h World Congresses of Srucural and Mulidiscilinary Oimizaion Rio de Janeiro, 30 May - 03 June 200, Brazil Maerial Resisance and Fricion in Cold Rolling A.K. Tieu, C. You, H.T. Zhu, C. Lu, Z.Y. Jiang and Giovanni D Alessio 2 School of Mechanical, Maerials and Mecharonics Engineering, Universiy of Wollongong, Wollongong, Ausralia, kieu@uow.edu.au 2 Packaging Producs, BlueScoe Seel, Por Kembla, Ausralia Absrac The rolling force model requires an accurae knowledge of he deformaion resisance of he maerial and a roer fricion coefficien calculaion. The main ineres of he aer is o invesigae he relaionshi of he facors involved in he roll force calculaion based on he Bland-Ford-Hill model. The deformaion resisance and fricion coefficien are deermined simulaneously wihou relying on emirical daa of maerial resisance. Eigh coefficiens in he maerial resisance and fricion coefficien models are derived by minimizing he error of he measured and calculaed rolling forces based on nonlinear leas squares oimizaion mehod. The effec of work roll wear and rolling seed on fricion coefficien in he roll bie were analyzed. The resuls already obained shows ha fricion decreases wih roll wear, and he lower he rolling seed, he higher is he fricion. Keywords: cold rolling, fricion, maerial resisance, oimizaion. Inroducion The oimizaion of manufacuring and roduc qualiy is ossible when he effec of each rocess sage and is influence on he rocess arameers are known. To roduce high qualiy roducs in andem cold rolling wih he rend owards hinner rolled sris and higher srengh maerials, i is necessary o deermine accuraely rocess conrol arameers in he modelling. Several mahemaical rolling force models such as Bland-Ford model [], Sone model [2] have already been roosed and widely alied in cold sri rolling. All of he rolling force models have a similar funcion form as follows, P = f ( R,H,h 0,h, f, b,k, µ,b,...) () in which he work roll radius R, he sri widh B, he slab hickness H, enry hickness h 0 and exi hickness h, fron and back ension sress f and b, maerial deformaion resisance K, fricion coefficien µ and oher relevan arameers are involved. From Eq.(), i is clear ha he calculaion of rolling force deends on he accurae and robus models of fricion and deformaion resisance. The deformaion resisance in cold rolling is deermined by he work hardening characerisics of maerial, which is a funcion of he rocess condiions in revious rolling and meallurgical comosiion of maerial. Many facors can affec he fricion in he roll bie, such as sri surface condiion, surface roughness of he rolls and sri, lubricaion condiions, rolling seed and ec [3]. The rolling force needs o be deermined from he couled effecs of maerial deformaion resisance and fricion coefficien according o real roducion daa in cold rolling A resen, he mahemaical models of maerial deformaion resisance and fricion coefficien are based mainly on emirical and saisical equaions ha are develoed searaely. Firs, he lane srain ess were carried ou o deermine he maerial deformaion resisance. Then, an inverse mehod was carried ou o mach he calculaed rolling force agains he measured values based on assumed fricion coefficiens. The main ineres of he aer is o invesigae he relaionshi of he facors involved in he roll force calculaion based on he Bland-Ford-Hill model. The deformaion resisance and fricion coefficien are deermined wihou relying on emirical daa of maerial resisance. Eigh coefficiens involved in he maerial resisance and fricion coefficien models are deermined simulaneously by minimizing he error of he measured and calculaed rolling forces based on nonlinear leas squares oimizaion algorihm. 2. Princiles of Calculaion 2.. Bland-Ford-Hill Rolling Force Equaion The general form of Bland-Ford-Hill equaion [4] is P = B K R' h Q n (2) in which B is he sri widh, K is he maerial deformaion resisance, R is he deformed work roll radius from Hichcock model [], h=h 0 -h, h 0 is he enry hickness, h is he exi hickness, n is he ension facor and can derived by C f + ( C ) b n = (3) K where C is he coefficien, f and b are he fron and back ension sress resecively. The srain facor Q in Eq.(2) can be described by a saisical model R Q =. 0 +. µ r. 02 r (4) h where r is he reducion and r = h h0, µ is he fricion coefficien. According o reference [3], he work roll surface roughness and rolling seed have an enormous effec on he fricion coefficien in he roll bie. Yang [4] also concluded ha he maerial
deformaion behaviors in he roll bie lay an imoran role on he fricion erformance. Regarding he above facors, he following equaion is roosed o describe he fricion coefficien in he roll bie, d µ = (a+ b h+ c r) () + e Nr where a, b, c, d, and e are coefficiens, N r is he lengh of rolled coils a each sand ha relaes o roll wear. The hickness h was inroduced o calculae he fricion in he Eq.(). However, i can be easily ransformed o he form of rolling seed because he mass flow should kee consan during he sable rolling rocess hi+ vi+ Cons hi vi = hi+ vi+ = Cons => hi = = () vi vi The maerial deformaion resisance can be defined as a funcion of srain and srain rae, m n K = l ε ε () where l, m and n are coefficiens, ε and ε are he srain and srain rae resecively. 2.2. Leas Square Oimizaion An oimizaion is a roblem ha can always be u ino he form as follows [, ], Find X, which minimizes F(X) Subjec o: g j(x)= 0, j =, 2,..., m h(x) k 0, k =, 2,..., l X X u X, i =,2,., n i i i where X is an n-dimensional design vecor, while he various oher funcions and equaion l X i is he lower bound and g j(x), k u X i is he uer bound, F(X) is called he "objecive funcion", h(x)are called he "consrains". All funcions involved are he linear or nonlinear funcions of he design variables X, and are exlici or imlici in X. Some analyical or numerical mehods have already been used in comuing he values of he design vecor X. One of he widely used aroaches is he leas squares heory. When esimaing he arameers of he nonlinear funcion wih leas square mehod, he sandard aroach using derivaives is no always ossible. Insead, ieraive mehods are very ofen inroduced o esablish he search direcion during each ieraion. These mehods search in a sewise way for he bes values of he esimaion. Usually, hey roceed by using a each se a linear aroximaion of he funcion and refine his aroximaion by successive correcions. Mos of hese sors of rocedures require secifying an iniial value of he design vecor. Beginning from his saring oin, he design vecor is udaed ieraively. Recen oular alicaions of hese echniques are known as gradien descen mehod, Gauss-Newon and Levenberg-Marquard aroximaions. 2.3. Objecive Funcion in he Oimizaion To calculae he coefficiens involved in he maerial resisance and fricion coefficien models by he leas squares oimizaion, an objecive funcion, which will be minimized, was buil as F( φ ) = f ( µ,k ) Pm () where P m is he measured rolling force, f ( µ,k ) is he imlici equaion o calculae he rolling force. Subsiuion of Eq.(4), Eq.() and Eq.() ino Eq.(2) leads o m n 0 d R f( µ,k) = Bl ε ε R h.. (a bh cr) + + + r 02. r n ( N r e) () + h 3. Oimizaion Calculaion Based on he nonlinear leas square oimizaion heory and Levenberg-Marquard imlemenaion, a Malab rogram has been develoed []. Table shows he imoran arameers used in he oimizaion. Table. Calculaion condiions Slab Work roll Young s Seel grade Coil number Poisson raio Sri widh hickness diameer modulus AK20 24 0.3.~2.2 mm 40~0 mm 220kN/mm 2 ~00 mm An ieraive rocedure is used o deduce eigh oimized coefficien, i.e. l, m, n, a, b, c, d and e involved in he he maerial deformaion resisance and fricion coefficien equaions. From he iniial guessed vecor Φ = [l, m, n, a, b, c, d, e], he objecive funcion F( φ ) = f ( µ,k ) P is minimized by nonlinear leas square oimizaion. During he oimizaion, he vecor Φ is limied in m he lower and uer bound consrains o make sure he final resuls wih clear hysical meaning. The final resul is as follows Φ = [40,, 0.,,, 0.00,.0, 0.0002] Hence, 0. K = 40 ε ε
.0 µ = ( + h+ 0.00r) + 0.0002 Nr 4. Resuls and Discussion The comarison of measured rolling forces and calculaed rolling forces can be found in Figure, in which Rc reresens he saisical confidence level. I can be seen ha he lowes confidence level is 0. a sand. The average confidence value of five sands in he cold sri mill is 0.34, which means he calculaed rolling forces are in good agreemen wih he measured values.. 0. 0... Rc = 0.... Rc =.. 0 (a) sand (b)sand 2 0 0. 0 Rc = 0.3... Rc = 0...4.3.2. 0. 0. 0 2 x 0 Rc = 0.3 (c) sand 3 (d)sand 4 0 0. 0. 0. 0. 0. 0. 0...2.3.4. x 0 (e) sand Figure. Comarison of measured rolling forces and calculaed values Figure 2 shows he comarison of deformaion resisance in cold sri mill beween oimizaion mehod and emirical daa. I can be seen ha he oimizaion values are in good agreemen wih he emirical daa.
200 Based on Emirical Eq Resuls - x Based on Oimizaion Resuls - o Deformaion Resisance Curve ---- Deformaion Resisance (N/mm 2 ) 000 00 00 400 2nd 3rd 4h h s 200 0 0.. 2 2. Srain Figure 2. Calculaed deformaion resisance Figure 3 and Table 2 show he fricion coefficiens disribuion calculaed by oimizaion mehod from sand o sand in cold sri mill. I can be seen ha he fricion coefficien decrease significanly when he rolling seed increases, aricularly in he earlier firs and second sands. The fricion coefficiens for five rolling sands ranged from 0.0 for sand o 0.0 for sand. 0.0 0.0 s sand 2nd sand 3rd sand 4h sand h sand Fricion Coefficien 0.0 0.03 0.0 0 200 400 00 00 000 200 400 Rolling Seed (m/min) Figure 3. Calculaed fricion coefficiens Table 2. The fricion coefficien values on differen sands Sand Sand 2 Sand 3 Sand 4 Sand Max. 0.0 0.03 0.030 Min. 0.0 0.032 4 3 0.0 The effec of roll wear on he fricion coefficien for sand can be found in Figure 4. The more coils are rolled (or lengh of rolled coils), he amoun of roll wear will be increased. I can be seen from Figure 4 ha he fricion is very much deenden on he roll wear. The resuls shows ha fricion decreases wih roll wear. Afer he work rolls change, he fricion coefficien decreases quickly. However, i will say sable afer he lengh of rolled coils reaches a cerain value. 0.0 0.0 v=2 (m/min) v=3. (m/min) v=0. (m/min 0.0 Fricin Coefficien 0.03 0.0 0 0 000 2000 3000 4000 000 000 Lengh of Rolled Coils (Km) Figure 4. Effec of roll wear on fricion coefficien
The rolling seed is a crucial arameers affecing fricion. Figure shows he variaion of he fricion coefficien wih he rolling seed for sand for a given schedule. I can be seen ha he lower he rolling seed, he higher is he fricion. When he seed is increased from a low value, he fricion coefficien dros significanly. Bu he endency of dro becomes lower when he rolling seed is higher. 0.2 0. 0. 0.4 Fricin Coefficien 0.2 0. 0.0 0.0 v=2 v=3. v=0. 0 0 200 400 00 00 000 200 Rolling Seed (m/min) Figure. Variaion of he fricion coefficien wih rolling seed. Conclusions An oimizaion of maerial deformaion resisance and fricion was develoed according o he real roducion daa in cold rolling. The calculaed rolling forces based on he roosed maerial resisance and fricion coefficien are in good agreemen wih he measured rolling forces. The fricion coefficien in he roll bie is much deenden on he work roll wear and rolling seed. The resuls already obained shows ha fricion decreases wih roll wear, and he lower he rolling seed, he higher is he fricion. Acknowledgemens This work is suored by an Ausralian Research Council Linkage-Projec gran. References. H. Ford, F. Ellis and D. R. Bland. Cold Rolling wih Sri Tension Par Ⅰ- A New Aroximae Mehod of Calculaion and a Comarison wih Oher Mehods. Journal of he Iron and Seel Insiue, 2, : 24-24. 2. M.D. Sone. Rolling of Thin Sri. Iron and Seel Engineer, 3, 30(2): -3. 3. W.Y.D. Yuen, Y. Poelianski and M. Prouen. Variaions of Fricion in he Roll Bie and Their Effecs on Cold Sri Rolling. Proceedings of ISS 3h Mechanical Working and Seel Processing Conference,, 33-3 4. J.Yang. Mahemaical model of rolling rocess. Beijing: Press of Meallurgical Indusry, 3. (in Chinese). J.H. Hichcock, ASME Publicaion, Aendix I, 3.. S.C. Chara, R.P. Canale. Numerical Mehods for Engineers wih sofware and Programming Alicaions. 4h ed. McGraw Hill, 2002.. G.N. Vanderlaas. Numerical Oimizaion Techniques for Engineering Design wih alicaions. McGraw-Hill, 4.. Malab. The Mahworks Inc. 2000.