Improved Billet Shape Modeling in Optimization of the Hot Rod and Wire Rolling Process

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Institutionen för systemteknik

Department of Electrical Engineering

Examensarbete

Improved Billet Shape Modeling in Optimization of

the Hot Rod and Wire Rolling Process

Examensarbete utfört i Reglerteknik vid Tekniska högskolan i Linköping

av

Jan Betshammar

LiTH-ISY-EX--06/3906--SE

Linköping 2006

Department of Electrical Engineering Linköpings tekniska högskola Linköpings universitet Linköpings universitet SE-581 83 Linköping, Sweden 581 83 Linköping

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Improved Billet Shape Modeling in Optimization of

the Hot Rod and Wire Rolling Process

Examensarbete utfört i Reglerteknik

vid Tekniska högskolan i Linköping

av

Jan Betshammar

LiTH-ISY-EX--06/3906--SE

Handledare: Daniel Axehill

Linköpings universitet

Rickard Lindkvist

ABB AB,Corporate Research

Examinator: Alf Isaksson

Linköpings universitet

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Avdelning, Institution

Division, Department

Division of Automatic Control Department of Electrical Engineering Linköpings universitet

SE-581 83 Linköping, Sweden

Datum Date 2006-12-08 Språk Language Svenska/Swedish Engelska/English Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats Övrig rapport

URL för elektronisk version

http://www.control.isy.liu.se http://www.ep.liu.se/2006/3906 ISBN — ISRN LiTH-ISY-EX--06/3906--SE

Serietitel och serienummer

Title of series, numbering

ISSN

—

Titel

Title

Förbättrad geometrimodellering av heta och valsspår för optimering inom profil-valsningsprocessen

Improved Billet Shape Modeling in Optimization of the Hot Rod and Wire Rolling Process

Författare

Author

Jan Betshammar

Sammanfattning

Abstract The hot rod and wire rolling process is widely used to produce rolled iron alloys in different shapes and dimensions. This industry has been under a constant development during the last decades. Today, complex geometries are produced at a high speed since it is possible to use several stands in each mill at the same time. A reason for the development is rising demands from customers. The most important demands are to save energy, to get better material properties and higher dimension accuracy. To meet these demands on speed and accuracy, a better control of how the material behaves in the process is needed. There is also a need to be able to quickly find a new setup of the mill in order to be able to produce other geometries.

The purpose with this Master Thesis is to model and simulate the hot rod and wire rolling process with the modeling language Modelica. The model is given the known inputs and the desired final result in order to compute the unknown inputs to the mill. To meet these goals, a model that depends on for example the gap between the rolls, the roll speeds and the tensions between different stands is needed. It should be possible to make simulations to find roll speeds or to calculate the tensions caused by known roll speeds.

With the help of models of the steps in the process, a model has been developed in Modelica. The model can be expanded to a mill with an arbitrary number of stands. In the search for the best way of modeling a hot rod and wire rolling mill, several algorithms have been simulated and analyzed in Modelica. The results from all simulations show that the billet and the groove should be described by different functions for the upper and the lower half. Furthermore, it is not a good solution to use only polynomials to describe the shapes in the process. A function with infinite derivative in the endpoints is needed to describe the billet in an acceptable way. The problem has also been solved using Matlab. In this work it is shown that the Modelica solution is preferred, compared to solving the optimization problems in Matlab. An advantage with the Modelica solution is that the model can be split into several easily connected sub models. Unfortunately it was even hard for Modelica to solve general problems. The describing functions made it hard to find the intersections and to keep the area constant during the rotation. The least square method could lead to bad approximations of the shapes.

Nyckelord

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Abstract

The hot rod and wire rolling process is widely used to produce rolled iron alloys in different shapes and dimensions. This industry has been under a constant development during the last decades. Today, complex geometries are produced at a high speed since it is possible to use several stands in each mill at the same time. A reason for the development is rising demands from customers. The most important demands are to save energy, to get better material properties and higher dimension accuracy. To meet these demands on speed and accuracy, a better control of how the material behaves in the process is needed. There is also a need to be able to quickly find a new setup of the mill in order to be able to produce other geometries.

The purpose with this Master Thesis is to model and simulate the hot rod and wire rolling process with the modeling language Modelica. The model is given the known inputs and the desired final result in order to compute the unknown inputs to the mill. To meet these goals, a model that depends on for example the gap between the rolls, the roll speeds and the tensions between different stands is needed. It should be possible to make simulations to find roll speeds or to calculate the tensions caused by known roll speeds.

With the help of models of the steps in the process, a model has been developed in Modelica. The model can be expanded to a mill with an arbitrary number of stands. In the search for the best way of modeling a hot rod and wire rolling mill, several algorithms have been simulated and analyzed in Modelica. The results from all simulations show that the billet and the groove should be described by different functions for the upper and the lower half. Furthermore, it is not a good solution to use only polynomials to describe the shapes in the process. A function with infinite derivative in the endpoints is needed to describe the billet in an acceptable way. The problem has also been solved using Matlab. In this work it is shown that the Modelica solution is preferred, compared to solving the optimization problems in Matlab. An advantage with the Modelica solution is that the model can be split into several easily connected sub models. Unfortunately it was even hard for Modelica to solve general problems. The describing functions made it hard to find the intersections and to keep the area constant during the rotation. The least square method could lead to bad approximations of the shapes.

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Sammanfattning

Profilvalsning används idag globalt som en process för att tillverka metaller i olika former och har under de senaste årtiondena varit i en lång utvecklingsfas. Numera valsas komplicerade geometrier samtidigt som produktiviteten har ökat drastiskt tack vare möjligheten att valsa med flera valspar samtidigt. Under utvecklings-processen har även kraven från kunderna ökat, främst vad gäller energibesparing och noggrannhet på den slutgiltiga produktens dimensioner och materialegenska-per. För att kunna möta dessa krav på snabbhet och noggrannhet krävs större kontroll av hur materialet beter sig i valsningsprocessen och att det snabbt går att bestämma hur valsverket ska ställas in när en ny serie ska tillverkas.

Syftet med examensarbetet som redovisas i denna rapport är främst att mo-dellera och simulera profilvalsningsprocessen med modelleringsspråket Modelica. Modellen ska sedan kunna användas för att utifrån givna indata till processen be-stämma olika inparametrar till valsverket för att uppnå önskat slutresultat. För att nå dessa mål krävs en modell som bland annat beror på spelet mellan valsarna, valshastigheterna och dragen mellan olika valspar. Simuleringar ska kunna göras för att både bestämma önskade valshastigheter och för att beräkna vilka drag givna valshastigheter ger upphov till i valsverket.

Med hjälp av modeller och beskrivningar av de olika stegen i valsningsproces-sen har en modell tagits fram i Modelica. Modellen har en given inprofil och kan sedan byggas ut till ett valsverk med önskat antal valspar. I Modelica har olika modeller och algoritmer simulerats och analyserats för att kunna svara på hur ett profilvalsverk modelleras på bästa sätt. Resultaten från de olika simule-ringarna visar att heta och valsspår bör beskrivas med olika funktioner för den övre och för den undre delen av profilen. Det är ingen bra lösning att endast an-vända polynom för att beskriva de olika formerna. En basfunktion med oändlig derivata i ändpunkterna är nödvändig för att beskriva hetan på ett bra sätt. En fördel med Modelica är även att modellen enkelt kan delas upp i flera enkelt sammankopplade delmodeller. Tyvärr var det svårt även för Modelica att lösa generella problem. Funktionerna som användes för att beskriva de olika formerna gjorde det svårt att hitta skärningspunkter och att hålla arean konstant under rotationen. Minsta kvadratmetoden resulterade ibland i dåliga approximationer av de olika formerna.

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Acknowledgments

First of all I would like to thank ABB AB, Corporate Research which has made it possible for me to do this Master Thesis work. During my work atABB, I have got a lots of technical experiences, but also other important experiences like getting insight into how projects are run in a large company.

I would also specifically direct a thank to my supervisor Rickard Lindkvist and the project members Jens Pettersson and Anders Daneryd atABBfor all coaching and all interesting discussions during these weeks. These have contributed to this Master Thesis in an important way, and have further enriched my stay at the company.

I would also like to thank my supervisor at Linköping University Daniel Axehill and my examiner Alf Isaksson for the support, motivation and guiding through the work. Also a huge thank to my opponent Johan Bergström for putting time and effort into reading my report and give me valuable feedback.

Finally, I want to thank all others making their Master Thesis at ABBand all employees at the department for a pleasant time and exciting football matches.

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Introduction

The hot rod and wire rolling process is widely used to produce rolled iron alloys in different shapes and dimensions. This industry has been under a constant development during the last decades driven by the competition between companies and the constantly rising demands from customers. Some of these demands are to save energy and money and other are about material properties and dimension accuracy. Another important optimization parameter when rolling long products is to maximize the production rate and minimize the shutdown time between two rolled products. During the change between two products, a complex optimization problem has to be solved. To minimize the shutdown time, this problem has to be solved in a faster and more reliable way. One part of this problem is to find the intersection between rolls and the input billet. The task atABB has been to model and optimize this cross-section and the model of the hot rod and wire rolling process.

1.1 Background

ABBis a leading long-term supplier of Industrial IT solutions, electrical equipment, drives and control systems for hot rod rolled steel. ABB Industrial IT solutions cover the complete scope from electrical equipment, drives and motors, automation solutions and production control, as well as provide a full range of professional services for the complete plant life cycle. ABB has products from power supply, drive systems to plant level control, with control that covers the whole range of long product rolling. In 1894,ASEA, nowABB, supplied the world’s first electrically driven rolling mill, at a time when competitors refrained from quoting electrical drive systems. Since this early start,ABBis known to be a technology leader that today has more than 150 digitally controlled mills all over the world.

Today, mills use several stands (pair of rolls) and geometries to produce ad-vanced shapes. The demands on the final products are very high. For example accuracy and tolerances on the shapes and material properties are important. An

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incorrect calculation can result in big losses. For example, compression between two stands which causes buckling. If the buckling is too large the billet will be compressed to a tangle and the process has to stop for several hours before the production can start again, see Figure 1.1. This and many other reasons lead to the obvious fact that it is important to describe the geometries of grooves and billet in a good way in the production of long products. To be able to handle the customer’s requirements on good and accurate results and at the same time maintain a high production speed, more knowledge is needed. Knowledge about the rolling process and the way of describing and modeling the roll geometries in an analytical way is essential to speed up the process. One important part of the models used byABBtoday is the way in which the circumference of the rolled material is mathematically described.

Figure 1.1. Example of buckling in hot rod and wire rolling

1.2 Purpose

The purpose with this Master Thesis is to improve the mathematical descriptions of the cross-section in hot rolling of long products in order to significantly speed up and stabilize various optimization and simulation tasks in Matlab [6] and Modelica [7]. During these tasks the cross-section in hot rod and wire rolling is recomputed a great number of times and therefore these computations have to be very fast. The optimization tasks involve for example, maximizing production rate, minimizing energy consumption or specifying a billet target width at each pass. After several hundred iterations the optimization converges, and in all iterations the shape of the billet has to be recomputed. Important questions are;

• Which is the best way to describe the geometries of rolls, one or several polynomials, circles, ellipses or in some other way?

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1.3 Scope 3

• In which way should the geometries be computed? • In which way should the optimization be done?

• How should the bulging of the free side of the billet be described?

• How should the billet area be calculated somewhere at the production line?

Several tests should be carried out and their purposes are to create clear results to evaluate. The evaluations will later on serve as a recommendation forABB. The work is primarily connected to a specific project but the outcome of this report may also be used in future projects.

1.3 Scope

Today ABB has a model that describes and models the hot rod and wire rolling process. In this model the cross-sections and all different shapes are described with discrete points. This Master Thesis shall result in an analytical work on billet shapes that describes and models a selection of rolling cases in a new way. The model will not model temperature variations or other material properties, but the geometries of the billet, roll speeds and tensions between different stands. With help of these new models introduced in Matlab and Modelica, this work will hopefully lead to improvements in future hot rod and wire rolling projects at

ABB.

1.4 Problem Description

Hot rolling of long products is a very complex process where many factors gradually change the final geometry and the properties of the rolled material. This process can lead to a general non-linear optimization problem with equality and inequality constraints. When changing the mill to produce another type of shape this complex problem has to be solved to find new control signals. Some of these signals are roll speed (change the production rate) and the gap between two rolls (change the reduction at each stand). The most important parts of the problem are to find the intersections between rolls and billet in every stand, calculate the area of the cross-section before and after each stand and rotate the billet between two stands, see Figure 1.2. The mills today are constructed as automatic lines where every other pair of rolls in most cases are placed in vertical and horizontal position to keep the shape of the billet. In our model this is instead handled by rotating the billet to make it possible to use the same model for every stand.

The major problem and the big task in this Master Thesis is to find an eas-ier and more efficient way to model the cross-section in hot rod and wire rolling processes. The model will be depending on the number of stands in the mill, dimensions of each roll, the size and geometries of the input billet, tensions and roll speeds. The second problem is to optimize this model for all stands in a mill with help of Matlab [6], Dymola [2] and the object oriented modeling language

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Figure 1.2. The cross-section in hot rod and wire rolling

Modelica [7] where analysis of different sets of grooves will be carried out. By modeling several stands and grooves in Matlab and Modelica, simulations will be made to illustrate that it is possible to model the hot rolling of long products with a higher rolling speed and with a more stable result than today. It will hope-fully lead to some answers in which way the geometries of rolls shall be described in the future.

1.5 Definitions

In this section, definitions of all quantities, terms, constants and abbreviations that are used in this report are presented, see Figure 1.3 and Table 1.1–1.3.

Oval Groove Round Groove Vertical Stand Horizontal Stand Rod/Billet/Material Rolls

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1.6 Thesis Structure 5

Quantity [SI unit] Description

A1, A2, Ai [m2] Cross-sectional area

A1e, A2e [m2] Equivalent cross-sectional area

AH [m2] The part of the cross-section area of the billet before

a stand that is outside the groove b1, b2, bi [m] Width of billet

be [m] Equivalent width of billet

D [m] Roll diameter

Dt[m] Theoretical roll diameter

Dwm[m] Mean/average/equivalent effective roll diameter

gap [m] Roll gap h1, h2, hi [m] Height of billet

h1e, h2e[m] Equivalent height of billet

h1m, h2m, him[m] Mean height of billet

hmax [m] Maximum height of billet

l1, l2, li [m] Length of billet

Ldm [m] Mean projected arc of contact

˙

mi [m3/s] Mass/Volume flow (density is constant)

R [m] Roll radius

Re [m] Equivalent roll radius

Rm [m] Mean roll radius

v1, v2, vi [m/sec] Speed of billet

vN [m/sec] Speed at the neutral plane

vr [m/sec] Peripheral speed of rolls

vp [m/sec] Horizontal component of vr

V1, V2, Vi [m3] Volume of billet

Table 1.1. Definition of quantities, where subscript 1 refers to a section of the billet

before a stand, subscript 2 to a section of the billet after a stand and subscript i to a section of the billet

1.6 Thesis Structure

Chapter 2, Preliminaries: Briefly describes rolling from an industrial point of

view and explains important mechanical mechanisms that have to be con-sidered during the modeling of the hot rod and wire rolling process. The chapter also presents the software and the programming languages that are used in this Master Thesis.

Chapter 3, Model: Describes the model of a stand and how it works. Presents

different models and algorithms.

Chapter 4, Simulations and Results: Presents different simulations and

re-sults during the development of the hot rod and wire rolling model.

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T erm [SI unit] Description α [rad] Bite angle

β [dim. less] Coefficient of spread γ [dim. less] Coefficient of draught

γm [dim. less] Mean/average/equivalent coefficient of draught

δ [rad] Neutral angle δw [dim. less] Form factor

δwm[dim. less] Mean/average/equivalent form factor

w[dim. less] Roll factor/thickness ratio

wm[dim. less] Mean/average/equivalent roll factor/thickness ratio

λ [dim. less] Coefficient of elongation

κ [dim. less] Coefficient in the spread formula investigated by Shi-nokura and Takai

W [dim. less] Coefficient in the spread formula investigated by Wusatowski

Wm[dim. less] Mean/average/equivalent coefficient in the spread

formula investigated by Wusatowski

Table 1.2. Definition of terms

Abbreviation Description

F F T Fast Fourier Transform QP Quadric Programming

Table 1.3. Definition of abbreviations

Chapter 6, Future Work: Discusses how the models in this Master Thesis can

be developed to improve future hot rod and wire rolling modeling and opti-mization.

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Chapter 2

Preliminaries

This chapter will present the basics of the hot rolling theory for long products, programs and programming languages and other theory that is necessary to un-derstand the main part of this report.

2.1 Basic Theory for the Hot Rolling Process

The hot rod and wire rolling process is a very complex process to produce bars, rods and wires in different geometries and dimensions because it is depending on many physical quantities and mechanisms. For further investigation it is important to understand the basic hot rolling theory. The most important variables and mechanisms in this process will be described and presented in this chapter.

Figure 2.1. Example from a hot rolling processes

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2.1.1 The Industrial Hot Rod and Wire Rolling Process

The rolling industry can be divided into two main parts, cold rolling (tempera-tures below the recrystallization temperature) and hot rolling (tempera(tempera-tures above the recrystallization temperature). In this Master Thesis only hot rolling will be discussed, see Figure 2.1. Hot rolling can also be divided into two categories, flat rolling and rolling of long products. In the flat rolling process, different sheets of metal are produced and in the production of long products, wires, rods and other shapes are produced. The primary part in this report will be the production of long products, i.e., rods and wires, see Figure 2.2.

Figure 2.2. View over a rod roll mill

There are many advantages with hot rolling compared to cold rolling. During the hot working of metals, the phenomena of strain-hardening and recrystallization take place simultaneously. With increased temperature, the billet becomes softer and more ductile since yield stress decreases [10]. This leads to that the material is easier to deform with a more accurate and stable result and that the billet will get better final geometry and material properties. Unfortunately, hot rolling has some negative effects to; the material has to be heated up in a furnace which requires a lot of energy and the work has to be done carefully while the temperature in the process is very high.

In the hot rolling industry it is very expensive and time-consuming to make a mistake. For example wires can reach velocities above 100 m/s at the end of the production. For that reason it is important that the settings of all parts in the mill have been calculated correctly before the process starts. One of these settings in

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2.1 Basic Theory for the Hot Rolling Process 9

the mill is the amount of reduction of metal in each pass. This has to be controlled in a way that the forces and torques on the rolls and their motors do not become too high.

When the mill is configured and is ready to start, the first step is to heat the metal in the furnace to give it the initial rolling temperature. The next step is to lead the billet into the roll stands. After it has passed one or several stands, see Figure 2.2, it is finished and has its desired shape, which is completely different from the input shape. In many cases, also the material properties are changed during the process, which often is undesirable. The fact is that the dimensions after the final stand will change and shrink when the iron is cooling down.

Today the rolling mills have several and often up to 20 stands. As a consequence of the rising number of passes in the mill, the risk for failures is increasing. To reduce this risk, more careful calculations and preparations have to be done. Most of the mills today are working by the principle “trial and error” and are using the operator’s knowledge and experience in their notebooks to setup the mill. This approach is ineffective and the knowledge takes a long time to build up, and might be hard to maintain. Therefore, an efficient and user friendly tool is really wanted to help operators when changing between two products in the mill.

Rod and Wire Rolling

In the part of hot rolling when rods, wires and other long products are produced the billet is formed between rolls with different grooves, see Figure 2.3. Shapes are designed such that the billet gets the desired geometry after the final stand. The purpose of most of the rolls in the mill is to quickly and carefully reduce the area of the cross-section of the billet. The final forming of the billet is actually done by a few passes at the end, often the 2–4 last stands. The rolls in these stands are designed to get a desired final result with certain geometries, dimensions, material properties and demands on the surface.

(a) Rod (b) Wire

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There are many combinations of rolls in mills today, called rolling-series. The most common rolling-series are the square-oval and the round-oval series, see Fig-ure 2.4. As a result of today’s modern rolling compared to the old processes, where everything was made by hand, there is no longer any need to reduce the dimen-sions of the billet in as few passes as possible. For that reason it is more common to use the more stable and flexible round-oval series. This series requires more stands but the result is better and the security during the production is higher. The mills today are constructed as automatic lines where every other pair of rolls in the most cases is placed in vertical and horizontal position to reduce the billet from both sides. This design suits the round-oval series very well because there is no need to turn the billet between the passes. A disadvantage with this design is still that the round passes are unstable and it requires that the oval passes are in vertical position during the whole process. Square passes, on the other hand, have the ability to turn the billet automatically between two stands.

(a) Oval (b) Round

(c) Bastard-round (d) Square

Figure 2.4. Different types of grooves in hot rod and wire rolling

2.1.2 Fundamentals of Rolling

There are two ways in which metals can be deformed, elastic or plastic deformation. The elastic deformation is reversible, which means that once the forces are no longer applied, the object returns to its original shape. The plastic deformation is non-reversible. An object in the plastic deformation range will first have undergone elastic deformation, which is reversible, so the object will return partly to its

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original shape [9]. The way in which the metal is deformed in the longitudinal hot rolling is by plastic deformation. It consists in applying compressive forces of appropriate magnitude to the metal being deformed.

The hot rod and wire rolling theory is often based on flat rolling and because of this all long product problems have to be transformed into a flat rolling problem. This theory requires many symbols and quantities to describe the process and the best way to introduce and illustrate some of them is to show a figure, see Figure 2.5. The meaning of these symbols can be seen in Section 1.5 and they will be discussed further on in this chapter.

R

v

1

_h

1

b

1

l

1

α

δ

v

2

h

₂

b

2

l

2

v

r

v

r

Figure 2.5. Schematic representation of flat rolling, where symbols with subscript 1 is

related to conditions before and 2 after the deformation zone in a stand

Conservations of Volume and Mass Flow

When the classical hot rolling theory was derived many assumptions were made. One of the most important when plastically deforming a metal is that the volume and the mass flow remains constant during successive stages of deformation. The conservation of volume can mathematically be formulated

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where

subscript 1 – refers to a section of the billet before a stand, subscript 2 – refers to a section of the billet after a stand,

subscript i – refers to some section of the billet before or after a stand. The fact is that this is not entirely correct, but it matches the reality very well [10]. On dividing the relation, it becomes

V2 V1 = h2b2l2 h1b1l1 = γβλ = 1 (2.2) where

γ is the coefficient of draught, β is the coefficient of spread, λ is the coefficient of elongation.

The product of the deformation coefficients must always be equal to 1 if the condition of constant volume is to be satisfied.

Since the density of the material is constant in the rolling process it could be removed from all mass flow expressions and the conservation in the mass flow can be expressed as

˙

mi= hibivi (2.3)

where subscript i – refers to some section of the billet.

Neutral Plane

The deformation zone is the most critical part of the hot rolling process. During this zone the billet speed increases from v1 to v2, see Figure 2.5. At the plane of

entry (at the bite angle α) the peripheral speed of rolls vris greater than the speed

of the billet v1, see Figure 2.6(a). This difference is called the forward slip (the rolls

slip forward). The corresponding difference in speed at the exit plane is denoted the backward slip. In one point the billet has the same speed as the rolls. This is called the neutral plane and is given by the neutral angle δ, see Figure 2.6. At the neutral plane the speed of the billet, vN, is equal to the horizontal component, vp,

of the peripheral speed, vr, of the rolls, see Figure 2.6(a).

Since the speeds are equal only at the neutral plane, this point is very im-portant. With use of this relation the entry and exit speeds of the billet can be calculated with help of the mass flow . With the principle of conservation of volume shown in Equation 2.1 and 2.2 things like spread and elongation can be expressed. The problem is how to calculate the neutral angle δ which depends on many quantities like the rolling speed vr, entry speed v1, reduction γ of the billet,

bite angle α etc. The derivation of δ has been made by for example Koncewicz and is presented by Wusatowski [10].

Another conclusion regarding the variation of billet speed in the deformation zone is that different friction mechanisms occur, see Figure 2.6(b). The frictional coefficient varies and depends on temperature, roll pressure, speed of rolls and

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v

1

α

δ

v

_r

v

_p

v

2

(a) Velocities and angles in rolling

v

₁

δ

Neutral plane

v

₂

Direction of

roll rotation

frictional forces

Direction of

(b) Frictional forces during the rolling

Figure 2.6. Quantities and terms in the longitudinal rolling process

conditions at the surfaces in contact. Often, a certain mean value of the friction for the whole roll gap is calculated [10].

Transforming Rod and Wire Rolling to Flat Rolling

When rolling rods, wires and other non-rectangular sections the rolls are not plane as in flat rolling. In this process the rolls have a groove, see Figure 1.3, Figure 2.4 and Appendix A. To maintain the principle of constant volume in Equation 2.1, the problem has to be translated into a flat rolling case. One approach to handle this is to add an additional term, the mean height hm of the billet. This can be

expressed as him= Ai bi (2.4) where

Ai is the cross-section area at position i between two stands,

biis the maximum width of the billet at this position.

When calculating this mean height it is important to know the geometries of the billet before and after the deformation zone. It is also important how the

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cross-sectional area and the breadth of the billet will be calculated. Equations 2.1 and 2.2 can using this mean height be expressed as

V1= V2= Vi= h1mb1l1= h2mb2l2= himbili (2.5) and V2 V1 =h2mb2l2 h1mb1l1 = γmβλ = 1 (2.6) where h2m h1m = A2 b2 b1

A1 = γm– denotes the mean coefficient of draught [10].

To solve these equations h1m and h2m must be calculated first. It is often

difficult and sometimes even impossible to calculate the areas that are needed. To find the mean height of the roll pass and billet in an easier way, Wusatowski developed another method to determine the average height hm of a given shape

as [10]

hm= hmaxm (2.7)

where

hmax is the maximum height of the section,

m is the coefficient determined for different shapes.

This method only works for simple shapes and geometries, and is not as general as desired.

A third approach that could be used to transform the rod and wire rolling problems into flat rolling problems instead of mean and average values is Lendl’s method of equivalent dimensions. With help of this method it is possible to use all flat rolling theories in the industry of long products. In Lendl’s method, equiv-alent dimensions are used instead of true and mean values in the calculations. In this theory a general geometry is transformed to an, in some sense, equivalent rectangular cross-section for which flat rolling relations are employed. The Lendl approach does not rely on thorough analytical or experimental work and an error of several percents in width estimation may occur according to Lendl’s original paper [4]. The equivalent dimensions are width be, height hie, area Aie and roll

radius Re, see Figure 2.7. The calculation of them follows the steps below

1. be is the distance between the intersections of the cross-section of the input

billet and the groove.

2. A1e and A2e is the part of the input billet(1)/groove(2) cross-section area

within be. These areas belong to so-called equivalent rectangles, dashed lines

in Figure 2.7.

3. h1e and h2eare calculated by dividing corresponding area by the equivalent

width be, i.e. h1e= A1e/beand h2e= A2e/be.

4. Re= R − h2e/2 + gap/2 where gap denotes the gap and R is the roll radius.

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2.1 Basic Theory for the Hot Rolling Process 15 b 1e h 1e h2e gap R R e A 1e A2e

Figure 2.7. Definition of equivalent dimensions

2.1.3 Theory of Spread

The increased breadth of the billet during rolling is called spread, see Figure 2.8. The amount of spread depends on many factors, such as the tension, temperature, friction, rolling speed, initial shape of the billet, type of steel, ratio of bar diame-ter to roll diamediame-ter, condition of roll and metal surface, etc. Many investigators attempted to consider all these factors and derive a general formula for the spread ∆b = b2−b1, as for example Sieble, Tafel, Sedlaczek, Zolotnikov, Tselikov and

oth-ers presented by Wusatowski [10]. Othoth-ers are for example Wusatowski himself [10] and, Shinokura and Takai [8].

Groove Input Output

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Wusatowski

Wusatowski derived an expression to calculate the spread in flat rolling that have been very popular and often used in different applications. In this formulation a form factor

δw=

b1

h1

(2.8)

and a roll factor, or thickness ratio,

w=

h1

D (2.9)

have been introduced.

The final equation to calculate the spread in flat rolling is found in the form [10]

β = γ−W (2.10)

where log10(W ) = −1.2690.56w δw

Using Equation 2.2, the elongation can be calculated as

λ = γ−(1−W ) (2.11) or the spread as a function of the known elongation

β = λW/(1−W ) (2.12) In rod and wire rolling the mean/average/equivalent coefficient of draught, γm,

is used to transform (2.10)–(2.12) to a flat rolling problem. Equation (2.10) will be as

β = γ−Wm

m (2.13)

where

log10(Wm) = −1.2690.56wmδwm,

wm= h1m/Dwmis the mean/average/equivalent roll factor,

δwm= b1/h1mis the mean/average/equivalent form factor,

Dwm= Dt− h2mis the mean/average/equivalent effective roll diameter,

Dt is the theoretical roll diameter, the distance between the roll axes

to-gether with the roll gap. The (2.11) will take form as

λ = γ−(1−Wm)

m (2.14)

and (2.12) as

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Shinokura and Takai

Another popular theory to calculate the spread in rod and wire rolling was exper-imentally investigated by Shinokura and Takai [8]. They made experiments for four different series; Square-Oval, Round-Oval, Square-Diamond and Diamond-Diamond. The geometry of the new output surface differs for these types of passes. For the Square-Oval pass it was assumed to be a straight line, in the Round-Oval pass it was approximated by two circular arcs, in the Square-Diamond pass the surface was parallel to the input and in the Diamond-Diamond pass the output has a smaller angle to the x-axis than the input.

The new intersections between the output billet and the groove as well as the pass area could be calculated exactly for all the passes. The spread formula is very simple and has only one coefficient. The formulation can predict the spread for all four types of passes mentioned above with high accuracy. The spread formula of Shinokura and Takai can be expressed as [8]

b2− b1 b1 = κ · Ldm b1+ 0.5h1 ·AH A1 (2.16) where

b1 is the width of the billet before rolling,

b2 is the width of the billet after rolling,

κ is a coefficient,

Ldmis the mean projected arc of contact =pRm(h1m− h2m),

h1mis the mean height of the billet before rolling,

h2mis the mean height of the billet after rolling,

Rmis the mean roll radius,

h1 is the maximum height of the billet before rolling,

AH is the part of the cross-section area of the billet before rolling that is

outside the groove,

A1is the cross-section area of the billet before rolling.

The common value of κ for all four passes is 0.83. Practical values for κ could be seen in Table 2.1. Pass κ Square-Oval 0.92 Round-Oval 0.97 Square-Diamond 0.83 Diamond-Diamond 0.95

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2.2 Software Tools

To be able to develop and evaluate different models and approaches it is necessary to use some programs with different functionalities and properties. This section will present these programs and programming languages and how they will be used in this Master Thesis.

2.2.1 The Use of Different Programs

The main objective with this work is to develop a model in Modelica that han-dles different grooves and the spread in hot rod and wire rolling. The choice of Modelica has been made because it should be easy to expand a mill with a new stand or to use other geometries. In Modelica each stand can be represented with a block and to expand a mill with a new stand only requires copying a block and connecting it to the others through connectors, see Figure 2.9. To obtain this goal several models and algorithms have to be modeled and evaluated. An easy way to do this is to use Matlab during the development for easy modifications and visualization. Thereafter, the models are transformed into Modelica and then visualized and verified with the help of the graphics in Matlab.

Figure 2.9. Examples of some blocks in Modelica

Matlab

Matlab is a high-performance language for technical computing. It integrates computation, visualization, and programming in an environment where problems and solutions are expressed in familiar mathematical notation. It is possible to include math, to do simulation and to visualize scientific and engineering graph-ics. The program also include graphical user interface building. In the industry, Matlab is used for high-productivity research, development, and analysis.

The Matlab system includes, e.g., functions, programming language and graphics. The Matlab mathematical function library is a collection of computa-tional algorithms ranging from elementary functions, like sum, sine, cosine, and complex arithmetic, to more sophisticated functions like matrix inverse and matrix eigenvalues. The Matlab language is a high-level matrix/array language with a

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2.2 Software Tools 19

possibility to write own functions and other. Graphics in Matlab has extensive facilities for displaying vectors and matrices as graphs [6].

Modelica

Modelica is an object-oriented language for modeling of large, complex physical systems and problems. The free Modelica language, libraries and simulation tools are available and have been utilized in demanding industrial applications. It is suited for multi-domain modeling, for example for modeling of mechatronic systems within automotive and robotics applications. Such systems are composed of mechanical, electrical and hydraulic subsystems, as well as control systems. General equations are used for modeling of the physical phenomena [2].

While Modelica resembles object-oriented programming languages, such as C++ or Java, it differs in two important respects. First, Modelica is a modeling language rather than a true programming language. Modelica classes are not compiled in a usual sense, they are translated into objects which are then exercised by a simulation engine. Second, although classes may contain algorithmic com-ponents similar to statements or blocks in programming languages, their primary content is a set of equations. In contrast to a typical assignment statement as for example Matlab, such as

x := 2 + y;

where the left-hand side of the statement is assigned a value calculated from the expression on the right-hand side, an equation in Modelica may have expressions on both its right- and left-hand sides, for example,

x + y = 3 * z;

Equations do not describe assignment but equality. In Modelica terms, equa-tions have no pre-defined causality. The simulation engine must manipulate the equations symbolically to determine their order of execution and which are inputs and outputs to solve the whole equation system [9].

Dymola

An editor and simulator for Modelica models is Dymola, Dynamic Modeling Laboratory. This is a program for modeling of various kinds of physical sys-tems and problems. It supports hierarchical model composition, libraries of truly reusable components, connectors and composite a casual connections. Model li-braries are available in many engineering domains and it is possible to do own models and libraries in other domains. Dymola uses a modeling methodology based on object orientation and equations. The usual need for manual conversion of equations to a block diagram is removed by the use of automatic formula ma-nipulation. Dymola also has an open interface that makes it possible to connect the model to other programs, e.g., Matlab [2].

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Chapter 3

Model

The hot rod and wire rolling model and its development will be presented in this chapter. First necessary parts in every stand in a mill are presented, and thereafter, different steps in the development process are described.

3.1 Model of a Stand

When modeling the billet shape it is important to be aware of the characteristics of different grooves and which quantities the model should calculate. How does the shape look like? In most cases it looks like a circle or an ellipse, see Figure 2.4. A general roll shape is shown and described in Appendix A. The most important quantities that the model should calculate is different areas, intersections between rolls and billet. Furthermore, it should rotate the billet between two stands. To be able to do this in a fast and accurate way the shapes have to be described with an analytical function with a known primitive. An important question that will be answered in this chapter is if ordinary polynomials is sufficient to describe roll shapes or if other functions as for example circles and ellipses are necessary.

3.1.1 Necessary Operations During a Calculation

The modeling of the mill starts at the first stand where the desired quantities and some inputs to the next stand are calculated. After that the calculation goes to the next stand and does the same. The procedure goes on to the last stand and can be expressed in the steps below

1. Find an analytic expression for all the grooves and the input billet. Set k = 1 and Inputk to the input shape of the mill

2. Set Rollkto the groove for stand k. Calculate the spread and find an analytic

expression for the output of stand k

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3. Rotate the Outputk and set Inputk+1 to the new analytical expression

re-ceived from the rotation

4. Set k = k + 1. If k ≤ number of stands then go to Step 2, else you are finished

3.1.2 Ways to Describe Grooves Analytically

Several analytical functions have been tested. They have all been evaluated to see in which extent they fit to the original shape and its properties.

Polynomials

First an ordinary polynomial, see Equation 3.1, was evaluated.

PN(x) = f (a, x) = N X i=0 aixi (3.1) where:

a is the polynomial coefficients, N is the degree of the polynomial.

In order to reduce the numerical errors and increase the accuracy when calcu-lating xi_{for big x and high degree of the polynomial, it is necessary to scale all}

x-and y-values to a value less than 1 as below

x = x/S (3.2a)

y = y/S (3.2b)

where S is the scale factor.

Chebyshev Polynomials

Chebyshev polynomials are important in approximation theory because the roots of the Chebyshev polynomials of the first kind are used as nodes in polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge’s phenomenon and provides an approximation that is close to the polyno-mial of best approximation to a continuous function under the maximum norm. The Chebyshev polynomials are defined by the recurrence relation

T0(x) = 1

T1(x) = x

Ti+1= 2x · Ti(x) − Ti−1(x)

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3.1 Model of a Stand 23 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1

Figure 3.1. The first few Chebyshev polynomials, the flat T0 and then T1, T2, T3, T4

and T5

Polynomial in Chebyshev form of degree N is a polynomial PN(x) of the form

PN(x) = N

X

i=0

aiTi(x) (3.4)

where Ti is the ith Chebyshev polynomial [9].

Circles and Ellipses

Since most of the grooves look like a circle or ellipse with high derivatives at the endpoints and no derivative at the midpoint an ellipse was added to the polyno-mial. A general ellipse can be expressed as

x b 2 + y h 2 = 1 (3.5) where

b is the width from the origin, h is the height from the origin. A function y(x) can be derived as

C(x) = h s 1 − x b 2 (3.6)

The final analytical function including an ellipse then becomes

P CN(x) = C(x) + PN(x) (3.7)

where

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PN(x) is the polynomial of degree N describing the difference between the

original shape and the circle C(x).

Superellipses and Hyperellipses

The superellipse is the geometric figure defined in the Cartesian coordinate system as the set of all points (x, y) with

x b m + y h n = 1 (3.8) where m, n > 0

b is the width from the origin, h is the height from the origin.

The case when m = n = 2 is an ordinary ellipse; increasing m and n beyond 2 yields the hyperellipses, which increasingly resemble rectangles, see Figure 3.2 [9].

−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1

m = 4

n = 4

h = 1

b = 1

−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1

m = 4

n = 6

h = 1

b = 1

Figure 3.2. Examples of superellipses

Radius and Angle Description

Another approach to model different shapes with an analytic function is to model the radius as a function of the angle, i.e., ρ(φ). Most shapes then result in smooth graphs without large derivatives. The exception is squared shapes with sharp corners. A polynomial is one way that could be used to express the analytic function. Another is Fourier series that are periodical and easily can be rotated between two stands just by a simple phase shift.

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3.1 Model of a Stand 25

3.1.3 Ways to Fit Analytical Functions to Points

To fit the original points to the analytic functions in Section 3.1.2, different meth-ods have to be analyzed.

Least Squares Method

A simple method to fit points to polynomial is the least squares method. This method can be formulated as below and minimizes the squared difference between actual y-values and the values from the analytical function f . The function f is a polynomial with coefficients a, hence, f is linear in a.

z(a) = min a M X k=1 (f (a, xk) − yk)2 (3.9) where

M is the number of points,

xk, yk are the real x- and y-values which are used to find a polynomial.

The least squares method can in an algebraic way be expressed as [1]

XT(y − Xa) = 0 (3.10) where

X is the matrix with all known x-values, row k = (x0 k, x

1 k, . . . , x

N k),

y is the vector with all known y-values, a is the coefficients of the polynomial.

This equation could be expressed as an equation system

XTXa = XTy (3.11) or as an assignment of the coefficients as a function of known x- and y-values that could be used in Matlab

a = (XTX)−1XTy (3.12) If XTX is non-singular, then the solution a is unique. When deriving the Equation 3.10 it could be expressed as below and in this way be used in Modelica, this formulation is actually the same formulation of the normal equations as Equa-tion 3.10 J (x)T(f (a, x) − y) = 0 (3.13) where J (x) =      ∂ f ∂a₀ x=x1 . . . _∂a∂ f N x=x1 .. . . .. ... ∂ f ∂a₀ x=xM . . . _∂a∂ f N x=xM      = X(x),

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N is the degree of the polynomial, M is the number of known points,

∂ f ∂a_i x=xk = xi_k,

subscript i – refers to the polynomial coefficient i, subscript k – refers to the known x- or y-value k,

(f (a, x) − y) =     f (a, x1) − y1 .. . f (a, xM) − yM     .

The least squares method could be expanded by introducing different weights to the coefficients that are searched for. When these weights are used it is possible to make some coefficients more important than other.

Quadric Programming Optimization

A more flexible solution is the quadric programming (QP) formulation, see (3.14a) and (3.14b). With this method, it is possible to add constraints on the derivative or on the function value in points of interest [5].

z(a) = min a 1 2a T_{Qa + c}T_a _(3.14a) subject to Aa ≥ b (3.14b) where

a is the polynomial coefficients, c = −PM

k=1ykxk,

Q =PM

k=1xkxkT,

A – constraints, if constraint l is related to a value then row l in A is xkT, if constraint l is related to a derivative then row l in A is x0k

T

and if constraint l is related to a second derivative then row l in A is x00_kT, b = 0, xk= (1, x1k, . . . , xNk )T, x0_k= (0, 1 · x0 k, . . . , N · x N −1 k )T, x00 k = (0, 0, 1 · 2 · x 0 k, . . . , (N − 1) · N · x N −2 k ) T_.

Fast Fourier Transform

One way to find polynomials describing the radius function, ρ(φ), is the least squares method but a better way is to user Fourier-series and a Fast Fourier Transform (FFT) the functions are periodic. In that case, the FFT can find a function that describes the original points exactly.

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3.1 Model of a Stand 27

3.1.4 Finding the Intersection Between Groove and Billet

To calculate the equivalent areas and the spread of the output billet it is necessary to find the intersection between rolls and billet and the x-values that solve (3.15).

f (c1, x) − f (c2, x) = 0 (3.15)

where

f is an analytic function describing a groove or a billet, ci– coefficients for the function f .

In Modelica this problem could be solved with this formulation. An easy way to find the intersection between the billet and the rolls described with analytical functions in Matlab is the Newton-Raphson method below [3]

xk+1= xk− f (xk) f0_(x k) (3.16a) |xk+1− xk| ≤ (3.16b)

which starts with an initial guess and stops when Equation 3.16b is satisfied.

3.1.5 Area Calculation

The areas could be calculated through the primitive functions to any analytical function. Some examples of primitive functions are shown below, for polynomials see (3.17) and for circles see (3.18).

Z PN(x) dx = N X i=0 ai i + 1x i+1 (3.17) Z C(x) dx =h 2  b arcsin x b + x s 1 − x b 2   (3.18)

With help of the width of the billet and the intersections in Section 3.1.4 different area can be calculated as

xp Z xn f (x) dx =F (x)x_xp n= F (xp) − F (xn) (3.19) where

xn is the negative intersection,

xp is the positive intersection,

f is an analytic function describing the roll, F is the primitive function to f .

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3.1.6 Spread Calculation

The spread in each stand can be calculated by using the equations and models presented by, e.g., Wusatowski [10] and Shinokura [8] in Section 2.1.3.

3.1.7 Rotation of Billet Between Two Stands

Before the output billet can be used in the next stand it has to be rotated so that the same model can be used there to calculate intersections, areas etc. During the rotation process it is important to find the new maximum and minimum x-values that describes the billet and have the y-value 0, see Figure 3.3. To solve this problem, the intersections between the analytic function and the new x-axis have to be found. The new x-axis can be expressed as

x

y

φ

y

_{= − tan(φ) · x}

Billet before rotation

Billet after rotation x−axis after rotation

Figure 3.3. Finding new y = 0 when rotating a billet

y = − tan(φ) · x (3.20) where φ is the rotational angle, and the equation that has to be solved is

f (c, x) − (− tan(φ) · x) = 0 (3.21) To get the new shape, the x- and y-values have to be rotated by the rotational matrix, Q(φ), as below Q(φ) = cos φ − sin φ sin φ cos φ (3.22)

and the rotated values could be found by using the equation below

xr yr = Q(φ) x y (3.23)

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3.2 Implementation in Matlab and Modelica 29

During this rotation it is important to choose the different points well, espe-cially the x-values that get the largest absolute y-values after the rotation. This operation has to be done to reduce the risk for oscillations at the middle of each shape, i.e., around the y-axis. One way to solve this problem is to increase the number of points and decrease the distance between those in the middle of each shape.

3.2 Implementation in Matlab and Modelica

The model in Section 3.1 was first implemented and verified in Matlab and thereafter in Modelica. In this section important details necessary to consider during the implementation are discussed.

3.2.1 Modeling in Modelica

While Modelica solves an equation system it is important how the system is for-mulated to help Modelica to solve the problem. In this section some functions in Modelica are presented and other aspects that are important during simulation and when formulating equation systems.

Formulation of equations

During a simulation different problems that stop the solving process can occur. One such thing is division by zero and therefore it is important to reformulate equations in a way that this problem will not occur. Other problems could be results from formulations involving square roots. The fact that the function is returning complex numbers for negative values is also a problem.

Starting Guess

When Modelica solves an equation system each variable has the initial value 0. To increase the probability to find a solution, and to do it in a more efficient way, it is possible to set another initial value in Modelica. To find the solution as fast as possible it is important to have a good initial value of as many variables as possible in the model. For example, a good value of the billet dimensions could be input dimensions at the first stand, the desired dimensions at the last stand and something between these for the other stands.

Maximum and Minimum Values

As a help during testing and debugging, it is possible to set minimum and maxi-mum values for all variables in the model. During the simulation, it is possible to stop the process if any of these limits are reached.

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Solving the Least Squares Problem

While simulations in Modelica are based on an equation system, the least squares problem can be formulated with help of the normal equations, see Equation 3.13, and in this way be solved by Modelica.

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Chapter 4

Simulations and Results

The chapter presents different simulations and results during the development of the hot rod and wire rolling model. During the initial simulations the spread factor was constant and the model was independent of all roll speeds and tensions between the stands. Speeds and tensions were then gradually introduced in the model.

4.1 Quadrants and Upper/Lower Half

The first attempt was to model different shapes with ordinary polynomials and the least squares method in Matlab. First, every quadrant was modeled separately to handle all different geometries and asymmetrical shapes. Thereafter the upper and lower parts were modeled separately since all evaluated shapes were symmetrical with respect to the x-axis. When the upper half first is rotated around the x-axis and after that around the y-axis the resulting shape is the lower half.

Results

To model each quadrant separately was not a good idea, since it often resulted in big oscillations at the endpoints (x-values). When the assumption was made that all shapes were symmetric around the x-axis the result was improved. However, to get an acceptable result the degree of the polynomial had to be large. A large degree of a polynomial was not good either. Large x-values were resulting in a polynomial that grows very fast and therefore all x- and y-values had to be scaled to get x-values below one. A high degree of polynomial in combination with bad choices of x-points in the rotation lead to big oscillations at the middle of the billet, see Figure 4.1. This is because the least squares method fits the few points there very well, but between the points the difference between the polynomial and the original shape can become large.

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Groove Input Output

Figure 4.1. Example of oscillations in the simulation

4.2 Quadric Programming

The second approach was to find the polynomials with quadric programming in Matlab instead of the least squares method. First constraints on the first deriva-tive were defined, e.g., the upper function had to increase for negaderiva-tive x-values and decrease for positive x-values and vice versa for the lower half of the shape. There-after, constraints on the second derivative were defined. The upper half should have a negative second derivative for all x-values and vice versa for the lower part. Constraints on the values in the endpoints were also added to guarantee that the function passes through these points.

Results

The result of this approach was better than the previous models with the least squares method. The degree of the polynomials could be decreased since con-straints on the first and second derivatives of the shape could now be included. This approach with a high order polynomial together with large distances between x-values at the middle of the shape could also result in bad results. The constraints are satisfied at all points when the billet is rotated, but not between the points where no constraints are defined. The result could be an oscillating curve like in Section 4.1.

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4.3 Describing the Spread 33

4.3 Describing the Spread

After that several approaches to model the spread and the new output were made by defining and evaluating different polynomials in Matlab. The output was created by combining the spread-polynomial and the groove to find a new analytic function describing these points. Spread-polynomial of first, second, third and fifth order were evaluated. They were found with help of y-values and their derivatives in the endpoints. The x- and y-values were also switched to find an analytic function of y, i.e., P (y) = x(y), to get an infinite derivative at y = 0. Another approach is to use only the new endpoints and the part of the groove that describes the output to find the analytic function for the output.

Results

The result of polynomials of degree 3 and 5 was not acceptable, the derivative of the shape was varying around zero and the shape was not just increasing or decreasing but a combination of this. The only ways that guaranteed no oscillations in the valid range was to use a second order polynomial or the approach using only the new endpoints. The second order polynomial should go through two points, the new endpoint and the new intersection between billet and groove, and has an infinite derivative at the endpoint. This function is formulated as f (y) = x(y) to get the derivative zero for y = 0 instead.

4.4 Least Squares Method in Modelica

In Modelica, the least square problem was solved by explicitly formulating and solving the normal equations given in Section 3.2.1. A model with polynomials and the least squares method was developed as in Section 4.1. Different blocks were implemented and tested and then assembled to a stand-block. Different stand-blocks were then connected in series to a mill, see Figure 2.9.

In the first model, all dimensions and parameters of the mill were read from a mat-file and the results were written back to other mat-files. These files were used to plot and visualize the result in Matlab. This solution was not flexible and not either fast to simulate. The next model had all dimensions and param-eters as paramparam-eters in Modelica which all can be changed easily in Dymola. When a model is simulated in Dymola, a stand alone program dymosim.exe is generated. This program could be run from Matlab. When calling the program in Matlab it returns all resulting variables from the simulation in Modelica. The result can then be used to visualize the simulation. This method was better and faster to simulate. The visualization was flexible but very slow because every single parameter and variable had to be read separately from the matrix after the simulation.

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Results

The result was bad, the degree of each polynomial had to be low to find inter-sections between input and output and the resulting polynomials were oscillating. The reasons to this were bad or no staring guess of the variables in Modelica, see Section 3.2.1, and the fact that the polynomial was growing very fast outside its valid limits. Another reason was that the low degree of the polynomials was leading to oscillations in the function fGroove(x) − fIn(x) which Modelica was

trying to find the roots to.

4.5 Descriptions with Circles

A new approach was developed and evaluated in Matlab. Since all of the shapes,

i.e., input, output and grooves, in reality much look like a circle or ellipse the

polynomials were not considered any more. The analytic function then was a circle/ellipse together with another analytic function, see Section 3.1.2. The coef-ficients for the ellipse were found by calculating the maximum height and width of each shape. The analytic function was found by the least squares method evaluating the difference between real points that describe the shapes and the circle-points. Examples of these functions for grooves and billets can be found in Figure 4.2. The primitive function for the circle was also derived to calculate the area of the shapes in an easy way, see Section 3.1.5.

(a) Oval groove

Groove Input Output

(b) Bastard-round groove

Figure 4.2. Analytic functions for different grooves

Results

The results were improved in many ways, e.g., no large oscillations and only a polynomial of low degree was necessary to reach good results. Unfortunately, there where small oscillations at the endpoints of the grooves while there derivatives

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4.6 Circles in Modelica 35

were not infinite in these points, which were assumed in the model. There were also some problems to evaluate the circle/ellipse function and its area since the functions generated complex values for invalid x-values.

4.6 Circles in Modelica

The model in Section 4.5 has here been developed and evaluated in Modelica with help of some new blocks that calculated the coefficients for the circles.

Results

The result was the same as in Matlab, but there were small problems to find intersections between the groove and the billet. This problem was finally solved by formulating the intersection problem with two equations like Equation 3.5 and with the polynomials included in the y-value like Equation 4.1.

x b 2 + y − P (x) h 2 = 1 (4.1)

4.7 Spread Formulation

In this simulation the model in Modelica was extended to calculate the spread as in Section 2.1.3 and the mass flow as in Section 2.1.2. The problem is how to calculate the cross-section area at the neutral plane that is needed to calculate the mass flow. Therefore some assumptions had to be made. The area and the geometries of the billet before and after each stand are known and the question is how these could be used to calculate the area at the neutral plane. The as-sumptions were related to the area and how this could be calculated in different ways. One of these was that the area is assumed to be the output area, A2, plus

two half-moons with the height hn− h2at the middle, see Figure 4.3(a). Another

was to use the pass area and the quotient between the height at the neutral plane and the height at the output, see Figure 4.3(b). The heights could be the real dimensions or the equivalent dimensions calculated with the equivalent rectangle method in Section 2.1.2. By scaling the output area with this quotient the neutral area could be calculated.

Results

The results were good and it was easy to simulate and find the dimensions of the billet through the whole rolling process. The spread was simulated in different ways and all results were trustworthy. During the first simulation, the wider output was parallel to the original shape. In the next simulation, the spread was more like in reality where the billet spreads more than the spread-factor at the widest point of the billet and vice versa at the intersection between the billet and the groove.

Improved Billet Shape Modeling in Optimization of the Hot Rod and Wire Rolling Process

Institutionen för systemteknik

Department of Electrical Engineering

Examensarbete

Improved Billet Shape Modeling in Optimization of

the Hot Rod and Wire Rolling Process

Improved Billet Shape Modeling in Optimization of

the Hot Rod and Wire Rolling Process

Examensarbete utfört i Reglerteknik

vid Tekniska högskolan i Linköping

av

Abstract

Sammanfattning

Acknowledgments

Contents

List of Figures

List of Tables

Chapter 1

Introduction

1.1

Background

1.2

Purpose

1.3

Scope

1.4

Problem Description

1.5

Definitions

1.6

Thesis Structure

Chapter 2

Preliminaries

2.1

Basic Theory for the Hot Rolling Process

2.1.1

The Industrial Hot Rod and Wire Rolling Process

2.1.2

Fundamentals of Rolling

R

v

h

b

l

α

δ

v

h

b

l

v

v

v

α

δ

v

v

v

v

δ

Neutral plane

v

Direction of

roll rotation

frictional forces

Direction of

2.1.3

Theory of Spread

2.2

Software Tools

2.2.1

The Use of Different Programs

Chapter 3

Model

3.1

Model of a Stand

3.1.1

Necessary Operations During a Calculation

3.1.2

Ways to Describe Grooves Analytically

_h

_{= − tan(φ) · x}