Development of Strip Lateral Movement Simulator for Temper Rolling+1

Shinichiro Aoe1,+2, Takuya Kitamura2, Tomoyoshi Ogasahara1 and Masaru Miyake3

1Steel Research Laboratory, JFE Steel Corporation, Kurashiki 712-8511, Japan2Steel Research Laboratory, JFE Steel Corporation, Chiba 260-0835, Japan3Steel Research Laboratory, JFE Steel Corporation, Handa 475-8611, Japan

The temper rolling process of hot-rolled strips is the final rolling step to improve the flatness of the strips and shape slippage of the coiledstrips. Poor flatness of a hot-rolled strip causes lateral movement of the strip during the temper rolling process. Manual leveling operations tocontrol this movement result in a much lower line speed and productivity. Moreover, the quality of manual leveling is not consistent, as itdepends on the operator’s experience. Lack of understanding of the lateral movement phenomenon has sustained manual operation anddiscouraged the development of an automatic control system. To solve this problem, the authors propose both a new lateral movement model anda theoretical method. A lateral movement model for temper rolling and a large-deflection strip model are important components in the newmodel. In the method, lateral movement stability is equivalent to an eigenvalue problem with lateral movement static equations. Their usefulnessis confirmed by comparing the results of experimental rolling in the laboratory with those of numerical calculations. The simulation resultsobtained using the proposed models confirm that actual problems can be solved more exactly than with the conventional linear model. Thus,simulation using the proposed models can support the investigation of lateral movement problems and the development of an automatic controlsystem. [doi:10.2320/matertrans.P-M2021832]

(Received December 21, 2020; Accepted April 4, 2021; Published July 25, 2021)

Keywords: temper rolling, strip lateral movement, rolling theory, buckling, hot-rolled steel strip, simulation, phase diagram

1. Introduction

The temper rolling process of hot-rolled steel strips, whichimproves the flatness of the strips and shape slippage of thecoiled strips, prevents lateral movement of hot-rolled stripsin downstream process lines. Figure 1 shows a developedcontrol system of a temper rolling line. The automatic controlsystem provides elongation setup based on a shape predictionmodel,1) dynamic control of flatness and elongation2) andcontrol of lateral movement. The shape prediction modelpredicts the strip shape at the mill entry side by usingphysical models and sensing data in the hot rolling process.Elongation setup outputs the tolerance range of elongationbased on the predicted strip shape at the entry side. Dynamiccontrol of flatness and elongation controls the strip flatnessto zero within the tolerance range of elongation. In thisdevelopment, a strip lateral movement simulator for thetemper rolling line, as shown in Fig. 2, was also developed.This numerical simulator makes it possible to assess theperformance of the logic and the robustness of lateralmovement control, and also enables assessment of opera-tional conditions such as a strip tension. Its backgroundtheories are described in detail in the following.

A strip sometimes appears to move in the lateral directionduring rolling with a rolling mill.3) Because this lateralmovement causes lower productivity and the work-rolldamage, many researchers and engineers have studied andanalyzed lateral movement in hot-strip4,5) and cold-strip3,6)

rolling. A lateral movement analysis for strip rolling requiresa coupled analysis of the strip lateral movement at the rollingmill and a model of strip deflection at the mill entry anddelivery sides. In general, a beam model is used in the stripdeflection model.3,6) The beam model is tuned to represent

Optimumcontroller

・Control range of elongation・Target flatness

Flatnessprediction

model

Lateralmovementcontroller

・Control range of elongation・Limitation of actuators

Quadratic programmingObjective function

Constraints・Flatness deviation

Shape meter

Roll speed

Lateral position

meter

・Elongation

Rolling direction

Payoff reel Tension reelRolling mill

Lateral position

meter

Roll speed Width meter

Thickness meter

・Roll separating force・Roll bending force

Optimal solutions

・Levelingposition

・Flatness

・Differentialtension

Fig. 1 Dynamic control system of temper rolling line.

CPCcontrol

Payoff reel

Tension reel

Shape meter

Rolling direction(maximum velocity 800 mpm)

EPC control

Out-of-plane deflectionof entry strip

Entrytension

Deliverytension

Temper rolling mill

Fig. 2 Strip lateral movement simulator for temper rolling line onADAMS.

+1This Paper was Originally Published in Japanese in J. Jpn. Soc. Technol.Plast. 61 (2020) 107­114. The captions of figures were slightly modified.

+2Corresponding author, E-mail: s-aoe@jfe-steel.co.jp

Materials Transactions, Vol. 62, No. 8 (2021) pp. 1168 to 1176©2021 The Japan Society for Technology of Plasticity

the actual results considering strip buckling. For example,bending stiffness is tuned to be smaller. On the other hand,from the viewpoints of calculation time and convergence, thecoupled analysis7) of microscopic rolling phenomena and thestandard elastic shell model is practically difficult.

A large-deflection strip model8) and a large-deflectiondiscrete strip model9) have been developed to solve the aboveproblems. The first model can be applied to a straight pathand elastic bending and is formulated by approximating anelastic plate model. The second model can be applied toa curved path and elasto-plastic bending. The simulatorshown in Fig. 2 represents the simulator based on the large-deflection discrete strip model. In the present study,a new lateral movement model (LMM) for temper rollingis derived from the approximation model for cold-striprolling, considering the temper rolling reduction is verysmall. A LMM for a temper rolling line with payoff, rollingand coiling processes, is proposed, and its usefulness isconfirmed by comparing the theoretical results, the resultsof experimental rolling in the laboratory and the results ofnumerical calculations. A lateral movement simulation ofthe simplex coiling process calculated by a general FEMhas already been presented,10) but lateral movementsimulation of a complex process such as a temper rollingline, including the payoff, rolling and coiling processes, isthe first trial.

2. Lateral Movement Model of Temper Rolling Line

2.1 Outline of modelFigure 3 shows the experimental rolling line to clarify the

proposed LMM. The line comprises a 2-high rolling mill,a payoff reel, a tension reel and deflector rolls at the entryand delivery sides. The LMM is solved by using eitherMATLAB/Simulink or MSC.ADAMS. The former has bettercompatibility with the large-deflection strip model,8) whereasthe latter is a general analysis application for multi-bodydynamics and has better compatibility with the large-deflection discrete strip model.9) The simulator in Fig. 2was built on MSC.ADAMS. The modeling and calculationtime of a simulator on MATLAB/Simulink is shorter. Eachmodel on Simulink is represented as a block diagram. Theblock provides the model calculation and interface ports

which input and output information such as forces andvelocities. The block is represented by a state-space modeland is described as a S-Function. The blocks represent theLMM at a payoff reel, referring to Appendix A2, the stripmodel at the entry side, the LMM of the temper mill, thestrip model at the delivery side and the LMM at a tensionreel with a boundary condition of simple support consideringthe deflector roll. These blocks are arranged in the order ofstrip movement. The unit system in this paper is the SI unitsystem.

2.2 Lateral movement model of temper rollingFigure 4 shows the schematic of strip lateral movement in

temper rolling. The x-axis shows the strip rolling direction,and the y-axis shows the lateral direction. The LMM for coldrolling is formulated by using the Bland & Ford formula,which is an approximation of cold rolling. The LMM fortemper rolling is formulated by simplifying the LMM forcold rolling, considering the very small reduction of the stripin temper rolling.

The work rolls of a rolling mill grip a strip. Assumingthe strip velocity in the lateral direction is zero (non-slip), theconstraint3,6) is given by

Dvn=Dt ¼ dvn=dtþ Uªn ¼ 0 ð1Þwhere t is time, vn is the strip lateral displacement at the mill,ªn is the strip orientation angle and U is the rolling velocity.When rolling an oriented strip, the strip appears to be movingin the lateral direction.

Considering the very small reduction in temper rolling andapproximating the forward slip equation of Bland & Ford11)

according to the formulation described in Appendix A1, thestrip orientation angle velocity is given by

dªndt

¼ U � 3

4Aðh0df � h1dfÞ þ

¢M0

kIM0 �

¢M1

kIM1 � ¬1

� �ð2Þ

where h0df and h1df are the wedges of the delivery and entrystrip, M0 and M1 are the in-plane moment acting on thedelivery and entry strip, ¬1 is the camber curvature of theentry strip, A is the strip cross-sectional area and I is thesecond moment of strip area. A and I are approximated ashb and hb3/12 when the wedge is very much smaller thanthe strip thickness h, where b is the strip width. k is the stripdeformation resistance. ¢M0 and ¢M1 show the influencecoefficients of the delivery moment M0 and the entry momentM1. They are represented respectively by

1750

620

600

300

φ 410

500△

1250 1250 1750

1000

Payoff reel Tension reel

Rolling millφ 410

φ 300φ 300Rolling direction

18 mm/s

620

Drive side

Operation sideLateralposition meter

Delivery unittension 49, 98 MPa

Entry unit tension9.8 to 98 MPa

Fig. 3 Experimental rolling line.

x

y

nvnθ1T 0T

Rolling direction

Velocity UWork roll

Entry side

Rolling mill

Payoff reel Tension reel

L LDelivery side

Fig. 4 Schematic of strip lateral movement in temper rolling.

Development of Strip Lateral Movement Simulator for Temper Rolling 1169

¢M0 ¼ffiffiffiffiffiffiffiffiffiffih=R0p

4®f1� ðT0 � 2®Pe0Þ=ðkAÞg

ffiffiffir

p;

¢M1 ¼ffiffiffiffiffiffiffiffiffiffih=R0p

4®f1� ðT1 � 2®Pe1Þ=ðkAÞg

ffiffiffir

p

where ® is the friction coefficient, RA is the radius of theflattening work roll, r is the reduction in thickness, T0 and T1are the tensions acting on the delivery and entry strip and Pe

0

and Pe1 are the roll separating force11) at the elastic recovery

and compression zones.Considering the plastic action on the rolled strip and the

mill stiffness, the equation for the delivery wedge4) is givenby

h0df ¼ 1

Kl þMl

�2bP

bl2vn þ

bKl

blSdf þMlh1df

� 16bhrKl

br2vn þ

P

khbl2ð1� ¡cÞðM0 þM1Þ

�ð3Þ

where Sdf is the difference in mill leveling, and P is the rollseparating force which contributes to strip rolling based onHill’s formula11) and is represented by

P ¼ 1:08þ r 1:79®

ffiffiffiffiffiffiR0

h

r� 1:02

!( )kð1� ¡cÞb

ffiffiffiffiffiffiffiffiffiffiR0hr

p:

bl is the distance between the work-roll chocks, br is thewidth of the work-roll barrel, hr is the center deflection of thework-roll barrel, Kl is the mill leveling stiffness and Ml is thestrip leveling stiffness represented by

Ml ¼ 0:54þ 3r

21:79®

ffiffiffiffiffiffiR0

h

r� 1:02

!( )kð1� ¡cÞb3

6bl2

ffiffiffiffiffiffiR0

rh

r:

¡c is the coefficient of the tension effect and is representedby

¡c ¼ ðT0 þ T1 � 2®Pe0 � 2®Pe

1Þ=ð2kAÞ:Adding the camber curvature by rolling to the entry

camber curvature ¬1, the delivery camber curvature4) ¬0 isgiven by

¬0 ¼ ðh0df � h1dfÞ=Aþ ¬1: ð4Þ

2.3 Strip deflection modelThe large-deflection strip model,8) which is an approxi-

mated model of the large-deflection plate model, is appliedto simulations on MATLAB/Simulink as the strip deflectionmodel. To shorten the calculation time, it is assumed that thestrip model does not flow in the strip rolling direction withinthe analysis space (pseudo-Euler form). A FEM analysis ofthe plate model makes the process of camber flow difficult,and the many elements necessary in order to secure theconvergence of a large-deflection FEM analysis result in alonger calculation time. On the other hand, using a basicbeam model shortens the calculation time and improvesthe convergence of calculation, but the beam model preventsprecise simulations. Therefore, we applied the large-deflection strip model, which is a specialized model forlateral movement problems and calculates out-of-planeelastic deflection with bending and twist, as shown inFig. 5. The functional of the large-deflection strip modelshown in Fig. 6 and the boundary conditions are given by

� ¼Z L

0

�EI

2ðcos½ð@xxvÞ þ sin½ð@xxwÞ � ¬Þ2

þ T

2ð@xvÞ2

�dx

þZ L

0

�Db

2ðsin½ð@xxvÞ � cos½ð@xxwÞÞ2

þ T

2ð@xwÞ2

�dx

þZ L

0

�Db3

24ð@xx½Þ2 þ ð1� ¯ÞDbþ Tb2

24

� �ð@x½Þ2

þ Ehb5

1440ð@x½Þ4

�dx

þMEz ð@xvðt; 0Þ � ªEÞ �MD

z ð@xvðt; LÞ � ªDÞ� FE

y ðvðt; 0Þ � vEÞ þ FDy ðvðt; LÞ � vDÞ;

wðt; 0Þ ¼ wðt; LÞ ¼ @xwðt; 0Þ ¼ @xwðt; LÞ ¼ 0;

½ðt; 0Þ ¼ ½ðt; LÞ ¼ @x½ðt; 0Þ ¼ @x½ðt; LÞ ¼ 0 ð5Þwhere L is the strip length, v is lateral deflection, w is verticaldeflection, ½ is the torsion angle, ¯ is the Poisson ratio, E isYoung’s modulus. D is the bending rigidity and is defined as

D ¼ Eh3=12� ð1� ¯2Þ�1:

T is tension, ¬ is the camber curvature that flows within thestrip, ME

z and MDz are the in-plane moments at the entry and

delivery sides, FEy and FD

y are the lateral shearing forces atthe entry and delivery sides, ªE and ªD are the orientationangles at the entry and delivery sides and vE and vD are thelateral displacements at the entry and delivery sides. In thispaper, the first-order partial derivative of a function v withrespect to the variable x is denoted by @xv. Equation (5)shows that deflections v and w are independent (in-planedeflections) if the torsion angle ½ is constrained to be zero,whereas the deflection v interacts nonlinearly with deflectionw by the medium of the torsion angle ½ (out-of-planedeflection).

Fig. 5 Torsional strip with unstable lateral movement.

Uy

x

TT

Ev DvEθ

EzM

DzM

EyF

DyF

( )xv

y

z

( )xw( )xω

Dθ

Fig. 6 Large-deflection strip model.

S. Aoe, T. Kitamura, T. Ogasahara and M. Miyake1170

The large-deflection strip model shown in Fig. 6 iscalculated nonlinearly by using the Rayleigh-Ritz method.12)

This calculation process is implemented by using a functionof the MEX S-Function, where the calculation of the tangentstiffness matrix and a solver of the system of linear equationsare programmed by using C/C++ language to shorten thecalculation time.

The block diagram of the strip deflection model interactswith the block diagrams of the LMM at the mill or on therolls through their information of forces and velocities. Theadvection equation of the camber curvature within a strip

D¬=Dt ¼ @t¬þ U@x¬ ¼ 0

is calculated by using the CIP method, which enables fastconvergence of calculation.

3. Experimental Rolling in Laboratory and Clarifica-tion of Models

3.1 Experimental rolling in laboratoryThe bifurcation point, where the path of lateral movement

changes from convergence to divergence with a decrease ofentry tension, is known.13) Here, the validity of the LMM ofthe temper rolling line is clarified by performing experimentalrolling in the laboratory and comparing the experimentalresults of the bifurcation point with the theoretical results.

Table 1 shows the experimental rolling conditions. Theunderlined values are the values of the experimentalconditions. First, steel strip rolling was stabilized withoutlateral movement in the initial state of the experiments. Theentry and delivery unit tensions were set to be 49MPa, andthe lateral displacement at the mill in manual leveling was setso as to achieve a convergence of within 3mm. Experimentalrolling was started after the entry unit tension was set to29MPa, and the entry tension was then decreased stepwiseduring rolling while monitoring the strip lateral movement.Rolling was stopped when the lateral displacement at themill exceeded a specified value. Figure 7 shows the resultsof this experiment, where the vertical axis shows the lateraldisplacement at the mill and the horizontal axis shows time.The path of the lateral displacement converged when theentry unit tension was above 20MPa, and diverged when the

entry unit tension was below 10MPa. The elongation in thisexperiment was about 1.5%. Figure 8 shows the exper-imental results when the delivery unit tension was set to98MPa and the other experimental conditions were the sameas those shown in Fig. 7. The path of the lateral displacementconverged when the entry unit tension was above 20MPa,and diverged when the entry unit tension was below 10MPa.Lateral displacement seemed to increase gradually when theentry unit tension was 20MPa. The elongation in thisexperiment was about 2.0%.

These experiments confirmed that the critical entry tension,which is the smallest tension for maintaining stable lateralmovement, existed at the bifurcation point. The critical entrytension slightly increased and lateral movement seemed tobecome slightly unstable when the delivery tension waslarger. Hence, it is considered that the delivery tension did notseriously affect the phenomena of lateral movement in theseexperiments.

3.2 Clarification of LMM by eigenvalue analysis forsteady state of lateral movement

In this section, the mechanism of the experimentalphenomena is theoretically analyzed by using lateral move-ment equations. It is considered that lateral deflectionconverges if the steady state of lateral movement3) existsand diverges if it does not exist. Figure 4 shows theschematic of the analyzed lateral movement. The strips atthe entry and delivery sides are assumed to be an elastic beam(restricted to in-plane bending) with the length of L.Considering the bias errors are zero, i.e.,

Table 1 Experimental rolling conditions.

0

20

40

60

80

100

-20

-15

-10

-5

0

5

0 200 400 600 800

Uni

t ten

sion

/ M

Pa

Late

ral d

ispl

acem

ent /

mm

Time / s

Lateral displacement Entry tension Delivery tension

Fig. 7 Experimental results (delivery unit tension 49MPa).

0

20

40

60

80

100

-20

-15

-10

-5

0

5

0 200 400 600 800

Uni

t ten

sion

/ M

Pa

Late

ral d

ispl

acem

ent /

mm

Time / s

Lateral displacement Entry tension Delivery tension

Fig. 8 Experimental results (delivery unit tension 98MPa).

Development of Strip Lateral Movement Simulator for Temper Rolling 1171

h1df ¼ 0; ¬1 ¼ 0; Sdf ¼ 0

and the equations are steady (i.e., the time derivative is zero),the lateral velocity constraint of eq. (1) and the orientationangle velocity of eq. (2) become

ªn ¼ @xv0ð0Þ ¼ 0; ð6Þ

� 3

4h0df þ

12¢M0

kb2M0 �

12¢M1

kb2M1 ¼ 0 ð7Þ

where v0 is the lateral deflection of the delivery beam.Considering the bias errors to be zero, eq. (3) of the deliverywedge becomes

h0df ¼ £vn þ ¸ðM0 þM1Þ ð8Þwhere £ and ¸ are the influence coefficient of the lateraldisplacement vn and the sum of the entry and deliverymoments, respectively. As eq. (8) agrees with eq. (3), £

and ¸ become

£ ¼ 2b

Kl þMl

P

bl2� 8hrKl

br2

� �;

¸ ¼ P

khbl2ð1� ¡cÞðKl þMlÞ:

Substituting eq. (8) in eq. (7), eq. (7) becomes

� £vn � S0M0 � S1M1 ¼ 0 ð9Þwhere

S0 ¼ ¸ � 16¢M0=ðkb2Þ; S1 ¼ ¸ þ 16¢M1=ðkb2Þ:The lateral deflection equation of the entry beam is given

by

EI@xxxxv1 � T1@xxv1 ¼ 0 ð�L � x � 0Þ ð10Þwhere v1 is the lateral deflection of the entry beam. Thelateral deflection equation of the delivery beam is given by

EI@xxxxv0 � T0@xxv0 ¼ 0 ð0 � x � LÞ: ð11ÞConsidering the fact that the coiled strips on the payoff

reel are rubbed together with a very small slip rate, theboundary conditions at the payoff reel are formulated asdescribed in Appendix A2 and are represented by

v1ð�LÞ � S2@xv1ð�LÞ ¼ 0; ð12Þ�S3T1@xv1ð�LÞ � S2EI@xxxv1ð�LÞ

þ EI@xxv1ð�LÞ ¼ 0 ð13Þwhere

S2 ¼ ºPORRPOR; S3 ¼ º2PORRPOR®POR=ð2­PORÞ;

®POR is the friction coefficient, ­POR is the critical slip ratio,14)

RPOR is the coil radius and ºPOR (= ³) is the effective windingangle.

The boundary conditions at the tension reel are representedby

v0ðLÞ ¼ 0; ð14ÞEIð@xxv0ðLÞ � ¬0Þ ¼ 0: ð15Þ

Substituting eq. (8) in eq. (4), the camber curvature ¬0 ofthe delivery strip becomes

¬0 ¼ 1þ ¸EI

A

� ��1£

Av0ð0Þ þ

¸EI

Að@xxv0ð0Þ þ @xxv1ð0ÞÞ

� �:

ð16ÞSubstituting eq. (16) in eq. (15), eq. (15) becomes

� 1þ ¸EI

A

� �@xxv0ðLÞ þ

£

Av0ð0Þ

þ ¸EI

Að@xxv0ð0Þ þ @xxv1ð0ÞÞ ¼ 0: ð17Þ

The connection conditions between the entry strip and thedelivery strip at the mill are given by

v1ð0Þ � v0ð0Þ ¼ 0; @xv1ð0Þ � @xv0ð0Þ ¼ 0: ð18ÞAccording to eq. (9), the balance equation of the in-planemoments at the mill is represented by

�£vn � S0EIð@xxv0ð0Þ � ¬0Þ � S1EI@xxv1ð0Þ ¼ 0: ð19ÞSubstituting eq. (16) in eq. (19), eq. (19) becomes

EI S1 þ ðS1 � S0Þ¸EI

A

� �@xxv1ð0Þ þ S0EI@xxv0ð0Þ

þ £ 1� S0EI

Aþ ¸EI

A

� �v0ð0Þ ¼ 0: ð20Þ

Appling an eigenvalue analysis, where the eigenfunctionsare the lateral deflection v1 and v0, to the linear systemrepresented by the field equations of eqs. (10) and (11) andthe boundary conditions of eqs. (6), (12), (13), (14), (17),(18) and (20) of the steady state, the condition if deflectionsolutions exist or not is formulated15) (eigenvalue problem).The general solutions of eqs. (10) and (11) are represented by

v1ðxÞ ¼ C1 þ C2xþ C3 sinhðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT1=ðEIÞ

pxÞ

þ C4 coshðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT1=ðEIÞ

pxÞ; ð21Þ

v0ðxÞ ¼ D1 þD2xþD3 sinhðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT0=ðEIÞ

pxÞ

þD4 coshðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT0=ðEIÞ

pxÞ ð22Þ

where C1, C2, C3, C4, D1, D2, D3 and D4 are unknownparameters. Substituting eqs. (21) and (22) in the 8 boundaryconditions of eqs. (6), (12), (13), (14), (17), (18) and (20), thelinear system is given by

½J�T ½C1 C2 C3 C4 D1 D2 D3 D4� ¼ 0: ð23ÞEquation (23) shows that all unknown parameters in thesteady state can become zero. On the other hand, all unknownparameters are considered to become indefinite if lateralmovement diverges. Lateral movement bifurcates when thedeterminant «J « of the matrix [J ] in eq. (23) is zero, i.e.,

jJðT0; T1; S0; S1; £ ; ¸Þj ¼ 0: ð24ÞEquation (24) is the discriminant of lateral movementbifurcation.

Figure 9 shows the numerical solution of eq. (24) obtainedby using the experimental conditions in Table 1. Based onthe calculation results of a discrete rolling model, the centerdeflection hr of the work-roll barrel was set to be

hr ¼ 9:92� 10�13 ½m=N� � P:

The true work-roll radius RA was calculated by Hitchcock’sequation.11) The friction coefficient, etc. were set to empirical

S. Aoe, T. Kitamura, T. Ogasahara and M. Miyake1172

values. The vertical axis shows the delivery unit tension, andthe horizontal axis shows the entry unit tension. The curvesshow the bifurcation curves. Lateral movement diverges(converges) in the left-side (right-side) region of these curves.The values in the legend show the reductions in thickness.The arrows show the stable and unstable regions of theexperimental lateral movements. The calculation results inFig. 9 show that the critical entry unit tension in case of thedelivery unit tension of 49MPa and reduction of 1.5% isabout 10.6MPa, and the critical entry unit tension in case ofthe delivery unit tension of 98MPa and reduction of 2.0% isabout 12.6MPa. Larger delivery tension increases the criticalentry tension or causes lateral movement to become moreunstable, but it seems that lateral movement does not dependsignificantly on the delivery tension under these experimentalconditions. Comparing the calculation results in Fig. 9 withthe experimental results in Figs. 7 and 8, it can be understoodthat the discriminant of eq. (24) is accurate, thereby showingthat the LMMs applied to formulating the discriminant arealso accurate. Figure 9 shows that smaller reductiondecreases the stable region of lateral movement or causeslateral movement to become more unstable. The calculatedvalue of the roll separating force (P + Pe

0 + Pe1) of 355 kN

roughly agreed with the experimental value of 392 kN.

3.3 Clarification and consideration of lateral movementsimulation

The lateral movement in experimental rolling is calculatedby using the simulation built on MATLAB/Simulink.Figure 10 shows the simulation results of the lateraldisplacement at the mill in case of the delivery unit tension

of 48MPa. The horizontal axis and vertical axis show timeand lateral displacement, respectively. A very small entrywedge of ¹0.45 µm was provided as the bias error so that theexperimental results would agree with the simulation results.An entry unit tension of 22.05MPa or more causes lateralmovement to converge. On the other hand, the entry unittension of under 17.15MPa makes lateral movement diverge.Thus, the validity of the simulation is shown by the closeagreement between the simulation results and the numericalresults of eq. (24).

Figure 10 shows that the lateral displacement in case theentry unit tension of 7.35MPa rapidly increases in about700 s. Figure 11 shows the vertical deflections of the entryand delivery strips in 700 s. The z-axis shows the verticaldirection. Out-of-plane deflection with torsion (lateralbuckling) at the entry side of the mill, corresponding toFig. 5, decreases the structural stiffness of a strip, and lowertension causes lateral buckling9) to occur more easily. On theother hand, the out-of-plane deflection at the delivery side ofthe mill in Fig. 11 does not appear, and also did not appearin the experiments.

Figure 12 shows the simulation results of the lateraldeflection in case of the entry unit tension of 22.05MPashown in Fig. 10. The horizontal axis shows the position inthe strip rolling direction, and the vertical axis shows lateraldisplacement. With progressing time, lateral deflectionconverges in the steady state governed by eqs. (6), (13) and(17) of the boundary conditions. Lateral displacement showsits maximum at the mill entry side in the initial state of the

0

2

4

6

8

10

0 1000 2000 3000 4000 5000

Late

ral d

ispl

acem

ent /

mm

Time / s

7.35 MPa12.25 MPa17.15 MPa22.05 MPa

Fig. 10 Simulation results of lateral displacement at rolling mill.

Rollingmill↓

Payoffreel

Tensionreel

Entry strip

Delivery strip

x / my / m

z/ m

m

Fig. 11 Strip deflection.

0

1

2

3

4

5

-3 -2 -1 0 1 2 3

Late

ral d

ispl

acem

ent /

mm

x / m

500 s

1000 s

2000 s5000 s4000 s3000 s

Fig. 12 Simulation results of lateral deflection (entry unit tension22.05MPa).

0

20

40

60

80

100

0 10 20 30 40 50

Del

iver

y un

it te

nsio

n / M

Pa

Entry unit tension / MPa

0.5%1.0%1.5%2.0%

StableUnstable

StableUnstable

Fig. 9 Phase diagram of strip lateral movement.

Development of Strip Lateral Movement Simulator for Temper Rolling 1173

calculation. Its maximum point approaches the mill in thesteady state governed by eq. (6). Figure 13 shows thesimulation results of the lateral deflection for the entry unittension of 7.35MPa in Fig. 10. In this case, lateral deflectiondiverges with progressing time. The point initially ap-proaches the mill and departs from the mill after 600 s. Thepoint appears at the mill entry side when lateral deflectiondiverges and entry tension is smaller than delivery tension.From eq. (1), the strip orientation angle at the mill becomesnegative and the lateral movement velocity becomes positivein this case. Therefore, lateral movement continues toincrease, and finally, lateral buckling occurs in the entrystrip after 600 s.

The simulation results when the strip deflection model ofthe entry and delivery sides is a beam model constrainedto in-plane deflection must agree with the discriminant ofeq. (24). The broken curves in Fig. 14 show the simulationresults of the lateral displacement at the mill when using abeam model. As in Fig. 10, the delivery tension is 49MPa.The entry wedge, which is a bias error, is a relatively largevalue of ¹2.25 µm (that is, 5 times the value of ¹0.45mmof the above-mentioned entry wedge, which is very small).The entry unit tensions of 9.8MPa and 19.6MPa causelateral movement to diverge and converge, respectively. Thislinear dynamical system makes these results agree with thediscriminant of eq. (24) regardless of the magnitude of biaserrors.

The solid curves in Fig. 14 show the simulation results ofthe lateral displacement at the mill when using the large-deflection strip model. The solid curves agree with thebroken curves in the initial term of the calculations, but thesolid curves in the case of smaller entry tension graduallydepart from the broken curves with progressing time.

Figure 15 shows the phase diagram for the entry tensionand the entry wedge based on the simulation results forreduction of 1.5%. The horizontal axis and vertical axisshow the entry unit tension and the magnitude of the entrywedge, respectively. The solid curve and the broken lineshow the results obtained by using the large-deflection stripmodel and a beam model, respectively. The critical entryunit tension of 41.1MPa obtained by using the large-deflection strip model is very different from the discriminantof eq. (24). Large values of the orientation angle ªn orlateral displacement vn cause the lateral buckling in the entrystrip to decrease the structural stiffness of the entry strip.Asymmetry error factors such as the entry wedge causelateral movement. As mentioned above, this type of errorcannot be ignored even if good operation is maintained in anactual temper rolling line. Applying the large-deflection stripmodel makes it possible to simulate lateral movement moresafely. On the other hand, the discriminant of eq. (24) showsthat asymmetry errors do not cause lateral movement, butthe cause of lateral movement is an unstable dynamicalsystem. Both a stable dynamical system and very small errormake lateral movement converge. Facilities and operationmust be managed in a stable condition for a dynamicalsystem for lateral movement. It is considered that methodssuch as the discriminant of eq. (24) are valid as tools for afirst-step study because these methods enable very simplecalculation.

4. Conclusions

In this paper, the validity of lateral movement simulationwas clarified by experimental rolling in the laboratory and atheoretical analysis with the aim of developing an automaticcontrol system for the temper rolling line. The resultsobtained in this study are shown below.(1) A practical lateral movement simulator based on a

newly-formulated LMM of a temper rolling and a stripdeflection model were proposed.

0

5

10

15

20

25

30

0 500 1000 1500 2000 2500 3000

Late

ral d

ispl

acem

ent /

mm

Time / s

29.4 MPa

39.2 MPa

49.0 MPa

9.8 MPa

19.6 MPa29.4 MPa

39.2 MPa

9.8 MPa

19.6 MPa

Fig. 14 Simulation results of lateral displacement at rolling mill.

0

0.5

1

1.5

2

2.5

3

0 10 20 30 40 50

Entry

wed

ge |h 1df

| / μ

m

Entry unit tension / MPa

Large-deflection strip model

Beam model

Fig. 15 Phase diagram based on simulation results.

0

1

2

3

4

5

-3 -2 -1 0 1 2 3

Late

ral d

ispl

acem

ent /

mm

x / m

300 s400 s

500 s

600 s

700 s800 s

200 s

Fig. 13 Simulation results of lateral deflection (entry unit tension7.35MPa).

S. Aoe, T. Kitamura, T. Ogasahara and M. Miyake1174

(2) The lateral movement in experimental rolling becameexcessive when the entry tension of the mill becamesmaller than a critical value. The mechanism of thisbehavior was clarified by the newly-formulatedtheoretical analysis (eigenvalue analysis) for evaluationof lateral movement.

(3) It was confirmed that the experimental results within therange of experimental conditions in this study agreedwith the simulation results, and the LMMs based on thequasi-static theoretical model were appropriate.

(4) The newly-developed method provides the optimalrolling conditions for preventing lateral movement bytheoretical analysis, calculations and experiments.

It is considered that the proposed methods can be appliedgenerally to lateral movement analysis problems other thanthe temper rolling line and are useful for investigating lateralmovement problems in further detail. However, the proposedstrip deflection model is designed specifically for strips withcamber and cannot be applied to strips with various defectiveshapes. Therefore, it will be necessary to design a morepractical strip deflection model for these strips.

Appendix

A1 Formulation of eq. (2)The strip rotational velocity ¼0 at the mill delivery side is

defined as

¼0 ¼ �@yU (A1-1)

where U is the strip velocity at the delivery side. Appling thedefinition of forward-slip fs, eq. (A1-1) is deformed to

¼0=U ¼ �ð1þ fsÞ�1@yfs

¼ ¢h0h0dfA

� ¢h1ð1� rÞh1df

Aþ ¢M0

M0

kI

� ¢M1

ð1� rÞM1

kI(A1-2)

where

¢h0 ¼ �hð1þ fsÞ�1@hfs; (A1-3)

¢h1 ¼ hð1� rÞ�1ð1þ fsÞ�1@h1fs;

¢M0 ¼ kð1þ fsÞ�1@·0fs; (A1-4)

¢M1 ¼ �kð1þ fsÞ�1@·1fs;

h and h1 are the thickness of the delivery and entry strip, and·0 and ·1 are the delivery and entry unit tension. Applingthe relationship between the entry and delivery wedges16) andeq. (A1-2), the strip rotational velocity ¼1 at the entry side isrepresented by

¼1

ð1� rÞU ¼ �ð1� ¢h0Þh0dfA

þ ð1� ¢h1Þð1� rÞh1df

A

þ ¢M0

M0

kI� ¢M1

ð1� rÞM1

kI: (A1-5)

The entry rotational velocity ¼1 is also represented by3,6)

Dªn=Dt ¼ dªn=dtþ ð1� rÞU¬1 ¼ ¼1 (A1-6)

with the entry orientation angle ªn. The following equationis formulated by substituting eq. (A1-5) in eq. (A1-6) andlimiting strip reduction r to zero:

dªndt

¼ U

�� ð1� ¢h0Þ

h0dfA

þ ð1� ¢h1Þh1dfA

þ ¢M0

M0

kI� ¢M1

M1

kI� ¬1

�: (A1-7)

Substituting Bland & Ford’s forward-slip equation11) ineq. (A1-3), the influence coefficient ¢h0 is represented by

¢h0 ¼

1

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� rÞ�1 � 1

p �ffiffiffi²

p

2®

þffiffiffi²

p

4®log

1

1� r

k� ·0

k� ·1

� �� �!

� tan

�1

2atan

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� rÞ�1 � 1

q �

�ffiffiffi²

p

4®log

1

1� r

k� ·0

k� ·1

� �� ��(A1-8)

where ² is h/RA. Limiting reduction r in eq. (A1-8) to zero,the influence coefficient ¢h0 is 1/4. The influence coefficient¢h1 is 1/4 in same operation.

Substituting Bland & Ford’s forward-slip equation11) ineq. (A1-4), the influence coefficient ¢M0 is represented by

¢M0 ¼kffiffiffi²

p

2®ðk� ·0Þtan

1

2atanð ffiffiffi

rp Þ �

ffiffiffi²

p

4®logðrþ 1Þ

� �:

(A1-9)

The following equation is formulated by expandingeq. (A1-9) with respect to

ffiffiffir

p:

¢M0 ¼kffiffiffi²

p

4®ðk� ·0Þffiffiffir

p þ o2ð ffiffiffir

p Þ: (A1-10)

Considering reduction r to be negligible, the second termon the right side of eq. (A1-10) is approximated as zero andthe influence coefficient ¢M0 is approximated as the firstterm on the right side. The influence coefficient ¢M1 is alsoapproximated similarly in operation. Substituting theapproximation value 1/4 of the influence coefficient ¢h0and ¢h1 in eq. (A1-7), eq. (2) is formulated.

A2 LMM and boundary conditions of payoff reelFigure A2-1 shows the force and moment acting on a strip

that winds around a coil at a payoff reel. It is considered thatthe winding strip is developed to become a rigid body, andthis rigid body moves on the developed plane of the coil. Thesurface of the rigid body moves from the left side to theright side. The entry endpoint (left endpoint) of the rigid bodyis fixed against the lateral direction, and the delivery endpoint

x

y

PORθ PORv

fPORF

PORPORRφ

1T1TEPORF

DPORF

PORMfPORM

U

Fig. A2-1 Rigid body on coil.

Development of Strip Lateral Movement Simulator for Temper Rolling 1175

(right endpoint) is connected with an elastic beam. The lateralfriction force Ff

POR

FfPOR ¼ � ºPOR®PORT1

­PORU

dvPORdt

þ UªPOR

� �(A2-1)

and the friction moment MfPOR

MfPOR ¼ � ºPOR®PORT1

12­PORUðº2

PORR2POR þ b2Þ dªPOR

dt(A2-2)

derived from rubbing between the surface of the rigid bodyand the plane of the coil act on the center of the rigid body.14)

In eqs. (A2-1) and (A2-2), vPOR is the lateral displacementof the delivery endpoint of the rigid body, and ªPOR is theorientation angle of the rigid body. The relationship betweenthe displacement vPOR and the orientation angle ªPOR isformulated as follows:

vPOR ¼ ºPORRPORªPOR: (A2-3)

The balance equation of moments is given by

ºPORRPORFfPOR=2þMf

POR þ ºPORRPORFDPOR

� T1vPOR þMPOR ¼ 0 (A2-4)

where FDPOR andMPOR are the lateral force and moment acting

on the delivery endpoint of the rigid body.Lateral force FD

POR is represented by

FDPOR ¼ �ðEI@xxxv1ð�LÞ � T1@xv1ð�LÞÞ (A2-5)

by using the balance condition of the shear force of the elasticbeam at the delivery endpoint. The connection conditionsbetween the rigid body and the elastic beam are representedby

ªPOP ¼ @xv1ð�LÞ; vPOR ¼ v1ð�LÞ: (A2-6)

Considering eqs. (A2-3) and (A2-5) as the boundaryconditions of the elastic beam and substituting the staticequations shown as eqs. (A2-1), (A2-2), (A2-5) and (A2-6)in eqs. (A2-3) and (A2-4), the boundary conditions ofeqs. (12) and (13) are formulated.

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S. Aoe, T. Kitamura, T. Ogasahara and M. Miyake1176