Analysis on Width Spread of Cold Rolled Strip

图学美Computer Aided Drafting, Design and ManufacturingCADDMVolume 23, Number 4, December 2013, Page 58GDGRAPHICSAnalysis on Width Spread of Cold Rolled StripYANG Guang-hui, DENG ChiSchool of Mechanical Engineering, University of Science and Technology Bejjing, Bejjing 100083, China.Abstract: Width spread is usually considered in hot rolling, nevertheless, it can be ignore in cold rolling. In order to control theCrown and flatness better simultaneously and realize the decoupling of crown and flatness, the transverse displacement function ofstrip is defined and deduced. The factors that affects the width spread are analyzing with the function. According to the unchangedprinciple of volume before and after rolling, the flatness model which considers the width spread is deduced. Through the model, thetransverse metal flowing of different stands and different thickness is analyzing, it is found that the width spread is biggest in coldrolling at the No.1 stand because of the maximum thickness and reduction and the minimum deformation resistance. It gives a goodfoundation for the decoupling of crown and flatness.Key words: width spread; decoupling; cold rolling; transverse metal flowing; flatness1 Introductionaccount in order to analyze the shape of strip .correctlyl-.])Shape control of strip include crown control and2 Proportion Crown of Differentflatness, and crown is the direct target of control,while it is a condition of realizing flatness target thatConditionscontrolling proportion crown equivalent between theneighboring stands. During controlling， it is2.1 Establishment of Flatness Modelimpossible to meet the target values of crown anproportion crown simultaneously， so there existsH+0Hcoupling relationship between crown and flatness, andit must be decoupled. When the shapes of rawmaterials and final products are both good, it mustabide by the principle of proportion crownequivalence, and that is:h+ OhhCn_ CHHOb + OB2Ob+ 0b2where CH and Ch are the entrance crown and exitcrown of strip respectively, and H and h are theentrance thickness and exit thickness of stripFig.1. Transverse section of strip before and after rolling.respectively.Actually，the principle of proportion crownSuppose take a longitudinal fiber from middle andequivalence does not consider the transverse metaledge parts of strip respectively as shown in Fig.1. Theflowing in the deformation area, when the strip iswidth, thickness and length of fiber in the middle partrolled, the deformations of width spread are differentbefore rolling are △b, H and L, and the width,and non-uniform in the different transverse regions,thickness and length of fiber in the edge part beforeand it directly affects the elongation change of strip,rolling are△b,中国煤化工orrespondingso the transverse metal flowing must be taken intowidth, thicknesYHCN M H G middle partProject Item: Supported by Beijing Higher Education Young Elite Teacher Project (No. YETP0369).Corresponding author: Y ANG Guang-hui, Male, Doctor, Associate Professor, E-mail: yanggh@ ustb. edu.cnYANG Guang-hui et al, Analyzing on Width Spread of Cold Rolled Strip59after rolling are△b+b), h and l, and the corresponding(Ob +△b)width, thickness and length of fiber in the edge par(Ob+Ob,)after rolling are△b+ b2, h +△h andl +△I. Accordingto the unchanged principle of volume before and afterDefine the proportion crown coefficient:rolling, it can be achieved:Nb.H .L= (Nb + Nb).h.l(1+△H.H、△Ob(H + AH).(L +△L)=(Ob + Ob2).(h +△h).(l +△l)hwhere△H and△h are transverse thickness deviationbefore and after rolling;△L and△l are fiber lengthDefine the flatness coefficient:deviation before and after rolling; Ob1 is width spreadof fiber in the edge part of strip after rolling; △b2 isε=1+width spread of fiber in the middle part of strip afterLrolling.The above two equations can be described:The flatness of strip can be described by thefollowing formula:Nb.H.L(Ob +△b).h△x10 =(2.k.ε-1)x10'Nb.(H + AH).(L +△L)(l+ NI)=(Ob +△b2).(h +△h)So it can be seen from above equation that the finalshape of strip is connected with the width spreadcoefficient h, the proportion crown coefficient k andSo the flatness of strip can be described by thethe flatness coefficient 8.2.2 Analysis on Proportion Crown(Qb +△b)1+△HH'In order to achieve a good shape of strip and make“x10s =5-.(1+)-1x10*the flatness△ll minimum, the proportion crowns of(△b + Ob2)different conditions are analyzed in Table.1, and it canbe shown that proportion crown should change withthe shape of strip before rolling so as to achieve goodDefine the width spread coefficient:shapes of strip.Table 1. Analyzing on Proportion Crown of Different ConditionsWidth spreadFlatness cofficient Shape before rollingProportion crowncoefficientHShmiddle bucklek<1AH Nc=1no buckle defect<号λ>10H、Shk2ledge buckleHhAH<△hAH、Ohk≥18>1sH. Sh2<1AH、Shk>1ε<1AH。Nε>1λ=1k=1AH_ Sh中国煤化工TYHCNM HG_Whether the final shape of strip or not, it isdetermined by many factors such as shape of strip60Computer Aided Drafing, Design and Manufacturing (CADDM), VoL.23. No.4, Dec. 2013before rolling, width spread of strip, width spread andThe transverse sections of strip entrance and exitso on.can be described and fit by quadratic and quarticfunction:3 Defining of Transverse DisplacementFunctionH(y)= H(O)+ B,bThe transverse displacement of metal in the rollingdeformation regions is the function of x and y (inFig.2), and it can be described by the following'-8:h()=h(O)+6b2(2)2 +b.(jy)4zSo the transverse displacement function u(y) can beachieved:?Hu(y)='AbshKy. a,b 2y_ shKy.2shNN2~ bshN2a,b r_ 6 _)shKyN2“bN2 bN2' shNwhere, N= Kb47.2μm。,b22 Vη22E Ah.ξ=1+-3kh.k" h。'2ENhFig.2. Geometrical dimensions of rolling deformationa2:hH'regions.μ is friction coefficient of contact surface; no is ratiobetween average unit pressure p and the value2k=1.15os; H and h are the average thicknesses ofW(x,y)= u(y)(1-Nhentrance and exit sections respectively; l is contactingarc; he is average thickness of deformation;△b iswhere△h is the reduction of the rolling pass; H is thewidth spread value; E is elastic module of strip.entrance thickness of deformation region; u() is4 Efects of Different Factors on Widthtransverse displacement function.At the entrance of deformation region, h,=H,SpreadW(x,y)=0. At the exit of deformation region, h,=h, .(1) Effect of reductionW(x,y)=u(y). The middle parts change with thechanging of hx.0.5The total power in the deformation region can be0.4achieved with the energy method:0.30.2N=vh[。F(y,u(y),u ())dywhere V| is exit average velocity of strip; b is theC1(12030width of strip.Pressing /9According to the minimum energy principle, theFig.3. Width spread change with reduction.total deformation work should be the smallest, so u(y)must meet the following Euler difference equation,i.e.,It is shown in Fig.3 that the reduction is bigger, sodoes the widt中国煤化工reduction is8Fd. 8Fincreased, thetregion is)= 0i YHCNMHGCudyaincreased, and w rougiuuiui iNoisuance will beYANG Guang-hui et al, Analyzing on Width Spread of Cold Rolled Strip61enlarged and it leads to the width spread bigger;It is shown in Fig.6 that the strip width has a closemeanwhile, with the pressure increasing， the metal relationship with lubricant. The friction coefficient isvolume of thickness direction will increase, and itbigger, the strip does not deform easily, and it affectsmakes the width spread increasing.the transverse metal flowing, so the width spread issmaller.(2) Effect of roll diameter0. 4080. 4g 0.6,0. 3960. 392三0.3880.38420 540 560 580 600 620 6400. 020.04 .0.060.08.1Roll diameter/mmFriction coefficientFig.4. Width spread change with roll diameter.Fig.6. Width spread change with friction cofficient.It is shown in Fig.4 that the roll diameter is bigger, 5 Construction of Finite Element Model ofso does the width spread. Because the roll diameter isRoll and Stripincreased, the length of deformation region isincreased, and the longitudinal resistance will beThe sketch map of a quarter model of the roll andenlarged and the metal flows easily to the transversestrip system is shown in Fig.7. According to thedirection. If the roll diameter is decreased, the contactGeometrical relationship of roll and strip， thearc is increased, the horizontal component of forces ofcoordinates of points A, B,C and D can be achieved aspressure is enlarged, so the actual deformationfollowed:resistance is decreased, meanwhile, the length ofdeformation region is decreased, and sliding distancex,=-Rsinθis shortened and it decreases the effect of frictionyA =-Rcosθresistance force. It helps the longitudinal metalflowing, so it decreases the width spread.(zA=0(3) Effect of strip widthxp=-Rsinθ-L0.5yp=-Rcosθ0.450.40.35[xc=-Rsinθ-L三0.3yc =-Rcosθ-0.5hj昌0.25(Zc=0600110016002100Strip width/m[xD = -RsinθFig.5. Width spread change with strip width.yp=-R cosθ -0.5h,( Zzp=0It is shown in Fig.5 that the strip width is bigger,the width spread is smaller. Because the width iswhereθ is the biting angle, 0= arccos( 1-△h/2R)=increased, the contact area between strip and roll is arccos(1-(ho-h)/2R); L is the length of strip; hois theenlarged, and the transverse resistance of metalentrance thickness of strip; h1 is the exit thickness offlowing will be enlarged, most of metals will flow instrip.the longitudinal direction and the width spread isThe Finite (中国煤化工model of rolldecreased.and strip is b:MHCNMHGsoftwareas(4) Effect of friction coefficientshown in Fig.8.62 .Computer Aided Drafting, Design and Manufacturing (CADDM), VoL.23, No.4. Dec. 2013yNo1 syrmetrical plane .?-R/θstrpBDNo.2 srametiraa plane(a) z direction(b) x directionFig.7. Sketch map of a quarter model of roll and strip.昌0.8+ 4.0mm0.6-3.0mm.0.40.5mm0.2-0.280400800-0.4-0.6人-0.8Distance referring to strip center/mmFig.8. Finite element model of a quarter model of roll andFig.10. Strip metal transverse flowing of differentstrip.thicknesses.6 Metal Transverse Flowing0. 40.5- +5#昌-0.2900-4000.3喜0.1Fig.11. Strip metal transverse flowing of different stands.Stand No. of tandem cold rolling millIt is shown in Fig.10 and Fig.11 that strip metaltransverse flowing of different thicknesses and stands.Fig.9. Width spread change of different stands.It can be seen that with the same reduction of strip, thestrip is thicker, the strip metal transverse flowing isIt is shown in Fig.9 that the width spread willlarger, so the metal transverse flowing of No.1 stand isthe maximum, nevertheless, that of No.5 stand is thedecrease in turn from No.1 stand to No.5 stand of theminimum because of the smaller reduction, the thinnertandem cold rolling mills. Because the reduction ofthickness and the bigger deformation resistance forceNo.1 stand is the largest, and the thickness of strip ofof strip when the trin i中国煤化工stand of the .the No.1 stand is thicker relatively. With the rollingtandem cold_vide a goodpass increasing, the reduction, thickness and widthfoundation forfYHC N M H Gd flatness.spread will decrease.7 ConclusionsYANG Guang-hui et al, Analyzing on Width Spread of Cold Rolled Strip63(1) The transverse displacement function of strip isanalysis of shape control for continuous hot rolling millsdefined and deduced. The factors such as theof super wide strip steel [J]. Computer Aided Drafting,Design and Manfacturing, 2013, 23(1): 79-83. .reduction, width of strip, roll diameter and frictioncoefficient that affects the width spread are analyzing[6] A.Seilinger, A.Mayrhofer, A Kainz. SmartCrown- -A newsystem for improved profile & flatness control in stripwith the function.mills []. Steel Times International, 2002, 11: 11-12.(2) The flatness model which considers the width7] Lian Jiachuang， Duan Zhenyong. Study on the widthspread of rolled strip [J]. lron and Steel, 1984, 19(11):spread is deduced according to the unchanged15-19 (in Chinese).principle of volume before and after rolling. The8] Huang Qinglu. The function relationship between thetransverse metal flowing of different stands andflatness, transverse thickness deviation and width spreadthickness is analyzing with the model, it is found that[J]. Alumimun Processing, 1999, 22(2): 9-11 (in Chinese).the width spread is biggest in cold rolling on the No.1YANG Guang-hui received the Ph.D.stand because of the maximum thickness, thdegree in the Mechanical Design andmaximum reduction and the minimum deformationTheory from University of Scienceresistance. It gives a good foundation for theand Technology Beijing， China indecoupling of crown and flatness.2007. He is currently an associateReferencesprofessor of School of MechanicalEngineering，University of Science[1] Ginzburg V B, Azzam M. Selection of optimum stripprofiles and flatness technology for rolling mills [J]. lronand Technology Beijing, China. His research interestsand Steel Engineer, 1997, 32(7): 30-38.include shape detection and control of strip, and[2] J.S.Wang, Z.YJjiang, A.K.Tieu. Adaptive calculation ofmechanical design.deformation resistance model of online process control inDENG Chi is currently a Mastertandem cold mill [J]. Journal of Material ProcessingTechnology, 2005: 585-590.candidate in School of Mechanical[3] Yang Guanghui, Cao Jianguo, Zhang Jie, et al. Profile andflatness control technology with a long shifting stroke onand Technology Beijing， Beijing,wide non-oriented electrical steel sheets []. Journal ofChina. His research interests includeIron and Steel Research International, 2012, 19(1): 31-35.shape detection and control of strip,4] N.Venkata Reddy, G.Suryanarayana. A set-up model formechanical design and manufacturing.tandem cold rolling mills [J]. Journal of MaterialProcessing Technology, 2001: 269-277.[5] YANG Guanghui, LI Yuqi, GAO Yang. Finite element中国煤化工MYHCNMH G