Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

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

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1986_A7 S46.pdf [ 6.14MB ]
Metadata
JSON: 831-1.0096924.json
JSON-LD: 831-1.0096924-ld.json
RDF/XML (Pretty): 831-1.0096924-rdf.xml
RDF/JSON: 831-1.0096924-rdf.json
Turtle: 831-1.0096924-turtle.txt
N-Triples: 831-1.0096924-rdf-ntriples.txt
Original Record: 831-1.0096924-source.json
Full Text
831-1.0096924-fulltext.txt
Citation
831-1.0096924.ris

Full Text

COMPUTER-AIDED ROLLING OF PARTS WITH VARIABLE RECTANGULAR CROSS-SECTION  by  NARIMAN SEPEHRI B . A . S c , 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 t h i s t h e s i s as conforming tc the r e q u i r e d  standard  UNIVERSITY OF BRITISH COLUMBIA APRIL, 1986 ©  NARIMAN SEPEHRI, 1986  &  In  presenting  this  thesis  in  partial  requirements f o r an advanced degree BRITISH  COLUMBIA,  I  agree  that  f r e e l y a v a i l a b l e f o r reference that  permission  scholarly Department  or  may  by  be  gain  shall  not  her  be  by  DEPARTMENT OF MECHANICAL ENGINEERING  Date: APRIL, 1986  s h a l l make i t further  the  Head  representatives.  allowed  permi s s i o n .  UNIVERSITY OF BRITISH COLUMBIA 2075 Wesbrook Place Vancouver, Canada V6T 1W5  I  OF  agree  copying of t h i s t h e s i s f o r  understood that copying or p u b l i c a t i o n of financial  UNIVERSITY  the L i b r a r y  granted  h i s or  the  and study.  for extensive  purposes  at  f u l f i l m e n t of the  this  of  It i s  thesis  without my  my  for  written  ABSTRACT A  computer-aided  rolling  of  symmetric  cross-section Sparling's  process parts  formula  distinct  f o r spread was  criteria  namely: kinematic  control  with  scheme  variable  the  rectangular  used  were  and  i n d e v e l o p i n g the method.  considered  dynamic  in  the a n a l y s i s ,  constraints.  In  convexity  Numerical  computer  was  were  used  solved  and  the  algorithm  of  the  passes  resulting  by an i t e r a t i v e method.  was  roll  solution,  then developed to determine  number of r o l l i n g passes r e q u i r e d , as w e l l variation  to  constraints.  formulation  equations  finishing  Based on the process c o n s t r a i n t s and the numerical a  order  p r e c i s i o n of the r o l l e d p a r t s , p r o v i s i o n s were  form of  non-linear  flat  found to be s u i t a b l e f o r  made f o r s p e c i f y i n g the t o l e r a n c e s of the i n the  for  i s proposed. As a s t a r t i n g p o i n t , E l - K a l a y and  t h i s a p p l i c a t i o n and thus was Two  planning  gap  as  a  as  the  the  dynamic  f u n c t i o n of the  rolled  length. Preliminary  l a b o r a t o r y experiments  to v e r i f y the v a l i d i t y of applicability  of  the  the  spread  predicted formula  process behaviour. These experiments of  the  spread  formula.  results in  a  control  and  the  determining the  l e d to the m o d i f i c a t i o n  between  the  predicted  experiments.  Operating aspects were a l s o c o n s i d e r e d . I t was that  conducted  Using the m o d i f i e d formula i t was  found that a good agreement e x i s t e d r e s u l t s and those of the  were then  proposed  system based on the r o l l e d l e n g t h would be i i  both  simple  rectangular reasonable  and parts  suitable. with  I t was then concluded  moderate  complexity,  formed-die  die-forging  are  considerable  advantages as i t r e p l a c e s  with the r o l l i n g  also  where  variation  software.  applicable,  this  in  that f o r  shape  and  rolling  and  method  has  the f o r g i n g  hardware  ACKNOWLEDGEMENT The  author  wishes  to  thank  Dr.  generous a s s i s t a n c e d u r i n g the course His  help  and  F. Sassani  of t h i s r e s e a r c h work.  the s h a r i n g of h i s experience  a p p r e c i a t e d d u r i n g the p r e p a r a t i o n of t h i s The  author  i s also  M e t a l l u r g i c a l Engineering The  financial  grateful  for h i s  to  were very much  thesis. the Department of  f o r the use of the f a c i l i t i e s .  support  was  provided  by the N a t i o n a l  Sciences and E n g i n e e r i n g Research C o u n c i l of Canada.  iv  Table of Contents ABSTRACT  i i  ACKNOWLEDGEMENT  iv  NOMENCLATURE  viii  LIST OF TABLES  ix  LIST OF FIGURES  .x  1.  INTRODUCTION  1  2.  LITERATURE SURVEY  7  2.1 Fundamentals  of the R o l l i n g Process  7  2.1.1 Basic Concepts and Geometric R e l a t i o n s  7  2.1.2 C a l c u l a t i o n of R o l l i n g Speed  9  2.1.3 Mechanism of B i t e 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 i n F l a t R o l l i n g 2.2.1 I n t r o d u c t i o n  15  2.2.2 Formulae  17  f o r Spread  2.3 Summary and E v a l u a t i o n 3.  15  ROLLING OF VARIABLE RECTANGULAR  22 CROSS-SECTIONS  3.1 B a s i c Formulation f o r Deformation  25 25  3.1.1 Uniform to Uniform Deformation  25  3.1.1.1 Method  25  3.1.1.2 Number of P o s s i b l e S o l u t i o n s  27  3.1.1.3 E x i s t e n c e of the S o l u t i o n s  27  3.1.2 Uniform to Non-uniform 3.1.3 Non-uniform  Deformation  t o Non-uniform  Deformation  30 ....32  3.2 M u l t i - P a s s Design Concept  33  3.2.1 Process C o n s t r a i n t s  33  3.2.1.1 Kinematic C o n s t r a i n t  v  34  3.2.1.2 Dynamic C o n s t r a i n t  35  3.2.1.3 Convexity C o n s t r a i n t  35  3.2.2 M u l t i - P a s s Design Procedure  36  3.3 Computer Software  4.  37  3.3.1 Data Generator  38  3.3.2 M u l t i - P a s s Planner  39  3.3.3 Graph Generator  45  3.3.4 Other Routines  45  3.4 O p e r a t i n g Aspects  46  3.5 Summary  49  and E v a l u a t i o n  EXPERIMENTAL EVALUATION  52  4.1 I n t r o d u c t i o n  52  4.2 Experimental Arrangements  53  4.2.1 Instrumentation  53  4.2.2 Specimens and Measurement  54  4.3 Steady S t a t e R o l l i n g  .......55  4.3.1 E f f e c t s of S e q u e n t i a l R o l l i n g on Spread 4.3.2 The Evaluation Algorithm 4.4 Unsteady  of  the  Multi-Pass  State R o l l i n g on Spread on Spread  4.4.3 E f f e c t s of R o l l Gap V a r i a t i o n 4.5 Summary 5.  on Spread  and E v a l u a t i o n  58 63 ...65 ....67  SAMPLE ILLUSTRATIONS  71  5. 1 Example No. 1  71  5.2 Example No. 2 6.  56 57  4.4.1 E f f e c t s of Height V a r i a t i o n 4.4.2 E f f e c t s of Width V a r i a t i o n  ...55  CONCLUSIONS  '  AND SCOPE FOR FUTURE WORK  vi  73 75  REFERENCES  118  APPENDIX A  123  APPENDIX B  124  vii  NOMENCLATURE  h,H  Height(Thickness)  w,W  Width(Breadth)  1,L  Length  A  Cross-sectional  V  Volume  Ah  Draft  S  Spread(an i n c r e a s e  R  Roll's  D  R o l l ' s diameter  i n width)  radius  Projected  length  a  Angle of b i t e  <fi  Rolling  angle  5  Neutral  angle  V  Linear  t  Rolling  f  Coefficient Local  area  of a r c of contact  v e l o c i t y of the r o l l e d temperature of f r i c t i o n  pressure  F  Local horizontal  P  Separating  M  Rolling  force  force  torque  viii  material  LIST OF TABLES  TABLE  PAGE  2-1  Value of c o r r e c t i o n  2-2  Average d e v i a t i o n s of p r e d i c t e d spread experimental  2-3  factor  i n Bachtinov's  formula.77  from  v a l u e s i n Wusatowki's s t u d i e s  E f f e c t of s t e e l composition  77  on spread r a t i o i n  Wusatowski's formula....  78  2-4  Values of the c o n s t a n t s i n S p a r l i n g ' s formula  79  2-5  Comparison between spread formulae  in Sparling's  experiments 2-6  79  Values of the c o n s t a n t s i n E l - k a l a y and formula  Sparling's 79  ix  LIST OF FIGURES  FIGURE 1-1  PAGE Taper  l e a f s p r i n g superimposed  on m u l t i - l e a f  s p r i n g assembly  1-2  Schematic  80  r e p r e s e n t a t i o n of an  eccentric-die  rolling  80  1- 3  Sequence of f l a t  rolling  of taper l e a f  2- 1  Schematic  2-2  V e l o c i t y diagram  2-3  T y p i c a l 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  2-6  Pressure d i s t r i b u t i o n  2-7  Schematic  r e p r e s e n t a t i o n of r o l l i n g  springs....80  process  i n a r o l l i n g process  82  contact....83  i n the r o l l gap  r e p r e s e n t a t i o n of geometric  in . r o l l i n g  81  83 deformation 84  x  3-1  Forming a uniform block  85  3-2  Different  85  3-3  Comparison between values of spread i n two  of  sequences of r o l l i n g  types  rolling  86  3-4  An example of a uniform  initial  block  3-5  An example of a non-uniform  3-6  An example of an i n t e r m e d i a t e s t a t e  3-7  T y p i c a l v a r i a t i o n of spread versus r o l l ' s  3-8  Schematic  86  f i n a l part  87  87 radius..88  r e p r e s e n t a t i o n of i n v e r s e c a l c u l a t i o n  method  88  3-9  General flow-diagram  of the computer software  3-10  General flow-chart of the m u l t i - p a s s design routine  89  90  3-11  Exaggerated  r e p r e s e n t a t i o n of r o l l i n t e r f e r e n c e . . . 9 1  3-12  Block diagram  of an analogue  xi.  c o n t r o l system  92  3- 13  Block diagram  of a D.D.C. system  93  4- 1  D i f f e r e n t modes of deformation  for r o l l e d steels..94  4-2  A p o s s i b l e mode of deformation  for rolled  aluminum stocks  4-3  Comparison between two c r o s s - s e c t i o n a l views under d i f f e r e n t  4-4  Deformation and  4-5  94  r o l l i n g procedures  95  of a uniform block w i t h i n two  four passes  96  T y p i c a l output of the computer program, two-pass deformation of a uniform block  4-6  97  T y p i c a l output of the computer program, deformation of a uniform block w i t h i n more than two passes  98  4-7  P r o f i l e s of three experimental s t e e l  4-8  Rate of height v a r i a t i o n  specimens....99  f o r three d i f f e r e n t  experimental s t e e l specimens  4-9  100  Comparison of experimental and p r e d i c t e d v a l u e s of spread f o r a h e i g h t v a r i a b l e specimen(i)  xii  100  4-10  Comparison of experimental and p r e d i c t e d values of  4-11  4-15  spread; specimen ( i ) ,  width i n c r e a s i n g  spread; specimen  (ii),  width i n c r e a s i n g .  Comparison of experimental  and p r e d i c t e d values  of  width i n c r e a s i n g  102  spread; specimen  (iii),  102  103  Comparison of experimental and p r e d i c t e d values of  4-16  f o r a height v a r i a b l e s p e c i m e n ( i i i ) . . . . 1 0 1  Comparison of experimental and p r e d i c t e d values of  4-14  spread  Comparison of experimental and p r e d i c t e d values of  4-13  101  Comparison of experimental and p r e d i c t e d values of  4-12  spread f o r a height v a r i a b l e s p e c i m e n ( i i )  spread; specimen ( i i ) ,  width d e c r e a s i n g  Comparison of experimental  and p r e d i c t e d values  of  width d e c r e a s i n g  spread; specimen ( i i i ) ,  4-17  Geometry of an aluminum specimen  4-18  Comparison of v a l u e s of spread f o r two modes of width v a r i a t i o n spec imen  103  104  104  different  i n the aluminum 105  xiii  4-19  V a r i a t i o n of r o l l gap v e r s u s r o l l e d l e n g t h i n a typical closing roll  4-20  Comparison of  4-21  Comparison  of e x p e r i m e n t a l and c a l c u l a t e d  o f s p r e a d f o r two  different  values  similar  106  specimens  r a t e s of r o l l gap c l o s i n g  107  V a r i a t i o n of r o l l gap v e r s u s r o l l e d l e n g t h i n an opening  4-23  105  spread i n a c l o s i n g r o l l gap p r o c e s s  under  4-22  gap p r o c e s s  r o l l gap p r o c e s s  Comparison for  107  of e x p e r i m e n t a l and c a l c u l a t e d  a u n i f o r m b l o c k i n an o p e n i n g  roll  spread  gap  process..  108  4-24  Appearance  of the e x p e r i m e n t a l r o l l i n g m i l l  4-25  P o s i t i o n of the furnace with r e s p e c t t o  the  rolling mill  4-26  4-27  109  S t e e l specimens of  unsteady  109  used to simulate c o n d i t i o n s  rolling  Aluminum specimens  110  used f o r c r i t i c a l  experiments  110  xiv  5-1  A part  with costant  w i d t h and  linearly  variable  height  5-2  Two  111  d i m e n s i o n a l v i e w s of  being  rolled  variable  5-3  Two  roll  possible  part  with  5-4  R o l l gap  5-5  . Variation  first  i n an  operation  with  after  linearly  gap  111  intermediate  linearly  variation  of  a uniform m a t e r i a l ,  shapes  variable  for  the  p e r c e n t a g e of  in r o l l i n g  a  height  first  112  solution  reduction  for  113  the  solution  5-6  R o l l gap  5-7  Variation second  113  variation  of  for  the  p e r c e n t a g e of  second  solution  reduction  for  114  the  solution  114 i  5-8  5-9  5-10  Initial  and  example  2  Typical  example of  rolling  process  Variation  final  geometry  of  the  part  in 115  of  roll  s e q u e n c e of  a  multi-pass 116  gap  versus  xv  rolled  length  for  the example of f i g u r e  5-11  Variation  5-9  of d r a f t and torque versus  l e n g t h f o r the example of f i g u r e  B-1  116  Curve-fitting  through  rolled  5-9  known data p o i n t s  xvi  117  125  1.  With  the  rapid  INTRODUCTION  development  mechanization and automation, are  becoming  all  aspects  more of  of new  new  materials,  the processes of manufacturing  v a r i e d . The a p p l i c a t i o n of computers i n  engineering,  (CAD/CAM),  manufacturing  of  has  specially  i n a way  in  design  l e d to the development  processes whereby a product can f r e q u e n t l y be made in  several  ways.  It  i s therefore  important to understand  many ways i n which a product can be processed , the that  and  these  the  effects  processes have on the product p r o p e r t i e s ,  their  advantages  and l i m i t a t i o n s as w e l l as the accuracy  that  expected.  A fundamental  suitable  methods  of  producing  c r i t e r i o n which determines  manufacturing,  is  the i n d i v i d u a l part  more a c c u r a t e l y  than  the  correct  so that  necessary  and  is  process  of  i t i s manufactured at  the  lowest  no  cost  possible . 1  Amongst a l l the manufacturing p r o c e s s e s , forming one  which i s becoming more important  by which the s i z e or shape of application  of  a  part  is  i s a method,  changed  f o r c e on the p a r t . Forming  change the shape of p a r t s . G e n e r a l l y and  . Forming  speaking  then forming  most  2  f i n e - g r a i n s t r u c t u r e s which  to  be  used .  increase  consequently prolong t h e i r working  1  the to  , i f the shape  made by one of the forming o p e r a t i o n s , process  by  i s a f a s t way  the r e q u i r e d accuracy of a p a r t are such that  economical  i s the  i t can is  be the  Formed p a r t s have  their  lives . 3  toughness  and  2  Industrial practice such  as rolling  forging,  various  pressing,  forming  techniques  or  stamping  extrusion.  i s known as the the most economic method amongst the  Rolling  other  ,  uses  forming  techniques.  I t has been p r a c t i c e d s i n c e the  f o u r t e e n t h century" and s t i l l manufacturing.  Flat  The mechanism  of  rotating  cylinders  rolling  and  i s simple;  the  friction  work-piece  c r o s s - s e c t i o n a l area of the p a r t . temperature material  two  draw the work-piece through  between them, by means of the cylinders  application  above  the  , i t is called  in  i s the simplest form of r o l l i n g .  rolling  flat  has e x t e n s i v e  force  , If  thus this  the opening between  the  reducing  the  i s done  recrystallization  hot  point  Hot r o l l i n g  rolling.  circular  in a of the  i s commonly  used f o r s t e e l p a r t s . T h i s t h e s i s presents a for  manufacturing  cross-section. equipments  and  computer-aided  rolling  symmetric p a r t s with v a r i a b l e r e c t a n g u l a r  This  form  of  parts  machinery. A taper  i s often  leaf  spring  used  road suspension  Fig. more  leads  s p r i n g can r e p l a c e a stack of f i v e or  c o n v e n t i o n a l constant  to  carrying Moreover,  thickness leaf  and mechanical  substantial capacity,  the means  system f o r t r u c k s and vans. According t o  1-1, a taper l e a f  c e r t a i n economical  in  i sa typical  example which i s becoming i n c r e a s i n g l y common as of  scheme  weight and  the c o n t r o l l e d  performance advantages. I t  saving  better  springs, offering  f o r the  ride  same  load  characteristics.  p a t t e r n of the taper l e a f s p r i n g  permits higher working s t r e s s e s t o be used  ,  and  gives  a  3 longer working  life.  A common method to produce such s p r i n g s i s based on use of d i e s e c c e n t r i c a l l y  f i x e d to a p a i r of r o l l s ,  of  forms  which  progressively  blanks  ( F i g . 1-2).  limit  the  rotation  the t h i c k n e s s of the s p r i n g  Closed d i e s are used  l a t e r a l spread  in  an  attempt  of the stock. G r i n d i n g  cost of m a i n t a i n i n g  of  separate In  producing  the  dies  example, can  be  rolling;  blank  a  produced  The  spring sizes.  here, a taper l e a f s p r i n g ,for  within  of uniform  through a v a r i a b l e r o l l gap,  frequent.  i s a l s o h i g h . Furthermore, a  t o o l set i s r e q u i r e d f o r d i f f e r e n t the method presented  rolled  the machine i s high, s i n c e  under the heavy side loads d i e breakages are cost  to  operations  are o f t e n needed to remove the f o r g i n g f l a s h from the m a t e r i a l . The  the  two  operations  cross-section is f i r s t  of  flat  rolled,  i n i t s width ( F i g . 1-3a), then,  retapered  from  its  o r i g i n a l t h i c k n e s s s i d e ( F i g . 1-3b)  such a way  that  the  initial  leaving  the  part  formed  constant  only  width  is  in  regained  in i t s t h i c k n e s s . The  main  f e a t u r e s of t h i s method can be summarized as below: (i)  The  blank  different  is  rolled  shapes  by  may  two  plain  rolls,  whereby  be produced from the same t o o l  set. (ii)  P a r t s with v a r i a t i o n both  in width and height can  be  produced by t h i s method. (iii)  Two  operations  i s the minimum r e q u i r e d to achieve a  c e r t a i n shape, however, frequently  more  than  two  passes  are  needed depending on the shape, appearance  (iv)  of the f i n i s h e d p a r t and the process  constraints.  Unlike  process  conventional  unsteady  state  continuously  rolling,  nature,  changes  the  i . e . , the  Using  a  suitable  Mi cro-processor  used.  ,  Simplicity  roll  gap  to form the d e s i r e d shape. The  ingoing m a t e r i a l c o u l d be non-uniform (v)  i s of an  control  as w e l l .  system,  such  as  a  permits f a s t o p e r a t i n g speeds to be i n r e s e t t i n g the system  parameters,  when design changes occur, can e a s i l y be accomplished through  the m i c r o - p r o c e s s o r . T h i s would r e s u l t  in a  higher p r o d u c t i o n r a t e and lower down time. (vi)  Parts,  produced  f u r t h e r treatment  by  this  method,  usually  and can be used d i r e c t l y ,  need no although  f u r t h e r p r o c e s s i n g c o u l d be done, i f necessary.  The (a)  o b j e c t i v e s of t h i s t h e s i s are as f o l l o w s : To study modify  the  theory  of  conventional  i t for application  rolling  i n the unsteady  and  rolling  process. (b)  To develop procedures  and s t r a t e g i e s f o r  determining  the p a r t i c u l a r s of the r o l l i n g of p a r t s with v a r i a b l e rectangular c r o s s - s e c t i o n . (c)  To i d e n t i f y and apply the process c o n s t r a i n t s to the above s t r a t e g i e s .  (d)  To evaluate the a p p l i c a b i l i t y of the developed experimentally  and  to  apply  method  corrections  and  5  modifications  on  the b a s i s  of  the  experimental  evidence. A l i t e r a t u r e survey i s presented i n chapter important  r e l a t i o n s with regard t o the geometry, kinematics  and dynamics of f l a t the geometric  r o l l i n g are f i r s t d e s c r i b e d . A study of  deformation of s t e e l under hot f l a t  presented next. T h i s covers a which  two. Some  also  includes  a  major  comparison  part  of  between  formlae used t o p r e d i c t the spread i n hot f l a t  rolling is  the chapter the d i f f e r e n t rolling.  The  chapter i s concluded by d i s c u s s i o n s l e a d i n g t o the s e l e c t i o n of  a s u i t a b l e spread formula In  chapter  presenting  a  deformation.  t h r e e , the proposed method i s d e s c r i b e d by  solution This  for  is  deformation  of  constraints  are applied  constraint  is  The  parts.  algorithm  of  f o r use i n t h i s work.  then  non-uniform  course  the  extended  parts.  to  to  uniform  i n c l u d e the  A l l known  practical  a t t h i s stage; however, the  dictated  numerical  uniform  by  approach  i s programmed and  the  desired  based  on  i s described  shape  the  next.  main  of  the  proposed Operating  aspects a r e b r i e f l y d i s c u s s e d a t the end of the c h a p t e r . Chapter discussions  four  presents  some  on b a s i c experiments  typical  which were performed  experimental r o l l i n g m i l l . The f i r s t part illustrates  the r e s u l t s  of  t o v e r i f y the  the  part  The  second  of  the experiments,  s t a t e c o n d i t i o n s , conducted method.  results  presents  this  and on an  chapter  under steady  applicability an  of  experimental  e v a l u a t i o n of the geometric deformation of the m a t e r i a l i n  6 unsteady  r o l l i n g . T h i s chapter i s completed  leading  to  the  introduction  of  an  by  discussions  improved  formula f o r  spread i n the general case. Two  examples  illustrated  in  with regard to the use of the method are  chapter  five.  Finally,  c o n c l u s i o n s are presented along with scope There are two appendices. method  for  finding  d e s c r i b e d . Appendix which  was  developed  of the v a r i a t i o n  the B  In appendix  roots  describes  in  chapter  six  f o r f u t u r e work. A  the  numerical  of n o n - l i n e a r equations i s a  curve-fitting  method,  to o b t a i n an a n a l y t i c a l r e p r e s e n t a t i o n  of r o l l  gap f o r each r o l l i n g  pass.  LITERATURE  2.  2.1  FUNDAMENTALS  2.1.1 BASIC  OF THE ROLLING  CONCEPTS  SURVEY  PROCESS  AND GEOMETRIC  RELATIONS  A schematic r e p r e s e n t a t i o n of f l a t r o l l i n g 2-1.  Fig.  In  the  successive  stages  of  i s shown rolling,  dimensions of a r e c t a n g u l a r bar a r e changed but  the  in the  volume  constancy holds, i . e , V =V =V =...=V 0  1  2  n  where ^n A /jl =h w l =  n  K,  l,  n  h  n  n  and  w  n  rt  n  n  a r e the c r o s s - s e c t i o n a l area,  height and the width of the stock a f t e r the n The  l  n  length,  stage.  i n c r e a s e i n l e n g t h of the stock a f t e r each pass i s  u s u a l l y g r e a t e r than the i n c r e a s e i n width. The i n c r e a s e width or l a t e r a l  elongation  i s called  spread.  R e f e r r i n g to F i g . 2 - 1 , when a uniform thickness the r o l l angle  h, enters the r o l l s  the plane a,  i s called  The denotes  of  rolls  at  a  one  arm  axes. The i n c l u d e d  leaving  the  rolls  angle,  is h .  p r o j e c t e d l e n g t h of the a r c of contact  distance  of the  bite.  of the bar  the r o l l s and the metal,  initial  a x i s , and the other arm i n  through the r o l l  I he angle  height the  passes  i t s apex on the r o l l  passing  bar with  , the edge of the bar touches  at a p o i n t through which  having  in  h i s the height of the bar x  from  7  the e x i t  2  1^  between i n the  s i d e of the r o l l s ,  8 corresponding the  to a r o l l i n g angle  incoming  dr af t ,  and  outgoing  4>. The  difference  between  t h i c k n e s s e s , i . e . , the  absolute  is Ah=h,-h  With  reference  to  (2-1)  2  F i g . 2-1, a geometric  r o l l i n g with r o l l s of the same diameter,  D,  relation for  (radius  of  R)  can be d e r i v e d as RCosa=R-(h,-h )/2 2  The  expression  for calculating  the angle  of b i t e  i s then  found as Cosa=1-(h -h )/2R=1-Ah/D 1  (2-2)  2  From t h i s , the a b s o l u t e d r a f t can be c a l c u l a t e d as Ah=DO-Cosa)  (2-3)  The p r o j e c t e d arc of c o n t a c t between the metal and rolls  i s c a l c u l a t e d from the g e o m e t r i c a l  relationship  lj=v/R -(R-Ah/2)=v/R-Ah-(Ah) /4 2  Equation  2  a simplified  (2-4)  2  (2-4) may be assumed without  a significant  error in  form 1^/R-Ah  This s i m p l i f i c a t i o n When  the  Ah<0.08R,  i s allowable  the  error  is  e x p r e s s i o n can be deduced f o r  (2-5)  f o r small angles  of  less  A similar  the  than  rolling  1% . 5  angle,  bite.  0,  and  t h i c k n e s s , h, Cos0=1-(h-h )/D 2  hence h=h +D(1-cos0) 2  (2-6)  9  The  angle  <f> i s c a l c u l a t e d  (2-7) the f o l l o w i n g r e l a t i o n s h i p  from  Sin^=x/R  2. 1. 2 CALCULATION  Rolled  OF ROLLING  stock enters the gap with a speed l e s s than the  peripheral  roll  the  i s g r e a t e r than  stock  SPEED  speed. On the other hand, the e x i t  at  which  in  called  the  angle a t  this  equation  holds  the  neutral plane  plane, is  stock. This  specified  5  is  the  plane  and the value of the r o l l i n g by  5.  The  v =v Cos5 V5  deformation  the h o r i z o n t a l component of the p e r i p h e r a l  speed i s equal t o the speed of the r o l l e d is  of  the p e r i p h e r a l speed of the r o l l s  (see F i g . 2-2). Thus, there i s a plane zone,  speed  following  (2-8)  r  speed of the r o l l e d  stock at the n e u t r a l  plane, v  r  i s the p e r i p h e r a l speed of the r o l l s .  A p p l y i n g the constancy  of volume  V=h , w , v , =h w v =h§w§vg = h5W§v .Cos6 2  v, and v and  2  6  2  /  planes,  respectively.  used the f o l l o w i n g r e l a t i o n s h i p s  of m i l d s t e e l under hot f l a t w /w 2  1  rolling = (h /h )2  f f  1  and w /w,=(h /h )-^ 6  where  (2-9)  are the speeds of the m a t e r i a l at the e n t r y  the e x i t  Wusatowski  2  6  1  f o r spread  .  ^ The  1  (-1.269(w,/h )(h,/D) ' 0  = 1 0  5 5 6  1  0  )  value of the n e u t r a l angle, 6, can now be r e l a t e d to the  outgoing  velocity (wg/w,)(hg/h,)v Cos6=(w /wT)(h /h,)v r  Equation  2  2  2  (2-7) can be w r i t t e n at <j>=8 as, hg/h,=[D(1-Cos6)+h ]/h. 2  Using the above two e q u a t i o n s , then  the  following  relationship  holds v Cos6/[D(1-Cos6)+h ] ( r  r  2  1  )=v /[h 2  ( J f _ 1 2  >]  from which, the value of the e x i t v e l o c i t y can be determined p r o v i d i n g t h a t the value of the n e u t r a l angle similar  neutral  be d i v i d e d zone  known.  A  r e l a t i o n s h i p c o u l d be w r i t t e n between v, and 5.  Koncewicz the  is  of  7  d e r i v e d a formula  f o r the  determination  angle; r e f e r r i n g to F i g . 2-3, the r o l l  i n t o two zones: backward  the zone of forward  gap can  s l i p and  s l i p . For f r e e r o l l i n g , without  of  the  f r o n t or  back t e n s i o n , ; dF +/ dF =0 a  6  2  o  dF,=p0R'W0(Sin0-fCos0)dtf>, to 2  to  f o r c e due  order to s o l v e equation  contact.  the  He a l s o suggested  the r o l l e d  gap, i . e . ,  i s the h o r i z o n t a l  f o r c e due  forward s l i p when 0<tf><6.  f to be constant along  of  i s the h o r i z o n t a l  backward s l i p when 5<<p<a;  dF =p0R'W0(Sin$+fCos<£)d0  In  (2-10)  6  1  (2-10), Konsewicz assumed p^ and whole  length  of  a linear variation  the  a r c of  f o r the width  stock along the h o r i z o n t a l l e n g t h of the  roll  11 W0=w (w -w )Sin0/Sina _  2  The  value  of  the  i n t e g r a t i n g equation 175).  Neither  Koncewicz's  2  neutral  Wusatowski's  approach  maximum  prediction  for finding  MECHANISM OF BITE The  angle can then be estimated by  (2-10) ( f o r d e t a i l e s see 6, pp  best way of f i n d i n g v, or v  2.1.3  1  f o r spread  nor  5 are a c c u r a t e enough.  The  i s through d i r e c t measurement.  2  AND  FRICTION  angle of a at which free r o l l i n g can take  p l a c e , i . e . , without u s i n g f o r c e to push the metal roll Fig.  170 t o  gap, i s c a l l e d  the maximum angle  into  the  of b i t e . R e f e r r i n g t o  2-3, at the p o i n t of e n t r y , f o r the element ( p ^ , S i n a ) d A = ( f •p^Cosa max  mflJC  of area dA,  )dA  Then Tana =i  (2-11)  max  So,  i f a^Tan f, 1  then the r o l l s b i t e the metal, without any  backforce or forward t e n s i o n . T h i s i s r e f e r r e d r o l l i n g . From the geometry of r o l l i n g , ' max Tana  and  d/  =  1  to  as  free  one can w r i t e  max  RCosa  approximately Tana  m w  - •R. &h /(R-&h /2 max  max  )  or T a n a  m a x =* v/Ah /R wcx  So the maximum d r a f t  for free r o l l i n g i s AlW=R-f  R e f e r r i n g to F i g . 2-4, direction  at  the  neutral  the  (2-12)  2  frictional  forces  plane, FF. When the r o l l  change gap i s  12 completely frictional be  filled,  the  pressure.  free r o l l i n g  the c o n d i t i o n  have  This  takes p l a c e , the  at the sector b-c should  than the sum of the f r i c t i o n a l  investigations  126  that r o l l i n g  i n the s e c t o r a-b and the  roll  allows  order  forces a s s i s t i n g r o l l i n g  greater  rolling  in  forces  horizontal  hindering  component  of  r e s u l t s i n a new c o n d i t i o n which  with higher  d r a f t s to take p l a c e . Recent  confirmed  the f o l l o w i n g i n e q u a l i t y as  for free r o l l i n g  ( f o r d e t a i l s see 6, pp 120 to  and 16, pp 204 to 206) 0<a During r o l l i n g  coefficient  of  coefficient With  both  ideal  <2Tan'f  the  friction  of f r i c t i o n ,  max  (2-13)  frictional  vary along  the  and the  the a r c of c o n t a c t . The  f, increases  lubrication,  forces  with the normal f o r c e .  coefficient  of  friction  decreases as the v e l o c i t y of the body i n c r e a s e s . Since  i t is  not  along  p o s s i b l e to c a l c u l a t e the c o e f f i c i e n t  the a r c of c o n t a c t ,  of f r i c t i o n  the average c o e f f i c i e n t  of  friction  is  used. There are many ways to f i n d 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 i s to measure roll  load  while  can  be achieved  to  the m a t e r i a l  207).  Ekelund  expresses the  a c e r t a i n amount of back  modification  suggested  temperature.  an  emperical  speed as w e l l  formula  of f r i c t i o n  Bachtinov ' 6  9  to Ekelund's formula to allow  of the r o l l i n g  of the This  tension  ( f o r d e t a i l s see 6, pp 128 to 130 and 16, p  the mean c o e f f i c i e n t  rolling  value  the plane of no s l i p i s at the e x i t .  by a p p l y i n g  8  the  which  as a f u n c t i o n of  then  proposed  a  f o r the i n f l u e n c e  13 f =a/<( 1.05-0.0005t) a=l. 0 f o r cast  i r o n or rough s t e e l  rolls,  a=0.8 f o r c h i l l e d and smooth s t e e l  rolls,  a=0. 55 f o r ground s t e e l t  i s the r o l l i n g  temperature(°C),  K i s a factor relating the  rolls,  f to the p e r i p h e r a l  r o l l , v,., a c c o r d i n g  A distinction  should  be  made  between  the s l i p p i n g  and the s t i c k i n g of metal t o the r o l l  specially  i n hot r o l l i n g  the  of  to t a b l e 2-1 .  friction  208)  speed  which occurs  (see 16, p 205). Korolev  (see 6,  p  has developed a formula which determines the length of s t i c k i n g zone.  2.1.4 PRESSURE DISTRIBUTION,  FORCE AND TORQUE  R e f e r r i n g t o F i g . 2-5, the v a r i a t i o n of along  the  ,ADGEC,  roll  shows  deformation  gap  work-hardening  two for  pressure  parts.  The lower p a r t  ideal  (frictionless)  ( i t i s almost h o r i z o n t a l i n hot r o l l i n g )  the upper p a r t DFEGD, r  overcome  contains  roll  the  additional  forces. A theoretical roll-pressure  shows the r o l l  from  necessary  to  c o n s t r a i n t caused by the f r i c t i o n  approach local  pressure  , and  f o r the  determination  stress-evaluation  of  i s now b r i e f l y  described. The  starting  representing roll  point  is  to  develop  the h o r i z o n t a l e q u i b l i b r i u m  gap. C o n s i d e r i n g  i n s e t diagram of  of  an  equation  forces  the elemental s l i c e of m a t e r i a l  F i g . 2-1,  the  horizontal  stress  i n the i n the o is  1 4  assumed t o  be  distributed  s e c t i o n . The h o r i z o n t a l in  equilibrium  uniformely  over  the  vertical  f o r c e s a c t i n g on the element w i l l be  i f the f o l l o w i n g  r e l a t i o n s h i p holds  d( ah)/dtf>+2R(p0Sintf>±rCos0) = O where r i s the shear s t r e s s due to whether zone,  the the  element  is  resultant  in  friction;  depending  on  the s l i p p i n g zone or s t i c k i n g  frictional  i s dependent  or  independent of the p r e s s u r e , r e s p e c t i v e l y . The negative  sign  r e f e r s to the c o n d i t i o n point, exit  force  on the entry  and the p o s i t i v e s i g n  side  of  the  neutral  r e f e r s to the c o n d i t i o n  on the  side. A p p l y i n g the p l a s t i c i t y c r i t e r i o n  using the  to the element and by  a numerical method, the d i s t r i b u t i o n of p r e s s u r e along roll  gap  Recently,  can  some  efforts  three-dimensional amongst whom, typical  be determined have  ( r e f e r to 10, 11 and 12).  been  made  to  find  the  d i s t r i b u t i o n of pressure i n the r o l l gap,  Lalli  1 3  and  Kobayashi '  can  1  be  named.  A  t h r e e - d i m e n s i o n a l d i s t r i b u t i o n of p r e s s u r e i s shown  in F i g . 2-6. Having as  well  as  integrating 111).  the over  acting the  torque  area  Due t o the d i f f i c u l t y  distribution to d e r i v e the  the p r e s s u r e d i s t r i b u t i o n , the s e p a r a t i n g  theoretically  of  can  be  contact  calculated  '  1 5  by  (see 16, pp 208 to  i n f i n d i n g the 1 0  force  actual  pressure  some attempts have been made  p r a c t i c a l and easy-to-use formulae f o r c a l c u l a t i n g  separating  force  in rolling  (see 6, pp 229 to 266). A  15  good  estimate  obtained  by  of  roll  considering  compression  between  1^ and width of w block  the  before  the  in  process  f l a t r o l l i n g can be as  a  homogenous  two p l a t e n s . The p l a t e n s are of l e n g t h  mea/7  and  load  , the mean value  after  rolling  of the  width  . The y i e l d along  this  The s e p a r a t i n g  P, necessary f o r the r o l l i n g  2 5  .  the  yield  the  value  for  15  the  gap; force,  (2-14)  f o r 1^ and i n c r e a s i n g the value of  s t r e s s by the amount of 20% (which  Orawan )  roll  i s then  P^-ld'"mean Substituting  the  s t r e s s of the  m a t e r i a l , Y, i s assumed to be constant i s almost true f o r hot r o l l i n g  of  was  suggested  c o n t r i b u t i o n of f r i c t i o n ,  the r o l l  by load  w i l l then be P=( 1 .2)Y-w It  is  true  resultant  with  good  approximation  f o r c e a c t s at the c e n t r e  hot  rolling  will  be  /R Ah  (2-15)  T  mea/lV  to  assume t h a t the  of the arc of c o n t a c t  in  (see 6, p 270). In that case the a p p l i e d torque  M=P(1 /2) = ( 0 . 6 ) Y • w d  m e an  R•Ah  (2-16)  2.2 GEOMETRIC DEFORMATION IN FLAT ROLLING  2. 2. 1  INTRODUCTION  In r o l l i n g , by pressure the  draft  (the change of t h i c k n e s s )  of the r o l l s . T h i s  increase  of the l e n g t h ,  the width, spread,  is  normally  i s produced  accompanied  by  e l o n g a t i o n , and the i n c r e a s e of  of the m a t e r i a l being  rolled.  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  i n p r e d i c t i n g t h e o r e t i c a l l y the  t h r e e - d i m e n s i o n a l p l a s t i c deformation of m a t e r i a l , the  existing  formulae  for predicting  the  plane  are edge  the block u s u a l l y becomes convex i n shape (other p o s s i b l e  modes of deformation w i l l these formulae  be d i s c u s s e d l a t e r ) .  m  width w  of  been  6  m fc"( fc" /^ = w  w  w  Generally, factors a f f e c t i n g  -  some  a c c o r d i n g to the f o l l o w i n g formula has  used as the r o l l e d  three main  In  the maximum v a l u e , w^, and i n some o t h e r s the  mean v a l u e , w ,  (i)  of  spread  e m p i r i c a l . R e f e r r i n g t o F i g . 2-7 , the i n i t i a l l y of  most  (2-17)  / / 3  spread can be d i v i d e d  into  groups ' : 1 7  Geometric  1 8  f a c t o r s such as aspect r a t i o , d r a f t and  the bar width to the l e n g t h of a r c of c o n t a c t r a t i o . (ii)  ~ F a c t o r s r e l a t i n g to f r i c t i o n a l c o n d i t i o n s roll  (iii)  such  as  s u r f a c e and s c a l e formation .  -Factors  affecting  such as s t r a i n  the y i e l d s t r e s s of the m a t e r i a l  r a t e , m a t e r i a l composition and r o l l i n g  temperature. The  following  comparison  between  section the  most  p r e d i c t i n g the spread of s t e e l interest at  deals  with  recent in  the e n t r y and e x i t  points.  study  formulae  hot  i s of course i n the geometric  a  flat  and  a  used  for  rolling.  The  shape of the m a t e r i a l  17 2, 2. 2 FORMULAE FOR SPREAD S i g n i f i c a n t work on spread has been ongoing  s i n c e 1955.  Prior to t h i s ,  some approximate formulae had been i n use i n  practice.  this  In  (1923), S e i b e l ' s Bechmann's  regard,  formula  formula  Tafel  (1932),  and Sedlaczek's formula  Trinks's  (1950) and Ekelund's  formula formula  (1933),  (1953) can  be named as the most well-known ( f o r d e t a i l s see 6, pp 84 to 86 and 4, pp 836 to 836). In  1955, Z. W u s a t o w s k i  following  published  19  a paper i n which the  formula f o r spread was proposed  where  H/ -1 .269(w,/h, ) ( h ^ D ) 0  5 5 6  =10  7=h /h 2  Wusatowski rolling  indicated  low  (0.5<7<0.9). comparison  this  carbon He  His  table,  h i s formula was a c c u r a t e while  steel  also  between  previously. to  that  1  with  conducted his  light  some experiments  formula  and  r e s u l t s are l i s t e d Wusatowski's  draft  ratios to make a  those  proposed  i n t a b l e 2-2. According  formula  shows  no  larger  deviation  than the other formulae except f o r the Ekelund's.  Wusatowski  suggested  could  reduced  for  be  rolling  surface  20  that  by i n t r o d u c i n g  temperature,  condition,  the mean e r r o r  a,  his  formula  some c o r r e c t i o n a l  factors  rolling  in  velocity,  b,  roll  c, and the type of s t e e l 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 f o r c a s t  i r o n and rough s t e e l  rolls.  c=i.000  f o r c h i l l e d cast  i r o n and smooth s t e e l  c=0.980  f o r ground  rolls.  d i s chosen  steel  rolls.  from t a b l e 2-3.  By using the new  formula,  total  from 4.59% to 2.24% which was s a t i s f a c t o r y  mean e r r o r  Wusatowski  at that time. Wusatowski a l s o would  be  valid  could  suggested  decrease  that  even f o r heavy d r a f t r a t i o s  his  the  formula  (0.1<7<0.5)  if  values of 3.954 and 0.967 were s u b s t i t u t e d f o r the constants 1.269 and 0.556 i n h i s formula, In  the  same  year,  respectively . 6  R.  Hill  1  7  '  1  8  '  2  proposed  0  an  a l t e r n a t i v e formula f o r spread  Ln(w /w )/Ln(h,/h )=0.5 EXP[-X(w / /D-Ah) ] 6  X  is a  2  1  l  constant which i s s e l e c t e d to f i t the experimental  data. H i l l A.W. to  t  suggested a value of 0.5 f o r X. McCrum  compare  the  Wusatowski's  18  i n 1956, c a r r i e d out some c r i t i c a l  three  and  most  recent  formulae:  H i l l ' s . H i s experiments  constant  width,  c o n d i t i o n s and  rolling  speed,  heights.  results  deviated  The  draft,  roll but  radius,  with  temperature frictional  different  considerably  p r e d i c t e d by the formulae given by Ekelund  Ekelund's,  were performed on  r o l l i n g of bars with the same m a t e r i a l , the same and  tests  and  from  stock those  Wusatowski,  19  but were more agreeable with those of H i l l ' s . McCrum that  a  value  r o l l i n g mild In the  a  showed  of 0.525 f o r X would give b e t t e r r e s u l t s when steel  1 8  .  growing  need f o r more p r e c i s e  r o l l i n g process,  information  about  a s e r i e s of experiments was c a r r i e d out  by L.G.M. S p a r l i n g , under c a r e f u l l y c o n t r o l l e d c o n d i t i o n s , 1 8  to see the e f f e c t s of v a r i o u s recommended the f o l l o w i n g  f a c t o r s governing  spread.  He  formula  S=C-EXP[-K(w /h ) (h,/R) (Ah/h ) ] A  1  B  G  1  1  where S=Ln(w /w!)/Ln(h /h ) m  w  m  and  1  i s c a l c u l a t e d from equation G are l i s t e d In  experiments,  were changed at a constant strain  (2-17). Constants C, K, A , B  i n t a b l e 2-4.  Sparling's  constant  2  rate  experiments was m i l d  a l l the geometric f a c t o r s  temperature  (5Sec" ). 1  (1100±10°C)  The m a t e r i a l  and  a  in Sparling's  steel.  There were 35 s a t i s f a c t o r y t e s t s . The r e s u l t s e x t r a c t e d from h i s  experiments  Wusatowski  show  nor that of H i l l  that  neither  the  formula  of  a c c u r a t e l y p r e d i c t s spread over  a wide range of experimental  conditions.  Table  2-5  shows  some r e l e v a n t r e s u l t s . In  1968,  investigations then; f r i c t i o n  El-Kalay to  an  and area  and its effect  Sparling  2 2  extended  which had been n e g l e c t e d on spread.  They  modified  their until the  20 formula d e r i v e d e a r l i e r by S p a r l i n g to g i v e the best f i t f o r t h e i r experimental data. The formula  i s as f o l l o w s  Ln(w /w,)/Ln(h /h )=A.EXP[-B(w,/h ) (h /R) (Ah/h ) ] C  m  1  2  D  1  the v a l u e s of c o n s t a n t s are l i s t e d  E  1  l  i n t a b l e 2-6 a c c o r d i n g to  the f r i c t i o n a l c o n d i t i o n s . In  the  same  J.M. A l e x a n d e r . 1 7  steel  (scale  rolling  year a paper was w r i t t e n by A. Helmi and More than 200 t e s t s were performed  free  steel  temperature  (9m/min).  The  with  (1000°C)  0.18%  and  carbon) at constant  constant  following  conclusions  formula  is basically  on m i l d  rolling  were  speed  d e r i v e d by the  authors: (i)-Wusatowski's predict  spread under  variables  a  very  in error,  limited  range  (the same deduction as S p a r l i n g  (i / ^ - H i l l ' s slightly  formula  is  basically  1 8  sound;  i n e r r o r as i t n e g l e c t s the e f f e c t  i t can only  of  geometric  and M c C r u m ) . 17  However,  i t is  of w,/!^  on the  spread. (i ii)-El-Kalay soundest  and  Sparling's  seems  to  be  the  existing.  Helmi  and  Alexander  which S p a r l i n g ' s formula not  formula  been  alternative  proved  by  formula  proposed  a range of c o n d i t i o n s i n  i s more a c c u r a t e , however,  other  works.  i t has  They a l s o developed an  f o r spread  S=0.95(h /w ) - EXP[-0.707( /(RAh) - )(h,/w,) • 1  1  1  1  0  W l  5  _ 0  9 7 1  ]  21  where S=Ln(w^/w )/Ln(h,/h ) 1  According  to the authors' paper t h i s formula can be a p p l i e d  over a very  wide  application  is  unity.  2  It  of  unlimited  was  temperature,  range  also  roll  the experiments  geometric when  except  verified  s u r f a c e and  would a f f e c t  variables  that  is  w^h, the  and  its  less  than  deviation  of  s c a l e c o n d i t i o n from those of  the value of 0 . 7 0 7  i n the above  expression. In 1 9 7 2 , J.G. to  make  a  Beese  comparison  performed  2 3  some  Alexander's  tests  between the E l - K a l a y and S p a r l i n g ' s  formula with that of Helmi and Alexander. that  industrial  The  r e s u l t s showed  estimates of maximum spread were a l l high  ( o v e r e s t i m a t e d ) , while S p a r l i n g ' s estimates of  mean  spread  on small s l a b s were s a t i s f a c t o r y . A new  formula was  developed  by Beese which was  claimed  between 3 to 1 6 .  to be a c c u r a t e f o r v a l u e s of w,/h,  S=0.61(h,/w,) • EXP[-0.32h /(R°- Ah - )] 1  3  5  0  5  1  "S"  has  the  same  definition  have  also  been  as  that  of  Helmi  and  Alexander's. There geometrical  deformation  theoretically,  developed  few attempts  the  material  f o r s p e c i f i c c a s e s ' " . One 1 8  i s the computer-aided was  of  a  modular upper  2  bound  to p r e d i c t under  rolling  typical  example  approach,  which  at B a t t e l l e Columbus l a b o r a t o r i e s , under  s p o n s o r s h i p to p r e d i c t the metal  the  flow i n the r o l l i n g  of  NASA an  22 airfoil  s e c t i o n . In t h i s method  rolls,  as  well  as  the  the  lateral  spread spread  profile and  under  longitudinal  e l o n g a t i o n are p r e d i c t e d .  2.3  SUMMARY  The  AND  first  fundamentals  EVALUATION  part of t h i s with  regard  concepts namely, r o l l i n g calculation  were  chapter  described  some  basic  to the process of r o l l i n g .  speed, mechanism of b i t e and  discussed.  These  Three force  concepts w i l l be  later  used to formulate the p r o c e s s c o n s t r a i n t s i n r o l l i n g . The  literature  survey on spread formulae  c o n s i d e r a b l e amount of study has been done e f f e c t s of the geometric  of f r i c t i o n a l  about  effects  2 2  predict  the  f a c t o r s on spread. However, d e s p i t e  the importance their  to  shows that a  .  conditions,  This  study  little  also  is  known  revealed  although there i s no s i n g l e formula  that  prediction  E l - K a l a y and S p a r l i n g ' s  under  all  conditions,  gives  that  formula p r o v i d e s s a t i s f a c t o r y estimates over a range of o p e r a t i n g c o n d i t i o n s . Helmi  reasonable  fairly  and Alexander's  has been found to overestimate the value of spread, the  fact  that  experiments; and  short  blocks  were  so, the r e s u l t s were g r e a t l y  f r o n t - e n d spread  incapable of  specimens  giving  used  only  formula due  in  to  their  i n f l u e n c e d by back  (see a l s o 23). B e s i d e s , the formula i s reasonable  estimates  of  spread  for  with aspect r a t i o s l e s s than u n i t y . No e v a l u a t i o n of  the Beese's formula has been r e p o r t e d . However, h i s is  wide  formula  capable of p r e d i c t i n g the spread of the s l a b s with  23 the  range  of  aspect  ratios  from  S p a r l i n g ' s formula can p r e d i c t the when  the aspect  ratio  effects  of  t h i s formula. folmula  spread  is still  of  of  deformations  and  spread  which, as w i l l  be seen  later,  deformation f o r the s t e e l  the spread  for different  i s not  the  only  the  formula  that  and  has  been  generalized  the  nature  S p a r l i n g ' s d i s c u s s i o n s ; he 1 8  industrial  would be expected to i n c r e a s e  processes,  the  spread  with  (1) - d e c r e a s i n g  temperature  (2) - d e c r e a s i n g  amount of s c a l e s  (3) ~ medium carbon s t e e l s used i n s t e a d of m i l d (4) - i n c r e a s i n g s t r a i n  of  i t c o u l d be adapted f o r d i f f e r e n t  itself,  suggested that for most  mode  stocks.  c o n d i t i o n s . T h i s m o d i f i c a t i o n has been based on of  formulae  f o r the rhombic c r o s s - s e c t i o n  E l - K a l a y and S p a r l i n g ' s formula such  of the spread.  t o Beese or Alexander were developed to  maximum  modified  defined.  Sparling's  (see s e c t i o n 4.2.2). Other  the  and  not very w e l l  t h a t , i t p r e d i c t s the mean value  l i k e those belonging estimate  cases  than one, although  El-Kalay  T h i s w i l l be u s e f u l when e s t i m a t i n g modes  f o r both  f r i c t i o n a l conditions are a l s o included in  Another f e a t u r e  is  t o 16. E l - K a l a y and  i s l e s s or g r e a t e r  the range of t h i s c a p a b i l i t y The  3  steel  rates  (5) - i n c r e a s i n g roughness of the r o l l e s . Conditions formula.  (2)  and  (5)  To s a t i s f y other  have already three  been i n c l u d e d  conditions,  the  formula has been suggested by S p a r l i n g as f o l l o w  i n the  modified  24 S=A.EXP[-B(w,/h ) (h /R) (Ah/h ) .f.g.j] c  D  1  (2-18)  E  1  l  S=Ln(w /w,)/Ln(h /h ) OT  1  2  where f i n d i c a t e s the e f f e c t s of rolled  material;  manganese mild  f=1  steel,  the for  f<1  composition 0.13%  for  of  carbon,  medium  the 0.55%  carbon  and  stainless steels. g  i n d i c a t e s the e f f e c t s of temperature of the  material;  g=1  f o r 1100°C, g<1  rolled  f o r 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 d u r i n g rolling;  j=1  5 s e c " , j<1 1  Therefore, provisions be  used  f o r an approximate mean s t r a i n  for higher s t r a i n El-Kalay  for modification as  starting  in  rate of  rates.  Sparling's  f a c t o r s , was  point  r o l l i n g p a r t s with v a r i a b l e will  and  the  deemed s u i t a b l e  developing  rectangular  formula,  with to  the method of  cross-section.  It  be seen l a t e r that the approach towards improvement, as  shown above, i s s u i t a b l e f o r the more rolling,  i.e.,  t he  unsteady  rolling.  generalized  form  of  3.  3.1  3.1.1  ROLLING  BASIC  OF  VARIABLE  FORMULATION  UNIFORM  TO  3.1.1.1  FOR  DEFORMATION  hod  A uniform block to  be  deformed  dimensions HF, involved (i)  (ii)  of dimensions HI,  block  of  and  LF  (Fig.  3-1a).  The  initial  The  rolled  blank  HM,  WM  ( F i g . 3-1b).  The  intermediate  the  required  LM  d e s i r e d shape The  have  steps  HF. the  to  as  Wl. the  have dimensions  i s then r o l l e d The  rolled  width  WF  into  material and  the  ( F i g . 3-1c) .  problem of  will  material  thickness  desired  on the s i d e  referred  material,  i s expected to  above.  is rolled  material  is  follows:  intermediate  dimensions  LI  uniform  The  length LF  and  a  WF  and  Wl  into  in the process are as  The  CROSS-SECTIONS  DEFORMATION  UNIFORM  Met  RECTANGULAR  the is  is  to  find  intermediate formed  within  generalized  this  pass h,=WI h =WM 2  w HI 1=  appropriate  m a t e r i a l so that the  form of the  (2-18), i s used. Applying  25  the  steps spread  formula to  the  stated formula,  the  first  26 w =HM 2  Ah=(WI-WM) R=RW  Ln(HM/HI)_ Ln(WI/WM) A  - (HT/WT) (WI/RW) ((WI-WM)/WI) f.g.j C  e  D  E  B  (3-1 ) and  f o r the second pass h,=HM h =HF 2  w , =WM w =WF 2  Ah=(HM-HF) R=RH  Ln(WF/WM) Ln(HM/HF)  - (WM/HM)C(HM/RH) ((HM-HF)/HM) f.g.j D  A  e  E  B  (3-2) where  RW  the f i r s t  and RH r e f e r to the r a d i i  of the r o l l s i n  and the second o p e r a t i o n , r e s p e c t i v e l y . The  volume constancy  a l s o e x i s t s during these  processes  HI.Wl.LI=HM.WM.LM=HF.WF.LF Two n o n - l i n e a r equations solved  simultaneously  with  (3"3)  (3-1) and (3-2) can be eqution  (3-3) t o o b t a i n  the i n t e r m e d i a t e dimensions as w e l l as the l e n g t h of the  f i n a l p a r t or the i n i t i a l  m a t e r i a l , depending on  whichever i s unknown. The  incremental  search  method  was  used i n  s o l v i n g the above e q u a t i o n s . The g e n e r a l a l g o r i t h m i s  27 described 3.1.1.2  i n appendix A.  Number  In a which  of Possible  two-pass  side  rolling  of the i n i t i a l  i . e . , named Wl, and which termed  Solutions  process, material  depending  i srolled  s i d e i s formed  (i)  distinct  (see F i g . 3-2): Case  one:  Side of Wl i s r o l l e d Convexity  first,  last,i . e . ,  HF, there would e x i s t , at most, four  solutions  on  first,  HM i s then reduced t o HF.  (out of f l a t n e s s ) appears on the s i d e of  HF. (i i)  Case  t wo:  Side of Wl i s r o l l e d  first,  HM i s then reduced t o WF.  Convexity appears on the s i d e of WF. (iii)  Case  t hr ee:  Side of HI i s r o l l e d  first,  HM i s then reduced t o HF.  Convexity appears on the s i d e of HF. (i v)  Case  four:  Side of HI i s r o l l e d  first,  HM i s then reduced t o WF.  Convexity appears on the s i d e of WF. It  i s necessary  shape  for  3.1.1.3  each  Existence  It  cas e is of  that  the  intermediate  di f f e r e nt .  the  Solutions  i s c l e a r that 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 area  t o mention  blank i s l e s s than the c r o s s - s e c t i o n a l  of the d e s i r e d p a r t , the proposed method i s not  28  a p p l i c a b l e . T h i s i s due t o t h e p h y s i c a l n a t u r e of t h e r o l l i n g p r o c e s s which always l e a d s t o a r e d u c t i o n the  cross-sectional  area.  Also,  c r o s s - s e c t i o n a l a r e a of the i n i t i a l blank than  that  the  i s larger  of t h e d e 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 t o e v a l u a t e Referring  when  in  the e x i s t e n c e  of the s o l u t i o n s .  t o F i g . 3-1a, f o u r cases may occur i n t h i s  regard: (i)  WI>WF and HI>HF the p r o c e s s i s p o s s i b l e .  (ii)  WI=WF , HI <HF or WKWF , HI <HF the p r o c e s s i s i m p o s s i b l e .  (iii)  WI>WF , HI<HF the p r o c e s s i s p o s s i b l e i f HF<HS, where  HS i s  the spread v a l u e of HI a f t e r a s i n g l e f r e e r o l l i n g i n which Wl i s reduced t o WF. (iv)  WKWF , HI >HF the p r o c e s s i s p o s s i b l e i f WF<WS, where  the spread v a l u e  WS i s  of Wl a f t e r a s i n g l e f r e e r o l l i n g i n  which HI i s reduced t o HF. The  physical explanation  i s that the extent rolling  i s less  of l a t e r a l than  f o r t h e l a s t two cases elongation  that  be  achieved  by a p p l y i n g side  by  of 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 , so t h e maximum e l o n g a t i o n only  caused  a  on one s i d e can single  rather  pass of  reduction  t o the other  than  two  successive  passes i n v o l v i n g a r e d u c t i o n on both s i d e s  29  or  on one s i d e .  above deduction Consider  An a n a l y t i c a l e x p l a n a t i o n f o r the  i s as f o l l o w s : a  block with dimensions  which undergoes a dimensions  width  of WM,  HM  reduction  HI, Wl and LI  to intermediate  and LM ( F i g . 3-3a.1>. Spread  occurs along the s i d e HI. The f o l l o w i n g r e l a t i o n s can then be w r i t t e n 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 where 0<X<1 relating  i s defined lateral  as a  elongation  (3-4d)  general  coefficient  t o the l o n g i t u d i n a l  e l o n g a t i o n . X=0, where no spread occurs plain  of  strain).  The in  (condition  intermediate m a t e r i a l then undergoes  rolling  such a way that i t s h e i g h t , HM, i s reduced  Spread  occurs on the s i d e of WM  (Fig.  t o HF.  3-3a.2). The  following r e l a t i o n s hold WF,.LF,.HF=WM.LM.HM WM.HM>WF,.HF  (3~5b)  WF,>WM , HF<HM  (3-5c)  (WF -WM)/WM=X(LF -LM)/LM  (3~5d)  1  Using equation  (3-5a)  1  (3-5a), equation  (3~5d) becomes:  (WF,-WM)/WM=X(WM.HM-WF,.HF)/(WF .HF) 1  or WF,={(1-X)WM.HF+/[(1-X) WM ,HF +4XWM .HM.HF]}/2HF 2  2  2  2  30 (3-6) WF,  i s now  value  compared to WF  which  2  is  of Wl when the same i n i t i a l  a single free r o l l i n g reduced  to HF  Using  a  that  elongated  m a t e r i a l undergoes  its  height,  HI,  is  ( F i g . 3-3b).  similar  calculated  so  the  approach  as  for  WF,;  WF  is  2  as WF .HF.LF =WI.HI.LI 2  (3-7a)  2  Wl.HI>WF .HF  (3-7b)  WF >WI , HF<HI  (3-7c)  (WF -WI)/WI=X(LF -LI )/LI  (3-7d)  2  2  2  2  or WF = { ( 1-X)WI .HFVt ( 1 -X) WI 2  2  2  .HF + 4XWI .HI .HF] 2  2  }/2HF  (3-8) A  comparison  that WF  2  The  between equations  (3-6) and  (3-8)  i s equal to or g r e a t e r than WF, .  e f f e c t of a subsequent r e d u c t i o n of one  the value  shows  of  the  spread  for  the  other  side,  on  side  is  i n v e s t i g a t e d e x p e r i m e n t a l l y in s e c t i o n 4.3.1.  3. J. 2 UNIFORM TO  A  typical  NON-UN I FORM DEFORMATION  example  rectangular cross-section material  which  of is  a shown  part in  having Fig.  non-uniform 3-5.  Initial  i s a uniform block i s shown i n F i g . 3-4.  formulate the problem the d e s i r e d part  is  divided  into  To N  equal l e n g t h segments ( F i g . 3-5). Lf7=Lf2=Lf3=...=Lfn=LF/N Each  segment  is  now  assumed  to  have  a  uniform  31 c r o s s - s e c t i o n a l area of H f j . W f j , formed by means segment  in  of  the  (3 = 1, 2,...,n),  two  operations  initial  material.  from  each  segment  is a  c o r r e s p o n d i n g segment  corresponding  T h e r e f o r e , the i n i t i a l  m a t e r i a l should a l s o be d i v i d e d i n t o N that  a  which can be  segments  volumetric  in  equivalent  a  way  of i t s  i n the d e s i r e d p a r t , i . e . ,  Hij.Wij.Lij=Hfj.Wfj.Lfj (j = l, 2, 3, . . . , n)  where i n t h i s case Hij,  ( j = 1, 2, . . , n) = c o ns t a nt  Wij,  (2 = l,2,..,n)  and  The in  lengths  =  constant  of the segments i n the i n i t i a l  m a t e r i a l would,  g e n e r a l , be unequal. Applying  3.1.1.1  the  to  each  same pair  procedure of  as d e s c r i b e d i n s e c t i o n  corresponding  segments,  the  f o l l o w i n g r e l a t i o n s h i p s can be w r i t t e n  Ln(Hmj/Hij) Ln(Wi j/Wmj)  -B(Hij/Wij) (Wij/RW) ((Wij-Wmj)/Wij) f.g.j c  e  D  E  (3=1,  (3-9)  2, 3,.., n)  Ln(Wfj/Wmj) , -B(Wmj/Hmj) (Hmj/RH) ((Hmj-Hfj)/Hmj) f.g.j Ln(Hmj/Hfj) ( j = l, 2, 3, . . , n) (3-10) c  D  E  e  Wij.Hij.Lij=Wmj.Hmj.Lmj=Wfj.Hfj.Lfj ( 3 = 1, 2, 3, . . . , n)  By s o l v i n g equations (3-9), for  a l l segments,  the  (3-11)  (3-10) and (3-11) s i m u l t a n e o u s l y  dimensions  of the segments of the  32 intermediate intermediate length,  m a t e r i a l are found. R e f e r r i n g to F i g . 3-6, the material consists  uniform  of  N,  generally  c r o s s - s e c t i o n segments with dimensions  Hmj and Lmj (j=1, 2,.., n). The incremental variation Hfj  f o r the f i r s t  first  part  by means of f l a t  and the  second  v a r i a t i o n of r o l l  values  of r o l l  Wmj, gap  and the second pass are then Wmj and  (j=l, 2,. . , n), r e s p e c t i v e l y . So, i n  desired  non-equal  pass  order  rolling,  should  to  form  the r o l l  vary  the  gap i n the  accordingly.  The  gap can be c o n t r o l l e d v i a the displacement  of the r o l l e d m a t e r i a l d u r i n g  each pass  (refer  to  section  3.4) .  3.1. 3 NON-UNIFORM TO NON-UNIFORM Normally  the  initial  c r o s s - s e c t i o n . However, deform  a  DEFORMATION material  sometimes  i t may  is be  of  uniform  desired  to  non-uniform m a t e r i a l to a non-uniform p a r t , e.g.,  some pre-manufactured p a r t s are r e q u i r e d t o be reshaped. The formulation  derived  in  3.1.2  i s applicable in this  except that H i j and Wij (j=1, 2 ..,n) are no longer ;  However,  a  control  occurs  In  this  regard,  if  condition  to  section  ( i i ) i n 3.1.1.3  f o r even one p a i r , then a f e a s i b l e s o l u t i o n does  e x i s t . For the case where c o n d i t i o n s the e x i s t e n c e the  constant.  i s necessary to check the e x i s t e n c e of  the s o l u t i o n s f o r every p a i r of segments ( r e f e r 3.1.1.3).  regard  not  ( i i i ) or ( i v ) occur and  of a s o l u t i o n i s i n q u e s t i o n ,  an  increase  in  r o l l ' s r a d i u s can l e a d to s a t i s f y the requirements. The  v a r i 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 i n F i g . 3-7. I t i s seen that the i n c r e a s e  radius  leads t o the i n c r e a s e of the l a t e r a l  elongtion. This  p h y s i c a l l y means that by i n c r e a s i n g the r o l l ' s projected  length  causes the value increase that  increase is  the  to  section  rolls'  of  arc  of e l o n g a t i o n  (refer  larger  of  radii  of  contact  radius,  4.4.1). F i g u r e smaller  direction  other  parameters which are d i s c u s s e d  i n t h e ensuing  3.2 MULTI-PASS  was  non-uniform  shown  theoretically,  rectangular  possible  operations  due  Satisfying  these  design.  practical  constraints  in different  to  that  cross-section  to  complete  some  a  part  The multi-pass  of  PROCESS  physical  the  process  process  within  two  constraints.  n e c e s s i t a t e s to increase the  passes  design  CONSTRAINTS  a  can be made through  i n v o l v e d . The procedure of  number  with  determining  i s termed the m u l t i - p a s s  i s a scheme which  is  of s a t i s f y i n g the p r a c t i c a l c o n s t r a i n t s at a l l times.  3.2.1  places  P r a c t i c a l l y , however, i t i s not  constraints  number of o p e r a t i o n s the  radius  DESIGN CONCEPT  two passes of f l a t r o l l i n g . always  the  chapters.  t  It  on  in r o l l ' s  l i m i t e d because of. i t s e f f e c t s on the process  and  to  3-7 a l s o shows  effects  spread. However, the i n c r e a s e  the  i n c r e a s e s which  i n the l a t e r a l  have  in roll's  capable  34 3.2.1.1  Ki nemat i c  The  Constraint  kinematic c o n s t r a i n t  continuous material  free into  rolling. the r o l l  r e f e r s t o the l i m i t of  In  order  inequality  at the onset of r o l l i n g Ah<R«f  I t has a l s o been shown that filled  c o n d i t i o n of f r e e r o l l i n g  rolling filled is  the r o l l  gap i s  (3-12) i s r e l a x e d i n  i s p e r m i t t e d . So, the  becomes  where 'b' i s a c o e f f i c i e n t to  (see s e c t i o n 2.1.3)  once  Ah<b-R-f  rolling  be  (3-12)  with m a t e r i a l , c o n d i t i o n  of  the  should  2  a way that more a b s o l u t e d r a f t  onset  draw  gap without back p r e s s u r e or  forward t e n s i o n , the f o l l o w i n g satisfied  to  (3-13)  2  that varies  almost  1 i n the  2 (recommended  of s t e e l ) , when the r o l l with m a t e r i a l . Normally,  from  gap  f o r hot  i s completely  the incoming m a t e r i a l  tapered a t i t s f r o n t head so the c o n d i t i o n of the  filled  gap can be assumed a t a l l times. The value of  'b' can be i n c r e a s e d by a p p l y i n g forward back p r e s s u r e . Rearranging  equation  t e n s i o n or  (3-13)  Ah/R<(b-f =C1)  (3-14)  2  CI  i s termed the  is  r e l a t e d mainly During  satisfied;  kinematic  constraint  to the f r i c t i o n a l  rolling,  inequality  otherwise,  s t o p and the r o l l e r s  and  i t s value  conditions. (3-14)  should  the m a t e r i a l deformation will  s u r f a c e of the m a t e r i a l .  begin  to  slip  be will  on the  35 3.2.1.2  Dynami c Cons t r ai nt  The dynamic c o n s t r a i n t the  machinery  r e f e r s to the c a p a c i t y of  i n performing  the p r o c e s s . Maximum  torque a v a i l a b l e by the r o l l i n g convenient that  criterion.  the r e q u i r e d  machine,  T,  is a  I t was shown i n s e c t i o n 2.1.4  torque  for  hot  rolling  is  approximately M=(0.6)R.Ah.W which  should  mea  „.Y  be l e s s than or equal t o the a v a i l a b l e  torque by the machine, i . e . , (0.6)R.Ah.W  mefln  .Y<T  or R. Ah.VI C2  i s called  the  mean  <{ (T/( 0 . 6Y) )=C2)  dynamic  the  i t i s specified  constraint;  a c c o r d i n g t o the maximum nominal  torque a v a i l a b l e  during  rolling,  (3-15)  i s not  3.2.1.3  Convexity  the term an  or  stall.  Constraint  Most r o l l e d b l o c k s end up convexity  satisfied  the machine can no longer form the  m a t e r i a l due t o the o v e r l o a d , and w i l l  out  convexity  of  flatness  constraint  acceptable  degree  with  some  form  of  on t h e i r s i d e s . Here, has been  adopted  to  of out of f l a t n e s s on  f i n i s h e d p a r t s . The l e s s the m a t e r i a l i s reduced one  by  r o l l i n g m i l l and the composition of the m a t e r i a l  to be r o l l e d . I f c o n d i t i o n  mean  (3-15)  on  s i d e , the l e s s the c o n v e x i t y occurs on the other  s i d e . I t i s t h e r e f o r e s u i t a b l e t o have  a' c o n s t r a i n t  36  on  the value of absolute d r a f t s  the  rolling  operations.  then  formulated  f o rthe f i n i s h i n g  in  the  (3-17)  2nd finishing  CC1  and  of t h e d e s i r e d  ,  will  developed  include  the determination  required  as  part  convexity  dimensions  taken  as  their  constraint  will  operations  but  t h e need  clear  decrease on t h e  f o r having  during the f i n i s h i n g  so of  f a r can  passes.  be  to  of  operations  specifications  in  forming  isfirst  of p a s s e s . These  as the f i n a l  applied  a  to  determine  w h i c h must go t h r o u g h t h e  pre-finish  dimensions  dimensions  are  w h i c h have t o be g a i n e d  blank. the  finishing  passes  above, t h e p r o c e s s o f m a n u f a c t u r i n g by  extended  t h e number  the m a t e r i a l  separating  explored  is  ingeneral.  of  the i n i t i a l After  is  well  pair  It  machinery  increase  on t h e d r a f t  part.  according to  DESIGN PROCEDURE  concepts  finishing  CC2 a r e s p e c i f i e d  of the r e q u i r e d  hand  3. 2. 2 MJLTI-PASS  non-uniform  pas s = c o ns t ant =CC 2  t h e use of h i g h c a p a c i t y  constraints  described  a s below  Draft  other  from  passes  (3-16)  t h e number  then  constraints are  t he 1 s I fi ni s hi ng pas s = cons t ant =CC1  that  the  of  in  the requirements  The  Convexity  pair  Dr aft  Values  The  f o rthe last  means  of  only  two  i n t h e manner such  a  shape  o p e r a t i o n s . Both t h e  37  and  kinematic  the  constraints  dynamic  are c o n s i d e r e d to  check the f e a s i b i l i t y of the assumed p r o c e s s . The c o n t r o l i s based  on determining the r o l l  in such a way exceed  the  maximum  closer,  material to  f o r both passes  that the incremental v a l u e s of  l e a d to a new be  gap v a r i a t i o n  allowed  rolled in  do  not  by the c o n s t r a i n t s . T h i s  will  s t a t e which may  shape  to  the  draft  be d i f f e r e n t , but  desired  geometry.  will  The  new  i s then s u b j e c t e d to the next p a i r of passes aiming  reach  the  continues  until  constraints  are  desired  shape.  a l l the  The  described  requirements  satisfied.  This  design of a number of passes which  procedure  related  eventually  to  the  leads to the  i s frequently  more  than  two.  3.3  COMPUTER  A  SOFTWARE  computer program has been developed to c a l c u l a t e the  information variable  for  the  rectangular  output i n d i f f e r e n t graphical  multi-pass  rolling  cross-section.  of  The  parts  with  r e s u l t s are then  formats f o r d i f f e r e n t purposes  such  as  v i s u a l i z a t i o n , o p e r a t i n g c o n t r o l , user r e f e r e n c e ,  e t c . R e f e r r i n g to F i g . 3 - 9 ,  the computer program c o n s i s t s of  four major modules: Data  Gene  Multi-Pass Two  and  Curve-Fit  r at  or  Process  Planner  Three-Dimensi and  onal  Micro-Processor  There are a l s o a number  of  user  Graph  Generator Data  directed  Generator  programs  which  38  serve to p r o v i d e o r g a n i z e d  data  as  input  for  the  above  modules. T h i s chapter d e s c r i b e s very b r i e f l y a  general  determined  case,  to  of a non-uniform  part, i s  data  for  GENERATOR  data  multi-pass  rolling  the s o l u t i o n  by the program.  3. 3. 1 DATA  The  i.e.,  how  generator process  provides planner.  necessary  t he  These data are c a t e g o r i z e d as  follows: (i)  Information about the p r o p e r t i e s of the m a t e r i a l : coefficient  of thermal  modulus of e l a s t i c i t y Y i e l d p o i n t at r o l l i n g (ii)  expansion. at r o l l i n g  temperature.  temperature.  Information about the r o l l i n g c o n d i t i o n : Rolling  temperature.  Ambient  temperature.  Frictional  condition  m a t e r i a l and (///^Information  the  between  the  surface  of  the  rollers.  about the i n i t i a l  and  final  shapes:  V a r i a t i o n of width and height of the i n i t i a l m a t e r i a l versus  i t s length.  Variation versus (iv)  of  width  and  height of the d e s i r e d p a r t  i t s length.  Information about the process Minimum, the r o l l s '  maximum radii.  and  constraints:  incremental v a l u e s allowed f o r  39  Values of c o n v e x i t y c o n s t r a i n t s . . Values of kinematic and dynamic (v)  Information about the format  constraints.  of the output  results:  P r e c i s i o n of c a l c u l a t i o n s . Number of d i s c r e t e data p o i n t s i n the output. The  user inputs the above i n f o r m a t i o n a c c o r d i n g to the  c o n d i t i o n of the manipulates,  underlying  organizes  problem.  and  The  passes  data  these  generator  data  m u l t i - p a s s planner's data-read f i l e . For example, generator  uses  the  values  of  the c o e f f i c i e n t  expansion, modulus of e l a s t i c i t y and determine  should  be  multiplied,  e f f e c t s of the thermal expansion of  rolled in  data  of thermal stress  length  of  to roll the  order to c o n s i d e r the  and the e l a s t i c  deformation  the m a t e r i a l .  3. 3< 2 MULTI-PASS  The  uses  radii during  phase  in  the  program  hierarchy  i s the  of the process d e s i g n . T h i s i s the main part which  the  determines values  PLANNER  next  procedure  to  yield  the  the c o r r e c t i o n f a c t o r s by which the computed  gap v a r i a t i o n and the corresponding material  the  to the  data  generated  the number  for  the  of  variation  by  the  passes of  data  needed, the r o l l  generator, the  and  incremental  gap and the r o l l s '  f o r each pass. Messages are a l s o provided to be output the  execution i n d i f f e r e n t  the designed  s i t u a t i o n s or as a guide  f e a t u r e s at the end of the e x e c u t i o n .  40 As  was  shown  earlier,  there  exist  at  most  four  s o l u t i o n s to each problem. The program s e l e c t s one case at a time  and  performs  computations  start  the  related  with  working  computations.  These  on the f i n i s h i n g passes.  Knowing the v a l u e s of the c o n v e x i t y c o n s t r a i n t s , the program first  calculates  the  appropriate  c a l c u l a t e s the dimensions pass. R e c a l l i n g  of  inequalities  the  rolls'  radii.  material  (3-14) and  entering  k  (3-19)  k  REDf /R <Cl k  W ( =I k  2)' ^  k  s  ^  fc  each  (3-15)  R .REDf .Vl <C2 k  I t then  (3-20)  k  l a r g e s t p o s s i b l e value of the width  e  of the m a t e r i a l f o r each pass, REDf / -j k  2)i i s the a l l o c a t e d  k  each one of the f i n i s h i n g Rk(k=l,2)>  ^  s  fc  ^  roll's  e  reduction  (draft)  for  passes, radius.  Rearranging the above i n e q u a l i t i e s f o r R^ g i v e s R >RED /Cl k  fk  R <C2/(RED Vl ) k  R/ = ( R E D f / C j ) k  and  fk  R =C2/(REDf U  which bracket the r o l l ' s  k  W)  k  are  k  the two  boundaries  r a d i u s . I t i s c l e a r that R/  be l e s s than or equal to R . u  The  roll's  should  r a d i u s f o r each pass  i s then found as the minimum p o s s i b l e v a l u e , i . e . , R =RED /Cl k  fk  Note that small r o l l e r s l e a d to a decrease i n torque and  the  required  load.  The c a l c u l a t e d r o l l ' s r a d i u s i s then m o d i f i e d 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 . T h i s i s done  41  by  rounding  up  the  value  of R^ to the nearest a v a i l a b l e  s i z e . A c o n t r o l i s necessary to check whether the new ^  still  value  of  ^k{k=l,2)'  any  stage of c a l c u l a t i o n s , the a p p l i e d c o n d i t i o n s c o u l d  be  fully  s  satisfied,  between the s t a t e d boundaries. I f at not  the program r e t u r n s with an a p p r o p r i a t e  messages and some g u i d e l i n e s . Knowing the  the  finishing  entering  Referring  passes,  each  calculation  values  pass  of the d r a f t s and r o l l s ' the  can  be  dimensions determined  of  radii for  the  using  material  the  inverse  method:  to  F i g . 3-8,  f o r the  second of the f i n i s h i n g  passes the f o l l o w i n g r e l a t i o n s h i p s hold  Hmj=Hf j+REDf  (3-21'a)  2  LnjWfj/Wmj) Ln(Hmj/Hf j )  -B(Wmj/Hmj) (Hmj/R ) ((Hmj-Hfj)/Hmj) f.g.j c  e  D  E  2  (3-21b) Lmj.Hmj.Wmj=Lfj.Hfj.Wfj  (3-21c)  ( 3 = 1, 2, 3, . . . , n)  T h i s set of equations i s s o l v e d simultaneously the  dimensions  to  find  of the m a t e r i a l e n t e r i n g the second pass or  l e a v i n g the f i r s t  pass. The same procedure can  be  followed  to f i n d the dimensions of the m a t e r i a l e n t e r i n g the f i r s t of the  finishing  passes: Wi 3=Vlm3+REDf  Ln(Hmj/Hij) Ln(Wi j/Wmj)  -B(Hij/Wij) (Wij/R,) ((Wij-Wmj)/Wij) f.g.j C  e  (3-22a)  }  D  E  (3-22b)  42 Lij.Hij.Wij=Lmj.Hmj.Wmj  (3-22c)  (j = l, 2, 3, . . . , n)  The  material  with  the incremental dimensions  of W i j ,  H i j , L i j , i s now assumed to be the d e s i r e d part which i s t o be  shaped  initial  w i t h i n an unknown number of passes from the known material.  determination  of  The the  numerical  above  approach  for  the  process w i t h i n two passes was  shown i n s e c t i o n 3.1. The r o l l s ' p a i r of passes are determined  radii  necessary  f o r the  by r e a p p l y i n g the i n e q u a l i t i e s  (3-19) and (3-20) R >DRAFT /CI k  (3-23)  kJ  R <C2/(DRAFT j k  stands  k  (3-24)  k  2) , j = U, 2, . . , n)  k=(l,  DRAFT j  .VI j )  k  for  the  incremental  value  of the  a b s o l u t e d r a f t d u r i n g r o l l i n g and i s unknown. Vl j  stands f o r the incremental value of the  k  the m a t e r i a l under the r o l l e r s  radius  following  and  inequality  the  larger  2  k  which g i v e s the lower l i m i t R  for The  2) ^  s  is  the  d e s i r a b l e , the  .Vl ) k  on  as  k= C2/(C1  M)  /  v  ^k{k=l  draft  that  i s derived R >C2/(Cl  where  of  at each pass.  Combining the two i n e q u a l i t i e s and having i n mind smaller  side  k  (3-25)  chosen as the maximum value of the width  each pass. values  of  rolls'  radii  then s e l e c t e d as mentioned  f o r each pass, ^ k ( k = l , 2 ) ,  earlier.  a  r  e  43 A f t e r the process has been determined 3.1.2),  it  dynamic  constraints.  calculated  is  e v a l u a t e d with r e s p e c t to the kinematic and This  incremental  p e r m i t t e d by the permissible  ( r e f e r to s e c t i o n  is  done  values  kinematic  incremental  by  for  and  comparing  the  drafts  dynamic  reduction  the  to  those  constraints.  The  can  then  be  The v a l u e s of the c a l c u l a t e d d r a f t s should be l e s s  than  or  s p e c i f i e d by the f o l l o w i n g  (PRED ) kj  inequalities  PRED <C2/(R .Vl kj  k  PRED <Cl.R kj  k=U,  equal are  2)  )  kj  k  , j = (l, 2,. . . ,n)  to the p e r m i s s i b l e v a l u e s . The v i o l a t i n g d r a f t values  reduced to meet t h i s A f t e r the f i r s t  constraint.  pass i s m o d i f i e d , dimensions  i n t e r m e d i a t e m a t e r i a l are then used to  of the  new  both  the  re-define  p e r m i s s i b l e incremental and c a l c u l a t e d d r a f t s f o r the pass. M o d i f i c a t i o n modified  values  of d r a f t s f o r a pass are too s m a l l ,  not be worthwhile unless the  it  is  to  to a l l o c a t e a  the  pass  is  next  major  t o t a l number of the passes as i t might  deleted  for  that  and  the  the  i t may  operation, case  process  is  pass. T h i s i m p l i e s that  the  i s not n e c e s s a r i l y an even number  be deduced at the f i r s t i n s t a n c e .  D i r e c t or forward s o l u t i o n dimensions  pass  If  one of the l a s t p a i r of passes. In t h i s  corresponding  continued  i s then done on the second pass.  second  is  used  to  determine  the  of the m a t e r i a l l e a v i n g the m o d i f i e d passes. T h i s  i s done by s o l v i n g sets of equations l i k e  those  in  (3-21)  44  and  (3-22) r e p e c t i v e l y  substituted  for  ,  and  REDJ-J  (3-22a). The dimensions then  providing REDf  that  RED  satisfied,  is  repeated  i.e., until  modification.  until  there  is  in  are v a r i a b l e ,  that  program  conditions  (iii)  satisfied.  In  both  initial  a  pair  of  cases,  of  section the  than  and the f i n a l  increments  until  execution  segments f o r which 3.1.1.3  program  are  not  automatically  roll's  those  s a t i s f i e d . Then, i f the new value of the less  pass  gap c a l c u l a t e d  i t may happen d u r i n g the  (iv)  these  the  i n c r e a s e s the value of the corresponding allowable  for  of the m a t e r i a l .  detects or  fully  expansion  which  dimensions  are  the e f f e c t s of the thermal  and the e l a s t i c deformation  some  need  passes.  factor  take  account  of  correction  to  the  pair  no  each pass are then m u l t i p l i e d by the  cases  (3-21a) and  constraints  for  For  e  of a new i n i t i a l m a t e r i a l  The incremental v a l u e s of r o l l  into  D  of the new i n t e r m e d i a t e m a t e r i a l are  i n t e r p r e t e d as the dimensions  procedure  (k=1,2)  i n equations  2  which i s to be r o l l e d w i t h i n a subsequent The  kj  radius  c o n d i t i o n s are  roll's  radius  the maximum a v a i l a b l e , the execution s t a r t s  the beginning using the new  roll's  by  radius;  is from  otherwise,  it  r e t u r n s back with a message with regard to the s i t u a t i o n and some g u i d e l i n e s . The  multi-pass  design r o u t i n e completes  by p a s s i n g the i n f o r m a t i o n and the proper  format  f o r the  related  i t s execution  messages  in a  user's r e f e r e n c e . A l s o , i t outputs  some r e l e v a n t data i n t o other f i l e s f o r d i f f e r e n t  uses  such  45 as  graph  generations.  multi-pass  design  3. 3. 3 GRAPH  information  of  the r o l l e d  ,three~dimensional  ,two-dimensi  as  onal  visualization  incremental  coordinates projected  the  during  the  operations  Three-dimensional graphic  values  the  of  of  visual  graphics visualization  two  plotting  width  and  these  Two-dimensional  3. 3. 4 OTHER  and v a r i a t i o n of  uses  serves t o provide a a  each  c o n s i s t s of three r o u t i n e s . The f i r s t  the shape.  values  dimensional  gap  after  by the three-dimensional  material(s) of  on  visualization,  roll  i s generated  intermediate  parts  visualization.  which  generator  are  of  i s shown i n F i g . 3-10.  algorithm  some parameters such  either  flow-chart  a r e a l s o s e v e r a l r o u t i n e s which h e l p to generate  visualized  the  general  GENERATOR  There  operation  The  and  height  generates  The  next  of  the  incremental  routine  f i n d s the  c o o r d i n a t e s . The t h i r d  routine  representation  of  the o b j e c t  terminal  on  a  or  plotter.  i s a l s o p o s s i b l e by u s i n g  any  package.  ROUTINES  Some  miscellaneous  routines  used  f o r the purpose  of  have been p r o v i d e d operation  a p p l i c a t i o n and the a l g o r i t h m i c procedures are d e s c r i b e d i n the f o l l o w i n g s e c t i o n .  control. of these  which The  routines  46  3.4  OPERATING ASPECTS  In  the  previous  v a r i a t i o n of the r o l l for  each  pass  sections,  i t was  shown  gap as a f u n c t i o n of the r o l l e d  the  and  roll  used.  gap  form f o r  The  to simulate the a c t u a l c o n d i t i o n s . Appendix B curve-fitting  algorithm.  The  curve  (position a  3-11). P a r t s with a high degree of geometric while the r o l l s are f o l l o w i n g a  over-rolled  order  to  prevent  t h i s , the r o l l e r s  path. However, t h i s w i l l  result  describes  r o l l e r s a r e supposed to  f o l l o w t h i s curve during the o p e r a t i o n  in  length  a c o n t i n u i t y up to the second order t o be smooth enough  the  In  of  r o l l e d l e n g t h . To d e r i v e an a n a l y t i c a l  t h i s r e l a t i o n s h i p , c u r v e - f i t t i n g has been has  the  i s determined i n the form of d i s c r e t e data  p o i n t s . F i g u r e 3-11 shows a t y p i c a l v a r i a t i o n versus  how  in F i g .  changes, may be certain  curve.  should a d j u s t  i n under-rolIing  their  (position b  F i g . 3-11). I t i s c l e a r that smaller r o l l s permit  with high v a r i a t i o n Roll's limiting program  i n geometry to be manufactured.  r a d i u s can be decreased  the r o l l ' s or  by  radii  the  i n many ways; by e i t h e r f o r the  kinematic  use  and/or  the value of the dynamic  by the dynamic  constraint  t o the s e l e c t i o n of a s m a l l e r r a d i u s but on the other  hand, i t may Provision  has  increase  the  number  of  operations  needed.  been made i n the program so that t h i s can be  done a r t i f i c i a l l y value  available  changing  c o n s t r a i n t s . Decreasing leads  parts  and i n an o p t i m i z e d  of the kinematic  of s m a l l e r r o l l ' s  way.  Increasing  the  c o n s t r a i n t w i l l a l s o l e a d to the use  r a d i u s but i t may  lead  to  a  need f o r  47  forward t e n s i o n . The  r o l l i n g machine should have a f a c i l i t y  f o r changing  the r o l l  gap as a c c u r a t e l y and as f a s t as p o s s i b l e . R o l l gap  control  based  a convenient based of  on the displacement  of the r o l l e d m a t e r i a l i s  way t o c o n t r o l the p r o c e s s . C o n t r o l  on other parameters such as time w i l l  lead to the use  e m p i r i c a l r e l a t i o n s which are not suggested calculation  2.1.2, roll  of  gap and the r o l l e d  during  rolling  speed).  scales.  (see s e c t i o n  Both v a l u e s of the  l e n g t h can be d e t e c t e d and read  the r o l l i n g . b y v a r i o u s d i g i t a l  as linear  strategies  out  read out systems such  The a p p l i c a t i o n of two  different  control  each i n d i v i d u a l output  variable  systems i s b r i e f l y d i s c u s s e d here.  (i)  Analog  With  analog  (actual r o l l are of  Control  System  control,  gap and r o l l e d length)  i s monitored.  then made i n the corresponding the servomotor or opening  maintain  the  output  at  a  of  input v a r i a b l e  the h y d r a u l i c  desired  level.  Changes (rotation  valve)  This  i s done  c o n t i n u o u s l y . The general block diagram of such a system shown  to  is  i n F i g . 3-12. As i s seen, the d i f f e r e n c e between the  d e s i r e d value and the a c t u a l value i s used  by  the  analog  c o n t r o l l e r . The mechanism by which the process v a r i a b l e s are a l t e r e d , depends on the p a r t i c u l a r the  machine.  variable during  in  The  desired  system and the design  value  itself.  It  should  the process  as  a  i s not be  function  a s e t p o i n t but  continuously of  of  followed  the a c t u a l  rolled  48  length. gap  So,  a  template  variation  and  proportionally  to  a  with the shape of the d e s i r e d r o l l  mechanical  follower  the l e n g t h  (Fig.3-12) are needed.  The  of  the  movement  of  which rolled  the  moves material  follower i s  s c a l e d and read out by a l i n e a r s c a l e measuring d e v i c e . As  i t can be deduced, i n the analog c o n t r o l ,  process control  a  different  i s hardwired  template  f o r each  i s needed. Besides, analog  and can not be e a s i l y changed when the  design i s changed. T h i s s t r a t e g y can a l s o be c a l l e d model  a mechanical  of  because a template  control,  of the d e s i r e d p r o f i l e and  f o l l o w e r a r e used to c o n t r o l the r o l l gap.  (ii)  Direct  Due  to  Digital  Control  (D.D.C.)  the i n c r e a s e d complexity and the i n f l e x i b i l i t y  the model  control  micro-computers,  systems,  namely  D irect  new Digital  f  methods  based  a  micro-computer.  In  a  digital  (actual  rolled  a  periodically  with  roll  gap and  sufficient  sampling  frequency. F i g . 3-13 shows the g e n e r a l block t y p i c a l D.D.C. each time and  The micro-computer  interval,  the r o l l  diagram  of a  i s programmed such that at  the a c t u a l v a l u e s of the  rolled  length  gap a r e read out. The recorded v a r i a b l e s a r e  then used by the micro-computer roll  controlled  control strategy a  micro-computer samples the v a r i a b l e s length)  on  have become  Cot r ol  i n c r e a s i n g l y r e l e v a n t . More than one loop may be with  adaptive  gap f o r the next  to  determine  the d e s i r e d  increments. The c u r v e - f i t t i n g program  i s used t o organize data i n a f a s h i o n s u i t a b l e f o r  the use  49 by the  micro-computer.  The adaptive  data  continuous data  o n D.D.C i s s o m e t i m e s  c o n t r o l s t r a t e g y based because  control  data  i nt h i s  transmission  from  i s fed to the c o n t r o l system.  programming  (software  and m o d e l l i n g computer system,  possible. Micro-computers  Moreover,  can b e e a s i l y  process a sthe design changes  3.5  SUMMARY  method  unconstrained  o f rolling  two-pass  a s (3-9)  the dimensions (3-9)  LnjWfj/Wmj) Ln(Hmj/Hf j)  reprogrammed  occur  2 7  A  i s seen  study  fast  for  a  .  o f parts  with  first described for  process.  This  and  variable a  simple  was followed b y equations  (3-10), to f i n d the incremental values o f  of  t h e intermediate  and  material.  -B(Hij/wij) (Wij/RW) ((Wij-Wmj)/wij) f.g.j c  e  D  E  -B(Wmj/Hmj) (Hmj/RH) ((Hmj-Hfj)/Hmj) f.g.j c  e  Rearranging  (3-10) a s f o l l o w s  (1 = 1,2,  a n d Wmj,  a  and  Ln(Hmj/Hij) L n ( W i j/Wmj)  it  wiring-up  with  numerically solving a resultant set o fnon-linear  equations  D.D.C,  AND EVALUATION  r e c t a n g u l a r c r o s s - s e c t i o n was  such  discrete  c o n t r o l of s e v e r a l machines i s  new  The  the template,  By t h e u s e o f  design).  simultaneous  instead o f  is substituted for  design)  (hardware  method  called  D  . . .  E  ,n)  t h a t x a n d $ a r e f u n c t i o n s o f two four v a r i a b l e parameters  unknowns, Hmj  H i j , W i j , H f j and  Wfj.  o f the b e h a v i o r of the above f u n c t i o n s shows t h a t  50 they and  both  are  continuous with respect  the four parameters  existence. section i.e,  Having  rolled  physical  solution  and Hmj, (3=1,  length  (3 = 1, 2, ...,/?),  This  the  for  3.1.1.3), i t i s deduced that  Wmj  length  a  in  the set  providing  that  are  continuous  that  the  of  each segment  their  ( r e f e r to  of  solutions,  W i j , H i j , Wfj with  (see 26, pp 209 to 213, "inverse implies  range  are continuous versus the  2 , . . . , n),  also  t o the two unknowns  solution  and  Hfj,  respect  to the  function  of  theorem").  the deformation of a  continuous shape to another, i . e . , the i n t e r m e d i a t e  material  i s a l s o continuous i n shape. The  method was then extended to i n c l u d e  physical constraints. l e a d t o an increase the  I t was d i s c u s s e d  i n the r e q u i r e d  and s a t i s f y the  how these  number  of  constraints passes  from  i d e a l minimum of two. For example, the i n c l u s i o n of the  convexity c o n s t r a i n t s ,  alone,  immediately  increases  this  number to f o u r . Normally, the i n i t i a l block.  A  suitable  material  initial  used would be a  material  cross-sectional  dimensions  are  cross-sectional  dimensions  of  equal the  f i n i s h i n g passes. R o l l ' s r a d i u s p l a y s the  magnitude  of  spread.  It  is  may to  material  be  uniform  one  the  whose largest  p r i o r to the  an important  role  in  also a f l e x i b l e tool in  making the deformation of more complex shapes p o s s i b l e . The discussion described.  last  part  of the chapter was a l l o c a t e d to a b r i e f  of o p e r a t i n g The  a s p e c t s . Two c o n t r o l s t r a t e g i e s were  control strategy  based on the r o l l e d  length  51 i m p l i e s the f a c t that the m a t e r i a l pass before between  entering  the  next,  should  otherwise  but  considerably 16) .  the  difference  the e x i t and the entrance v e l o c i t y , not only  a v a r i a b l e t e n s i o n or compression along block,  completely leave a  also difficult  makes  the  the  process  ( f o r more d e t a i l s ,  length  causes of the  of measurement  see  33  pp  9  to  4. EXPERIMENTAL EVALUATION 4.1 INTRODUCTION The  t h e o r e t i c a l a n a l y s i s of r o l l i n g p a r t s with 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 presented t h i s stage the method to  is  need  i n chapter t h r e e . At  f o r experimental  evaluation  of the  sensed. The purpose of doing the experiments  was  examine the p h y s i c a l aspects of the method d i s c u s s e d , and  to  evaluate  its  accuracy  in  predicting  the  process  behaviour. The o b j e c t i v e s were: 1.  To e v a l u a t e the c a p a b i l i t y of the spread formula, used,  i n p r e d i c t i n g the geometric  m a t e r i a l d u r i n g unsteady implement the necessary 2.  of the  r o l l i n g and t o determine and corrections.  To e v a l u a t e the v a l i d i t y of the computer a l g o r i t h m i n determining The  of  deformation  first  the process  specifications.  o b j e c t i v e r e l a t e s t o the f a c t that f o r most  the o p e r a t i o n s of the process  incoming  material  is  under  non-uniform  and  investigation, the  roll  the  gap i s  s u b j e c t e d to v a r i a t i o n . E l - K a l a y and S p a r l i n g ' s formula, stated,  i s best s u i t e d f o r the e s t i m a t i o n of spread  that  the  material  known  r e s u l t s of the d i r e c t a p p l i c a t i o n of t h i s  the g e n e r a l s i t u a t i o n where the dimensions and  the r o l l  formula  the  input  gap are v a r i a b l e would be i n c e r t a i n  e r r o r , and that some c o r r e c t i v e from experiments  of  as  i n the  c o n v e n t i o n a l constant t h i c k n e s s r o l l i n g . Thus, i t was  to  as  factors  could  and i n c o r p o r a t e d i n the formula.  52  be  derived  53  The second o b j e c t i v e i s b a s i c a l l y validation  of  experimenets different  the computer  were  rolling  although,  for  the  program. In t h i s regard, some  performed  on  uniform  blocks  under  conditions.  Most of the experiments steel;  the t e s t  there  were  were  conducted  some  on  hot m i l d  experiments which were  performed on warm aluminum.  4 . 2  E X P E R I M E N T A L  4. 2. 1  ARRANGEMENTS  INSTRUMENTATION  The  rolling  mill  used  f o r experiments  was a small  l a b o r a t o r y unit, (see F i g . 4-24). The nominal torque motor  used,  was  4.77Kgf-m.  available, specifically: The  rolls'  radii  c o n d i t i o n s . The upper adjusting and a  the r o l l  power  translated  50.8mm  roll  could  68.5rev/min.  and with d i f f e r e n t be  moved  surface  vertically for  gap. T h i s was done through a hand-wheel  screw. into  and four angular speeds were  17.2, 34.3, 51.4 and  were  of the  one  Sixteen inch  rotation vertical  r o l l . Due to wear on the s i d e s of  of  the hand-wheel  movement of the upper  the threads,  the upper  i n the v e r t i c a l  direction.  Due t o the misalignment of the b e a r i n g s , a s l i g h t  horizontal  roll  slack  had some slack or backlash,  also  existed.  P r e l i m i n a r y experiments with aluminum  were performed t o measure the amount of the v e r t i c a l  slack.  T h i s was found t o be 0.6mm i n the range of r o l l  gaps between  10mm t o 25mm. The e x i s t a n c e of the h o r i z o n t a l  slack  could  54 have caused  the r o l l e d m a t e r i a l to twist  plane;  prevent  to  this,  a guide way  in  the  horizontal  or j i g was  built  and  i n s t a l l e d at the e n t r y p o i n t . The a b s o l u t e maximum r e d u c t i o n that  a heated s t e e l  s i n g l e pass, was wide  experimented  to be about  5mm  for  a  25mm  specimen. The  dimensions  "600x400x400mm". been  s l a b c o u l d have been s u b j e c t e d t o , in a  reached  distance  of  The  with  the  maximum this  furnace  used  temperature  furnace  was  were  that c o u l d have  about  1200°C.  between the furnace and the r o l l i n g m i l l was  The about  two meters (see F i g . 4-25).  4. 2. 2 SPECIMENS  AND  MEASUREMENT  S u i t a b l y long specimens were used the  effect  of  the  modes  experiments (regular  of with  of  for  deformations hot  steel;  small  isolate  were double  specimens .  Two  23  observed bulged  or i r r e g u l a r ) c r o s s - s e c t i o n s . F i g . 4-1  two and the procedure  to  e x c e s s i v e spread at the f r o n t and back  ends which can be s i g n i f i c a n t major  in order  during and  rhombic  shows these  f o r the c a l c u l a t i o n of the mean  spread. P r e l i m i n a r y experiments  the  value  on the constant t h i c k n e s s  r o l l i n g of s t e e l showed that a good agreement e x i s t s between the  results  formula  of  the experiments  and those p r e d i c t e d by the  (2-18) p r o v i d i n g that c o n d i t i o n s of rough r o l l s  and  heavy s c a l e s are assumed (see t a b l e 2-6). To o b t a i n the  roll  settings,  the  m a t e r i a l was  required  final  thickness  of  the  rolled  m o d i f i e d by adding the thermal expansion of the  55 bar.  The thermal expansion  working  temperature  thickness.  The  of  from the room temperature  1100°C  effects  of  under the r o l l s , and the m i l l was  was  1.48%  t o the  of ,the  exit  e l a s t i c deformation of the bar s p r i n g on the e x i t  known t o be very small f o r hot s t e e l  thickness,  ( l e s s than 0.1% of  the e x i t t h i c k n e s s ) and thus was n e g l e c t e d . 2 2  4.3  STEADY STATE ROLLING  The constant  r o l l i n g of uniform p a r t s when the r o l l i s r e f e r r e d t o as the steady  s e c t i o n some t y p i c a l experiments d e s c r i b e d . The purpose  state  gap i s kept In t h i s  rolling.  on steady s t a t e r o l l i n g a r e  of these experiments  were t o e v a l u a t e  the computer a l g o r i t h m and t o 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", r o l l e d on one s i d e down re-heated of  18mm.  The  specimen  rolled  19.70mm  ( F i g . 4-3a).  Next,  i n a s i n g l e pass t o the same  i n c r e a s e d width was measured  operations  different  then  same  s e t of  was  a s i m i l a r block was  thickness  of  15.55mm.  19.78mm ( F i g . 4-3b), which  was g r e a t e r than the value o b t a i n e d i n the f i r s t The  was  and r e - r o l l e d on the same side t o a new t h i c k n e s s  15.55mm. The i n c r e a s e d width a f t e r these  measured  The  to  180mm long, was  experiments  were  experiment.  performed  s i z e s of b l o c k s and s i m i l a r r e s u l t s were  with  observed  which i n d i c a t e d t h a t , f o r a given t o t a l d r a f t , the magnitude  56  of  spread  decreases  as  the number  P h y s i c a l l y , t h i s i s due t o the f a c t d r a f t , the higher  shows  that  a reproduction  increases.  the higher  the spread; a l s o , the smaller  the m a t e r i a l under the r o l l s , 4-3  of passes  the  the width of  the l a r g e r the spread.  Figure  of the r e a l c r o s s - s e c t i o n s  of the  specimens at each stage of the above experiments. I t i s seen that  when  the d r a f t i s s m a l l , the d i s t o r t i o n on the spread  side i s also small. This have  a  passes  flatter  surfaces  in the multi-pass  4. 3. 2 THE EVALUATION  A uniform block rolled  within  i m p l i e s that the r o l l e d  two  // the d r a f t s  process  parts  in the last  are kept  OF THE MULTI-PASS  ALGORITHM  of  "25.4mmx25.4mm"  cross-section  operations  and  of  small.  was  reduced  c r o s s - s e c t i o n a l dimensions t o "20mmx20mm". The value roll  pair  can  was  i n new of the  gap f o r each pass was p r e d i c t e d by the program and the  roll  gap was s e t t o that value  and  the c r o s s - s e c t i o n a l views of the m a t e r i a l a t each stage  are reproduced i n F i g . 4~4a. is  illustrated  beforehand.  The  dimensions  A l s o , the output of the program  i n F i g . 4-5.  In the second experiment, the m u l t i - p a s s  design  concept  was  a p p l i e d and two f i n i s h i n g passes with a 2mm of r e d u c t i o n  for  each, were s p e c i f i e d . The c r o s s - s e c t i o n a l shapes of the  rolled  material  prediction  after  a r e provided  respect i v e l y .  each in  pass, as w e l l as the computer figures  4-4b  and  4-6,  57  It  i s seen that a p p l y i n g the concept  c o n s t r a i n t s t o the process  of the c o n v e x i t y  improves the appearance  of the  f i n i s h e d part noticeably.  4.4  UNSTEADY STATE ROLLING When  the m a t e r i a l  t o be r o l l e d  the r o l l  gap changes d u r i n g  termed  unsteady  operations non-uniform  the r o l l i n g ,  state  involved  rolling.  It  of  to  the  of  a r e of an unsteady  No p r e v i o u s work has been done  evaluate  state the  f o r the deformation  rolled  within  not only the r o l l rolled, and  nature.  geometric  of  non-uniform  a certain gap v a r i e s  roll  planning  parts, block  each which  gap. In r e a l i t y , however,  while  the segment  i s being  but a l s o the segment i s a f f e c t e d by the succeeding  the preceding segments  segment  in question.  material  encounters  different  from  experiments unsteadiness  In a  that  which  are d i f f e r e n t  of  deformation  the steady  were needed t o e v a l u a t e on  the spread.  following situations The width  of  any  from  the  other words, each p o r t i o n of the  history  of  takes p l a c e , whenever  1.  with  of the m a t e r i a l i n t h i s type of r o l l i n g .  segment i s assumed to be a member of a uniform is  is  rolling  parts  was d e t a i l e d e a r l i e r that i n the process  scheme  and/or  the process  Most  i n the p r o d u c t i o n  cross-section  deformation  i s non-uniform  one  state  r o l l i n g . Some  the e f f e c t s  Thus, unsteady or  which i s  a  of  state  combination  such  rolling of the  exist the  material  being  rolled  is  variable.  58  2.  The height  3.  The r o l l The  each,  experiments.  4. 4. 1 EFFECTS To of  each  The  ON  v a r i e d from  specimen;  specimen;  of  concave  Sides laterally  225mm  were  and  the  types same  The h e i g h t  in  different types of  the l e n g t h as f o l l o w s  5 x 1 0 " ( L E N G T H ) + 12 4  (mm)  2  8(LENGTH) + 12  (mm)  parabolic 1 . 2|/ (LENGTH) + 1 2  from t h e narrower  5mm  specimens  three  for the three  ( s e e F i g . 4-26), w i t h  three  the  taper  convex  f o r each  with  (see F i g . 4 - 7 ) .  marked  variation  constant  variation,  o f t h e s p e c i m e n s were  height  along  i n the  parabolic  straight  i s measured  constant  the  SPREAD  of  height  versus  HEIGHT= length  c a n be  evaluate  12mm t o 30mm b u t  HEIGHT=0.0  The  variable.  i n the following s e c t i o n s .  of height  HEIGHT^3.5  3rd  is  to  experiments  t h e same l e n g t h  variation  order  two were kept  VARIATION  the e f f e c t  specimen;  2nd  in  other  s p e c i m e n s were f o r m u l a t e d 1st  rolled  rolling.  o f 12.8mm were u s e d  specimen  patterns.  Thus,  important  OF HEIGHT  evaluate  width  during  are presented  specimens with  width of  the  These  obtained  being  o f t h e above c o n d i t i o n s on s p r e a d ,  independently.  of  results  gap changes  effects  evaluated effect  of the material  specimen  rolled  head. logitudinally  intervals.  i s shown  from  (mm)  The r a t e o f  i n F i g . 4-8. A l l  t h e n a r r o w e r head  t h i c k n e s s o f 12.9mm. The p l o t s  of  and  the  to a  experiments  59 and  the p r e d i c t e d spread versus  the corresponding  initial  height a r e shown i n f i g u r e s 4-9, 4-10 and 4-11. R e f e r r i n g to f i g u r e s 4-8 and 4-10, which it  has  a constant r a t e of height v a r i a t i o n  i s seen that when the i n i t i a l  the  absolute  draft  simulated v a l u e s absolute d r a f t from  conventionl  It  in a  height i s s m a l l , i . e . , when  good  calculated  (unmodified)  agreement,  unmodified  by  spread  El-Kalay  formula. The maximum  (/)  shows  values  larger than  even  that  to  from  the  is  seen  from  Fig.  4-11,  f o r small d r a f t s , the measured v a l u e s a r e l a r g e r predicted values. This  f o r the specimen  (iii),  i s due  the  rate  to the  of height  i s l a r g e r on the head end.  i s deduced that f o r low d r a f t s ,  low r a t e of height v a r i a t i o n s ,  El-kalay  deviation  r a t e of height i n c r e a s e i s l a r g e r f o r specimen  than the unmodified  for  error  does the specimen (//),  (/' ) ( r e f e r t o F i g . 4-8). I t i s a l s o  It  and S p a r l i n g ' s  f o r the d r a f t v a l u e s g r e a t e r than 3mm. T h i s  the  variation  as the  from f i g u r e s 4-9 and 4-10 t h a t , f o r the same  predicted  Particularly  fact  but  to be more than 3%.  d r a f t , the specimen  that  (0.08mm/mm),  i n c r e a s e s , the a c t u a l v a l u e s begin t o d e v i a t e  i s seen  because  (//)  i s s m a l l , experimental values and the  are  the values  was found  f o r specimen  l e s s than 2.0mm, and  less  than  0.05 mm/mm,  and S p a r l i n g ' s p r e d i c t i o n s a r e reasonably  be used, but as the a b s o l u t e d r a f t  height v a r i a t i o n  and/or  i n c r e a s e s , the a c t u a l spread  accurate  the r a t e  of  i n c r e a s e s such  that the c o n v e n t i o n a l formula begins t o d e t e r i o r a t e  in  its  60 p r e d i c t i v e , a c c u r a t e l y . The d e v i a t i o n i s than  5% i n some An  is  that  the a r c of  lateral  f o r the effect of  the  spread  be  more  height  on  variation  the height v a r i a t i o n affects the length of  contact  determining  to  instances.  explanation  spread  seen  which  value  is  of  an  important  spread.  Hill  d e p o n d s i n v e r s e l y on t h e  parameter  in  s h o w e d how t h e  1 8  length  of  arc  of  contact: (i) -  when  i / R . A h — t h e  lateral  elongation  is  maximum. (ii)  -  when  occurs,  i/R.Ah—»0,  condition  i.e., the lateral  elongation  P h y s i c a l l y , when t h e l e n g t h o f t h e a r c the  roll's  increase material elongate  surface  in friction, becomes  of  and the material the  more on t h e l a t e r a l  actual lateral elongation  such  plane  strain  i s zero. contact  between  increases, due t o the  longitudinal  difficult,  of  that  direction.  i s l a r g e r than  elongation  of  the  material tends This  is  why  the predicted  to the  value  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 . Another  effect  of  height  v a r i a t i o n i s on t h e s t r a i n  rate which i n d i r e c t l y a f f e c t s the spread. the value  o f mean s t r a i n  MEAN  STRAIN  rate as  Sim  1 2  formulated  follows  RATE=v(R.Ah)~°' Ln(l/(l-r)) 5  v i s the peripheral v e l o c i t y of r o l l s , r i s the reduction A  study  of  the  in  behaviour  thicknessCAh/h,) of t h e above formula  shows t h a t  61 draft  i s d i r e c t l y r e l a t e d to the mean s t r a i n r a t e ; as d r a f t  i n c r e a s e s , mean s t r a i n r a t e a s s o c i a t e d with the process a l s o increases. be  hardened  hardening the  in  in  respective  more s t r a i n most  S t r a i n r a t e causes the m a t e r i a l The  degree  of  strain  l o n g i t u d i n a l and l a t e r a l d i r e c t i o n depends on s t r a i n s . Under the c o n d i t i o n s  in elongation  rolling  greater  a l l directions.  under r o l l i n g to  purposes,  resistance  to  which  than spread, which  is  strain  would  hardening  deformation  in  promote  true  for  the  cause  elongation  a  than i n  spread. A modification  of the spread formula was then needed to  improve the t h e o r e t i c a l p r e d i c t i o n by c o n s i d e r i n g of  height  the e f f e c t  v a r i a t i o n . In doing so, the f o l l o w i n g p o i n t s are  presented (a) - S p a r l i n g proposed  1 8  included  formula,  some  (2-18),  temperature  variation,  composition  (see s e c t i o n  assigning  values  coefficients, vice  less  would  f o r the  strain  rate  2.3). than  predict  of  material  suggested  unity the  effects  and  He  in his  for  that these  l a r g e r spread and  versa.  (b) - M o d i f i c a t i o n t o the such  coefficients  spread  t h a t , f o r a uniform block,  process,  i t becomes  formula  i . e . , a steady  v a r i a t i o n should  improved  Also  the a r c of contact  be  state  ineffective.  (c) - Rate of height formula.  should  should  appear  i n the  i t s e f f e c t on the l e n g t h of be i n c l u d e d .  62  (d)-  The e f f e c t of height v a r i a t i o n on spread depends  on the f r i c t i o n a l c o n d i t i o n ; the lower 2 2  between  the  rolls  and  the f r i c t i o n  the m a t e r i a l , the l e s s e r i t  a f f e c t s the value of spread and v i c e v e r s a . The f o l l o w i n g formula  i s then suggested  to s a t i s f y the above  requirements  Ln(w /w,) =A.EXP[-B(w /h ) (h /R) (Ah/h ) -T] Ln(h,/h ) m  c  1  1  D  E  1  1  2  T i s a parametric c o e f f i c i e n t which was d e f i n e d as f o l l o w s T=(1+^.Hv/R.Ah/w,)  Hv i s the a b s o l u t e r a t e of height material  (4-1 )  -D  variation  of  at the c r o s s - s e c t i o n where the spread  the i s to  be p r e d i c t e d . £ i s a c o n s t a n t ; the value of which was found 6. 15 and  to  be  f o r minimum d i f f e r e n c e between the experimental the p r e d i c t e d r e s u l t s . The method of l e a s t  f i t t i n g was used  in this  As i t i s seen, the necessary the improved  square  regard.  requirements  are  satisfied  in  formula:  (i) - When the block i s uniform,  i . e . , when Hv=0,  T=1  to  and the  formula  converts  then  i t s conventional  form. (ii) -  For  higher  r a t e s of height v a r i a t i o n , or f o r  l a r g e r l e n g t h s of the a r c of c o n t a c t , the value of becomes  smaller  and  consequently,  spread w i l l become l a r g e r .  the  T  calculated  63 (iii) in  The e f f e c t of h e i g h t v a r i a t i o n  the  El-Kalay  and  Sparling's  parameter D ( t a b l e 2-6) (iv) -  For  very  i s weighted as  formula,  using  as the power.  l a r g e v a l u e s of w  where the p l a i n  1f  s t r a i n c o n d i t i o n 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 values  formula  was  then  used  to  predict  of spread f o r the experiments performed. The  are shown i n f i g u r e s 4-9, predictions  are  in  4-10 a  and 4-11.  very  experimental r e s u l t s . The mean predictions  and  the  good  results  As i s seen, the agreement  deviation  values  VARIATION  SPREAD  was  new  with  the  the  new  between  experimental  the  found to be  w i t h i n ±0.8% .  4. 4. 2 EFFECTS  OF  In • the parameters  WIDTH  following  were  kept  ON  experiments, constant  all  expect  the  m a t e r i a l being r o l l e d . I d e n t i c a l specimens as were  width.  All  the  three  types  of  10.3mm.  4-12,  4-13  and 4-14  Fig.  4-7  interpreted as  the  thickness  of  show the r e s u l t s . For  each specimen, the v a r i a t i o n of the i n c r e a s e d initial  of the  specimens were heated to  from the narrower head to a  width  versus  width has been p l o t t e d . The p r e d i c t e d v a l u e s of  the spread width u s i n g E l - K a l a y and S p a r l i n g ' s also  in  selected  1150°C and r o l l e d Figures  relevant  width  used. The constant dimension t h i s time was  as the height and the non-uniform s i d e was  the  the  superimposed  formula  are  on each graph. I t i s seen t h a t , there i s  64 an  agreement  experiments for  and  a slight  was  within In  between  the  those p r e d i c t e d  constant s h i f t .  the  the  second  set  wider  against their  with  predicted  those  results.  m e a s u r e d and  by  larger than  was  a  the  for  4-13  4-15,  and  having  linearly  variable  effect, after  was  a  then it  v a l u e s of  width  were  compared  the p r e d i c t i o n  was  the  I t i s seen  is  A  that  slightly  smaller widths,  Figure  constant  affects width  rolled  is rolled.  i n c r e a s e and  however,  comparison shows t h a t  primary  the e x t e n t the  spread i s width  variation  specimen  a  symmetric  of t h i s  other  i s t o have a  should was  of  aluminum  to which the  s p r e a d . In  the  an  and  purpose  for  between  i t i s t o the  shows  thickness  specimen  The  4-17  the  greater  on aluminum p i e c e s showed t h e p r o o f  to observe  the  this  4-16  i n c r e a s e than  w i d t h . The  variation of  10.3mm but  and  both width  f o r example,  assertion.  specimen  experiment  from  change.  t o the width  decrease. Experiments  sign  and  type  f o r m u l a . F i g u r e s 4-15  the  figures  the  except  same  v a l u e s and  the d e v i a t i o n  doesn't  width  the  measured  the  The  reduction in spread.  deviation  recent  first.  initial  show t h a t  v a l u e s of w i d t h ,  more s e n s i t i v e  experiments,  rolled  deviation  of  was  t o a t h i c k n e s s of  relative  this  deviation  from  1.2%.  cause  the  simulation,  the  Maximum d e v i a t i o n  These experiments decrease  by  The  of  head was  were p l o t t e d  show t h e  obtained  1.7%.  s p e c i m e n s were h o t - r o l l e d time  results  rolling  condition words,  if  significant  lose  its  symmetry  rolled  t o an  absolute  65  r e d u c t i o n of about 5mm specimen maintained  at a temperature its  symmetry.  of 450°C. The  Fig.  4-18  rolled  shows  this  r e s u l t g r a p h i c a l l y . I t i s seen that the p o i n t s corresponding to  the two  diviation  symmetric near  ends  the middle  overlap  except  for  a  region where the aspect  slight ratio is  large. The spread  above experiments formula  and  cross-section, 1.7%. of this  show that using the c o n v e n t i o n a l  applying  overstimates  the  A parametric c o r r e c t i o n  height v a r i a t i o n , was  it  for  each  value  of spread by about  f a c t o r , A,  s i m i l a r to the case  then  introduced i n order to  reduce  error A= ( 1 . 0 + T J . W V )  i s the absolute  Wv  C i s chosen from  (4-2)  c  r a t e of width table  variation,  2-6,  77 i s a c o n s t a n t ; value of which was The  individual  implementation  e r r o r to l e s s than  4. 4. 3 EFFECTS  The  OF  purpose  evaluate  the  spread. As  of A i n the spread formula decreased  ROLL  GAP  of  through  VARIATION  the of  ON  experiments  the  gap  roll  and  was  to  v a r i a t i o n on the  while  developing  segment of the m a t e r i a l was  an element from a uniform  a predetermined  SPREAD  following  mentioned e a r l i e r ,  method, each lengthwise as i f i t was  the  0.5%.  effects  i t was  7.75  found to be  block  f i x e d r o l l gap.  assumed f o r a l l the segments i n a s e q u e n t i a l  the  imagined  being  rolled  In f a c t t h i s i s manner.  Thus,  66  for  any e l e m e n t  continuously  being  changing,  known. Two c a s e s  For  the  although entry  c a n be i d e n t i f i e d  (i)  The r o l l  gap i s o p e n i n g  (ii)  The r o l l  gap i s c l o s i n g  either  spread  case,  isstill  different are  rolled,  r a t e s of r o l l  long  roll  rod with  is  gap  along  in  the  predicted  which  each  of the r o l l  were  measured  4-20b d e p i c t s an a l t e r n a t i v e  block 9.1mm  a complementary was  to  hot-rolled 12.7mm  incremental  varied  during  values  .corresponding  indicated  examples  that both  through  from  facility  a  12.7mm t o  i n doing  so,  a s u n i f o r m as  gap d u r i n g t h i s  process  f o r some c r o s s - s e c t i o n s and  the  compared  to  those  as a p p l i e d to  results,  while F i g .  r e p r e s e n t a t i o n of t h e same.  experiment, while the  the  an roll  operation  identical  predicted the opening  uniform  gap was opened (Fig.  of the i n c r e a s e d width  l e n g t h were m e a s u r e d a n d a r e shown the  with  dimensions  rolled  E l - k a l a y and S p a r l i n g ' s formula  segment. F i g u r e 4-20a shows  In  Some t y p i c a l  w i t h a speed  F i g . 4-19. The s p r e a d  length by  gap v a r i a t i o n on  were p e r f o r m e d  t o 1150°C a n d  operation  The v a r i a t i o n  shown  dimensions a r e  cross-sectional  g a p was c h a n g e d m a n u a l l y  possible.  gap i s  regard:  of the r o l l  9.95mm. Due t o t h e l a c k o f m e c h a n i c a l the  roll  here.  square  roll  in this  gap v a r i a t i o n .  "12.7mmx12.7mm" was h e a t e d closing  and e x i t  unknown. The e x p e r i m e n t s  illustrated A  the e f f e c t s  the  4-22).  along  i n F i g . 4-23  v a l u e s . The above and the c l o s i n g  from The  the r o l l e d along  with  experiments of the  roll  67  gap have an The  i n c r e a s i n g e f f e c t on  effects  more s i g n i f i c a n t verified  and  of  roll  spread.  gap opening  f o r higher d r a f t s . T h i s was  increase  draft,  higher  Figure  4-21  as the d r a f t  rate  of  roll  gap  and  the  experimental  i n c r e a s e s . A l s o , f o r the same changing  causes  more  spread.  shows t h i s assesment; r e f e r r i n g to t h i s  the higher v a l u e s of spread correspond was  experimentally  the f i n d i n g s are shown i n f i g u r e 4-20b, where  the d i f f e r e n c e between the p r e d i c t e d values  and c l o s i n g becomes  rolled  with  comparison was  the  based  to the specimen which  higher r a t e of r o l l  on measuring  figure,  the  gap c l o s i n g .  increased  width  The at  equal t h i c k n e s s c r o s s - s e c t i o n s .  4.5  SUMMARY AND  This  chapter  conducted of The  EVALUATION  described  experiments  to e v a l u a t e the d i f f e r e n t  rolling  parts  experiments  non-uniform  which  aspects of  the  were  process  with 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 . were  performed  on  b l o c k s . Some experiments  p r i m a r i l y conducted to  the  both  uniform  and  on uniform b l o c k s were  to c o n f i r m the b a s i c concepts. These l e d  a p p r o p r i a t e arrangements to be set up f o r the subsequent  experiments. Handling m i l l was  done  minutes.  The  or  the  m a t e r i a l from the furnace to the  manually.  heating  time  was  about  specimens which should have been r o l l e d  more were immediately  re-heating  The  rolling  time  was  re-heated  then  reduced  before to  re-rolling. almost  half.  20  twice The Large  68  temperature  variations  was  found  to  have  a significant  e f f e c t on the magnitude of spread. Although the furnace placed  as  close  temperature large  to  rolling  mill  as  p o s s i b l e , the  drop during the h a n d l i n g , was p o s t u l a t e d  enough  to f a l l  Sparling's  formula  1100±10°C.  This  4.3.2  the  where  below the range  was  was  the  underestimated.  accepted  observed  computer In  to  be  i n which E l - k a l a y and  to  be  accurate, i . e . ,  from the r e s u l t s of s e c t i o n  predicted  order  was  to  values  take  account  the  drop during the  heated  1150°C. Good agreement was then observed between  the r e s u l t s from the theory and those The  strain  higher  temperatures with  (>1100°C). different  the  rollers  s a t i s f i e d the c o n d i t i o n Sparling's  formula.  the  were  experiments.  namely,  assertion speed  experiments,  was  set  of  strain  however,  the q u a l i t a t i v e  at  was  a  the  fixed angular  to the lowest value which rate  A l l the quantitative  done on hot m i l d s t e e l , aluminum,  this  rolling  temperature. However, f o r other of  of  specimens  r a t e was found to be l e s s e f f e c t i v e on spread at  experimented  speed  the  all  temperature to  handling,  into  were  some  in  El-kalay  and were  experiments  were  or c r i t i c a l  performed  experiments.  F i g u r e 4-27 shows the aluminum specimens which were used these  experiments.  their different happened  during  The  modes  problem  of  with  deformation  the experiments  on  in  aluminum s t o c k s was which  quite  often  (see F i g . 4-2). T h e r e f o r e ,  aluminum was found u n s u i t a b l e f o r s i m u l a t i n g the hot r o l l i n g of  steel.  69  Whenever also  taken  thickness  necessary, into  of  the t h i c k n e s s of the s c a l e s were  account  during  the s c a l e s  0.8mm, depending had been heated  the measurements.  were found t o vary from 0.2mm t o  mostly on the number of times the  in  specimen  i n the furnace.  The e f f e c t s of width, height and the r o l l on  The  gap v a r i a t i o n  spread were i n v e s t i g a t e d . I t was found that the i n c r e a s e the h e i g h t of the incoming  material  has a  significant  e f f e c t on the spread. A method of c o r r e c t i o n was proposed i n the  form  of equation  consider  This  correction  can a l s o  the e f f e c t of n e g a t i v e r a t e of h e i g h t v a r i a t i o n by  changing (4- 1) ,  (4-1).  the sign to  of  the  coefficient  D,  in  the  formula  positive.  Width  variation,  either  positive  or n e g a t i v e , was  observed t o have a d e c r e a s i n g e f f e c t on spread, however, i t s e f f e c t was not as much s i g n i f i c a n t as the e f f e c t variation. equation A roll  A  parametric  correction  (4-2) was then proposed  factor  of height  i n the form of  t o implement these e f f e c t s .  q u a l i t a t i v e experimental study on the e f f e c t s of the  gap changing on spread showed that both the opening and  the c l o s i n g of the r o l l  gap have an i n c r e a s i n g e f f e c t on the  magnitude  This  of  proportional  spread.  appropriate the r o l l  i s probable  to  be  t o the l e n g t h of the arc of c o n t a c t , a b s o l u t e  d r a f t and the r o l l i n g  of  effect  facility  speed. Due t o the l a c k of access t o an f o r automatic and continuous  gap, an adequate  not p o s s i b l e .  quantitative  changing  examination  was  70  The extend with  capacity  of  the  rolling  the experiments to the r o l l i n g high  rates  of  variation  in  mill of  d i d not allow to larger  height  e x t e n s i v e range of experiments in a v a r i e t y  and of  specimens width. real  An  cases  are needed to enhance the r e s u l t s e x t r a c t e d i n t h i s work.  5. SAMPLE  ILLUSTRATIONS  5.1 EXAMPLE N O . l  A uniform 70mmx250mm height  block  with  cross-sectional  i s t o be r o l l e d t o a part with l i n e a r l y  ( F i g . 5-1). The width of  remain constant Figure  the part  the s i d e  though  single  predictable,  pass, , the  are  c o n t r o l l a b l e process,  passes should  be used. The r o l l s '  arbitrary  computer  initial  program  material  intermediate  pass.  should  initial  side. Figure  material  Reduction  least  So,  for a  two  rolling  of these passes were  by  an  first  5~3a shows  has  gap  run of  s i d e of by the  the shape  of  5-4 and 5-5  the r e d u c t i o n  for  been d e f i n e d as the incremental thickness. that the l a r g e r s i d e of the  i . e . , the height operation  and  should  be  rolled  on the l a t e r a l dimension.  5-3b shows the shape of the intermediate Also,  followed  f o r t h i s case. F i g u r e s  second s o l u t i o n i m p l i e d material,  followed  radii  be r o l l e d  r a t i o of d r a f t over the i n i t i a l The  at  dimensions,  produced two s o l u t i o n s f o r the above  show the v a r i a t i o n of the r o l l  case.  constant  s o l u t i o n i m p l i e d that the smaller  r o l l i n g of the other  each  block.  set t o 175mm and 180mm, r e s p e c t i v e l y . One  problem. The f i r s t  the  final  uncontrollable.  dimensionally  the  the  to  i f only one r o l l i n g pass with l i n e a r l y v a r i a b l e gap  i s performed. With a  the  i s required  of  of  variable  and equal t o the width of the i n i t i a l  5-2 shows the spread on  width,  dimensions  material  first, Figure  for this  f i g u r e s 5-6 and 5-7 show the v a r i a t i o n of r o l l  71  72  gap and  the r e d u c t i o n d u r i n g each  pass  for  this  feasible  solution. In  the  variable  second  roll  case,  gap.  On  there  is  pass  with  the other hand, i n the f i r s t  case,  both passes have v a r i a b l e r o l l gap; has  been  assigned  to  geometry on t h i s s i d e barrelling  occurs  only  one  because the second  form the height of the p a r t , b e t t e r is  on  obtained  and  consequently,  one  or  the  other  sequence  s e l e c t e d . However, i t i s seen process  constraints  were  impossible.  The  i l l u s t r a t e s the r o l l i n g of a process c o n s t r a i n t s i n e f f e c t .  a  better  of o p e r a t i o n s may  be  that s i n c e i n t h i s example the  not  r e d u c t i o n reach to more than practically  the  the uniform s i d e . Thus, depending on  which s i d e of the m a t e r i a l i s p r e f e r r e d to a t t a i n finish,  pass  considered,  80%,  a  second more  the  proportion  values of which  is  example which f o l l o w s ,  complex  part  with  the  73 5.2  EXAMPLE N O . 2  In the  this  width  illustrative of which  varies parabolically the  process  following  example, the p r o d u c t i o n of a  i s constant  part  and the height of which  i s c o n s i d e r e d . U n l i k e example No.1, a l l  constraints  are applied  in this  case. The  i n f o r m a t i o n were s u p p l i e d t o the computer  program  as the input d a t a : Dimensions  of the i n i t i a l  block ( F i g . 5-8a):  width, 155mm h e i g h t , 11Omm Dimensions  of the f i n a l p a r t ( F i g .  width, constant,  150mm  height,  par ab ol i c al I y  maximum  varies value,  5~8b):  {minimum  value,  50mm/  100mm)  l e n g t h , 2000mm Yield  stress  temperature,  of  the m a t e r i a l  a t the working  15Kgf/mm  2  Modulus of e l a s t i c i t y  of the m a t e r i a l a t the  temperature,  15x10 Kgf/mm  Coefficient  of thermal  3  working  2  expansion  of the m a t e r i a l ,  3.3x10 (1/°C) 6  Working temperature,  1100°C  Ambient temperature,  30°C  C o e f f i c i e n t of f r i c t i o n  between  and the m a t e r i a l , assuming Maximum 5000Kgf-m  torque  the r o l l ' s  surface  rough r o l l s , 0.35  available  by  the r o l l i n g  mill,  74  roll  sizes  available:  Minimum r a d i u s , 100mm Maximum r a d i u s , 500mm (with increments of 5mm) A p p l i e d d r a f t s f o r the f i n i s h i n g First  passes:  pass, 5mm  Second pass, 5mm P r e c i s i o n of computation, 0.001 Number of segments that the f i n a l  part  i s divided  i n t o , 25 F i v e r o l l i n g passes were Figure  5-9  shows  shows the v a r i a t i o n length.  Figure  determined  by  the program.  the sequence of these passes. F i g . 5-10 of  5-11  the r o l l  provides  gap versus  more  p a r t i c u l a r s of the planned p r o c e s s .  the  rolled  i n f o r m a t i o n about the  6. CONCLUSIONS AND SCOPE FOR FUTURE WORK A computer-aided parts  having  developed. adopted These  process planning scheme f o r r o l l i n g of  variable  El-Kalay  and  a  were  evaluations  of  p r o c e s s . These variation.  and  number  done  Sparling's  cross-section  spread  was  formula  was  of m o d i f i c a t i o n s were i n c o r p o r a t e d .  through the  analytical  particular  included  The  rectangular  results  width,  and  characteristics height  and  of the experiments  a g a i n s t those p r e d i c t e d by the  experimental  modified  of  the  the  roll-gap  were c o n t r a s t e d  formula  and  good  agreements were seen. An important o b s e r v a t i o n a r i s i n g that  from  this  the process c o n s t r a i n t s and the unsteady  process p l a y e d the key computer  algorithm  roles  in  planning  study  was  nature of the  the  scheme.  A  has been developed which determines the  number of r o l l i n g passes r e q u i r e d and the s p e c i f i c a t i o n s f o r each pass. The  flat  die-forging  r o l l i n g p r o c e s s , where formed-die are  cost advantage rolling  also  different This  This  hardware  with  the  implying that the same t o o l i n g set  f o r the p r o d u c t i o n  of  an  infinite  number  of  developing  a  shapes. work  computer-aided  was  a  system  first for  attempt rolling  in parts  with  r e c t a n g u l a r c r o s s - s e c t i o n . More work, both theoretically experimentally,  and  can have a c o n s i d e r a b l e  as i t r e p l a c e s the forging  software.  may.be used  applicable,  rolling  in  d i f f e r e n t aspects are suggested  75  variable and f o r the  76  f u t u r e work. The non-uniform continued  study  blocks  on  through  extensively.  evaluate  further,  the  variable  This  the  geometric  is  deformation  roll  gaps should be  essential  accuracy  of  not  only  was  and expand i t  for  i n c l u s i o n of the e f f e c t s of the r o l l  gap v a r i a t i o n  was  not completed  this  to  the formula which  developed d u r i n g t h i s work, but a l s o to modify  in  of  which  study  due  to  the  laboratory  complexity  is  not  yet  adequately  constraints. The known. all  l i m i t of part An  algorithmic  the parameters  whether  the  approach  in e f f e c t  production  of  may  and a  be developed  to part  determine  to r e l a t e beforehand  with a given shape i s  f e a s i b l e by t h i s method. Finally,  the  method  continuous o p e r a t i o n s . T h i s can relating  can  start  to the design of machinery  important and  be  relatively difficult  extended with  to  permit  considerations  (and c o n t r o l s ) and task to do.  i s an  Table 2-1 Value roll  of c o r r e c t i o n factor as v e l o c i t y in Bachtinov's  v (nv/sec.) to2 3 r  k  1.  .9  4  5  .8  .72 .66 .6  6  7  8  function of peripheral  formula(6)  9  10  12  14  16  18  20to22  .57 .55 .52 .47 .45 .43 .42  .41  Table 2-2 Summary of the average deviations of predicted spread from the experimental values in Wusatowski s  studies  1  the  average d e v i a t i o n  from r e a l v a l u e s  spread  formulae  -  +  %  %  %  1.88  1.24  3.12  Ekelund  3.45  1.14  4.59  V7usatowski  5.32  0.50  5.82  Seible  4.84  2.19  7.03  Tafle  6.79  2.90  9.69  Trinks  total  deviation  & Ssdlaczek  Table 2-3 Effect of steel composition on spread ratio in Wusatowski's formula(20) composition c%  Si%  0.06  Mn%  of Ni%  steel Cr%  W%  correctoin type o f s t e e l factor d  0.22  1.00000  B a s i c Bessemer s t .  0.20  0.20  0.50  1.02026  0.25% carbon  steel  0.30  0.25  0.50  1.02338  0.35% carbon  steel  1.04  0.30  0.45  1.00734  Tool  steel  1.25  0.20  0.25  1.01454  Tool  steel  0.35  0.50  0.60  1.01636  Manganese  steel  1.00  0.30  1.50  1.01068  Manganese  steel  0.50  1.70  0.70  1.01410  Spring  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 r e s i s t a n t s t .  0.06  0.20  0.25  1.01034  Case hardening s t .  1.30  0.25  0.30  1.00902  Alloy  tool  steel  0.40  1.90  0.60  1.02719  Alloy  tool  steel  3.50  0.40 0.50  2.00  0.30  1.80  steel  79  Table 2-4 Values of the constants in Sparling's formula(18)  c  K  A  B  0.981  1.615  0.900  0.550  G -0.250  Table 2-5 Comparison between three spread formulae in Sparling's experiments formula  No. o f t e s t s  No. o f t e s t s w i t h e r r o r more than 15% 10% 21 23  Hill  26  Wusatowski  26  13  8  Sparling  26  3  0  Hill  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  0.993 0.980  2.186 2.105  0.569 0.569  0.402 0.402  -0.123 -0.123  rough  rolls light scales heavy scales  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 forming, (b) height forming  leaf springs; (a) width  Figure 2-1 Schematic representation of a rolling prtfcess(6)  82  Figure 2-2 V e l o c i t y diagram i n a r o l l i n g process(6)  83  F i  Figure 2-5 Pressure distribution along  the  arc  of  contact(6)  Figure 2-6 A typical example of true pressure d i s t r i bution  in  the roll  gap over a half width of the rolled stock(6)  .«*•  1 Friction  .  Figure 2-7 Schematic representation of geometric deformation in rolling(32)  Figure 3-1 Forming a uniform block; (a) i n i t i a l and final dimensions (b) pass one (c) pass two  Figure 3-2~Different sequences of rolling  86  (b)  Figure 3-3 Comparison between values o f spread 1n two types of r o l l i n g ; (a) double pass (b) s i n g l e 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 A D  15  18  5  8  a  tee  288  388  488  see  6ee  Figure 3-7 Typical variation of spread versus the r o l l ' s radius for a constant draft  Figure 3-8 Schematic representation of the inverse calculation method  user DATA  data  GENERATOR  for 3 - D I M . presentation  ^  3-DIM  »  |~  GRAPH GENERATOR  input  r->i  file  11  2-DIM  J  GRAPH GENERATOR  data for torque variation presentatj onj' MULTI-PASS  PROCESS  PROCESS  data for r o l l gap | variation presentation  PLANNER  £  iM>  file  CONTROLLER  ±  output; results, message  DATA FOR  ORGANIZOR  MICRO-PROCESS  gure 3-9 General flow-diagram of the computer software  OR)">  computer memory  90  ^taxt^^ read the initial  data  .apply the convexity constraint, .calculate the dimensions of the.material 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.  Y — i  .store the results print out •design the next the pair of results. passes. 7fT .modify the answer, .calculate the dimensions of the new material coming out of the modifend ied passes. •assume the above material as an initial material.  Figure 3-10  General flow-chart of the multi-pass design routine  91  388  258 .  R 0 L L G A P  208 .  158 .  188 .  under rotting over rolling  58 .  8  I 58  I 108  I 158  I 288  I 258  I 388  I 358  I 488  I 458  588  DISPLACEMENT  Figure 3-11 An exaggerated representation of roll interference  roll-gap changing mechanism; servo, hydro,  rolling process  actual height of rolled material  « • •'  1  Ant-a^f^f^a  with the machine.  tranducer  t7  Figure 3-12 Block diagram of an analogue control system  93  roll-gap changing mechanism; servo, hydro,  actual height of rolled rolling process  material. actual  rolled length. >•  interface with machine  interface with micro-computer  analog to digital convertor  digital to analog convertor -  DD C micro computer operator interface  |T  computer memory  Figure 3-13 Block diagram of a D.D.C. system  94  wf 1  -  "mean^b-^K^f) w  b  = ( w  bl  + w  b2^  2  w =(w +w )/2 f  Wf2_  f1  f2  Wb2-  w  mean b-  w  f=(w i+w )12  = w  f  1 / 3 ( w  b- f) w  f2  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,  "  "  "  width ,height(mm)  state of material initial  25.40,25.40  intermediate  19.05,26.85  final  20.30,20.05  imi  1  imi  state of material_  inw  fa  width,height(mm)  initial  25.40,25.40  1  intermediate  21.30,26.18  2 d intermediate  21.80,21.95  3 " intermediate  19.80,22.30  final  20.13,20.05  s t  n  ra  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 2 .000, 2 . ND PASS=2.000 0  RESULTS; CASE  i.*»••••••• VERY FIRST PASS-WIOTH REDUCTION NEXT PASS=HEIGHT REDUCTION . e t c .  •••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 DRAFT  IN 1.ST PASS=4.32112 IN 2 N D 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 (i)  concave  (ii)  linear  (iii)  convex  in mm)  VO VD  8.175 .  I 0  N  .025 .  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 .  PREVIOUS PREDICTION *•"* EXPERIMENTAL VALUES NEW PREDICTION  25  28  15  18  i—i—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 4-12 Comparison of experimental and predicted values of spread, specimen 1, width increasing 35 — PREVIOUS PREDICTION * — * EXPERIMENTAL VALUES — NEW PREDICTION  38  25  28  15 . //  18 18  12  i  i  i  i  i  i  14  16  18  28  22  24  INITIAL  i—i 26  28  r 38  32  34  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 18  i  T  i  i  i—i—i—i—I—I—I—r  1  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 16  1 1 1 18  28  22  1 1 1 1 1 1 24  26  28  38  32  34  1 1 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 58  188  158  288  1  258  1  380  1  358  r 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 0  1  1  58  i38  r-—i——i 150  280  2S0  1  1  r  388  3S8  408  ROLLED LENGTH  13.8 S R E A D  13.4 J  J 13.2 . D T H 13 . *—* EXPERIMENT * — * PREDICTED VALUE  12.8  8  1  1  8.5 *  I  1 1.5  1 2  1 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 .  • • * t  12.8 .  8  1 I  1 2  1 3  1 4  I I 5  6  1 7  1 8  1 9  18  I I 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  •UNSTEADY ROLLING EXPERIMENT « — ° SIMULATED SPREAD  13.8  13.8  13.4 .  13.2 .  13.  •  12.8  —I 8  58  1  1  188  158  1  r  288258  388  1  1  r  350488  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) f i r s t solution v  b) second solution  258  1*1. PASS '2nd. PASS  268 .  158  tee .  58  188  -r  ise 288  ~ r -r~ m -  258  388  458  358  see  ROLLED LENGTH  Figure 5-4 Process behaviour of the f i r s t 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 f i r s t possible solution: percentage of reduction versus rolled length(mm)  8  1—' I 8  56  1  1  168  156  1 208  1  1  2S8  366  1 356  T 468  1 456  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 PASS I PASS 2 PASS 3 — PASS 4 PASS 5  28  —I 8  288  r 488  T  -  888  T  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 8  r—I 288 488  1 688  1 888  1  1  1  1  1  1  1868 1286 1488 1688 1868 2888  ROLLED LEN6TH  a THOUSANDS 6 -  PASS I 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, "Manufacturing  New  B.H.  Processes.",  York,  John Wiley and  Sons Inc.,  1969, pp 1 to 5.  Datsko, J . "Material  John  Properties  Wiley  and  Manufacturing  and Sons Inc., New  Processes.",  York,  1966,  pp 280 to  306.  Alexandro, J.M.  and Brewer,  "Manufacturing  R.C.  Properties  of  Nostrand Company L t d . , London,  Roberts,  D.Van  1963.  W.L.  "Hot  Rolling  York,  Siebel,  Materials.",  of  Steel.",  Marcel  Dekker Inc.,  New  1983.  E.  " Formabi  Wusatowski,  I i I y in Metal  of  Braunchweig,  1932.  Rolling.",  Pergamon  Press,  1969.  S.  "Forward  Slip  Occuring  Spread."  and  Selesia University  Underwood,  Dusseldorf,  Z.  "Fundamentals  Koncewicz,  V/orking.",  L.R.  Neutral  ,  Angle  Doctoral  in Hot  Thesis,  of Technology, Poland,  Rolling  with  Manuscript,  1961.  119 "The  9.  Rolling  "Research  of Metals.",  on the  V o l . 1, London,  Rolling  Strip.",  A  1952.  symposium  of  s e l e c t e d papers 1948-1958, B.I.S.R.A., London, 1960.  10. Orowan, E. "The  Flat  Calculation  of  Roll  Pressure  in Hot  R o l l i n g . " , Proc. I n s t . Mech. Engrs.,  and  Cold  150,  1943,  pp 140 to 167.  11. Alexander, J.M. "On  the  Theory  of  Rolling.",  Proc. Royal  Society  London, S e r i e s A, V o l . 326, 1972, pp 535 t o 568.  12. Sims, R.B. "The  Calculation  Rolling  of the Roll  Mills.",  Force  and Torque  Proc. I n s t . Mech. Engrs.,  in  Hot  168, 1954,  pp 191 t o 200.  13. La H i , L.A. "An  Analytical  Thickness  Shear  Rolling  Model  Stress  Including  Through  distributions.",  J o u r n a l of  E n g i n e e r i n g M a t e r i a l s and Technology, V o l . 106, J a n . 1984, pp 1 to 8.  14. Kobayashi, S. and Oh, S.I. "An  Approximate  Analysis  Method  of R o l l i n g . "  for  a  Thr e e-Di mens i onal  , I n t . J . Mech.  S c i . , Pergman  Press, V o l . 17, 1975, pp 293 to 305.  15. Orowan, E. and Pascoe, K.J. "A Power  Simple  Method  Consumption  of  Calculating  in Hot  Flat  Roll  Pressure  Rolling.",  Iron  S t e e l I n s t . , S p e c i a l Rept., No. 34, 1946, p 124.  and and  1 20 16. Rowe,  G.W.  "An  Introduction  to the P r i n c i p l e of Metal  Working.",  Edward A r n o l d L t d . , London, 1968.  17. Helmi, A. and Alexander, "Geometric  Factors  Rolling  I,M. Affecting  of Steel.",  Spread  J . Iron S t e e l  in  Inst.,  Hot  Flat  206,- Nov.  1968, pp 1110 t o 1117.  18. S p a r l i n g ,  L.G.M.  "Formula Inst.  for  Spread  in  Hot  Flat  Rolling.",  Proc.  Mech. Eng., 175, 1961, pp 604 t o 640.  19. Wusatowski, Z. "Hot  Rolling:  Elongation.",  a  Study  Iron  of  Draught,  Spread  and  S t e e l , London, V o l . 28, Feb. 1955,  pp 49 t o 54.  20. Wusatowski, Z. "Hot  Rolling:  Elongation  Study  Draught,  Iron  Spread  Steel,  and  London, V o l .  1955, pp 89 t o 94.  G.D.,  et a l .  "Comput e r-Ai de d Analysis in  of  (continued).",  28, March  21. L a h o t i ,  A  Plate  Rolling.",  of Metal  Flow  and  Stresses  J . Mech. Work and Tech., V o l . 4,  1980, pp 105 to 110.  22. E l . K a l a y , A.K.E.H.A. and S p a r l i n g , "Factors Load, Iron  Affecting Torque  Steel  Friction,  and  Inst.,  Spread  L.G.M. and  Their  in Hot Flat  Effect  on  R o l l i n g . " , J.  206, Feb. 1968, pp 152 t o 163.  23. Beese, J.G. "Ratio  of Lateral  Strain  to Thickness  Strain  During  121 Hot  Rolling  June 1972,  24.  of Steel  pp 433  Ishikawa, T.,  to  on  Steel  Inst.,  Profile  pp 772  and  Shape  Conference  to  of  on  the  Steel  783.  J.M.  "Machine  Tool  Eighteenth  Design  on Reasearch.",  International  Research, MacMillan  Proceedings  Machine  Tool  Press L t d . London,  Design  of  and  1978.  R.C.  "Advanced  calculus.",  New  1956.  York,  27. Groover,  McGraw H i l l  Book Company  Inc.,  M.P.  "Automation  Production  Manufacturing.",  Cliffs,  28. Tomilson,  New  and  1959,  29. Canahan B.,  Jersey,  1980,  Stringer,  Elongation  pp  157  Numerical  York, N.Y.,  and  pp 414  to  Computer-Ai  Inc.,  to  ded  Englewood  485.  J.D. in  Flat  Tool  Forging.",  JISI,  160.  Luther, H.A.  "Applied  and Wilkes  J.O.  Methods.",  Willey  &  Sons,  New  1969.  30. F o r s y t h e , G.E., "Computer  System  Prentice-Hall  A. and  "Spread  Oct.  the  International  R o l l i n g , Tokyo, 1980,  26. Buck,  Iron  436.  Study  Strip.",  25. Alexander,  J.  et a l .  "Fundamental  Rolled  Slabs.",  Malcolm, M.A.  Methods  for  and Moler  Mathematical  P r e n t i c - H a l l , Englewood C l i f f s ,  N.J.,  CB. Computations.",  1977.  1 22 31. Duncan, J.P. and Mair, "Sculptured  S.G.  Surfaces  in Engineering  and  Medicine.",  Cambridge U n i v e r s i t y Press, Cambridge, 1983.  32.  "Taper  Leaf  Spring  Rolling  Machine.",  File  H i l l e E n g i n e e r i n g Company L t d . , S h e f f i e l d ,  No.  1LS/1,  England.  33. Bryant, G.F. "Automation  Institute,  of Tandem  Mills.",  London, 1973.  The  Iron  and  Steel  APPENDIX  Method  of  Incremental  Search  in  A  Root  Finding  General Problem: Given  an  algebraic  the v a l u e ( s ) of x,  equation  i . e . , the  of the form f(x)=0,  root(s),  that  find  satisfy  the  equation.  Algorithm ' : 2 9  3 0  given s t a r t i n g Values  0  of f ( x ) f o r s u c c e s s i v e values  x +2Ax,..., 0  f(x)  p o i n t , x , an increment  are  determined  until  reverted  s i g n change i n f ( x ) occurs above  smaller  the  x , 0  x +Ax, 0  a s i g n change i n . sign  change  is  back and the incremental search i s repeated  u s i n g a smaller increment The  of  occurs, i . e . , when f(x)•f(x+Ax)<0  The l a s t value of x, preceding  Ax i s chosen.  procedure  increments,  ( e.g.,  AX=AX/10)  until  a  again.  i s repeated using p r o g r e s s i v e l y until  a  value of the root i s o b t a i n e d .  123  sufficiently  accurate  APPENDIX B  Cur v e  The  main  analytical points.  purpose  form  A  fitting  in  curve-fitting  for describing  curve  with  a  i s to  set of  continuity  up  to be s u i t a b l e f o r a p p l i c a t i o n  study.  conical  arc i n t e r p o l a t i o n  Conical arc interpolation avoid  unwanted  discrete  an data  to second order was  considered A  find  in  technique  the was  present  selected.  i s an approach which i s d e v i s e d to  oscillation  .  3 1  The general  form of a conic  curve i s Ax +Bxy+Cy +Dx+Ey+F=0 2  2  with the slope •y ' =-( 2Ax+By+D)/( 2Cy+Bx+E) and  the second d e r i v a t i v e y"=-2(y' +By'+A)/(2Cy+Bx+E) 2  The  expression  f o r the conic curve may be r e w r i t t e n as x +B'xy+C'y +D'x+E'y+F'=0 2  2  where A*0, B'=B/A, C'=C/A,...etc. The  above  expression,  thus,  c o n s t a n t s which can be found e.g.  illustrate  routine, curve  the  consider  is first  way  that  the  points  the  five  conditions,  2  algorithm given  slopes  3  as  passes  repeated  of  the  points  at  mean  points  slope  curve-fitting  i n F i g . B-1. A conic three p o i n t s i n such 2  and  3 s a t i s f y the  between p o i n t s / ,  , 4 ( f o r d e t a i l s on weighted mean slope slope  coefficients  of  through p o i n t s 3 and 4.  f o r p o i n t s 4,5  passes  of  and  3 and  concepts,  curvature  at  w e l l as the weighted mean slope at p o i n t 4 are  then used t o f i n d the  curve  satisfy  to  see 31, pp 130 t o 135). Values  which  independent  passed through the f i r s t  c o n d i t i o n s of weighted  point  five  two p o s i t i o n s , two slopes and one c u r v a t u r e . To  a  involves  through  and three  5, 6...,etc. points 124  the  conical  curve  The same procedure i s The (n-2),  last (n-1)  conical and n,  125  s a t i s f y i n g the c o n d i t i o n s of slope and  curvature  at  point  (n-2).  Using the weighted smoothness  mean slope  concept  of the j o i n e d c u r v e s . A l s o , due  the c o n i c a l curves, o s c i l l a t i o n p o i n t s does not  between  guarantees  to the nature of  the  two  occur . 3 1  Figure B-l C u r v e - f i t t i n g  the  through known data points  adjecent  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0096924/manifest

Comment

Related Items