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Computer-aided rolling of parts with variable rectangular cross-section Sepehri, Nariman 1986

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COMPUTER-AIDED ROLLING OF PARTS WITH VARIABLE RECTANGULAR CROSS-SECTION by NARIMAN SEPEHRI B.A.Sc, TEHRAN UNIVERSITY OF TECHNOLOGY, IRAN, 1984 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER*OF APPLIED SCIENCE in FACULTY OF GRADUATE STUDIES DEPARTMENT OF MECHANICAL ENGINEERING We accept this thesis as conforming tc the required standard UNIVERSITY OF BRITISH COLUMBIA APRIL, 1986 © NARIMAN SEPEHRI, 1986 & In presenting t h i s thesis in p a r t i a l fulfilment of the requirements for an advanced degree at the UNIVERSITY OF BRITISH COLUMBIA, I agree that the Library s h a l l make i t fre e l y available for reference and study. I further agree that permission for extensive copying of th i s thesis for scholarly purposes may be granted by the Head of my Department or by his or her representatives. It i s understood that copying or publication of th i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permi ssion. DEPARTMENT OF MECHANICAL ENGINEERING UNIVERSITY OF BRITISH COLUMBIA 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: APRIL, 1986 ABSTRACT A computer-aided process planning scheme for f l a t r o l l i n g of symmetric parts with variable rectangular cross-section i s proposed. As a starting point, El-Kalay and Sparling's formula for spread was found to be suitable for th i s application and thus was used in developing the method. Two d i s t i n c t c r i t e r i a were considered in the analysis, namely: kinematic and dynamic constraints. In order to control the precision of the r o l l e d parts, provisions were made for specifying the tolerances of the f i n i s h i n g passes in the form of convexity constraints. Numerical formulation was used and the res u l t i n g non-linear equations were solved by an i t e r a t i v e method. Based on the process constraints and the numerical solution, a computer algorithm was then developed to determine the number of r o l l i n g passes required, as well as the dynamic variation of the r o l l gap as a function of the r o l l e d length. Preliminary laboratory experiments were then conducted to v e r i f y the v a l i d i t y of the predicted results and the a p p l i c a b i l i t y of the spread formula in determining the process behaviour. These experiments led to the modification of the spread formula. Using the modified formula i t was found that a good agreement existed between the predicted results and those of the experiments. Operating aspects were also considered. It was proposed that a control system based on the r o l l e d length would be i i both simple and suitable. It was then concluded that for rectangular parts with moderate va r i a t i o n in shape and reasonable complexity, where formed-die r o l l i n g and die-forging are also applicable, t h i s method has considerable advantages as i t replaces the forging hardware with the r o l l i n g s o f t w a r e . ACKNOWLEDGEMENT The author wishes to thank Dr. F. Sassani for his generous assistance during the course of thi s research work. His help and the sharing of his experience were very much appreciated during the preparation of this thesis. The author i s also grateful to the Department of Metal l u r g i c a l Engineering for the use of the f a c i l i t i e s . The f i n a n c i a l support was provided by the National Sciences and Engineering Research Council of Canada. iv Table of Contents ABSTRACT i i ACKNOWLEDGEMENT iv NOMENCLATURE v i i i LIST OF TABLES ix LIST OF FIGURES .x 1. INTRODUCTION 1 2. LITERATURE SURVEY 7 2.1 Fundamentals of the Rolling Process 7 2.1.1 Basic Concepts and Geometric Relations 7 2.1.2 Calculation of Rolling Speed 9 2.1.3 Mechanism of Bite and F r i c t i o n 11 2.1.4 Pressure D i s t r i b u t i o n , Force and Torque ...13 2.2 Geometric Deformation in Flat R o l l i n g 15 2.2.1 Introduction 15 2.2.2 Formulae for Spread 17 2.3 Summary and Evaluation 22 3. ROLLING OF VARIABLE RECTANGULAR CROSS-SECTIONS 25 3.1 Basic Formulation for Deformation 25 3.1.1 Uniform to Uniform Deformation 25 3.1.1.1 Method 25 3.1.1.2 Number of Possible Solutions 27 3.1.1.3 Existence of the Solutions 27 3.1.2 Uniform to Non-uniform Deformation 30 3.1.3 Non-uniform to Non-uniform Deformation ....32 3.2 Multi-Pass Design Concept 33 3.2.1 Process Constraints 33 3.2.1.1 Kinematic Constraint 34 v 3.2.1.2 Dynamic Constraint 35 3.2.1.3 Convexity Constraint 35 3.2.2 Multi-Pass Design Procedure 36 3.3 Computer Software 37 3.3.1 Data Generator 38 3.3.2 Multi-Pass Planner 39 3.3.3 Graph Generator 45 3.3.4 Other Routines 45 3.4 Operating Aspects 46 3.5 Summary and Evaluation 49 4. EXPERIMENTAL EVALUATION 52 4.1 Introduction 52 4.2 Experimental Arrangements 53 4.2.1 Instrumentation 53 4.2.2 Specimens and Measurement 54 4.3 Steady State Roll i n g .......55 4.3.1 Effects of Sequential Roll i n g on Spread ...55 4.3.2 The Evaluation of the Multi-Pass Algorithm 56 4.4 Unsteady State Roll i n g 57 4.4.1 Eff e c t s of Height Variation on Spread 58 4.4.2 Effects of Width Variation on Spread 63 4.4.3 Eff e c t s of R o l l Gap Variation on Spread ...65 4.5 Summary and Evaluation ....67 5. SAMPLE ILLUSTRATIONS 71 5. 1 Example No. 1 71 5.2 Example No. 2 ' 73 6. CONCLUSIONS AND SCOPE FOR FUTURE WORK 75 v i REFERENCES 118 APPENDIX A 123 APPENDIX B 124 v i i NOMENCLATURE h,H Height(Thickness) w,W Width(Breadth) 1,L Length A Cross-sectional area V Volume Ah Draft S Spread(an increase in width) R Roll's radius D Roll's diameter Projected length of arc of contact a Angle of bite <fi Rolling angle 5 Neutral angle V Linear v e l o c i t y of the r o l l e d material t R o l l i n g temperature f C o e f f i c i e n t of f r i c t i o n Local pressure F Local horizontal force P Separating force M Rolling torque v i i i LIST OF TABLES TABLE PAGE 2-1 Value of correction factor in Bachtinov's formula.77 2-2 Average deviations of predicted spread from experimental values in Wusatowki's studies 77 2-3 Effect of steel composition on spread r a t i o in Wusatowski's formula.... 78 2-4 Values of the constants in Sparling's formula 79 2-5 Comparison between spread formulae in Sparling's experiments 79 2-6 Values of the constants in El-kalay and Sparling's formula 79 i x LIST OF FIGURES FIGURE PAGE 1-1 Taper leaf spring superimposed on multi-leaf spring assembly 80 1-2 Schematic representation of an eccentric-die r o l l i n g 80 1- 3 Sequence of f l a t r o l l i n g of taper leaf springs....80 2- 1 Schematic representation of r o l l i n g process 81 2-2 Velocity diagram in a r o l l i n g process 82 2-3 Typical v a r i a t i o n of width in r o l l i n g . . . 82 2-4 D i s t r i b u t i o n of f r i c t i o n a l forces 83 2-5 Pressure d i s t r i b u t i o n along the arc of contact....83 2-6 Pressure d i s t r i b u t i o n in the r o l l gap 83 2-7 Schematic representation of geometric deformation in .rolling 84 x 3-1 Forming a uniform block 85 3-2 Different sequences of r o l l i n g 85 3-3 Comparison between values of spread in two types of r o l l i n g 86 3-4 An example of a uniform i n i t i a l block 86 3-5 An example of a non-uniform f i n a l part 87 3-6 An example of an intermediate state 87 3-7 Typical v a r i a t i o n of spread versus r o l l ' s radius..88 3-8 Schematic representation of inverse c a l c u l a t i o n method 88 3-9 General flow-diagram of the computer software 89 3-10 General flow-chart of the multi-pass design routine 90 3-11 Exaggerated representation of r o l l interference...91 3-12 Block diagram of an analogue control system 92 xi. 3- 13 Block diagram of a D.D.C. system 93 4- 1 Different modes of deformation for r o l l e d steels..94 4-2 A possible mode of deformation for r o l l e d aluminum stocks 94 4-3 Comparison between two cross-sectional views under d i f f e r e n t r o l l i n g procedures 95 4-4 Deformation of a uniform block within two and four passes 96 4-5 Typical output of the computer program, two-pass deformation of a uniform block 97 4-6 Typical output of the computer program, deformation of a uniform block within more than two passes 98 4-7 P r o f i l e s of three experimental steel specimens....99 4-8 Rate of height v a r i a t i o n for three d i f f e r e n t experimental steel specimens 100 4-9 Comparison of experimental and predicted values of spread for a height variable specimen(i) 100 x i i 4-10 Comparison of experimental and predicted values of spread for a height variable specimen(ii) 101 4-11 Comparison of experimental and predicted values of spread for a height variable specimen(iii)....101 4-12 Comparison of experimental and predicted values of spread; specimen ( i ) , width increasing 102 4-13 Comparison of experimental and predicted values of spread; specimen ( i i ) , width increasing. 102 4-14 Comparison of experimental and predicted values of spread; specimen ( i i i ) , width increasing 103 4-15 Comparison of experimental and predicted values of spread; specimen ( i i ) , width decreasing 103 4-16 Comparison of experimental and predicted values of spread; specimen ( i i i ) , width decreasing 104 4-17 Geometry of an aluminum specimen 104 4-18 Comparison of values of spread for two d i f f e r e n t modes of width va r i a t i o n in the aluminum spec imen 105 x i i i 4-19 V a r i a t i o n o f r o l l g a p v e r s u s r o l l e d l e n g t h i n a t y p i c a l c l o s i n g r o l l g a p p r o c e s s 105 4-20 C o m p a r i s o n o f e x p e r i m e n t a l a n d c a l c u l a t e d v a l u e s o f s p r e a d i n a c l o s i n g r o l l g a p p r o c e s s 106 4-21 C o m p a r i s o n o f s p r e a d f o r t w o s i m i l a r s p e c i m e n s u n d e r d i f f e r e n t r a t e s o f r o l l g a p c l o s i n g 107 4-22 V a r i a t i o n o f r o l l g a p v e r s u s r o l l e d l e n g t h i n a n o p e n i n g r o l l g a p p r o c e s s 107 4-23 C o m p a r i s o n o f e x p e r i m e n t a l a n d c a l c u l a t e d s p r e a d f o r a u n i f o r m b l o c k i n a n o p e n i n g r o l l g a p p r o c e s s . . 108 4-24 A p p e a r a n c e o f t h e e x p e r i m e n t a l r o l l i n g m i l l 109 4-25 P o s i t i o n o f t h e f u r n a c e w i t h r e s p e c t t o t h e r o l l i n g m i l l 109 4-26 S t e e l s p e c i m e n s u s e d t o s i m u l a t e c o n d i t i o n s o f u n s t e a d y r o l l i n g 110 4-27 A l u m i n u m s p e c i m e n s u s e d f o r c r i t i c a l e x p e r i m e n t s 110 xiv 5-1 A p a r t with c o s t a n t width and l i n e a r l y v a r i a b l e h e i g h t 111 5-2 Two dimensional views of a uniform m a t e r i a l , a f t e r being r o l l e d i n an o p e r a t i o n with l i n e a r l y v a r i a b l e r o l l gap 111 5-3 Two p o s s i b l e i n t e r m e d i a t e shapes i n r o l l i n g a p a r t with l i n e a r l y v a r i a b l e height 112 5-4 R o l l gap v a r i a t i o n f o r the f i r s t s o l u t i o n 113 5-5 . V a r i a t i o n of percentage of r e d u c t i o n f o r the f i r s t s o l u t i o n 113 5-6 R o l l gap v a r i a t i o n f o r the second s o l u t i o n 114 5-7 V a r i a t i o n of percentage of r e d u c t i o n f o r the second s o l u t i o n 114 i 5-8 I n i t i a l and f i n a l geometry of the p a r t i n example 2 115 5-9 T y p i c a l example of sequence of a m u l t i - p a s s r o l l i n g process 116 5-10 V a r i a t i o n of r o l l gap versus r o l l e d l e n g t h f o r xv the example of figure 5-9 116 5-11 Variation of draft and torque versus r o l l e d length for the example of figure 5-9 117 B-1 C u r v e - f i t t i n g through known data points 125 xvi 1. INTRODUCTION With the rapid development of new materials, mechanization and automation, the processes of manufacturing are becoming more varied. The application of computers in a l l aspects of engineering, s p e c i a l l y in design and manufacturing (CAD/CAM), has in a way led to the development of new processes whereby a product can frequently be made in several ways. It i s therefore important to understand the many ways in which a product can be processed , the e f f e c t s that these processes have on the product properties, their advantages and l i m i t a t i o n s as well as the accuracy that is expected. A fundamental c r i t e r i o n which determines suitable methods of manufacturing, i s the correct process of producing the individual part so that i t is manufactured no more accurately than necessary and at the lowest cost possible 1. Amongst a l l the manufacturing processes, forming i s the one which i s becoming more important . Forming i s a method, by which the size or shape of a part is changed b y the application of force on the part. Forming i s a fast way to change the shape of parts. Generally speaking , i f the shape and the required accuracy of a part are such that i t can be made by one of the forming operations, then forming i s the most economical process to be used 2. Formed parts have fine-grain structures which increase their toughness and consequently prolong their working l i v e s 3 . 1 2 Industrial practice uses various forming techniques such as rolling , forging, pressing, stamping or extrusion. Rolling is known as the the most economic method amongst the other forming techniques. It has been practiced since the fourteenth century" and s t i l l has extensive application in manufacturing. Flat rolling i s the simplest form of r o l l i n g . The mechanism of f l a t r o l l i n g i s simple; two c i r c u l a r rotating cylinders draw the work-piece through the opening between them, by means of the f r i c t i o n force between the cylinders and the work-piece , thus reducing the cross-sectional area of the part. If thi s i s done in a temperature above the r e c r y s t a l l i z a t i o n point of the material , i t is c a l l e d hot rolling. Hot r o l l i n g i s commonly used for steel parts. This thesis presents a computer-aided r o l l i n g scheme for manufacturing symmetric parts with variable rectangular cross-section. This form of parts i s often used in equipments and machinery. A taper leaf spring i s a t y p i c a l example which is becoming increasingly common as the means of road suspension system for trucks and vans. According to Fig . 1-1, a taper leaf spring can replace a stack of five or more conventional constant thickness leaf springs, of f e r i n g certain economical and mechanical performance advantages. It leads to substantial weight saving for the same load carrying capacity, and better ride c h a r a c t e r i s t i c s . Moreover, the controlled pattern of the taper leaf spring permits higher working stresses to be used , and gives a 3 longer working l i f e . A common method to produce such springs i s based on the use of dies e c c e n t r i c a l l y fixed to a pair of r o l l s , rotation of which progressively forms the thickness of the spring blanks (Fig. 1-2). Closed dies are used in an attempt to l i m i t the l a t e r a l spread of the stock. Grinding operations are often needed to remove the forging fla s h from the r o l l e d material. The cost of maintaining the machine i s high, since under the heavy side loads die breakages are frequent. The cost of producing the dies i s also high. Furthermore, a separate tool set i s required for d i f f e r e n t spring siz e s . In the method presented here, a taper leaf spring ,for example, can be produced within two operations of f l a t r o l l i n g ; a blank of uniform cross-section i s f i r s t r o l l e d , through a variable r o l l gap, in i t s width (Fig. 1-3a), then, retapered from i t s o r i g i n a l thickness side (Fig. 1-3b) in such a way that the i n i t i a l constant width is regained leaving the part formed only in i t s thickness. The main features of t h i s method can be summarized as below: (i) The blank i s r o l l e d by two pla i n r o l l s , whereby d i f f e r e n t shapes may be produced from the same tool set. (ii) Parts with variation both in width and height can be produced by t h i s method. (iii) Two operations is the minimum required to achieve a c e r t a i n shape, however, more than two passes are frequently needed depending on the shape, appearance of the finished part and the process constraints. (iv) Unlike conventional r o l l i n g , the process i s of an unsteady state nature, i . e . , the r o l l gap continuously changes to form the desired shape. The ingoing material could be non-uniform as well. (v) Using a suitable control system, such as a Mi cro-processor , permits fast operating speeds to be used. Simplicity in resetting the system parameters, when design changes occur, can e a s i l y be accomplished through the micro-processor. This would result in a higher production rate and lower down time. (vi) Parts, produced by t h i s method, usually need no further treatment and can be used d i r e c t l y , although further processing could be done, i f necessary. The objectives of t h i s thesis are as follows: (a) To study the theory of conventional r o l l i n g and modify i t for application in the unsteady r o l l i n g process. (b) To develop procedures and strategies for determining the p a r t i c u l a r s of the r o l l i n g of parts with variable rectangular cross-section. (c) To ide n t i f y and apply the process constraints to the above strategies. (d) To evaluate the a p p l i c a b i l i t y of the developed method experimentally and to apply corrections and 5 modifications on the basis of the experimental evidence. A l i t e r a t u r e survey is presented in chapter two. Some important relations with regard to the geometry, kinematics and dynamics of f l a t r o l l i n g are f i r s t described. A study of the geometric deformation of steel under hot f l a t r o l l i n g i s presented next. This covers a major part of the chapter which also includes a comparison between the d i f f e r e n t formlae used to predict the spread in hot f l a t r o l l i n g . The chapter i s concluded by discussions leading to the selection of a suitable spread formula for use in thi s work. In chapter three, the proposed method i s described by presenting a solution for the uniform to uniform deformation. This i s then extended to include the deformation of non-uniform parts. A l l known p r a c t i c a l constraints are applied at t h i s stage; however, the main constraint is of course dictated by the desired shape of the parts. The numerical approach based on the proposed algorithm i s programmed and is described next. Operating aspects are b r i e f l y discussed at the end of the chapter. Chapter four presents some t y p i c a l results and discussions on basic experiments which were performed on an experimental r o l l i n g m i l l . The f i r s t part of this chapter i l l u s t r a t e s the results of the experiments, under steady state conditions, conducted to v e r i f y the a p p l i c a b i l i t y of the method. The second part presents an experimental evaluation of the geometric deformation of the material in 6 unsteady r o l l i n g . This chapter i s completed by discussions leading to the introduction of an improved formula for spread in the general case. Two examples with regard to the use of the method are i l l u s t r a t e d in chapter f i v e . F i n a l l y , in chapter six conclusions are presented along with scope for future work. There are two appendices. In appendix A the numerical method for finding the roots of non-linear equations i s described. Appendix B describes a c u r v e - f i t t i n g method, which was developed to obtain an a n a l y t i c a l representation of the va r i a t i o n of r o l l gap for each r o l l i n g pass. 2. L I T E R A T U R E S U R V E Y 2.1 F U N D A M E N T A L S O F T H E R O L L I N G P R O C E S S 2.1.1 BASIC CONCEPTS AND GEOMETRIC RELATIONS A schematic representation of f l a t r o l l i n g i s shown in Fi g . 2 - 1 . In the successive stages of r o l l i n g , the dimensions of a rectangular bar are changed but the volume constancy holds, i.e, V 0 = V 1 = V 2 = . . . = V n where ^n =A /jl n=h rtw nl n Kn, ln, hn and wn are the cross-sectional area, length, height and the width of the stock after the nl n stage. The increase in length of the stock af t e r each pass i s usually greater than the increase in width. The increase in width or l a t e r a l elongation i s c a l l e d spread. Referring to F i g . 2 - 1 , when a uniform bar with i n i t i a l thickness h, enters the r o l l s , the edge of the bar touches the r o l l at a point through which passes one arm of the angle having i t s apex on the r o l l axis, and the other arm in the plane passing through the r o l l axes. The included angle, a, i s c a l l e d I he angle of bite. The height of the bar leaving the r o l l s i s h 2. 1^ denotes the projected length of the arc of contact between the r o l l s and the metal, h is the height of the bar in the r o l l s at a distance x from the exit side of the r o l l s , 7 8 corresponding to a r o l l i n g angle 4>. The difference between the incoming and outgoing thicknesses, i . e . , the absolute dr af t , i s Ah=h,-h2 (2-1) With reference to F i g . 2-1, a geometric r e l a t i o n for r o l l i n g with r o l l s of the same diameter, D, (radius of R) can be derived as RCosa=R-(h,-h2)/2 The expression for c a l c u l a t i n g the angle of bite i s then found as Cosa=1-(h1-h2)/2R=1-Ah/D (2-2) From t h i s , the absolute draft can be calculated as Ah=DO-Cosa) (2-3) The projected arc of contact between the metal and the r o l l s i s calculated from the geometrical r e l a t i o n s h i p lj=v/R2-(R-Ah/2) 2=v/R-Ah-(Ah) 2/4 (2-4) Equation (2-4) may be assumed without a s i g n i f i c a n t error in a s i m p l i f i e d form 1^/R-Ah (2-5) This s i m p l i f i c a t i o n is allowable for small angles of b i t e . When Ah<0.08R, the error i s less than 1%5. A similar expression can be deduced for the r o l l i n g angle, 0, and thickness, h, Cos0=1-(h-h 2)/D (2-6) hence h=h2+D(1-cos0) 9 (2-7) The angle <f> i s calculated from the following r e l a t i o n s h i p Sin^=x/R 2. 1. 2 CALCULATION OF ROLLING SPEED Rolled stock enters the gap with a speed less than the peripheral r o l l speed. On the other hand, the exit speed of the stock i s greater than the peripheral speed of the r o l l s (see F i g . 2-2). Thus, there i s a plane in the deformation zone, at which the horizontal component of the peripheral speed i s equal to the speed of the r o l l e d stock. This plane is c a l l e d the neutral plane, and the value of the r o l l i n g angle at t h i s plane i s sp e c i f i e d by 5. The following equation holds v 5=v rCos5 (2-8) V 5 i s the speed of the r o l l e d stock at the neutral plane, v r i s the peripheral speed of the r o l l s . Applying the constancy of volume V=h , w , v , =h2 w 2 v 2 =h§w§vg = h5W§v/.Cos6 (2-9) v, and v 2 are the speeds of the material at the entry and the exit planes, respectively. Wusatowski 6 used the following relationships for spread of mild steel under hot f l a t r o l l i n g w2/w1 = ( h 2 / h 1 ) - f f and w 6/w,=(h 6/h 1)-^ where . 1 0 ^ = 1 0(-1.269(w,/h 1)(h,/D) 0' 5 5 6) The value of the neutral angle, 6, can now be related to the outgoing v e l o c i t y (wg/w,)(hg/h,)v rCos6=(w 2/wT )(h 2/h,)v 2 Equation (2-7) can be written at <j>=8 as, hg/h,=[D(1-Cos6)+h 2]/h. Using the above two equations, the following r e l a t i o n s h i p then holds v r C o s 6 / [ D ( 1 - C o s 6 ) + h 2 ] ( r - 1 ) = v 2 / [ h 2 ( J f _ 1 >] from which, the value of the exit v e l o c i t y can be determined providing that the value of the neutral angle is known. A similar r e l a t i o n s h i p could be written between v, and 5. Koncewicz 7 derived a formula for the determination of the neutral angle; r e f e r r i n g to F i g . 2-3, the r o l l gap can be divided into two zones: the zone of forward s l i p and the zone of backward s l i p . For free r o l l i n g , without front or back tension, ; adF 1+/ 6dF 2=0 (2-10) 6 o dF,=p0R'W0(Sin0-fCos0)dtf>, is the horizontal force due to backward s l i p when 5<<p<a; dF 2=p0R'W0(Sin$+fCos<£)d0 is the horizontal force due to forward s l i p when 0<tf><6. In order to solve equation (2-10), Konsewicz assumed p^ and f to be constant along the whole length of the arc of contact. He also suggested a linear v a r i a t i o n for the width of the r o l l e d stock along the horizontal length of the r o l l gap, i . e . , 11 W0=w 2 _(w 2-w 1)Sin0/Sina The value of the neutral angle can then be estimated by integrating equation (2-10) (for detailes see 6, pp 170 to 175). Neither Wusatowski's prediction for spread nor Koncewicz's approach for finding 5 are accurate enough. The best way of finding v, or v 2 i s through dir e c t measurement. 2.1.3 MECHANISM OF BITE AND FRICTION The maximum angle of a at which free r o l l i n g can take place, i . e . , without using force to push the metal into the r o l l gap, i s c a l l e d the maximum angle of b i t e . Referring to Fi g . 2-3, at the point of entry, for the element of area dA, (p^,Sina m a x)dA=(f •p^Cosa m f l J C)dA Then Tanamax=i (2-11) So, i f a^Tan 1f, then the r o l l s bite the metal, without any backforce or forward tension. This is referred to as free r o l l i n g . From the geometry of r o l l i n g , one can write 'Tanamax = 1d/RCosamax and approximately T a n a m w - •R. &hmax/(R-&hmax/2 ) or T a n amax =* v/Ahwcx/R So the maximum draft for free r o l l i n g i s AlW=R-f 2 (2-12) Referring to F i g . 2-4, the f r i c t i o n a l forces change di r e c t i o n at the neutral plane, FF. When the r o l l gap i s 12 completely f i l l e d , in order that r o l l i n g takes place, the f r i c t i o n a l forces a s s i s t i n g r o l l i n g at the sector b-c should be greater than the sum of the f r i c t i o n a l forces hindering r o l l i n g in the sector a-b and the horizontal component of the r o l l pressure. This results in a new condition which allows free r o l l i n g with higher drafts to take place. Recent investigations have confirmed the following inequality as the condition for free r o l l i n g (for d e t a i l s see 6, pp 120 to 126 and 16, pp 204 to 206) 0<a m a x<2Tan'f (2-13) During r o l l i n g both the f r i c t i o n a l forces and the c o e f f i c i e n t of f r i c t i o n vary along the arc of contact. The c o e f f i c i e n t of f r i c t i o n , f, increases with the normal force. With ideal l u b r i c a t i o n , the c o e f f i c i e n t of f r i c t i o n decreases as the ve l o c i t y of the body increases. Since i t is not possible to calculate the c o e f f i c i e n t of f r i c t i o n along the arc of contact, the average c o e f f i c i e n t of f r i c t i o n i s used. There are many ways to find the average c o e f f i c i e n t of f r i c t i o n . A common method is to measure the value of the r o l l load while the plane of no s l i p i s at the e x i t . This can be achieved by applying a certa i n amount of back tension to the material (for d e t a i l s see 6, pp 128 to 130 and 16, p 207). Ekelund 8 suggested an emperical formula which expresses the mean c o e f f i c i e n t of f r i c t i o n as a function of the r o l l i n g temperature. Bachtinov 6' 9 then proposed a modification to Ekelund's formula to allow for the influence of the r o l l i n g speed as well 1 3 f =a/<( 1.05-0.0005t) a=l. 0 for cast iron or rough steel r o l l s , a=0.8 for c h i l l e d and smooth steel r o l l s , a=0. 55 for ground steel r o l l s , t i s the r o l l i n g temperature(°C), K i s a factor r e l a t i n g f to the peripheral speed of the r o l l , v,., according to table 2-1 . A d i s t i n c t i o n should be made between the sli p p i n g f r i c t i o n and the st i c k i n g of metal to the r o l l which occurs s p e c i a l l y in hot r o l l i n g (see 16, p 205). Korolev (see 6, p 208) has developed a formula which determines the length of the s t i c k i n g zone. 2.1.4 PRESSURE DISTRIBUTION, FORCE AND TORQUE Referring to Fig . 2-5, the variation of r o l l pressure along the r o l l gap contains two parts. The lower part ,ADGEC, shows work-hardening for ideal ( f r i c t i o n l e s s ) deformation ( i t i s almost horizontal in hot r o l l i n g ) , and the upper part rDFEGD, shows the r o l l pressure necessary to overcome the additional constraint caused by the f r i c t i o n forces. A theoreti c a l approach for the determination of roll-pressure from l o c a l stress-evaluation i s now b r i e f l y described. The s t a r t i n g point i s to develop an equation representing the horizontal equiblibrium of forces in the r o l l gap. Considering the elemental s l i c e of material in the inset diagram of F i g . 2-1, the horizontal stress o i s 1 4 assumed to be di s t r i b u t e d uniformely over the v e r t i c a l section. The horizontal forces acting on the element w i l l be in equilibrium i f the following relationship holds d( ah)/dtf>+2R(p0Sintf>±rCos0) = O where r i s the shear stress due to f r i c t i o n ; depending on whether the element i s in the slipping zone or stic k i n g zone, the resultant f r i c t i o n a l force i s dependent or independent of the pressure, respectively. The negative sign refers to the condition on the entry side of the neutral point, and the positive sign refers to the condition on the exit side. Applying the p l a s t i c i t y c r i t e r i o n to the element and by using a numerical method, the d i s t r i b u t i o n of pressure along the r o l l gap can be determined (refer to 10, 11 and 12). Recently, some e f f o r t s have been made to find the three-dimensional d i s t r i b u t i o n of pressure in the r o l l gap, amongst whom, L a l l i 1 3 and Kobayashi 1' can be named. A t y p i c a l three-dimensional d i s t r i b u t i o n of pressure is shown in F i g . 2-6. Having the pressure d i s t r i b u t i o n , the separating force as well as the acting torque can be calculated by integrating over the area of contact (see 16, pp 208 to 111). Due to the d i f f i c u l t y in finding the actual pressure d i s t r i b u t i o n t h e o r e t i c a l l y 1 0 ' 1 5 some attempts have been made to derive p r a c t i c a l and easy-to-use formulae for ca l c u l a t i n g the separating force in r o l l i n g (see 6, pp 229 to 266). A 15 good estimate of the r o l l load in f l a t r o l l i n g can be obtained by considering the process as a homogenous compression between two platens. The platens are of length 1^ and width of wmea/7, the mean value of the width of the block before and after r o l l i n g . The y i e l d stress of the material, Y, i s assumed to be constant along the r o l l gap; th i s i s almost true for hot r o l l i n g 2 5 . The separating force, P, necessary for the r o l l i n g i s then P^-ld'"mean (2-14) Substituting the value for 1^ and increasing the value of y i e l d stress by the amount of 20% (which was suggested by Orawan 1 5) for the contribution of f r i c t i o n , the r o l l load w i l l then be P=( 1 .2)Y-w m e a / l V/R TAh (2-15) It is true with good approximation to assume that the resultant force acts at the centre of the arc of contact in hot r o l l i n g (see 6, p 270). In that case the applied torque w i l l be M=P(1d/2) = (0.6)Y•w m e anR•Ah (2-16) 2.2 GEOMETRIC DEFORMATION IN FLAT ROLLING 2. 2. 1 INTRODUCTION In r o l l i n g , d r a f t (the change of thickness) i s produced by pressure of the r o l l s . This i s normally accompanied by the increase of the length, e l o n g a t i o n , and the increase of the width, spread, of the material being r o l l e d . These are 16 connected and s t r i c t l y dependent on one another. Due to the d i f f i c u l t y in predicting t h e o r e t i c a l l y the three-dimensional p l a s t i c deformation of material, most of the exi s t i n g formulae for predicting the spread are empirical. Referring to F i g . 2-7 , the i n i t i a l l y plane edge of the block usually becomes convex in shape (other possible modes of deformation w i l l be discussed l a t e r ) . In some of these formulae the maximum value, w^ , and in some others the mean value, wm, according to the following formula has been used as the r o l l e d width 6 wm = wfc"( wfc" w/^ / / 3 (2-17) Generally, factors a f f e c t i n g spread can be divided into three main g r o u p s 1 7 ' 1 8 : (i) - Geometric factors such as aspect r a t i o , draft and the bar width to the length of arc of contact r a t i o . (ii) ~ Factors r e l a t i n g to f r i c t i o n a l conditions such as r o l l surface and scale formation . (iii) -Factors a f f e c t i n g the y i e l d stress of the material such as s t r a i n rate, material composition and r o l l i n g temperature. The following section deals with a study and a comparison between the most recent formulae used for predicting the spread of steel in hot f l a t r o l l i n g . The interest i s of course in the geometric shape of the material at the entry and exit points. 1 7 2, 2. 2 FORMULAE FOR SPREAD S i g n i f i c a n t work on spread has been ongoing since 1955. Prior to t h i s , some approximate formulae had been in use in prac t i c e . In th i s regard, Tafel and Sedlaczek's formula (1923), Seibel's formula (1932), Trinks's formula (1933), Bechmann's formula (1950) and Ekelund's formula (1953) can be named as the most well-known (for d e t a i l s see 6, pp 84 to 86 and 4, pp 836 to 836). In 1955, Z. Wusatowski 1 9 published a paper in which the following formula for spread was proposed where H/=10-1 .269(w,/h, ) ( h ^ D ) 0 - 5 5 6 7=h 2/h 1 Wusatowski indicated that his formula was accurate while r o l l i n g low carbon steel with l i g h t draft ratios (0.5<7<0.9). He also conducted some experiments to make a comparison between his formula and those proposed previously. His results are l i s t e d in table 2-2. According to this table, Wusatowski's formula shows no larger deviation than the other formulae except for the Ekelund's. Wusatowski suggested 2 0 that the mean error in his formula could be reduced by introducing some correc t i o n a l factors for r o l l i n g temperature, a, r o l l i n g v e l o c i t y , b, r o l l surface condition, c, and the type of steel to be r o l l e d , d, /3=a . b.c .d. 7""^ 18 a=1.005 while r o l l i n g at 750° to 900°C. a=1.000 while r o l l i n g above 900°C. c=7.020 for cast iron and rough steel r o l l s . c=i.000 for c h i l l e d cast iron and smooth steel r o l l s . c=0.980 for ground steel r o l l s . d is chosen from table 2-3. By using the new formula, Wusatowski could decrease the t o t a l mean error from 4.59% to 2.24% which was sati s f a c t o r y at that time. Wusatowski also suggested that his formula would be v a l i d even for heavy draft r a t i o s (0.1<7<0.5) i f values of 3.954 and 0.967 were substituted for the constants 1.269 and 0.556 in his formula, r e s p e c t i v e l y 6 . In the same year, R. H i l l 1 7 ' 1 8 ' 2 0 proposed an alternative formula for spread Ln(w6/wt )/Ln(h,/h 2 )=0.5 EXP[-X(w1/l/D-Ah) ] X i s a constant which is selected to f i t the experimental data. H i l l suggested a value of 0.5 for X. A.W. McCrum18 in 1956, ca r r i e d out some c r i t i c a l tests to compare the three most recent formulae: Ekelund's, Wusatowski's and H i l l ' s . His experiments were performed on r o l l i n g of bars with the same material, the same temperature and constant width, draft, r o l l radius, f r i c t i o n a l conditions and r o l l i n g speed, but with d i f f e r e n t stock heights. The results deviated considerably from those predicted by the formulae given by Ekelund and Wusatowski, 19 but were more agreeable with those of H i l l ' s . McCrum showed that a value of 0.525 for X would give better results when r o l l i n g mild s t e e l 1 8 . In a growing need for more precise information about the r o l l i n g process, a series of experiments was carried out by L.G.M. S p a r l i n g 1 8 , under c a r e f u l l y controlled conditions, to see the e f f e c t s of various factors governing spread. He recommended the following formula S=C-EXP[-K(w 1/h 1) A(h,/R) B(Ah/h 1) G] where S=Ln(w m/w!)/Ln(h 1/h 2) wm is calculated from equation (2-17). Constants C, K, A , B and G are l i s t e d in table 2-4. In Sparling's experiments, a l l the geometric factors were changed at a constant temperature (1100±10°C) and a constant s t r a i n rate (5Sec" 1). The material in Sparling's experiments was mild s t e e l . There were 35 s a t i s f a c t o r y t e s t s . The results extracted from his experiments show that neither the formula of Wusatowski nor that of H i l l accurately predicts spread over a wide range of experimental conditions. Table 2-5 shows some relevant r e s u l t s . In 1968, El-Kalay and S p a r l i n g 2 2 extended their investigations to an area which had been neglected u n t i l then; f r i c t i o n and its effect on spread. They modified the 20 formula derived e a r l i e r by Sparling to give the best f i t for their experimental data. The formula is as follows Ln(w m/w,)/Ln(h 1/h 2)=A.EXP[-B(w,/h 1) C(h 1/R) D(Ah/h l) E] the values of constants are l i s t e d in table 2-6 according to the f r i c t i o n a l conditions. In the same year a paper was written by A. Helmi and J.M. Alexander 1 7. More than 200 tests were performed on mild steel (scale free steel with 0.18% carbon) at constant r o l l i n g temperature (1000°C) and constant r o l l i n g speed (9m/min). The following conclusions were derived by the authors: (i)-Wusatowski's formula i s b a s i c a l l y in error, i t can only predict spread under a very limited range of geometric variables (the same deduction as S p a r l i n g 1 8 and McCrum 1 7). (i / ^ - H i l l ' s formula is b a s i c a l l y sound; However, i t i s s l i g h t l y in error as i t neglects the eff e c t of w,/!^  on the spread. (i ii)-El-Kalay and Sparling's formula seems to be the soundest e x i s t i n g . Helmi and Alexander proposed a range of conditions in which Sparling's formula is more accurate, however, i t has not been proved by other works. They also developed an alt e r n a t i v e formula for spread S=0.95(h 1/w 1) 1- 1EXP[-0.707( W l/(RAh) 0- 5)(h,/w,) _ 0• 9 7 1] 21 where S=Ln(w^/w1)/Ln(h,/h2) According to the authors' paper t h i s formula can be applied over a very wide range of geometric variables and i t s application i s unlimited except when w^h, is less than unity. It was also v e r i f i e d that the deviation of temperature, r o l l surface and scale condition from those of the experiments would a f f e c t the value of 0 . 7 0 7 in the above expression. In 1 9 7 2 , J.G. Beese 2 3 performed some i n d u s t r i a l tests to make a comparison between the El-Kalay and Sparling's formula with that of Helmi and Alexander. The results showed that Alexander's estimates of maximum spread were a l l high (overestimated), while Sparling's estimates of mean spread on small slabs were s a t i s f a c t o r y . A new formula was developed by Beese which was claimed to be accurate for values of w,/h, between 3 to 1 6 . S=0.61(h,/w,) 1• 3EXP[-0.32h 1/(R°- 5Ah 0- 5)] "S" has the same d e f i n i t i o n as that of Helmi and Alexander's. There have also been a few attempts to predict the geometrical deformation of the material under r o l l i n g t h e o r e t i c a l l y , for s p e c i f i c c a s e s 1 8 ' 2 " . One t y p i c a l example i s the computer-aided modular upper bound approach, which was developed at B a t t e l l e Columbus laboratories, under NASA sponsorship to predict the metal flow in the r o l l i n g of an 22 a i r f o i l section. In t h i s method the spread p r o f i l e under r o l l s , as well as the l a t e r a l spread and longitudinal elongation are predicted. 2 . 3 SUMMARY A N D E V A L U A T I O N The f i r s t part of t h i s chapter described some basic fundamentals with regard to the process of r o l l i n g . Three concepts namely, r o l l i n g speed, mechanism of bite and force c a l c u l a t i o n were discussed. These concepts w i l l be later used to formulate the process constraints in r o l l i n g . The l i t e r a t u r e survey on spread formulae shows that a considerable amount of study has been done to predict the effects of the geometric factors on spread. However, despite the importance of f r i c t i o n a l conditions, l i t t l e i s known about their e f f e c t s 2 2 . This study also revealed that although there i s no single formula that gives reasonable prediction under a l l conditions, El-Kalay and Sparling's formula provides s a t i s f a c t o r y estimates over a f a i r l y wide range of operating conditions. Helmi and Alexander's formula has been found to overestimate the value of spread, due to the fact that short specimens were used in their experiments; so, the results were greatly influenced by back and front-end spread (see also 23). Besides, the formula is incapable of giving reasonable estimates of spread for blocks with aspect r a t i o s less than unity. No evaluation of the Beese's formula has been reported. However, his formula is only capable of predicting the spread of the slabs with 23 the range of aspect r a t i o s from 3 to 16. El-Kalay and Sparling's formula can predict the spread for both cases when the aspect r a t i o i s less or greater than one, although the range of t h i s c a p a b i l i t y i s s t i l l not very well defined. The e f f e c t s of f r i c t i o n a l conditions are also included in this formula. Another feature of El-Kalay and Sparling's folmula i s that, i t predicts the mean value of the spread. This w i l l be useful when estimating the spread for di f f e r e n t modes of deformations (see section 4.2.2). Other formulae l i k e those belonging to Beese or Alexander were developed to estimate the maximum spread for the rhombic cross-section which, as w i l l be seen l a t e r , i s not the only mode of deformation for the steel stocks. El-Kalay and Sparling's formula has been generalized and modified such that i t could be adapted for d i f f e r e n t conditions. This modification has been based on the nature of the formula i t s e l f , and Sparling's d i s c u s s i o n s 1 8 ; he suggested that for most i n d u s t r i a l processes, the spread would be expected to increase with (1) - decreasing temperature (2) - decreasing amount of scales (3) ~ medium carbon steels used instead of mild steel (4) - increasing s t r a i n rates (5) - increasing roughness of the r o l l e s . Conditions (2) and (5) have already been included in the formula. To s a t i s f y other three conditions, the modified formula has been suggested by Sparling as follow 24 S=A.EXP[-B(w,/h 1) c(h 1/R) D(Ah/h l) E.f.g.j] (2-18) S=Ln(wOT/w,)/Ln(h1/h2) where f indicates the e f f e c t s of the composition of the r o l l e d material; f=1 for 0.13% carbon, 0.55% manganese mild s t e e l , f<1 for medium carbon and s t a i n l e s s s t e e l s . g indicates the e f f e c t s of temperature of the r o l l e d material; g=1 for 1100°C, g<1 for lower temperatures, j r e f l e c t s the e f f e c t s of mean s t r a i n rate during the r o l l i n g ; j=1 for an approximate mean st r a i n rate of 5sec" 1, j<1 for higher s t r a i n rates. Therefore, El-Kalay and Sparling's formula, with provisions for modification factors, was deemed suitable to be used as s t a r t i n g point in developing the method of r o l l i n g parts with variable rectangular cross-section. It w i l l be seen l a t e r that the approach towards improvement, as shown above, is suitable for the more generalized form of r o l l i n g , i . e . , t he unsteady rolling. 3 . R O L L I N G O F V A R I A B L E R E C T A N G U L A R C R O S S - S E C T I O N S 3 . 1 B A S I C F O R M U L A T I O N F O R D E F O R M A T I O N 3.1.1 UNIFORM TO UNIFORM DEFORMATION 3.1.1.1 Met hod A uniform block of dimensions HI, Wl and LI i s to be deformed into a uniform block of desired dimensions HF, WF and LF (Fig. 3-1a). The steps involved in the process are as follows: (i) The i n i t i a l blank i s r o l l e d on the side Wl. The r o l l e d material referred to as the i n t e r m e d i a t e m a t e r i a l , w i l l have dimensions HM, WM and LM (Fig. 3-1b). ( i i ) The intermediate material i s then r o l l e d into the required thickness HF. The r o l l e d material i s expected to have the width WF and the length LF (Fig. 3-1c) . The problem is to find the appropriate dimensions of the intermediate material so that the desired shape is formed within the steps stated above. The generalized form of the spread formula, (2-18), i s used. Applying t h i s formula to the f i r s t pass h,=WI h2=WM w 1 =HI 25 26 w2=HM Ah=(WI-WM) R=RW Ln(HM/HI)_ -B(HT/WT)C(WI/RW)D((WI-WM)/WI)Ef.g.j Ln(WI/WM) A e (3-1 ) and for the second pass h,=HM h2=HF w , =WM w2=WF Ah=(HM-HF) R=RH Ln(WF/WM) -B(WM/HM)C(HM/RH)D((HM-HF)/HM)Ef.g.j Ln(HM/HF) A e (3-2) where RW and RH refer to the r a d i i of the r o l l s in the f i r s t and the second operation, respectively. The volume constancy also exists during these processes HI.Wl.LI=HM.WM.LM=HF.WF.LF (3"3) Two non-linear equations (3-1) and (3-2) can be solved simultaneously with eqution (3-3) to obtain the intermediate dimensions as well as the length of the f i n a l part or the i n i t i a l material, depending on whichever is unknown. The incremental search method was used in solving the above equations. The general algorithm i s 27 described in appendix A. 3.1.1.2 Number of Possible Solutions In a two-pass r o l l i n g process, depending on which side of the i n i t i a l material i s r o l l e d f i r s t , i . e . , named Wl, and which side i s formed l a s t , i . e . , termed HF, there would e x i s t , at most, four d i s t i n c t solutions (see F i g . 3-2): (i) Case one: Side of Wl i s r o l l e d f i r s t , HM i s then reduced to HF. Convexity (out of flatness) appears on the side of HF. (i i) Case t wo: Side of Wl i s r o l l e d f i r s t , HM i s then reduced to WF. Convexity appears on the side of WF. (iii) Case t hr ee: Side of HI i s r o l l e d f i r s t , HM i s then reduced to HF. Convexity appears on the side of HF. (i v) Case four: Side of HI i s r o l l e d f i r s t , HM is then reduced to WF. Convexity appears on the side of WF. It i s necessary to mention that the intermediate shape for each cas e is di f f e r e nt . 3.1.1.3 Existence of the Solutions It i s clear that when the cross-sectional area of the i n i t i a l blank i s less than the cross-sectional area of the desired part, the proposed method i s not 28 a p p l i c a b l e . This i s due to the p h y s i c a l nature of the r o l l i n g process which always leads to a reduction i n the c r o s s - s e c t i o n a l area. A l s o , when the c r o s s - s e c t i o n a l area of the i n i t i a l blank i s l a r g e r than that of the de s i r e d p a r t , a c o n t r o l i s s t i l l needed to evaluate the existence of the s o l u t i o n s . R e f e r r i n g to F i g . 3-1a, four cases may occur i n t h i s regard: (i) WI>WF and HI>HF the process i s p o s s i b l e . (ii) WI=WF , HI <HF or WKWF , HI <HF the process i s impossible. (iii) WI>WF , HI<HF the process i s p o s s i b l e i f HF<HS, where HS i s the spread value of HI a f t e r a s i n g l e free r o l l i n g i n which Wl i s reduced to WF. (iv) WKWF , HI >HF the process i s p o s s i b l e i f WF<WS, where WS i s the spread value of Wl a f t e r a s i n g l e free r o l l i n g i n which HI i s reduced to HF. The p h y s i c a l explanation f o r the l a s t two cases i s that the extent of l a t e r a l elongation caused by r o l l i n g i s l e s s than that of the l o n g i t u d i n a l e l o n g a t i o n , so the maximum elongation on one side can only be achieved by applying a s i n g l e pass of reduction to the other side rather than two successive passes i n v o l v i n g a reduction on both sides 29 or on one side. An a n a l y t i c a l explanation for the above deduction i s as follows: Consider a block with dimensions HI, Wl and LI which undergoes a width reduction to intermediate dimensions of WM, HM and LM (Fig. 3-3a.1>. Spread occurs along the side HI. The following relations can then be written Wl.HI.LI=WM.HM.LM (3-4a) WI.HI>WM.HM (3-4b) WM<WI , HM>HI (3-4c) (HM-HI)/HI=X(LM-LI)/LI (3-4d) where 0<X<1 is defined as a general c o e f f i c i e n t r e l a t i n g l a t e r a l elongation to the longitudinal elongation. X=0, where no spread occurs (condition of plain s t r a i n ) . The intermediate material then undergoes r o l l i n g in such a way that i t s height, HM, i s reduced to HF. Spread occurs on the side of WM (Fig. 3-3a.2). The following relations hold WF,.LF,.HF=WM.LM.HM (3-5a) WM.HM>WF,.HF (3~5b) WF,>WM , HF<HM (3-5c) (WF1-WM)/WM=X(LF1-LM)/LM (3~5d) Using equation (3-5a), equation (3~5d) becomes: (WF,-WM)/WM=X(WM.HM-WF,.HF)/(WF1.HF) or WF,={(1-X)WM.HF+/[(1-X)2WM2,HF2+4XWM2.HM.HF]}/2HF 30 (3-6) WF, i s now compared to WF2 which i s the elongated value of Wl when the same i n i t i a l material undergoes a single free r o l l i n g so that i t s height, HI, i s reduced to HF (Fig. 3-3b). Using a similar approach as for WF,; WF2 is calculated as WF2.HF.LF2=WI.HI.LI (3-7a) Wl.HI>WF2.HF (3-7b) WF2>WI , HF<HI (3-7c) (WF2-WI)/WI=X(LF2-LI )/LI (3-7d) or WF2 = { ( 1-X)WI .HFVt ( 1 -X) 2WI 2 .HF2 + 4XWI 2 .HI .HF] }/2HF (3-8) A comparison between equations (3-6) and (3-8) shows that WF2 i s equal to or greater than WF, . The effect of a subsequent reduction of one side, on the value of the spread for the other side is investigated experimentally in section 4.3.1. 3. J. 2 UNIFORM TO NON-UN I FORM DEFORMATION A t y p i c a l example of a part having non-uniform rectangular cross-section i s shown in F i g . 3-5. I n i t i a l material which is a uniform block is shown in F i g . 3-4. To formulate the problem the desired part i s divided into N equal length segments (Fig. 3-5). Lf7=Lf2=Lf3=...=Lfn=LF/N Each segment is now assumed to have a uniform 31 cross-sectional area of Hfj.Wfj, (3 = 1, 2,...,n), which can be formed by means of two operations from a corresponding segment in the i n i t i a l material. Therefore, the i n i t i a l material should also be divided into N segments in a way that each segment i s a volumetric equivalent of i t s corresponding segment in the desired part, i . e . , Hij.Wij.Lij=Hfj.Wfj.Lfj (j = l, 2, 3, . . . , n) where in t h i s case H i j , ( j = 1, 2, . . , n) = c o ns t a nt and Wij, (2 = l,2,..,n) = constant The lengths of the segments in the i n i t i a l material would, in general, be unequal. Applying the same procedure as described in section 3.1.1.1 to each pair of corresponding segments, the following relationships can be written Ln(Hmj/Hij) e-B(Hij/Wij) c(Wij/RW) D((Wij-Wmj)/Wij) Ef.g.j Ln(Wi j/Wmj) (3=1, 2, 3,.., n) (3-9) Ln(Wfj/Wmj) , e-B(Wmj/Hmj) c(Hmj/RH) D((Hmj-Hfj)/Hmj) Ef.g.j Ln(Hmj/Hfj) ( j = l, 2, 3, . . , n) (3-10) Wij.Hij.Lij=Wmj.Hmj.Lmj=Wfj.Hfj.Lfj ( 3 = 1, 2, 3, . . . , n) (3-11) By solving equations (3-9), (3-10) and (3-11) simultaneously for a l l segments, the dimensions of the segments of the 32 intermediate material are found. Referring to F i g . 3-6, the intermediate material consists of N, generally non-equal length, uniform cross-section segments with dimensions Wmj, Hmj and Lmj (j=1, 2,.., n). The incremental values of r o l l gap variation for the f i r s t and the second pass are then Wmj and Hfj (j=l, 2,. . , n), respectively. So, in order to form the desired part by means of f l a t r o l l i n g , the r o l l gap in the f i r s t and the second pass should vary accordingly. The variation of r o l l gap can be controlled via the displacement of the r o l l e d material during each pass (refer to section 3.4) . 3.1. 3 NON-UNIFORM TO NON-UNIFORM DEFORMATION Normally the i n i t i a l material i s of uniform cross-section. However, sometimes i t may be desired to deform a non-uniform material to a non-uniform part, e.g., some pre-manufactured parts are required to be reshaped. The formulation derived in 3.1.2 i s applicable in th i s regard except that Hij and Wij (j=1, 2; ..,n) are no longer constant. However, a control is necessary to check the existence of the solutions for every pair of segments (refer to section 3.1.1.3). In t h i s regard, i f condition ( i i ) in 3.1.1.3 occurs for even one pair, then a feasible solution does not exi s t . For the case where conditions ( i i i ) or ( i v ) occur and the existence of a solution i s in question, an increase in the r o l l ' s radius can lead to s a t i s f y the requirements. The vari a t i o n of spread versus r o l l ' s radius for a t y p i c a l case 33 is shown in F i g . 3-7. It i s seen that the increase in r o l l ' s radius leads to the increase of the l a t e r a l elongtion. This p h y s i c a l l y means that by increasing the r o l l ' s radius, the projected length of the arc of contact increases which causes the value of elongation in the l a t e r a l d i r e c t i o n to increase (refer to section 4.4.1). Figure 3-7 also shows that larger r o l l s ' r a d i i have smaller e f f e c t s on the increase of spread. However, the increase in r o l l ' s radius is l imited because of. i t s e f f e c t s on the process constraints and other parameters which are discussed in d i f f e r e n t places i n t t h e ensuing chapters. 3.2 MULTI-PASS DESIGN CONCEPT It was shown t h e o r e t i c a l l y , that a part with a non-uniform rectangular cross-section can be made through two passes of f l a t r o l l i n g . P r a c t i c a l l y , however, i t i s not always possible to complete the process within two operations due to some physical process constraints. Sat i s f y i n g these constraints necessitates to increase the number of operations involved. The procedure of determining the p r a c t i c a l number of passes is termed the m u l t i - p a s s design. The multi-pass design is a scheme which is capable of s a t i s f y i n g the p r a c t i c a l constraints at a l l times. 3.2.1 PROCESS CONSTRAINTS 34 3.2.1.1 Ki nemat i c Constraint The kinematic constraint refers to the l i m i t of continuous free r o l l i n g . In order to draw the material into the r o l l gap without back pressure or forward tension, the following inequality should be s a t i s f i e d at the onset of r o l l i n g (see section 2.1.3) Ah<R«f 2 (3-12) It has also been shown that once the r o l l gap i s f i l l e d with material, condition (3-12) i s relaxed in a way that more absolute draft i s permitted. So, the condition of free r o l l i n g becomes Ah<b-R-f2 (3-13) where 'b' is a c o e f f i c i e n t that varies from 1 in the onset of r o l l i n g to almost 2 (recommended for hot r o l l i n g of s t e e l ) , when the r o l l gap i s completely f i l l e d with material. Normally, the incoming material i s tapered at i t s front head so the condition of the f i l l e d gap can be assumed at a l l times. The value of 'b' can be increased by applying forward tension or back pressure. Rearranging equation (3-13) Ah/R<(b-f 2=C1) (3-14) CI i s termed the kinematic constraint and i t s value i s related mainly to the f r i c t i o n a l conditions. During r o l l i n g , inequality (3-14) should be s a t i s f i e d ; otherwise, the material deformation w i l l stop and the r o l l e r s w i l l begin to s l i p on the surface of the material. 35 3.2.1.2 Dynami c Cons t r ai nt The dynamic constraint refers to the capacity of the machinery in performing the process. Maximum torque available by the r o l l i n g machine, T, i s a convenient c r i t e r i o n . It was shown in section 2.1.4 that the required torque for hot r o l l i n g i s approximately M=(0.6)R.Ah.W m e a„.Y which should be less than or equal to the available torque by the machine, i . e . , (0.6)R.Ah.Wmefln.Y<T or R. Ah.VIm e a n<{ (T/( 0 . 6Y) )=C2) (3-15) C2 i s c a l l e d the dynamic constraint; i t i s spe c i f i e d according to the maximum nominal torque available by the r o l l i n g m i l l and the composition of the material to be r o l l e d . If condition (3-15) is not s a t i s f i e d during r o l l i n g , the machine can no longer form the material due to the overload, and w i l l s t a l l . 3.2.1.3 Convexity Constraint Most r o l l e d blocks end up with some form of convexity or out of flatness on their sides. Here, the term convexity constraint has been adopted to mean an acceptable degree of out of flatness on finished parts. The less the material i s reduced on one side, the less the convexity occurs on the other side. It is therefore suitable to have a' constraint 36 on the value of ab s o l u t e d r a f t s f o r the l a s t p a i r of the r o l l i n g o p e r a t i o n s . Convexity c o n s t r a i n t s are then formulated f o r the f i n i s h i n g passes as below Dr aft in t he 1 s I fi ni s hi ng pas s = cons t ant =CC1 ( 3 - 1 6 ) Draft in the 2nd finishing pas s = c o ns t ant =CC 2 ( 3 - 1 7 ) Values CC1 and CC2 are s p e c i f i e d a c c o r d i n g to the requirements of the d e s i r e d p a r t . I t i s c l e a r that the use of high c a p a c i t y machinery w i l l decrease the number of the r e q u i r e d o p e r a t i o n s but on the other hand , w i l l i n c r e a s e the need f o r having c o n s t r a i n t s on the d r a f t d u r i n g the f i n i s h i n g passes. 3. 2. 2 MJLTI-PASS DESIGN PROCEDURE The concepts developed so f a r can be extended to in c l u d e the det e r m i n a t i o n of the number of o p e r a t i o n s r e q u i r e d as w e l l as t h e i r s p e c i f i c a t i o n s i n forming a non-uniform p a r t i n g e n e r a l . The convexity constraint i s f i r s t a p p l i e d to determine the dimensions of the m a t e r i a l which must go through the f i n i s h i n g p a i r of passes. These p r e - f i n i s h dimensions are then taken as the f i n a l dimensions which have to be gained from the i n i t i a l blank. A f t e r s e p a r a t i n g the f i n i s h i n g passes i n the manner d e s c r i b e d above, the process of manufacturing such a shape i s e x p l o r e d by means of only two o p e r a t i o n s . Both the 37 kinematic and the dynamic constraints are considered to check the f e a s i b i l i t y of the assumed process. The control is based on determining the r o l l gap variation for both passes in such a way that the incremental values of draft do not exceed the maximum allowed by the constraints. This w i l l lead to a new r o l l e d state which may be d i f f e r e n t , but w i l l be closer, in shape to the desired geometry. The new material is then subjected to the next pair of passes aiming to reach the desired shape. The described procedure continues u n t i l a l l the requirements related to the constraints are s a t i s f i e d . This eventually leads to the design of a number of passes which i s frequently more than two. 3 . 3 COMPUTER SOFTWARE A computer program has been developed to calculate the information for the multi-pass r o l l i n g of parts with variable rectangular cross-section. The results are then output in d i f f e r e n t formats for d i f f e r e n t purposes such as graphical v i s u a l i z a t i o n , operating control, user reference, etc. Referring to F i g . 3 -9 , the computer program consists of four major modules: Data Gene r at or Multi-Pass Process Planner Two and Three-Dimensi onal Graph Generator Curve-Fit and Micro-Processor Data Generator There are also a number of user directed programs which 38 serve to provide organized data as input for the above modules. This chapter describes very b r i e f l y how the solution to a general case, i . e . , r o l l i n g of a non-uniform part, i s determined by the program. 3. 3. 1 DATA GENERATOR The data generator provides data necessary for t he m u l t i - p a s s process planner. These data are categorized as follows: ( i ) Information about the properties of the material: c o e f f i c i e n t of thermal expansion. modulus of e l a s t i c i t y at r o l l i n g temperature. Yiel d point at r o l l i n g temperature. ( i i ) Information about the r o l l i n g condition: Rolling temperature. Ambient temperature. F r i c t i o n a l condition between the surface of the material and the r o l l e r s . (///^Information about the i n i t i a l and f i n a l shapes: Variation of width and height of the i n i t i a l material versus i t s length. Variation of width and height of the desired part versus i t s length. ( i v ) Information about the process constraints: Minimum, maximum and incremental values allowed for the r o l l s ' r a d i i . 39 Values of convexity constraints.. Values of kinematic and dynamic constraints. (v) Information about the format of the output r e s u l t s : Precision of cal c u l a t i o n s . Number of discrete data points in the output. The user inputs the above information according to the condition of the underlying problem. The data generator manipulates, organizes and passes these data to the multi-pass planner's data-read f i l e . For example, the data generator uses the values of the c o e f f i c i e n t of thermal expansion, modulus of e l a s t i c i t y and the y i e l d stress to determine the correction factors by which the computed r o l l gap v a r i a t i o n and the corresponding r o l l e d length of the material should be multiplied, in order to consider the effects of the thermal expansion and the e l a s t i c deformation of the material. 3. 3< 2 MULTI-PASS PLANNER The next phase in the program hierarchy is the procedure of the process design. This i s the main part which uses the data generated by the data generator, and determines the number of passes needed, the incremental values for the variation of the r o l l gap and the r o l l s ' r a d i i for each pass. Messages are also provided to be output during the execution in d i f f e r e n t situations or as a guide to the designed features at the end of the execution. 40 As was shown e a r l i e r , there exist at most four solutions to each problem. The program selects one case at a time and performs the related computations. These computations start with working on the f i n i s h i n g passes. Knowing the values of the convexity constraints, the program f i r s t calculates the appropriate r o l l s ' r a d i i . It then calculates the dimensions of the material entering each pass. Recalling i n e q u a l i t i e s (3-14) and (3-15) Rk.REDfk.Vlk<C2 (3-19) REDfk/Rk<Cl (3-20) Wk(k=I 2)' ^ s fc^e largest possible value of the width of the material for each pass, REDfk/k-j 2)i is the allocated reduction (draft) for each one of the f i n i s h i n g passes, Rk(k=l,2)> ^ s fc^e r o l l ' s radius. Rearranging the above i n e q u a l i t i e s for R^ gives Rk>REDfk/Cl Rk<C2/(REDfkVlk) R/ = (REDf k/Cj) and RU=C2/(REDf kWk) are the two boundaries which bracket the r o l l ' s radius. It i s clear that R/ should be less than or equal to Ru. The r o l l ' s radius for each pass is then found as the minimum possible value, i . e . , Rk=REDfk/Cl Note that small r o l l e r s lead to a decrease in the required torque and load. The calculated r o l l ' s radius i s then modified to s a t i s f y the conditions of the a v a i l a b i l i t y of the r o l l e r s . This i s done 41 by rounding up the value of R^ to the nearest available si z e . A control i s necessary to check whether the new value of ^k{k=l,2)' ^ s s t i l l between the stated boundaries. If at any stage of cal c u l a t i o n s , the applied conditions could not be f u l l y s a t i s f i e d , the program returns with an appropriate messages and some guidelines. Knowing the values of the drafts and r o l l s ' r a d i i for the f i n i s h i n g passes, the dimensions of the material entering each pass can be determined using the inverse calculation method: Referring to F i g . 3-8, for the second of the f i n i s h i n g passes the following relationships hold Hmj=Hf j+REDf2 (3-21'a) LnjWfj/Wmj) e-B(Wmj/Hmj) c(Hmj/R 2) D((Hmj-Hfj)/Hmj) Ef.g.j Ln(Hmj/Hf j) (3-21b) Lmj.Hmj.Wmj=Lfj.Hfj.Wfj (3-21c) ( 3 = 1, 2, 3, . . . , n) This set of equations is solved simultaneously to find the dimensions of the material entering the second pass or leaving the f i r s t pass. The same procedure can be followed to f i n d the dimensions of the material entering the f i r s t of the f i n i s h i n g passes: Wi 3=Vlm3+REDf } (3-22a) Ln(Hmj/Hij) e-B(Hij/Wij) C(Wij/R,) D((Wij-Wmj)/Wij) Ef.g.j Ln(Wi j/Wmj) (3-22b) 42 Lij.Hij.Wij=Lmj.Hmj.Wmj (3-22c) (j = l, 2, 3, . . . , n) The material with the incremental dimensions of Wij, H i j , L i j , i s now assumed to be the desired part which i s to be shaped within an unknown number of passes from the known i n i t i a l material. The numerical approach for the determination of the above process within two passes was shown in section 3.1. The r o l l s ' r a d i i necessary for the pair of passes are determined by reapplying the i n e q u a l i t i e s (3-19) and (3-20) Rk>DRAFTkJ/CI (3-23) Rk<C2/(DRAFTkj .VIkj ) (3-24) k=(l, 2) , j = U, 2, . . , n) DRAFTkj stands for the incremental value of the absolute draft during r o l l i n g and i s unknown. Vlkj stands for the incremental value of the side of the material under the r o l l e r s at each pass. Combining the two in e q u a l i t i e s and having in mind that the smaller radius and the larger draft is desirable, the following inequality i s derived Rk2>C2/(Cl .Vlk) which gives the lower l i m i t on as Rk=v/C2/(C1 Mk) (3-25) where ^k{k=l 2) ^s chosen as the maximum value of the width for each pass. The values of r o l l s ' r a d i i for each pass, ^ k ( k = l , 2 ) , a r e then selected as mentioned e a r l i e r . 43 After the process has been determined (refer to section 3.1.2), i t i s evaluated with respect to the kinematic and dynamic constraints. This i s done by comparing the calculated incremental values for the drafts to those permitted by the kinematic and dynamic constraints. The permissible incremental reduction (PREDkj) can then be sp e c i f i e d by the following i n e q u a l i t i e s PREDkj<C2/(Rk.Vlkj ) PREDkj<Cl.Rk k=U, 2) , j = (l, 2,. . . ,n) The values of the calculated drafts should be less than or equal to the permissible values. The v i o l a t i n g draft values are reduced to meet th i s constraint. After the f i r s t pass i s modified, dimensions of the new intermediate material are then used to re-define both the permissible incremental and calculated drafts for the second pass. Modification i s then done on the second pass. If the modified values of drafts for a pass are too small, i t may not be worthwhile to allocate a pass for that operation, unless i t is one of the l a s t pair of passes. In this case the corresponding pass i s deleted and the process i s continued to the next major pass. This implies that the t o t a l number of the passes is not necessarily an even number as i t might be deduced at the f i r s t instance. Direct or forward solution is used to determine the dimensions of the material leaving the modified passes. This is done by solving sets of equations l i k e those in (3-21) 44 and (3-22) repectively , providing that REDkj (k=1,2) D e substituted for REDJ-J and REDf2 in equations (3-21a) and (3-22a). The dimensions of the new intermediate material are then interpreted as the dimensions of a new i n i t i a l material which is to be r o l l e d within a subsequent pair of passes. The procedure i s repeated u n t i l constraints are f u l l y s a t i s f i e d , i . e . , u n t i l there i s no need for pass modification. The incremental values of r o l l gap calculated for each pass are then m u l t i p l i e d by the correction factor to take into account the ef f e c t s of the thermal expansion and the e l a s t i c deformation of the material. For cases in which both the i n i t i a l and the f i n a l dimensions are variable, i t may happen during the execution that the program detects a pair of segments for which conditions (iii) or (iv) of section 3.1.1.3 are not s a t i s f i e d . In these cases, the program automatically increases the value of the corresponding r o l l ' s radius by some allowable increments u n t i l those conditions are s a t i s f i e d . Then, i f the new value of the r o l l ' s radius i s less than the maximum avail a b l e , the execution starts from the beginning using the new r o l l ' s radius; otherwise, i t returns back with a message with regard to the situation and some guidelines. The multi-pass design routine completes i t s execution by passing the information and the related messages in a proper format for the user's reference. Also, i t outputs some relevant data into other f i l e s for d i f f e r e n t uses such 45 as graph generations. The general flow-chart of the multi-pass design algorithm i s shown in F i g . 3-10. 3. 3. 3 GRAPH GENERATOR There are also several routines which help to generate v i s u a l i z e d information of the r o l l e d parts after each operation , t h r e e ~ d i m e n s i o n a l visualization, and v a r i a t i o n of some parameters such as r o l l gap during the operations ,two-dimensi onal visualization. Three-dimensional v i s u a l i z a t i o n is generated by the three-dimensional graphic generator which consists of three routines. The f i r s t uses the incremental values of width and height of the intermediate material(s) and generates the incremental coordinates of the shape. The next routine finds the projected values of these coordinates. The t h i r d routine serves to provide a vi s u a l representation of the object either on a graphics terminal or on a pl o t t e r . Two-dimensional visualization i s also possible by using any two dimensional p l o t t i n g package. 3. 3. 4 OTHER ROUTINES Some miscellaneous routines have been provided which are used for the purpose of operation c o n t r o l . The application and the algorithmic procedures of these routines are described in the following section. 46 3.4 OPERATING ASPECTS In the previous sections, i t was shown how the var i a t i o n of the r o l l gap as a function of the r o l l e d length for each pass i s determined in the form of discrete data points. Figure 3-11 shows a t y p i c a l v a r i a t i o n of r o l l gap versus the r o l l e d length. To derive an a n a l y t i c a l form for th i s relationship, c u r v e - f i t t i n g has been used. The curve has a continuity up to the second order to be smooth enough and to simulate the actual conditions. Appendix B describes the c u r v e - f i t t i n g algorithm. The r o l l e r s are supposed to follow t h i s curve during the operation (position a in F i g . 3-11). Parts with a high degree of geometric changes, may be over-rolled while the r o l l s are following a certain curve. In order to prevent t h i s , the r o l l e r s should adjust their path. However, this w i l l result in under-rolIing (position b in F i g . 3-11). It i s clear that smaller r o l l s permit parts with high variation in geometry to be manufactured. Roll's radius can be decreased in many ways; by either l i m i t i n g the r o l l ' s r a d i i available for the use by the program or by changing the kinematic and/or dynamic constraints. Decreasing the value of the dynamic constraint leads to the selection of a smaller radius but on the other hand, i t may increase the number of operations needed. Provision has been made in the program so that t h i s can be done a r t i f i c i a l l y and in an optimized way. Increasing the value of the kinematic constraint w i l l also lead to the use of smaller r o l l ' s radius but i t may lead to a need for 47 forward tension. The r o l l i n g machine should have a f a c i l i t y for changing the r o l l gap as accurately and as fast as possible. R o l l gap control based on the displacement of the r o l l e d material is a convenient way to control the process. Control strategies based on other parameters such as time w i l l lead to the use of empirical relations which are not suggested (see section 2.1.2, calculation of rolling speed). Both values of the r o l l gap and the r o l l e d length can be detected and read out during the rolling.by various d i g i t a l read out systems such as linear scales. The application of two d i f f e r e n t control systems i s b r i e f l y discussed here. (i) Analog Control System With analog control, each individual output variable (actual r o l l gap and r o l l e d length) i s monitored. Changes are then made in the corresponding input variable (rotation of the servomotor or opening of the hydraulic valve) to maintain the output at a desired l e v e l . This is done continuously. The general block diagram of such a system is shown in F i g . 3-12. As is seen, the difference between the desired value and the actual value is used by the analog c o n t r o l l e r . The mechanism by which the process variables are altered, depends on the p a r t i c u l a r system and the design of the machine. The desired value i s not a set point but variable in i t s e l f . It should be continuously followed during the process as a function of the actual r o l l e d 48 length. So, a template with the shape of the desired r o l l gap variation and a mechanical follower which moves proportionally to the length of the r o l l e d material (Fig.3-12) are needed. The movement of the follower i s scaled and read out by a linear scale measuring device. As i t can be deduced, in the analog control, for each process a d i f f e r e n t template i s needed. Besides, analog control i s hardwired and can not be ea s i l y changed when the design i s changed. This strategy can also be c a l l e d adaptive model control, because a template of the desired p r o f i l e and a mechanical follower are used to control the r o l l gap. (ii) Direct Digital Control (D.D.C.) Due to the increased complexity and the i n f l e x i b i l i t y of the model control systems, new methods based on micro-computers, namely Dfirect Digital Cot r ol have become increasingly relevant. More than one loop may be controlled with a micro-computer. In a d i g i t a l control strategy a micro-computer samples the variables (actual r o l l gap and r o l l e d length) p e r i o d i c a l l y with a s u f f i c i e n t sampling frequency. F i g . 3-13 shows the general block diagram of a ty p i c a l D.D.C. The micro-computer i s programmed such that at each time i n t e r v a l , the actual values of the r o l l e d length and the r o l l gap are read out. The recorded variables are then used by the micro-computer to determine the desired r o l l gap for the next increments. The c u r v e - f i t t i n g program is used to organize data in a fashion suitable for the use 49 b y t h e m i c r o - c o m p u t e r . T h e c o n t r o l s t r a t e g y b a s e d o n D.D.C i s s o m e t i m e s c a l l e d adaptive data control b e c a u s e i n t h i s m e t h o d i n s t e a d o f c o n t i n u o u s d a t a t r a n s m i s s i o n f r o m t h e t e m p l a t e , d i s c r e t e d a t a i s f e d t o t h e c o n t r o l s y s t e m . By t h e u s e o f D.D.C, p r o g r a m m i n g (software design) i s s u b s t i t u t e d f o r w i r i n g - u p a n d m o d e l l i n g (hardware design). M o r e o v e r , w i t h a f a s t c o m p u t e r s y s t e m , s i m u l t a n e o u s c o n t r o l o f s e v e r a l m a c h i n e s i s p o s s i b l e . M i c r o - c o m p u t e r s c a n b e e a s i l y r e p r o g r a m m e d f o r a new p r o c e s s a s t h e d e s i g n c h a n g e s o c c u r 2 7 . 3.5 SUMMARY A N D E V A L U A T I O N T h e m e t h o d o f r o l l i n g o f p a r t s w i t h v a r i a b l e r e c t a n g u l a r c r o s s - s e c t i o n was f i r s t d e s c r i b e d f o r a s i m p l e u n c o n s t r a i n e d t w o - p a s s p r o c e s s . T h i s w as f o l l o w e d b y n u m e r i c a l l y s o l v i n g a r e s u l t a n t s e t o f n o n - l i n e a r e q u a t i o n s s u c h a s ( 3 - 9 ) a n d ( 3 - 1 0 ) , t o f i n d t h e i n c r e m e n t a l v a l u e s o f t h e d i m e n s i o n s o f t h e i n t e r m e d i a t e m a t e r i a l . R e a r r a n g i n g e q u a t i o n s ( 3 - 9 ) a n d ( 3 - 1 0 ) a s f o l l o w s L n ( H m j / H i j ) e - B ( H i j / w i j ) c ( W i j / R W ) D ( ( W i j - W m j ) / w i j ) E f . g . j L n ( W i j/Wmj) L n j W f j / W m j ) e - B ( W m j / H m j ) c ( H m j / R H ) D ( ( H m j - H f j ) / H m j ) E f . g . j L n ( H m j / H f j ) (1 = 1,2, . . . ,n) i t i s s e e n t h a t x a n d $ a r e f u n c t i o n s o f t w o u n k n o w n s , Hmj a n d Wmj, a n d f o u r v a r i a b l e p a r a m e t e r s H i j , W i j , H f j a n d W f j . A s t u d y o f t h e b e h a v i o r o f t h e a b o v e f u n c t i o n s s h o w s t h a t 50 they both are continuous with respect to the two unknowns and the four parameters in the physical range of their existence. Having a solution for each segment (refer to section 3.1.1.3), i t i s deduced that the set of solutions, i. e , Wmj and Hmj, (3=1, 2 , . . . , n), are continuous versus the r o l l e d length providing that Wij, H i j , Wfj and H f j , (3 = 1, 2, ...,/?), are also continuous with respect to the length (see 26, pp 209 to 213, "inverse function theorem"). This implies that the solution of the deformation of a continuous shape to another, i . e . , the intermediate material i s also continuous in shape. The method was then extended to include and s a t i s f y the physical constraints. It was discussed how these constraints lead to an increase in the required number of passes from the ideal minimum of two. For example, the inclusion of the convexity constraints, alone, immediately increases this number to four. Normally, the i n i t i a l material used would be a uniform block. A suitable i n i t i a l material may be one whose cross-sectional dimensions are equal to the largest cross-sectional dimensions of the material prior to the f i n i s h i n g passes. Roll's radius plays an important role in the magnitude of spread. It is also a f l e x i b l e tool in making the deformation of more complex shapes possible. The l a s t part of the chapter was allocated to a brief discussion of operating aspects. Two control strategies were described. The control strategy based on the r o l l e d length 51 implies the fact that the material should completely leave a pass before entering the next, otherwise the difference between the exit and the entrance v e l o c i t y , not only causes a variable tension or compression along the length of the block, but also makes the process of measurement considerably d i f f i c u l t (for more d e t a i l s , see 33 pp 9 to 16) . 4. EXPERIMENTAL EVALUATION 4.1 INTRODUCTION The theore t i c a l analysis of r o l l i n g parts with variable rectangular cross-section was presented in chapter three. At th i s stage the need for experimental evaluation of the method i s sensed. The purpose of doing the experiments was to examine the physical aspects of the method discussed, and to evaluate i t s accuracy in predicting the process behaviour. The objectives were: 1. To evaluate the c a p a b i l i t y of the spread formula, as used, in predicting the geometric deformation of the material during unsteady r o l l i n g and to determine and implement the necessary corrections. 2. To evaluate the v a l i d i t y of the computer algorithm in determining the process s p e c i f i c a t i o n s . The f i r s t objective relates to the fact that for most of the operations of the process under investigation, the incoming material is non-uniform and the r o l l gap is subjected to v a r i a t i o n . El-Kalay and Sparling's formula, as stated, i s best suited for the estimation of spread in the conventional constant thickness r o l l i n g . Thus, i t was known that the results of the direct application of t h i s formula to the general s i t u a t i o n where the dimensions of the input material and the r o l l gap are variable would be in certain error, and that some corrective factors could be derived from experiments and incorporated in the formula. 52 53 The second objective i s b a s i c a l l y the test for the val i d a t i o n of the computer program. In thi s regard, some experimenets were performed on uniform blocks under d i f f e r e n t r o l l i n g conditions. Most of the experiments were conducted on hot mild s t e e l ; although, there were some experiments which were performed on warm aluminum. 4 . 2 E X P E R I M E N T A L A R R A N G E M E N T S 4. 2. 1 INSTRUMENTATION The r o l l i n g m i l l used for experiments was a small laboratory unit, (see F i g . 4-24). The nominal torque of the motor used, was 4.77Kgf-m. and four angular speeds were availa b l e , s p e c i f i c a l l y : 17.2, 34.3, 51.4 and 68.5rev/min. The r o l l s ' r a d i i were 50.8mm and with d i f f e r e n t surface conditions. The upper r o l l could be moved v e r t i c a l l y for adjusting the r o l l gap. This was done through a hand-wheel and a power screw. Sixteen rotation of the hand-wheel translated into one inch v e r t i c a l movement of the upper r o l l . Due to wear on the sides of the threads, the upper r o l l had some slack or backlash, in the v e r t i c a l d i r e c t i o n . Due to the misalignment of the bearings, a s l i g h t horizontal slack also existed. Preliminary experiments with aluminum were performed to measure the amount of the v e r t i c a l slack. This was found to be 0.6mm in the range of r o l l gaps between 10mm to 25mm. The existance of the horizontal slack could 54 have caused the r o l l e d material to twist in the horizontal plane; to prevent t h i s , a guide way or j i g was b u i l t and i n s t a l l e d at the entry point. The absolute maximum reduction that a heated steel slab could have been subjected to, in a single pass, was experimented to be about 5mm for a 25mm wide specimen. The dimensions of the furnace used were "600x400x400mm". The maximum temperature that could have been reached with t h i s furnace was about 1200°C. The distance between the furnace and the r o l l i n g m i l l was about two meters (see F i g . 4-25). 4. 2. 2 SPECIMENS AND MEASUREMENT Suitably long specimens were used in order to iso l a t e the e f f e c t of the excessive spread at the front and back ends which can be s i g n i f i c a n t for small specimens 2 3. Two major modes of deformations were observed during the experiments with hot s t e e l ; double bulged and rhombic ( r e g u l a r or i r r e g u l a r ) cross-sections. F i g . 4-1 shows these two and the procedure for the c a l c u l a t i o n of the mean value of spread. Preliminary experiments on the constant thickness r o l l i n g of steel showed that a good agreement exists between the r e s u l t s of the experiments and those predicted by the formula (2-18) providing that conditions of rough r o l l s and heavy scales are assumed (see table 2-6). To obtain the r o l l s ettings, the required f i n a l thickness of the r o l l e d material was modified by adding the thermal expansion of the 55 bar. The thermal expansion from the room temperature to the working temperature of 1100°C was 1.48% of ,the exit thickness. The effects of e l a s t i c deformation of the bar under the r o l l s , and the m i l l spring on the exit thickness, was known to be very small for hot steel (less than 0.1% of the exit t h i c k n e s s 2 2 ) and thus was neglected. 4.3 STEADY STATE ROLLING The r o l l i n g of uniform parts when the r o l l gap i s kept constant i s referred to as the steady state rolling. In thi s section some t y p i c a l experiments on steady state r o l l i n g are described. The purpose of these experiments were to evaluate the computer algorithm and to study the shapes of the r o l l e d cross-sect ions. 4. 3.1 EFFECTS OF SEQUENTIAL ROLLING ON SPREAD A heated uniform block "19.1mmx19.1 mm", 180mm long, was ro l l e d on one side down to 18mm. The specimen was then re-heated and r e - r o l l e d on the same side to a new thickness of 15.55mm. The increased width after these operations was measured 19.70mm (Fig. 4-3a). Next, a similar block was r o l l e d in a single pass to the same thickness of 15.55mm. The increased width was measured 19.78mm (Fig. 4-3b), which was greater than the value obtained in the f i r s t experiment. The same set of experiments were performed with d i f f e r e n t sizes of blocks and similar results were observed which indicated that, for a given t o t a l d r a f t , the magnitude 56 of spread decreases as the number of passes increases. Physically, t h i s i s due to the fact that the higher the draft, the higher the spread; also, the smaller the width of the material under the r o l l s , the larger the spread. Figure 4-3 shows a reproduction of the real cross-sections of the specimens at each stage of the above experiments. It i s seen that when the draft i s small, the d i s t o r t i o n on the spread side is also small. This implies that the r o l l e d parts can have a f l a t t e r surfaces // the d r a f t s in the last pair of passes in the multi-pass process are kept small. 4. 3. 2 THE EVALUATION OF THE MULTI-PASS ALGORITHM A uniform block of cross-section "25.4mmx25.4mm" was ro l l e d within two operations and was reduced in new cross-sectional dimensions to "20mmx20mm". The value of the r o l l gap for each pass was predicted by the program and the r o l l gap was set to that value beforehand. The dimensions and the cross-sectional views of the material at each stage are reproduced in Fig . 4~4a. Also, the output of the program is i l l u s t r a t e d in F i g . 4-5. In the second experiment, the multi-pass design concept was applied and two f i n i s h i n g passes with a 2mm of reduction for each, were s p e c i f i e d . The cross-sectional shapes of the ro l l e d material after each pass, as well as the computer prediction are provided in figures 4-4b and 4-6, respect i v e l y . 57 It i s seen that applying the concept of the convexity constraints to the process improves the appearance of the finished part noticeably. 4 . 4 UNSTEADY STATE ROLLING When the material to be r o l l e d i s non-uniform and/or the r o l l gap changes during the r o l l i n g , the process i s termed unsteady state rolling. Most of the r o l l i n g operations involved in the production of parts with non-uniform cross-section are of an unsteady state nature. No previous work has been done to evaluate the geometric deformation of the material in th i s type of r o l l i n g . It was detailed e a r l i e r that in the process planning scheme for the deformation of non-uniform parts, each segment i s assumed to be a member of a uniform block which is r o l l e d within a certain r o l l gap. In r e a l i t y , however, not only the r o l l gap varies while the segment i s being r o l l e d , but also the segment i s affected by the succeeding and the preceding segments which are di f f e r e n t from the segment in question. In other words, each portion of the material encounters a history of deformation which i s dif f e r e n t from that of the steady state r o l l i n g . Some experiments were needed to evaluate the ef f e c t s of such unsteadiness on the spread. Thus, unsteady state r o l l i n g takes place, whenever any one or a combination of the following situations exist 1. The width of the material being rolled is variable. 58 2. The height of the material being r o l l e d is variable. 3. The r o l l gap changes during r o l l i n g . The e f f e c t s of the above c o n d i t i o n s on spread, can be eva l u a t e d independently. Thus, i n order to ev a l u a t e the e f f e c t of each, the other two were kept constant i n the experiments. These important experiments along with the r e s u l t s o b tained are presented i n the f o l l o w i n g s e c t i o n s . 4. 4. 1 EFFECTS OF HEIGHT VARIATION ON SPREAD To e v a l u a t e the e f f e c t of height v a r i a t i o n , three types of specimens with the same l e n g t h of 225mm and the same width width of 12.8mm were used (see F i g . 4-7). The height of each specimen v a r i e d from 12mm to 30mm but i n d i f f e r e n t p a t t e r n s . The v a r i a t i o n of height f o r the three types of specimens were formulated versus the le n g t h as f o l l o w s 1st specimen; concave parabolic HEIGHT^3.5 5x10" 4(LENGTH) 2 + 12 (mm) 2nd specimen; straight taper HEIGHT=0.0 8(LENGTH) + 12 (mm) 3rd specimen; convex parabolic HEIGHT= 1 . 2|/ (LENGTH) + 1 2 (mm) The l e n g t h i s measured from the narrower head. Sides of the specimens were marked l o g i t u d i n a l l y and l a t e r a l l y (see F i g . 4-26), with 5mm i n t e r v a l s . The ra t e of height v a r i a t i o n f o r each specimen i s shown i n F i g . 4-8. A l l three specimens were r o l l e d from the narrower head to a constant t h i c k n e s s of 12.9mm. The p l o t s of the experiments 59 and the predicted spread versus the corresponding i n i t i a l height are shown in figures 4-9, 4-10 and 4-11. Referring to figures 4-8 and 4-10, for specimen (//) which has a constant rate of height variation (0.08mm/mm), i t i s seen that when the i n i t i a l height i s small, i . e . , when the absolute draft i s small, experimental values and the simulated values are in a good agreement, but as the absolute draft increases, the actual values begin to deviate from the values calculated by El-Kalay and Sparling's conventionl (unmodified) spread formula. The maximum error was found to be more than 3%. It i s seen from figures 4-9 and 4-10 that, for the same draft, the specimen (/) shows larger deviation from the unmodified predicted values than does the specimen (//), P a r t i c u l a r l y for the draft values greater than 3mm. This i s because the rate of height increase is larger for specimen (/' ) (refer to Fig . 4-8). It is also seen from F i g . 4-11, that even for small dr a f t s , the measured values are larger than the unmodified predicted values. This i s due to the fact that for the specimen ( i i i ) , the rate of height variation i s larger on the head end. It i s deduced that for low draf t s , less than 2.0mm, and for low rate of height variations, less than 0.05 mm/mm, El-kalay and Sparling's predictions are reasonably accurate to be used, but as the absolute draft and/or the rate of height v a r i a t i o n increases, the actual spread increases such that the conventional formula begins to deteriorate in i t s 60 p r e d i c t i v e , a c c u r a t e l y . T h e d e v i a t i o n i s s e e n t o b e m o r e t h a n 5% i n some i n s t a n c e s . A n e x p l a n a t i o n f o r t h e e f f e c t o f height v a r i a t i o n o n s p r e a d i s t h a t t h e h e i g h t v a r i a t i o n a f f e c t s t h e l e n g t h o f t h e a r c o f c o n t a c t w h i c h i s a n i m p o r t a n t p a r a m e t e r i n d e t e r m i n i n g t h e v a l u e o f s p r e a d . H i l l 1 8 s h o w e d how t h e l a t e r a l s p r e a d d e p o n d s i n v e r s e l y o n t h e l e n g t h o f a r c o f c o n t a c t : (i) - when i / R . A h — t h e l a t e r a l e l o n g a t i o n i s m a x i m u m . (ii) - when i / R . A h — » 0 , c o n d i t i o n o f p l a n e s t r a i n o c c u r s , i . e . , t h e l a t e r a l e l o n g a t i o n i s z e r o . P h y s i c a l l y , w hen t h e l e n g t h o f t h e a r c o f c o n t a c t b e t w e e n t h e r o l l ' s s u r f a c e a n d t h e m a t e r i a l i n c r e a s e s , d u e t o t h e i n c r e a s e i n f r i c t i o n , t h e l o n g i t u d i n a l e l o n g a t i o n o f t h e m a t e r i a l b e c o m e s d i f f i c u l t , s u c h t h a t m a t e r i a l t e n d s t o e l o n g a t e m o r e o n t h e l a t e r a l d i r e c t i o n . T h i s i s why t h e a c t u a l l a t e r a l e l o n g a t i o n i s l a r g e r t h a n t h e p r e d i c t e d v a l u e when n e g l e c t i n g t h e e f f e c t o f h e i g h t v a r i a t i o n . A n o t h e r e f f e c t o f h e i g h t v a r i a t i o n i s o n t h e s t r a i n r a t e w h i c h i n d i r e c t l y a f f e c t s t h e s p r e a d . S i m 1 2 f o r m u l a t e d t h e v a l u e o f mean s t r a i n r a t e a s f o l l o w s MEAN STRAIN R A T E = v ( R . A h ) ~ ° ' 5 L n ( l / ( l - r ) ) v i s t h e p e r i p h e r a l v e l o c i t y o f r o l l s , r i s t h e r e d u c t i o n i n t h i c k n e s s C A h / h , ) A s t u d y o f t h e b e h a v i o u r o f t h e a b o v e f o r m u l a s h o w s t h a t 61 draft i s d i r e c t l y related to the mean st r a i n rate; as draft increases, mean st r a i n rate associated with the process also increases. Strain rate causes the material under r o l l i n g to be hardened in a l l d i r e c t i o n s . The degree of st r a i n hardening in longitudinal and l a t e r a l d i r e c t i o n depends on the respective s t r a i n s . Under the conditions which promote more s t r a i n in elongation than spread, which is true for the most rolling purposes, s t r a i n hardening would cause a greater resistance to deformation in elongation than in spread. A modification of the spread formula was then needed to improve the theoretical prediction by considering the effect of height v a r i a t i o n . In doing so, the following points are presented (a) - S p a r l i n g 1 8 included some c o e f f i c i e n t s in his proposed formula, (2-18), for the e f f e c t s of temperature v a r i a t i o n , s t r a i n rate and material composition (see section 2 . 3 ) . He suggested that assigning values less than unity for these c o e f f i c i e n t s , would predict the larger spread and vice versa. (b) - Modification to the spread formula should be such that, for a uniform block, i . e . , a steady state process, i t becomes i n e f f e c t i v e . (c) - Rate of height v a r i a t i o n should appear in the improved formula. Also i t s effect on the length of the arc of contact should be included. 62 (d)- The e f f e c t of height variation on spread depends on the f r i c t i o n a l c o n d i t i o n 2 2 ; the lower the f r i c t i o n between the r o l l s and the material, the lesser i t af f e c t s the value of spread and vice versa. The following formula i s then suggested to s a t i s f y the above requirements T i s a parametric c o e f f i c i e n t which was defined as follows Hv i s the absolute rate of height v a r i a t i o n of the material at the cross-section where the spread i s to be predicted. £ i s a constant; the value of which was found to be 6. 15 for minimum difference between the experimental and the predicted r e s u l t s . The method of least square f i t t i n g was used in t h i s regard. As i t is seen, the necessary requirements are s a t i s f i e d in the improved formula: (i) - When the block i s uniform, i . e . , when Hv=0, then T=1 and the formula converts to i t s conventional form. ( i i ) - For higher rates of height v a r i a t i o n , or for larger lengths of the arc of contact, the value of T becomes smaller and consequently, the calculated spread w i l l become larger. Ln(wm/w,) Ln(h,/h 2) =A.EXP[-B(w 1/h 1)c(h 1/R) D(Ah/h 1) E-T] T=(1+^.Hv/R.Ah/w,)-D (4-1 ) 63 ( i i i ) - The ef f e c t of height variation is weighted as in the El-Kalay and Sparling's formula, using parameter D (table 2-6) as the power. ( i v ) - For very large values of w1f where the plai n s t r a i n condition holds, e f f e c t of height v a r i a t i o n on spread becomes n e g l i g i b l e . The improved formula was then used to predict the values of spread for the experiments performed. The results are shown in figures 4-9, 4-10 and 4-11. As i s seen, the new predictions are in a very good agreement with the experimental r e s u l t s . The mean deviation between the new predictions and the experimental values was found to be within ±0.8% . 4. 4. 2 EFFECTS OF WIDTH VARIATION ON SPREAD In • the following experiments, a l l the relevant parameters were kept constant expect the width of the material being r o l l e d . Identical specimens as in F i g . 4-7 were used. The constant dimension th i s time was interpreted as the height and the non-uniform side was selected as the width. A l l the three types of specimens were heated to 1150°C and r o l l e d from the narrower head to a thickness of 10.3mm. Figures 4-12, 4-13 and 4-14 show the r e s u l t s . For each specimen, the va r i a t i o n of the increased width versus the i n i t i a l width has been plotted. The predicted values of the spread width using El-Kalay and Sparling's formula are also superimposed on each graph. It i s seen that, there i s 64 an agreement between the r e s u l t s obtained from the experiments and those p r e d i c t e d by the s i m u l a t i o n , except fo r a s l i g h t constant s h i f t . The d e v i a t i o n was measured and was w i t h i n 1.7%. In the second set of experiments, the same type specimens were h o t - r o l l e d t o a t h i c k n e s s of 10.3mm but t h i s time the wider head was r o l l e d f i r s t . The values of width were p l o t t e d a g a i n s t t h e i r i n i t i a l v a l u e s and were compared with those p r e d i c t e d by the formula. F i g u r e s 4-15 and 4-16 show the r e s u l t s . Maximum d e v i a t i o n from the p r e d i c t i o n was measured and was 1.2%. These experiments show that both width i n c r e a s e and the decrease cause a r e d u c t i o n i n spread. I t i s seen that f o r l a r g e r v a l u e s of width, the d e v i a t i o n i s s l i g h t l y g r e a t e r than the d e v i a t i o n f o r the s m a l l e r widths, however, the r e l a t i v e d e v i a t i o n doesn't change. A comparison between f i g u r e s 4-13 and 4-15, f o r example, shows that spread i s more s e n s i t i v e to the width i n c r e a s e than i t i s to the width decrease. Experiments on aluminum p i e c e s showed the proof of t h i s recent a s s e r t i o n . F i g u r e 4-17 shows an aluminum specimen having a constant t h i c k n e s s and a symmetric l i n e a r l y v a r i a b l e width. The primary purpose of t h i s r o l l i n g experiment was to observe the extent to which the c o n d i t i o n of width v a r i a t i o n a f f e c t s the spread. In other words, i f the s i g n of the width v a r i a t i o n i s to have a s i g n i f i c a n t e f f e c t , then the r o l l e d specimen should l o s e i t s symmetry a f t e r i t i s r o l l e d . The specimen was r o l l e d to an a b s o l u t e 65 reduction of about 5mm at a temperature of 450°C. The r o l l e d specimen maintained i t s symmetry. F i g . 4-18 shows this result graphically. It i s seen that the points corresponding to the two symmetric ends overlap except for a s l i g h t d i v i a t i o n near the middle region where the aspect r a t i o is large. The above experiments show that using the conventional spread formula and applying i t for each individual cross-section, overstimates the value of spread by about 1.7%. A parametric correction factor, A, similar to the case of height v a r i a t i o n , was then introduced in order to reduce th i s error A= ( 1 . 0 + T J . W V ) c (4-2) Wv i s the absolute rate of width v a r i a t i o n , C is chosen from table 2-6, 77 i s a constant; value of which was found to be 7.75 The implementation of A in the spread formula decreased the error to less than 0.5%. 4. 4. 3 EFFECTS OF ROLL GAP VARIATION ON SPREAD The purpose of the following experiments was to evaluate the e f f e c t s of the r o l l gap v a r i a t i o n on the spread. As i t was mentioned e a r l i e r , while developing the method, each lengthwise segment of the material was imagined as i f i t was an element from a uniform block being r o l l e d through a predetermined and fixed r o l l gap. In fact t h i s i s assumed for a l l the segments in a sequential manner. Thus, 66 f o r any element being r o l l e d , although the r o l l gap i s co n t i n u o u s l y changing, the en t r y and e x i t dimensions are known. Two cases can be i d e n t i f i e d i n t h i s regard: (i) The r o l l gap i s opening ( i i ) The r o l l gap i s c l o s i n g For e i t h e r case, the e f f e c t s of the r o l l gap v a r i a t i o n on spread i s s t i l l unknown. The experiments were performed with d i f f e r e n t r a t e s of r o l l gap v a r i a t i o n . Some t y p i c a l examples are i l l u s t r a t e d here. A long square rod with c r o s s - s e c t i o n a l dimensions "12.7mmx12.7mm" was heated to 1150°C and r o l l e d through a c l o s i n g r o l l gap o p e r a t i o n which v a r i e d from 12.7mm to 9.95mm. Due to the lack of mechanical f a c i l i t y i n doing so, the r o l l gap was changed manually with a speed as uniform as p o s s i b l e . The v a r i a t i o n of the r o l l gap durin g t h i s process i s shown i n F i g . 4-19. The spread f o r some c r o s s - s e c t i o n s along the le n g t h were measured and compared to those p r e d i c t e d by E l - k a l a y and S p a r l i n g ' s formula as a p p l i e d to each segment. F i g u r e 4-20a shows the r e s u l t s , while F i g . 4-20b d e p i c t s an a l t e r n a t i v e r e p r e s e n t a t i o n of the same. In a complementary experiment, an i d e n t i c a l uniform block was h o t - r o l l e d while the r o l l gap was opened from 9.1mm to 12.7mm du r i n g the o p e r a t i o n ( F i g . 4-22). The incremental values of the i n c r e a s e d width along the r o l l e d l e n g t h were measured and are shown i n F i g . 4-23 along with the .corresponding p r e d i c t e d v a l u e s . The above experiments i n d i c a t e d that both the opening and the c l o s i n g of the r o l l 67 gap have an increasing e f f e c t on spread. The e f f e c t s of r o l l gap opening and closing becomes more s i g n i f i c a n t for higher d r a f t s . This was experimentally v e r i f i e d and the findings are shown in figure 4-20b, where the difference between the predicted and the experimental values increase as the draft increases. Also, for the same draft, higher rate of r o l l gap changing causes more spread. Figure 4-21 shows this assesment; r e f e r r i n g to t h i s figure, the higher values of spread correspond to the specimen which was r o l l e d with the higher rate of r o l l gap c l o s i n g . The comparison was based on measuring the increased width at equal thickness cross-sections. 4.5 SUMMARY AND EVALUATION This chapter described the experiments which were conducted to evaluate the d i f f e r e n t aspects of the process of r o l l i n g parts with variable rectangular cross-section. The experiments were performed on both uniform and non-uniform blocks. Some experiments on uniform blocks were primarily conducted to confirm the basic concepts. These led to appropriate arrangements to be set up for the subsequent experiments. Handling the material from the furnace to the r o l l i n g m i l l was done manually. The heating time was about 20 minutes. The specimens which should have been r o l l e d twice or more were immediately re-heated before r e - r o l l i n g . The re-heating time was then reduced to almost h a l f . Large 68 temperature variations was found to have a s i g n i f i c a n t e f f e c t on the magnitude of spread. Although the furnace was placed as close to the r o l l i n g m i l l as possible, the temperature drop during the handling, was postulated to be large enough to f a l l below the range in which El-kalay and Sparling's formula was accepted to be accurate, i . e . , 1100±10°C. This was observed from the results of section 4.3.2 where the computer predicted values were a l l underestimated. In order to take into account the temperature drop during the handling, the specimens were heated to 1150°C. Good agreement was then observed between the results from the theory and those of the experiments. The s t r a i n rate was found to be less e f f e c t i v e on spread at higher temperatures (>1100°C). t h i s assertion was experimented with d i f f e r e n t r o l l i n g speed at a fixed temperature. However, for other experiments, the angular speed of the r o l l e r s was set to the lowest value which s a t i s f i e d the condition of s t r a i n rate in El-kalay and Sparling's formula. A l l the quantitative experiments were done on hot mild s t e e l , however, some were performed on aluminum, namely, the q u a l i t a t i v e or c r i t i c a l experiments. Figure 4-27 shows the aluminum specimens which were used in these experiments. The problem with aluminum stocks was their d i f f e r e n t modes of deformation which quite often happened during the experiments (see Fig . 4-2). Therefore, aluminum was found unsuitable for simulating the hot r o l l i n g of s t e e l . 69 Whenever necessary, the thickness of the scales were also taken into account during the measurements. The thickness of the scales were found to vary from 0.2mm to 0.8mm, depending mostly on the number of times the specimen had been heated in the furnace. The effects of width, height and the r o l l gap variation on spread were investigated. It was found that the increase in the height of the incoming material has a s i g n i f i c a n t e f f e c t on the spread. A method of correction was proposed in the form of equation (4-1). This correction can also consider the effect of negative rate of height variation by changing the sign of the c o e f f i c i e n t D, in the formula (4- 1) , to p o s i t i v e . Width v a r i a t i o n , either p o s i t i v e or negative, was observed to have a decreasing ef f e c t on spread, however, i t s eff e c t was not as much s i g n i f i c a n t as the ef f e c t of height v a r i a t i o n . A parametric correction factor in the form of equation (4-2) was then proposed to implement these e f f e c t s . A q u a l i t a t i v e experimental study on the effects of the r o l l gap changing on spread showed that both the opening and the closing of the r o l l gap have an increasing effect on the magnitude of spread. This ef f e c t i s probable to be proportional to the length of the arc of contact, absolute draft and the r o l l i n g speed. Due to the lack of access to an appropriate f a c i l i t y for automatic and continuous changing of the r o l l gap, an adequate quantitative examination was not possible. 70 The capacity of the r o l l i n g m i l l did not allow to extend the experiments to the r o l l i n g of larger specimens with high rates of variation in height and width. An extensive range of experiments in a variety of real cases are needed to enhance the results extracted in t h i s work. 5. SAMPLE ILLUSTRATIONS 5.1 EXAMPLE NO.l A uniform block with cross-sectional dimensions of 70mmx250mm is to be r o l l e d to a part with l i n e a r l y variable height (Fig. 5-1). The width of the part i s required to remain constant and equal to the width of the i n i t i a l block. Figure 5-2 shows the spread on the side of the constant width, i f only one r o l l i n g pass with l i n e a r l y variable gap is performed. With a single pass, , the f i n a l dimensions, though predictable, are uncontrollable. So, for a dimensionally controllable process, at least two r o l l i n g passes should be used. The r o l l s ' r a d i i of these passes were ar b i t r a r y set to 175mm and 180mm, respectively. One run of the computer program produced two solutions for the above problem. The f i r s t solution implied that the smaller side of the i n i t i a l material should be r o l l e d f i r s t followed by the r o l l i n g of the other side. Figure 5~3a shows the shape of the intermediate material for t h i s case. Figures 5-4 and 5-5 show the var i a t i o n of the r o l l gap and the reduction for each pass. Reduction has been defined as the incremental r a t i o of draft over the i n i t i a l thickness. The second solution implied that the larger side of the i n i t i a l material, i . e . , the height should be r o l l e d f i r s t , followed by an operation on the l a t e r a l dimension. Figure 5-3b shows the shape of the intermediate material for thi s case. Also, figures 5-6 and 5-7 show the var i a t i o n of r o l l 71 72 gap and the reduction during each pass for t h i s feasible solution. In the second case, there is only one pass with variable r o l l gap. On the other hand, in the f i r s t case, both passes have variable r o l l gap; because the second pass has been assigned to form the height of the part, better geometry on this side is obtained and consequently, the b a r r e l l i n g occurs on the uniform side. Thus, depending on which side of the material i s preferred to a t t a i n a better f i n i s h , one or the other sequence of operations may be selected. However, i t i s seen that since in t h i s example the process constraints were not considered, the values of reduction reach to more than 80%, a proportion which i s p r a c t i c a l l y impossible. The second example which follows, i l l u s t r a t e s the r o l l i n g of a more complex part with the process constraints in e f f e c t . 73 5 . 2 E X A M P L E N O . 2 In t h i s i l l u s t r a t i v e example, the production of a part the width of which i s constant and the height of which varies p a r a b o l i c a l l y i s considered. Unlike example No.1, a l l the process constraints are applied in th i s case. The following information were supplied to the computer program as the input data: Dimensions of the i n i t i a l block (Fig. 5-8a): width, 155mm height, 11Omm Dimensions of the f i n a l p a r t ( F i g . 5~8b): width, constant, 150mm height, varies par ab ol i c al I y {minimum value, 50mm/ maximum value, 100mm) length, 2000mm Yi e l d stress of the material at the working temperature, 15Kgf/mm2 Modulus of e l a s t i c i t y of the material at the working temperature, 15x103Kgf/mm2 C o e f f i c i e n t of thermal expansion of the material, 3.3x10 6(1/°C) Working temperature, 1100°C Ambient temperature, 30°C Co e f f i c i e n t of f r i c t i o n between the r o l l ' s surface and the material, assuming rough r o l l s , 0.35 Maximum torque available by the r o l l i n g m i l l , 5000Kgf-m 74 r o l l sizes a v a i l a b l e : Minimum radius, 100mm Maximum radius, 500mm (with increments of 5mm) Applied drafts for the f i n i s h i n g passes: F i r s t pass, 5mm Second pass, 5mm Precision of computation, 0.001 Number of segments that the f i n a l part i s divided into, 25 Five r o l l i n g passes were determined by the program. Figure 5-9 shows the sequence of these passes. F i g . 5-10 shows the var i a t i o n of the r o l l gap versus the r o l l e d length. Figure 5-11 provides more information about the pa r t i c u l a r s of the planned process. 6. CONCLUSIONS AND SCOPE FOR FUTURE WORK A computer-aided process planning scheme for r o l l i n g of parts having variable rectangular cross-section was developed. El-Kalay and Sparling's spread formula was adopted and a number of modifications were incorporated. These were done through a n a l y t i c a l and experimental evaluations of the p a r t i c u l a r c h a r a c t e r i s t i c s of the process. These included width, height and the roll-gap v a r i a t i o n . The results of the experiments were contrasted against those predicted by the modified formula and good agreements were seen. An important observation a r i s i n g from t h i s study was that the process constraints and the unsteady nature of the process played the key roles in planning the scheme. A computer algorithm has been developed which determines the number of r o l l i n g passes required and the s p e c i f i c a t i o n s for each pass. The f l a t r o l l i n g process, where formed-die r o l l i n g and die-forging are also applicable, can have a considerable cost advantage as i t replaces the forging hardware with the rolling software. This implying that the same tooling set may.be used for the production of an i n f i n i t e number of di f f e r e n t shapes. This work was a f i r s t attempt in developing a computer-aided system for r o l l i n g parts with variable rectangular cross-section. More work, both theoretically and experimentally, in d i f f e r e n t aspects are suggested for the 75 76 future work. The study on the geometric deformation of non-uniform blocks through variable r o l l gaps should be continued extensively. This is essential not only to evaluate further, the accuracy of the formula which was developed during t h i s work, but also to modify and expand i t for inclusion of the effects of the r o l l gap variation which was not completed in this study due to the laboratory constraints. The l i m i t of part complexity i s not yet adequately known. An algorithmic approach may be developed to relate a l l the parameters in e f f e c t and to determine beforehand whether the production of a part with a given shape i s feasible by t h i s method. F i n a l l y , the method can be extended to permit continuous operations. This can start with considerations r e l a t i n g to the design of machinery (and controls) and is an important and r e l a t i v e l y d i f f i c u l t task to do. Table 2-1 Value of correct ion factor as function of peripheral r o l l ve loc i ty in Bachtinov's formula(6) vr(nv/sec.) to2 3 4 5 6 7 8 9 10 12 14 16 18 20to22 k 1. .9 .8 .72 .66 .6 .57 .55 .52 .47 .45 .43 .42 .41 Table 2-2 Summary of the average deviations of predicted spread from the experimental values in Wusatowski1s studies t h e average d e v i a t i o n from r e a l v a l u e s s p r e a d formulae - + t o t a l d e v i a t i o n % % % 1.88 1.24 3.12 E k e l u n d 3.45 1.14 4.59 V7usatowski 5.32 0.50 5.82 S e i b l e 4.84 2.19 7.03 T a f l e & Ssdlaczek 6.79 2.90 9.69 T r i n k s Table 2-3 Effect of steel composition on spread ratio in Wusatowski's formula(20) composition of steel correctoin factor d type of s t e e l c% S i % Mn% Ni% Cr% W% 0.06 0.22 1.00000 Basic Bessemer s t . 0.20 0.20 0.50 1.02026 0.25% carbon s t e e l 0.30 0.25 0.50 1.02338 0.35% carbon s t e e l 1.04 0.30 0.45 1.00734 Tool s t e e l 1.25 0.20 0.25 1.01454 Tool s t e e l 0.35 0.50 0.60 1.01636 Manganese s t e e l 1.00 0.30 1.50 1.01068 Manganese s t e e l 0.50 1.70 0.70 1.01410 Spring s t e e l 0.50 0.40 24.0 0.99741 Wear r e s i s t a n t s t . 1.20 0.35 13.0 1.00887 Wear re s i s t a n t s t . 0.06 0.20 0.25 3.50 0.40 1.01034 Case hardening s t . 1.30 0.25 0.30 0.50 1.80 1.00902 Alloy t o o l s t e e l 0.40 1.90 0.60 2.00 0.30 1.02719 Alloy t o o l s t e e l 79 Table 2-4 Values of the constants in Sparling's formula(18) c K A B G 0.981 1.615 0.900 0.550 -0.250 Table 2-5 Comparison between three spread formulae in Sparling's experiments formula No. of tests No. of tests with error more than 10% 15% H i l l 26 23 21 Wusatowski 26 13 8 Sparling 26 3 0 H i l l 9 9 9 Wusatowski 9 5 2 Sparling 9 3 1 Table 2-6 Values of the constants in El-Kalay and Sparling's formula(18) condition A B C D E smooth r o l l s light scales heavy scales 0.851 0.955 1.766 1.844 0.643 0.643 0.386 0.386 -0.104 -0.104 rough r o l l s light scales heavy scales 0.993 0.980 2.186 2.105 0.569 0.569 0.402 0.402 -0.123 -0.123 Figure 1-1 Taper leaf spring superimposed on multi-leaf spring of similar load bearing characteristics(32) Figure 1-3 Sequence of flat rolling of taper leaf springs; (a) width forming, (b) height forming Figure 2-1 Schematic representation of a rolling prtfcess(6) 82 Figure 2-2 Veloc i ty diagram in a r o l l i n g process(6) 83 Figure 2-5 Pressure distribution along the arc of contact(6) F i . .«*• 1 Friction Figure 2-6 A typical example of true pressure distr i -bution in the roll gap over a half width of the rolled stock(6) Figure 2-7 Schematic representation of geometric deformation in ro l l ing(32) Figure 3-1 Forming a uniform block; (a) initial and final dimensions (b) pass one (c) pass two Figure 3-2~Different sequences of rolling 86 (b) Figure 3-3 Comparison between values of spread 1n two types of r o l l i n g ; (a) double pass (b) s ingle pass Wij Figure 3-4 Typical example of a uniform i n i t i a l block 87 88 38 25 28 S P R E 15 A D 18 5 8 a tee 288 388 488 see 6ee Figure 3-7 Typical variation of spread versus the roll 's radius for a constant draft Figure 3-8 Schematic representation of the inverse calculation method D A T A G E N E R A T O R input f i le M U L T I - P A S S P R O C E S S P L A N N E R £ user data for 3 - D I M . ^ » presentation |~ r->i 1 1 data for torque variation presentatj onj' J data for rol l gap | iM> variation presentation output; results, message 3 - D I M G R A P H G E N E R A T O R 2 - D I M G R A P H G E N E R A T O R P R O C E S S C O N T R O L L E R ± file D A T A O R G A N I Z O R F O R M I C R O - P R O C E S S OR)"> computer memory gure 3-9 General flow-diagram of the computer software 90 ^ t a x t ^ ^ read the initial data .apply the convexity constraint, .calculate the dimensions of the.mat-erial which is to be rolled in the--last pair of passes, .take the above material as desired part. .apply the kinematic and dynamic constraints. .calculate the rolls' radii and the permissible drafts. \ f .calculate the incremental drafts for the case, in which the desired part is formed within two passes. .store the results •design the next pair of passes. 7fT Y — i print out the results. .modify the answer, .calculate the dimensions of the new material coming out of the modif-ied passes. •assume the above material as an ini-tial material. end Figure 3-10 General flow-chart of the multi-pass design routine 91 R 0 L L G A P 388 258 . 208 . 158 . 188 . 58 . under rotting over rolling I I I I I I I I I 8 58 108 158 288 258 388 358 488 458 588 DISPLACEMENT Figure 3-11 An exaggerated representation of roll interference roll-gap changing mechanism; servo, hydro, « • • ' rolling process actual height of 1 Ant-a^f^f^a rolled material with the machine. tranducer t7 Figure 3-12 Block diagram of an analogue control system 93 roll-gap changing mechanism; servo, hydro, interface with machine digital to analog convertor -operator interface rolling process actual height of rolled material. actual rolled length. >• interface with micro-computer analog to digital convertor D D C micro computer |T computer memory Figure 3-13 Block diagram of a D.D.C. system 94 wf 1 -W f 2 _ W b 2 -" m e a n ^ b - ^ K ^ f ) w b = ( w b l + w b 2 ^ 2 wf=(wf1+wf2)/2 wmean = wb- 1 / 3 ( wb- wf) wf=(w fi+w f 2)12 Figure 4-1 Two different modes of deformation for rolled steel stocks; (a) double bulged cross-section (b) rhombic cross-section aluminum Figure 4-2 A possible mode of deformation for aluminum stocks 95 Figure 4-3 Comparison between two cross-sectional views of a material, under different rolling procedures; (a) two pass process, total draft=3.55mm (b) one pass process, " " " state of material initial intermediate final width ,height(mm) 25.40,25.40 19.05,26.85 20.30,20.05 1 imi imi inw fa state of material_ initial 1 s t intermediate 2nd intermediate 3ra" intermediate final width,height(mm) 25.40,25.40 21.30,26.18 21.80,21.95 19.80,22.30 20.13,20.05 Figure 4-4 Deformation of a uniform block within: (a) two passes (b) four passes 97 INITIAL DIMENSIONS OF MATERIAL: WIDTH=25.40000 HEIGHT=25.40000 LENGTH=40.00000 FINAL DIMENSIONS OF THE PRODUCT: WIDTH=20.00000 HEIGHT=20.00000 LENGTH=64.51600 RADIUS OF ROLLERS: IN THE FIRST PASS=50.80000 IN THE SECOND PASS=50.80000 RESULTS; CASE 1 FIRST PASS=WIDTH REDUCTION SECOND PASS=HEIGHT REDUCTION CONVEXITY ON THE WIDTH OF THE FINAL PRODUCT DIMENSIONS OF THE INTERMEDIATE CROSS SECTION: WIDTH=18.89746 HEIGHT=26.54404 LENGTH = 51 .44663 FIRST PASS ROLL GAP=19.17714 SECOND PASS ROLL GAP=20.29600 REDUCTION IN FIRST PASS(%)=25.60055 REDUCTION INSECOND PASS(%)=24.65351 END OF EXECUTIONS Figure 4-5 Typical output of the computer program, two pass deformation of a uniform block(dimensions in mm) 98 INITIAL DATA INITIAL DIMENSIONS OF MATERIAL: WIDTH=25.40000 HEIGHT = 25.40000 LENGTH=40.00000 FINAL DIMENSIONS OF THE PRODUCT: W1DTH=20.OOOOO HEIGHT=20.OOOOO LENGTH=64.5160O KINEMATIC AND DYNAMIC CONSTRAINT:C1=0.2000.C2 = 300000.000 REDUCTION FOR THE LAST PAIR OF PASSES : 1 . ST PASS 0 2 .000, 2 . ND PASS=2.000 RESULTS; CASE i.*»••••••• VERY FIRST PASS-WIOTH REDUCTION NEXT PASS=HEIGHT REDUCTION .etc. •••DIMENSIONS OF MATR. COMING OUT OF THE t.ST PASS: WIDTH=21.07888 HEIGHT=26.04848 LENGTH=46.99996 •••DIMENSIONS OF MATR. COMING OUT OF THE 2.ND PASS: WI0TH=21.72186 HEIGHT=21.92156 LENGTH=54.19496 •FIRST PASS ROLL GAP=21.39084 SECOND PASS ROLL GAP=22.24599 •DRAFT IN 1.ST PASS=4.32112 DRAFT IN 2ND PASS=4.12692 •RADIUS OF ROLLERS IN THIS PAIR OF PASSES IN THE FIRST PASS=50.80000 IN THE SECOND PASS=50.80000 •••DIMENSIONS OF MATR. COMING OUT OF THE 1.ST PASS: WIDTH=19.72186 HEIGHT=22.00000 LENGTH=59.47808 ••DIMENSIONS OF MATR. COMING OUT OF THE 2.ND PASS: WIDTH = 20. OOOOO HEIGHT=20.00000 LENGTH=64.51600 •FIRST PASS ROLL GAP-20.O13740 . SECOND PASS ROLL GAP=20.29600 •DRAFT IN 1.ST PASS=2.OOOOO DRAFT IN 2.NO PA5S-2.00000 •RADIUS OF ROLLERS IN THIS PAIR OF PASSES IN THE FIRST PASS=50.80000 IN THE SECOND PASS=50.80000 FINAL PRODUCT IS ACHIEVED AT THIS STAGE NO; OF PAIR OF PASSES: 2 aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa Figure 4-6 Typical output of the computer program, deformation of a uniform block, considering the process constraints (dimensions in mm) 225 igure 4-7 Profiles of the three experimental steel specimens(dimensions in mm) (i) concave (ii) linear ( i i i ) convex VO VD 8.175 . I 0 .025 . N te 12 14 18 18 28 22 24 28 28 38 32 34 INITIAL HEIGHT Figure 4-8 Rate of height variation for three different experimental specimens 18 12 14 16 18 28 22 24 26 28 38 32 34 INITIAL HEIGHT Figure 4-9 Comparison of experimental results, showing both the previous and new prediction of spread (specimen i) 17 12 14 16 18 28 22 24 26 28 38 32 34 INITIAL HEIGHT ure 4-10 Comparison of experimental results, showing both the previous and new prediction of spread (specimen i i ) 18 12 14 16 18 28 22 24 26 28 38 32 34 INITIAL HEIGHT ure 4-11 Comparison of experimental results, showing both the previous and new prediction of spread (specimen i i i ) 35 38 . 25 28 15 18 PREVIOUS PREDICTION *•"* EXPERIMENTAL VALUES NEW PREDICTION i—i—i—i—i—i—i—i—i—i—r 18 12 14 16 18 28 22 24 26 28 38 32 INITIAL WIDTH 34 Figure 4-12 Comparison of experimental and predicted values of spread, specimen 1, width increasing 35 38 25 28 15 . 18 — PREVIOUS PREDICTION *—* EXPERIMENTAL VALUES — NEW PREDICTION // i i i i i i i—i r 18 12 14 16 18 28 22 24 26 28 38 32 34 INITIAL WIDTH Figure 4-13 Comparison of experimental and predicted values of spread, specimen i i , width increasing 35 — PREVIOUS PREDICTION * * EXPERIMENTAL VALUES — NEW PREDICTION 1 i T i i i—i—i—i—I—I—I—r 1 18 12 14 16 18 28 22 24 26 28 38 32 34 INITIAL WIDTH Figure 4-14 Comparison of experimental and calculated values of spread, specimen i i i , width increasing 32 INITIAL WIDTH Figure 4-15 Comparison of experimental and calculated values of spread, specimen i i , width decreasing 32 PREVIOUS PREDICTION *—* EXPERIMENTAL VALUES —NEW PREDICTION 25 26 27 28 29 30 3! 32 INITIAL WIDTH Figure 4-16 Comparison of experimental and calculated values of spread, specimen i i i , width decreasing Figure 4-17 The geometry of the constant height, variable width aluminum specimen, before and after rolling (dimensions in mm) 1 05 45 48 35 38 25 28 . 15 1 1 1 1 1 1 1 1 1 1 1 1 16 18 28 22 24 26 28 38 32 34 36 38 48 INITIAL WIDTH Figure 4-18 Comparison of values of spread for two different modes of width variation in the aluminum specimen 13 12.5 . 11.5 . 18.5 . 9.5 . i 1 1 1 r 58 188 158 288 258 380 358 480 ROLLED LENGTH Figure 4-19 Variation of roll gap(mm) versus the rolled length in a typical closing roll gap process 106 * * UNSTEADY ROLLING EXPERIMENTS 1ST EXPERIMENT) *™* SIMULATED SPREAD PREDICTIONCIST EXPERIMENT) I 1 1 r-—i——i 1 1 r 0 58 i38 150 280 2S0 388 3S8 408 ROLLED LENGTH 13.8 S R E A D 13.4 J J 13.2 . D T H 13 . 12.8 *—* EXPERIMENT * — * PREDICTED VALUE 1 1 1 1 1 8 8.5 * I 1.5 2 2.5 3 DRAFT «b» Figure 4-20 Comparison of experimental and calculated values of spread(mm) for a uniform block in a closing roll gap process; (a) versus rolled length (b) versus draft 107 •—•UNSTEADY ROLLINGCIST EXPERIMENT) 13.8 . 9 — 0 UNSTEADY ROLLING (2ND EXPERIMENT) 13.6 . 13.4 . 13.2 . 13 . • 12.8 . • * • t 1 1 1 1 I I 1 1 1 I I 8 I 2 3 4 5 6 7 8 9 18 II 12 CROSS-SECTION NUMBER Figure 4-21 Comparison of spread for two similar specimens under different rates of roll gap closing 13 , 12.S . 8 58 188 158 2 8 8 2 5 8 3 8 0 3 5 8 4 8 8 ROLLED LENGTH Figure 4-22 Variation of roll gap(mm) versus rolled length(mm) in an opening roll gap process 13.8 13.8 13.4 . 13.2 . 13. • 12.8 •UNSTEADY ROLLING EXPERIMENT «—° SIMULATED SPREAD —I 1 1 1 1 r 1 r 8 58 188 158 2 8 8 2 5 8 388 3 5 0 4 8 8 ROLLED LENGTH Figure 4-23 Comparison of experimental and calculated spread(mm) for a uniform block in an opening roll gap process 109 Figure 4-25 Position of the heating furnace with respect to the rolling mil 1 Figure 4-27 The aluminum specimens used for the cr experiments 70mm Figure 5-1 A part with constant width and linearly variable height Figure 5-2 Two dimensional views of a uniform material, after being rolled in an operation with linearly variable roll gap 1 12 Figure 5-3 Two possible intermediate shapes in rolling a part with linearly variable height, from a uniform block; (a) first solution vb) second solution 258 268 . 158 tee . 1*1. PASS '2nd. PASS 58 188 ise - r 288 258 388 ROLLED LENGTH ~r -358 -r~ m 458 see Figure 5-4 Process behaviour of the first possible solution: roll gap(mm) versus rolled 1ength(mm) ' U i . PASS '2nd. PASS I 468 450 sea ROLLED LEN6TH Figure 5-5 Process behaviour of the first possible solution: percentage of reduction versus rolled length(mm) 8 1—' I 1 1 1 1 1 1 T 1 8 56 168 156 208 2S8 366 356 468 456 568 ROLLED LfNBTH Figure 5-6 Process behaviour of the second possible solution: roll gap(mm) versus rolled length(mm) 8 58 188 156 288 258 386 358 468 456 588 ROLLED LEN6TH Figure 5-7 Process behaviour of the second possible solution: percentage of reduction versus rolled length(mm) Figure 5-8 Initial and final geometry of the part in example 2 PASS 1 PASS 2 PASS 5 Figure 5-9 Sequence of rolling for a typical example of a multi-pass rolling process(dimensions in mm) 166 128 . 188 68 . 48 28 —I r -8 288 488 PASS I PASS 2 PASS 3 — PASS 4 PASS 5 T T 888 1888 1288 1488 1688 1808 2000 ROLLED LENGTH Figure 5-10 Variation of roll gap(mm) versus rolled length for the example of figure 5-9 58 PASS 1 •— PASS 2 I r—I 1 1 1 1 1 1 1 1 8 288 488 688 888 1868 1286 1488 1688 1868 2888 ROLLED LEN6TH a THOUSANDS PASS I 6 - PASS 2 PASS 3 — PASS 4 PASS 5 8 288 488 688 888 1888 1288 1468 1688 1888 2888 ROLLED LENGTH b Figure 5-11 Variation of draft(mm),a, and torque(kg.m versus rolled 1ength(mm) for the example figure 5-9 1 18 REFERENCES Begeman, M.J. and Amstead, B.H. "Manufacturing Processes.", John Wiley and Sons Inc., New York, 1969, pp 1 to 5. Datsko, J. "Material Properties and Manufacturing Processes.", John Wiley and Sons Inc., New York, 1966, pp 280 to 306. Alexandro, J.M. and Brewer, R.C. "Manufacturing Properties of Materials.", D.Van Nostrand Company Ltd., London, 1963. Roberts, W.L. "Hot R o l l i n g of Steel.", Marcel Dekker Inc., New York, 1983. Siebel, E. " Formabi I i I y in Metal V/orking.", Dusseldorf, 1932. Wusatowski, Z. "Fundamentals of R o l l i n g . " , Pergamon Press, Braunchweig, 1969. Koncewicz, S. "Forward S l i p and Neutral Angle in Hot R o l l i n g with Occuring Spread." , Doctoral Thesis, Manuscript, Selesia University of Technology, Poland, 1961. Underwood, L.R. 119 "The R o l l i n g of Metals.", Vol. 1, London, 1952. 9. "Research on the R o l l i n g S t r i p . " , A symposium of selected papers 1948-1958, B.I.S.R.A., London, 1960. 10. Orowan, E. "The Calculation of Roll Pressure in Hot and Cold F l a t R o l l i n g . " , Proc. Inst. Mech. Engrs., 150, 1943, pp 140 to 167. 11. Alexander, J.M. "On the Theory of R o l l i n g . " , Proc. Royal Society London, Series A, Vol. 326, 1972, pp 535 to 568. 12. Sims, R.B. "The Calculation of the Roll Force and Torque in Hot R o l l i n g M i l l s . " , Proc. Inst. Mech. Engrs., 168, 1954, pp 191 to 200. 13. La H i , L.A. "An A n a l y t i c a l R o l l i n g Model Including Through Thickness Shear Stress d i s t r i b u t i o n s . " , Journal of Engineering Materials and Technology, Vol. 106, Jan. 1984, pp 1 to 8. 14. Kobayashi, S. and Oh, S.I. "An Approximate Method for a Thr e e-Di mens i onal A n a l y s i s of R o l l i n g . " , Int. J. Mech. S c i . , Pergman Press, Vol. 17, 1975, pp 293 to 305. 15. Orowan, E. and Pascoe, K.J. "A Simple Method of Calculating Roll Pressure and Power Consumption in Hot Flat R o l l i n g . " , Iron and Steel Inst., Special Rept., No. 34, 1946, p 124. 1 20 16. Rowe, G.W. "An Introduction to the P r i n c i p l e of Metal Working.", Edward Arnold Ltd., London, 1968. 17. Helmi, A. and Alexander, I,M. "Geometric Factors A f f e c t i n g Spread in Hot Flat R o l l i n g of Steel.", J. Iron Steel Inst., 206,- Nov. 1968, pp 1110 to 1117. 18. Sparling, L.G.M. "Formula for Spread in Hot Flat R o l l i n g . " , Proc. Inst. Mech. Eng., 175, 1961, pp 604 to 640. 19. Wusatowski, Z. "Hot R o l l i n g : a Study Elongation.", Iron Steel, pp 49 to 54. of Draught, Spread and London, Vol. 28, Feb. 1955, 20. Wusatowski, Z. "Hot R o l l i n g : A Study of Draught, Spread and Elongation (continued).", Iron Steel, London, Vol. 28, March 1955, pp 89 to 94. 21. Lahoti, G.D., et a l . "Comput e r-Ai de d Analysis of Metal Flow and Stresses in Plate R o l l i n g . " , J. Mech. Work and Tech., Vol. 4, 1980, pp 105 to 110. 22. El.Kalay, A.K.E.H.A. and Sparling, L.G.M. "Factors A f f e c t i n g F r i c t i o n , and Their Effect on Load, Torque and Spread in Hot Flat R o l l i n g . " , J. Iron Steel Inst., 206, Feb. 1968, pp 152 to 163. 23. Beese, J.G. "Ratio of Lateral Strain to Thickness Strain During 121 Hot R o l l i n g of Steel Slabs.", J. Iron Steel Inst., June 1972, pp 433 to 436. 24. Ishikawa, T., et a l . "Fundamental Study on the P r o f i l e and Shape of the Rolled S t r i p . " , International Conference on Steel R o l l i n g , Tokyo, 1980, pp 772 to 783. 25. Alexander, J.M. "Machine Tool Design on Reasearch.", Proceedings of Eighteenth International Machine Tool Design and Research, MacMillan Press Ltd. London, 1978. 26. Buck, R.C. "Advanced c a l c u l u s . " , McGraw H i l l Book Company Inc., New York, 1956. 27. Groover, M.P. "Automation Production System and Computer-Ai ded Manufacturing.", Prentice-Hall Inc., Englewood C l i f f s , New Jersey, 1980, pp 414 to 485. 28. Tomilson, A. and Stringer, J.D. "Spread and Elongation in Flat Tool Forging.", JISI, Oct. 1959, pp 157 to 160. 29. Canahan B., Luther, H.A. and Wilkes J.O. "Applied Numerical Methods.", Willey & Sons, New York, N.Y., 1969. 30. Forsythe, G.E., Malcolm, M.A. and Moler C B . "Computer Methods for Mathematical Computations.", P r e n t i c - H a l l , Englewood C l i f f s , N.J., 1977. 1 22 31. Duncan, J.P. and Mair, S.G. "Sculptured Surfaces in Engineering and Medicine.", Cambridge University Press, Cambridge, 1983. 32. "Taper Leaf Spring R o l l i n g Machine.", F i l e No. 1LS/1, H i l l e Engineering Company Ltd., S h e f f i e l d , England. 33. Bryant, G.F. "Automation of Tandem M i l l s . " , The Iron and Steel I n s t i t u t e , London, 1973. APPENDIX A Method of Incremental Search in Root Finding General Problem: Given an algebraic equation of the form f(x)=0, find the value(s) of x, i . e . , the root(s), that s a t i s f y the equation. A l g o r i t h m 2 9 ' 3 0 : given sta r t i n g point, x 0, an increment Ax is chosen. Values of f(x) for successive values of x 0, x0+Ax, x0+2Ax,..., are determined u n t i l a sign change in f(x) occurs, i . e . , when f(x)•f(x+Ax)<0 . The la s t value of x, preceding the sign change is reverted back and the incremental search is repeated using a smaller increment ( e.g., A X = A X / 1 0 ) u n t i l a sign change in f(x) occurs again. The above procedure i s repeated using progressively smaller increments, u n t i l a s u f f i c i e n t l y accurate value of the root i s obtained. 123 APPENDIX B Cur v e f i t t i n g The main purpose in c u r v e - f i t t i n g i s to find an a n a l y t i c a l form for describing a set of discrete data points. A curve with continuity up to second order was considered to be suitable for application in the present study. A conical arc i n t e r p o l a t i o n technique was selected. Conical arc interpolation is an approach which i s devised to avoid unwanted o s c i l l a t i o n 3 1 . The general form of a conic curve i s Ax2+Bxy+Cy2+Dx+Ey+F=0 with the slope •y ' =-( 2Ax+By+D)/( 2Cy+Bx+E) and the second derivative y"=-2(y'2+By'+A)/(2Cy+Bx+E) The expression for the conic curve may be rewritten as x2+B'xy+C'y2+D'x+E'y+F'=0 where A*0, B'=B/A, C'=C/A,...etc. The above expression, thus, involves fiv e independent constants which can be found to s a t i s f y f i v e conditions, e.g. two positions, two slopes and one curvature. To i l l u s t r a t e the algorithm of the c u r v e - f i t t i n g routine, consider the given points in F i g . B-1. A conic curve i s f i r s t passed through the f i r s t three points in such a way that the slopes at points 2 and 3 s a t i s f y the conditions of weighted mean s l o p e between points / , 3 and points 2 , 4 (for d e t a i l s on weighted mean slope concepts, see 31, pp 130 to 135). Values of slope and curvature at point 3 as well as the weighted mean slope at point 4 are then used to fin d the c o e f f i c i e n t s of the conical curve which passes through points 3 and 4. The same procedure is repeated for points 4,5 and 5, 6...,etc. The la s t conical curve passes through three points (n-2), (n-1) and n, 1 2 4 125 s a t i s f y i n g the conditions of slope and curvature at point (n-2). Using the weighted mean slope concept guarantees the smoothness of the joined curves. Also, due to the nature of the conical curves, o s c i l l a t i o n between the two adjecent points does not o c c u r 3 1 . Figure B-l Curve - f i t t i ng through known data points 

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