Non-linear Dynamics of Inlet Film Thickness during Unsteady Rolling Process

Transcription

1 5 CHINESE JOUNAL OF MECHANICAL ENGINEEING Vol 9aNo a16 DOI: 191/CJME available online at wwwspringerlinkcom; wwwcjmenetcom Non-linear Dynamics of Inlet Film Tickness during Unsteady olling Process FU Kuo ZANG Yong* GAO Ziying QIN Qin and WU Diping Scool of Mecanical Engineering University of Science and Tecnology Beijing Beijing 18 Cina eceived July 15; revised November17 15; accepted January Abstract: Te inlet film tickness directly affects film and stress distribution of rolling interfaces Unsteady factors suc as unsteady back tension may disturb te inlet film tickness However te current models of unsteady inlet film tickness lack unsteady disturbance factors and do not take surface topograpy into consideration In tis paper based on te ydrodynamic analysis of inlet zone an unsteady rolling film model wic concerns te direction of surface topograpy is built up Considering te small fluctuation of inlet angle absolute reduction reduction ratio inlet strip tickness and roll radius as te input variables and te fluctuation of inlet film tickness as te output variable te non-linear relationsip between te input and output is discussed Te discussion results sow tat tere is 18 pase difference between te inlet film tickness and te input variables suc as te fluctuant absolute reduction te fluctuant reduction ratio and non-uniform inlet strip tickness but tere is no pase difference between unsteady roll radius and te output Te inlet angle te steady roll radius and te direction of surface topograpy ave significant influence on te fluctuant amplitude of unsteady inlet film tickness Tis study proposes an analysis metod for unsteady inlet film tickness wic takes surface topograpy and new disturbance factors into consideration Keywords: cold rolling lubricant film inlet zone surface topograpy unsteady non-linear 1 Introduction Cold rolling is one of te most important and comple metal forming processes wic involves surface topograpy [1 7] rolling mill vibration [8 1] and friction eat [1 14] Lubrication in cold strip rolling takes a very important part in reducing rolling force and roll wear and improving te surface quality of products [1 ] Te rolling lubrication caracteristic parameters include te rolling pressure te rolling friction stress surface topograpy te film distribution te inlet film tickness and so on Te previous researces [4 7] sow tat te surface topograpy of te rolls and te strip as obvious effect on te rolling process Based on te average flow eynolds equation wic was proposed by PATI et al [15 16] in te late 197s LIU et al [1 ] focused on mied lubrication and obtained te relationsip between te rolling process parameters and te stress distribution of te deformation zone by numerical calculation Similar researces were also done by WILSON et al [4] ASP et al [5] JESWIET [6] and TSAO et al [7] Te lubricant volume of te inlet zone directly affects te film distribution in te deformation zone and te film tickness at inlet point [9] were te plastic deformation of * Corresponding autor yzang@ustbeducn Supported by National Natural Science Foundation of Cina (Grant No ) PD Program Foundation of Ministry of Education of Cina (Grant No 16114) and Beijing Higer Education Young Elite Teacer Project of Cina (Grant No YETP67) Cinese Mecanical Engineering Society and Springer-Verlag Berlin Heidelberg 16 metal begins At a certain degree te inlet film tickness indicates te amount of lubricant brougt into deformation zone by te moving rolls and strip Based on WILSON s classic inlet film tickness equation [17] SUN et al [] proposed an inlet film tickness model in wic te surface topograpy was taken into account LU et al [18] studied te inlet zone of mied lubrication and focused on te inlet zone lengt Te inlet zone models of SUN et al [] WILSON et al [17] and LU et al [18] only concern te steady condition in tis case te caracteristic parameters of rolling lubrication like inlet film tickness keep constant in time However in practical rolling process te unsteady factors suc as fluctuant tension and non-uniform strip tickness may cause variation of te inlet film tickness WANG et al [8 1] developed te unsteady model of inlet film tickness under te fluctuation of roll speed and back tension Toug WANG s model can reflect te non-steady inlet film tickness but it ignores te surface topograpy and te pressure gradient wic may induce inaccurate Te unsteady factors in rolling process suc as unsteady reduction non-uniform inlet strip tickness and roll radius were not concerned also olling process is a transient state wic canges overtime [1 1] Tere are lots of production losses caused by te unsteady rolling process [1] Unsteady factors suc as rolling mill vibration non-uniform strip tickness and fluctuation of back tension may result in te unsteady rolling interface Tese unsteady factors may also cause

2 CHINESE JOUNAL OF MECHANICAL ENGINEEING 5 unsteady rolling lubrication caracteristic parameters suc as te inlet film tickness etc Te ecessive vibration of film tickness may cause roll wear and poor product quality [9] terefore tere is important teoretical significance and use value to do researc on te unsteady inlet film tickness Based on te geometry relationsip of film tickness at inlet zone and te ydrodynamic analysis tis paper developed a model of unsteady inlet film tickness Te surface topograpy factors suc as te direction of surface topograpy were concerned Te effect of oter unsteady factors suc as fluctuant absolute reduction and reduction ratio non-uniform inlet strip tickness and fluctuant roll radius on te inlet film tickness was studied in detail Te relationsip of pase position between te input and output variables te fluctuant amplitude of te inlet film tickness were also included wic is affected by te factors suc as inlet angle direction of surface topograpy and roll radius Film Tickness Geometry elationsip of Inlet zone Te basic rolling process geometry is illustrated in Fig 1 Because of symmetry only alf configuration is considered Te -ais is set as te direction along te inlet zone wic is opposite to te rolling direction te direction perpendicular to te paper is set as te y-ais and te z-ais is vertical to te -ais in te paper Te overall contact area is customarily divided into tree zones [ 9] : te inlet zone te work zone and te outlet zone At te inlet zone te lubricant is drawn into te space between te roll and te strip by te motion of tem Opposite to te inlet zone te lubricant pressure decreases quickly at te outlet zone Work zone is te area were metal plastic deformation appens Fig 1 Scematic of rolling gage wit lubricant According to Fig 1 te circle of te top roll can be described as ( + l) + ( z- z ) (1) were is te inlet film tickness G is te inletstrip tickness Suppose te top roll does not deform during te rolling process and te inlet angle is very tiny [8 9] and approimately tan» According to Fig 1 and Eq (1) te outline of top roll can be epressed as z- -( + l) + z () Epand Eq () at by Taylor series and cut it off at triple function te tickness of lubricant film at inlet zone can be described as G 1 z Tis is te geometry relationsip of film tickness at inlet zone Similar model was also used in oter inlet zone researces [ 8 1] Average Flow eynolds Equation In te late 197s te average flow eynolds equation wic concerns tree dimensional rougness structure was proposed by PATI et al [15 16] In teir model te unit flow in te and y directions are given by ì p æ 1+ ö ç u u u u q T - s ç 1 çè ø p qy y î 1 y were p is te mean pressure of te unit and wen te unit is small enoug p can be considered as te ydrodynamic pressure of a point is te viscosity of te lubricating fluid u 1 and u are te speeds of te roll and te rolled seet at inlet zone is te nominal film tickness T is te average gap and it also can be regarded as te matematical epectation of te actual local film tickness T (Fig ) s is te sear flow factor and y are te pressure flow factors of and y direction q and q y are te local flows of and y direction In above definition q is assumed to consist of tree terms [16] Te first term is te mean flow due to mean pressure gradient in te direction can be tougt as a correction factor Te second term represents te flow transported due to te entrainment velocity ( u1+ u)/ Te tird term represents additional flow transported due to sliding in roug interface Mean flow balance of te control volume results in an average eynolds equation [16] : (4) (5) were is te roll radius l is te work zone lengt z is te centre position of te top roll at vertical direction: z + cos + G () æ ö æ ö p p + ç 1 y è ø yçè 1 y ø u + u u -u t 1 T 1 s T (6)

3 54 Y FU Kuo et al: Non-linear Dynamics of Inlet Film Tickness during Unsteady olling Process Here is te combined rougness eigt [1 4] : 1 + (7) were 1 and are te random rougness eigts of te roll and strip surfaces At direction te pressure flow factor can be set up as follows [1 15] : -rh ì 1 -Ce 1 -r î 1 + Ce > 1 were is te surface pattern parameter as sown in Fig H is te ratio of to r and c are te parameters associated wit Wen > 1(vertical stripe) C < 1 r > 1 > 1; wen < 1 (orizontal stripe) C > 1 r < 1 < 1 [15 16] Te relationsip between y and is [15] (8) æ 1ö ( ) y H H ç çè (9) ø can be tougt as a correction factor wic reflects te influence of te direction of surface topograpy on te fluid flow Wen > 1 te vertical stripe elps te lubricant flow and it increases te flow generated by pressure gradient Wereas wen < 1 te orizontal stripe prevents te lubricant flow and it decreases te flow generated by pressure gradient flow transport will be impeded due to te stagnant fluid in te valleys of te stationary In tis condition te s term will be negative If te two surfaces ave te same rougness configuration tese two effects cancel eac oter and results in s Te surfaces of rolling interface are usually not completely smoot in actual rolling process Assuming te surfaces of te rolls and te strip ave te same combined rougness and same surface [ pattern parameter terefore s 16] In most analysis of rolling process it is usually considered te strip as no epansion at y direction [1] and tere is no side discarge of lubricant in te ais direction of te roll [] ( y direction) wic means te same pressure and no pressure gradient at y direction Based on te above assumption we can get: æ ö p y yçè 1 y ø (1) Te mean velocity of te strip and te roll at inlet zone is Define contact factor c [] 1+ u 1 u u (11) T c (1) c represents te non-contact area ratio in oter word it is te non-contact probability of a point c usually obeys Gaussian distribution linear distribution rectangular distribution and eponential distribution [ 5-6] Based on Eqs (1) (1) and Eq (6) becomes s æ ö ç p - cu 1 + c çè 1 ø t (1) Fig Scematic of surface topograpy Te film of inlet zone is usually considered to be so tick tat c can be regarded as 1 [] terefore Eq (1) turns into 4 Simplification of Average Flow eynolds Equation æ ö p ç - u1 + çè 1 ø t (14) [1 Similar to te sear flow factor 16] s wic is te coefficient of te tird term of q can be tougt as a coefficient to correct additional flow transported due to sliding in roug interface PATI as given a reasonable eplanation for [16] s : in te case of a roug surface moving against a stationary smoot surface te moving roug surface elps to transport te fluid into te gap between te two surfaces On te oter and if a smoot surface is sliding against a stationary roug surface te Neglect te time term Eq (14) can be epressed as æ ö p ç -u1 çè 1 ø (15) 5 Unsteady Hydrodynamic Analysis Vast majority derivation of eynolds equation is under

4 CHINESE JOUNAL OF MECHANICAL ENGINEEING 55 te steady condition wic ignores te t term suc as efs [1 7] However rolling process is a transient state wic sould connect wit time [1] From te previous section te first order Taylor s epansion of te film tickness s geometry relationsip at inlet zone is set as + (16) Take Eq (16) into t and define te time ratio of inlet film tickness + t (17) were is te time ratio of inlet angle Take Eq (17) into Eq (14) and ten integrate we get 1 p + - u + 1 f1 ( ) (18) Te left and of Eq (18) represents te flow generated by pressure gradient in te direction f 1 is a constant generated by te integration At te boundary between inlet zone and work zone ( ) te first and second terms of rigt and of Eq (18) become zeros f1 1 p (19) Te pressure gradient in Eq (19) can be approimately obtained from Eq (15) According to Tresca yield criterion [8 19] te inlet pressure can be epressed as p() - s () were is te yield stress of te strip s is te back tension At infinity of p ( ) (1) From te boundary condition () (1) and Eq (15) we can get te pressure gradient ( ) p 1 u ( ) ( s) + At te inlet point ( ) p u ( s) Take Eq () into Eq (19) we get () () ( - ) s f u (4) 6 Suppose te control unit is small enoug [15] te mean pressure p can be regard as te fluid pressure p of a point [15 16] p p (5) Take Eqs (4) and (5) into Eq (18) we get p ( - u ) + 1 æ ( ) ö - s çu 1-1 ç çè 6 ø (6) According to te power of Eq (6) can be decomposed into tree parts: ì 1 é ( ) ù pa -s 1 u - 6 ê ú ë û pb 1 ( - u 1) p 6 c î (7) were p a p b and p c are separately te corresponding part of 1 power of Take Eq (16) into pa and pb and ten use te boundary condition to integrate tem pa and p b will converge at However tere is quadratic power term of in pc and it does not converge at using te first order film geometry relationsip Te iger power of te larger numerator of te rigt and of Eq (7) and te more impossible to converge pc will be convergent wen using te second order Taylor s epansion of film geometry relationsip We also can compreend tis by te actual geometrical relationsip of roll gage: compared wit te first order Taylor s epansion of te geometry relationsip of film tickness at inlet zone tere eists te etra quadratic term of in te second order Taylor s epansion wic means not only te larger space but also te quicker decrease of te film pressure and easier convergence of te equations as increases Integrate Eq (7) tere is ì 6 é ( ) ù - s pa - 1 u f ê ú ë û 1 æ 1 ö ( - 1) - p b u + ç f çè b ø 6 pc C + fc î a (8)

5 56 Y FU Kuo et al: Non-linear Dynamics of Inlet Film Tickness during Unsteady olling Process were f a f b f c are te constants generated by integration C is a parameter wic includes Te pressure of inlet zone can be epressed as C p p + p + p (9) a b c Use te boundary condition (1) we get Take Eq () into Eq (9) we get fa + fb + f c () 6 1 æ 1 ö + ( 1) p C - u ç - - çè ø é ( - ) ù s 6 1 u 1-6 êë úû Use te boundary condition () Eq (1) becomes (1) C () were C is te parameter wic includes and ( + ) ( ) ( ) were DG G are te steady absolute reduction steady reduction ratio steady inlet strip tickness and steady roll radius δ DG δ δ G δ are te wave amplitudes of teir unsteady terms is te frequency Based on Eqs (4) and (6) te unsteady inlet angle separately caused by unsteady absolute reduction unsteady reduction ratio unsteady inlet strip tickness and unsteady roll radius can be epressed as ì G + δgsin( t) [ + δsin( t) ] G [ G + δg sin( t) ] G î + δsin( t) (7) Te unsteady inlet angle can be unified and epressed as + δ () t (8) were is te steady inlet angle and can be calculated by Eq (4) δ () t is te fluctuant part caused by te fluctuation of unsteady factors - - ln + - () Eq () is te epression of te inlet film tickness s time ratio Assume tat te terms and relate to time and oter parameters are steady according to te rolling teory [19] te inlet angle can be epressed as l G G (4) ì æ δ sin( ) ö δ () G t 1 1 t + - çè G ø æ δsin( ) ö δ () t t ç çè ø æ δ sin( ) ö δ () G t t ç çè G ø æ 1 ö δ () t - 1 ç 1+ δ sin( ) / è t î ø (9) were is te reduction ratio D G is te absolute reduction According to te rolling teory [19] ì G G-G1 G G-G1 î G G (5) Wen tere is small fluctuation of sine waves of te absolute reduction te reduction ratio te inlet strip tickness and te roll radius ì G G + δgsin( t) + δsin( t) G G + δgsin( t) î + δsin( t) (6) Under te special condition of sine wave of δ () t te inlet angle can be epressed as + δsin( t ) (4) Take Eqs (8) and (4) into Eq () and ten integrate tem we can get C C t (41) 1 + δ () C + C δ sin( t ) (4) were is te steady inlet film tickness C is te steady form of C C 1 and C are te constants generated by integration Based on average flow eynolds

6 CHINESE JOUNAL OF MECHANICAL ENGINEEING 57 equation a steady inlet film tickness model was set up in ef [] ere 1 (4) a were a is te corresponding inlet film tickness not concerning te surface topograpy [] and it can be calculated by te famous inlet film tickness model proposed by WILSON et al [17] Te lubricant viscosity and te pressure coefficient of viscosity of lubricant ave a significant influence on a As te lubricant viscosity increases a becomes ticker Based on te assumption of te initial condition t we can get C1 C and Eqs (41) and (4) become C t (44) + δ () + C δ sin( t ) (45) 6 Nondimensionalization To simplify te calculation non-dimensional variables sould be adopted Define te non-dimensional time and non-dimensional frequency as [8 1] ì t t u T / u1 î u1 1 (46) were T is te non-dimensional time is te non-dimensional frequency T t u1 is te steady roll speed Based on te assumption is very small u 1 can be regarded as te vertical speed of te roll and /u 1 can be regarded as te time of passing te inlet film tickness in te vertical direction Eq (44) can be nondimensionalized as H H + M sin( T ) (49) q were M is te fluctuant amplitude of te inlet film tickness ì Hq q C M î 7 Eample and esults q δ (5) Eqs (47) and (49) can be solved by numerical calculation wit Matlab Parameters in te calculation come from a process of rolling aluminum seet on a two-roll mill wit roll radius of mm rolling speed ranging from m/s to 48 m/s tickness of aluminum seet is 1 mm initial inlet film tickness of 5 m reduction ratio of back tension stress ranging from to 684 MPa lubricant viscosity of 8 Pa s yield stress of aluminum of 9775 MPa and te combined surface rougness is 1 m 71 Pase position analysis of inlet film tickness Based on Eqs (7) (41) wen te fluctuant amplitude as no difference te influence of te unsteady factors on te inlet angle and inlet film tickness will be te same q 5 μm 1 δ G/ G δ / δ G / G 1 Fig sows tat wen te unsteady factors get sine wave (input) te inlet angle and te inlet film tickness will ave non-linear cycle fluctuation wic fluctuates in te same frequency of te input and is similar to te sine wave Te maimum value of inlet angle is and minimum value is Te maimum value of inlet film tickness is and minimum value is 918 H H H (47) q + were Hq is te initial non-dimensional inlet film tickness H is te unsteady term of te non-dimensional inlet film tickness ì Hq q C H î q δ () t (48) Fig Influence of unsteady factors on te inlet angle and inlet film tickness q is a definitive inlet film tickness under te steady condition Eq (45) can be nondimensionalized as In Fig tere is no pase difference between te input and inlet angle wereas tere is 18 pase difference

7 58 Y FU Kuo et al: Non-linear Dynamics of Inlet Film Tickness during Unsteady olling Process between te input variable and unsteady inlet film tickness It coincides wit te facts: wen tere is instantaneous decrease of te absolute reduction or te reduction ratio te top roll will ave a downtrend wic means a tinner inlet film tickness Due to te etrusion wen tere is instantaneous increase of te inlet strip tickness te inlet film tickness will become tinner Te influence of unsteady roll radius on te inlet angle and inlet film tickness is sown in Fig 4 Wen tere is sine wave of unsteady roll radius (input) te inlet angle and te inlet film tickness will ave non-linear cycle fluctuation and tey fluctuates in te same frequency of te input q 5 μm 1 δ / 1 Te maimum value of inlet angle is and te minimum value is 95 5 Te maimum value of inlet film tickness is and te minimum value is 99 5 orizontal stripe increases it Fig 5 Influence of surface stripe direction on te inlet film tickness 7 Influence of inlet angle and roll radius on te fluctuant amplitude of inlet film tickness Te influence of inlet angle on te fluctuant amplitude of unsteady inlet film tickness is sown in Fig 6 ere δ / 1 Te steady inlet angle is separately and 6 Te corresponding fluctuant amplitude of unsteady inlet film tickness M is and 16 7 Fig 6 sows tat te fluctuant amplitude of unsteady inlet film tickness decreases as te steady inlet angle increases Fig 4 Influence of unsteady roll radius on te inlet angle and inlet film tickness In Fig 4 tere is 18 pase difference between te unsteady roll radius and te inlet angle wereas tere is no pase difference between te input (unsteady roll radius) and te inlet film tickness In te practical rolling process te deformation zone is very small compared to te roll radius Te fluctuation of roll radius can be regarded as te concave and te bulge wic eist in te surface of te roll Wen te roll radius increases it can be regarded as te bulge of te roll entering te deformation zone gradually wic brings more lubricant and increases te inlet film tickness instantaneously 7 Influence of stripe direction on te inlet film tickness Fig 5 sows te influence of surface topograpy (stripe direction) on te inlet film tickness δ / 1 is separately 1/9 1/ 1 9 Wen > 1 te vertical stripe elps te lubricant flow and it decreases te steady inlet film tickness; wen < 1 te orizontal stripe prevents te lubricant to flow and it increases te steady inlet film tickness Te vertical stripe decreases te fluctuant amplitude of unsteady inlet film tickness wereas te Fig 6 Influence of inlet angle on te fluctuant amplitude of unsteady inlet film tickness Wen te steady roll radius is separately 18 mm 19 mm mm 1 mm mm mm te corresponding fluctuant amplitude of unsteady inlet film tickness M is Fig 7 sows tat te fluctuant amplitude of unsteady inlet film tickness increases as te steady roll radius increases 74 Comparison and validation Te inlet film tickness calculated separately by tis paper and WANG s model [8 1] is sown in Fig 8 Te corresponding fluctuant amplitude of unsteady inlet film

8 CHINESE JOUNAL OF MECHANICAL ENGINEEING 59 tickness in tis study is In WANG s model M18 and tere is eponential damping in it orizontal stripe increases it (4) As te inlet angle increases te fluctuant amplitude of inlet film tickness decreases Wen te steady roll radius increases te fluctuant amplitude of inlet film tickness increases wic induced by oter unsteady factors like nonuniform inlet strip tickness unsteady absolute reduction and reduction ratio Fig 7 Influence of roll radius on te fluctuant amplitude of unsteady inlet film tickness 8 Conclusions Fig 8 Comparison of inlet film tickness (1) Based on geometry relationsip of film tickness at inlet zone and ydrodynamic analysis tis paper developed an unsteady inlet film tickness model in wic te direction of surface topograpy was concerned Te effect of unsteady factors suc as fluctuant reduction non-uniform inlet strip tickness and fluctuant roll radius on te inlet film tickness was studied in detail Te relationsips of pase position between te input and output variables and te fluctuant amplitude of te inlet film tickness were also discussed () Tere is 18 pase difference between input variables (absolute reduction reduction ratio inlet strip tickness) and unsteady inlet film tickness Wen te fluctuant amplitudes of absolute reduction reduction ratio and inlet strip tickness increase te inlet film tickness instantaneously decreases () Te vertical stripe elps te lubricant flow and decreases te steady inlet film tickness; te orizontal stripe prevents te lubricant to flow and increases te steady inlet film tickness Te vertical stripe decreases te fluctuant amplitude of inlet film tickness wereas te eferences [1] LIU Lemin ZANG Yong CHEN Yuanyuan Hydrodynamic analysis of partial film lubrication in te cold rolling process[j] International Journal of Advanced Manufacturing Tecnology 1 54: [] LIU Lemin ZANG Yong CHEN Yuanyuan Study on te ydrodynamic lubrication beavior in cold rolling process[j] Advanced Materials esearc : [] SUN Jianlin KANG Yonglin ZHANG Xinming Model for inlet film tickness in mied-lubrication rolling[j] Te Cinese Journal of Nonferrous Metals 1 11(1): 18 1 [4] WILSON W D LEE W Mecanics of surface rougening in metal forming processes[j] Journal of Manufacturing Science and Engineering 1 1: 79 8 [5] ASP W WICHEN C M Effects of surface-topograpy directionality and lubrication condition on frictional beavior during plastic deformation[j] Journal of Materials Processing Tecnology 15 16: [6] JESWIET J A comparison of friction coefficients in cold rolling[j] Journal of Materials Processing Tecnology : 9 44 [7] TSAO Y H SAGENT L B A mied lubrication model for cold rolling of metals[j] ASLE Transactions 1977 : 55 6 [8] WANG Qiaoyi YU Dejie esearc on dynamic property of metal-forming processes based on an unsteady lubrication teory[j] Journal of Hunan University 6 (): [9] WANG Qiaoyi LI Ziua esearc on caracteristics of oil film for work interface in metal rolling process[j] Cina Mecanical Engineering 8 (15): [1] WANG Qiaoyi HUANG Haijun LI Ziua Control of vibration for unsteady lubrication based on metal-forming process[j] Journal of Central Sout University 1 41(4): [11] HU P H EHMANN K F A dynamic model of te rolling process PartI: omogeneous model[j] International Journal of Macine Tools & Manufacture 4: 1 19 [1] HU P H EHMANN K F A dynamic model of te rolling process Part II: inomogeneous model[j] International Journal of Macine Tools & Manufacture 4: 1 1 [1] LIU Y J TIEU A K A termal mied film lubrication model in cold rolling[j] Journal of Materials Processing Tecnology 1: 7 [14] TIEU A K KOSASIH P B GODBOLE A A termal analysis of strip-rolling in mied-film lubrication wit O/W emulsions[j] Tribology International 6 9: [15] PATI N CHENG HS An average flow model for determining effects of tree dimensional rougness on partial ydrodynamic lubrication[j] ASME Journal of Lubrication Tecnology : 1 17 [16] PATI N CHENG H S Application of average flow model to lubrication between roug s1iding surfaces[j] ASME Journal of Lubrication Tecnology : 9 [17] WILSON WD WALOWIT J A An isotermal ydrodynamic lubrication teory for strip rolling wit front and back tension[c]//tribology Convention London United Kingdom 1971: [18] LU C TIEU A K JIANG Z Y Modeling of te inlet zone in te

9 5 Y FU Kuo et al: Non-linear Dynamics of Inlet Film Tickness during Unsteady olling Process mied lubrication situation of cold strip rolling[j] Journal of Materials Processing Tecnology 14: [19] ZHAO Ziye Metal plasticity deformation and roll teory[m] Beijing: Metallurgical Industry Press 1996 Biograpical notes FU Kuo born in1986 is currently a PD candidate at Scool of Mecanical Engineering University of Science and Tecnology Beijing Cina He received is bacelor degree from Taiyuan University of Science and Tecnology Cina in 9 His researc interests include unsteady rolling process and rolling lubrication Tel: ; fukuosince1986@16com ZANG Yong born in 196 is currently a professor and a PD candidate supervisor at Scool of Mecanical Engineering University of Science and Tecnology Beijing Cina His researc interests include analysis and control of mecanical beavior of industry equipment yzang@ustbeducn GAO Ziying born in 1979 is currently an associate professor at Scool of Mecanical Engineering University of Science and Tecnology Beijing Cina Her researc interests include mill vibration and nonlinear vibration analysis gaoziying@meustbeducn QIN Qin born in 197 is currently an associate professor at Scool of Mecanical Engineering University of Science and Tecnology Beijing Cina His researc interests include simulation and control beavior of industrial macinery qinqin@meustbeducn WU Diping born in 197 is currently an associate professor at Scool of Mecanical Engineering University of Science and Tecnology Beijing Cina His researc interests include computer simulation and field test wudiping@ustbeducn