance in the calibration and in the design of Digital Twins is the mathematical definition of the deformation zone geometry and, in particular, digitizing the incoming and exiting contours of the rolled solids.

The proposed method for digitizing the pass morphometry opens the gate for mobilizing the existing computing potentials and analytic theories and apply them for improving the sustainability of rolling systems that lie in the heart of the largest known industries: the intermediate processing of engineering materials.

It is significant to note that the proposed method conforms also to the simple double symmetrical geometries, thus eliminating the barriers for analysing the wide range of calibration cases within the same space of observations.

The outlook of needing the vectors that comprise over hundred components is arguably repulsing in the eyes of a human analyst. However, these vectors are not designed for processing by the human minds—they are intended for the analyses performed by the Artificial Intelligence. Although, we, the humans, are able of visualizing only four or at most six dimensional phenomena, we are capable of visualizing, designing and employing, the machine learning tools that can analyse multi-dimensional vectors and unravel the new knowledge buried in the myriad of the industrial records.

References

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2. Смирнов В.К., Шилов Ю.А., Инатович Ю.В. Калибровка прокатных валков. М.: Металлургия, 1987. 368 с.

3. Берковский В.С., Жадан В.Т., Шишко В.Б. Аналитическое описание формы калибров // Тр. ин-та / Мос. ин-т стали и сплавов. 1979. Вып. 118. С. 13-18.

4. Эффективность деформации сортовых профилей / С.А.Тулупов, Г.С.Гун, В.Д.Онискив и др. М.: Металлургия, 1990. 280 с.

5. Spuzic S. Translating Morphometric Information About the Rolling Passes Into a Structured Database // Калибровочное бюро. 2020. №17. С. 5-12. URL: http://passdesign.ru/number17 (дата обращения: 30.06.2021).

6. Lu Y., Liu C., I-Kai Wang K., Huang H., Xu X. Digital Twin-driven smart manufacturing: Connotation, reference model, applications and research issues // Robotics and Computer-Integrated Manufacturing, vol 61, 2020. 101837, ISSN 0736-5845. URL: https://doi.org/10.1016/j.rcim.2019.101837.

7. Fahle S., Prinz C., Kuhlenkotter B. Systematic review on machine learning (ML) methods for manufacturing processes - Identifying artificial intelligence (AI) methods for field application // Procedia CIRP, vol 93, 2020. P. 413-418, ISSN 2212-8271. URL: https://doi.org/10.1016/j.procir.2020.04.109.

8. Kotsiopoulos T., Sarigiannidis P., Ioannidis D., Tzovaras D. Machine Learning and Deep Learning in smart manufacturing: The Smart Grid paradigm // Computer Science Review, vol 40, 2021. 100341. ISSN 1574-0137. URL: https://doi.org/10.1016/j.cosrev.2020.100341.

9. Левандовский С.А. К вопросу о дискретизации описания формы калибров // Калибровочное Бюро. 2013. №1. С. 56-65. URL: http://passdesign.ru/number1 (дата обращения: 30.06.2021).

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Acknowledgement: The work on this manuscript has been supported by the STEM organizational unit at the University of South Australia.

Appendix

Figs. I to IV illustrate the pass orientation procedure for the third pass.

S.a 8.8 6,6 8.7 7.4 10.3 M Caliber for

pass 2

o>

N.L.

a

b

Fig. I. The entry contour (a) and the 3rd calibre (b)

Note that the coordinates in Fig. Ia coincide with the coordinates of the initial double symmetrical contour (shown in the Figs. 5b and 9 in the manuscript body). The solid cross-section produced in the first asymmetrical calibre (Fig. 6 in the manuscript body) is entered into the asymmetrical calibre №3 as shown in Fig. II.

Fig. II. The position of the entry cross-section at the instant of the first contact with the calibre.

Note that the complete Fig. II is yet to be rotated in order to assume the orientation required by the axes of the initial double symmetrical contour

For the 3rd pass, the orientation of the coordinates needs to be identical to the pass 1 and pass 2 coordinates (shown in the Fig. 5b in the manuscript body). The initial horizontal axis of the calibre 3 coincides with so-called «calibre neutral line». Since the initial position of the entry contour is defined by the points of the first contact, the counter-clock rotation of the configuration shown in Fig. II is needed to orient the x-axis into the horizontal position (and the_y-axis into the vertical position).

<

С

N.L.

Fig. III. First contact at the entry of the calibre 3 deformation zone; the orientation of both entry contour and the calibre 3 is rotated to match the orientation

of the initial double symmetrical contour

This leads to uncovering a feasible solution for the orientation of the exit cross-section shown in Fig. IV in red colour. The pass coordinate system then allows for partitioning the exit contour as specified in reference [5]. The orientation (angle) for the calibre 3 roll diameter is defined relative to the vertical coordinate.

Fig. IV. Contour at the entry and the exit of the deformation zone showing the groove inclination to fit the orientation of the initial double symmetrical contour

The point on the grove contour at which the roll diameter is measured, is selected as the point of the first contact resulting in the smallest diameter value. When the nominal roll diameters are not all equal, then the analogous procedure is required to define each roll diameter.

Digitizing the light-blue (cyan) coloured segments is not described in the above procedure. The basic calibration design routines allow for deciding about such details. Notwithstanding, the analogous procedure allows for digitizing the complete groove including the roll gap if the analysis requires so.