Research ArticleStability Analysis of a Nonlinear Coupled Vibration Model in aTandem Cold Rolling Mill

Xing Lu Jie Sun Guangtao Li Zhenhua Wang and Dianhua Zhang

e State Key Laboratory of Rolling and Automation Northeastern University Shenyang 110819 Liaoning China

Correspondence should be addressed to Jie Sun sunjieralneueducn

Received 24 October 2018 Revised 21 January 2019 Accepted 24 January 2019 Published 17 February 2019

Academic Editor Roger Serra

Copyright copy 2019 Xing Lu et al)is is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Mill chatter in tandem cold rolling mill is a major rejection to the quality and production of the strips In most mill vibrationmodels either the roll mass is usually limited to vibrate in vertical direction and vertical-horizontal directions or the multiple rollssystem is simplified to a single mass system However the torsional chatter is also a typical type of mill chatter and the presence ofintermediate roll and backup roll will affect the overall vibration of themill structure system In this paper a newly cold rollingmillvibration model coupled with the dynamic rolling processing model and nonlinear vibration model is proposed with theconsideration of dynamic coupling and nonlinear characteristics of the rolling process multiroll equilibrium and roll movementin both vertical-horizontal-torsional directions By using Hopf bifurcation theorem and RouthndashHurwitz determinant the ex-istence of the Hopf bifurcation point of the mill vibration system and bifurcation characteristics are analyzed At last the influenceof different rolling conditions on the stability of the coupled mill system is investigated and these results can also be used to designan optimum rolling schedule and determine the appearance of mill chatter under certain rolling conditions

1 Introduction

With the optimization and upgrading in the steel industry theurgent demands for high strip quality and strip yield havebecome more stringent However due to the abnormal millvibration the quality and production of the strip have beenrestricted Abnormal vibration in the tandem cold rollingmill isa common and complicated phenomenon Researchers tend toconsider thatmill chatter is a typical self-excited vibration [1ndash3]which consists of third-octave mode chatter (150ndash250Hz)fifth-octave mode chatter (500ndash700Hz) and torsional chatter

In the mill chatter analysis the most typical models arefinite element method (FEM) models multiple degrees offreedom structure model and mill drive system torsionalstructure models Niroomand et al [4ndash6] studied the third-octave mode chatter and fifth-octave mode chatter in thevertical direction by using FEM analysis models In thesepapers the backup roll was set as a mass point and a con-nector was used to represent the displacement of the backuproll and mill stand housing In FEM models the structuremode coupling characteristics cannot be fully reflected Formost vertical vibration models the mill stand was modeled

with multiple degrees of freedom Kapil et al [7] used thefurther simplified structure model to analyze the frequencyresponse in detail Kimura et al [8 9] and Heidari et al[10ndash13] illustrated the influence of lubrication characteristicon mill chatter However in these studies only the verticalvibration phenomena were investigated But in the actualrolling process the roll vibrated in more than one directionSo it is necessary to establish a coupled chatter model whichallows roll to vibrate in two directions Yun et al presented adynamic coupled chatter model that allows roll vibrate in twodirections and investigated the relationship between mill vi-bration and the rolling conditions [2] Kim et al [14] proposeda vertical and horizontal coupling chatter model and analyzedthe dynamic response of roll displacement under differentrolling pressures In the research of torsional vibration on therotating shaft Krot [15] analyzed the nonlinear torsionalvibration caused by angular and radial wear in gear Liu et al[16 17] established a multiple degree of freedom of mill maindrive system and studied the torsional chatter characteristicsin the main drive system of the mill stand

Bifurcation was an important theory of nonlinear fieldwhich implied the sudden change of the topological property

HindawiShock and VibrationVolume 2019 Article ID 4358631 14 pageshttpsdoiorg10115520194358631

of the dynamic system when the parameters change in thedynamic characteristic system Hopf bifurcation was a typicaltype of bifurcation system It was named after the name of theGerman mathematician Eberhard Hopf who proposed theconcept It referred to the bifurcation of the periodic oscil-lation and the appearance of a stable limit cycle which iscaused by the dynamic change of the parameter [18] Hopfbifurcation had theoretical value not only in the dynamicbifurcation research but also closely related to the generationof self-excited vibration in engineering Lee found that theHopf bifurcation phenomenon in the wing flutter system wasthemain cause of the casing vibration [19] and themagnitudeof the pitch angle played an important role in this vibrationproblem )e characteristics and localization of multiplependulum-type centrifugal pendulum vibration absorberswere investigated to suppress torsional vibrations of shaft byusing Hopf bifurcation theory [20] Zhang and Dai usednonlinear creep theory to study bifurcation phenomenon andstability of China Railways High-speed vehicle (CRH-3) [21]

Many researchers have found that the dynamic rollingmill system was often accompanied with the generation oflimit cycle when unstable oscillation occurs during rollingSo Hopf bifurcation theorem was also introduced into themill stand vibration analysis Gao et al [22 23] established adynamic model of rolling mill system and analyzed the Hopfbifurcation and mill stability Liu et al [24] established adynamic vibration considering the torsional vibration ofgear and the horizontal vibration of the roll and the stabilityof the mill drive system was analyzed Peng et al [25]proposed a swing chatter model coupling the vertical-horizontal-torsional vibration and studied the influence ofmeshing impact force on the roll system stability

Researchers have shown the existence of negativedamping effects in mill structure But the relationship be-tween dynamic rolling conditions and damping coefficient isnot clearly illustrated Besides the Hopf bifurcation phe-nomenon in the rolling mill drive vibration system is animportant cause of system instability )e occurrence of theHopf bifurcation limit cycle determines the stability andstable range of the mill )en vibration problem caused bynegative damping effect can be transformed into the analysisof the Hopf bifurcation phenomenon in the vibration modelScholars have done plenty scientific works about vibrationin cold rolling However most of those theses were eitherconcentrated on vertical vibration or torsional vibrationor the whole roll system was simplified by one single roll)erefore a further multiroll and nonlinear vertical-horizontal-torsional coupled vibration model of the univer-sal crown control mill (UCM mill) needs to be proposed andanalyzed And based on this proposed rolling mill vibrationmodel the bifurcation phenomenon and stability of the millsystem is analyzed to illustrate the dynamic characteristics ofthe UCM mill more comprehensively

2 Dynamic Rolling Process Model

)e geometry of roll gap in the cold rolling is presented inFigure 1 O is the original center of the work roll and Oprime is

the new center of the flattened roll xin xn and xout are theentry position neutral position and exit position α and c

are the bite angle and neutral angle vin and vout are the stripspeed at the entry and exit point hin hn and hout are the stripthickness at the entry neutral and exit point xw is thedynamic vibration displacement of work roll in the hori-zontal direction yw is the dynamic vibration displacementof work roll in the vertical direction Tf and Tb are the frontand back tension

In cold rolling Hitchcock indicated that the flattenedwork roll radius Rprime can be evaluated as

Rprime R 1 +16 1minus υ2( 1113857

πE1

P

w middot hin minus hout( 11138571113896 1113897 (1)

where υ is Poissonrsquos ratio in this research υ 03 w is thewidth of the strip E1 is Youngrsquos modulus of the work roll P

is the total rolling force R and Rprime are the work roll originalradius and roll flattened radius

Considering work roll elastic deformation the contactarc length in the roll gap is

lprime

R hin minus hout( 1113857 + 81minus υ2

πE1Rp1113888 1113889

2

11139741113972

+ 81minus υ2

πE1Rp (2)

where p is the average rolling forceAssuming the contour of the roll gap can be written as a

parabolic curve the strip thickness at any point in the rollgap can be determined as [11]

h(x) hc +x2

Rprime (3)

where hc is the roll gap spacing distance along the centerlineof the work roll and x is the position of the strip along therolling direction in the roll gap

From equation (3) the entry position of the strip can begiven as

xin xw minus

Rprime hin minus hc( 1113857

1113969

(4)

So the dynamic roll gap spacing of the roll centerline canbe related as [11 22]

hout hc + 2yw (5)

Based on equation (5) and considering mill elasticdeformation the variation of strip thickness at exit isexpressed as

dhout 2yw +dP

Kw

(6)

where dP is the rolling force variation and kw is theequivalent spring constant of the work roll

And strip thickness at the neutral point can be calculatedby [8]

hn hin + hout

2minus

12μ

loghin

houttimes1 minus Tfkhout

1113872 1113873

1 minus Tbkhin1113872 1113873

⎛⎝ ⎞⎠ (7)

2 Shock and Vibration

)e neutral angle is calculated by

c

hout

Rprime

1113971

times tan

hout

Rprime

1113971

timeshn

2⎛⎝ ⎞⎠ (8)

where khinand khout

are the deformation resistance at stripentry and exit position and μ is the friction coefficient in theroll gap

Based on equation (7) strip thickness fluctuation at theneutral point can be expressed as

dhn zhn

zhindhin +

zhn

zhoutdhout +

zhn

zTf

dTf +zhn

zTb

dTb

12minus

12μhin

1113888 1113889dhin +12

+1

2μhout1113888 1113889dhout

+1

2μ khoutminusTf1113872 1113873

dTf minus1

2μ khinminusTb1113872 1113873

dTb

(9)

Considering the mass conservation law to the mass flowcontinuity equation

Vrhn vinhin vouthout (10)

where Vr is the work roll circumference velocity)e strip speed variation at the entry and exit can be

expressed as [8]

dvin Vr

hindhn

dvout Vr

h2out

houtdhn minus hndhout( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(11)

According to Hookersquos law the variation of tension isproportional to integral of speed difference between entry ofi stand and exit of iminus 1 stand [26] Assuming the exit speed ofiminus 1 stand and entry speed of i+ 1 stand are always keptconstant the tension variation at the entry and exit can beexpressed as

dTfi minusE2

550251113946 dvoutidt

dTbi E2

550251113946 dvinidt

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(12)

where subscript i means i stand And in this research thestrip length between i and i + 1 stand is 55025m

In the most rolling mill vibration models the frictioncoefficient was usually set as a constant value However inthe cold rolling the lubrication characteristics in the rollgap were significantly different under the high rolling speedand low speed Sims and Arthur [27] found that when therolling speed was higher than 025ms the friction co-efficient showed a nonlinear characteristic which is causedby unsteady rolling conditions As a result it is necessary toinvestigate the dynamic vibration stability in the UCM millwith the consideration of nonlinear friction characteristicsConsidering the roll wear and other conditions into con-sideration the friction coefficient in this research is obtainedby [28]

μ μ0 + μV middot eminus VrV0( )1113874 1113875 middot 1 + cR middot Ra minusRa0

1113872 11138731113960 1113961

middot 1 +cW

1 + LL0( 11138571113888 1113889

(13)

where μ0 and μV are the basic friction coefficient that relatedto lubrication and lubrication coefficient related to rollingspeed V0 is the basic rolling speed Ra and Ra0

are actualwork roll roughness and roughness of unused work roll CR

is the work roll roughness coefficient cW is the coefficientrelated to cumulative rolling strip length and L and L0 arethe cumulative rolling strip length of work roll and datumrolling length

)e rolling force is calculated by Bland-Ford-Hillequation [29 30]

P wlprimekQpnt (14)

A

BNeutralpointC

Rprime α

γ

Oprime

O

hout2hn2

hin2

xin xn xout

vin

voutTb

Tf

xw

yw

Figure 1 Vibration geometry model of the roll gap

Shock and Vibration 3

where

Qp 108 + 179εμ

Rprimehin

1113971

minus 102ε

nt 1minus1minus μt( 1113857Tf + μtTb1113960 1113961

k μt 07

ε hin minus hout

hin

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)

and k is the deformation resistance of the rolling strip In thisresearch deformation resistance of the strip is expressed bythe following equation

k A + B middot εΣ( 1113857 1minusC middot eminusDmiddotεΣ1113872 1113873 (16)

where εΣ is average strain which can be given by εΣ

(23

radic) middot ln(H0h) h (hin + 2hout3) A B C and D are

references in deformation resistance model In this researchA 50000MPa B 17900MPa C 020 and D 500 H0is the thickness of the raw material

)e investigation of UCMmill showed that No 4 and No5 stand chatter frequently so a comprehensive mill vibrationmonitoring test was implemented in a 1450mm UCM plantby using CTC-type accelerometer sensors )e accelerometersensors were mounted on the work roll bearings of No 5stand When rolling the most common products of035mmstrip in this plant the vibration signals during thestable rolling stage are shown in Figure 2 It illustrated thatalthough the horizontal rolling force was relatively smallcompared with the vertical rolling force the horizontal rollingforce was still big enough )erefore it was sufficiently sig-nificant to establish a comprehensive model which allows therolls to vibrate in both horizontal and vertical directions

Considering the dynamic equilibrium of strip in x di-rection the horizontal rolling force can be calculated asfollows

Px 1113946a

0P sin θdθ + 1113946

a

cPμ cos θdθ minus 1113946

c

0Pμ cos θdθ

w

2Tfhout minusTbhin1113872 1113873

(17)

)e vertical rolling force can be written as

Py 1113946α

0P cos θdθ + 1113946

α

cPμ sin θdθ minus 1113946

c

0Pμ sin θdθ

wlprimekQpnt 1113946α

0cos θdθ + 1113946

α

cμ sin θdθ minus 1113946

c

0μ sin θdθ1113888 1113889

wlprimekQpnt 1113946α

0cos θdθ minus μ cos α + 2μ cos cminus μ1113874 1113875

(18)

Considering that the bite angle α and neutral angle c areextremely small it assumed that the vertical rolling force isequal to the total rolling force [14 22] so equation (18) canbe expressed as

Py asymp wlprimekQpnt 1113946α

0cos θdθ asymp wlprimekQpnt P (19)

Assuming the thickness fluctuation and tension varia-tion are caused by roll gap vibration as separated distur-bances the deviation of the rolling force in the x directioncan be written as

dPx zPx

zhindhin +

zPx

zhoutdhout +

zPx

zTb

dTb +zPx

zTf

dTf (20)

Based on equation (17) the deviation of the rolling forcein the x direction can be also expressed as

dPx minuswTb

2dhin + wTfdyw minus

whin

2dTb +

whout

2dTf

(21)

)e variation of the rolling force in the y direction can bedefined as

dPy zP

zhindhin +

zP

zhoutdhout +

zP

zTf

dTf +zP

zTb

dTb (22)

where

zP

zhin wlprimeknt minus

1792

μ

Rprime

h3in

1113971

+32

times 179μhout

Rprime

h5in

1113971

minus 102hout

h2in

⎛⎝ ⎞⎠

zP

zhout wlprimeknt minus179

μhin

Rprimehin

1113971

+ 1021

hin

⎛⎝ ⎞⎠

zP

zTb

minus07wlprimeQp

zP

zTf

minus03wlprimeQp

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(23)

)e rolling torque of the upper roll system can be de-duced as [23 25]

M w 1113946c

0μpR dθminusw 1113946

α

cμpR dθ

wμpR(2cminus α)

PμR(2cminus α)

(24)

Considering the bite angle α and neutral angle c areextremely small α asymp

(hin minus houtRprime)1113969

c asymp sin c (xw minus

xnRprime) and xn

Rprime(hn minus hc)

1113969

Also the variation of the rolling torque can be calcu-

lated by

dM zM

zhindhin +

zM

zhoutdhout +

zM

zTf

dTf +zM

zTb

dTb +zM

zxw

dxw

(25)

where

4 Shock and Vibration

zM

zhin wlprimeknt(2cminus α)1113888minus

1792

μ

Rprime

h3in

1113971

+32

times 179μhout

Rprime

h5in

1113971

minus 102hout

h2in

1113889minus12

wlprimekQpntμR1

hin minus hout( 1113857Rprime1113969

zM

zhout wlprimeknt(2cminus α) minus179

μhin

Rprimehin

1113971

+ 1021

hin

⎛⎝ ⎞⎠

+12

wlprimekQpntμR1

hin minus hout( 1113857Rprime1113969

zM

zTb

minus07wlprimeQpμR(2cminus α)

zM

zTf

minus03wlprimeQpμR(2cminus α)

zM

zxw

wlprimekQpntμR 21Rprimeminus α1113874 1113875

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)

)e calculated rolling conditions in this research areshown in Table 1)e grade of rolling strip is MRT5 and theexit strip width is 900mm

Figure 3 shows the comparison of calculated rolling forcein this rolling process model and actual rolling force in theexperiment )e rolling conditions are shown in Table 1 Asshown in Figure 3 the maximum relative error betweenactual rolling force and calculated rolling force is 723)ecomparative results of the simulation and experimental datashow that this chatter model has a high accuracy

3 Dynamic Nonlinear Vibration Model of theUCM Mill

In most vibration research studies the rolling mill standwas simplified as a mass-spring system To investigate the

displacements of each roll and interaction between rolls amodel with multiple degrees of freedom is necessaryHowever with the consideration of the symmetry of themill structure along the rolling pass line the multipledegrees of freedom chatter model can be simplifiedFigure 4 is the simplified nonlinear vertical-horizontal-torsional coupled vibration model considering the rollmovement in more than one direction

As shown in Figure 4 the top work roll (WR) wouldvibrate in both vertical and horizontal directions While theintermediate roll (IMR) and backup roll (BUR) were notallowed to vibrate in horizontal direction due to the as-sembly and restriction of the roll bearings [25]

A horizontal roll vibration displacement of WR isgenerated due to the installation offset of e between WR andIMR and the equivalent spring constant in horizontal di-rection can be calculated by

ndash010

ndash005

000

005

010

Vibr

atio

n ac

cele

ratio

n (g

)

00 02 04 06 08 10Time (s)

(a)

ndash010

ndash005

000

005

010

Vib

ratio

n ac

cele

ratio

n (g

)

00 02 04 06 08 10Time (s)

(b)

Figure 2 Time domain vibration signal of the work roll (a) vibration acceleration in the vertical direction (b) vibration acceleration in thehorizontal direction

Table 1 Rolling conditions

Parameters 1 stand 2 stand 3 stand 4 stand 5 standEntrythickness hin(mm)

2000 1269 0762 0460 0300

Exit thicknesshout (mm) 1269 0762 0460 0300 0200

Rolling forceP (kN) 7838863 8455166 7781567 7500996 7722711

Exit tension Tf(MPa) 140554 165252 188160 205575 68000

Entry tensionTb (MPa) 55000 140554 165252 188160 205575

Frictioncoefficient μ 0065 0060 0040 0030 0020

Roll velocityVr (m sminus1) 3381 5652 9425 14441 22121

Strip entryspeed vin(m sminus1)

2250 3546 5903 9774 14977

Shock and Vibration 5

Kwx e

Rw + RimKw (27)

where Rw and Rim are the roll radius of work roll and in-termediate roll and Kw is the contact spring constant be-tween upper work roll and intermediate roll

With the consideration of the coupling movement of therolls in both vertical horizontal and torsional directions thedynamic equilibrium equation of the upper roll system wasgiven by

Mw euroxw + Cwx _xw + Kwxxw ΔPx

Mw euroyw + Cwy _yw minus _yim( 1113857 + Kwy yw minusyim( 1113857 ΔPy

Mim euroyim minusCwy _yw minus _yim( 1113857 + Cim _yim minus _yb( 1113857

minusKwy yw minusyim( 1113857 + Kim yim minusyb( 1113857 0

Mb euroyb minusCim _yim minus _yb( 1113857 + Cb _yb minusKim yim minusyb( 1113857

+ Kbyb 0

Jweuroθw + Cwj

_θw + Kwjθw ΔM

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(28)

where xw is the WR vibration displacement in horizontaldirection yw yim and yb are the vibration displacement ofWR IMR and BUR in vertical direction θw is the WRtorsional vibration angular displacement ΔPx and ΔPy arethe fluctuation of rolling force in horizontal and verticaldirections ΔM is the variation of rolling torque and MwMim and Mb are the mass of work roll intermediate rolland backup roll respectively

)e mill stand spring constant is the total stiffness of thewhole rolling mill consisting of all the rolls )e mill springconstant (K) is measured by a stiffness calibration test )espring constants of work roll (Kw) intermediate roll (Kim)and backup roll (Kb) are calculated by an FEM model of theUCM mill [5 6 12] )e roll vibration displacement underdifferent rolling forces was obtained by a simulation ex-periment )e spring constants of the work roll in-termediate roll and backup roll were obtained by analyzingthe dynamic roll vibration displacement under differentrolling forces

Damping coefficients of the intermediate and backuprolls are calibrated by a simulation program [31 32] First

Rolling direction

BUR

IMR

WR

Kb CbCb

Kim Cim

Kb

Mb

Mim

MwCwyKwy

KimCim

CwyKwy

ΔP Kwx

Kwx Cwx

CwxJw

Kwj Cwj

(a)

Cim

Mim

Mb

Mw

Kim

Kb Cb

Kw Cw

yb

yim

Kwx e

TfTb

xw

yw

(b)

Figure 4 Geometry of the vertical-horizontal-torsional vibration model (a) simplified vertical-horizontal-torsional coupled vibrationmodel (b) geometry of coupled vibration model in x-y direction

0

2000

4000

6000

8000

10000

Rolli

ng fo

rce (

kN)

1 2 3 4 5Stand number

Actual rolling forceCalculated rolling force

Figure 3 Comparison between the actual rolling force and calculated rolling force

6 Shock and Vibration

the dynamic differential equation of the system is regularizedto obtain the equivalent damping matrix )en the dampingregular matrix is transformed into the original coordinatesystem and let the damping ratio be 003 Finally theequivalent damping value of the roll system is calculatedWork roll rotational inertia mass work roll rotational springconstant and work roll rotational damping coefficient areobtained from the work of Zeng et al [23] as the design andmill structure of rolling stands are similar to one another)e roll diameters and work roll installation offset are ob-tained from the UCM rolling mill installation drawingsTable 2 shows the structure specifications of the UCMmill inequation (28)

Equation (28) is the nonlinear dynamic equation of thevertical-horizontal-torsional coupled vibration modelConvert the above differential equation into the followingstate matrix form

_X F(X μ) + Hdhin (29)

whereX xw _xw yw _yw yim _yim yb _yb θw _θw dTf dTb1113960 1113961

T

x1 x2 x3 x121113858 1113859T

H 0minuswTb

2Mw

0zP

zhinMw

0 0 0 0 0zM

zhinJw

0 01113890 1113891

T

F

0 1 0 0 0 0 0 0 0 0 0 0F21 F22 F23 0 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 00 0 F43 F44 F45 F46 0 0 0 0 0 00 0 0 0 0 1 0 0 0 0 0 00 0 F63 F64 F65 F66 F67 F68 0 0 0 00 0 0 0 0 0 0 1 0 0 0 00 0 0 0 F85 F86 F87 F88 0 0 0 00 0 0 0 0 0 0 0 0 1 0 00 F102 F103 0 0 0 0 0 F109 F1010 F1011 F10120 0 F113 0 0 0 0 0 0 0 F1111 F11120 0 F123 0 0 0 0 0 0 0 F1211 F1212

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

F21 minusKwx

Mw

F22 minusCwx

Mw

F23 wTf

Mw

F43 2 zPzhout( 1113857minusKwy

Mw

F44 minusCwy

Mw

F45 Kwy

Mw

F46 Cwy

Mw

F63 Kwy

Mim

F64 Cwy

Mim

F65 minusKwy + Kim

Mim

F66 minusCwy + Cim

Mim

F67 Kim

Mim

F68 Cim

Mim

F85 Kim

Mb

F86 Cim

Mb

F87 minusKim + Kb

Mb

F88 minusCim + Cb

Mb

F102 zM

Jwzxw

F103 2zM

Jwzhout

F109 minusKwj

Jw

F1010 minusCwj

Jw

F1011 zM

JwzTf

F1012 zM

JwzTb

F113 minus2E2Vr zhnzhout( 1113857hout minus hn( 1113857

55025h2out

F1111 minusE2Vr zhnzTf1113872 1113873

55025hout

F1112 minusE2Vr zhnzTb( 1113857

55025hout

F123 2E2Vr zhnzhout( 1113857

55025hin

F1211 E2Vr zhnzTf1113872 1113873

55025hin

F1212 E2Vr zhnzTf1113872 1113873

55025hin

(30)

)en equation (29) is the nonlinear dynamic stateequation of the vertical-horizontal-torsional coupled vi-bration system considering the nonlinear characteristicduring the rolling process the movement of rolls in bothvertical-horizontal directions and the torsional vibration

4 Analysis of Vibration Stability of theRoll System

41 Hopf Bifurcation Parameter Calculation )e frictioncharacteristic would cause the instability of the dynamicstate equation so in this research μ has been chosen as the

Shock and Vibration 7

Hopf bifurcation parameter Assuming the entry thicknessof each stand is constant the equilibrium point is Xwhere F(X) 0 )e system equilibrium point can betransformed to the origin by coordinate transformation X0

[0 0 0 0 0 0 0 0 0 0 0 0]T Let JF(X) be the Jacobianmatrix of F(X) So the Hopf bifurcation can be obtained byjudging all the eigenvalues of JF(X) However it is difficult tocalculate the eigenvalues of JF(X) since the calculationamount is quite large )en the RouthndashHurwitz determinantis used to calculate the Hopf bifurcation point [33]

)e characteristic equation of Jacobian matrix JF(X) ofthe dynamic vertical-horizontal-torsional coupling vibrationmodel is assumed to be

a12λ12

+ a11λ11

+ middot middot middot + a2λ2

+ a1λ + a0 0 (31)

)e main determinant

D1 a1 D2 a1 a0

a3 a2

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868 middot middot middot Di

a1 a0 middot middot middot 0

a3 a2 middot middot middot 0

middot middot middot middot middot middot middot middot middot middot middot middot

a2iminus1 a2iminus2 middot middot middot ai

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(32)

where ai 0 if ilt 0 or igt n and a12 1Assuming μlowast is the Hopf bifurcation the Hopf bi-

furcation conditions are [23 24 34]

a0 μlowast( 1113857gt 0 D1 μlowast( 1113857gt 0 middot middot middot Dnminus2 μlowast( 1113857gt 0 Dnminus1 μlowast( 1113857 0

dΔiminus1 μlowast( 1113857

dμne 0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(33)

According to these assumptions the Hopf bifurcationpoint of μlowast occurs at μlowast 00048 and μlowast 00401

42Analysis of StableRangeofFriction When the value of μlowastis smaller than the first bifurcation point of μlowast 00048such as μlowast 00040 the response of work roll displacementand Hopf bifurcation diagram are shown in Figure 5 )eroll displacement will diverge rapidly and the equilibriumpoint of the equilibrium equation is unstable Besides asshown in Figure 5(b) the phase trajectory will disperseoutward of the origin in a ring shape and the radius cur-vature of the phase trajectory increases gradually It isconcluded that the roll vibration system is unstable and millchatter appears

As shown in Figure 6(a) when μlowast 00048 the work rolldisplacement exhibits as an equal amplitude vibration andthe roll vibrates at its balanced position )is steady regionrefers to the maximum and minimum fluctuation range ofwork roll Meanwhile Figure 6(b) shows that the phasediagram will begin to converge toward a stable limit cycle)e strip will produce a slight thickness fluctuation withinthe quality requirements and the mill system is stable underthese rolling conditions

When μlowast 00300 the roll systemrsquos time history responseand the phase diagram are shown in Figure 7 It can be seenfrom these figures that the phase trajectory will convergestoward a smaller and smaller circle near the origin point )eequilibrium point of work roll is asymptotically stable

)e roll system stability about the critical parameter ofμlowast 00401 is shown in Figure 8 When at the critical Hopfbifurcation parameter displacement of work roll firstlybecomes larger and finally converges to an equal amplitudeoscillation )e equilibrium point is unstable and the Hopfbifurcation diagram presents a ring shape However dy-namic range of the fixed limit cycle in Figure 8 is a littlelarger compared with Figure 6 )is is mainly because thefriction is bigger and as a result the fluctuation of theresponse rolling force ΔPi under the friction coefficient of00401 is much bigger

)e numerical solution of work roll displacement underthe Hopf bifurcation point of μlowast 00420 is shown in Fig-ure 9)e roll displacement will disperse outward all the timeBesides the phase diagram of the work roll will disperseoutward of the cycle and the radius curvature of the phasetrajectory increases gradually It illustrates that the mill vi-bration system is unstable and mill chatter is produced

When μlowast lt 00048 the roll system is unstable and whenμlowast 00048 the system is stable So the Hopf bifurcation ofμlowast 00048 is the subcritical bifurcation point and μlowast

00401 is the supercritical bifurcation pointMany researchers attribute the change of stability in

rolling mill system to the sudden change of the rolling forcefluctuation in dynamic equilibrium equation [11 25 35])e optimal setting range of friction coefficient is 00048ndash00401 and mill chatter appears when the friction is eithertoo high or too low )is is mainly due to the fluctuation ofrolling force When the friction is high with the increase offriction coefficient the neutral angle becomes bigger andthen the neutral point shifts to the entry side so the distancebetween the entry and neutral point becomes shorter andthe variation of the entry speed decreases [11] However theexit forward slip and exit speed fluctuation will increase

Table 2 Structure specifications of the UCM mill

Parameters ValueWork roll mass Mw (Kg) 3914Intermediate roll mass Mim (Kg) 5410Backup roll mass Mb (Kg) 31330Work roll contact spring constant Kw (N mminus1) 091times 1010

Intermediate roll spring constant Kim (N mminus1) 103times1010

Backup roll spring constant Kb (N mminus1) 585times1010

Work roll damping coefficient x directionCwx (Nmiddots mminus1) 500times104

Work roll damping coefficient in y directionCwy (Nmiddots mminus1) 164times105

Intermediate roll damping coefficient Cim (Nmiddots mminus1) 570times105

Backup roll damping coefficient Cb (Nmiddots mminus1) 510times106

Work roll rotational inertia mass Jw (Kgmiddotm2) 1381Work roll rotational spring constant Kwj (Nmiddotmmiddotradminus1) 790times106

Work roll rotational damping coefficientCwj (Nmiddotmmiddots radminus1) 4178

Work roll diameter Rw (mm) 2125Intermediate roll diameter Rim (mm) 245Backup roll diameter Rb (mm) 650Work roll installation offset e (mm) 5

8 Shock and Vibration

ndash80 times 10ndash5

ndash40 times 10ndash5

00

40 times 10ndash5

80 times 10ndash5W

ork

roll

disp

lace

men

t (m

)

00 02 04 06 08 10Time (s)

(a)

ndash002

ndash001

000

001

002 First point

x 4 (m

s)

00 80 times 10ndash540 times 10ndash5ndash40 times 10ndash5ndash80 times 10ndash5

x3 (m)

Last point

(b)

Figure 5 Dynamic response of the work roll displacement and the phase diagram under μlowast 00040 (a) time history of work rolldisplacement (b) phase diagram

00

Wor

k ro

ll di

spla

cem

ent (

m)

00 02 04 06 08 10Time (s)

10 times 10ndash7

50 times 10ndash8

ndash50 times 10ndash8

ndash10 times 10ndash7

(a)

00

12 times 10ndash7

60 times 10ndash8

ndash60 times 10ndash8

ndash12 times 10ndash7

ndash60 times 10ndash9 60 times 10ndash9ndash30 times 10ndash9 30 times 10ndash9

x 4 (m

s)

00x3 (m)

Last

poin

t

(b)

Figure 6 Dynamic response of the work roll displacement and the phase diagram under μlowast 00048 (a) time history of work rolldisplacement (b) phase diagram

ndash10 times 10ndash6

ndash50 times 10ndash7

00

50 times 10ndash7

10 times 10ndash6

Wor

k ro

ll di

spla

cem

ent (

m)

00 02 04 06 08 10Time (s)

(a)

ndash10 times 10ndash6 ndash50 times 10ndash7 00 50 times 10ndash7 10 times 10ndash6ndash80 times 10ndash4

ndash40 times 10ndash4

00

40 times 10ndash4

x 4 (m

s)

x3 (m)

Firs

t poi

nt

(b)

Figure 7 Dynamic response of the work roll displacement and the phase diagram under μlowast 00300 (a) time history of work rolldisplacement (b) phase diagram

Shock and Vibration 9

Based on equation (12) the tension variation at the entryposition (dTb) and exit position (dTf) is smaller comparedwith low friction coefficient rolling condition so the fluc-tuation of the response rolling force ΔPi increases and thestability of the stand decreases Meanwhile when the frictionis quite small zPzhout will increase with the decrease offriction coefficient F43 in equation (29) decreases so theinstantaneous dynamic structure spring constant decreasesand the fluctuation of the response rolling force ΔPi causedby the strip exit thickness variation will increase so thestability of the stand is decreased

43 Stable Range Analysis of Other Rolling Parameters

431 Calculation of Limited Rolling Speed Based on thiscoupled vertical-horizontal-torsional vibration analysismodel rolling speed has been chosen as the Hopf bifurcationparameter to obtain the limited stability rolling speed of the

UCM tandem cold rolling mill and other rolling conditionsare shown in Table 1 )e actual bifurcation point of rollingspeed is 251ms Figure 10 shows the time history of workroll displacement under the rolling speed of 252msAccording to Figure 10 the displacement amplitude ofthe work roll shows a divergence trend and the self-excitedvibration happens )e critical rolling speed calculated bythe proposed chatter model is 251ms

Figure 11 shows the tension and roll gap variation in No4 and No 5 stands obtained from the Production DataAcquisition (PDA) system As is shown in Figure 11(b)when the rolling speed is 221ms the back and front tensionof the No 5 stand fluctuate within a small range and the millstand is stable However when the rolling speed is raised to230ms the back and front tension of the No 5 standfluctuate more violently and the front tension of the No 5stand begins to diverge as shown in Figure 11(a) It is foundthat the roll gap of No 4 and No 5 stands fluctuates moreviolently by comparing Figures 11(c) and 11(d) Meanwhile

00 02 04 06 08 10

00

20 times 10ndash7

ndash20 times 10ndash7Wor

k ro

ll di

spla

cem

ent (

m)

Time (s)

(a)

00

00

10 times 10ndash4

50 times 10ndash5

ndash50 times 10ndash5

ndash10 times 10ndash4

ndash20 times 10ndash7 20 times 10ndash7

Last point

x 4 (m

s)

x3 (m)

First point

(b)

Figure 8 Dynamic response of the work roll displacement and the phase diagram under μlowast 00401 (a) time history of work rolldisplacement (b) phase diagram

00 02 04 06 08 10

00

20 times 10ndash6

ndash20 times 10ndash6Wor

k ro

ll di

spla

cem

ent (

m)

Time (s)

(a)

ndash20 times 10ndash6 00 20 times 10ndash6

00

10 times 10ndash3

50 times 10ndash4

ndash50 times 10ndash4

ndash10 times 10ndash3

x 4 (m

s)

x3 (m)

Last point

(b)

Figure 9 Dynamic response of the work roll displacement and the phase diagram under μlowast 00420 (a) time history of work rolldisplacement (b) phase diagram

10 Shock and Vibration

a sudden and harsh vibration noise will be heard in the No 5stand and finally the strip breaks It shows mill chatterappears under the speed of 230ms and the actual criticalrolling speed is 230ms

)e calculated critical rolling speed is 251ms while theactual critical rolling speed is 230ms )ere is a certainrelative error between the calculated critical speed and actualcritical rolling speed )is is due to the small coefficientdifference in Jacobian matrix With the change of rollingspeed the actual value of the tension and friction is changedAnd the calculated value of tension factor nt and frictioncoefficient in zPzh and zMzh is a little different with theactual value So the calculated limited rolling speed is a littlebigger than the actual value However by comparing thecritical rolling speed calculated by the proposed model withthe actual critical rolling speed it can be seen that theproposed model is still accurate

)e vibration is largely influenced by the friction co-efficient and rolling speed Based on equation (13) Figure 12shows the friction coefficient response of No 5 stand versusspeed Despite the changes of speed the friction coefficient isstill in the stability range of 00048ndash00401 It is divided intotwo regions In region 1 both the friction coefficient androlling speed are in the stable range In region 2 the frictioncoefficient is still in the stable range while the speed exceedsthe limited speed rangeWhen the rolling speed is lower than251ms the mill system is stable without changing thelubrication conditions And when the rolling speed is higherthan 251ms the stand is unstable However the frictioncoefficient is in the middle value of stable range under thespeed is 251ms so mill chatter might be eliminated byreducing the basic friction coefficient (by improving thelubrication characteristics such as increasing the concen-tration of emulsion increasing spray flow and improvingthe property of the base oil)

432 Calculation of Maximum Reduction Assuming theentry thickness is constant the exit thickness is variable Andthe exit thickness has been chosen as the Hopf bifurcationparameter Other calculated rolling conditions are shown in

Table 1 )en the calculated bifurcation point of exitthickness is 0168mm)e optimal setting range of reductionfor No 5 stand is 0ndash44When the reduction of No 5 stand is44 the increase of reduction leads to a lower mill standstability )e reason is also related to the increase of rollingforce variation under the larger reduction rolling conditions

As can be seen in Table 1 in normal rolling schedule thereduction rate is 333 And the max reduction is up to 44In reality considering there are other factors that influencethe stand stability such as rolling speed and lubricationconditions the max reduction is lower than 44 In generalthe effect of the reduction in the mill stability is lower thanthe improvement of friction coefficient

433 Front Tension Stable Range When the work roll ve-locity is 221ms the calculated bifurcation point of exittension is 15ndash133MPa Small front tension means that therolling force will be quite large So under the same excitationinterferences the variation of rolling force will be muchbigger and the stand stability is decreased When the fronttension is big enough tension factor nt decreases leading toa larger value in zPzh and zMzh As a result the fluc-tuation of rolling force and roll torque increases and thestability of the system is reduced as well So the influence oftension on the mill stability is smaller than other parameters

)ese results can be used to estimate mill chatter in acertain stand And the stable range of friction reduction andtension can be used to adjust the rolling schedule in theUCM mill to avoid mill chatter under high rolling speed

5 Conclusion

(1) A new multiple degrees of freedom and nonlineardynamic vertical-horizontal-torsional coupled millchatter model for UCM cold rolling mill is estab-lished In the dynamic rolling process model thetime-varying coupling effect between rolling forceroll torque variation and fluctuation of stripthickness and tension is formulated In the vibrationmodel the movement of work roll in both vertical-horizontal-torsional directions and the multirollcharacteristics in the UCM rolling mill are consid-ered Finally the dynamic relationship of rollmovement responses and dynamic changes in theroll gap is formulated

(2) Selecting the friction coefficient as the Hopf bi-furcation parameter and by using RouthndashHurwitzdeterminant the Hopf bifurcation point of the millvibration system is calculated and the bifurcationcharacteristics are analyzed Research shows thateither too low or too high friction coefficient in-creases the probability of mill chatter When thefriction coefficient is extremely low the mill systemproduces a stable limit cycle With the frictioncontinuing to increase the phase trajectory willconverge toward the origin point However the limitcycle of phase diagram disappears and dispersesoutward under high friction rolling conditions

00 02 04 06 08 10ndash2 times 10ndash7

ndash1 times 10ndash7

0

1 times 10ndash7

2 times 10ndash7

Wor

k ro

ll di

spla

cem

ent (

m)

Time (s)

Figure 10 Work roll time history displacement under the rollingspeed of 252ms

Shock and Vibration 11

0 1 2 30

20

40

60

80

Fron

t ten

sion

(kN

)

Back tension

Back

tens

ion

(kN

)

Time (s)

0

20

40

60

Front tension

(a)

0 1 2 30

20

40

60

80

Fron

t ten

sion

(kN

)

Back tension

Back

tens

ion

(kN

)

Time (s)

0

20

40

60

Front tension

(b)

0 1 2 3

015

018

021

024

Roll

gap

of N

o 5

stan

d (m

m)

Roll gap of No 4 stand

Roll

gap

of N

o 4

stan

d (m

m)

Time (s)

268

272

276

280

Roll gap of No 5 stand

(c)

0 1 2 3

030

033

036

039

Roll

gap

of N

o 5

stan

d (m

m)

Roll gap of No 4 stand

Roll

gap

of N

o4

stand

(mm

)

Time (s)

272

276

280

284

Roll gap of No 5 stand

(d)

Figure 11 Tension and roll gap time history (a) back and front tension of No 5 stand under the speed of 230ms (b) back and fronttension of No 5 stand under the speed of 221ms (c) roll gap of No 4 and No 5 stands under the speed of 230ms (d) roll gap of No 4 andNo 5 stands under the speed of 221ms

0 10 20 30

0020

0025

0030

0035

0040

0045

(2)

(25 002101)

Fric

tion

coeffi

cien

t

Speed (ms)

x = 251

(1)

Figure 12 Friction coefficient response of No 5 stand versus speed

12 Shock and Vibration

(3) Based on the coupled mill chatter model the millsystemrsquos stable limited rolling speed stable range ofreduction and front tension are investigatedWithout changing the rolling speed the most ef-fective way to eliminate mill chatter is to change thelubrication conditions and adjust the reductionschedule And the effect of tension on the vibration isquite limited )ese methods can be used to improvethe chatter phenomena in rolling mill under highrolling speed )e proposed vibration model canalso be used to design an optimum rolling schedulein the UCM tandem cold rolling mill and to de-termine the appearance of mill chatter under certainrolling conditions

Data Availability

)e rolling conditions data used to support the findings ofthis study are included within the article

Conflicts of Interest

)e authors declare that they have no conflicts of interestregarding the publication of this manuscript

Acknowledgments

)is work was supported by the National Natural ScienceFoundation of China (51774084 and 51634002) the NationalKey RampD Program of China (2017YFB0304100) the Fun-damental Research Funds for the Central Universities(N170708020) and the Open Research Fund from the StateKey Laboratory of Rolling and Automation NortheasternUniversity (2017RALKFKT006)

References

[1] I S Yun W R D Wilson and K F Ehmann ldquoReview ofchatter studies in cold rollingrdquo International Journal ofMachine Tools and Manufacture vol 38 no 12pp 1499ndash1530 1998

[2] I-S Yun W R D Wilson and K F Ehmann ldquoChatter in thestrip rolling process Part 1 dynamic model of rollingrdquoJournal of Manufacturing Science and Engineering vol 120no 2 pp 330ndash336 1998

[3] J Tlusty G Chandra S Critchley and D Paton ldquoChatter incold rollingrdquo CIRP Annals vol 31 no 1 pp 195ndash199 1982

[4] N M Reza F M Reza M Salimi and H Shojaei ldquoExper-imental investigations and ALE finite element methodanalysis of chatter in cold strip rollingrdquo ISIJ Internationalvol 52 no 12 pp 2245ndash2253 2012

[5] M R Niroomand M R Forouzan M Salimi and H ShojaeildquoExperimental investigations and ALE finite element methodanalysis of chatter in cold strip rollingrdquo ISIJ Internationalvol 52 no 12 pp 2245ndash2253 2012

[6] M R Niroomand M R Forouzan and M Salimi ldquo)eo-retical and experimental analysis of chatter in tandem coldrolling mills based on wave propagation theoryrdquo ISIJ In-ternational vol 55 no 3 pp 637ndash646 2015

[7] S Kapil P Eberhard and S K Dwivedy ldquoNonlinear dynamicanalysis of a parametrically excited cold rolling millrdquo Journal

of Manufacturing Science and Engineering vol 136 no 4p 041012 2014

[8] Y Kimura Y Sodani N Nishiura N Ikeuchi and Y MiharaldquoAnalysis of chaffer in tandem cold rolling millsrdquo ISIJ In-ternational vol 43 no 1 pp 77ndash84 2003

[9] N Fujita Y Kimura K Kobayashi et al ldquoDynamic control oflubrication characteristics in high speed tandem cold rollingrdquoJournal of Materials Processing Technology vol 229pp 407ndash416 2016

[10] A Heidari and M R Forouzan ldquoOptimization of cold rollingprocess parameters in order to increasing rolling speedlimited by chatter vibrationsrdquo Journal of Advanced Researchvol 4 no 1 pp 27ndash34 2013

[11] A Heidari M R Forouzan and S Akbarzadeh ldquoEffect offriction on tandem cold rolling mills chatteringrdquo ISIJ In-ternational vol 54 no 10 pp 2349ndash2356 2014

[12] A Heidari M R Forouzan and S Akbarzadeh ldquoDevelop-ment of a rolling chatter model considering unsteady lubri-cationrdquo ISIJ International vol 54 no 1 pp 165ndash170 2014

[13] A Heidari M R Forouzan and M R Niroomand ldquoDe-velopment and evaluation of friction models for chattersimulation in cold strip rollingrdquo International Journal ofAdvanced Manufacturing Technology vol 96 no 4 pp 1ndash212018

[14] Y Kim C-W Kim S Lee and H Park ldquoDynamic modelingand numerical analysis of a cold rolling millrdquo InternationalJournal of Precision Engineering and Manufacturing vol 14no 3 pp 407ndash413 2013

[15] P V Krot ldquoNonlinear vibrations and backlashes diagnosticsin the rolling mills drive trainsrdquo in Proceedings of the 6thEUROMECH Nonlinear Dynamics Conference Saint Peters-burg Russia July 2008

[16] S Liu B Sun S Zhao J Li and W Zhang ldquoStronglynonlinear dynamics of torsional vibration for rolling millrsquoselectromechanical coupling systemrdquo Steel Research In-ternational vol 86 no 9 pp 984ndash992 2015

[17] S Liu J Wang J Liu and Y Li ldquoNonlinear parametricallyexcited vibration and active control of gear pair system withtime-varying characteristicrdquo Chinese Physics B vol 24 no 10pp 303ndash311 2015

[18] B D Hassard N D Kazarinoff and Y H Wan eory andApplications of Hopf Bifurcation Cambridge University PressCambridge UK 1981

[19] B H K Lee S J Price and Y SWong ldquoNonlinear aeroelasticanalysis of airfoils bifurcation and chaosrdquo Progress inAerospace Sciences vol 35 no 3 pp 205ndash334 1999

[20] K Nishimura T Ikeda and Y Harata ldquoLocalization phe-nomena in torsional rotating shaft systems with multiplecentrifugal pendulum vibration absorbersrdquo Nonlinear Dy-namics vol 83 no 3 pp 1705ndash1726 2015

[21] T Zhang and H Dai ldquoBifurcation analysis of high-speedrailway wheel-setrdquo Nonlinear Dynamics vol 83 no 3pp 1511ndash1528 2015

[22] Z Y Gao Y Zang and T Han ldquoHopf bifurcation andstability analysis on the mill drive systemrdquo Applied Mechanicsamp Materials vol 121-126 pp 1514ndash1520 2012

[23] L Zeng Y Zang and Z Gao ldquoHopf bifurcation control forrolling mill multiple-mode-coupling vibration under non-linear frictionrdquo Journal of Vibration and Acoustics vol 139no 6 article 061015 2017

[24] S Liu Z Wang J Wang and H Li ldquoSliding bifurcationresearch of a horizontal-torsional coupled main drive systemof rolling millrdquo Nonlinear Dynamics vol 83 pp 441ndash4552015

Shock and Vibration 13

[25] Y Peng M Zhang J-L Sun and Y Zhang ldquoExperimentaland numerical investigation on the roll system swing vibra-tion characteristics of a hot rolling millrdquo ISIJ Internationalvol 57 no 9 pp 1567ndash1576 2017

[26] J Pittner and M A Simaan ldquoOptimal control of continuoustandem cold metal rollingrdquo in Proceedings of 2008 AmericanControl Conference Seattle WA USA June 2008

[27] R B Sims and D F S Arthur ldquoSpeed dependent variables incold strip rollingrdquo Journal of Iron and Steel Institute vol 172no 3 pp 285ndash295 1952

[28] S Z Chen D H Zhang J Sun J S Wang and J SongldquoOnline calculation model of rolling force for cold rolling millbased on numerical integrationrdquo in Proceedings of the 24thChinese Control and Decision Conference (CCDC) TaiyuanChina May 2012

[29] D R Bland and H Ford ldquo)e calculation of roll force andtorque in cold strip rolling with tensionsrdquo Proceedings of theInstitution of Mechanical Engineers vol 159 no 1 pp 144ndash163 2006

[30] R Hill ldquoRelations between roll-force torque and the appliedtensions in strip-rollingrdquo Proceedings of the Institution ofMechanical Engineers vol 163 no 1 pp 135ndash140 2006

[31] R E Johnson and Q Qi ldquoChatter dynamics in sheet rollingrdquoInternational Journal of Mechanical Sciences vol 36 no 7pp 617ndash630 1994

[32] Z Gao L Bai and Q Li ldquoResearch on critical rolling speed ofself-excited vibration in the tandem rolling process of thinstriprdquo Journal of Mechanical Engineering vol 53 no 12pp 118ndash132 2017

[33] R C Dorf and R H Bishop Modern Control SystemsUniversity of California Davis CA USA 12th edition 2010

[34] W M Liu ldquoCriterion of hopf bifurcations without usingeigenvaluesrdquo Journal of Mathematical Analysis and Appli-cations vol 182 no 1 pp 250ndash256 1994

[35] L Zeng Y Zang and Z Gao ldquoMultiple-modal-couplingmodeling and stability analysis of cold rolling mill vibra-tionrdquo Shock and Vibration vol 2016 Article ID 234738626 pages 2016

14 Shock and Vibration

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)e neutral angle is calculated by

c

hout

Rprime

1113971

times tan

hout

Rprime

1113971

timeshn

2⎛⎝ ⎞⎠ (8)

where khinand khout

are the deformation resistance at stripentry and exit position and μ is the friction coefficient in theroll gap

Based on equation (7) strip thickness fluctuation at theneutral point can be expressed as

dhn zhn

zhindhin +

zhn

zhoutdhout +

zhn

zTf

dTf +zhn

zTb

dTb

12minus

12μhin

1113888 1113889dhin +12

+1

2μhout1113888 1113889dhout

+1

2μ khoutminusTf1113872 1113873

dTf minus1

2μ khinminusTb1113872 1113873

dTb

(9)

Considering the mass conservation law to the mass flowcontinuity equation

Vrhn vinhin vouthout (10)

where Vr is the work roll circumference velocity)e strip speed variation at the entry and exit can be

expressed as [8]

dvin Vr

hindhn

dvout Vr

h2out

houtdhn minus hndhout( 1113857

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(11)

According to Hookersquos law the variation of tension isproportional to integral of speed difference between entry ofi stand and exit of iminus 1 stand [26] Assuming the exit speed ofiminus 1 stand and entry speed of i+ 1 stand are always keptconstant the tension variation at the entry and exit can beexpressed as

dTfi minusE2

550251113946 dvoutidt

dTbi E2

550251113946 dvinidt

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(12)

where subscript i means i stand And in this research thestrip length between i and i + 1 stand is 55025m

In the most rolling mill vibration models the frictioncoefficient was usually set as a constant value However inthe cold rolling the lubrication characteristics in the rollgap were significantly different under the high rolling speedand low speed Sims and Arthur [27] found that when therolling speed was higher than 025ms the friction co-efficient showed a nonlinear characteristic which is causedby unsteady rolling conditions As a result it is necessary toinvestigate the dynamic vibration stability in the UCM millwith the consideration of nonlinear friction characteristicsConsidering the roll wear and other conditions into con-sideration the friction coefficient in this research is obtainedby [28]

μ μ0 + μV middot eminus VrV0( )1113874 1113875 middot 1 + cR middot Ra minusRa0

1113872 11138731113960 1113961

middot 1 +cW

1 + LL0( 11138571113888 1113889

(13)

where μ0 and μV are the basic friction coefficient that relatedto lubrication and lubrication coefficient related to rollingspeed V0 is the basic rolling speed Ra and Ra0

are actualwork roll roughness and roughness of unused work roll CR

is the work roll roughness coefficient cW is the coefficientrelated to cumulative rolling strip length and L and L0 arethe cumulative rolling strip length of work roll and datumrolling length

)e rolling force is calculated by Bland-Ford-Hillequation [29 30]

P wlprimekQpnt (14)

A

BNeutralpointC

Rprime α

γ

Oprime

O

hout2hn2

hin2

xin xn xout

vin

voutTb

Tf

xw

yw

Figure 1 Vibration geometry model of the roll gap

Shock and Vibration 3

where

Qp 108 + 179εμ

Rprimehin

1113971

minus 102ε

nt 1minus1minus μt( 1113857Tf + μtTb1113960 1113961

k μt 07

ε hin minus hout

hin

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)

and k is the deformation resistance of the rolling strip In thisresearch deformation resistance of the strip is expressed bythe following equation

k A + B middot εΣ( 1113857 1minusC middot eminusDmiddotεΣ1113872 1113873 (16)

where εΣ is average strain which can be given by εΣ

(23

radic) middot ln(H0h) h (hin + 2hout3) A B C and D are

references in deformation resistance model In this researchA 50000MPa B 17900MPa C 020 and D 500 H0is the thickness of the raw material

)e investigation of UCMmill showed that No 4 and No5 stand chatter frequently so a comprehensive mill vibrationmonitoring test was implemented in a 1450mm UCM plantby using CTC-type accelerometer sensors )e accelerometersensors were mounted on the work roll bearings of No 5stand When rolling the most common products of035mmstrip in this plant the vibration signals during thestable rolling stage are shown in Figure 2 It illustrated thatalthough the horizontal rolling force was relatively smallcompared with the vertical rolling force the horizontal rollingforce was still big enough )erefore it was sufficiently sig-nificant to establish a comprehensive model which allows therolls to vibrate in both horizontal and vertical directions

Considering the dynamic equilibrium of strip in x di-rection the horizontal rolling force can be calculated asfollows

Px 1113946a

0P sin θdθ + 1113946

a

cPμ cos θdθ minus 1113946

c

0Pμ cos θdθ

w

2Tfhout minusTbhin1113872 1113873

(17)

)e vertical rolling force can be written as

Py 1113946α

0P cos θdθ + 1113946

α

cPμ sin θdθ minus 1113946

c

0Pμ sin θdθ

wlprimekQpnt 1113946α

0cos θdθ + 1113946

α

cμ sin θdθ minus 1113946

c

0μ sin θdθ1113888 1113889

wlprimekQpnt 1113946α

0cos θdθ minus μ cos α + 2μ cos cminus μ1113874 1113875

(18)

Considering that the bite angle α and neutral angle c areextremely small it assumed that the vertical rolling force isequal to the total rolling force [14 22] so equation (18) canbe expressed as

Py asymp wlprimekQpnt 1113946α

0cos θdθ asymp wlprimekQpnt P (19)

Assuming the thickness fluctuation and tension varia-tion are caused by roll gap vibration as separated distur-bances the deviation of the rolling force in the x directioncan be written as

dPx zPx

zhindhin +

zPx

zhoutdhout +

zPx

zTb

dTb +zPx

zTf

dTf (20)

Based on equation (17) the deviation of the rolling forcein the x direction can be also expressed as

dPx minuswTb

2dhin + wTfdyw minus

whin

2dTb +

whout

2dTf

(21)

)e variation of the rolling force in the y direction can bedefined as

dPy zP

zhindhin +

zP

zhoutdhout +

zP

zTf

dTf +zP

zTb

dTb (22)

where

zP

zhin wlprimeknt minus

1792

μ

Rprime

h3in

1113971

+32

times 179μhout

Rprime

h5in

1113971

minus 102hout

h2in

⎛⎝ ⎞⎠

zP

zhout wlprimeknt minus179

μhin

Rprimehin

1113971

+ 1021

hin

⎛⎝ ⎞⎠

zP

zTb

minus07wlprimeQp

zP

zTf

minus03wlprimeQp

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(23)

)e rolling torque of the upper roll system can be de-duced as [23 25]

M w 1113946c

0μpR dθminusw 1113946

α

cμpR dθ

wμpR(2cminus α)

PμR(2cminus α)

(24)

Considering the bite angle α and neutral angle c areextremely small α asymp

(hin minus houtRprime)1113969

c asymp sin c (xw minus

xnRprime) and xn

Rprime(hn minus hc)

1113969

Also the variation of the rolling torque can be calcu-

lated by

dM zM

zhindhin +

zM

zhoutdhout +

zM

zTf

dTf +zM

zTb

dTb +zM

zxw

dxw

(25)

where

4 Shock and Vibration

zM

zhin wlprimeknt(2cminus α)1113888minus

1792

μ

Rprime

h3in

1113971

+32

times 179μhout

Rprime

h5in

1113971

minus 102hout

h2in

1113889minus12

wlprimekQpntμR1

hin minus hout( 1113857Rprime1113969

zM

zhout wlprimeknt(2cminus α) minus179

μhin

Rprimehin

1113971

+ 1021

hin

⎛⎝ ⎞⎠

+12

wlprimekQpntμR1

hin minus hout( 1113857Rprime1113969

zM

zTb

minus07wlprimeQpμR(2cminus α)

zM

zTf

minus03wlprimeQpμR(2cminus α)

zM

zxw

wlprimekQpntμR 21Rprimeminus α1113874 1113875

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)

)e calculated rolling conditions in this research areshown in Table 1)e grade of rolling strip is MRT5 and theexit strip width is 900mm

Figure 3 shows the comparison of calculated rolling forcein this rolling process model and actual rolling force in theexperiment )e rolling conditions are shown in Table 1 Asshown in Figure 3 the maximum relative error betweenactual rolling force and calculated rolling force is 723)ecomparative results of the simulation and experimental datashow that this chatter model has a high accuracy

3 Dynamic Nonlinear Vibration Model of theUCM Mill

In most vibration research studies the rolling mill standwas simplified as a mass-spring system To investigate the

displacements of each roll and interaction between rolls amodel with multiple degrees of freedom is necessaryHowever with the consideration of the symmetry of themill structure along the rolling pass line the multipledegrees of freedom chatter model can be simplifiedFigure 4 is the simplified nonlinear vertical-horizontal-torsional coupled vibration model considering the rollmovement in more than one direction

As shown in Figure 4 the top work roll (WR) wouldvibrate in both vertical and horizontal directions While theintermediate roll (IMR) and backup roll (BUR) were notallowed to vibrate in horizontal direction due to the as-sembly and restriction of the roll bearings [25]

A horizontal roll vibration displacement of WR isgenerated due to the installation offset of e between WR andIMR and the equivalent spring constant in horizontal di-rection can be calculated by

ndash010

ndash005

000

005

010

Vibr

atio

n ac

cele

ratio

n (g

)

00 02 04 06 08 10Time (s)

(a)

ndash010

ndash005

000

005

010

Vib

ratio

n ac

cele

ratio

n (g

)

00 02 04 06 08 10Time (s)

(b)

Figure 2 Time domain vibration signal of the work roll (a) vibration acceleration in the vertical direction (b) vibration acceleration in thehorizontal direction

Table 1 Rolling conditions

Parameters 1 stand 2 stand 3 stand 4 stand 5 standEntrythickness hin(mm)

2000 1269 0762 0460 0300

Exit thicknesshout (mm) 1269 0762 0460 0300 0200

Rolling forceP (kN) 7838863 8455166 7781567 7500996 7722711

Exit tension Tf(MPa) 140554 165252 188160 205575 68000

Entry tensionTb (MPa) 55000 140554 165252 188160 205575

Frictioncoefficient μ 0065 0060 0040 0030 0020

Roll velocityVr (m sminus1) 3381 5652 9425 14441 22121

Strip entryspeed vin(m sminus1)

2250 3546 5903 9774 14977

Shock and Vibration 5

Kwx e

Rw + RimKw (27)

where Rw and Rim are the roll radius of work roll and in-termediate roll and Kw is the contact spring constant be-tween upper work roll and intermediate roll

With the consideration of the coupling movement of therolls in both vertical horizontal and torsional directions thedynamic equilibrium equation of the upper roll system wasgiven by

Mw euroxw + Cwx _xw + Kwxxw ΔPx

Mw euroyw + Cwy _yw minus _yim( 1113857 + Kwy yw minusyim( 1113857 ΔPy

Mim euroyim minusCwy _yw minus _yim( 1113857 + Cim _yim minus _yb( 1113857

minusKwy yw minusyim( 1113857 + Kim yim minusyb( 1113857 0

Mb euroyb minusCim _yim minus _yb( 1113857 + Cb _yb minusKim yim minusyb( 1113857

+ Kbyb 0

Jweuroθw + Cwj

_θw + Kwjθw ΔM

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(28)

where xw is the WR vibration displacement in horizontaldirection yw yim and yb are the vibration displacement ofWR IMR and BUR in vertical direction θw is the WRtorsional vibration angular displacement ΔPx and ΔPy arethe fluctuation of rolling force in horizontal and verticaldirections ΔM is the variation of rolling torque and MwMim and Mb are the mass of work roll intermediate rolland backup roll respectively

)e mill stand spring constant is the total stiffness of thewhole rolling mill consisting of all the rolls )e mill springconstant (K) is measured by a stiffness calibration test )espring constants of work roll (Kw) intermediate roll (Kim)and backup roll (Kb) are calculated by an FEM model of theUCM mill [5 6 12] )e roll vibration displacement underdifferent rolling forces was obtained by a simulation ex-periment )e spring constants of the work roll in-termediate roll and backup roll were obtained by analyzingthe dynamic roll vibration displacement under differentrolling forces

Damping coefficients of the intermediate and backuprolls are calibrated by a simulation program [31 32] First

Rolling direction

BUR

IMR

WR

Kb CbCb

Kim Cim

Kb

Mb

Mim

MwCwyKwy

KimCim

CwyKwy

ΔP Kwx

Kwx Cwx

CwxJw

Kwj Cwj

(a)

Cim

Mim

Mb

Mw

Kim

Kb Cb

Kw Cw

yb

yim

Kwx e

TfTb

xw

yw

(b)

Figure 4 Geometry of the vertical-horizontal-torsional vibration model (a) simplified vertical-horizontal-torsional coupled vibrationmodel (b) geometry of coupled vibration model in x-y direction

0

2000

4000

6000

8000

10000

Rolli

ng fo

rce (

kN)

1 2 3 4 5Stand number

Actual rolling forceCalculated rolling force

Figure 3 Comparison between the actual rolling force and calculated rolling force

6 Shock and Vibration

the dynamic differential equation of the system is regularizedto obtain the equivalent damping matrix )en the dampingregular matrix is transformed into the original coordinatesystem and let the damping ratio be 003 Finally theequivalent damping value of the roll system is calculatedWork roll rotational inertia mass work roll rotational springconstant and work roll rotational damping coefficient areobtained from the work of Zeng et al [23] as the design andmill structure of rolling stands are similar to one another)e roll diameters and work roll installation offset are ob-tained from the UCM rolling mill installation drawingsTable 2 shows the structure specifications of the UCMmill inequation (28)

Equation (28) is the nonlinear dynamic equation of thevertical-horizontal-torsional coupled vibration modelConvert the above differential equation into the followingstate matrix form

_X F(X μ) + Hdhin (29)

whereX xw _xw yw _yw yim _yim yb _yb θw _θw dTf dTb1113960 1113961

T

x1 x2 x3 x121113858 1113859T

H 0minuswTb

2Mw

0zP

zhinMw

0 0 0 0 0zM

zhinJw

0 01113890 1113891

T

F

0 1 0 0 0 0 0 0 0 0 0 0F21 F22 F23 0 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 00 0 F43 F44 F45 F46 0 0 0 0 0 00 0 0 0 0 1 0 0 0 0 0 00 0 F63 F64 F65 F66 F67 F68 0 0 0 00 0 0 0 0 0 0 1 0 0 0 00 0 0 0 F85 F86 F87 F88 0 0 0 00 0 0 0 0 0 0 0 0 1 0 00 F102 F103 0 0 0 0 0 F109 F1010 F1011 F10120 0 F113 0 0 0 0 0 0 0 F1111 F11120 0 F123 0 0 0 0 0 0 0 F1211 F1212

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

F21 minusKwx

Mw

F22 minusCwx

Mw

F23 wTf

Mw

F43 2 zPzhout( 1113857minusKwy

Mw

F44 minusCwy

Mw

F45 Kwy

Mw

F46 Cwy

Mw

F63 Kwy

Mim

F64 Cwy

Mim

F65 minusKwy + Kim

Mim

F66 minusCwy + Cim

Mim

F67 Kim

Mim

F68 Cim

Mim

F85 Kim

Mb

F86 Cim

Mb

F87 minusKim + Kb

Mb

F88 minusCim + Cb

Mb

F102 zM

Jwzxw

F103 2zM

Jwzhout

F109 minusKwj

Jw

F1010 minusCwj

Jw

F1011 zM

JwzTf

F1012 zM

JwzTb

F113 minus2E2Vr zhnzhout( 1113857hout minus hn( 1113857

55025h2out

F1111 minusE2Vr zhnzTf1113872 1113873

55025hout

F1112 minusE2Vr zhnzTb( 1113857

55025hout

F123 2E2Vr zhnzhout( 1113857

55025hin

F1211 E2Vr zhnzTf1113872 1113873

55025hin

F1212 E2Vr zhnzTf1113872 1113873

55025hin

(30)

)en equation (29) is the nonlinear dynamic stateequation of the vertical-horizontal-torsional coupled vi-bration system considering the nonlinear characteristicduring the rolling process the movement of rolls in bothvertical-horizontal directions and the torsional vibration

4 Analysis of Vibration Stability of theRoll System

41 Hopf Bifurcation Parameter Calculation )e frictioncharacteristic would cause the instability of the dynamicstate equation so in this research μ has been chosen as the

Shock and Vibration 7

Hopf bifurcation parameter Assuming the entry thicknessof each stand is constant the equilibrium point is Xwhere F(X) 0 )e system equilibrium point can betransformed to the origin by coordinate transformation X0

[0 0 0 0 0 0 0 0 0 0 0 0]T Let JF(X) be the Jacobianmatrix of F(X) So the Hopf bifurcation can be obtained byjudging all the eigenvalues of JF(X) However it is difficult tocalculate the eigenvalues of JF(X) since the calculationamount is quite large )en the RouthndashHurwitz determinantis used to calculate the Hopf bifurcation point [33]

)e characteristic equation of Jacobian matrix JF(X) ofthe dynamic vertical-horizontal-torsional coupling vibrationmodel is assumed to be

a12λ12

+ a11λ11

+ middot middot middot + a2λ2

+ a1λ + a0 0 (31)

)e main determinant

D1 a1 D2 a1 a0

a3 a2

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868 middot middot middot Di

a1 a0 middot middot middot 0

a3 a2 middot middot middot 0

middot middot middot middot middot middot middot middot middot middot middot middot

a2iminus1 a2iminus2 middot middot middot ai

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(32)

where ai 0 if ilt 0 or igt n and a12 1Assuming μlowast is the Hopf bifurcation the Hopf bi-

furcation conditions are [23 24 34]

a0 μlowast( 1113857gt 0 D1 μlowast( 1113857gt 0 middot middot middot Dnminus2 μlowast( 1113857gt 0 Dnminus1 μlowast( 1113857 0

dΔiminus1 μlowast( 1113857

dμne 0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(33)

According to these assumptions the Hopf bifurcationpoint of μlowast occurs at μlowast 00048 and μlowast 00401

42Analysis of StableRangeofFriction When the value of μlowastis smaller than the first bifurcation point of μlowast 00048such as μlowast 00040 the response of work roll displacementand Hopf bifurcation diagram are shown in Figure 5 )eroll displacement will diverge rapidly and the equilibriumpoint of the equilibrium equation is unstable Besides asshown in Figure 5(b) the phase trajectory will disperseoutward of the origin in a ring shape and the radius cur-vature of the phase trajectory increases gradually It isconcluded that the roll vibration system is unstable and millchatter appears

As shown in Figure 6(a) when μlowast 00048 the work rolldisplacement exhibits as an equal amplitude vibration andthe roll vibrates at its balanced position )is steady regionrefers to the maximum and minimum fluctuation range ofwork roll Meanwhile Figure 6(b) shows that the phasediagram will begin to converge toward a stable limit cycle)e strip will produce a slight thickness fluctuation withinthe quality requirements and the mill system is stable underthese rolling conditions

When μlowast 00300 the roll systemrsquos time history responseand the phase diagram are shown in Figure 7 It can be seenfrom these figures that the phase trajectory will convergestoward a smaller and smaller circle near the origin point )eequilibrium point of work roll is asymptotically stable

)e roll system stability about the critical parameter ofμlowast 00401 is shown in Figure 8 When at the critical Hopfbifurcation parameter displacement of work roll firstlybecomes larger and finally converges to an equal amplitudeoscillation )e equilibrium point is unstable and the Hopfbifurcation diagram presents a ring shape However dy-namic range of the fixed limit cycle in Figure 8 is a littlelarger compared with Figure 6 )is is mainly because thefriction is bigger and as a result the fluctuation of theresponse rolling force ΔPi under the friction coefficient of00401 is much bigger

)e numerical solution of work roll displacement underthe Hopf bifurcation point of μlowast 00420 is shown in Fig-ure 9)e roll displacement will disperse outward all the timeBesides the phase diagram of the work roll will disperseoutward of the cycle and the radius curvature of the phasetrajectory increases gradually It illustrates that the mill vi-bration system is unstable and mill chatter is produced

When μlowast lt 00048 the roll system is unstable and whenμlowast 00048 the system is stable So the Hopf bifurcation ofμlowast 00048 is the subcritical bifurcation point and μlowast

00401 is the supercritical bifurcation pointMany researchers attribute the change of stability in

rolling mill system to the sudden change of the rolling forcefluctuation in dynamic equilibrium equation [11 25 35])e optimal setting range of friction coefficient is 00048ndash00401 and mill chatter appears when the friction is eithertoo high or too low )is is mainly due to the fluctuation ofrolling force When the friction is high with the increase offriction coefficient the neutral angle becomes bigger andthen the neutral point shifts to the entry side so the distancebetween the entry and neutral point becomes shorter andthe variation of the entry speed decreases [11] However theexit forward slip and exit speed fluctuation will increase

Table 2 Structure specifications of the UCM mill

Parameters ValueWork roll mass Mw (Kg) 3914Intermediate roll mass Mim (Kg) 5410Backup roll mass Mb (Kg) 31330Work roll contact spring constant Kw (N mminus1) 091times 1010

Intermediate roll spring constant Kim (N mminus1) 103times1010

Backup roll spring constant Kb (N mminus1) 585times1010

Work roll damping coefficient x directionCwx (Nmiddots mminus1) 500times104

Work roll damping coefficient in y directionCwy (Nmiddots mminus1) 164times105

Intermediate roll damping coefficient Cim (Nmiddots mminus1) 570times105

Backup roll damping coefficient Cb (Nmiddots mminus1) 510times106

Work roll rotational inertia mass Jw (Kgmiddotm2) 1381Work roll rotational spring constant Kwj (Nmiddotmmiddotradminus1) 790times106

Work roll rotational damping coefficientCwj (Nmiddotmmiddots radminus1) 4178

Work roll diameter Rw (mm) 2125Intermediate roll diameter Rim (mm) 245Backup roll diameter Rb (mm) 650Work roll installation offset e (mm) 5

8 Shock and Vibration

ndash80 times 10ndash5

ndash40 times 10ndash5

00

40 times 10ndash5

80 times 10ndash5W

ork

roll

disp

lace

men

t (m

)

00 02 04 06 08 10Time (s)

(a)

ndash002

ndash001

000

001

002 First point

x 4 (m

s)

00 80 times 10ndash540 times 10ndash5ndash40 times 10ndash5ndash80 times 10ndash5

x3 (m)

Last point

(b)

Figure 5 Dynamic response of the work roll displacement and the phase diagram under μlowast 00040 (a) time history of work rolldisplacement (b) phase diagram

00

Wor

k ro

ll di

spla

cem

ent (

m)

00 02 04 06 08 10Time (s)

10 times 10ndash7

50 times 10ndash8

ndash50 times 10ndash8

ndash10 times 10ndash7

(a)

00

12 times 10ndash7

60 times 10ndash8

ndash60 times 10ndash8

ndash12 times 10ndash7

ndash60 times 10ndash9 60 times 10ndash9ndash30 times 10ndash9 30 times 10ndash9

x 4 (m

s)

00x3 (m)

Last

poin

t

(b)

Figure 6 Dynamic response of the work roll displacement and the phase diagram under μlowast 00048 (a) time history of work rolldisplacement (b) phase diagram

ndash10 times 10ndash6

ndash50 times 10ndash7

00

50 times 10ndash7

10 times 10ndash6

Wor

k ro

ll di

spla

cem

ent (

m)

00 02 04 06 08 10Time (s)

(a)

ndash10 times 10ndash6 ndash50 times 10ndash7 00 50 times 10ndash7 10 times 10ndash6ndash80 times 10ndash4

ndash40 times 10ndash4

00

40 times 10ndash4

x 4 (m

s)

x3 (m)

Firs

t poi

nt

(b)

Figure 7 Dynamic response of the work roll displacement and the phase diagram under μlowast 00300 (a) time history of work rolldisplacement (b) phase diagram

Shock and Vibration 9

Based on equation (12) the tension variation at the entryposition (dTb) and exit position (dTf) is smaller comparedwith low friction coefficient rolling condition so the fluc-tuation of the response rolling force ΔPi increases and thestability of the stand decreases Meanwhile when the frictionis quite small zPzhout will increase with the decrease offriction coefficient F43 in equation (29) decreases so theinstantaneous dynamic structure spring constant decreasesand the fluctuation of the response rolling force ΔPi causedby the strip exit thickness variation will increase so thestability of the stand is decreased

43 Stable Range Analysis of Other Rolling Parameters

431 Calculation of Limited Rolling Speed Based on thiscoupled vertical-horizontal-torsional vibration analysismodel rolling speed has been chosen as the Hopf bifurcationparameter to obtain the limited stability rolling speed of the

UCM tandem cold rolling mill and other rolling conditionsare shown in Table 1 )e actual bifurcation point of rollingspeed is 251ms Figure 10 shows the time history of workroll displacement under the rolling speed of 252msAccording to Figure 10 the displacement amplitude ofthe work roll shows a divergence trend and the self-excitedvibration happens )e critical rolling speed calculated bythe proposed chatter model is 251ms

Figure 11 shows the tension and roll gap variation in No4 and No 5 stands obtained from the Production DataAcquisition (PDA) system As is shown in Figure 11(b)when the rolling speed is 221ms the back and front tensionof the No 5 stand fluctuate within a small range and the millstand is stable However when the rolling speed is raised to230ms the back and front tension of the No 5 standfluctuate more violently and the front tension of the No 5stand begins to diverge as shown in Figure 11(a) It is foundthat the roll gap of No 4 and No 5 stands fluctuates moreviolently by comparing Figures 11(c) and 11(d) Meanwhile

00 02 04 06 08 10

00

20 times 10ndash7

ndash20 times 10ndash7Wor

k ro

ll di

spla

cem

ent (

m)

Time (s)

(a)

00

00

10 times 10ndash4

50 times 10ndash5

ndash50 times 10ndash5

ndash10 times 10ndash4

ndash20 times 10ndash7 20 times 10ndash7

Last point

x 4 (m

s)

x3 (m)

First point

(b)

Figure 8 Dynamic response of the work roll displacement and the phase diagram under μlowast 00401 (a) time history of work rolldisplacement (b) phase diagram

00 02 04 06 08 10

00

20 times 10ndash6

ndash20 times 10ndash6Wor

k ro

ll di

spla

cem

ent (

m)

Time (s)

(a)

ndash20 times 10ndash6 00 20 times 10ndash6

00

10 times 10ndash3

50 times 10ndash4

ndash50 times 10ndash4

ndash10 times 10ndash3

x 4 (m

s)

x3 (m)

Last point

(b)

Figure 9 Dynamic response of the work roll displacement and the phase diagram under μlowast 00420 (a) time history of work rolldisplacement (b) phase diagram

10 Shock and Vibration

a sudden and harsh vibration noise will be heard in the No 5stand and finally the strip breaks It shows mill chatterappears under the speed of 230ms and the actual criticalrolling speed is 230ms

)e calculated critical rolling speed is 251ms while theactual critical rolling speed is 230ms )ere is a certainrelative error between the calculated critical speed and actualcritical rolling speed )is is due to the small coefficientdifference in Jacobian matrix With the change of rollingspeed the actual value of the tension and friction is changedAnd the calculated value of tension factor nt and frictioncoefficient in zPzh and zMzh is a little different with theactual value So the calculated limited rolling speed is a littlebigger than the actual value However by comparing thecritical rolling speed calculated by the proposed model withthe actual critical rolling speed it can be seen that theproposed model is still accurate

)e vibration is largely influenced by the friction co-efficient and rolling speed Based on equation (13) Figure 12shows the friction coefficient response of No 5 stand versusspeed Despite the changes of speed the friction coefficient isstill in the stability range of 00048ndash00401 It is divided intotwo regions In region 1 both the friction coefficient androlling speed are in the stable range In region 2 the frictioncoefficient is still in the stable range while the speed exceedsthe limited speed rangeWhen the rolling speed is lower than251ms the mill system is stable without changing thelubrication conditions And when the rolling speed is higherthan 251ms the stand is unstable However the frictioncoefficient is in the middle value of stable range under thespeed is 251ms so mill chatter might be eliminated byreducing the basic friction coefficient (by improving thelubrication characteristics such as increasing the concen-tration of emulsion increasing spray flow and improvingthe property of the base oil)

432 Calculation of Maximum Reduction Assuming theentry thickness is constant the exit thickness is variable Andthe exit thickness has been chosen as the Hopf bifurcationparameter Other calculated rolling conditions are shown in

Table 1 )en the calculated bifurcation point of exitthickness is 0168mm)e optimal setting range of reductionfor No 5 stand is 0ndash44When the reduction of No 5 stand is44 the increase of reduction leads to a lower mill standstability )e reason is also related to the increase of rollingforce variation under the larger reduction rolling conditions

As can be seen in Table 1 in normal rolling schedule thereduction rate is 333 And the max reduction is up to 44In reality considering there are other factors that influencethe stand stability such as rolling speed and lubricationconditions the max reduction is lower than 44 In generalthe effect of the reduction in the mill stability is lower thanthe improvement of friction coefficient

433 Front Tension Stable Range When the work roll ve-locity is 221ms the calculated bifurcation point of exittension is 15ndash133MPa Small front tension means that therolling force will be quite large So under the same excitationinterferences the variation of rolling force will be muchbigger and the stand stability is decreased When the fronttension is big enough tension factor nt decreases leading toa larger value in zPzh and zMzh As a result the fluc-tuation of rolling force and roll torque increases and thestability of the system is reduced as well So the influence oftension on the mill stability is smaller than other parameters

)ese results can be used to estimate mill chatter in acertain stand And the stable range of friction reduction andtension can be used to adjust the rolling schedule in theUCM mill to avoid mill chatter under high rolling speed

5 Conclusion

(1) A new multiple degrees of freedom and nonlineardynamic vertical-horizontal-torsional coupled millchatter model for UCM cold rolling mill is estab-lished In the dynamic rolling process model thetime-varying coupling effect between rolling forceroll torque variation and fluctuation of stripthickness and tension is formulated In the vibrationmodel the movement of work roll in both vertical-horizontal-torsional directions and the multirollcharacteristics in the UCM rolling mill are consid-ered Finally the dynamic relationship of rollmovement responses and dynamic changes in theroll gap is formulated

(2) Selecting the friction coefficient as the Hopf bi-furcation parameter and by using RouthndashHurwitzdeterminant the Hopf bifurcation point of the millvibration system is calculated and the bifurcationcharacteristics are analyzed Research shows thateither too low or too high friction coefficient in-creases the probability of mill chatter When thefriction coefficient is extremely low the mill systemproduces a stable limit cycle With the frictioncontinuing to increase the phase trajectory willconverge toward the origin point However the limitcycle of phase diagram disappears and dispersesoutward under high friction rolling conditions

00 02 04 06 08 10ndash2 times 10ndash7

ndash1 times 10ndash7

0

1 times 10ndash7

2 times 10ndash7

Wor

k ro

ll di

spla

cem

ent (

m)

Time (s)

Figure 10 Work roll time history displacement under the rollingspeed of 252ms

Shock and Vibration 11

0 1 2 30

20

40

60

80

Fron

t ten

sion

(kN

)

Back tension

Back

tens

ion

(kN

)

Time (s)

0

20

40

60

Front tension

(a)

0 1 2 30

20

40

60

80

Fron

t ten

sion

(kN

)

Back tension

Back

tens

ion

(kN

)

Time (s)

0

20

40

60

Front tension

(b)

0 1 2 3

015

018

021

024

Roll

gap

of N

o 5

stan

d (m

m)

Roll gap of No 4 stand

Roll

gap

of N

o 4

stan

d (m

m)

Time (s)

268

272

276

280

Roll gap of No 5 stand

(c)

0 1 2 3

030

033

036

039

Roll

gap

of N

o 5

stan

d (m

m)

Roll gap of No 4 stand

Roll

gap

of N

o4

stand

(mm

)

Time (s)

272

276

280

284

Roll gap of No 5 stand

(d)

Figure 11 Tension and roll gap time history (a) back and front tension of No 5 stand under the speed of 230ms (b) back and fronttension of No 5 stand under the speed of 221ms (c) roll gap of No 4 and No 5 stands under the speed of 230ms (d) roll gap of No 4 andNo 5 stands under the speed of 221ms

0 10 20 30

0020

0025

0030

0035

0040

0045

(2)

(25 002101)

Fric

tion

coeffi

cien

t

Speed (ms)

x = 251

(1)

Figure 12 Friction coefficient response of No 5 stand versus speed

12 Shock and Vibration

(3) Based on the coupled mill chatter model the millsystemrsquos stable limited rolling speed stable range ofreduction and front tension are investigatedWithout changing the rolling speed the most ef-fective way to eliminate mill chatter is to change thelubrication conditions and adjust the reductionschedule And the effect of tension on the vibration isquite limited )ese methods can be used to improvethe chatter phenomena in rolling mill under highrolling speed )e proposed vibration model canalso be used to design an optimum rolling schedulein the UCM tandem cold rolling mill and to de-termine the appearance of mill chatter under certainrolling conditions

Data Availability

)e rolling conditions data used to support the findings ofthis study are included within the article

Conflicts of Interest

)e authors declare that they have no conflicts of interestregarding the publication of this manuscript

Acknowledgments

)is work was supported by the National Natural ScienceFoundation of China (51774084 and 51634002) the NationalKey RampD Program of China (2017YFB0304100) the Fun-damental Research Funds for the Central Universities(N170708020) and the Open Research Fund from the StateKey Laboratory of Rolling and Automation NortheasternUniversity (2017RALKFKT006)

References

[1] I S Yun W R D Wilson and K F Ehmann ldquoReview ofchatter studies in cold rollingrdquo International Journal ofMachine Tools and Manufacture vol 38 no 12pp 1499ndash1530 1998

[2] I-S Yun W R D Wilson and K F Ehmann ldquoChatter in thestrip rolling process Part 1 dynamic model of rollingrdquoJournal of Manufacturing Science and Engineering vol 120no 2 pp 330ndash336 1998

[3] J Tlusty G Chandra S Critchley and D Paton ldquoChatter incold rollingrdquo CIRP Annals vol 31 no 1 pp 195ndash199 1982

[4] N M Reza F M Reza M Salimi and H Shojaei ldquoExper-imental investigations and ALE finite element methodanalysis of chatter in cold strip rollingrdquo ISIJ Internationalvol 52 no 12 pp 2245ndash2253 2012

[5] M R Niroomand M R Forouzan M Salimi and H ShojaeildquoExperimental investigations and ALE finite element methodanalysis of chatter in cold strip rollingrdquo ISIJ Internationalvol 52 no 12 pp 2245ndash2253 2012

[6] M R Niroomand M R Forouzan and M Salimi ldquo)eo-retical and experimental analysis of chatter in tandem coldrolling mills based on wave propagation theoryrdquo ISIJ In-ternational vol 55 no 3 pp 637ndash646 2015

[7] S Kapil P Eberhard and S K Dwivedy ldquoNonlinear dynamicanalysis of a parametrically excited cold rolling millrdquo Journal

of Manufacturing Science and Engineering vol 136 no 4p 041012 2014

[8] Y Kimura Y Sodani N Nishiura N Ikeuchi and Y MiharaldquoAnalysis of chaffer in tandem cold rolling millsrdquo ISIJ In-ternational vol 43 no 1 pp 77ndash84 2003

[9] N Fujita Y Kimura K Kobayashi et al ldquoDynamic control oflubrication characteristics in high speed tandem cold rollingrdquoJournal of Materials Processing Technology vol 229pp 407ndash416 2016

[10] A Heidari and M R Forouzan ldquoOptimization of cold rollingprocess parameters in order to increasing rolling speedlimited by chatter vibrationsrdquo Journal of Advanced Researchvol 4 no 1 pp 27ndash34 2013

[11] A Heidari M R Forouzan and S Akbarzadeh ldquoEffect offriction on tandem cold rolling mills chatteringrdquo ISIJ In-ternational vol 54 no 10 pp 2349ndash2356 2014

[12] A Heidari M R Forouzan and S Akbarzadeh ldquoDevelop-ment of a rolling chatter model considering unsteady lubri-cationrdquo ISIJ International vol 54 no 1 pp 165ndash170 2014

[13] A Heidari M R Forouzan and M R Niroomand ldquoDe-velopment and evaluation of friction models for chattersimulation in cold strip rollingrdquo International Journal ofAdvanced Manufacturing Technology vol 96 no 4 pp 1ndash212018

[14] Y Kim C-W Kim S Lee and H Park ldquoDynamic modelingand numerical analysis of a cold rolling millrdquo InternationalJournal of Precision Engineering and Manufacturing vol 14no 3 pp 407ndash413 2013

[15] P V Krot ldquoNonlinear vibrations and backlashes diagnosticsin the rolling mills drive trainsrdquo in Proceedings of the 6thEUROMECH Nonlinear Dynamics Conference Saint Peters-burg Russia July 2008

[16] S Liu B Sun S Zhao J Li and W Zhang ldquoStronglynonlinear dynamics of torsional vibration for rolling millrsquoselectromechanical coupling systemrdquo Steel Research In-ternational vol 86 no 9 pp 984ndash992 2015

[17] S Liu J Wang J Liu and Y Li ldquoNonlinear parametricallyexcited vibration and active control of gear pair system withtime-varying characteristicrdquo Chinese Physics B vol 24 no 10pp 303ndash311 2015

[18] B D Hassard N D Kazarinoff and Y H Wan eory andApplications of Hopf Bifurcation Cambridge University PressCambridge UK 1981

[19] B H K Lee S J Price and Y SWong ldquoNonlinear aeroelasticanalysis of airfoils bifurcation and chaosrdquo Progress inAerospace Sciences vol 35 no 3 pp 205ndash334 1999

[20] K Nishimura T Ikeda and Y Harata ldquoLocalization phe-nomena in torsional rotating shaft systems with multiplecentrifugal pendulum vibration absorbersrdquo Nonlinear Dy-namics vol 83 no 3 pp 1705ndash1726 2015

[21] T Zhang and H Dai ldquoBifurcation analysis of high-speedrailway wheel-setrdquo Nonlinear Dynamics vol 83 no 3pp 1511ndash1528 2015

[22] Z Y Gao Y Zang and T Han ldquoHopf bifurcation andstability analysis on the mill drive systemrdquo Applied Mechanicsamp Materials vol 121-126 pp 1514ndash1520 2012

[23] L Zeng Y Zang and Z Gao ldquoHopf bifurcation control forrolling mill multiple-mode-coupling vibration under non-linear frictionrdquo Journal of Vibration and Acoustics vol 139no 6 article 061015 2017

[24] S Liu Z Wang J Wang and H Li ldquoSliding bifurcationresearch of a horizontal-torsional coupled main drive systemof rolling millrdquo Nonlinear Dynamics vol 83 pp 441ndash4552015

Shock and Vibration 13

[25] Y Peng M Zhang J-L Sun and Y Zhang ldquoExperimentaland numerical investigation on the roll system swing vibra-tion characteristics of a hot rolling millrdquo ISIJ Internationalvol 57 no 9 pp 1567ndash1576 2017

[26] J Pittner and M A Simaan ldquoOptimal control of continuoustandem cold metal rollingrdquo in Proceedings of 2008 AmericanControl Conference Seattle WA USA June 2008

[27] R B Sims and D F S Arthur ldquoSpeed dependent variables incold strip rollingrdquo Journal of Iron and Steel Institute vol 172no 3 pp 285ndash295 1952

[28] S Z Chen D H Zhang J Sun J S Wang and J SongldquoOnline calculation model of rolling force for cold rolling millbased on numerical integrationrdquo in Proceedings of the 24thChinese Control and Decision Conference (CCDC) TaiyuanChina May 2012

[29] D R Bland and H Ford ldquo)e calculation of roll force andtorque in cold strip rolling with tensionsrdquo Proceedings of theInstitution of Mechanical Engineers vol 159 no 1 pp 144ndash163 2006

[30] R Hill ldquoRelations between roll-force torque and the appliedtensions in strip-rollingrdquo Proceedings of the Institution ofMechanical Engineers vol 163 no 1 pp 135ndash140 2006

[31] R E Johnson and Q Qi ldquoChatter dynamics in sheet rollingrdquoInternational Journal of Mechanical Sciences vol 36 no 7pp 617ndash630 1994

[32] Z Gao L Bai and Q Li ldquoResearch on critical rolling speed ofself-excited vibration in the tandem rolling process of thinstriprdquo Journal of Mechanical Engineering vol 53 no 12pp 118ndash132 2017

[33] R C Dorf and R H Bishop Modern Control SystemsUniversity of California Davis CA USA 12th edition 2010

[34] W M Liu ldquoCriterion of hopf bifurcations without usingeigenvaluesrdquo Journal of Mathematical Analysis and Appli-cations vol 182 no 1 pp 250ndash256 1994

[35] L Zeng Y Zang and Z Gao ldquoMultiple-modal-couplingmodeling and stability analysis of cold rolling mill vibra-tionrdquo Shock and Vibration vol 2016 Article ID 234738626 pages 2016

14 Shock and Vibration

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where

Qp 108 + 179εμ

Rprimehin

1113971

minus 102ε

nt 1minus1minus μt( 1113857Tf + μtTb1113960 1113961

k μt 07

ε hin minus hout

hin

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(15)

and k is the deformation resistance of the rolling strip In thisresearch deformation resistance of the strip is expressed bythe following equation

k A + B middot εΣ( 1113857 1minusC middot eminusDmiddotεΣ1113872 1113873 (16)

where εΣ is average strain which can be given by εΣ

(23

radic) middot ln(H0h) h (hin + 2hout3) A B C and D are

references in deformation resistance model In this researchA 50000MPa B 17900MPa C 020 and D 500 H0is the thickness of the raw material

)e investigation of UCMmill showed that No 4 and No5 stand chatter frequently so a comprehensive mill vibrationmonitoring test was implemented in a 1450mm UCM plantby using CTC-type accelerometer sensors )e accelerometersensors were mounted on the work roll bearings of No 5stand When rolling the most common products of035mmstrip in this plant the vibration signals during thestable rolling stage are shown in Figure 2 It illustrated thatalthough the horizontal rolling force was relatively smallcompared with the vertical rolling force the horizontal rollingforce was still big enough )erefore it was sufficiently sig-nificant to establish a comprehensive model which allows therolls to vibrate in both horizontal and vertical directions

Considering the dynamic equilibrium of strip in x di-rection the horizontal rolling force can be calculated asfollows

Px 1113946a

0P sin θdθ + 1113946

a

cPμ cos θdθ minus 1113946

c

0Pμ cos θdθ

w

2Tfhout minusTbhin1113872 1113873

(17)

)e vertical rolling force can be written as

Py 1113946α

0P cos θdθ + 1113946

α

cPμ sin θdθ minus 1113946

c

0Pμ sin θdθ

wlprimekQpnt 1113946α

0cos θdθ + 1113946

α

cμ sin θdθ minus 1113946

c

0μ sin θdθ1113888 1113889

wlprimekQpnt 1113946α

0cos θdθ minus μ cos α + 2μ cos cminus μ1113874 1113875

(18)

Considering that the bite angle α and neutral angle c areextremely small it assumed that the vertical rolling force isequal to the total rolling force [14 22] so equation (18) canbe expressed as

Py asymp wlprimekQpnt 1113946α

0cos θdθ asymp wlprimekQpnt P (19)

Assuming the thickness fluctuation and tension varia-tion are caused by roll gap vibration as separated distur-bances the deviation of the rolling force in the x directioncan be written as

dPx zPx

zhindhin +

zPx

zhoutdhout +

zPx

zTb

dTb +zPx

zTf

dTf (20)

Based on equation (17) the deviation of the rolling forcein the x direction can be also expressed as

dPx minuswTb

2dhin + wTfdyw minus

whin

2dTb +

whout

2dTf

(21)

)e variation of the rolling force in the y direction can bedefined as

dPy zP

zhindhin +

zP

zhoutdhout +

zP

zTf

dTf +zP

zTb

dTb (22)

where

zP

zhin wlprimeknt minus

1792

μ

Rprime

h3in

1113971

+32

times 179μhout

Rprime

h5in

1113971

minus 102hout

h2in

⎛⎝ ⎞⎠

zP

zhout wlprimeknt minus179

μhin

Rprimehin

1113971

+ 1021

hin

⎛⎝ ⎞⎠

zP

zTb

minus07wlprimeQp

zP

zTf

minus03wlprimeQp

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(23)

)e rolling torque of the upper roll system can be de-duced as [23 25]

M w 1113946c

0μpR dθminusw 1113946

α

cμpR dθ

wμpR(2cminus α)

PμR(2cminus α)

(24)

Considering the bite angle α and neutral angle c areextremely small α asymp

(hin minus houtRprime)1113969

c asymp sin c (xw minus

xnRprime) and xn

Rprime(hn minus hc)

1113969

Also the variation of the rolling torque can be calcu-

lated by

dM zM

zhindhin +

zM

zhoutdhout +

zM

zTf

dTf +zM

zTb

dTb +zM

zxw

dxw

(25)

where

4 Shock and Vibration

zM

zhin wlprimeknt(2cminus α)1113888minus

1792

μ

Rprime

h3in

1113971

+32

times 179μhout

Rprime

h5in

1113971

minus 102hout

h2in

1113889minus12

wlprimekQpntμR1

hin minus hout( 1113857Rprime1113969

zM

zhout wlprimeknt(2cminus α) minus179

μhin

Rprimehin

1113971

+ 1021

hin

⎛⎝ ⎞⎠

+12

wlprimekQpntμR1

hin minus hout( 1113857Rprime1113969

zM

zTb

minus07wlprimeQpμR(2cminus α)

zM

zTf

minus03wlprimeQpμR(2cminus α)

zM

zxw

wlprimekQpntμR 21Rprimeminus α1113874 1113875

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)

)e calculated rolling conditions in this research areshown in Table 1)e grade of rolling strip is MRT5 and theexit strip width is 900mm

Figure 3 shows the comparison of calculated rolling forcein this rolling process model and actual rolling force in theexperiment )e rolling conditions are shown in Table 1 Asshown in Figure 3 the maximum relative error betweenactual rolling force and calculated rolling force is 723)ecomparative results of the simulation and experimental datashow that this chatter model has a high accuracy

3 Dynamic Nonlinear Vibration Model of theUCM Mill

In most vibration research studies the rolling mill standwas simplified as a mass-spring system To investigate the

displacements of each roll and interaction between rolls amodel with multiple degrees of freedom is necessaryHowever with the consideration of the symmetry of themill structure along the rolling pass line the multipledegrees of freedom chatter model can be simplifiedFigure 4 is the simplified nonlinear vertical-horizontal-torsional coupled vibration model considering the rollmovement in more than one direction

As shown in Figure 4 the top work roll (WR) wouldvibrate in both vertical and horizontal directions While theintermediate roll (IMR) and backup roll (BUR) were notallowed to vibrate in horizontal direction due to the as-sembly and restriction of the roll bearings [25]

A horizontal roll vibration displacement of WR isgenerated due to the installation offset of e between WR andIMR and the equivalent spring constant in horizontal di-rection can be calculated by

ndash010

ndash005

000

005

010

Vibr

atio

n ac

cele

ratio

n (g

)

00 02 04 06 08 10Time (s)

(a)

ndash010

ndash005

000

005

010

Vib

ratio

n ac

cele

ratio

n (g

)

00 02 04 06 08 10Time (s)

(b)

Figure 2 Time domain vibration signal of the work roll (a) vibration acceleration in the vertical direction (b) vibration acceleration in thehorizontal direction

Table 1 Rolling conditions

Parameters 1 stand 2 stand 3 stand 4 stand 5 standEntrythickness hin(mm)

2000 1269 0762 0460 0300

Exit thicknesshout (mm) 1269 0762 0460 0300 0200

Rolling forceP (kN) 7838863 8455166 7781567 7500996 7722711

Exit tension Tf(MPa) 140554 165252 188160 205575 68000

Entry tensionTb (MPa) 55000 140554 165252 188160 205575

Frictioncoefficient μ 0065 0060 0040 0030 0020

Roll velocityVr (m sminus1) 3381 5652 9425 14441 22121

Strip entryspeed vin(m sminus1)

2250 3546 5903 9774 14977

Shock and Vibration 5

Kwx e

Rw + RimKw (27)

where Rw and Rim are the roll radius of work roll and in-termediate roll and Kw is the contact spring constant be-tween upper work roll and intermediate roll

With the consideration of the coupling movement of therolls in both vertical horizontal and torsional directions thedynamic equilibrium equation of the upper roll system wasgiven by

Mw euroxw + Cwx _xw + Kwxxw ΔPx

Mw euroyw + Cwy _yw minus _yim( 1113857 + Kwy yw minusyim( 1113857 ΔPy

Mim euroyim minusCwy _yw minus _yim( 1113857 + Cim _yim minus _yb( 1113857

minusKwy yw minusyim( 1113857 + Kim yim minusyb( 1113857 0

Mb euroyb minusCim _yim minus _yb( 1113857 + Cb _yb minusKim yim minusyb( 1113857

+ Kbyb 0

Jweuroθw + Cwj

_θw + Kwjθw ΔM

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(28)

where xw is the WR vibration displacement in horizontaldirection yw yim and yb are the vibration displacement ofWR IMR and BUR in vertical direction θw is the WRtorsional vibration angular displacement ΔPx and ΔPy arethe fluctuation of rolling force in horizontal and verticaldirections ΔM is the variation of rolling torque and MwMim and Mb are the mass of work roll intermediate rolland backup roll respectively

)e mill stand spring constant is the total stiffness of thewhole rolling mill consisting of all the rolls )e mill springconstant (K) is measured by a stiffness calibration test )espring constants of work roll (Kw) intermediate roll (Kim)and backup roll (Kb) are calculated by an FEM model of theUCM mill [5 6 12] )e roll vibration displacement underdifferent rolling forces was obtained by a simulation ex-periment )e spring constants of the work roll in-termediate roll and backup roll were obtained by analyzingthe dynamic roll vibration displacement under differentrolling forces

Damping coefficients of the intermediate and backuprolls are calibrated by a simulation program [31 32] First

Rolling direction

BUR

IMR

WR

Kb CbCb

Kim Cim

Kb

Mb

Mim

MwCwyKwy

KimCim

CwyKwy

ΔP Kwx

Kwx Cwx

CwxJw

Kwj Cwj

(a)

Cim

Mim

Mb

Mw

Kim

Kb Cb

Kw Cw

yb

yim

Kwx e

TfTb

xw

yw

(b)

Figure 4 Geometry of the vertical-horizontal-torsional vibration model (a) simplified vertical-horizontal-torsional coupled vibrationmodel (b) geometry of coupled vibration model in x-y direction

0

2000

4000

6000

8000

10000

Rolli

ng fo

rce (

kN)

1 2 3 4 5Stand number

Actual rolling forceCalculated rolling force

Figure 3 Comparison between the actual rolling force and calculated rolling force

6 Shock and Vibration

the dynamic differential equation of the system is regularizedto obtain the equivalent damping matrix )en the dampingregular matrix is transformed into the original coordinatesystem and let the damping ratio be 003 Finally theequivalent damping value of the roll system is calculatedWork roll rotational inertia mass work roll rotational springconstant and work roll rotational damping coefficient areobtained from the work of Zeng et al [23] as the design andmill structure of rolling stands are similar to one another)e roll diameters and work roll installation offset are ob-tained from the UCM rolling mill installation drawingsTable 2 shows the structure specifications of the UCMmill inequation (28)

Equation (28) is the nonlinear dynamic equation of thevertical-horizontal-torsional coupled vibration modelConvert the above differential equation into the followingstate matrix form

_X F(X μ) + Hdhin (29)

whereX xw _xw yw _yw yim _yim yb _yb θw _θw dTf dTb1113960 1113961

T

x1 x2 x3 x121113858 1113859T

H 0minuswTb

2Mw

0zP

zhinMw

0 0 0 0 0zM

zhinJw

0 01113890 1113891

T

F

0 1 0 0 0 0 0 0 0 0 0 0F21 F22 F23 0 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 00 0 F43 F44 F45 F46 0 0 0 0 0 00 0 0 0 0 1 0 0 0 0 0 00 0 F63 F64 F65 F66 F67 F68 0 0 0 00 0 0 0 0 0 0 1 0 0 0 00 0 0 0 F85 F86 F87 F88 0 0 0 00 0 0 0 0 0 0 0 0 1 0 00 F102 F103 0 0 0 0 0 F109 F1010 F1011 F10120 0 F113 0 0 0 0 0 0 0 F1111 F11120 0 F123 0 0 0 0 0 0 0 F1211 F1212

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

F21 minusKwx

Mw

F22 minusCwx

Mw

F23 wTf

Mw

F43 2 zPzhout( 1113857minusKwy

Mw

F44 minusCwy

Mw

F45 Kwy

Mw

F46 Cwy

Mw

F63 Kwy

Mim

F64 Cwy

Mim

F65 minusKwy + Kim

Mim

F66 minusCwy + Cim

Mim

F67 Kim

Mim

F68 Cim

Mim

F85 Kim

Mb

F86 Cim

Mb

F87 minusKim + Kb

Mb

F88 minusCim + Cb

Mb

F102 zM

Jwzxw

F103 2zM

Jwzhout

F109 minusKwj

Jw

F1010 minusCwj

Jw

F1011 zM

JwzTf

F1012 zM

JwzTb

F113 minus2E2Vr zhnzhout( 1113857hout minus hn( 1113857

55025h2out

F1111 minusE2Vr zhnzTf1113872 1113873

55025hout

F1112 minusE2Vr zhnzTb( 1113857

55025hout

F123 2E2Vr zhnzhout( 1113857

55025hin

F1211 E2Vr zhnzTf1113872 1113873

55025hin

F1212 E2Vr zhnzTf1113872 1113873

55025hin

(30)

)en equation (29) is the nonlinear dynamic stateequation of the vertical-horizontal-torsional coupled vi-bration system considering the nonlinear characteristicduring the rolling process the movement of rolls in bothvertical-horizontal directions and the torsional vibration

4 Analysis of Vibration Stability of theRoll System

41 Hopf Bifurcation Parameter Calculation )e frictioncharacteristic would cause the instability of the dynamicstate equation so in this research μ has been chosen as the

Shock and Vibration 7

Hopf bifurcation parameter Assuming the entry thicknessof each stand is constant the equilibrium point is Xwhere F(X) 0 )e system equilibrium point can betransformed to the origin by coordinate transformation X0

[0 0 0 0 0 0 0 0 0 0 0 0]T Let JF(X) be the Jacobianmatrix of F(X) So the Hopf bifurcation can be obtained byjudging all the eigenvalues of JF(X) However it is difficult tocalculate the eigenvalues of JF(X) since the calculationamount is quite large )en the RouthndashHurwitz determinantis used to calculate the Hopf bifurcation point [33]

)e characteristic equation of Jacobian matrix JF(X) ofthe dynamic vertical-horizontal-torsional coupling vibrationmodel is assumed to be

a12λ12

+ a11λ11

+ middot middot middot + a2λ2

+ a1λ + a0 0 (31)

)e main determinant

D1 a1 D2 a1 a0

a3 a2

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868 middot middot middot Di

a1 a0 middot middot middot 0

a3 a2 middot middot middot 0

middot middot middot middot middot middot middot middot middot middot middot middot

a2iminus1 a2iminus2 middot middot middot ai

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(32)

where ai 0 if ilt 0 or igt n and a12 1Assuming μlowast is the Hopf bifurcation the Hopf bi-

furcation conditions are [23 24 34]

a0 μlowast( 1113857gt 0 D1 μlowast( 1113857gt 0 middot middot middot Dnminus2 μlowast( 1113857gt 0 Dnminus1 μlowast( 1113857 0

dΔiminus1 μlowast( 1113857

dμne 0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(33)

According to these assumptions the Hopf bifurcationpoint of μlowast occurs at μlowast 00048 and μlowast 00401

42Analysis of StableRangeofFriction When the value of μlowastis smaller than the first bifurcation point of μlowast 00048such as μlowast 00040 the response of work roll displacementand Hopf bifurcation diagram are shown in Figure 5 )eroll displacement will diverge rapidly and the equilibriumpoint of the equilibrium equation is unstable Besides asshown in Figure 5(b) the phase trajectory will disperseoutward of the origin in a ring shape and the radius cur-vature of the phase trajectory increases gradually It isconcluded that the roll vibration system is unstable and millchatter appears

As shown in Figure 6(a) when μlowast 00048 the work rolldisplacement exhibits as an equal amplitude vibration andthe roll vibrates at its balanced position )is steady regionrefers to the maximum and minimum fluctuation range ofwork roll Meanwhile Figure 6(b) shows that the phasediagram will begin to converge toward a stable limit cycle)e strip will produce a slight thickness fluctuation withinthe quality requirements and the mill system is stable underthese rolling conditions

When μlowast 00300 the roll systemrsquos time history responseand the phase diagram are shown in Figure 7 It can be seenfrom these figures that the phase trajectory will convergestoward a smaller and smaller circle near the origin point )eequilibrium point of work roll is asymptotically stable

)e roll system stability about the critical parameter ofμlowast 00401 is shown in Figure 8 When at the critical Hopfbifurcation parameter displacement of work roll firstlybecomes larger and finally converges to an equal amplitudeoscillation )e equilibrium point is unstable and the Hopfbifurcation diagram presents a ring shape However dy-namic range of the fixed limit cycle in Figure 8 is a littlelarger compared with Figure 6 )is is mainly because thefriction is bigger and as a result the fluctuation of theresponse rolling force ΔPi under the friction coefficient of00401 is much bigger

)e numerical solution of work roll displacement underthe Hopf bifurcation point of μlowast 00420 is shown in Fig-ure 9)e roll displacement will disperse outward all the timeBesides the phase diagram of the work roll will disperseoutward of the cycle and the radius curvature of the phasetrajectory increases gradually It illustrates that the mill vi-bration system is unstable and mill chatter is produced

When μlowast lt 00048 the roll system is unstable and whenμlowast 00048 the system is stable So the Hopf bifurcation ofμlowast 00048 is the subcritical bifurcation point and μlowast

00401 is the supercritical bifurcation pointMany researchers attribute the change of stability in

rolling mill system to the sudden change of the rolling forcefluctuation in dynamic equilibrium equation [11 25 35])e optimal setting range of friction coefficient is 00048ndash00401 and mill chatter appears when the friction is eithertoo high or too low )is is mainly due to the fluctuation ofrolling force When the friction is high with the increase offriction coefficient the neutral angle becomes bigger andthen the neutral point shifts to the entry side so the distancebetween the entry and neutral point becomes shorter andthe variation of the entry speed decreases [11] However theexit forward slip and exit speed fluctuation will increase

Table 2 Structure specifications of the UCM mill

Parameters ValueWork roll mass Mw (Kg) 3914Intermediate roll mass Mim (Kg) 5410Backup roll mass Mb (Kg) 31330Work roll contact spring constant Kw (N mminus1) 091times 1010

Intermediate roll spring constant Kim (N mminus1) 103times1010

Backup roll spring constant Kb (N mminus1) 585times1010

Work roll damping coefficient x directionCwx (Nmiddots mminus1) 500times104

Work roll damping coefficient in y directionCwy (Nmiddots mminus1) 164times105

Intermediate roll damping coefficient Cim (Nmiddots mminus1) 570times105

Backup roll damping coefficient Cb (Nmiddots mminus1) 510times106

Work roll rotational inertia mass Jw (Kgmiddotm2) 1381Work roll rotational spring constant Kwj (Nmiddotmmiddotradminus1) 790times106

Work roll rotational damping coefficientCwj (Nmiddotmmiddots radminus1) 4178

Work roll diameter Rw (mm) 2125Intermediate roll diameter Rim (mm) 245Backup roll diameter Rb (mm) 650Work roll installation offset e (mm) 5

8 Shock and Vibration

ndash80 times 10ndash5

ndash40 times 10ndash5

00

40 times 10ndash5

80 times 10ndash5W

ork

roll

disp

lace

men

t (m

)

00 02 04 06 08 10Time (s)

(a)

ndash002

ndash001

000

001

002 First point

x 4 (m

s)

00 80 times 10ndash540 times 10ndash5ndash40 times 10ndash5ndash80 times 10ndash5

x3 (m)

Last point

(b)

Figure 5 Dynamic response of the work roll displacement and the phase diagram under μlowast 00040 (a) time history of work rolldisplacement (b) phase diagram

00

Wor

k ro

ll di

spla

cem

ent (

m)

00 02 04 06 08 10Time (s)

10 times 10ndash7

50 times 10ndash8

ndash50 times 10ndash8

ndash10 times 10ndash7

(a)

00

12 times 10ndash7

60 times 10ndash8

ndash60 times 10ndash8

ndash12 times 10ndash7

ndash60 times 10ndash9 60 times 10ndash9ndash30 times 10ndash9 30 times 10ndash9

x 4 (m

s)

00x3 (m)

Last

poin

t

(b)

Figure 6 Dynamic response of the work roll displacement and the phase diagram under μlowast 00048 (a) time history of work rolldisplacement (b) phase diagram

ndash10 times 10ndash6

ndash50 times 10ndash7

00

50 times 10ndash7

10 times 10ndash6

Wor

k ro

ll di

spla

cem

ent (

m)

00 02 04 06 08 10Time (s)

(a)

ndash10 times 10ndash6 ndash50 times 10ndash7 00 50 times 10ndash7 10 times 10ndash6ndash80 times 10ndash4

ndash40 times 10ndash4

00

40 times 10ndash4

x 4 (m

s)

x3 (m)

Firs

t poi

nt

(b)

Figure 7 Dynamic response of the work roll displacement and the phase diagram under μlowast 00300 (a) time history of work rolldisplacement (b) phase diagram

Shock and Vibration 9

Based on equation (12) the tension variation at the entryposition (dTb) and exit position (dTf) is smaller comparedwith low friction coefficient rolling condition so the fluc-tuation of the response rolling force ΔPi increases and thestability of the stand decreases Meanwhile when the frictionis quite small zPzhout will increase with the decrease offriction coefficient F43 in equation (29) decreases so theinstantaneous dynamic structure spring constant decreasesand the fluctuation of the response rolling force ΔPi causedby the strip exit thickness variation will increase so thestability of the stand is decreased

43 Stable Range Analysis of Other Rolling Parameters

431 Calculation of Limited Rolling Speed Based on thiscoupled vertical-horizontal-torsional vibration analysismodel rolling speed has been chosen as the Hopf bifurcationparameter to obtain the limited stability rolling speed of the

UCM tandem cold rolling mill and other rolling conditionsare shown in Table 1 )e actual bifurcation point of rollingspeed is 251ms Figure 10 shows the time history of workroll displacement under the rolling speed of 252msAccording to Figure 10 the displacement amplitude ofthe work roll shows a divergence trend and the self-excitedvibration happens )e critical rolling speed calculated bythe proposed chatter model is 251ms

Figure 11 shows the tension and roll gap variation in No4 and No 5 stands obtained from the Production DataAcquisition (PDA) system As is shown in Figure 11(b)when the rolling speed is 221ms the back and front tensionof the No 5 stand fluctuate within a small range and the millstand is stable However when the rolling speed is raised to230ms the back and front tension of the No 5 standfluctuate more violently and the front tension of the No 5stand begins to diverge as shown in Figure 11(a) It is foundthat the roll gap of No 4 and No 5 stands fluctuates moreviolently by comparing Figures 11(c) and 11(d) Meanwhile

00 02 04 06 08 10

00

20 times 10ndash7

ndash20 times 10ndash7Wor

k ro

ll di

spla

cem

ent (

m)

Time (s)

(a)

00

00

10 times 10ndash4

50 times 10ndash5

ndash50 times 10ndash5

ndash10 times 10ndash4

ndash20 times 10ndash7 20 times 10ndash7

Last point

x 4 (m

s)

x3 (m)

First point

(b)

Figure 8 Dynamic response of the work roll displacement and the phase diagram under μlowast 00401 (a) time history of work rolldisplacement (b) phase diagram

00 02 04 06 08 10

00

20 times 10ndash6

ndash20 times 10ndash6Wor

k ro

ll di

spla

cem

ent (

m)

Time (s)

(a)

ndash20 times 10ndash6 00 20 times 10ndash6

00

10 times 10ndash3

50 times 10ndash4

ndash50 times 10ndash4

ndash10 times 10ndash3

x 4 (m

s)

x3 (m)

Last point

(b)

Figure 9 Dynamic response of the work roll displacement and the phase diagram under μlowast 00420 (a) time history of work rolldisplacement (b) phase diagram

10 Shock and Vibration

a sudden and harsh vibration noise will be heard in the No 5stand and finally the strip breaks It shows mill chatterappears under the speed of 230ms and the actual criticalrolling speed is 230ms

)e calculated critical rolling speed is 251ms while theactual critical rolling speed is 230ms )ere is a certainrelative error between the calculated critical speed and actualcritical rolling speed )is is due to the small coefficientdifference in Jacobian matrix With the change of rollingspeed the actual value of the tension and friction is changedAnd the calculated value of tension factor nt and frictioncoefficient in zPzh and zMzh is a little different with theactual value So the calculated limited rolling speed is a littlebigger than the actual value However by comparing thecritical rolling speed calculated by the proposed model withthe actual critical rolling speed it can be seen that theproposed model is still accurate

)e vibration is largely influenced by the friction co-efficient and rolling speed Based on equation (13) Figure 12shows the friction coefficient response of No 5 stand versusspeed Despite the changes of speed the friction coefficient isstill in the stability range of 00048ndash00401 It is divided intotwo regions In region 1 both the friction coefficient androlling speed are in the stable range In region 2 the frictioncoefficient is still in the stable range while the speed exceedsthe limited speed rangeWhen the rolling speed is lower than251ms the mill system is stable without changing thelubrication conditions And when the rolling speed is higherthan 251ms the stand is unstable However the frictioncoefficient is in the middle value of stable range under thespeed is 251ms so mill chatter might be eliminated byreducing the basic friction coefficient (by improving thelubrication characteristics such as increasing the concen-tration of emulsion increasing spray flow and improvingthe property of the base oil)

432 Calculation of Maximum Reduction Assuming theentry thickness is constant the exit thickness is variable Andthe exit thickness has been chosen as the Hopf bifurcationparameter Other calculated rolling conditions are shown in

Table 1 )en the calculated bifurcation point of exitthickness is 0168mm)e optimal setting range of reductionfor No 5 stand is 0ndash44When the reduction of No 5 stand is44 the increase of reduction leads to a lower mill standstability )e reason is also related to the increase of rollingforce variation under the larger reduction rolling conditions

As can be seen in Table 1 in normal rolling schedule thereduction rate is 333 And the max reduction is up to 44In reality considering there are other factors that influencethe stand stability such as rolling speed and lubricationconditions the max reduction is lower than 44 In generalthe effect of the reduction in the mill stability is lower thanthe improvement of friction coefficient

433 Front Tension Stable Range When the work roll ve-locity is 221ms the calculated bifurcation point of exittension is 15ndash133MPa Small front tension means that therolling force will be quite large So under the same excitationinterferences the variation of rolling force will be muchbigger and the stand stability is decreased When the fronttension is big enough tension factor nt decreases leading toa larger value in zPzh and zMzh As a result the fluc-tuation of rolling force and roll torque increases and thestability of the system is reduced as well So the influence oftension on the mill stability is smaller than other parameters

)ese results can be used to estimate mill chatter in acertain stand And the stable range of friction reduction andtension can be used to adjust the rolling schedule in theUCM mill to avoid mill chatter under high rolling speed

5 Conclusion

(1) A new multiple degrees of freedom and nonlineardynamic vertical-horizontal-torsional coupled millchatter model for UCM cold rolling mill is estab-lished In the dynamic rolling process model thetime-varying coupling effect between rolling forceroll torque variation and fluctuation of stripthickness and tension is formulated In the vibrationmodel the movement of work roll in both vertical-horizontal-torsional directions and the multirollcharacteristics in the UCM rolling mill are consid-ered Finally the dynamic relationship of rollmovement responses and dynamic changes in theroll gap is formulated

(2) Selecting the friction coefficient as the Hopf bi-furcation parameter and by using RouthndashHurwitzdeterminant the Hopf bifurcation point of the millvibration system is calculated and the bifurcationcharacteristics are analyzed Research shows thateither too low or too high friction coefficient in-creases the probability of mill chatter When thefriction coefficient is extremely low the mill systemproduces a stable limit cycle With the frictioncontinuing to increase the phase trajectory willconverge toward the origin point However the limitcycle of phase diagram disappears and dispersesoutward under high friction rolling conditions

00 02 04 06 08 10ndash2 times 10ndash7

ndash1 times 10ndash7

0

1 times 10ndash7

2 times 10ndash7

Wor

k ro

ll di

spla

cem

ent (

m)

Time (s)

Figure 10 Work roll time history displacement under the rollingspeed of 252ms

Shock and Vibration 11

0 1 2 30

20

40

60

80

Fron

t ten

sion

(kN

)

Back tension

Back

tens

ion

(kN

)

Time (s)

0

20

40

60

Front tension

(a)

0 1 2 30

20

40

60

80

Fron

t ten

sion

(kN

)

Back tension

Back

tens

ion

(kN

)

Time (s)

0

20

40

60

Front tension

(b)

0 1 2 3

015

018

021

024

Roll

gap

of N

o 5

stan

d (m

m)

Roll gap of No 4 stand

Roll

gap

of N

o 4

stan

d (m

m)

Time (s)

268

272

276

280

Roll gap of No 5 stand

(c)

0 1 2 3

030

033

036

039

Roll

gap

of N

o 5

stan

d (m

m)

Roll gap of No 4 stand

Roll

gap

of N

o4

stand

(mm

)

Time (s)

272

276

280

284

Roll gap of No 5 stand

(d)

Figure 11 Tension and roll gap time history (a) back and front tension of No 5 stand under the speed of 230ms (b) back and fronttension of No 5 stand under the speed of 221ms (c) roll gap of No 4 and No 5 stands under the speed of 230ms (d) roll gap of No 4 andNo 5 stands under the speed of 221ms

0 10 20 30

0020

0025

0030

0035

0040

0045

(2)

(25 002101)

Fric

tion

coeffi

cien

t

Speed (ms)

x = 251

(1)

Figure 12 Friction coefficient response of No 5 stand versus speed

12 Shock and Vibration

(3) Based on the coupled mill chatter model the millsystemrsquos stable limited rolling speed stable range ofreduction and front tension are investigatedWithout changing the rolling speed the most ef-fective way to eliminate mill chatter is to change thelubrication conditions and adjust the reductionschedule And the effect of tension on the vibration isquite limited )ese methods can be used to improvethe chatter phenomena in rolling mill under highrolling speed )e proposed vibration model canalso be used to design an optimum rolling schedulein the UCM tandem cold rolling mill and to de-termine the appearance of mill chatter under certainrolling conditions

Data Availability

)e rolling conditions data used to support the findings ofthis study are included within the article

Conflicts of Interest

)e authors declare that they have no conflicts of interestregarding the publication of this manuscript

Acknowledgments

)is work was supported by the National Natural ScienceFoundation of China (51774084 and 51634002) the NationalKey RampD Program of China (2017YFB0304100) the Fun-damental Research Funds for the Central Universities(N170708020) and the Open Research Fund from the StateKey Laboratory of Rolling and Automation NortheasternUniversity (2017RALKFKT006)

References

[1] I S Yun W R D Wilson and K F Ehmann ldquoReview ofchatter studies in cold rollingrdquo International Journal ofMachine Tools and Manufacture vol 38 no 12pp 1499ndash1530 1998

[2] I-S Yun W R D Wilson and K F Ehmann ldquoChatter in thestrip rolling process Part 1 dynamic model of rollingrdquoJournal of Manufacturing Science and Engineering vol 120no 2 pp 330ndash336 1998

[3] J Tlusty G Chandra S Critchley and D Paton ldquoChatter incold rollingrdquo CIRP Annals vol 31 no 1 pp 195ndash199 1982

[4] N M Reza F M Reza M Salimi and H Shojaei ldquoExper-imental investigations and ALE finite element methodanalysis of chatter in cold strip rollingrdquo ISIJ Internationalvol 52 no 12 pp 2245ndash2253 2012

[5] M R Niroomand M R Forouzan M Salimi and H ShojaeildquoExperimental investigations and ALE finite element methodanalysis of chatter in cold strip rollingrdquo ISIJ Internationalvol 52 no 12 pp 2245ndash2253 2012

[6] M R Niroomand M R Forouzan and M Salimi ldquo)eo-retical and experimental analysis of chatter in tandem coldrolling mills based on wave propagation theoryrdquo ISIJ In-ternational vol 55 no 3 pp 637ndash646 2015

[7] S Kapil P Eberhard and S K Dwivedy ldquoNonlinear dynamicanalysis of a parametrically excited cold rolling millrdquo Journal

of Manufacturing Science and Engineering vol 136 no 4p 041012 2014

[8] Y Kimura Y Sodani N Nishiura N Ikeuchi and Y MiharaldquoAnalysis of chaffer in tandem cold rolling millsrdquo ISIJ In-ternational vol 43 no 1 pp 77ndash84 2003

[9] N Fujita Y Kimura K Kobayashi et al ldquoDynamic control oflubrication characteristics in high speed tandem cold rollingrdquoJournal of Materials Processing Technology vol 229pp 407ndash416 2016

[10] A Heidari and M R Forouzan ldquoOptimization of cold rollingprocess parameters in order to increasing rolling speedlimited by chatter vibrationsrdquo Journal of Advanced Researchvol 4 no 1 pp 27ndash34 2013

[11] A Heidari M R Forouzan and S Akbarzadeh ldquoEffect offriction on tandem cold rolling mills chatteringrdquo ISIJ In-ternational vol 54 no 10 pp 2349ndash2356 2014

[12] A Heidari M R Forouzan and S Akbarzadeh ldquoDevelop-ment of a rolling chatter model considering unsteady lubri-cationrdquo ISIJ International vol 54 no 1 pp 165ndash170 2014

[13] A Heidari M R Forouzan and M R Niroomand ldquoDe-velopment and evaluation of friction models for chattersimulation in cold strip rollingrdquo International Journal ofAdvanced Manufacturing Technology vol 96 no 4 pp 1ndash212018

[14] Y Kim C-W Kim S Lee and H Park ldquoDynamic modelingand numerical analysis of a cold rolling millrdquo InternationalJournal of Precision Engineering and Manufacturing vol 14no 3 pp 407ndash413 2013

[15] P V Krot ldquoNonlinear vibrations and backlashes diagnosticsin the rolling mills drive trainsrdquo in Proceedings of the 6thEUROMECH Nonlinear Dynamics Conference Saint Peters-burg Russia July 2008

[16] S Liu B Sun S Zhao J Li and W Zhang ldquoStronglynonlinear dynamics of torsional vibration for rolling millrsquoselectromechanical coupling systemrdquo Steel Research In-ternational vol 86 no 9 pp 984ndash992 2015

[17] S Liu J Wang J Liu and Y Li ldquoNonlinear parametricallyexcited vibration and active control of gear pair system withtime-varying characteristicrdquo Chinese Physics B vol 24 no 10pp 303ndash311 2015

[18] B D Hassard N D Kazarinoff and Y H Wan eory andApplications of Hopf Bifurcation Cambridge University PressCambridge UK 1981

[19] B H K Lee S J Price and Y SWong ldquoNonlinear aeroelasticanalysis of airfoils bifurcation and chaosrdquo Progress inAerospace Sciences vol 35 no 3 pp 205ndash334 1999

[20] K Nishimura T Ikeda and Y Harata ldquoLocalization phe-nomena in torsional rotating shaft systems with multiplecentrifugal pendulum vibration absorbersrdquo Nonlinear Dy-namics vol 83 no 3 pp 1705ndash1726 2015

[21] T Zhang and H Dai ldquoBifurcation analysis of high-speedrailway wheel-setrdquo Nonlinear Dynamics vol 83 no 3pp 1511ndash1528 2015

[22] Z Y Gao Y Zang and T Han ldquoHopf bifurcation andstability analysis on the mill drive systemrdquo Applied Mechanicsamp Materials vol 121-126 pp 1514ndash1520 2012

[23] L Zeng Y Zang and Z Gao ldquoHopf bifurcation control forrolling mill multiple-mode-coupling vibration under non-linear frictionrdquo Journal of Vibration and Acoustics vol 139no 6 article 061015 2017

[24] S Liu Z Wang J Wang and H Li ldquoSliding bifurcationresearch of a horizontal-torsional coupled main drive systemof rolling millrdquo Nonlinear Dynamics vol 83 pp 441ndash4552015

Shock and Vibration 13

[25] Y Peng M Zhang J-L Sun and Y Zhang ldquoExperimentaland numerical investigation on the roll system swing vibra-tion characteristics of a hot rolling millrdquo ISIJ Internationalvol 57 no 9 pp 1567ndash1576 2017

[26] J Pittner and M A Simaan ldquoOptimal control of continuoustandem cold metal rollingrdquo in Proceedings of 2008 AmericanControl Conference Seattle WA USA June 2008

[27] R B Sims and D F S Arthur ldquoSpeed dependent variables incold strip rollingrdquo Journal of Iron and Steel Institute vol 172no 3 pp 285ndash295 1952

[28] S Z Chen D H Zhang J Sun J S Wang and J SongldquoOnline calculation model of rolling force for cold rolling millbased on numerical integrationrdquo in Proceedings of the 24thChinese Control and Decision Conference (CCDC) TaiyuanChina May 2012

[29] D R Bland and H Ford ldquo)e calculation of roll force andtorque in cold strip rolling with tensionsrdquo Proceedings of theInstitution of Mechanical Engineers vol 159 no 1 pp 144ndash163 2006

[30] R Hill ldquoRelations between roll-force torque and the appliedtensions in strip-rollingrdquo Proceedings of the Institution ofMechanical Engineers vol 163 no 1 pp 135ndash140 2006

[31] R E Johnson and Q Qi ldquoChatter dynamics in sheet rollingrdquoInternational Journal of Mechanical Sciences vol 36 no 7pp 617ndash630 1994

[32] Z Gao L Bai and Q Li ldquoResearch on critical rolling speed ofself-excited vibration in the tandem rolling process of thinstriprdquo Journal of Mechanical Engineering vol 53 no 12pp 118ndash132 2017

[33] R C Dorf and R H Bishop Modern Control SystemsUniversity of California Davis CA USA 12th edition 2010

[34] W M Liu ldquoCriterion of hopf bifurcations without usingeigenvaluesrdquo Journal of Mathematical Analysis and Appli-cations vol 182 no 1 pp 250ndash256 1994

[35] L Zeng Y Zang and Z Gao ldquoMultiple-modal-couplingmodeling and stability analysis of cold rolling mill vibra-tionrdquo Shock and Vibration vol 2016 Article ID 234738626 pages 2016

14 Shock and Vibration

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zM

zhin wlprimeknt(2cminus α)1113888minus

1792

μ

Rprime

h3in

1113971

+32

times 179μhout

Rprime

h5in

1113971

minus 102hout

h2in

1113889minus12

wlprimekQpntμR1

hin minus hout( 1113857Rprime1113969

zM

zhout wlprimeknt(2cminus α) minus179

μhin

Rprimehin

1113971

+ 1021

hin

⎛⎝ ⎞⎠

+12

wlprimekQpntμR1

hin minus hout( 1113857Rprime1113969

zM

zTb

minus07wlprimeQpμR(2cminus α)

zM

zTf

minus03wlprimeQpμR(2cminus α)

zM

zxw

wlprimekQpntμR 21Rprimeminus α1113874 1113875

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(26)

)e calculated rolling conditions in this research areshown in Table 1)e grade of rolling strip is MRT5 and theexit strip width is 900mm

Figure 3 shows the comparison of calculated rolling forcein this rolling process model and actual rolling force in theexperiment )e rolling conditions are shown in Table 1 Asshown in Figure 3 the maximum relative error betweenactual rolling force and calculated rolling force is 723)ecomparative results of the simulation and experimental datashow that this chatter model has a high accuracy

3 Dynamic Nonlinear Vibration Model of theUCM Mill

In most vibration research studies the rolling mill standwas simplified as a mass-spring system To investigate the

displacements of each roll and interaction between rolls amodel with multiple degrees of freedom is necessaryHowever with the consideration of the symmetry of themill structure along the rolling pass line the multipledegrees of freedom chatter model can be simplifiedFigure 4 is the simplified nonlinear vertical-horizontal-torsional coupled vibration model considering the rollmovement in more than one direction

As shown in Figure 4 the top work roll (WR) wouldvibrate in both vertical and horizontal directions While theintermediate roll (IMR) and backup roll (BUR) were notallowed to vibrate in horizontal direction due to the as-sembly and restriction of the roll bearings [25]

A horizontal roll vibration displacement of WR isgenerated due to the installation offset of e between WR andIMR and the equivalent spring constant in horizontal di-rection can be calculated by

ndash010

ndash005

000

005

010

Vibr

atio

n ac

cele

ratio

n (g

)

00 02 04 06 08 10Time (s)

(a)

ndash010

ndash005

000

005

010

Vib

ratio

n ac

cele

ratio

n (g

)

00 02 04 06 08 10Time (s)

(b)

Figure 2 Time domain vibration signal of the work roll (a) vibration acceleration in the vertical direction (b) vibration acceleration in thehorizontal direction

Table 1 Rolling conditions

Parameters 1 stand 2 stand 3 stand 4 stand 5 standEntrythickness hin(mm)

2000 1269 0762 0460 0300

Exit thicknesshout (mm) 1269 0762 0460 0300 0200

Rolling forceP (kN) 7838863 8455166 7781567 7500996 7722711

Exit tension Tf(MPa) 140554 165252 188160 205575 68000

Entry tensionTb (MPa) 55000 140554 165252 188160 205575

Frictioncoefficient μ 0065 0060 0040 0030 0020

Roll velocityVr (m sminus1) 3381 5652 9425 14441 22121

Strip entryspeed vin(m sminus1)

2250 3546 5903 9774 14977

Shock and Vibration 5

Kwx e

Rw + RimKw (27)

where Rw and Rim are the roll radius of work roll and in-termediate roll and Kw is the contact spring constant be-tween upper work roll and intermediate roll

With the consideration of the coupling movement of therolls in both vertical horizontal and torsional directions thedynamic equilibrium equation of the upper roll system wasgiven by

Mw euroxw + Cwx _xw + Kwxxw ΔPx

Mw euroyw + Cwy _yw minus _yim( 1113857 + Kwy yw minusyim( 1113857 ΔPy

Mim euroyim minusCwy _yw minus _yim( 1113857 + Cim _yim minus _yb( 1113857

minusKwy yw minusyim( 1113857 + Kim yim minusyb( 1113857 0

Mb euroyb minusCim _yim minus _yb( 1113857 + Cb _yb minusKim yim minusyb( 1113857

+ Kbyb 0

Jweuroθw + Cwj

_θw + Kwjθw ΔM

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(28)

where xw is the WR vibration displacement in horizontaldirection yw yim and yb are the vibration displacement ofWR IMR and BUR in vertical direction θw is the WRtorsional vibration angular displacement ΔPx and ΔPy arethe fluctuation of rolling force in horizontal and verticaldirections ΔM is the variation of rolling torque and MwMim and Mb are the mass of work roll intermediate rolland backup roll respectively

)e mill stand spring constant is the total stiffness of thewhole rolling mill consisting of all the rolls )e mill springconstant (K) is measured by a stiffness calibration test )espring constants of work roll (Kw) intermediate roll (Kim)and backup roll (Kb) are calculated by an FEM model of theUCM mill [5 6 12] )e roll vibration displacement underdifferent rolling forces was obtained by a simulation ex-periment )e spring constants of the work roll in-termediate roll and backup roll were obtained by analyzingthe dynamic roll vibration displacement under differentrolling forces

Damping coefficients of the intermediate and backuprolls are calibrated by a simulation program [31 32] First

Rolling direction

BUR

IMR

WR

Kb CbCb

Kim Cim

Kb

Mb

Mim

MwCwyKwy

KimCim

CwyKwy

ΔP Kwx

Kwx Cwx

CwxJw

Kwj Cwj

(a)

Cim

Mim

Mb

Mw

Kim

Kb Cb

Kw Cw

yb

yim

Kwx e

TfTb

xw

yw

(b)

Figure 4 Geometry of the vertical-horizontal-torsional vibration model (a) simplified vertical-horizontal-torsional coupled vibrationmodel (b) geometry of coupled vibration model in x-y direction

0

2000

4000

6000

8000

10000

Rolli

ng fo

rce (

kN)

1 2 3 4 5Stand number

Actual rolling forceCalculated rolling force

Figure 3 Comparison between the actual rolling force and calculated rolling force

6 Shock and Vibration

the dynamic differential equation of the system is regularizedto obtain the equivalent damping matrix )en the dampingregular matrix is transformed into the original coordinatesystem and let the damping ratio be 003 Finally theequivalent damping value of the roll system is calculatedWork roll rotational inertia mass work roll rotational springconstant and work roll rotational damping coefficient areobtained from the work of Zeng et al [23] as the design andmill structure of rolling stands are similar to one another)e roll diameters and work roll installation offset are ob-tained from the UCM rolling mill installation drawingsTable 2 shows the structure specifications of the UCMmill inequation (28)

Equation (28) is the nonlinear dynamic equation of thevertical-horizontal-torsional coupled vibration modelConvert the above differential equation into the followingstate matrix form

_X F(X μ) + Hdhin (29)

whereX xw _xw yw _yw yim _yim yb _yb θw _θw dTf dTb1113960 1113961

T

x1 x2 x3 x121113858 1113859T

H 0minuswTb

2Mw

0zP

zhinMw

0 0 0 0 0zM

zhinJw

0 01113890 1113891

T

F

0 1 0 0 0 0 0 0 0 0 0 0F21 F22 F23 0 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 00 0 F43 F44 F45 F46 0 0 0 0 0 00 0 0 0 0 1 0 0 0 0 0 00 0 F63 F64 F65 F66 F67 F68 0 0 0 00 0 0 0 0 0 0 1 0 0 0 00 0 0 0 F85 F86 F87 F88 0 0 0 00 0 0 0 0 0 0 0 0 1 0 00 F102 F103 0 0 0 0 0 F109 F1010 F1011 F10120 0 F113 0 0 0 0 0 0 0 F1111 F11120 0 F123 0 0 0 0 0 0 0 F1211 F1212

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

F21 minusKwx

Mw

F22 minusCwx

Mw

F23 wTf

Mw

F43 2 zPzhout( 1113857minusKwy

Mw

F44 minusCwy

Mw

F45 Kwy

Mw

F46 Cwy

Mw

F63 Kwy

Mim

F64 Cwy

Mim

F65 minusKwy + Kim

Mim

F66 minusCwy + Cim

Mim

F67 Kim

Mim

F68 Cim

Mim

F85 Kim

Mb

F86 Cim

Mb

F87 minusKim + Kb

Mb

F88 minusCim + Cb

Mb

F102 zM

Jwzxw

F103 2zM

Jwzhout

F109 minusKwj

Jw

F1010 minusCwj

Jw

F1011 zM

JwzTf

F1012 zM

JwzTb

F113 minus2E2Vr zhnzhout( 1113857hout minus hn( 1113857

55025h2out

F1111 minusE2Vr zhnzTf1113872 1113873

55025hout

F1112 minusE2Vr zhnzTb( 1113857

55025hout

F123 2E2Vr zhnzhout( 1113857

55025hin

F1211 E2Vr zhnzTf1113872 1113873

55025hin

F1212 E2Vr zhnzTf1113872 1113873

55025hin

(30)

)en equation (29) is the nonlinear dynamic stateequation of the vertical-horizontal-torsional coupled vi-bration system considering the nonlinear characteristicduring the rolling process the movement of rolls in bothvertical-horizontal directions and the torsional vibration

4 Analysis of Vibration Stability of theRoll System

41 Hopf Bifurcation Parameter Calculation )e frictioncharacteristic would cause the instability of the dynamicstate equation so in this research μ has been chosen as the

Shock and Vibration 7

Hopf bifurcation parameter Assuming the entry thicknessof each stand is constant the equilibrium point is Xwhere F(X) 0 )e system equilibrium point can betransformed to the origin by coordinate transformation X0

[0 0 0 0 0 0 0 0 0 0 0 0]T Let JF(X) be the Jacobianmatrix of F(X) So the Hopf bifurcation can be obtained byjudging all the eigenvalues of JF(X) However it is difficult tocalculate the eigenvalues of JF(X) since the calculationamount is quite large )en the RouthndashHurwitz determinantis used to calculate the Hopf bifurcation point [33]

)e characteristic equation of Jacobian matrix JF(X) ofthe dynamic vertical-horizontal-torsional coupling vibrationmodel is assumed to be

a12λ12

+ a11λ11

+ middot middot middot + a2λ2

+ a1λ + a0 0 (31)

)e main determinant

D1 a1 D2 a1 a0

a3 a2

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868 middot middot middot Di

a1 a0 middot middot middot 0

a3 a2 middot middot middot 0

middot middot middot middot middot middot middot middot middot middot middot middot

a2iminus1 a2iminus2 middot middot middot ai

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(32)

where ai 0 if ilt 0 or igt n and a12 1Assuming μlowast is the Hopf bifurcation the Hopf bi-

furcation conditions are [23 24 34]

a0 μlowast( 1113857gt 0 D1 μlowast( 1113857gt 0 middot middot middot Dnminus2 μlowast( 1113857gt 0 Dnminus1 μlowast( 1113857 0

dΔiminus1 μlowast( 1113857

dμne 0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(33)

According to these assumptions the Hopf bifurcationpoint of μlowast occurs at μlowast 00048 and μlowast 00401

42Analysis of StableRangeofFriction When the value of μlowastis smaller than the first bifurcation point of μlowast 00048such as μlowast 00040 the response of work roll displacementand Hopf bifurcation diagram are shown in Figure 5 )eroll displacement will diverge rapidly and the equilibriumpoint of the equilibrium equation is unstable Besides asshown in Figure 5(b) the phase trajectory will disperseoutward of the origin in a ring shape and the radius cur-vature of the phase trajectory increases gradually It isconcluded that the roll vibration system is unstable and millchatter appears

As shown in Figure 6(a) when μlowast 00048 the work rolldisplacement exhibits as an equal amplitude vibration andthe roll vibrates at its balanced position )is steady regionrefers to the maximum and minimum fluctuation range ofwork roll Meanwhile Figure 6(b) shows that the phasediagram will begin to converge toward a stable limit cycle)e strip will produce a slight thickness fluctuation withinthe quality requirements and the mill system is stable underthese rolling conditions

When μlowast 00300 the roll systemrsquos time history responseand the phase diagram are shown in Figure 7 It can be seenfrom these figures that the phase trajectory will convergestoward a smaller and smaller circle near the origin point )eequilibrium point of work roll is asymptotically stable

)e roll system stability about the critical parameter ofμlowast 00401 is shown in Figure 8 When at the critical Hopfbifurcation parameter displacement of work roll firstlybecomes larger and finally converges to an equal amplitudeoscillation )e equilibrium point is unstable and the Hopfbifurcation diagram presents a ring shape However dy-namic range of the fixed limit cycle in Figure 8 is a littlelarger compared with Figure 6 )is is mainly because thefriction is bigger and as a result the fluctuation of theresponse rolling force ΔPi under the friction coefficient of00401 is much bigger

)e numerical solution of work roll displacement underthe Hopf bifurcation point of μlowast 00420 is shown in Fig-ure 9)e roll displacement will disperse outward all the timeBesides the phase diagram of the work roll will disperseoutward of the cycle and the radius curvature of the phasetrajectory increases gradually It illustrates that the mill vi-bration system is unstable and mill chatter is produced

When μlowast lt 00048 the roll system is unstable and whenμlowast 00048 the system is stable So the Hopf bifurcation ofμlowast 00048 is the subcritical bifurcation point and μlowast

00401 is the supercritical bifurcation pointMany researchers attribute the change of stability in

rolling mill system to the sudden change of the rolling forcefluctuation in dynamic equilibrium equation [11 25 35])e optimal setting range of friction coefficient is 00048ndash00401 and mill chatter appears when the friction is eithertoo high or too low )is is mainly due to the fluctuation ofrolling force When the friction is high with the increase offriction coefficient the neutral angle becomes bigger andthen the neutral point shifts to the entry side so the distancebetween the entry and neutral point becomes shorter andthe variation of the entry speed decreases [11] However theexit forward slip and exit speed fluctuation will increase

Table 2 Structure specifications of the UCM mill

Parameters ValueWork roll mass Mw (Kg) 3914Intermediate roll mass Mim (Kg) 5410Backup roll mass Mb (Kg) 31330Work roll contact spring constant Kw (N mminus1) 091times 1010

Intermediate roll spring constant Kim (N mminus1) 103times1010

Backup roll spring constant Kb (N mminus1) 585times1010

Work roll damping coefficient x directionCwx (Nmiddots mminus1) 500times104

Work roll damping coefficient in y directionCwy (Nmiddots mminus1) 164times105

Intermediate roll damping coefficient Cim (Nmiddots mminus1) 570times105

Backup roll damping coefficient Cb (Nmiddots mminus1) 510times106

Work roll rotational inertia mass Jw (Kgmiddotm2) 1381Work roll rotational spring constant Kwj (Nmiddotmmiddotradminus1) 790times106

Work roll rotational damping coefficientCwj (Nmiddotmmiddots radminus1) 4178

Work roll diameter Rw (mm) 2125Intermediate roll diameter Rim (mm) 245Backup roll diameter Rb (mm) 650Work roll installation offset e (mm) 5

8 Shock and Vibration

ndash80 times 10ndash5

ndash40 times 10ndash5

00

40 times 10ndash5

80 times 10ndash5W

ork

roll

disp

lace

men

t (m

)

00 02 04 06 08 10Time (s)

(a)

ndash002

ndash001

000

001

002 First point

x 4 (m

s)

00 80 times 10ndash540 times 10ndash5ndash40 times 10ndash5ndash80 times 10ndash5

x3 (m)

Last point

(b)

Figure 5 Dynamic response of the work roll displacement and the phase diagram under μlowast 00040 (a) time history of work rolldisplacement (b) phase diagram

00

Wor

k ro

ll di

spla

cem

ent (

m)

00 02 04 06 08 10Time (s)

10 times 10ndash7

50 times 10ndash8

ndash50 times 10ndash8

ndash10 times 10ndash7

(a)

00

12 times 10ndash7

60 times 10ndash8

ndash60 times 10ndash8

ndash12 times 10ndash7

ndash60 times 10ndash9 60 times 10ndash9ndash30 times 10ndash9 30 times 10ndash9

x 4 (m

s)

00x3 (m)

Last

poin

t

(b)

Figure 6 Dynamic response of the work roll displacement and the phase diagram under μlowast 00048 (a) time history of work rolldisplacement (b) phase diagram

ndash10 times 10ndash6

ndash50 times 10ndash7

00

50 times 10ndash7

10 times 10ndash6

Wor

k ro

ll di

spla

cem

ent (

m)

00 02 04 06 08 10Time (s)

(a)

ndash10 times 10ndash6 ndash50 times 10ndash7 00 50 times 10ndash7 10 times 10ndash6ndash80 times 10ndash4

ndash40 times 10ndash4

00

40 times 10ndash4

x 4 (m

s)

x3 (m)

Firs

t poi

nt

(b)

Figure 7 Dynamic response of the work roll displacement and the phase diagram under μlowast 00300 (a) time history of work rolldisplacement (b) phase diagram

Shock and Vibration 9

Based on equation (12) the tension variation at the entryposition (dTb) and exit position (dTf) is smaller comparedwith low friction coefficient rolling condition so the fluc-tuation of the response rolling force ΔPi increases and thestability of the stand decreases Meanwhile when the frictionis quite small zPzhout will increase with the decrease offriction coefficient F43 in equation (29) decreases so theinstantaneous dynamic structure spring constant decreasesand the fluctuation of the response rolling force ΔPi causedby the strip exit thickness variation will increase so thestability of the stand is decreased

43 Stable Range Analysis of Other Rolling Parameters

431 Calculation of Limited Rolling Speed Based on thiscoupled vertical-horizontal-torsional vibration analysismodel rolling speed has been chosen as the Hopf bifurcationparameter to obtain the limited stability rolling speed of the

UCM tandem cold rolling mill and other rolling conditionsare shown in Table 1 )e actual bifurcation point of rollingspeed is 251ms Figure 10 shows the time history of workroll displacement under the rolling speed of 252msAccording to Figure 10 the displacement amplitude ofthe work roll shows a divergence trend and the self-excitedvibration happens )e critical rolling speed calculated bythe proposed chatter model is 251ms

Figure 11 shows the tension and roll gap variation in No4 and No 5 stands obtained from the Production DataAcquisition (PDA) system As is shown in Figure 11(b)when the rolling speed is 221ms the back and front tensionof the No 5 stand fluctuate within a small range and the millstand is stable However when the rolling speed is raised to230ms the back and front tension of the No 5 standfluctuate more violently and the front tension of the No 5stand begins to diverge as shown in Figure 11(a) It is foundthat the roll gap of No 4 and No 5 stands fluctuates moreviolently by comparing Figures 11(c) and 11(d) Meanwhile

00 02 04 06 08 10

00

20 times 10ndash7

ndash20 times 10ndash7Wor

k ro

ll di

spla

cem

ent (

m)

Time (s)

(a)

00

00

10 times 10ndash4

50 times 10ndash5

ndash50 times 10ndash5

ndash10 times 10ndash4

ndash20 times 10ndash7 20 times 10ndash7

Last point

x 4 (m

s)

x3 (m)

First point

(b)

Figure 8 Dynamic response of the work roll displacement and the phase diagram under μlowast 00401 (a) time history of work rolldisplacement (b) phase diagram

00 02 04 06 08 10

00

20 times 10ndash6

ndash20 times 10ndash6Wor

k ro

ll di

spla

cem

ent (

m)

Time (s)

(a)

ndash20 times 10ndash6 00 20 times 10ndash6

00

10 times 10ndash3

50 times 10ndash4

ndash50 times 10ndash4

ndash10 times 10ndash3

x 4 (m

s)

x3 (m)

Last point

(b)

Figure 9 Dynamic response of the work roll displacement and the phase diagram under μlowast 00420 (a) time history of work rolldisplacement (b) phase diagram

10 Shock and Vibration

a sudden and harsh vibration noise will be heard in the No 5stand and finally the strip breaks It shows mill chatterappears under the speed of 230ms and the actual criticalrolling speed is 230ms

)e calculated critical rolling speed is 251ms while theactual critical rolling speed is 230ms )ere is a certainrelative error between the calculated critical speed and actualcritical rolling speed )is is due to the small coefficientdifference in Jacobian matrix With the change of rollingspeed the actual value of the tension and friction is changedAnd the calculated value of tension factor nt and frictioncoefficient in zPzh and zMzh is a little different with theactual value So the calculated limited rolling speed is a littlebigger than the actual value However by comparing thecritical rolling speed calculated by the proposed model withthe actual critical rolling speed it can be seen that theproposed model is still accurate

)e vibration is largely influenced by the friction co-efficient and rolling speed Based on equation (13) Figure 12shows the friction coefficient response of No 5 stand versusspeed Despite the changes of speed the friction coefficient isstill in the stability range of 00048ndash00401 It is divided intotwo regions In region 1 both the friction coefficient androlling speed are in the stable range In region 2 the frictioncoefficient is still in the stable range while the speed exceedsthe limited speed rangeWhen the rolling speed is lower than251ms the mill system is stable without changing thelubrication conditions And when the rolling speed is higherthan 251ms the stand is unstable However the frictioncoefficient is in the middle value of stable range under thespeed is 251ms so mill chatter might be eliminated byreducing the basic friction coefficient (by improving thelubrication characteristics such as increasing the concen-tration of emulsion increasing spray flow and improvingthe property of the base oil)

432 Calculation of Maximum Reduction Assuming theentry thickness is constant the exit thickness is variable Andthe exit thickness has been chosen as the Hopf bifurcationparameter Other calculated rolling conditions are shown in

Table 1 )en the calculated bifurcation point of exitthickness is 0168mm)e optimal setting range of reductionfor No 5 stand is 0ndash44When the reduction of No 5 stand is44 the increase of reduction leads to a lower mill standstability )e reason is also related to the increase of rollingforce variation under the larger reduction rolling conditions

As can be seen in Table 1 in normal rolling schedule thereduction rate is 333 And the max reduction is up to 44In reality considering there are other factors that influencethe stand stability such as rolling speed and lubricationconditions the max reduction is lower than 44 In generalthe effect of the reduction in the mill stability is lower thanthe improvement of friction coefficient

433 Front Tension Stable Range When the work roll ve-locity is 221ms the calculated bifurcation point of exittension is 15ndash133MPa Small front tension means that therolling force will be quite large So under the same excitationinterferences the variation of rolling force will be muchbigger and the stand stability is decreased When the fronttension is big enough tension factor nt decreases leading toa larger value in zPzh and zMzh As a result the fluc-tuation of rolling force and roll torque increases and thestability of the system is reduced as well So the influence oftension on the mill stability is smaller than other parameters

)ese results can be used to estimate mill chatter in acertain stand And the stable range of friction reduction andtension can be used to adjust the rolling schedule in theUCM mill to avoid mill chatter under high rolling speed

5 Conclusion

(1) A new multiple degrees of freedom and nonlineardynamic vertical-horizontal-torsional coupled millchatter model for UCM cold rolling mill is estab-lished In the dynamic rolling process model thetime-varying coupling effect between rolling forceroll torque variation and fluctuation of stripthickness and tension is formulated In the vibrationmodel the movement of work roll in both vertical-horizontal-torsional directions and the multirollcharacteristics in the UCM rolling mill are consid-ered Finally the dynamic relationship of rollmovement responses and dynamic changes in theroll gap is formulated

(2) Selecting the friction coefficient as the Hopf bi-furcation parameter and by using RouthndashHurwitzdeterminant the Hopf bifurcation point of the millvibration system is calculated and the bifurcationcharacteristics are analyzed Research shows thateither too low or too high friction coefficient in-creases the probability of mill chatter When thefriction coefficient is extremely low the mill systemproduces a stable limit cycle With the frictioncontinuing to increase the phase trajectory willconverge toward the origin point However the limitcycle of phase diagram disappears and dispersesoutward under high friction rolling conditions

00 02 04 06 08 10ndash2 times 10ndash7

ndash1 times 10ndash7

0

1 times 10ndash7

2 times 10ndash7

Wor

k ro

ll di

spla

cem

ent (

m)

Time (s)

Figure 10 Work roll time history displacement under the rollingspeed of 252ms

Shock and Vibration 11

0 1 2 30

20

40

60

80

Fron

t ten

sion

(kN

)

Back tension

Back

tens

ion

(kN

)

Time (s)

0

20

40

60

Front tension

(a)

0 1 2 30

20

40

60

80

Fron

t ten

sion

(kN

)

Back tension

Back

tens

ion

(kN

)

Time (s)

0

20

40

60

Front tension

(b)

0 1 2 3

015

018

021

024

Roll

gap

of N

o 5

stan

d (m

m)

Roll gap of No 4 stand

Roll

gap

of N

o 4

stan

d (m

m)

Time (s)

268

272

276

280

Roll gap of No 5 stand

(c)

0 1 2 3

030

033

036

039

Roll

gap

of N

o 5

stan

d (m

m)

Roll gap of No 4 stand

Roll

gap

of N

o4

stand

(mm

)

Time (s)

272

276

280

284

Roll gap of No 5 stand

(d)

Figure 11 Tension and roll gap time history (a) back and front tension of No 5 stand under the speed of 230ms (b) back and fronttension of No 5 stand under the speed of 221ms (c) roll gap of No 4 and No 5 stands under the speed of 230ms (d) roll gap of No 4 andNo 5 stands under the speed of 221ms

0 10 20 30

0020

0025

0030

0035

0040

0045

(2)

(25 002101)

Fric

tion

coeffi

cien

t

Speed (ms)

x = 251

(1)

Figure 12 Friction coefficient response of No 5 stand versus speed

12 Shock and Vibration

(3) Based on the coupled mill chatter model the millsystemrsquos stable limited rolling speed stable range ofreduction and front tension are investigatedWithout changing the rolling speed the most ef-fective way to eliminate mill chatter is to change thelubrication conditions and adjust the reductionschedule And the effect of tension on the vibration isquite limited )ese methods can be used to improvethe chatter phenomena in rolling mill under highrolling speed )e proposed vibration model canalso be used to design an optimum rolling schedulein the UCM tandem cold rolling mill and to de-termine the appearance of mill chatter under certainrolling conditions

Data Availability

)e rolling conditions data used to support the findings ofthis study are included within the article

Conflicts of Interest

)e authors declare that they have no conflicts of interestregarding the publication of this manuscript

Acknowledgments

)is work was supported by the National Natural ScienceFoundation of China (51774084 and 51634002) the NationalKey RampD Program of China (2017YFB0304100) the Fun-damental Research Funds for the Central Universities(N170708020) and the Open Research Fund from the StateKey Laboratory of Rolling and Automation NortheasternUniversity (2017RALKFKT006)

References

[1] I S Yun W R D Wilson and K F Ehmann ldquoReview ofchatter studies in cold rollingrdquo International Journal ofMachine Tools and Manufacture vol 38 no 12pp 1499ndash1530 1998

[2] I-S Yun W R D Wilson and K F Ehmann ldquoChatter in thestrip rolling process Part 1 dynamic model of rollingrdquoJournal of Manufacturing Science and Engineering vol 120no 2 pp 330ndash336 1998

[3] J Tlusty G Chandra S Critchley and D Paton ldquoChatter incold rollingrdquo CIRP Annals vol 31 no 1 pp 195ndash199 1982

[4] N M Reza F M Reza M Salimi and H Shojaei ldquoExper-imental investigations and ALE finite element methodanalysis of chatter in cold strip rollingrdquo ISIJ Internationalvol 52 no 12 pp 2245ndash2253 2012

[5] M R Niroomand M R Forouzan M Salimi and H ShojaeildquoExperimental investigations and ALE finite element methodanalysis of chatter in cold strip rollingrdquo ISIJ Internationalvol 52 no 12 pp 2245ndash2253 2012

[6] M R Niroomand M R Forouzan and M Salimi ldquo)eo-retical and experimental analysis of chatter in tandem coldrolling mills based on wave propagation theoryrdquo ISIJ In-ternational vol 55 no 3 pp 637ndash646 2015

[7] S Kapil P Eberhard and S K Dwivedy ldquoNonlinear dynamicanalysis of a parametrically excited cold rolling millrdquo Journal

of Manufacturing Science and Engineering vol 136 no 4p 041012 2014

[8] Y Kimura Y Sodani N Nishiura N Ikeuchi and Y MiharaldquoAnalysis of chaffer in tandem cold rolling millsrdquo ISIJ In-ternational vol 43 no 1 pp 77ndash84 2003

[9] N Fujita Y Kimura K Kobayashi et al ldquoDynamic control oflubrication characteristics in high speed tandem cold rollingrdquoJournal of Materials Processing Technology vol 229pp 407ndash416 2016

[10] A Heidari and M R Forouzan ldquoOptimization of cold rollingprocess parameters in order to increasing rolling speedlimited by chatter vibrationsrdquo Journal of Advanced Researchvol 4 no 1 pp 27ndash34 2013

[11] A Heidari M R Forouzan and S Akbarzadeh ldquoEffect offriction on tandem cold rolling mills chatteringrdquo ISIJ In-ternational vol 54 no 10 pp 2349ndash2356 2014

[12] A Heidari M R Forouzan and S Akbarzadeh ldquoDevelop-ment of a rolling chatter model considering unsteady lubri-cationrdquo ISIJ International vol 54 no 1 pp 165ndash170 2014

[13] A Heidari M R Forouzan and M R Niroomand ldquoDe-velopment and evaluation of friction models for chattersimulation in cold strip rollingrdquo International Journal ofAdvanced Manufacturing Technology vol 96 no 4 pp 1ndash212018

[14] Y Kim C-W Kim S Lee and H Park ldquoDynamic modelingand numerical analysis of a cold rolling millrdquo InternationalJournal of Precision Engineering and Manufacturing vol 14no 3 pp 407ndash413 2013

[15] P V Krot ldquoNonlinear vibrations and backlashes diagnosticsin the rolling mills drive trainsrdquo in Proceedings of the 6thEUROMECH Nonlinear Dynamics Conference Saint Peters-burg Russia July 2008

[16] S Liu B Sun S Zhao J Li and W Zhang ldquoStronglynonlinear dynamics of torsional vibration for rolling millrsquoselectromechanical coupling systemrdquo Steel Research In-ternational vol 86 no 9 pp 984ndash992 2015

[17] S Liu J Wang J Liu and Y Li ldquoNonlinear parametricallyexcited vibration and active control of gear pair system withtime-varying characteristicrdquo Chinese Physics B vol 24 no 10pp 303ndash311 2015

[18] B D Hassard N D Kazarinoff and Y H Wan eory andApplications of Hopf Bifurcation Cambridge University PressCambridge UK 1981

[19] B H K Lee S J Price and Y SWong ldquoNonlinear aeroelasticanalysis of airfoils bifurcation and chaosrdquo Progress inAerospace Sciences vol 35 no 3 pp 205ndash334 1999

[20] K Nishimura T Ikeda and Y Harata ldquoLocalization phe-nomena in torsional rotating shaft systems with multiplecentrifugal pendulum vibration absorbersrdquo Nonlinear Dy-namics vol 83 no 3 pp 1705ndash1726 2015

[21] T Zhang and H Dai ldquoBifurcation analysis of high-speedrailway wheel-setrdquo Nonlinear Dynamics vol 83 no 3pp 1511ndash1528 2015

[22] Z Y Gao Y Zang and T Han ldquoHopf bifurcation andstability analysis on the mill drive systemrdquo Applied Mechanicsamp Materials vol 121-126 pp 1514ndash1520 2012

[23] L Zeng Y Zang and Z Gao ldquoHopf bifurcation control forrolling mill multiple-mode-coupling vibration under non-linear frictionrdquo Journal of Vibration and Acoustics vol 139no 6 article 061015 2017

[24] S Liu Z Wang J Wang and H Li ldquoSliding bifurcationresearch of a horizontal-torsional coupled main drive systemof rolling millrdquo Nonlinear Dynamics vol 83 pp 441ndash4552015

Shock and Vibration 13

[25] Y Peng M Zhang J-L Sun and Y Zhang ldquoExperimentaland numerical investigation on the roll system swing vibra-tion characteristics of a hot rolling millrdquo ISIJ Internationalvol 57 no 9 pp 1567ndash1576 2017

[26] J Pittner and M A Simaan ldquoOptimal control of continuoustandem cold metal rollingrdquo in Proceedings of 2008 AmericanControl Conference Seattle WA USA June 2008

[27] R B Sims and D F S Arthur ldquoSpeed dependent variables incold strip rollingrdquo Journal of Iron and Steel Institute vol 172no 3 pp 285ndash295 1952

[28] S Z Chen D H Zhang J Sun J S Wang and J SongldquoOnline calculation model of rolling force for cold rolling millbased on numerical integrationrdquo in Proceedings of the 24thChinese Control and Decision Conference (CCDC) TaiyuanChina May 2012

[29] D R Bland and H Ford ldquo)e calculation of roll force andtorque in cold strip rolling with tensionsrdquo Proceedings of theInstitution of Mechanical Engineers vol 159 no 1 pp 144ndash163 2006

[30] R Hill ldquoRelations between roll-force torque and the appliedtensions in strip-rollingrdquo Proceedings of the Institution ofMechanical Engineers vol 163 no 1 pp 135ndash140 2006

[31] R E Johnson and Q Qi ldquoChatter dynamics in sheet rollingrdquoInternational Journal of Mechanical Sciences vol 36 no 7pp 617ndash630 1994

[32] Z Gao L Bai and Q Li ldquoResearch on critical rolling speed ofself-excited vibration in the tandem rolling process of thinstriprdquo Journal of Mechanical Engineering vol 53 no 12pp 118ndash132 2017

[33] R C Dorf and R H Bishop Modern Control SystemsUniversity of California Davis CA USA 12th edition 2010

[34] W M Liu ldquoCriterion of hopf bifurcations without usingeigenvaluesrdquo Journal of Mathematical Analysis and Appli-cations vol 182 no 1 pp 250ndash256 1994

[35] L Zeng Y Zang and Z Gao ldquoMultiple-modal-couplingmodeling and stability analysis of cold rolling mill vibra-tionrdquo Shock and Vibration vol 2016 Article ID 234738626 pages 2016

14 Shock and Vibration

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Kwx e

Rw + RimKw (27)

where Rw and Rim are the roll radius of work roll and in-termediate roll and Kw is the contact spring constant be-tween upper work roll and intermediate roll

With the consideration of the coupling movement of therolls in both vertical horizontal and torsional directions thedynamic equilibrium equation of the upper roll system wasgiven by

Mw euroxw + Cwx _xw + Kwxxw ΔPx

Mw euroyw + Cwy _yw minus _yim( 1113857 + Kwy yw minusyim( 1113857 ΔPy

Mim euroyim minusCwy _yw minus _yim( 1113857 + Cim _yim minus _yb( 1113857

minusKwy yw minusyim( 1113857 + Kim yim minusyb( 1113857 0

Mb euroyb minusCim _yim minus _yb( 1113857 + Cb _yb minusKim yim minusyb( 1113857

+ Kbyb 0

Jweuroθw + Cwj

_θw + Kwjθw ΔM

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(28)

where xw is the WR vibration displacement in horizontaldirection yw yim and yb are the vibration displacement ofWR IMR and BUR in vertical direction θw is the WRtorsional vibration angular displacement ΔPx and ΔPy arethe fluctuation of rolling force in horizontal and verticaldirections ΔM is the variation of rolling torque and MwMim and Mb are the mass of work roll intermediate rolland backup roll respectively

)e mill stand spring constant is the total stiffness of thewhole rolling mill consisting of all the rolls )e mill springconstant (K) is measured by a stiffness calibration test )espring constants of work roll (Kw) intermediate roll (Kim)and backup roll (Kb) are calculated by an FEM model of theUCM mill [5 6 12] )e roll vibration displacement underdifferent rolling forces was obtained by a simulation ex-periment )e spring constants of the work roll in-termediate roll and backup roll were obtained by analyzingthe dynamic roll vibration displacement under differentrolling forces

Damping coefficients of the intermediate and backuprolls are calibrated by a simulation program [31 32] First

Rolling direction

BUR

IMR

WR

Kb CbCb

Kim Cim

Kb

Mb

Mim

MwCwyKwy

KimCim

CwyKwy

ΔP Kwx

Kwx Cwx

CwxJw

Kwj Cwj

(a)

Cim

Mim

Mb

Mw

Kim

Kb Cb

Kw Cw

yb

yim

Kwx e

TfTb

xw

yw

(b)

Figure 4 Geometry of the vertical-horizontal-torsional vibration model (a) simplified vertical-horizontal-torsional coupled vibrationmodel (b) geometry of coupled vibration model in x-y direction

0

2000

4000

6000

8000

10000

Rolli

ng fo

rce (

kN)

1 2 3 4 5Stand number

Actual rolling forceCalculated rolling force

Figure 3 Comparison between the actual rolling force and calculated rolling force

6 Shock and Vibration

the dynamic differential equation of the system is regularizedto obtain the equivalent damping matrix )en the dampingregular matrix is transformed into the original coordinatesystem and let the damping ratio be 003 Finally theequivalent damping value of the roll system is calculatedWork roll rotational inertia mass work roll rotational springconstant and work roll rotational damping coefficient areobtained from the work of Zeng et al [23] as the design andmill structure of rolling stands are similar to one another)e roll diameters and work roll installation offset are ob-tained from the UCM rolling mill installation drawingsTable 2 shows the structure specifications of the UCMmill inequation (28)

Equation (28) is the nonlinear dynamic equation of thevertical-horizontal-torsional coupled vibration modelConvert the above differential equation into the followingstate matrix form

_X F(X μ) + Hdhin (29)

whereX xw _xw yw _yw yim _yim yb _yb θw _θw dTf dTb1113960 1113961

T

x1 x2 x3 x121113858 1113859T

H 0minuswTb

2Mw

0zP

zhinMw

0 0 0 0 0zM

zhinJw

0 01113890 1113891

T

F

0 1 0 0 0 0 0 0 0 0 0 0F21 F22 F23 0 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 00 0 F43 F44 F45 F46 0 0 0 0 0 00 0 0 0 0 1 0 0 0 0 0 00 0 F63 F64 F65 F66 F67 F68 0 0 0 00 0 0 0 0 0 0 1 0 0 0 00 0 0 0 F85 F86 F87 F88 0 0 0 00 0 0 0 0 0 0 0 0 1 0 00 F102 F103 0 0 0 0 0 F109 F1010 F1011 F10120 0 F113 0 0 0 0 0 0 0 F1111 F11120 0 F123 0 0 0 0 0 0 0 F1211 F1212

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

F21 minusKwx

Mw

F22 minusCwx

Mw

F23 wTf

Mw

F43 2 zPzhout( 1113857minusKwy

Mw

F44 minusCwy

Mw

F45 Kwy

Mw

F46 Cwy

Mw

F63 Kwy

Mim

F64 Cwy

Mim

F65 minusKwy + Kim

Mim

F66 minusCwy + Cim

Mim

F67 Kim

Mim

F68 Cim

Mim

F85 Kim

Mb

F86 Cim

Mb

F87 minusKim + Kb

Mb

F88 minusCim + Cb

Mb

F102 zM

Jwzxw

F103 2zM

Jwzhout

F109 minusKwj

Jw

F1010 minusCwj

Jw

F1011 zM

JwzTf

F1012 zM

JwzTb

F113 minus2E2Vr zhnzhout( 1113857hout minus hn( 1113857

55025h2out

F1111 minusE2Vr zhnzTf1113872 1113873

55025hout

F1112 minusE2Vr zhnzTb( 1113857

55025hout

F123 2E2Vr zhnzhout( 1113857

55025hin

F1211 E2Vr zhnzTf1113872 1113873

55025hin

F1212 E2Vr zhnzTf1113872 1113873

55025hin

(30)

)en equation (29) is the nonlinear dynamic stateequation of the vertical-horizontal-torsional coupled vi-bration system considering the nonlinear characteristicduring the rolling process the movement of rolls in bothvertical-horizontal directions and the torsional vibration

4 Analysis of Vibration Stability of theRoll System

41 Hopf Bifurcation Parameter Calculation )e frictioncharacteristic would cause the instability of the dynamicstate equation so in this research μ has been chosen as the

Shock and Vibration 7

Hopf bifurcation parameter Assuming the entry thicknessof each stand is constant the equilibrium point is Xwhere F(X) 0 )e system equilibrium point can betransformed to the origin by coordinate transformation X0

[0 0 0 0 0 0 0 0 0 0 0 0]T Let JF(X) be the Jacobianmatrix of F(X) So the Hopf bifurcation can be obtained byjudging all the eigenvalues of JF(X) However it is difficult tocalculate the eigenvalues of JF(X) since the calculationamount is quite large )en the RouthndashHurwitz determinantis used to calculate the Hopf bifurcation point [33]

)e characteristic equation of Jacobian matrix JF(X) ofthe dynamic vertical-horizontal-torsional coupling vibrationmodel is assumed to be

a12λ12

+ a11λ11

+ middot middot middot + a2λ2

+ a1λ + a0 0 (31)

)e main determinant

D1 a1 D2 a1 a0

a3 a2

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868 middot middot middot Di

a1 a0 middot middot middot 0

a3 a2 middot middot middot 0

middot middot middot middot middot middot middot middot middot middot middot middot

a2iminus1 a2iminus2 middot middot middot ai

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(32)

where ai 0 if ilt 0 or igt n and a12 1Assuming μlowast is the Hopf bifurcation the Hopf bi-

furcation conditions are [23 24 34]

a0 μlowast( 1113857gt 0 D1 μlowast( 1113857gt 0 middot middot middot Dnminus2 μlowast( 1113857gt 0 Dnminus1 μlowast( 1113857 0

dΔiminus1 μlowast( 1113857

dμne 0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(33)

According to these assumptions the Hopf bifurcationpoint of μlowast occurs at μlowast 00048 and μlowast 00401

42Analysis of StableRangeofFriction When the value of μlowastis smaller than the first bifurcation point of μlowast 00048such as μlowast 00040 the response of work roll displacementand Hopf bifurcation diagram are shown in Figure 5 )eroll displacement will diverge rapidly and the equilibriumpoint of the equilibrium equation is unstable Besides asshown in Figure 5(b) the phase trajectory will disperseoutward of the origin in a ring shape and the radius cur-vature of the phase trajectory increases gradually It isconcluded that the roll vibration system is unstable and millchatter appears

As shown in Figure 6(a) when μlowast 00048 the work rolldisplacement exhibits as an equal amplitude vibration andthe roll vibrates at its balanced position )is steady regionrefers to the maximum and minimum fluctuation range ofwork roll Meanwhile Figure 6(b) shows that the phasediagram will begin to converge toward a stable limit cycle)e strip will produce a slight thickness fluctuation withinthe quality requirements and the mill system is stable underthese rolling conditions

When μlowast 00300 the roll systemrsquos time history responseand the phase diagram are shown in Figure 7 It can be seenfrom these figures that the phase trajectory will convergestoward a smaller and smaller circle near the origin point )eequilibrium point of work roll is asymptotically stable

)e roll system stability about the critical parameter ofμlowast 00401 is shown in Figure 8 When at the critical Hopfbifurcation parameter displacement of work roll firstlybecomes larger and finally converges to an equal amplitudeoscillation )e equilibrium point is unstable and the Hopfbifurcation diagram presents a ring shape However dy-namic range of the fixed limit cycle in Figure 8 is a littlelarger compared with Figure 6 )is is mainly because thefriction is bigger and as a result the fluctuation of theresponse rolling force ΔPi under the friction coefficient of00401 is much bigger

)e numerical solution of work roll displacement underthe Hopf bifurcation point of μlowast 00420 is shown in Fig-ure 9)e roll displacement will disperse outward all the timeBesides the phase diagram of the work roll will disperseoutward of the cycle and the radius curvature of the phasetrajectory increases gradually It illustrates that the mill vi-bration system is unstable and mill chatter is produced

When μlowast lt 00048 the roll system is unstable and whenμlowast 00048 the system is stable So the Hopf bifurcation ofμlowast 00048 is the subcritical bifurcation point and μlowast

00401 is the supercritical bifurcation pointMany researchers attribute the change of stability in

rolling mill system to the sudden change of the rolling forcefluctuation in dynamic equilibrium equation [11 25 35])e optimal setting range of friction coefficient is 00048ndash00401 and mill chatter appears when the friction is eithertoo high or too low )is is mainly due to the fluctuation ofrolling force When the friction is high with the increase offriction coefficient the neutral angle becomes bigger andthen the neutral point shifts to the entry side so the distancebetween the entry and neutral point becomes shorter andthe variation of the entry speed decreases [11] However theexit forward slip and exit speed fluctuation will increase

Table 2 Structure specifications of the UCM mill

Parameters ValueWork roll mass Mw (Kg) 3914Intermediate roll mass Mim (Kg) 5410Backup roll mass Mb (Kg) 31330Work roll contact spring constant Kw (N mminus1) 091times 1010

Intermediate roll spring constant Kim (N mminus1) 103times1010

Backup roll spring constant Kb (N mminus1) 585times1010

Work roll damping coefficient x directionCwx (Nmiddots mminus1) 500times104

Work roll damping coefficient in y directionCwy (Nmiddots mminus1) 164times105

Intermediate roll damping coefficient Cim (Nmiddots mminus1) 570times105

Backup roll damping coefficient Cb (Nmiddots mminus1) 510times106

Work roll rotational inertia mass Jw (Kgmiddotm2) 1381Work roll rotational spring constant Kwj (Nmiddotmmiddotradminus1) 790times106

Work roll rotational damping coefficientCwj (Nmiddotmmiddots radminus1) 4178

Work roll diameter Rw (mm) 2125Intermediate roll diameter Rim (mm) 245Backup roll diameter Rb (mm) 650Work roll installation offset e (mm) 5

8 Shock and Vibration

ndash80 times 10ndash5

ndash40 times 10ndash5

00

40 times 10ndash5

80 times 10ndash5W

ork

roll

disp

lace

men

t (m

)

00 02 04 06 08 10Time (s)

(a)

ndash002

ndash001

000

001

002 First point

x 4 (m

s)

00 80 times 10ndash540 times 10ndash5ndash40 times 10ndash5ndash80 times 10ndash5

x3 (m)

Last point

(b)

Figure 5 Dynamic response of the work roll displacement and the phase diagram under μlowast 00040 (a) time history of work rolldisplacement (b) phase diagram

00

Wor

k ro

ll di

spla

cem

ent (

m)

00 02 04 06 08 10Time (s)

10 times 10ndash7

50 times 10ndash8

ndash50 times 10ndash8

ndash10 times 10ndash7

(a)

00

12 times 10ndash7

60 times 10ndash8

ndash60 times 10ndash8

ndash12 times 10ndash7

ndash60 times 10ndash9 60 times 10ndash9ndash30 times 10ndash9 30 times 10ndash9

x 4 (m

s)

00x3 (m)

Last

poin

t

(b)

Figure 6 Dynamic response of the work roll displacement and the phase diagram under μlowast 00048 (a) time history of work rolldisplacement (b) phase diagram

ndash10 times 10ndash6

ndash50 times 10ndash7

00

50 times 10ndash7

10 times 10ndash6

Wor

k ro

ll di

spla

cem

ent (

m)

00 02 04 06 08 10Time (s)

(a)

ndash10 times 10ndash6 ndash50 times 10ndash7 00 50 times 10ndash7 10 times 10ndash6ndash80 times 10ndash4

ndash40 times 10ndash4

00

40 times 10ndash4

x 4 (m

s)

x3 (m)

Firs

t poi

nt

(b)

Figure 7 Dynamic response of the work roll displacement and the phase diagram under μlowast 00300 (a) time history of work rolldisplacement (b) phase diagram

Shock and Vibration 9

Based on equation (12) the tension variation at the entryposition (dTb) and exit position (dTf) is smaller comparedwith low friction coefficient rolling condition so the fluc-tuation of the response rolling force ΔPi increases and thestability of the stand decreases Meanwhile when the frictionis quite small zPzhout will increase with the decrease offriction coefficient F43 in equation (29) decreases so theinstantaneous dynamic structure spring constant decreasesand the fluctuation of the response rolling force ΔPi causedby the strip exit thickness variation will increase so thestability of the stand is decreased

43 Stable Range Analysis of Other Rolling Parameters

431 Calculation of Limited Rolling Speed Based on thiscoupled vertical-horizontal-torsional vibration analysismodel rolling speed has been chosen as the Hopf bifurcationparameter to obtain the limited stability rolling speed of the

UCM tandem cold rolling mill and other rolling conditionsare shown in Table 1 )e actual bifurcation point of rollingspeed is 251ms Figure 10 shows the time history of workroll displacement under the rolling speed of 252msAccording to Figure 10 the displacement amplitude ofthe work roll shows a divergence trend and the self-excitedvibration happens )e critical rolling speed calculated bythe proposed chatter model is 251ms

Figure 11 shows the tension and roll gap variation in No4 and No 5 stands obtained from the Production DataAcquisition (PDA) system As is shown in Figure 11(b)when the rolling speed is 221ms the back and front tensionof the No 5 stand fluctuate within a small range and the millstand is stable However when the rolling speed is raised to230ms the back and front tension of the No 5 standfluctuate more violently and the front tension of the No 5stand begins to diverge as shown in Figure 11(a) It is foundthat the roll gap of No 4 and No 5 stands fluctuates moreviolently by comparing Figures 11(c) and 11(d) Meanwhile

00 02 04 06 08 10

00

20 times 10ndash7

ndash20 times 10ndash7Wor

k ro

ll di

spla

cem

ent (

m)

Time (s)

(a)

00

00

10 times 10ndash4

50 times 10ndash5

ndash50 times 10ndash5

ndash10 times 10ndash4

ndash20 times 10ndash7 20 times 10ndash7

Last point

x 4 (m

s)

x3 (m)

First point

(b)

Figure 8 Dynamic response of the work roll displacement and the phase diagram under μlowast 00401 (a) time history of work rolldisplacement (b) phase diagram

00 02 04 06 08 10

00

20 times 10ndash6

ndash20 times 10ndash6Wor

k ro

ll di

spla

cem

ent (

m)

Time (s)

(a)

ndash20 times 10ndash6 00 20 times 10ndash6

00

10 times 10ndash3

50 times 10ndash4

ndash50 times 10ndash4

ndash10 times 10ndash3

x 4 (m

s)

x3 (m)

Last point

(b)

Figure 9 Dynamic response of the work roll displacement and the phase diagram under μlowast 00420 (a) time history of work rolldisplacement (b) phase diagram

10 Shock and Vibration

a sudden and harsh vibration noise will be heard in the No 5stand and finally the strip breaks It shows mill chatterappears under the speed of 230ms and the actual criticalrolling speed is 230ms

)e calculated critical rolling speed is 251ms while theactual critical rolling speed is 230ms )ere is a certainrelative error between the calculated critical speed and actualcritical rolling speed )is is due to the small coefficientdifference in Jacobian matrix With the change of rollingspeed the actual value of the tension and friction is changedAnd the calculated value of tension factor nt and frictioncoefficient in zPzh and zMzh is a little different with theactual value So the calculated limited rolling speed is a littlebigger than the actual value However by comparing thecritical rolling speed calculated by the proposed model withthe actual critical rolling speed it can be seen that theproposed model is still accurate

)e vibration is largely influenced by the friction co-efficient and rolling speed Based on equation (13) Figure 12shows the friction coefficient response of No 5 stand versusspeed Despite the changes of speed the friction coefficient isstill in the stability range of 00048ndash00401 It is divided intotwo regions In region 1 both the friction coefficient androlling speed are in the stable range In region 2 the frictioncoefficient is still in the stable range while the speed exceedsthe limited speed rangeWhen the rolling speed is lower than251ms the mill system is stable without changing thelubrication conditions And when the rolling speed is higherthan 251ms the stand is unstable However the frictioncoefficient is in the middle value of stable range under thespeed is 251ms so mill chatter might be eliminated byreducing the basic friction coefficient (by improving thelubrication characteristics such as increasing the concen-tration of emulsion increasing spray flow and improvingthe property of the base oil)

432 Calculation of Maximum Reduction Assuming theentry thickness is constant the exit thickness is variable Andthe exit thickness has been chosen as the Hopf bifurcationparameter Other calculated rolling conditions are shown in

Table 1 )en the calculated bifurcation point of exitthickness is 0168mm)e optimal setting range of reductionfor No 5 stand is 0ndash44When the reduction of No 5 stand is44 the increase of reduction leads to a lower mill standstability )e reason is also related to the increase of rollingforce variation under the larger reduction rolling conditions

As can be seen in Table 1 in normal rolling schedule thereduction rate is 333 And the max reduction is up to 44In reality considering there are other factors that influencethe stand stability such as rolling speed and lubricationconditions the max reduction is lower than 44 In generalthe effect of the reduction in the mill stability is lower thanthe improvement of friction coefficient

433 Front Tension Stable Range When the work roll ve-locity is 221ms the calculated bifurcation point of exittension is 15ndash133MPa Small front tension means that therolling force will be quite large So under the same excitationinterferences the variation of rolling force will be muchbigger and the stand stability is decreased When the fronttension is big enough tension factor nt decreases leading toa larger value in zPzh and zMzh As a result the fluc-tuation of rolling force and roll torque increases and thestability of the system is reduced as well So the influence oftension on the mill stability is smaller than other parameters

)ese results can be used to estimate mill chatter in acertain stand And the stable range of friction reduction andtension can be used to adjust the rolling schedule in theUCM mill to avoid mill chatter under high rolling speed

5 Conclusion

(1) A new multiple degrees of freedom and nonlineardynamic vertical-horizontal-torsional coupled millchatter model for UCM cold rolling mill is estab-lished In the dynamic rolling process model thetime-varying coupling effect between rolling forceroll torque variation and fluctuation of stripthickness and tension is formulated In the vibrationmodel the movement of work roll in both vertical-horizontal-torsional directions and the multirollcharacteristics in the UCM rolling mill are consid-ered Finally the dynamic relationship of rollmovement responses and dynamic changes in theroll gap is formulated

(2) Selecting the friction coefficient as the Hopf bi-furcation parameter and by using RouthndashHurwitzdeterminant the Hopf bifurcation point of the millvibration system is calculated and the bifurcationcharacteristics are analyzed Research shows thateither too low or too high friction coefficient in-creases the probability of mill chatter When thefriction coefficient is extremely low the mill systemproduces a stable limit cycle With the frictioncontinuing to increase the phase trajectory willconverge toward the origin point However the limitcycle of phase diagram disappears and dispersesoutward under high friction rolling conditions

00 02 04 06 08 10ndash2 times 10ndash7

ndash1 times 10ndash7

0

1 times 10ndash7

2 times 10ndash7

Wor

k ro

ll di

spla

cem

ent (

m)

Time (s)

Figure 10 Work roll time history displacement under the rollingspeed of 252ms

Shock and Vibration 11

0 1 2 30

20

40

60

80

Fron

t ten

sion

(kN

)

Back tension

Back

tens

ion

(kN

)

Time (s)

0

20

40

60

Front tension

(a)

0 1 2 30

20

40

60

80

Fron

t ten

sion

(kN

)

Back tension

Back

tens

ion

(kN

)

Time (s)

0

20

40

60

Front tension

(b)

0 1 2 3

015

018

021

024

Roll

gap

of N

o 5

stan

d (m

m)

Roll gap of No 4 stand

Roll

gap

of N

o 4

stan

d (m

m)

Time (s)

268

272

276

280

Roll gap of No 5 stand

(c)

0 1 2 3

030

033

036

039

Roll

gap

of N

o 5

stan

d (m

m)

Roll gap of No 4 stand

Roll

gap

of N

o4

stand

(mm

)

Time (s)

272

276

280

284

Roll gap of No 5 stand

(d)

Figure 11 Tension and roll gap time history (a) back and front tension of No 5 stand under the speed of 230ms (b) back and fronttension of No 5 stand under the speed of 221ms (c) roll gap of No 4 and No 5 stands under the speed of 230ms (d) roll gap of No 4 andNo 5 stands under the speed of 221ms

0 10 20 30

0020

0025

0030

0035

0040

0045

(2)

(25 002101)

Fric

tion

coeffi

cien

t

Speed (ms)

x = 251

(1)

Figure 12 Friction coefficient response of No 5 stand versus speed

12 Shock and Vibration

(3) Based on the coupled mill chatter model the millsystemrsquos stable limited rolling speed stable range ofreduction and front tension are investigatedWithout changing the rolling speed the most ef-fective way to eliminate mill chatter is to change thelubrication conditions and adjust the reductionschedule And the effect of tension on the vibration isquite limited )ese methods can be used to improvethe chatter phenomena in rolling mill under highrolling speed )e proposed vibration model canalso be used to design an optimum rolling schedulein the UCM tandem cold rolling mill and to de-termine the appearance of mill chatter under certainrolling conditions

Data Availability

)e rolling conditions data used to support the findings ofthis study are included within the article

Conflicts of Interest

)e authors declare that they have no conflicts of interestregarding the publication of this manuscript

Acknowledgments

)is work was supported by the National Natural ScienceFoundation of China (51774084 and 51634002) the NationalKey RampD Program of China (2017YFB0304100) the Fun-damental Research Funds for the Central Universities(N170708020) and the Open Research Fund from the StateKey Laboratory of Rolling and Automation NortheasternUniversity (2017RALKFKT006)

References

[1] I S Yun W R D Wilson and K F Ehmann ldquoReview ofchatter studies in cold rollingrdquo International Journal ofMachine Tools and Manufacture vol 38 no 12pp 1499ndash1530 1998

[2] I-S Yun W R D Wilson and K F Ehmann ldquoChatter in thestrip rolling process Part 1 dynamic model of rollingrdquoJournal of Manufacturing Science and Engineering vol 120no 2 pp 330ndash336 1998

[3] J Tlusty G Chandra S Critchley and D Paton ldquoChatter incold rollingrdquo CIRP Annals vol 31 no 1 pp 195ndash199 1982

[4] N M Reza F M Reza M Salimi and H Shojaei ldquoExper-imental investigations and ALE finite element methodanalysis of chatter in cold strip rollingrdquo ISIJ Internationalvol 52 no 12 pp 2245ndash2253 2012

[5] M R Niroomand M R Forouzan M Salimi and H ShojaeildquoExperimental investigations and ALE finite element methodanalysis of chatter in cold strip rollingrdquo ISIJ Internationalvol 52 no 12 pp 2245ndash2253 2012

[6] M R Niroomand M R Forouzan and M Salimi ldquo)eo-retical and experimental analysis of chatter in tandem coldrolling mills based on wave propagation theoryrdquo ISIJ In-ternational vol 55 no 3 pp 637ndash646 2015

[7] S Kapil P Eberhard and S K Dwivedy ldquoNonlinear dynamicanalysis of a parametrically excited cold rolling millrdquo Journal

of Manufacturing Science and Engineering vol 136 no 4p 041012 2014

[8] Y Kimura Y Sodani N Nishiura N Ikeuchi and Y MiharaldquoAnalysis of chaffer in tandem cold rolling millsrdquo ISIJ In-ternational vol 43 no 1 pp 77ndash84 2003

[9] N Fujita Y Kimura K Kobayashi et al ldquoDynamic control oflubrication characteristics in high speed tandem cold rollingrdquoJournal of Materials Processing Technology vol 229pp 407ndash416 2016

[10] A Heidari and M R Forouzan ldquoOptimization of cold rollingprocess parameters in order to increasing rolling speedlimited by chatter vibrationsrdquo Journal of Advanced Researchvol 4 no 1 pp 27ndash34 2013

[11] A Heidari M R Forouzan and S Akbarzadeh ldquoEffect offriction on tandem cold rolling mills chatteringrdquo ISIJ In-ternational vol 54 no 10 pp 2349ndash2356 2014

[12] A Heidari M R Forouzan and S Akbarzadeh ldquoDevelop-ment of a rolling chatter model considering unsteady lubri-cationrdquo ISIJ International vol 54 no 1 pp 165ndash170 2014

[13] A Heidari M R Forouzan and M R Niroomand ldquoDe-velopment and evaluation of friction models for chattersimulation in cold strip rollingrdquo International Journal ofAdvanced Manufacturing Technology vol 96 no 4 pp 1ndash212018

[14] Y Kim C-W Kim S Lee and H Park ldquoDynamic modelingand numerical analysis of a cold rolling millrdquo InternationalJournal of Precision Engineering and Manufacturing vol 14no 3 pp 407ndash413 2013

[15] P V Krot ldquoNonlinear vibrations and backlashes diagnosticsin the rolling mills drive trainsrdquo in Proceedings of the 6thEUROMECH Nonlinear Dynamics Conference Saint Peters-burg Russia July 2008

[16] S Liu B Sun S Zhao J Li and W Zhang ldquoStronglynonlinear dynamics of torsional vibration for rolling millrsquoselectromechanical coupling systemrdquo Steel Research In-ternational vol 86 no 9 pp 984ndash992 2015

[17] S Liu J Wang J Liu and Y Li ldquoNonlinear parametricallyexcited vibration and active control of gear pair system withtime-varying characteristicrdquo Chinese Physics B vol 24 no 10pp 303ndash311 2015

[18] B D Hassard N D Kazarinoff and Y H Wan eory andApplications of Hopf Bifurcation Cambridge University PressCambridge UK 1981

[19] B H K Lee S J Price and Y SWong ldquoNonlinear aeroelasticanalysis of airfoils bifurcation and chaosrdquo Progress inAerospace Sciences vol 35 no 3 pp 205ndash334 1999

[20] K Nishimura T Ikeda and Y Harata ldquoLocalization phe-nomena in torsional rotating shaft systems with multiplecentrifugal pendulum vibration absorbersrdquo Nonlinear Dy-namics vol 83 no 3 pp 1705ndash1726 2015

[21] T Zhang and H Dai ldquoBifurcation analysis of high-speedrailway wheel-setrdquo Nonlinear Dynamics vol 83 no 3pp 1511ndash1528 2015

[22] Z Y Gao Y Zang and T Han ldquoHopf bifurcation andstability analysis on the mill drive systemrdquo Applied Mechanicsamp Materials vol 121-126 pp 1514ndash1520 2012

[23] L Zeng Y Zang and Z Gao ldquoHopf bifurcation control forrolling mill multiple-mode-coupling vibration under non-linear frictionrdquo Journal of Vibration and Acoustics vol 139no 6 article 061015 2017

[24] S Liu Z Wang J Wang and H Li ldquoSliding bifurcationresearch of a horizontal-torsional coupled main drive systemof rolling millrdquo Nonlinear Dynamics vol 83 pp 441ndash4552015

Shock and Vibration 13

[25] Y Peng M Zhang J-L Sun and Y Zhang ldquoExperimentaland numerical investigation on the roll system swing vibra-tion characteristics of a hot rolling millrdquo ISIJ Internationalvol 57 no 9 pp 1567ndash1576 2017

[26] J Pittner and M A Simaan ldquoOptimal control of continuoustandem cold metal rollingrdquo in Proceedings of 2008 AmericanControl Conference Seattle WA USA June 2008

[27] R B Sims and D F S Arthur ldquoSpeed dependent variables incold strip rollingrdquo Journal of Iron and Steel Institute vol 172no 3 pp 285ndash295 1952

[28] S Z Chen D H Zhang J Sun J S Wang and J SongldquoOnline calculation model of rolling force for cold rolling millbased on numerical integrationrdquo in Proceedings of the 24thChinese Control and Decision Conference (CCDC) TaiyuanChina May 2012

[29] D R Bland and H Ford ldquo)e calculation of roll force andtorque in cold strip rolling with tensionsrdquo Proceedings of theInstitution of Mechanical Engineers vol 159 no 1 pp 144ndash163 2006

[30] R Hill ldquoRelations between roll-force torque and the appliedtensions in strip-rollingrdquo Proceedings of the Institution ofMechanical Engineers vol 163 no 1 pp 135ndash140 2006

[31] R E Johnson and Q Qi ldquoChatter dynamics in sheet rollingrdquoInternational Journal of Mechanical Sciences vol 36 no 7pp 617ndash630 1994

[32] Z Gao L Bai and Q Li ldquoResearch on critical rolling speed ofself-excited vibration in the tandem rolling process of thinstriprdquo Journal of Mechanical Engineering vol 53 no 12pp 118ndash132 2017

[33] R C Dorf and R H Bishop Modern Control SystemsUniversity of California Davis CA USA 12th edition 2010

[34] W M Liu ldquoCriterion of hopf bifurcations without usingeigenvaluesrdquo Journal of Mathematical Analysis and Appli-cations vol 182 no 1 pp 250ndash256 1994

[35] L Zeng Y Zang and Z Gao ldquoMultiple-modal-couplingmodeling and stability analysis of cold rolling mill vibra-tionrdquo Shock and Vibration vol 2016 Article ID 234738626 pages 2016

14 Shock and Vibration

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the dynamic differential equation of the system is regularizedto obtain the equivalent damping matrix )en the dampingregular matrix is transformed into the original coordinatesystem and let the damping ratio be 003 Finally theequivalent damping value of the roll system is calculatedWork roll rotational inertia mass work roll rotational springconstant and work roll rotational damping coefficient areobtained from the work of Zeng et al [23] as the design andmill structure of rolling stands are similar to one another)e roll diameters and work roll installation offset are ob-tained from the UCM rolling mill installation drawingsTable 2 shows the structure specifications of the UCMmill inequation (28)

Equation (28) is the nonlinear dynamic equation of thevertical-horizontal-torsional coupled vibration modelConvert the above differential equation into the followingstate matrix form

_X F(X μ) + Hdhin (29)

whereX xw _xw yw _yw yim _yim yb _yb θw _θw dTf dTb1113960 1113961

T

x1 x2 x3 x121113858 1113859T

H 0minuswTb

2Mw

0zP

zhinMw

0 0 0 0 0zM

zhinJw

0 01113890 1113891

T

F

0 1 0 0 0 0 0 0 0 0 0 0F21 F22 F23 0 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 00 0 F43 F44 F45 F46 0 0 0 0 0 00 0 0 0 0 1 0 0 0 0 0 00 0 F63 F64 F65 F66 F67 F68 0 0 0 00 0 0 0 0 0 0 1 0 0 0 00 0 0 0 F85 F86 F87 F88 0 0 0 00 0 0 0 0 0 0 0 0 1 0 00 F102 F103 0 0 0 0 0 F109 F1010 F1011 F10120 0 F113 0 0 0 0 0 0 0 F1111 F11120 0 F123 0 0 0 0 0 0 0 F1211 F1212

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

F21 minusKwx

Mw

F22 minusCwx

Mw

F23 wTf

Mw

F43 2 zPzhout( 1113857minusKwy

Mw

F44 minusCwy

Mw

F45 Kwy

Mw

F46 Cwy

Mw

F63 Kwy

Mim

F64 Cwy

Mim

F65 minusKwy + Kim

Mim

F66 minusCwy + Cim

Mim

F67 Kim

Mim

F68 Cim

Mim

F85 Kim

Mb

F86 Cim

Mb

F87 minusKim + Kb

Mb

F88 minusCim + Cb

Mb

F102 zM

Jwzxw

F103 2zM

Jwzhout

F109 minusKwj

Jw

F1010 minusCwj

Jw

F1011 zM

JwzTf

F1012 zM

JwzTb

F113 minus2E2Vr zhnzhout( 1113857hout minus hn( 1113857

55025h2out

F1111 minusE2Vr zhnzTf1113872 1113873

55025hout

F1112 minusE2Vr zhnzTb( 1113857

55025hout

F123 2E2Vr zhnzhout( 1113857

55025hin

F1211 E2Vr zhnzTf1113872 1113873

55025hin

F1212 E2Vr zhnzTf1113872 1113873

55025hin

(30)

)en equation (29) is the nonlinear dynamic stateequation of the vertical-horizontal-torsional coupled vi-bration system considering the nonlinear characteristicduring the rolling process the movement of rolls in bothvertical-horizontal directions and the torsional vibration

4 Analysis of Vibration Stability of theRoll System

41 Hopf Bifurcation Parameter Calculation )e frictioncharacteristic would cause the instability of the dynamicstate equation so in this research μ has been chosen as the

Shock and Vibration 7

Hopf bifurcation parameter Assuming the entry thicknessof each stand is constant the equilibrium point is Xwhere F(X) 0 )e system equilibrium point can betransformed to the origin by coordinate transformation X0

[0 0 0 0 0 0 0 0 0 0 0 0]T Let JF(X) be the Jacobianmatrix of F(X) So the Hopf bifurcation can be obtained byjudging all the eigenvalues of JF(X) However it is difficult tocalculate the eigenvalues of JF(X) since the calculationamount is quite large )en the RouthndashHurwitz determinantis used to calculate the Hopf bifurcation point [33]

)e characteristic equation of Jacobian matrix JF(X) ofthe dynamic vertical-horizontal-torsional coupling vibrationmodel is assumed to be

a12λ12

+ a11λ11

+ middot middot middot + a2λ2

+ a1λ + a0 0 (31)

)e main determinant

D1 a1 D2 a1 a0

a3 a2

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868 middot middot middot Di

a1 a0 middot middot middot 0

a3 a2 middot middot middot 0

middot middot middot middot middot middot middot middot middot middot middot middot

a2iminus1 a2iminus2 middot middot middot ai

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(32)

where ai 0 if ilt 0 or igt n and a12 1Assuming μlowast is the Hopf bifurcation the Hopf bi-

furcation conditions are [23 24 34]

a0 μlowast( 1113857gt 0 D1 μlowast( 1113857gt 0 middot middot middot Dnminus2 μlowast( 1113857gt 0 Dnminus1 μlowast( 1113857 0

dΔiminus1 μlowast( 1113857

dμne 0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(33)

According to these assumptions the Hopf bifurcationpoint of μlowast occurs at μlowast 00048 and μlowast 00401

42Analysis of StableRangeofFriction When the value of μlowastis smaller than the first bifurcation point of μlowast 00048such as μlowast 00040 the response of work roll displacementand Hopf bifurcation diagram are shown in Figure 5 )eroll displacement will diverge rapidly and the equilibriumpoint of the equilibrium equation is unstable Besides asshown in Figure 5(b) the phase trajectory will disperseoutward of the origin in a ring shape and the radius cur-vature of the phase trajectory increases gradually It isconcluded that the roll vibration system is unstable and millchatter appears

As shown in Figure 6(a) when μlowast 00048 the work rolldisplacement exhibits as an equal amplitude vibration andthe roll vibrates at its balanced position )is steady regionrefers to the maximum and minimum fluctuation range ofwork roll Meanwhile Figure 6(b) shows that the phasediagram will begin to converge toward a stable limit cycle)e strip will produce a slight thickness fluctuation withinthe quality requirements and the mill system is stable underthese rolling conditions

When μlowast 00300 the roll systemrsquos time history responseand the phase diagram are shown in Figure 7 It can be seenfrom these figures that the phase trajectory will convergestoward a smaller and smaller circle near the origin point )eequilibrium point of work roll is asymptotically stable

)e roll system stability about the critical parameter ofμlowast 00401 is shown in Figure 8 When at the critical Hopfbifurcation parameter displacement of work roll firstlybecomes larger and finally converges to an equal amplitudeoscillation )e equilibrium point is unstable and the Hopfbifurcation diagram presents a ring shape However dy-namic range of the fixed limit cycle in Figure 8 is a littlelarger compared with Figure 6 )is is mainly because thefriction is bigger and as a result the fluctuation of theresponse rolling force ΔPi under the friction coefficient of00401 is much bigger

)e numerical solution of work roll displacement underthe Hopf bifurcation point of μlowast 00420 is shown in Fig-ure 9)e roll displacement will disperse outward all the timeBesides the phase diagram of the work roll will disperseoutward of the cycle and the radius curvature of the phasetrajectory increases gradually It illustrates that the mill vi-bration system is unstable and mill chatter is produced

When μlowast lt 00048 the roll system is unstable and whenμlowast 00048 the system is stable So the Hopf bifurcation ofμlowast 00048 is the subcritical bifurcation point and μlowast

00401 is the supercritical bifurcation pointMany researchers attribute the change of stability in

rolling mill system to the sudden change of the rolling forcefluctuation in dynamic equilibrium equation [11 25 35])e optimal setting range of friction coefficient is 00048ndash00401 and mill chatter appears when the friction is eithertoo high or too low )is is mainly due to the fluctuation ofrolling force When the friction is high with the increase offriction coefficient the neutral angle becomes bigger andthen the neutral point shifts to the entry side so the distancebetween the entry and neutral point becomes shorter andthe variation of the entry speed decreases [11] However theexit forward slip and exit speed fluctuation will increase

Table 2 Structure specifications of the UCM mill

Parameters ValueWork roll mass Mw (Kg) 3914Intermediate roll mass Mim (Kg) 5410Backup roll mass Mb (Kg) 31330Work roll contact spring constant Kw (N mminus1) 091times 1010

Intermediate roll spring constant Kim (N mminus1) 103times1010

Backup roll spring constant Kb (N mminus1) 585times1010

Work roll damping coefficient x directionCwx (Nmiddots mminus1) 500times104

Work roll damping coefficient in y directionCwy (Nmiddots mminus1) 164times105

Intermediate roll damping coefficient Cim (Nmiddots mminus1) 570times105

Backup roll damping coefficient Cb (Nmiddots mminus1) 510times106

Work roll rotational inertia mass Jw (Kgmiddotm2) 1381Work roll rotational spring constant Kwj (Nmiddotmmiddotradminus1) 790times106

Work roll rotational damping coefficientCwj (Nmiddotmmiddots radminus1) 4178

Work roll diameter Rw (mm) 2125Intermediate roll diameter Rim (mm) 245Backup roll diameter Rb (mm) 650Work roll installation offset e (mm) 5

8 Shock and Vibration

ndash80 times 10ndash5

ndash40 times 10ndash5

00

40 times 10ndash5

80 times 10ndash5W

ork

roll

disp

lace

men

t (m

)

00 02 04 06 08 10Time (s)

(a)

ndash002

ndash001

000

001

002 First point

x 4 (m

s)

00 80 times 10ndash540 times 10ndash5ndash40 times 10ndash5ndash80 times 10ndash5

x3 (m)

Last point

(b)

Figure 5 Dynamic response of the work roll displacement and the phase diagram under μlowast 00040 (a) time history of work rolldisplacement (b) phase diagram

00

Wor

k ro

ll di

spla

cem

ent (

m)

00 02 04 06 08 10Time (s)

10 times 10ndash7

50 times 10ndash8

ndash50 times 10ndash8

ndash10 times 10ndash7

(a)

00

12 times 10ndash7

60 times 10ndash8

ndash60 times 10ndash8

ndash12 times 10ndash7

ndash60 times 10ndash9 60 times 10ndash9ndash30 times 10ndash9 30 times 10ndash9

x 4 (m

s)

00x3 (m)

Last

poin

t

(b)

Figure 6 Dynamic response of the work roll displacement and the phase diagram under μlowast 00048 (a) time history of work rolldisplacement (b) phase diagram

ndash10 times 10ndash6

ndash50 times 10ndash7

00

50 times 10ndash7

10 times 10ndash6

Wor

k ro

ll di

spla

cem

ent (

m)

00 02 04 06 08 10Time (s)

(a)

ndash10 times 10ndash6 ndash50 times 10ndash7 00 50 times 10ndash7 10 times 10ndash6ndash80 times 10ndash4

ndash40 times 10ndash4

00

40 times 10ndash4

x 4 (m

s)

x3 (m)

Firs

t poi

nt

(b)

Figure 7 Dynamic response of the work roll displacement and the phase diagram under μlowast 00300 (a) time history of work rolldisplacement (b) phase diagram

Shock and Vibration 9

Based on equation (12) the tension variation at the entryposition (dTb) and exit position (dTf) is smaller comparedwith low friction coefficient rolling condition so the fluc-tuation of the response rolling force ΔPi increases and thestability of the stand decreases Meanwhile when the frictionis quite small zPzhout will increase with the decrease offriction coefficient F43 in equation (29) decreases so theinstantaneous dynamic structure spring constant decreasesand the fluctuation of the response rolling force ΔPi causedby the strip exit thickness variation will increase so thestability of the stand is decreased

43 Stable Range Analysis of Other Rolling Parameters

431 Calculation of Limited Rolling Speed Based on thiscoupled vertical-horizontal-torsional vibration analysismodel rolling speed has been chosen as the Hopf bifurcationparameter to obtain the limited stability rolling speed of the

UCM tandem cold rolling mill and other rolling conditionsare shown in Table 1 )e actual bifurcation point of rollingspeed is 251ms Figure 10 shows the time history of workroll displacement under the rolling speed of 252msAccording to Figure 10 the displacement amplitude ofthe work roll shows a divergence trend and the self-excitedvibration happens )e critical rolling speed calculated bythe proposed chatter model is 251ms

Figure 11 shows the tension and roll gap variation in No4 and No 5 stands obtained from the Production DataAcquisition (PDA) system As is shown in Figure 11(b)when the rolling speed is 221ms the back and front tensionof the No 5 stand fluctuate within a small range and the millstand is stable However when the rolling speed is raised to230ms the back and front tension of the No 5 standfluctuate more violently and the front tension of the No 5stand begins to diverge as shown in Figure 11(a) It is foundthat the roll gap of No 4 and No 5 stands fluctuates moreviolently by comparing Figures 11(c) and 11(d) Meanwhile

00 02 04 06 08 10

00

20 times 10ndash7

ndash20 times 10ndash7Wor

k ro

ll di

spla

cem

ent (

m)

Time (s)

(a)

00

00

10 times 10ndash4

50 times 10ndash5

ndash50 times 10ndash5

ndash10 times 10ndash4

ndash20 times 10ndash7 20 times 10ndash7

Last point

x 4 (m

s)

x3 (m)

First point

(b)

Figure 8 Dynamic response of the work roll displacement and the phase diagram under μlowast 00401 (a) time history of work rolldisplacement (b) phase diagram

00 02 04 06 08 10

00

20 times 10ndash6

ndash20 times 10ndash6Wor

k ro

ll di

spla

cem

ent (

m)

Time (s)

(a)

ndash20 times 10ndash6 00 20 times 10ndash6

00

10 times 10ndash3

50 times 10ndash4

ndash50 times 10ndash4

ndash10 times 10ndash3

x 4 (m

s)

x3 (m)

Last point

(b)

Figure 9 Dynamic response of the work roll displacement and the phase diagram under μlowast 00420 (a) time history of work rolldisplacement (b) phase diagram

10 Shock and Vibration

a sudden and harsh vibration noise will be heard in the No 5stand and finally the strip breaks It shows mill chatterappears under the speed of 230ms and the actual criticalrolling speed is 230ms

)e calculated critical rolling speed is 251ms while theactual critical rolling speed is 230ms )ere is a certainrelative error between the calculated critical speed and actualcritical rolling speed )is is due to the small coefficientdifference in Jacobian matrix With the change of rollingspeed the actual value of the tension and friction is changedAnd the calculated value of tension factor nt and frictioncoefficient in zPzh and zMzh is a little different with theactual value So the calculated limited rolling speed is a littlebigger than the actual value However by comparing thecritical rolling speed calculated by the proposed model withthe actual critical rolling speed it can be seen that theproposed model is still accurate

)e vibration is largely influenced by the friction co-efficient and rolling speed Based on equation (13) Figure 12shows the friction coefficient response of No 5 stand versusspeed Despite the changes of speed the friction coefficient isstill in the stability range of 00048ndash00401 It is divided intotwo regions In region 1 both the friction coefficient androlling speed are in the stable range In region 2 the frictioncoefficient is still in the stable range while the speed exceedsthe limited speed rangeWhen the rolling speed is lower than251ms the mill system is stable without changing thelubrication conditions And when the rolling speed is higherthan 251ms the stand is unstable However the frictioncoefficient is in the middle value of stable range under thespeed is 251ms so mill chatter might be eliminated byreducing the basic friction coefficient (by improving thelubrication characteristics such as increasing the concen-tration of emulsion increasing spray flow and improvingthe property of the base oil)

432 Calculation of Maximum Reduction Assuming theentry thickness is constant the exit thickness is variable Andthe exit thickness has been chosen as the Hopf bifurcationparameter Other calculated rolling conditions are shown in

Table 1 )en the calculated bifurcation point of exitthickness is 0168mm)e optimal setting range of reductionfor No 5 stand is 0ndash44When the reduction of No 5 stand is44 the increase of reduction leads to a lower mill standstability )e reason is also related to the increase of rollingforce variation under the larger reduction rolling conditions

As can be seen in Table 1 in normal rolling schedule thereduction rate is 333 And the max reduction is up to 44In reality considering there are other factors that influencethe stand stability such as rolling speed and lubricationconditions the max reduction is lower than 44 In generalthe effect of the reduction in the mill stability is lower thanthe improvement of friction coefficient

433 Front Tension Stable Range When the work roll ve-locity is 221ms the calculated bifurcation point of exittension is 15ndash133MPa Small front tension means that therolling force will be quite large So under the same excitationinterferences the variation of rolling force will be muchbigger and the stand stability is decreased When the fronttension is big enough tension factor nt decreases leading toa larger value in zPzh and zMzh As a result the fluc-tuation of rolling force and roll torque increases and thestability of the system is reduced as well So the influence oftension on the mill stability is smaller than other parameters

)ese results can be used to estimate mill chatter in acertain stand And the stable range of friction reduction andtension can be used to adjust the rolling schedule in theUCM mill to avoid mill chatter under high rolling speed

5 Conclusion

(1) A new multiple degrees of freedom and nonlineardynamic vertical-horizontal-torsional coupled millchatter model for UCM cold rolling mill is estab-lished In the dynamic rolling process model thetime-varying coupling effect between rolling forceroll torque variation and fluctuation of stripthickness and tension is formulated In the vibrationmodel the movement of work roll in both vertical-horizontal-torsional directions and the multirollcharacteristics in the UCM rolling mill are consid-ered Finally the dynamic relationship of rollmovement responses and dynamic changes in theroll gap is formulated

(2) Selecting the friction coefficient as the Hopf bi-furcation parameter and by using RouthndashHurwitzdeterminant the Hopf bifurcation point of the millvibration system is calculated and the bifurcationcharacteristics are analyzed Research shows thateither too low or too high friction coefficient in-creases the probability of mill chatter When thefriction coefficient is extremely low the mill systemproduces a stable limit cycle With the frictioncontinuing to increase the phase trajectory willconverge toward the origin point However the limitcycle of phase diagram disappears and dispersesoutward under high friction rolling conditions

00 02 04 06 08 10ndash2 times 10ndash7

ndash1 times 10ndash7

0

1 times 10ndash7

2 times 10ndash7

Wor

k ro

ll di

spla

cem

ent (

m)

Time (s)

Figure 10 Work roll time history displacement under the rollingspeed of 252ms

Shock and Vibration 11

0 1 2 30

20

40

60

80

Fron

t ten

sion

(kN

)

Back tension

Back

tens

ion

(kN

)

Time (s)

0

20

40

60

Front tension

(a)

0 1 2 30

20

40

60

80

Fron

t ten

sion

(kN

)

Back tension

Back

tens

ion

(kN

)

Time (s)

0

20

40

60

Front tension

(b)

0 1 2 3

015

018

021

024

Roll

gap

of N

o 5

stan

d (m

m)

Roll gap of No 4 stand

Roll

gap

of N

o 4

stan

d (m

m)

Time (s)

268

272

276

280

Roll gap of No 5 stand

(c)

0 1 2 3

030

033

036

039

Roll

gap

of N

o 5

stan

d (m

m)

Roll gap of No 4 stand

Roll

gap

of N

o4

stand

(mm

)

Time (s)

272

276

280

284

Roll gap of No 5 stand

(d)

Figure 11 Tension and roll gap time history (a) back and front tension of No 5 stand under the speed of 230ms (b) back and fronttension of No 5 stand under the speed of 221ms (c) roll gap of No 4 and No 5 stands under the speed of 230ms (d) roll gap of No 4 andNo 5 stands under the speed of 221ms

0 10 20 30

0020

0025

0030

0035

0040

0045

(2)

(25 002101)

Fric

tion

coeffi

cien

t

Speed (ms)

x = 251

(1)

Figure 12 Friction coefficient response of No 5 stand versus speed

12 Shock and Vibration

(3) Based on the coupled mill chatter model the millsystemrsquos stable limited rolling speed stable range ofreduction and front tension are investigatedWithout changing the rolling speed the most ef-fective way to eliminate mill chatter is to change thelubrication conditions and adjust the reductionschedule And the effect of tension on the vibration isquite limited )ese methods can be used to improvethe chatter phenomena in rolling mill under highrolling speed )e proposed vibration model canalso be used to design an optimum rolling schedulein the UCM tandem cold rolling mill and to de-termine the appearance of mill chatter under certainrolling conditions

Data Availability

)e rolling conditions data used to support the findings ofthis study are included within the article

Conflicts of Interest

)e authors declare that they have no conflicts of interestregarding the publication of this manuscript

Acknowledgments

)is work was supported by the National Natural ScienceFoundation of China (51774084 and 51634002) the NationalKey RampD Program of China (2017YFB0304100) the Fun-damental Research Funds for the Central Universities(N170708020) and the Open Research Fund from the StateKey Laboratory of Rolling and Automation NortheasternUniversity (2017RALKFKT006)

References

[1] I S Yun W R D Wilson and K F Ehmann ldquoReview ofchatter studies in cold rollingrdquo International Journal ofMachine Tools and Manufacture vol 38 no 12pp 1499ndash1530 1998

[2] I-S Yun W R D Wilson and K F Ehmann ldquoChatter in thestrip rolling process Part 1 dynamic model of rollingrdquoJournal of Manufacturing Science and Engineering vol 120no 2 pp 330ndash336 1998

[3] J Tlusty G Chandra S Critchley and D Paton ldquoChatter incold rollingrdquo CIRP Annals vol 31 no 1 pp 195ndash199 1982

[4] N M Reza F M Reza M Salimi and H Shojaei ldquoExper-imental investigations and ALE finite element methodanalysis of chatter in cold strip rollingrdquo ISIJ Internationalvol 52 no 12 pp 2245ndash2253 2012

[5] M R Niroomand M R Forouzan M Salimi and H ShojaeildquoExperimental investigations and ALE finite element methodanalysis of chatter in cold strip rollingrdquo ISIJ Internationalvol 52 no 12 pp 2245ndash2253 2012

[6] M R Niroomand M R Forouzan and M Salimi ldquo)eo-retical and experimental analysis of chatter in tandem coldrolling mills based on wave propagation theoryrdquo ISIJ In-ternational vol 55 no 3 pp 637ndash646 2015

[7] S Kapil P Eberhard and S K Dwivedy ldquoNonlinear dynamicanalysis of a parametrically excited cold rolling millrdquo Journal

of Manufacturing Science and Engineering vol 136 no 4p 041012 2014

[8] Y Kimura Y Sodani N Nishiura N Ikeuchi and Y MiharaldquoAnalysis of chaffer in tandem cold rolling millsrdquo ISIJ In-ternational vol 43 no 1 pp 77ndash84 2003

[9] N Fujita Y Kimura K Kobayashi et al ldquoDynamic control oflubrication characteristics in high speed tandem cold rollingrdquoJournal of Materials Processing Technology vol 229pp 407ndash416 2016

[10] A Heidari and M R Forouzan ldquoOptimization of cold rollingprocess parameters in order to increasing rolling speedlimited by chatter vibrationsrdquo Journal of Advanced Researchvol 4 no 1 pp 27ndash34 2013

[11] A Heidari M R Forouzan and S Akbarzadeh ldquoEffect offriction on tandem cold rolling mills chatteringrdquo ISIJ In-ternational vol 54 no 10 pp 2349ndash2356 2014

[12] A Heidari M R Forouzan and S Akbarzadeh ldquoDevelop-ment of a rolling chatter model considering unsteady lubri-cationrdquo ISIJ International vol 54 no 1 pp 165ndash170 2014

[13] A Heidari M R Forouzan and M R Niroomand ldquoDe-velopment and evaluation of friction models for chattersimulation in cold strip rollingrdquo International Journal ofAdvanced Manufacturing Technology vol 96 no 4 pp 1ndash212018

[14] Y Kim C-W Kim S Lee and H Park ldquoDynamic modelingand numerical analysis of a cold rolling millrdquo InternationalJournal of Precision Engineering and Manufacturing vol 14no 3 pp 407ndash413 2013

[15] P V Krot ldquoNonlinear vibrations and backlashes diagnosticsin the rolling mills drive trainsrdquo in Proceedings of the 6thEUROMECH Nonlinear Dynamics Conference Saint Peters-burg Russia July 2008

[16] S Liu B Sun S Zhao J Li and W Zhang ldquoStronglynonlinear dynamics of torsional vibration for rolling millrsquoselectromechanical coupling systemrdquo Steel Research In-ternational vol 86 no 9 pp 984ndash992 2015

[17] S Liu J Wang J Liu and Y Li ldquoNonlinear parametricallyexcited vibration and active control of gear pair system withtime-varying characteristicrdquo Chinese Physics B vol 24 no 10pp 303ndash311 2015

[18] B D Hassard N D Kazarinoff and Y H Wan eory andApplications of Hopf Bifurcation Cambridge University PressCambridge UK 1981

[19] B H K Lee S J Price and Y SWong ldquoNonlinear aeroelasticanalysis of airfoils bifurcation and chaosrdquo Progress inAerospace Sciences vol 35 no 3 pp 205ndash334 1999

[20] K Nishimura T Ikeda and Y Harata ldquoLocalization phe-nomena in torsional rotating shaft systems with multiplecentrifugal pendulum vibration absorbersrdquo Nonlinear Dy-namics vol 83 no 3 pp 1705ndash1726 2015

[21] T Zhang and H Dai ldquoBifurcation analysis of high-speedrailway wheel-setrdquo Nonlinear Dynamics vol 83 no 3pp 1511ndash1528 2015

[22] Z Y Gao Y Zang and T Han ldquoHopf bifurcation andstability analysis on the mill drive systemrdquo Applied Mechanicsamp Materials vol 121-126 pp 1514ndash1520 2012

[23] L Zeng Y Zang and Z Gao ldquoHopf bifurcation control forrolling mill multiple-mode-coupling vibration under non-linear frictionrdquo Journal of Vibration and Acoustics vol 139no 6 article 061015 2017

[24] S Liu Z Wang J Wang and H Li ldquoSliding bifurcationresearch of a horizontal-torsional coupled main drive systemof rolling millrdquo Nonlinear Dynamics vol 83 pp 441ndash4552015

Shock and Vibration 13

[25] Y Peng M Zhang J-L Sun and Y Zhang ldquoExperimentaland numerical investigation on the roll system swing vibra-tion characteristics of a hot rolling millrdquo ISIJ Internationalvol 57 no 9 pp 1567ndash1576 2017

[26] J Pittner and M A Simaan ldquoOptimal control of continuoustandem cold metal rollingrdquo in Proceedings of 2008 AmericanControl Conference Seattle WA USA June 2008

[27] R B Sims and D F S Arthur ldquoSpeed dependent variables incold strip rollingrdquo Journal of Iron and Steel Institute vol 172no 3 pp 285ndash295 1952

[28] S Z Chen D H Zhang J Sun J S Wang and J SongldquoOnline calculation model of rolling force for cold rolling millbased on numerical integrationrdquo in Proceedings of the 24thChinese Control and Decision Conference (CCDC) TaiyuanChina May 2012

[29] D R Bland and H Ford ldquo)e calculation of roll force andtorque in cold strip rolling with tensionsrdquo Proceedings of theInstitution of Mechanical Engineers vol 159 no 1 pp 144ndash163 2006

[30] R Hill ldquoRelations between roll-force torque and the appliedtensions in strip-rollingrdquo Proceedings of the Institution ofMechanical Engineers vol 163 no 1 pp 135ndash140 2006

[31] R E Johnson and Q Qi ldquoChatter dynamics in sheet rollingrdquoInternational Journal of Mechanical Sciences vol 36 no 7pp 617ndash630 1994

[32] Z Gao L Bai and Q Li ldquoResearch on critical rolling speed ofself-excited vibration in the tandem rolling process of thinstriprdquo Journal of Mechanical Engineering vol 53 no 12pp 118ndash132 2017

[33] R C Dorf and R H Bishop Modern Control SystemsUniversity of California Davis CA USA 12th edition 2010

[34] W M Liu ldquoCriterion of hopf bifurcations without usingeigenvaluesrdquo Journal of Mathematical Analysis and Appli-cations vol 182 no 1 pp 250ndash256 1994

[35] L Zeng Y Zang and Z Gao ldquoMultiple-modal-couplingmodeling and stability analysis of cold rolling mill vibra-tionrdquo Shock and Vibration vol 2016 Article ID 234738626 pages 2016

14 Shock and Vibration

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Hopf bifurcation parameter Assuming the entry thicknessof each stand is constant the equilibrium point is Xwhere F(X) 0 )e system equilibrium point can betransformed to the origin by coordinate transformation X0

[0 0 0 0 0 0 0 0 0 0 0 0]T Let JF(X) be the Jacobianmatrix of F(X) So the Hopf bifurcation can be obtained byjudging all the eigenvalues of JF(X) However it is difficult tocalculate the eigenvalues of JF(X) since the calculationamount is quite large )en the RouthndashHurwitz determinantis used to calculate the Hopf bifurcation point [33]

)e characteristic equation of Jacobian matrix JF(X) ofthe dynamic vertical-horizontal-torsional coupling vibrationmodel is assumed to be

a12λ12

+ a11λ11

+ middot middot middot + a2λ2

+ a1λ + a0 0 (31)

)e main determinant

D1 a1 D2 a1 a0

a3 a2

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868 middot middot middot Di

a1 a0 middot middot middot 0

a3 a2 middot middot middot 0

middot middot middot middot middot middot middot middot middot middot middot middot

a2iminus1 a2iminus2 middot middot middot ai

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(32)

where ai 0 if ilt 0 or igt n and a12 1Assuming μlowast is the Hopf bifurcation the Hopf bi-

furcation conditions are [23 24 34]

a0 μlowast( 1113857gt 0 D1 μlowast( 1113857gt 0 middot middot middot Dnminus2 μlowast( 1113857gt 0 Dnminus1 μlowast( 1113857 0

dΔiminus1 μlowast( 1113857

dμne 0

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(33)

According to these assumptions the Hopf bifurcationpoint of μlowast occurs at μlowast 00048 and μlowast 00401

42Analysis of StableRangeofFriction When the value of μlowastis smaller than the first bifurcation point of μlowast 00048such as μlowast 00040 the response of work roll displacementand Hopf bifurcation diagram are shown in Figure 5 )eroll displacement will diverge rapidly and the equilibriumpoint of the equilibrium equation is unstable Besides asshown in Figure 5(b) the phase trajectory will disperseoutward of the origin in a ring shape and the radius cur-vature of the phase trajectory increases gradually It isconcluded that the roll vibration system is unstable and millchatter appears

As shown in Figure 6(a) when μlowast 00048 the work rolldisplacement exhibits as an equal amplitude vibration andthe roll vibrates at its balanced position )is steady regionrefers to the maximum and minimum fluctuation range ofwork roll Meanwhile Figure 6(b) shows that the phasediagram will begin to converge toward a stable limit cycle)e strip will produce a slight thickness fluctuation withinthe quality requirements and the mill system is stable underthese rolling conditions

When μlowast 00300 the roll systemrsquos time history responseand the phase diagram are shown in Figure 7 It can be seenfrom these figures that the phase trajectory will convergestoward a smaller and smaller circle near the origin point )eequilibrium point of work roll is asymptotically stable

)e roll system stability about the critical parameter ofμlowast 00401 is shown in Figure 8 When at the critical Hopfbifurcation parameter displacement of work roll firstlybecomes larger and finally converges to an equal amplitudeoscillation )e equilibrium point is unstable and the Hopfbifurcation diagram presents a ring shape However dy-namic range of the fixed limit cycle in Figure 8 is a littlelarger compared with Figure 6 )is is mainly because thefriction is bigger and as a result the fluctuation of theresponse rolling force ΔPi under the friction coefficient of00401 is much bigger

)e numerical solution of work roll displacement underthe Hopf bifurcation point of μlowast 00420 is shown in Fig-ure 9)e roll displacement will disperse outward all the timeBesides the phase diagram of the work roll will disperseoutward of the cycle and the radius curvature of the phasetrajectory increases gradually It illustrates that the mill vi-bration system is unstable and mill chatter is produced

When μlowast lt 00048 the roll system is unstable and whenμlowast 00048 the system is stable So the Hopf bifurcation ofμlowast 00048 is the subcritical bifurcation point and μlowast

00401 is the supercritical bifurcation pointMany researchers attribute the change of stability in

rolling mill system to the sudden change of the rolling forcefluctuation in dynamic equilibrium equation [11 25 35])e optimal setting range of friction coefficient is 00048ndash00401 and mill chatter appears when the friction is eithertoo high or too low )is is mainly due to the fluctuation ofrolling force When the friction is high with the increase offriction coefficient the neutral angle becomes bigger andthen the neutral point shifts to the entry side so the distancebetween the entry and neutral point becomes shorter andthe variation of the entry speed decreases [11] However theexit forward slip and exit speed fluctuation will increase

Table 2 Structure specifications of the UCM mill

Parameters ValueWork roll mass Mw (Kg) 3914Intermediate roll mass Mim (Kg) 5410Backup roll mass Mb (Kg) 31330Work roll contact spring constant Kw (N mminus1) 091times 1010

Intermediate roll spring constant Kim (N mminus1) 103times1010

Backup roll spring constant Kb (N mminus1) 585times1010

Work roll damping coefficient x directionCwx (Nmiddots mminus1) 500times104

Work roll damping coefficient in y directionCwy (Nmiddots mminus1) 164times105

Intermediate roll damping coefficient Cim (Nmiddots mminus1) 570times105

Backup roll damping coefficient Cb (Nmiddots mminus1) 510times106

Work roll rotational inertia mass Jw (Kgmiddotm2) 1381Work roll rotational spring constant Kwj (Nmiddotmmiddotradminus1) 790times106

Work roll rotational damping coefficientCwj (Nmiddotmmiddots radminus1) 4178

Work roll diameter Rw (mm) 2125Intermediate roll diameter Rim (mm) 245Backup roll diameter Rb (mm) 650Work roll installation offset e (mm) 5

8 Shock and Vibration

ndash80 times 10ndash5

ndash40 times 10ndash5

00

40 times 10ndash5

80 times 10ndash5W

ork

roll

disp

lace

men

t (m

)

00 02 04 06 08 10Time (s)

(a)

ndash002

ndash001

000

001

002 First point

x 4 (m

s)

00 80 times 10ndash540 times 10ndash5ndash40 times 10ndash5ndash80 times 10ndash5

x3 (m)

Last point

(b)

Figure 5 Dynamic response of the work roll displacement and the phase diagram under μlowast 00040 (a) time history of work rolldisplacement (b) phase diagram

00

Wor

k ro

ll di

spla

cem

ent (

m)

00 02 04 06 08 10Time (s)

10 times 10ndash7

50 times 10ndash8

ndash50 times 10ndash8

ndash10 times 10ndash7

(a)

00

12 times 10ndash7

60 times 10ndash8

ndash60 times 10ndash8

ndash12 times 10ndash7

ndash60 times 10ndash9 60 times 10ndash9ndash30 times 10ndash9 30 times 10ndash9

x 4 (m

s)

00x3 (m)

Last

poin

t

(b)

Figure 6 Dynamic response of the work roll displacement and the phase diagram under μlowast 00048 (a) time history of work rolldisplacement (b) phase diagram

ndash10 times 10ndash6

ndash50 times 10ndash7

00

50 times 10ndash7

10 times 10ndash6

Wor

k ro

ll di

spla

cem

ent (

m)

00 02 04 06 08 10Time (s)

(a)

ndash10 times 10ndash6 ndash50 times 10ndash7 00 50 times 10ndash7 10 times 10ndash6ndash80 times 10ndash4

ndash40 times 10ndash4

00

40 times 10ndash4

x 4 (m

s)

x3 (m)

Firs

t poi

nt

(b)

Figure 7 Dynamic response of the work roll displacement and the phase diagram under μlowast 00300 (a) time history of work rolldisplacement (b) phase diagram

Shock and Vibration 9

Based on equation (12) the tension variation at the entryposition (dTb) and exit position (dTf) is smaller comparedwith low friction coefficient rolling condition so the fluc-tuation of the response rolling force ΔPi increases and thestability of the stand decreases Meanwhile when the frictionis quite small zPzhout will increase with the decrease offriction coefficient F43 in equation (29) decreases so theinstantaneous dynamic structure spring constant decreasesand the fluctuation of the response rolling force ΔPi causedby the strip exit thickness variation will increase so thestability of the stand is decreased

43 Stable Range Analysis of Other Rolling Parameters

431 Calculation of Limited Rolling Speed Based on thiscoupled vertical-horizontal-torsional vibration analysismodel rolling speed has been chosen as the Hopf bifurcationparameter to obtain the limited stability rolling speed of the

UCM tandem cold rolling mill and other rolling conditionsare shown in Table 1 )e actual bifurcation point of rollingspeed is 251ms Figure 10 shows the time history of workroll displacement under the rolling speed of 252msAccording to Figure 10 the displacement amplitude ofthe work roll shows a divergence trend and the self-excitedvibration happens )e critical rolling speed calculated bythe proposed chatter model is 251ms

Figure 11 shows the tension and roll gap variation in No4 and No 5 stands obtained from the Production DataAcquisition (PDA) system As is shown in Figure 11(b)when the rolling speed is 221ms the back and front tensionof the No 5 stand fluctuate within a small range and the millstand is stable However when the rolling speed is raised to230ms the back and front tension of the No 5 standfluctuate more violently and the front tension of the No 5stand begins to diverge as shown in Figure 11(a) It is foundthat the roll gap of No 4 and No 5 stands fluctuates moreviolently by comparing Figures 11(c) and 11(d) Meanwhile

00 02 04 06 08 10

00

20 times 10ndash7

ndash20 times 10ndash7Wor

k ro

ll di

spla

cem

ent (

m)

Time (s)

(a)

00

00

10 times 10ndash4

50 times 10ndash5

ndash50 times 10ndash5

ndash10 times 10ndash4

ndash20 times 10ndash7 20 times 10ndash7

Last point

x 4 (m

s)

x3 (m)

First point

(b)

Figure 8 Dynamic response of the work roll displacement and the phase diagram under μlowast 00401 (a) time history of work rolldisplacement (b) phase diagram

00 02 04 06 08 10

00

20 times 10ndash6

ndash20 times 10ndash6Wor

k ro

ll di

spla

cem

ent (

m)

Time (s)

(a)

ndash20 times 10ndash6 00 20 times 10ndash6

00

10 times 10ndash3

50 times 10ndash4

ndash50 times 10ndash4

ndash10 times 10ndash3

x 4 (m

s)

x3 (m)

Last point

(b)

Figure 9 Dynamic response of the work roll displacement and the phase diagram under μlowast 00420 (a) time history of work rolldisplacement (b) phase diagram

10 Shock and Vibration

a sudden and harsh vibration noise will be heard in the No 5stand and finally the strip breaks It shows mill chatterappears under the speed of 230ms and the actual criticalrolling speed is 230ms

)e calculated critical rolling speed is 251ms while theactual critical rolling speed is 230ms )ere is a certainrelative error between the calculated critical speed and actualcritical rolling speed )is is due to the small coefficientdifference in Jacobian matrix With the change of rollingspeed the actual value of the tension and friction is changedAnd the calculated value of tension factor nt and frictioncoefficient in zPzh and zMzh is a little different with theactual value So the calculated limited rolling speed is a littlebigger than the actual value However by comparing thecritical rolling speed calculated by the proposed model withthe actual critical rolling speed it can be seen that theproposed model is still accurate

)e vibration is largely influenced by the friction co-efficient and rolling speed Based on equation (13) Figure 12shows the friction coefficient response of No 5 stand versusspeed Despite the changes of speed the friction coefficient isstill in the stability range of 00048ndash00401 It is divided intotwo regions In region 1 both the friction coefficient androlling speed are in the stable range In region 2 the frictioncoefficient is still in the stable range while the speed exceedsthe limited speed rangeWhen the rolling speed is lower than251ms the mill system is stable without changing thelubrication conditions And when the rolling speed is higherthan 251ms the stand is unstable However the frictioncoefficient is in the middle value of stable range under thespeed is 251ms so mill chatter might be eliminated byreducing the basic friction coefficient (by improving thelubrication characteristics such as increasing the concen-tration of emulsion increasing spray flow and improvingthe property of the base oil)

432 Calculation of Maximum Reduction Assuming theentry thickness is constant the exit thickness is variable Andthe exit thickness has been chosen as the Hopf bifurcationparameter Other calculated rolling conditions are shown in

Table 1 )en the calculated bifurcation point of exitthickness is 0168mm)e optimal setting range of reductionfor No 5 stand is 0ndash44When the reduction of No 5 stand is44 the increase of reduction leads to a lower mill standstability )e reason is also related to the increase of rollingforce variation under the larger reduction rolling conditions

As can be seen in Table 1 in normal rolling schedule thereduction rate is 333 And the max reduction is up to 44In reality considering there are other factors that influencethe stand stability such as rolling speed and lubricationconditions the max reduction is lower than 44 In generalthe effect of the reduction in the mill stability is lower thanthe improvement of friction coefficient

433 Front Tension Stable Range When the work roll ve-locity is 221ms the calculated bifurcation point of exittension is 15ndash133MPa Small front tension means that therolling force will be quite large So under the same excitationinterferences the variation of rolling force will be muchbigger and the stand stability is decreased When the fronttension is big enough tension factor nt decreases leading toa larger value in zPzh and zMzh As a result the fluc-tuation of rolling force and roll torque increases and thestability of the system is reduced as well So the influence oftension on the mill stability is smaller than other parameters

)ese results can be used to estimate mill chatter in acertain stand And the stable range of friction reduction andtension can be used to adjust the rolling schedule in theUCM mill to avoid mill chatter under high rolling speed

5 Conclusion

(1) A new multiple degrees of freedom and nonlineardynamic vertical-horizontal-torsional coupled millchatter model for UCM cold rolling mill is estab-lished In the dynamic rolling process model thetime-varying coupling effect between rolling forceroll torque variation and fluctuation of stripthickness and tension is formulated In the vibrationmodel the movement of work roll in both vertical-horizontal-torsional directions and the multirollcharacteristics in the UCM rolling mill are consid-ered Finally the dynamic relationship of rollmovement responses and dynamic changes in theroll gap is formulated

(2) Selecting the friction coefficient as the Hopf bi-furcation parameter and by using RouthndashHurwitzdeterminant the Hopf bifurcation point of the millvibration system is calculated and the bifurcationcharacteristics are analyzed Research shows thateither too low or too high friction coefficient in-creases the probability of mill chatter When thefriction coefficient is extremely low the mill systemproduces a stable limit cycle With the frictioncontinuing to increase the phase trajectory willconverge toward the origin point However the limitcycle of phase diagram disappears and dispersesoutward under high friction rolling conditions

00 02 04 06 08 10ndash2 times 10ndash7

ndash1 times 10ndash7

0

1 times 10ndash7

2 times 10ndash7

Wor

k ro

ll di

spla

cem

ent (

m)

Time (s)

Figure 10 Work roll time history displacement under the rollingspeed of 252ms

Shock and Vibration 11

0 1 2 30

20

40

60

80

Fron

t ten

sion

(kN

)

Back tension

Back

tens

ion

(kN

)

Time (s)

0

20

40

60

Front tension

(a)

0 1 2 30

20

40

60

80

Fron

t ten

sion

(kN

)

Back tension

Back

tens

ion

(kN

)

Time (s)

0

20

40

60

Front tension

(b)

0 1 2 3

015

018

021

024

Roll

gap

of N

o 5

stan

d (m

m)

Roll gap of No 4 stand

Roll

gap

of N

o 4

stan

d (m

m)

Time (s)

268

272

276

280

Roll gap of No 5 stand

(c)

0 1 2 3

030

033

036

039

Roll

gap

of N

o 5

stan

d (m

m)

Roll gap of No 4 stand

Roll

gap

of N

o4

stand

(mm

)

Time (s)

272

276

280

284

Roll gap of No 5 stand

(d)

Figure 11 Tension and roll gap time history (a) back and front tension of No 5 stand under the speed of 230ms (b) back and fronttension of No 5 stand under the speed of 221ms (c) roll gap of No 4 and No 5 stands under the speed of 230ms (d) roll gap of No 4 andNo 5 stands under the speed of 221ms

0 10 20 30

0020

0025

0030

0035

0040

0045

(2)

(25 002101)

Fric

tion

coeffi

cien

t

Speed (ms)

x = 251

(1)

Figure 12 Friction coefficient response of No 5 stand versus speed

12 Shock and Vibration

(3) Based on the coupled mill chatter model the millsystemrsquos stable limited rolling speed stable range ofreduction and front tension are investigatedWithout changing the rolling speed the most ef-fective way to eliminate mill chatter is to change thelubrication conditions and adjust the reductionschedule And the effect of tension on the vibration isquite limited )ese methods can be used to improvethe chatter phenomena in rolling mill under highrolling speed )e proposed vibration model canalso be used to design an optimum rolling schedulein the UCM tandem cold rolling mill and to de-termine the appearance of mill chatter under certainrolling conditions

Data Availability

)e rolling conditions data used to support the findings ofthis study are included within the article

Conflicts of Interest

)e authors declare that they have no conflicts of interestregarding the publication of this manuscript

Acknowledgments

)is work was supported by the National Natural ScienceFoundation of China (51774084 and 51634002) the NationalKey RampD Program of China (2017YFB0304100) the Fun-damental Research Funds for the Central Universities(N170708020) and the Open Research Fund from the StateKey Laboratory of Rolling and Automation NortheasternUniversity (2017RALKFKT006)

References

[1] I S Yun W R D Wilson and K F Ehmann ldquoReview ofchatter studies in cold rollingrdquo International Journal ofMachine Tools and Manufacture vol 38 no 12pp 1499ndash1530 1998

[2] I-S Yun W R D Wilson and K F Ehmann ldquoChatter in thestrip rolling process Part 1 dynamic model of rollingrdquoJournal of Manufacturing Science and Engineering vol 120no 2 pp 330ndash336 1998

[3] J Tlusty G Chandra S Critchley and D Paton ldquoChatter incold rollingrdquo CIRP Annals vol 31 no 1 pp 195ndash199 1982

[4] N M Reza F M Reza M Salimi and H Shojaei ldquoExper-imental investigations and ALE finite element methodanalysis of chatter in cold strip rollingrdquo ISIJ Internationalvol 52 no 12 pp 2245ndash2253 2012

[5] M R Niroomand M R Forouzan M Salimi and H ShojaeildquoExperimental investigations and ALE finite element methodanalysis of chatter in cold strip rollingrdquo ISIJ Internationalvol 52 no 12 pp 2245ndash2253 2012

[6] M R Niroomand M R Forouzan and M Salimi ldquo)eo-retical and experimental analysis of chatter in tandem coldrolling mills based on wave propagation theoryrdquo ISIJ In-ternational vol 55 no 3 pp 637ndash646 2015

[7] S Kapil P Eberhard and S K Dwivedy ldquoNonlinear dynamicanalysis of a parametrically excited cold rolling millrdquo Journal

of Manufacturing Science and Engineering vol 136 no 4p 041012 2014

[8] Y Kimura Y Sodani N Nishiura N Ikeuchi and Y MiharaldquoAnalysis of chaffer in tandem cold rolling millsrdquo ISIJ In-ternational vol 43 no 1 pp 77ndash84 2003

[9] N Fujita Y Kimura K Kobayashi et al ldquoDynamic control oflubrication characteristics in high speed tandem cold rollingrdquoJournal of Materials Processing Technology vol 229pp 407ndash416 2016

[10] A Heidari and M R Forouzan ldquoOptimization of cold rollingprocess parameters in order to increasing rolling speedlimited by chatter vibrationsrdquo Journal of Advanced Researchvol 4 no 1 pp 27ndash34 2013

[11] A Heidari M R Forouzan and S Akbarzadeh ldquoEffect offriction on tandem cold rolling mills chatteringrdquo ISIJ In-ternational vol 54 no 10 pp 2349ndash2356 2014

[12] A Heidari M R Forouzan and S Akbarzadeh ldquoDevelop-ment of a rolling chatter model considering unsteady lubri-cationrdquo ISIJ International vol 54 no 1 pp 165ndash170 2014

[13] A Heidari M R Forouzan and M R Niroomand ldquoDe-velopment and evaluation of friction models for chattersimulation in cold strip rollingrdquo International Journal ofAdvanced Manufacturing Technology vol 96 no 4 pp 1ndash212018

[14] Y Kim C-W Kim S Lee and H Park ldquoDynamic modelingand numerical analysis of a cold rolling millrdquo InternationalJournal of Precision Engineering and Manufacturing vol 14no 3 pp 407ndash413 2013

[15] P V Krot ldquoNonlinear vibrations and backlashes diagnosticsin the rolling mills drive trainsrdquo in Proceedings of the 6thEUROMECH Nonlinear Dynamics Conference Saint Peters-burg Russia July 2008

[16] S Liu B Sun S Zhao J Li and W Zhang ldquoStronglynonlinear dynamics of torsional vibration for rolling millrsquoselectromechanical coupling systemrdquo Steel Research In-ternational vol 86 no 9 pp 984ndash992 2015

[17] S Liu J Wang J Liu and Y Li ldquoNonlinear parametricallyexcited vibration and active control of gear pair system withtime-varying characteristicrdquo Chinese Physics B vol 24 no 10pp 303ndash311 2015

[18] B D Hassard N D Kazarinoff and Y H Wan eory andApplications of Hopf Bifurcation Cambridge University PressCambridge UK 1981

[19] B H K Lee S J Price and Y SWong ldquoNonlinear aeroelasticanalysis of airfoils bifurcation and chaosrdquo Progress inAerospace Sciences vol 35 no 3 pp 205ndash334 1999

[20] K Nishimura T Ikeda and Y Harata ldquoLocalization phe-nomena in torsional rotating shaft systems with multiplecentrifugal pendulum vibration absorbersrdquo Nonlinear Dy-namics vol 83 no 3 pp 1705ndash1726 2015

[21] T Zhang and H Dai ldquoBifurcation analysis of high-speedrailway wheel-setrdquo Nonlinear Dynamics vol 83 no 3pp 1511ndash1528 2015

[22] Z Y Gao Y Zang and T Han ldquoHopf bifurcation andstability analysis on the mill drive systemrdquo Applied Mechanicsamp Materials vol 121-126 pp 1514ndash1520 2012

[23] L Zeng Y Zang and Z Gao ldquoHopf bifurcation control forrolling mill multiple-mode-coupling vibration under non-linear frictionrdquo Journal of Vibration and Acoustics vol 139no 6 article 061015 2017

[24] S Liu Z Wang J Wang and H Li ldquoSliding bifurcationresearch of a horizontal-torsional coupled main drive systemof rolling millrdquo Nonlinear Dynamics vol 83 pp 441ndash4552015

Shock and Vibration 13

[25] Y Peng M Zhang J-L Sun and Y Zhang ldquoExperimentaland numerical investigation on the roll system swing vibra-tion characteristics of a hot rolling millrdquo ISIJ Internationalvol 57 no 9 pp 1567ndash1576 2017

[26] J Pittner and M A Simaan ldquoOptimal control of continuoustandem cold metal rollingrdquo in Proceedings of 2008 AmericanControl Conference Seattle WA USA June 2008

[27] R B Sims and D F S Arthur ldquoSpeed dependent variables incold strip rollingrdquo Journal of Iron and Steel Institute vol 172no 3 pp 285ndash295 1952

[28] S Z Chen D H Zhang J Sun J S Wang and J SongldquoOnline calculation model of rolling force for cold rolling millbased on numerical integrationrdquo in Proceedings of the 24thChinese Control and Decision Conference (CCDC) TaiyuanChina May 2012

[29] D R Bland and H Ford ldquo)e calculation of roll force andtorque in cold strip rolling with tensionsrdquo Proceedings of theInstitution of Mechanical Engineers vol 159 no 1 pp 144ndash163 2006

[30] R Hill ldquoRelations between roll-force torque and the appliedtensions in strip-rollingrdquo Proceedings of the Institution ofMechanical Engineers vol 163 no 1 pp 135ndash140 2006

[31] R E Johnson and Q Qi ldquoChatter dynamics in sheet rollingrdquoInternational Journal of Mechanical Sciences vol 36 no 7pp 617ndash630 1994

[32] Z Gao L Bai and Q Li ldquoResearch on critical rolling speed ofself-excited vibration in the tandem rolling process of thinstriprdquo Journal of Mechanical Engineering vol 53 no 12pp 118ndash132 2017

[33] R C Dorf and R H Bishop Modern Control SystemsUniversity of California Davis CA USA 12th edition 2010

[34] W M Liu ldquoCriterion of hopf bifurcations without usingeigenvaluesrdquo Journal of Mathematical Analysis and Appli-cations vol 182 no 1 pp 250ndash256 1994

[35] L Zeng Y Zang and Z Gao ldquoMultiple-modal-couplingmodeling and stability analysis of cold rolling mill vibra-tionrdquo Shock and Vibration vol 2016 Article ID 234738626 pages 2016

14 Shock and Vibration

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Submit your manuscripts atwwwhindawicom

ndash80 times 10ndash5

ndash40 times 10ndash5

00

40 times 10ndash5

80 times 10ndash5W

ork

roll

disp

lace

men

t (m

)

00 02 04 06 08 10Time (s)

(a)

ndash002

ndash001

000

001

002 First point

x 4 (m

s)

00 80 times 10ndash540 times 10ndash5ndash40 times 10ndash5ndash80 times 10ndash5

x3 (m)

Last point

(b)

Figure 5 Dynamic response of the work roll displacement and the phase diagram under μlowast 00040 (a) time history of work rolldisplacement (b) phase diagram

00

Wor

k ro

ll di

spla

cem

ent (

m)

00 02 04 06 08 10Time (s)

10 times 10ndash7

50 times 10ndash8

ndash50 times 10ndash8

ndash10 times 10ndash7

(a)

00

12 times 10ndash7

60 times 10ndash8

ndash60 times 10ndash8

ndash12 times 10ndash7

ndash60 times 10ndash9 60 times 10ndash9ndash30 times 10ndash9 30 times 10ndash9

x 4 (m

s)

00x3 (m)

Last

poin

t

(b)

Figure 6 Dynamic response of the work roll displacement and the phase diagram under μlowast 00048 (a) time history of work rolldisplacement (b) phase diagram

ndash10 times 10ndash6

ndash50 times 10ndash7

00

50 times 10ndash7

10 times 10ndash6

Wor

k ro

ll di

spla

cem

ent (

m)

00 02 04 06 08 10Time (s)

(a)

ndash10 times 10ndash6 ndash50 times 10ndash7 00 50 times 10ndash7 10 times 10ndash6ndash80 times 10ndash4

ndash40 times 10ndash4

00

40 times 10ndash4

x 4 (m

s)

x3 (m)

Firs

t poi

nt

(b)

Figure 7 Dynamic response of the work roll displacement and the phase diagram under μlowast 00300 (a) time history of work rolldisplacement (b) phase diagram

Shock and Vibration 9

Based on equation (12) the tension variation at the entryposition (dTb) and exit position (dTf) is smaller comparedwith low friction coefficient rolling condition so the fluc-tuation of the response rolling force ΔPi increases and thestability of the stand decreases Meanwhile when the frictionis quite small zPzhout will increase with the decrease offriction coefficient F43 in equation (29) decreases so theinstantaneous dynamic structure spring constant decreasesand the fluctuation of the response rolling force ΔPi causedby the strip exit thickness variation will increase so thestability of the stand is decreased

43 Stable Range Analysis of Other Rolling Parameters

431 Calculation of Limited Rolling Speed Based on thiscoupled vertical-horizontal-torsional vibration analysismodel rolling speed has been chosen as the Hopf bifurcationparameter to obtain the limited stability rolling speed of the

UCM tandem cold rolling mill and other rolling conditionsare shown in Table 1 )e actual bifurcation point of rollingspeed is 251ms Figure 10 shows the time history of workroll displacement under the rolling speed of 252msAccording to Figure 10 the displacement amplitude ofthe work roll shows a divergence trend and the self-excitedvibration happens )e critical rolling speed calculated bythe proposed chatter model is 251ms

Figure 11 shows the tension and roll gap variation in No4 and No 5 stands obtained from the Production DataAcquisition (PDA) system As is shown in Figure 11(b)when the rolling speed is 221ms the back and front tensionof the No 5 stand fluctuate within a small range and the millstand is stable However when the rolling speed is raised to230ms the back and front tension of the No 5 standfluctuate more violently and the front tension of the No 5stand begins to diverge as shown in Figure 11(a) It is foundthat the roll gap of No 4 and No 5 stands fluctuates moreviolently by comparing Figures 11(c) and 11(d) Meanwhile

00 02 04 06 08 10

00

20 times 10ndash7

ndash20 times 10ndash7Wor

k ro

ll di

spla

cem

ent (

m)

Time (s)

(a)

00

00

10 times 10ndash4

50 times 10ndash5

ndash50 times 10ndash5

ndash10 times 10ndash4

ndash20 times 10ndash7 20 times 10ndash7

Last point

x 4 (m

s)

x3 (m)

First point

(b)

Figure 8 Dynamic response of the work roll displacement and the phase diagram under μlowast 00401 (a) time history of work rolldisplacement (b) phase diagram

00 02 04 06 08 10

00

20 times 10ndash6

ndash20 times 10ndash6Wor

k ro

ll di

spla

cem

ent (

m)

Time (s)

(a)

ndash20 times 10ndash6 00 20 times 10ndash6

00

10 times 10ndash3

50 times 10ndash4

ndash50 times 10ndash4

ndash10 times 10ndash3

x 4 (m

s)

x3 (m)

Last point

(b)

Figure 9 Dynamic response of the work roll displacement and the phase diagram under μlowast 00420 (a) time history of work rolldisplacement (b) phase diagram

10 Shock and Vibration

a sudden and harsh vibration noise will be heard in the No 5stand and finally the strip breaks It shows mill chatterappears under the speed of 230ms and the actual criticalrolling speed is 230ms

)e calculated critical rolling speed is 251ms while theactual critical rolling speed is 230ms )ere is a certainrelative error between the calculated critical speed and actualcritical rolling speed )is is due to the small coefficientdifference in Jacobian matrix With the change of rollingspeed the actual value of the tension and friction is changedAnd the calculated value of tension factor nt and frictioncoefficient in zPzh and zMzh is a little different with theactual value So the calculated limited rolling speed is a littlebigger than the actual value However by comparing thecritical rolling speed calculated by the proposed model withthe actual critical rolling speed it can be seen that theproposed model is still accurate

)e vibration is largely influenced by the friction co-efficient and rolling speed Based on equation (13) Figure 12shows the friction coefficient response of No 5 stand versusspeed Despite the changes of speed the friction coefficient isstill in the stability range of 00048ndash00401 It is divided intotwo regions In region 1 both the friction coefficient androlling speed are in the stable range In region 2 the frictioncoefficient is still in the stable range while the speed exceedsthe limited speed rangeWhen the rolling speed is lower than251ms the mill system is stable without changing thelubrication conditions And when the rolling speed is higherthan 251ms the stand is unstable However the frictioncoefficient is in the middle value of stable range under thespeed is 251ms so mill chatter might be eliminated byreducing the basic friction coefficient (by improving thelubrication characteristics such as increasing the concen-tration of emulsion increasing spray flow and improvingthe property of the base oil)

432 Calculation of Maximum Reduction Assuming theentry thickness is constant the exit thickness is variable Andthe exit thickness has been chosen as the Hopf bifurcationparameter Other calculated rolling conditions are shown in

Table 1 )en the calculated bifurcation point of exitthickness is 0168mm)e optimal setting range of reductionfor No 5 stand is 0ndash44When the reduction of No 5 stand is44 the increase of reduction leads to a lower mill standstability )e reason is also related to the increase of rollingforce variation under the larger reduction rolling conditions

As can be seen in Table 1 in normal rolling schedule thereduction rate is 333 And the max reduction is up to 44In reality considering there are other factors that influencethe stand stability such as rolling speed and lubricationconditions the max reduction is lower than 44 In generalthe effect of the reduction in the mill stability is lower thanthe improvement of friction coefficient

433 Front Tension Stable Range When the work roll ve-locity is 221ms the calculated bifurcation point of exittension is 15ndash133MPa Small front tension means that therolling force will be quite large So under the same excitationinterferences the variation of rolling force will be muchbigger and the stand stability is decreased When the fronttension is big enough tension factor nt decreases leading toa larger value in zPzh and zMzh As a result the fluc-tuation of rolling force and roll torque increases and thestability of the system is reduced as well So the influence oftension on the mill stability is smaller than other parameters

)ese results can be used to estimate mill chatter in acertain stand And the stable range of friction reduction andtension can be used to adjust the rolling schedule in theUCM mill to avoid mill chatter under high rolling speed

5 Conclusion

(1) A new multiple degrees of freedom and nonlineardynamic vertical-horizontal-torsional coupled millchatter model for UCM cold rolling mill is estab-lished In the dynamic rolling process model thetime-varying coupling effect between rolling forceroll torque variation and fluctuation of stripthickness and tension is formulated In the vibrationmodel the movement of work roll in both vertical-horizontal-torsional directions and the multirollcharacteristics in the UCM rolling mill are consid-ered Finally the dynamic relationship of rollmovement responses and dynamic changes in theroll gap is formulated

(2) Selecting the friction coefficient as the Hopf bi-furcation parameter and by using RouthndashHurwitzdeterminant the Hopf bifurcation point of the millvibration system is calculated and the bifurcationcharacteristics are analyzed Research shows thateither too low or too high friction coefficient in-creases the probability of mill chatter When thefriction coefficient is extremely low the mill systemproduces a stable limit cycle With the frictioncontinuing to increase the phase trajectory willconverge toward the origin point However the limitcycle of phase diagram disappears and dispersesoutward under high friction rolling conditions

00 02 04 06 08 10ndash2 times 10ndash7

ndash1 times 10ndash7

0

1 times 10ndash7

2 times 10ndash7

Wor

k ro

ll di

spla

cem

ent (

m)

Time (s)

Figure 10 Work roll time history displacement under the rollingspeed of 252ms

Shock and Vibration 11

0 1 2 30

20

40

60

80

Fron

t ten

sion

(kN

)

Back tension

Back

tens

ion

(kN

)

Time (s)

0

20

40

60

Front tension

(a)

0 1 2 30

20

40

60

80

Fron

t ten

sion

(kN

)

Back tension

Back

tens

ion

(kN

)

Time (s)

0

20

40

60

Front tension

(b)

0 1 2 3

015

018

021

024

Roll

gap

of N

o 5

stan

d (m

m)

Roll gap of No 4 stand

Roll

gap

of N

o 4

stan

d (m

m)

Time (s)

268

272

276

280

Roll gap of No 5 stand

(c)

0 1 2 3

030

033

036

039

Roll

gap

of N

o 5

stan

d (m

m)

Roll gap of No 4 stand

Roll

gap

of N

o4

stand

(mm

)

Time (s)

272

276

280

284

Roll gap of No 5 stand

(d)

Figure 11 Tension and roll gap time history (a) back and front tension of No 5 stand under the speed of 230ms (b) back and fronttension of No 5 stand under the speed of 221ms (c) roll gap of No 4 and No 5 stands under the speed of 230ms (d) roll gap of No 4 andNo 5 stands under the speed of 221ms

0 10 20 30

0020

0025

0030

0035

0040

0045

(2)

(25 002101)

Fric

tion

coeffi

cien

t

Speed (ms)

x = 251

(1)

Figure 12 Friction coefficient response of No 5 stand versus speed

12 Shock and Vibration

(3) Based on the coupled mill chatter model the millsystemrsquos stable limited rolling speed stable range ofreduction and front tension are investigatedWithout changing the rolling speed the most ef-fective way to eliminate mill chatter is to change thelubrication conditions and adjust the reductionschedule And the effect of tension on the vibration isquite limited )ese methods can be used to improvethe chatter phenomena in rolling mill under highrolling speed )e proposed vibration model canalso be used to design an optimum rolling schedulein the UCM tandem cold rolling mill and to de-termine the appearance of mill chatter under certainrolling conditions

Data Availability

)e rolling conditions data used to support the findings ofthis study are included within the article

Conflicts of Interest

)e authors declare that they have no conflicts of interestregarding the publication of this manuscript

Acknowledgments

)is work was supported by the National Natural ScienceFoundation of China (51774084 and 51634002) the NationalKey RampD Program of China (2017YFB0304100) the Fun-damental Research Funds for the Central Universities(N170708020) and the Open Research Fund from the StateKey Laboratory of Rolling and Automation NortheasternUniversity (2017RALKFKT006)

References

[1] I S Yun W R D Wilson and K F Ehmann ldquoReview ofchatter studies in cold rollingrdquo International Journal ofMachine Tools and Manufacture vol 38 no 12pp 1499ndash1530 1998

[2] I-S Yun W R D Wilson and K F Ehmann ldquoChatter in thestrip rolling process Part 1 dynamic model of rollingrdquoJournal of Manufacturing Science and Engineering vol 120no 2 pp 330ndash336 1998

[3] J Tlusty G Chandra S Critchley and D Paton ldquoChatter incold rollingrdquo CIRP Annals vol 31 no 1 pp 195ndash199 1982

[4] N M Reza F M Reza M Salimi and H Shojaei ldquoExper-imental investigations and ALE finite element methodanalysis of chatter in cold strip rollingrdquo ISIJ Internationalvol 52 no 12 pp 2245ndash2253 2012

[5] M R Niroomand M R Forouzan M Salimi and H ShojaeildquoExperimental investigations and ALE finite element methodanalysis of chatter in cold strip rollingrdquo ISIJ Internationalvol 52 no 12 pp 2245ndash2253 2012

[6] M R Niroomand M R Forouzan and M Salimi ldquo)eo-retical and experimental analysis of chatter in tandem coldrolling mills based on wave propagation theoryrdquo ISIJ In-ternational vol 55 no 3 pp 637ndash646 2015

[7] S Kapil P Eberhard and S K Dwivedy ldquoNonlinear dynamicanalysis of a parametrically excited cold rolling millrdquo Journal

of Manufacturing Science and Engineering vol 136 no 4p 041012 2014

[8] Y Kimura Y Sodani N Nishiura N Ikeuchi and Y MiharaldquoAnalysis of chaffer in tandem cold rolling millsrdquo ISIJ In-ternational vol 43 no 1 pp 77ndash84 2003

[9] N Fujita Y Kimura K Kobayashi et al ldquoDynamic control oflubrication characteristics in high speed tandem cold rollingrdquoJournal of Materials Processing Technology vol 229pp 407ndash416 2016

[10] A Heidari and M R Forouzan ldquoOptimization of cold rollingprocess parameters in order to increasing rolling speedlimited by chatter vibrationsrdquo Journal of Advanced Researchvol 4 no 1 pp 27ndash34 2013

[11] A Heidari M R Forouzan and S Akbarzadeh ldquoEffect offriction on tandem cold rolling mills chatteringrdquo ISIJ In-ternational vol 54 no 10 pp 2349ndash2356 2014

[12] A Heidari M R Forouzan and S Akbarzadeh ldquoDevelop-ment of a rolling chatter model considering unsteady lubri-cationrdquo ISIJ International vol 54 no 1 pp 165ndash170 2014

[13] A Heidari M R Forouzan and M R Niroomand ldquoDe-velopment and evaluation of friction models for chattersimulation in cold strip rollingrdquo International Journal ofAdvanced Manufacturing Technology vol 96 no 4 pp 1ndash212018

[14] Y Kim C-W Kim S Lee and H Park ldquoDynamic modelingand numerical analysis of a cold rolling millrdquo InternationalJournal of Precision Engineering and Manufacturing vol 14no 3 pp 407ndash413 2013

[15] P V Krot ldquoNonlinear vibrations and backlashes diagnosticsin the rolling mills drive trainsrdquo in Proceedings of the 6thEUROMECH Nonlinear Dynamics Conference Saint Peters-burg Russia July 2008

[16] S Liu B Sun S Zhao J Li and W Zhang ldquoStronglynonlinear dynamics of torsional vibration for rolling millrsquoselectromechanical coupling systemrdquo Steel Research In-ternational vol 86 no 9 pp 984ndash992 2015

[17] S Liu J Wang J Liu and Y Li ldquoNonlinear parametricallyexcited vibration and active control of gear pair system withtime-varying characteristicrdquo Chinese Physics B vol 24 no 10pp 303ndash311 2015

[18] B D Hassard N D Kazarinoff and Y H Wan eory andApplications of Hopf Bifurcation Cambridge University PressCambridge UK 1981

[19] B H K Lee S J Price and Y SWong ldquoNonlinear aeroelasticanalysis of airfoils bifurcation and chaosrdquo Progress inAerospace Sciences vol 35 no 3 pp 205ndash334 1999

[20] K Nishimura T Ikeda and Y Harata ldquoLocalization phe-nomena in torsional rotating shaft systems with multiplecentrifugal pendulum vibration absorbersrdquo Nonlinear Dy-namics vol 83 no 3 pp 1705ndash1726 2015

[21] T Zhang and H Dai ldquoBifurcation analysis of high-speedrailway wheel-setrdquo Nonlinear Dynamics vol 83 no 3pp 1511ndash1528 2015

[22] Z Y Gao Y Zang and T Han ldquoHopf bifurcation andstability analysis on the mill drive systemrdquo Applied Mechanicsamp Materials vol 121-126 pp 1514ndash1520 2012

[23] L Zeng Y Zang and Z Gao ldquoHopf bifurcation control forrolling mill multiple-mode-coupling vibration under non-linear frictionrdquo Journal of Vibration and Acoustics vol 139no 6 article 061015 2017

[24] S Liu Z Wang J Wang and H Li ldquoSliding bifurcationresearch of a horizontal-torsional coupled main drive systemof rolling millrdquo Nonlinear Dynamics vol 83 pp 441ndash4552015

Shock and Vibration 13

[25] Y Peng M Zhang J-L Sun and Y Zhang ldquoExperimentaland numerical investigation on the roll system swing vibra-tion characteristics of a hot rolling millrdquo ISIJ Internationalvol 57 no 9 pp 1567ndash1576 2017

[26] J Pittner and M A Simaan ldquoOptimal control of continuoustandem cold metal rollingrdquo in Proceedings of 2008 AmericanControl Conference Seattle WA USA June 2008

[27] R B Sims and D F S Arthur ldquoSpeed dependent variables incold strip rollingrdquo Journal of Iron and Steel Institute vol 172no 3 pp 285ndash295 1952

[28] S Z Chen D H Zhang J Sun J S Wang and J SongldquoOnline calculation model of rolling force for cold rolling millbased on numerical integrationrdquo in Proceedings of the 24thChinese Control and Decision Conference (CCDC) TaiyuanChina May 2012

[29] D R Bland and H Ford ldquo)e calculation of roll force andtorque in cold strip rolling with tensionsrdquo Proceedings of theInstitution of Mechanical Engineers vol 159 no 1 pp 144ndash163 2006

[30] R Hill ldquoRelations between roll-force torque and the appliedtensions in strip-rollingrdquo Proceedings of the Institution ofMechanical Engineers vol 163 no 1 pp 135ndash140 2006

[31] R E Johnson and Q Qi ldquoChatter dynamics in sheet rollingrdquoInternational Journal of Mechanical Sciences vol 36 no 7pp 617ndash630 1994

[32] Z Gao L Bai and Q Li ldquoResearch on critical rolling speed ofself-excited vibration in the tandem rolling process of thinstriprdquo Journal of Mechanical Engineering vol 53 no 12pp 118ndash132 2017

[33] R C Dorf and R H Bishop Modern Control SystemsUniversity of California Davis CA USA 12th edition 2010

[34] W M Liu ldquoCriterion of hopf bifurcations without usingeigenvaluesrdquo Journal of Mathematical Analysis and Appli-cations vol 182 no 1 pp 250ndash256 1994

[35] L Zeng Y Zang and Z Gao ldquoMultiple-modal-couplingmodeling and stability analysis of cold rolling mill vibra-tionrdquo Shock and Vibration vol 2016 Article ID 234738626 pages 2016

14 Shock and Vibration

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Submit your manuscripts atwwwhindawicom

Based on equation (12) the tension variation at the entryposition (dTb) and exit position (dTf) is smaller comparedwith low friction coefficient rolling condition so the fluc-tuation of the response rolling force ΔPi increases and thestability of the stand decreases Meanwhile when the frictionis quite small zPzhout will increase with the decrease offriction coefficient F43 in equation (29) decreases so theinstantaneous dynamic structure spring constant decreasesand the fluctuation of the response rolling force ΔPi causedby the strip exit thickness variation will increase so thestability of the stand is decreased

43 Stable Range Analysis of Other Rolling Parameters

431 Calculation of Limited Rolling Speed Based on thiscoupled vertical-horizontal-torsional vibration analysismodel rolling speed has been chosen as the Hopf bifurcationparameter to obtain the limited stability rolling speed of the

UCM tandem cold rolling mill and other rolling conditionsare shown in Table 1 )e actual bifurcation point of rollingspeed is 251ms Figure 10 shows the time history of workroll displacement under the rolling speed of 252msAccording to Figure 10 the displacement amplitude ofthe work roll shows a divergence trend and the self-excitedvibration happens )e critical rolling speed calculated bythe proposed chatter model is 251ms

Figure 11 shows the tension and roll gap variation in No4 and No 5 stands obtained from the Production DataAcquisition (PDA) system As is shown in Figure 11(b)when the rolling speed is 221ms the back and front tensionof the No 5 stand fluctuate within a small range and the millstand is stable However when the rolling speed is raised to230ms the back and front tension of the No 5 standfluctuate more violently and the front tension of the No 5stand begins to diverge as shown in Figure 11(a) It is foundthat the roll gap of No 4 and No 5 stands fluctuates moreviolently by comparing Figures 11(c) and 11(d) Meanwhile

00 02 04 06 08 10

00

20 times 10ndash7

ndash20 times 10ndash7Wor

k ro

ll di

spla

cem

ent (

m)

Time (s)

(a)

00

00

10 times 10ndash4

50 times 10ndash5

ndash50 times 10ndash5

ndash10 times 10ndash4

ndash20 times 10ndash7 20 times 10ndash7

Last point

x 4 (m

s)

x3 (m)

First point

(b)

Figure 8 Dynamic response of the work roll displacement and the phase diagram under μlowast 00401 (a) time history of work rolldisplacement (b) phase diagram

00 02 04 06 08 10

00

20 times 10ndash6

ndash20 times 10ndash6Wor

k ro

ll di

spla

cem

ent (

m)

Time (s)

(a)

ndash20 times 10ndash6 00 20 times 10ndash6

00

10 times 10ndash3

50 times 10ndash4

ndash50 times 10ndash4

ndash10 times 10ndash3

x 4 (m

s)

x3 (m)

Last point

(b)

Figure 9 Dynamic response of the work roll displacement and the phase diagram under μlowast 00420 (a) time history of work rolldisplacement (b) phase diagram

10 Shock and Vibration

a sudden and harsh vibration noise will be heard in the No 5stand and finally the strip breaks It shows mill chatterappears under the speed of 230ms and the actual criticalrolling speed is 230ms

)e calculated critical rolling speed is 251ms while theactual critical rolling speed is 230ms )ere is a certainrelative error between the calculated critical speed and actualcritical rolling speed )is is due to the small coefficientdifference in Jacobian matrix With the change of rollingspeed the actual value of the tension and friction is changedAnd the calculated value of tension factor nt and frictioncoefficient in zPzh and zMzh is a little different with theactual value So the calculated limited rolling speed is a littlebigger than the actual value However by comparing thecritical rolling speed calculated by the proposed model withthe actual critical rolling speed it can be seen that theproposed model is still accurate

)e vibration is largely influenced by the friction co-efficient and rolling speed Based on equation (13) Figure 12shows the friction coefficient response of No 5 stand versusspeed Despite the changes of speed the friction coefficient isstill in the stability range of 00048ndash00401 It is divided intotwo regions In region 1 both the friction coefficient androlling speed are in the stable range In region 2 the frictioncoefficient is still in the stable range while the speed exceedsthe limited speed rangeWhen the rolling speed is lower than251ms the mill system is stable without changing thelubrication conditions And when the rolling speed is higherthan 251ms the stand is unstable However the frictioncoefficient is in the middle value of stable range under thespeed is 251ms so mill chatter might be eliminated byreducing the basic friction coefficient (by improving thelubrication characteristics such as increasing the concen-tration of emulsion increasing spray flow and improvingthe property of the base oil)

432 Calculation of Maximum Reduction Assuming theentry thickness is constant the exit thickness is variable Andthe exit thickness has been chosen as the Hopf bifurcationparameter Other calculated rolling conditions are shown in

Table 1 )en the calculated bifurcation point of exitthickness is 0168mm)e optimal setting range of reductionfor No 5 stand is 0ndash44When the reduction of No 5 stand is44 the increase of reduction leads to a lower mill standstability )e reason is also related to the increase of rollingforce variation under the larger reduction rolling conditions

As can be seen in Table 1 in normal rolling schedule thereduction rate is 333 And the max reduction is up to 44In reality considering there are other factors that influencethe stand stability such as rolling speed and lubricationconditions the max reduction is lower than 44 In generalthe effect of the reduction in the mill stability is lower thanthe improvement of friction coefficient

433 Front Tension Stable Range When the work roll ve-locity is 221ms the calculated bifurcation point of exittension is 15ndash133MPa Small front tension means that therolling force will be quite large So under the same excitationinterferences the variation of rolling force will be muchbigger and the stand stability is decreased When the fronttension is big enough tension factor nt decreases leading toa larger value in zPzh and zMzh As a result the fluc-tuation of rolling force and roll torque increases and thestability of the system is reduced as well So the influence oftension on the mill stability is smaller than other parameters

)ese results can be used to estimate mill chatter in acertain stand And the stable range of friction reduction andtension can be used to adjust the rolling schedule in theUCM mill to avoid mill chatter under high rolling speed

5 Conclusion

(1) A new multiple degrees of freedom and nonlineardynamic vertical-horizontal-torsional coupled millchatter model for UCM cold rolling mill is estab-lished In the dynamic rolling process model thetime-varying coupling effect between rolling forceroll torque variation and fluctuation of stripthickness and tension is formulated In the vibrationmodel the movement of work roll in both vertical-horizontal-torsional directions and the multirollcharacteristics in the UCM rolling mill are consid-ered Finally the dynamic relationship of rollmovement responses and dynamic changes in theroll gap is formulated

(2) Selecting the friction coefficient as the Hopf bi-furcation parameter and by using RouthndashHurwitzdeterminant the Hopf bifurcation point of the millvibration system is calculated and the bifurcationcharacteristics are analyzed Research shows thateither too low or too high friction coefficient in-creases the probability of mill chatter When thefriction coefficient is extremely low the mill systemproduces a stable limit cycle With the frictioncontinuing to increase the phase trajectory willconverge toward the origin point However the limitcycle of phase diagram disappears and dispersesoutward under high friction rolling conditions

00 02 04 06 08 10ndash2 times 10ndash7

ndash1 times 10ndash7

0

1 times 10ndash7

2 times 10ndash7

Wor

k ro

ll di

spla

cem

ent (

m)

Time (s)

Figure 10 Work roll time history displacement under the rollingspeed of 252ms

Shock and Vibration 11

0 1 2 30

20

40

60

80

Fron

t ten

sion

(kN

)

Back tension

Back

tens

ion

(kN

)

Time (s)

0

20

40

60

Front tension

(a)

0 1 2 30

20

40

60

80

Fron

t ten

sion

(kN

)

Back tension

Back

tens

ion

(kN

)

Time (s)

0

20

40

60

Front tension

(b)

0 1 2 3

015

018

021

024

Roll

gap

of N

o 5

stan

d (m

m)

Roll gap of No 4 stand

Roll

gap

of N

o 4

stan

d (m

m)

Time (s)

268

272

276

280

Roll gap of No 5 stand

(c)

0 1 2 3

030

033

036

039

Roll

gap

of N

o 5

stan

d (m

m)

Roll gap of No 4 stand

Roll

gap

of N

o4

stand

(mm

)

Time (s)

272

276

280

284

Roll gap of No 5 stand

(d)

Figure 11 Tension and roll gap time history (a) back and front tension of No 5 stand under the speed of 230ms (b) back and fronttension of No 5 stand under the speed of 221ms (c) roll gap of No 4 and No 5 stands under the speed of 230ms (d) roll gap of No 4 andNo 5 stands under the speed of 221ms

0 10 20 30

0020

0025

0030

0035

0040

0045

(2)

(25 002101)

Fric

tion

coeffi

cien

t

Speed (ms)

x = 251

(1)

Figure 12 Friction coefficient response of No 5 stand versus speed

12 Shock and Vibration

(3) Based on the coupled mill chatter model the millsystemrsquos stable limited rolling speed stable range ofreduction and front tension are investigatedWithout changing the rolling speed the most ef-fective way to eliminate mill chatter is to change thelubrication conditions and adjust the reductionschedule And the effect of tension on the vibration isquite limited )ese methods can be used to improvethe chatter phenomena in rolling mill under highrolling speed )e proposed vibration model canalso be used to design an optimum rolling schedulein the UCM tandem cold rolling mill and to de-termine the appearance of mill chatter under certainrolling conditions

Data Availability

)e rolling conditions data used to support the findings ofthis study are included within the article

Conflicts of Interest

)e authors declare that they have no conflicts of interestregarding the publication of this manuscript

Acknowledgments

)is work was supported by the National Natural ScienceFoundation of China (51774084 and 51634002) the NationalKey RampD Program of China (2017YFB0304100) the Fun-damental Research Funds for the Central Universities(N170708020) and the Open Research Fund from the StateKey Laboratory of Rolling and Automation NortheasternUniversity (2017RALKFKT006)

References

[1] I S Yun W R D Wilson and K F Ehmann ldquoReview ofchatter studies in cold rollingrdquo International Journal ofMachine Tools and Manufacture vol 38 no 12pp 1499ndash1530 1998

[2] I-S Yun W R D Wilson and K F Ehmann ldquoChatter in thestrip rolling process Part 1 dynamic model of rollingrdquoJournal of Manufacturing Science and Engineering vol 120no 2 pp 330ndash336 1998

[3] J Tlusty G Chandra S Critchley and D Paton ldquoChatter incold rollingrdquo CIRP Annals vol 31 no 1 pp 195ndash199 1982

[4] N M Reza F M Reza M Salimi and H Shojaei ldquoExper-imental investigations and ALE finite element methodanalysis of chatter in cold strip rollingrdquo ISIJ Internationalvol 52 no 12 pp 2245ndash2253 2012

[5] M R Niroomand M R Forouzan M Salimi and H ShojaeildquoExperimental investigations and ALE finite element methodanalysis of chatter in cold strip rollingrdquo ISIJ Internationalvol 52 no 12 pp 2245ndash2253 2012

[6] M R Niroomand M R Forouzan and M Salimi ldquo)eo-retical and experimental analysis of chatter in tandem coldrolling mills based on wave propagation theoryrdquo ISIJ In-ternational vol 55 no 3 pp 637ndash646 2015

[7] S Kapil P Eberhard and S K Dwivedy ldquoNonlinear dynamicanalysis of a parametrically excited cold rolling millrdquo Journal

of Manufacturing Science and Engineering vol 136 no 4p 041012 2014

[8] Y Kimura Y Sodani N Nishiura N Ikeuchi and Y MiharaldquoAnalysis of chaffer in tandem cold rolling millsrdquo ISIJ In-ternational vol 43 no 1 pp 77ndash84 2003

[9] N Fujita Y Kimura K Kobayashi et al ldquoDynamic control oflubrication characteristics in high speed tandem cold rollingrdquoJournal of Materials Processing Technology vol 229pp 407ndash416 2016

[10] A Heidari and M R Forouzan ldquoOptimization of cold rollingprocess parameters in order to increasing rolling speedlimited by chatter vibrationsrdquo Journal of Advanced Researchvol 4 no 1 pp 27ndash34 2013

[11] A Heidari M R Forouzan and S Akbarzadeh ldquoEffect offriction on tandem cold rolling mills chatteringrdquo ISIJ In-ternational vol 54 no 10 pp 2349ndash2356 2014

[12] A Heidari M R Forouzan and S Akbarzadeh ldquoDevelop-ment of a rolling chatter model considering unsteady lubri-cationrdquo ISIJ International vol 54 no 1 pp 165ndash170 2014

[13] A Heidari M R Forouzan and M R Niroomand ldquoDe-velopment and evaluation of friction models for chattersimulation in cold strip rollingrdquo International Journal ofAdvanced Manufacturing Technology vol 96 no 4 pp 1ndash212018

[14] Y Kim C-W Kim S Lee and H Park ldquoDynamic modelingand numerical analysis of a cold rolling millrdquo InternationalJournal of Precision Engineering and Manufacturing vol 14no 3 pp 407ndash413 2013

[15] P V Krot ldquoNonlinear vibrations and backlashes diagnosticsin the rolling mills drive trainsrdquo in Proceedings of the 6thEUROMECH Nonlinear Dynamics Conference Saint Peters-burg Russia July 2008

[16] S Liu B Sun S Zhao J Li and W Zhang ldquoStronglynonlinear dynamics of torsional vibration for rolling millrsquoselectromechanical coupling systemrdquo Steel Research In-ternational vol 86 no 9 pp 984ndash992 2015

[17] S Liu J Wang J Liu and Y Li ldquoNonlinear parametricallyexcited vibration and active control of gear pair system withtime-varying characteristicrdquo Chinese Physics B vol 24 no 10pp 303ndash311 2015

[18] B D Hassard N D Kazarinoff and Y H Wan eory andApplications of Hopf Bifurcation Cambridge University PressCambridge UK 1981

[19] B H K Lee S J Price and Y SWong ldquoNonlinear aeroelasticanalysis of airfoils bifurcation and chaosrdquo Progress inAerospace Sciences vol 35 no 3 pp 205ndash334 1999

[20] K Nishimura T Ikeda and Y Harata ldquoLocalization phe-nomena in torsional rotating shaft systems with multiplecentrifugal pendulum vibration absorbersrdquo Nonlinear Dy-namics vol 83 no 3 pp 1705ndash1726 2015

[21] T Zhang and H Dai ldquoBifurcation analysis of high-speedrailway wheel-setrdquo Nonlinear Dynamics vol 83 no 3pp 1511ndash1528 2015

[22] Z Y Gao Y Zang and T Han ldquoHopf bifurcation andstability analysis on the mill drive systemrdquo Applied Mechanicsamp Materials vol 121-126 pp 1514ndash1520 2012

[23] L Zeng Y Zang and Z Gao ldquoHopf bifurcation control forrolling mill multiple-mode-coupling vibration under non-linear frictionrdquo Journal of Vibration and Acoustics vol 139no 6 article 061015 2017

[24] S Liu Z Wang J Wang and H Li ldquoSliding bifurcationresearch of a horizontal-torsional coupled main drive systemof rolling millrdquo Nonlinear Dynamics vol 83 pp 441ndash4552015

Shock and Vibration 13

[25] Y Peng M Zhang J-L Sun and Y Zhang ldquoExperimentaland numerical investigation on the roll system swing vibra-tion characteristics of a hot rolling millrdquo ISIJ Internationalvol 57 no 9 pp 1567ndash1576 2017

[26] J Pittner and M A Simaan ldquoOptimal control of continuoustandem cold metal rollingrdquo in Proceedings of 2008 AmericanControl Conference Seattle WA USA June 2008

[27] R B Sims and D F S Arthur ldquoSpeed dependent variables incold strip rollingrdquo Journal of Iron and Steel Institute vol 172no 3 pp 285ndash295 1952

[28] S Z Chen D H Zhang J Sun J S Wang and J SongldquoOnline calculation model of rolling force for cold rolling millbased on numerical integrationrdquo in Proceedings of the 24thChinese Control and Decision Conference (CCDC) TaiyuanChina May 2012

[29] D R Bland and H Ford ldquo)e calculation of roll force andtorque in cold strip rolling with tensionsrdquo Proceedings of theInstitution of Mechanical Engineers vol 159 no 1 pp 144ndash163 2006

[30] R Hill ldquoRelations between roll-force torque and the appliedtensions in strip-rollingrdquo Proceedings of the Institution ofMechanical Engineers vol 163 no 1 pp 135ndash140 2006

[31] R E Johnson and Q Qi ldquoChatter dynamics in sheet rollingrdquoInternational Journal of Mechanical Sciences vol 36 no 7pp 617ndash630 1994

[32] Z Gao L Bai and Q Li ldquoResearch on critical rolling speed ofself-excited vibration in the tandem rolling process of thinstriprdquo Journal of Mechanical Engineering vol 53 no 12pp 118ndash132 2017

[33] R C Dorf and R H Bishop Modern Control SystemsUniversity of California Davis CA USA 12th edition 2010

[34] W M Liu ldquoCriterion of hopf bifurcations without usingeigenvaluesrdquo Journal of Mathematical Analysis and Appli-cations vol 182 no 1 pp 250ndash256 1994

[35] L Zeng Y Zang and Z Gao ldquoMultiple-modal-couplingmodeling and stability analysis of cold rolling mill vibra-tionrdquo Shock and Vibration vol 2016 Article ID 234738626 pages 2016

14 Shock and Vibration

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

a sudden and harsh vibration noise will be heard in the No 5stand and finally the strip breaks It shows mill chatterappears under the speed of 230ms and the actual criticalrolling speed is 230ms

)e calculated critical rolling speed is 251ms while theactual critical rolling speed is 230ms )ere is a certainrelative error between the calculated critical speed and actualcritical rolling speed )is is due to the small coefficientdifference in Jacobian matrix With the change of rollingspeed the actual value of the tension and friction is changedAnd the calculated value of tension factor nt and frictioncoefficient in zPzh and zMzh is a little different with theactual value So the calculated limited rolling speed is a littlebigger than the actual value However by comparing thecritical rolling speed calculated by the proposed model withthe actual critical rolling speed it can be seen that theproposed model is still accurate

)e vibration is largely influenced by the friction co-efficient and rolling speed Based on equation (13) Figure 12shows the friction coefficient response of No 5 stand versusspeed Despite the changes of speed the friction coefficient isstill in the stability range of 00048ndash00401 It is divided intotwo regions In region 1 both the friction coefficient androlling speed are in the stable range In region 2 the frictioncoefficient is still in the stable range while the speed exceedsthe limited speed rangeWhen the rolling speed is lower than251ms the mill system is stable without changing thelubrication conditions And when the rolling speed is higherthan 251ms the stand is unstable However the frictioncoefficient is in the middle value of stable range under thespeed is 251ms so mill chatter might be eliminated byreducing the basic friction coefficient (by improving thelubrication characteristics such as increasing the concen-tration of emulsion increasing spray flow and improvingthe property of the base oil)

432 Calculation of Maximum Reduction Assuming theentry thickness is constant the exit thickness is variable Andthe exit thickness has been chosen as the Hopf bifurcationparameter Other calculated rolling conditions are shown in

Table 1 )en the calculated bifurcation point of exitthickness is 0168mm)e optimal setting range of reductionfor No 5 stand is 0ndash44When the reduction of No 5 stand is44 the increase of reduction leads to a lower mill standstability )e reason is also related to the increase of rollingforce variation under the larger reduction rolling conditions

As can be seen in Table 1 in normal rolling schedule thereduction rate is 333 And the max reduction is up to 44In reality considering there are other factors that influencethe stand stability such as rolling speed and lubricationconditions the max reduction is lower than 44 In generalthe effect of the reduction in the mill stability is lower thanthe improvement of friction coefficient

433 Front Tension Stable Range When the work roll ve-locity is 221ms the calculated bifurcation point of exittension is 15ndash133MPa Small front tension means that therolling force will be quite large So under the same excitationinterferences the variation of rolling force will be muchbigger and the stand stability is decreased When the fronttension is big enough tension factor nt decreases leading toa larger value in zPzh and zMzh As a result the fluc-tuation of rolling force and roll torque increases and thestability of the system is reduced as well So the influence oftension on the mill stability is smaller than other parameters

)ese results can be used to estimate mill chatter in acertain stand And the stable range of friction reduction andtension can be used to adjust the rolling schedule in theUCM mill to avoid mill chatter under high rolling speed

5 Conclusion

(1) A new multiple degrees of freedom and nonlineardynamic vertical-horizontal-torsional coupled millchatter model for UCM cold rolling mill is estab-lished In the dynamic rolling process model thetime-varying coupling effect between rolling forceroll torque variation and fluctuation of stripthickness and tension is formulated In the vibrationmodel the movement of work roll in both vertical-horizontal-torsional directions and the multirollcharacteristics in the UCM rolling mill are consid-ered Finally the dynamic relationship of rollmovement responses and dynamic changes in theroll gap is formulated

(2) Selecting the friction coefficient as the Hopf bi-furcation parameter and by using RouthndashHurwitzdeterminant the Hopf bifurcation point of the millvibration system is calculated and the bifurcationcharacteristics are analyzed Research shows thateither too low or too high friction coefficient in-creases the probability of mill chatter When thefriction coefficient is extremely low the mill systemproduces a stable limit cycle With the frictioncontinuing to increase the phase trajectory willconverge toward the origin point However the limitcycle of phase diagram disappears and dispersesoutward under high friction rolling conditions

00 02 04 06 08 10ndash2 times 10ndash7

ndash1 times 10ndash7

0

1 times 10ndash7

2 times 10ndash7

Wor

k ro

ll di

spla

cem

ent (

m)

Time (s)

Figure 10 Work roll time history displacement under the rollingspeed of 252ms

Shock and Vibration 11

0 1 2 30

20

40

60

80

Fron

t ten

sion

(kN

)

Back tension

Back

tens

ion

(kN

)

Time (s)

0

20

40

60

Front tension

(a)

0 1 2 30

20

40

60

80

Fron

t ten

sion

(kN

)

Back tension

Back

tens

ion

(kN

)

Time (s)

0

20

40

60

Front tension

(b)

0 1 2 3

015

018

021

024

Roll

gap

of N

o 5

stan

d (m

m)

Roll gap of No 4 stand

Roll

gap

of N

o 4

stan

d (m

m)

Time (s)

268

272

276

280

Roll gap of No 5 stand

(c)

0 1 2 3

030

033

036

039

Roll

gap

of N

o 5

stan

d (m

m)

Roll gap of No 4 stand

Roll

gap

of N

o4

stand

(mm

)

Time (s)

272

276

280

284

Roll gap of No 5 stand

(d)

Figure 11 Tension and roll gap time history (a) back and front tension of No 5 stand under the speed of 230ms (b) back and fronttension of No 5 stand under the speed of 221ms (c) roll gap of No 4 and No 5 stands under the speed of 230ms (d) roll gap of No 4 andNo 5 stands under the speed of 221ms

0 10 20 30

0020

0025

0030

0035

0040

0045

(2)

(25 002101)

Fric

tion

coeffi

cien

t

Speed (ms)

x = 251

(1)

Figure 12 Friction coefficient response of No 5 stand versus speed

12 Shock and Vibration

(3) Based on the coupled mill chatter model the millsystemrsquos stable limited rolling speed stable range ofreduction and front tension are investigatedWithout changing the rolling speed the most ef-fective way to eliminate mill chatter is to change thelubrication conditions and adjust the reductionschedule And the effect of tension on the vibration isquite limited )ese methods can be used to improvethe chatter phenomena in rolling mill under highrolling speed )e proposed vibration model canalso be used to design an optimum rolling schedulein the UCM tandem cold rolling mill and to de-termine the appearance of mill chatter under certainrolling conditions

Data Availability

)e rolling conditions data used to support the findings ofthis study are included within the article

Conflicts of Interest

)e authors declare that they have no conflicts of interestregarding the publication of this manuscript

Acknowledgments

)is work was supported by the National Natural ScienceFoundation of China (51774084 and 51634002) the NationalKey RampD Program of China (2017YFB0304100) the Fun-damental Research Funds for the Central Universities(N170708020) and the Open Research Fund from the StateKey Laboratory of Rolling and Automation NortheasternUniversity (2017RALKFKT006)

References

[1] I S Yun W R D Wilson and K F Ehmann ldquoReview ofchatter studies in cold rollingrdquo International Journal ofMachine Tools and Manufacture vol 38 no 12pp 1499ndash1530 1998

[2] I-S Yun W R D Wilson and K F Ehmann ldquoChatter in thestrip rolling process Part 1 dynamic model of rollingrdquoJournal of Manufacturing Science and Engineering vol 120no 2 pp 330ndash336 1998

[3] J Tlusty G Chandra S Critchley and D Paton ldquoChatter incold rollingrdquo CIRP Annals vol 31 no 1 pp 195ndash199 1982

[4] N M Reza F M Reza M Salimi and H Shojaei ldquoExper-imental investigations and ALE finite element methodanalysis of chatter in cold strip rollingrdquo ISIJ Internationalvol 52 no 12 pp 2245ndash2253 2012

[5] M R Niroomand M R Forouzan M Salimi and H ShojaeildquoExperimental investigations and ALE finite element methodanalysis of chatter in cold strip rollingrdquo ISIJ Internationalvol 52 no 12 pp 2245ndash2253 2012

[6] M R Niroomand M R Forouzan and M Salimi ldquo)eo-retical and experimental analysis of chatter in tandem coldrolling mills based on wave propagation theoryrdquo ISIJ In-ternational vol 55 no 3 pp 637ndash646 2015

[7] S Kapil P Eberhard and S K Dwivedy ldquoNonlinear dynamicanalysis of a parametrically excited cold rolling millrdquo Journal

of Manufacturing Science and Engineering vol 136 no 4p 041012 2014

[8] Y Kimura Y Sodani N Nishiura N Ikeuchi and Y MiharaldquoAnalysis of chaffer in tandem cold rolling millsrdquo ISIJ In-ternational vol 43 no 1 pp 77ndash84 2003

[9] N Fujita Y Kimura K Kobayashi et al ldquoDynamic control oflubrication characteristics in high speed tandem cold rollingrdquoJournal of Materials Processing Technology vol 229pp 407ndash416 2016

[10] A Heidari and M R Forouzan ldquoOptimization of cold rollingprocess parameters in order to increasing rolling speedlimited by chatter vibrationsrdquo Journal of Advanced Researchvol 4 no 1 pp 27ndash34 2013

[11] A Heidari M R Forouzan and S Akbarzadeh ldquoEffect offriction on tandem cold rolling mills chatteringrdquo ISIJ In-ternational vol 54 no 10 pp 2349ndash2356 2014

[12] A Heidari M R Forouzan and S Akbarzadeh ldquoDevelop-ment of a rolling chatter model considering unsteady lubri-cationrdquo ISIJ International vol 54 no 1 pp 165ndash170 2014

[13] A Heidari M R Forouzan and M R Niroomand ldquoDe-velopment and evaluation of friction models for chattersimulation in cold strip rollingrdquo International Journal ofAdvanced Manufacturing Technology vol 96 no 4 pp 1ndash212018

[14] Y Kim C-W Kim S Lee and H Park ldquoDynamic modelingand numerical analysis of a cold rolling millrdquo InternationalJournal of Precision Engineering and Manufacturing vol 14no 3 pp 407ndash413 2013

[15] P V Krot ldquoNonlinear vibrations and backlashes diagnosticsin the rolling mills drive trainsrdquo in Proceedings of the 6thEUROMECH Nonlinear Dynamics Conference Saint Peters-burg Russia July 2008

[16] S Liu B Sun S Zhao J Li and W Zhang ldquoStronglynonlinear dynamics of torsional vibration for rolling millrsquoselectromechanical coupling systemrdquo Steel Research In-ternational vol 86 no 9 pp 984ndash992 2015

[17] S Liu J Wang J Liu and Y Li ldquoNonlinear parametricallyexcited vibration and active control of gear pair system withtime-varying characteristicrdquo Chinese Physics B vol 24 no 10pp 303ndash311 2015

[18] B D Hassard N D Kazarinoff and Y H Wan eory andApplications of Hopf Bifurcation Cambridge University PressCambridge UK 1981

[19] B H K Lee S J Price and Y SWong ldquoNonlinear aeroelasticanalysis of airfoils bifurcation and chaosrdquo Progress inAerospace Sciences vol 35 no 3 pp 205ndash334 1999

[20] K Nishimura T Ikeda and Y Harata ldquoLocalization phe-nomena in torsional rotating shaft systems with multiplecentrifugal pendulum vibration absorbersrdquo Nonlinear Dy-namics vol 83 no 3 pp 1705ndash1726 2015

[21] T Zhang and H Dai ldquoBifurcation analysis of high-speedrailway wheel-setrdquo Nonlinear Dynamics vol 83 no 3pp 1511ndash1528 2015

[22] Z Y Gao Y Zang and T Han ldquoHopf bifurcation andstability analysis on the mill drive systemrdquo Applied Mechanicsamp Materials vol 121-126 pp 1514ndash1520 2012

[23] L Zeng Y Zang and Z Gao ldquoHopf bifurcation control forrolling mill multiple-mode-coupling vibration under non-linear frictionrdquo Journal of Vibration and Acoustics vol 139no 6 article 061015 2017

[24] S Liu Z Wang J Wang and H Li ldquoSliding bifurcationresearch of a horizontal-torsional coupled main drive systemof rolling millrdquo Nonlinear Dynamics vol 83 pp 441ndash4552015

Shock and Vibration 13

[25] Y Peng M Zhang J-L Sun and Y Zhang ldquoExperimentaland numerical investigation on the roll system swing vibra-tion characteristics of a hot rolling millrdquo ISIJ Internationalvol 57 no 9 pp 1567ndash1576 2017

[26] J Pittner and M A Simaan ldquoOptimal control of continuoustandem cold metal rollingrdquo in Proceedings of 2008 AmericanControl Conference Seattle WA USA June 2008

[27] R B Sims and D F S Arthur ldquoSpeed dependent variables incold strip rollingrdquo Journal of Iron and Steel Institute vol 172no 3 pp 285ndash295 1952

[28] S Z Chen D H Zhang J Sun J S Wang and J SongldquoOnline calculation model of rolling force for cold rolling millbased on numerical integrationrdquo in Proceedings of the 24thChinese Control and Decision Conference (CCDC) TaiyuanChina May 2012

[29] D R Bland and H Ford ldquo)e calculation of roll force andtorque in cold strip rolling with tensionsrdquo Proceedings of theInstitution of Mechanical Engineers vol 159 no 1 pp 144ndash163 2006

[30] R Hill ldquoRelations between roll-force torque and the appliedtensions in strip-rollingrdquo Proceedings of the Institution ofMechanical Engineers vol 163 no 1 pp 135ndash140 2006

[31] R E Johnson and Q Qi ldquoChatter dynamics in sheet rollingrdquoInternational Journal of Mechanical Sciences vol 36 no 7pp 617ndash630 1994

[32] Z Gao L Bai and Q Li ldquoResearch on critical rolling speed ofself-excited vibration in the tandem rolling process of thinstriprdquo Journal of Mechanical Engineering vol 53 no 12pp 118ndash132 2017

[33] R C Dorf and R H Bishop Modern Control SystemsUniversity of California Davis CA USA 12th edition 2010

[34] W M Liu ldquoCriterion of hopf bifurcations without usingeigenvaluesrdquo Journal of Mathematical Analysis and Appli-cations vol 182 no 1 pp 250ndash256 1994

[35] L Zeng Y Zang and Z Gao ldquoMultiple-modal-couplingmodeling and stability analysis of cold rolling mill vibra-tionrdquo Shock and Vibration vol 2016 Article ID 234738626 pages 2016

14 Shock and Vibration

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

0 1 2 30

20

40

60

80

Fron

t ten

sion

(kN

)

Back tension

Back

tens

ion

(kN

)

Time (s)

0

20

40

60

Front tension

(a)

0 1 2 30

20

40

60

80

Fron

t ten

sion

(kN

)

Back tension

Back

tens

ion

(kN

)

Time (s)

0

20

40

60

Front tension

(b)

0 1 2 3

015

018

021

024

Roll

gap

of N

o 5

stan

d (m

m)

Roll gap of No 4 stand

Roll

gap

of N

o 4

stan

d (m

m)

Time (s)

268

272

276

280

Roll gap of No 5 stand

(c)

0 1 2 3

030

033

036

039

Roll

gap

of N

o 5

stan

d (m

m)

Roll gap of No 4 stand

Roll

gap

of N

o4

stand

(mm

)

Time (s)

272

276

280

284

Roll gap of No 5 stand

(d)

Figure 11 Tension and roll gap time history (a) back and front tension of No 5 stand under the speed of 230ms (b) back and fronttension of No 5 stand under the speed of 221ms (c) roll gap of No 4 and No 5 stands under the speed of 230ms (d) roll gap of No 4 andNo 5 stands under the speed of 221ms

0 10 20 30

0020

0025

0030

0035

0040

0045

(2)

(25 002101)

Fric

tion

coeffi

cien

t

Speed (ms)

x = 251

(1)

Figure 12 Friction coefficient response of No 5 stand versus speed

12 Shock and Vibration

(3) Based on the coupled mill chatter model the millsystemrsquos stable limited rolling speed stable range ofreduction and front tension are investigatedWithout changing the rolling speed the most ef-fective way to eliminate mill chatter is to change thelubrication conditions and adjust the reductionschedule And the effect of tension on the vibration isquite limited )ese methods can be used to improvethe chatter phenomena in rolling mill under highrolling speed )e proposed vibration model canalso be used to design an optimum rolling schedulein the UCM tandem cold rolling mill and to de-termine the appearance of mill chatter under certainrolling conditions

Data Availability

)e rolling conditions data used to support the findings ofthis study are included within the article

Conflicts of Interest

)e authors declare that they have no conflicts of interestregarding the publication of this manuscript

Acknowledgments

)is work was supported by the National Natural ScienceFoundation of China (51774084 and 51634002) the NationalKey RampD Program of China (2017YFB0304100) the Fun-damental Research Funds for the Central Universities(N170708020) and the Open Research Fund from the StateKey Laboratory of Rolling and Automation NortheasternUniversity (2017RALKFKT006)

References

[1] I S Yun W R D Wilson and K F Ehmann ldquoReview ofchatter studies in cold rollingrdquo International Journal ofMachine Tools and Manufacture vol 38 no 12pp 1499ndash1530 1998

[2] I-S Yun W R D Wilson and K F Ehmann ldquoChatter in thestrip rolling process Part 1 dynamic model of rollingrdquoJournal of Manufacturing Science and Engineering vol 120no 2 pp 330ndash336 1998

[3] J Tlusty G Chandra S Critchley and D Paton ldquoChatter incold rollingrdquo CIRP Annals vol 31 no 1 pp 195ndash199 1982

[4] N M Reza F M Reza M Salimi and H Shojaei ldquoExper-imental investigations and ALE finite element methodanalysis of chatter in cold strip rollingrdquo ISIJ Internationalvol 52 no 12 pp 2245ndash2253 2012

[5] M R Niroomand M R Forouzan M Salimi and H ShojaeildquoExperimental investigations and ALE finite element methodanalysis of chatter in cold strip rollingrdquo ISIJ Internationalvol 52 no 12 pp 2245ndash2253 2012

[6] M R Niroomand M R Forouzan and M Salimi ldquo)eo-retical and experimental analysis of chatter in tandem coldrolling mills based on wave propagation theoryrdquo ISIJ In-ternational vol 55 no 3 pp 637ndash646 2015

[7] S Kapil P Eberhard and S K Dwivedy ldquoNonlinear dynamicanalysis of a parametrically excited cold rolling millrdquo Journal

of Manufacturing Science and Engineering vol 136 no 4p 041012 2014

[8] Y Kimura Y Sodani N Nishiura N Ikeuchi and Y MiharaldquoAnalysis of chaffer in tandem cold rolling millsrdquo ISIJ In-ternational vol 43 no 1 pp 77ndash84 2003

[9] N Fujita Y Kimura K Kobayashi et al ldquoDynamic control oflubrication characteristics in high speed tandem cold rollingrdquoJournal of Materials Processing Technology vol 229pp 407ndash416 2016

[10] A Heidari and M R Forouzan ldquoOptimization of cold rollingprocess parameters in order to increasing rolling speedlimited by chatter vibrationsrdquo Journal of Advanced Researchvol 4 no 1 pp 27ndash34 2013

[11] A Heidari M R Forouzan and S Akbarzadeh ldquoEffect offriction on tandem cold rolling mills chatteringrdquo ISIJ In-ternational vol 54 no 10 pp 2349ndash2356 2014

[12] A Heidari M R Forouzan and S Akbarzadeh ldquoDevelop-ment of a rolling chatter model considering unsteady lubri-cationrdquo ISIJ International vol 54 no 1 pp 165ndash170 2014

[13] A Heidari M R Forouzan and M R Niroomand ldquoDe-velopment and evaluation of friction models for chattersimulation in cold strip rollingrdquo International Journal ofAdvanced Manufacturing Technology vol 96 no 4 pp 1ndash212018

[14] Y Kim C-W Kim S Lee and H Park ldquoDynamic modelingand numerical analysis of a cold rolling millrdquo InternationalJournal of Precision Engineering and Manufacturing vol 14no 3 pp 407ndash413 2013

[15] P V Krot ldquoNonlinear vibrations and backlashes diagnosticsin the rolling mills drive trainsrdquo in Proceedings of the 6thEUROMECH Nonlinear Dynamics Conference Saint Peters-burg Russia July 2008

[16] S Liu B Sun S Zhao J Li and W Zhang ldquoStronglynonlinear dynamics of torsional vibration for rolling millrsquoselectromechanical coupling systemrdquo Steel Research In-ternational vol 86 no 9 pp 984ndash992 2015

[17] S Liu J Wang J Liu and Y Li ldquoNonlinear parametricallyexcited vibration and active control of gear pair system withtime-varying characteristicrdquo Chinese Physics B vol 24 no 10pp 303ndash311 2015

[18] B D Hassard N D Kazarinoff and Y H Wan eory andApplications of Hopf Bifurcation Cambridge University PressCambridge UK 1981

[19] B H K Lee S J Price and Y SWong ldquoNonlinear aeroelasticanalysis of airfoils bifurcation and chaosrdquo Progress inAerospace Sciences vol 35 no 3 pp 205ndash334 1999

[20] K Nishimura T Ikeda and Y Harata ldquoLocalization phe-nomena in torsional rotating shaft systems with multiplecentrifugal pendulum vibration absorbersrdquo Nonlinear Dy-namics vol 83 no 3 pp 1705ndash1726 2015

[21] T Zhang and H Dai ldquoBifurcation analysis of high-speedrailway wheel-setrdquo Nonlinear Dynamics vol 83 no 3pp 1511ndash1528 2015

[22] Z Y Gao Y Zang and T Han ldquoHopf bifurcation andstability analysis on the mill drive systemrdquo Applied Mechanicsamp Materials vol 121-126 pp 1514ndash1520 2012

[23] L Zeng Y Zang and Z Gao ldquoHopf bifurcation control forrolling mill multiple-mode-coupling vibration under non-linear frictionrdquo Journal of Vibration and Acoustics vol 139no 6 article 061015 2017

[24] S Liu Z Wang J Wang and H Li ldquoSliding bifurcationresearch of a horizontal-torsional coupled main drive systemof rolling millrdquo Nonlinear Dynamics vol 83 pp 441ndash4552015

Shock and Vibration 13

[25] Y Peng M Zhang J-L Sun and Y Zhang ldquoExperimentaland numerical investigation on the roll system swing vibra-tion characteristics of a hot rolling millrdquo ISIJ Internationalvol 57 no 9 pp 1567ndash1576 2017

[26] J Pittner and M A Simaan ldquoOptimal control of continuoustandem cold metal rollingrdquo in Proceedings of 2008 AmericanControl Conference Seattle WA USA June 2008

[27] R B Sims and D F S Arthur ldquoSpeed dependent variables incold strip rollingrdquo Journal of Iron and Steel Institute vol 172no 3 pp 285ndash295 1952

[28] S Z Chen D H Zhang J Sun J S Wang and J SongldquoOnline calculation model of rolling force for cold rolling millbased on numerical integrationrdquo in Proceedings of the 24thChinese Control and Decision Conference (CCDC) TaiyuanChina May 2012

[29] D R Bland and H Ford ldquo)e calculation of roll force andtorque in cold strip rolling with tensionsrdquo Proceedings of theInstitution of Mechanical Engineers vol 159 no 1 pp 144ndash163 2006

[30] R Hill ldquoRelations between roll-force torque and the appliedtensions in strip-rollingrdquo Proceedings of the Institution ofMechanical Engineers vol 163 no 1 pp 135ndash140 2006

[31] R E Johnson and Q Qi ldquoChatter dynamics in sheet rollingrdquoInternational Journal of Mechanical Sciences vol 36 no 7pp 617ndash630 1994

[32] Z Gao L Bai and Q Li ldquoResearch on critical rolling speed ofself-excited vibration in the tandem rolling process of thinstriprdquo Journal of Mechanical Engineering vol 53 no 12pp 118ndash132 2017

[33] R C Dorf and R H Bishop Modern Control SystemsUniversity of California Davis CA USA 12th edition 2010

[34] W M Liu ldquoCriterion of hopf bifurcations without usingeigenvaluesrdquo Journal of Mathematical Analysis and Appli-cations vol 182 no 1 pp 250ndash256 1994

[35] L Zeng Y Zang and Z Gao ldquoMultiple-modal-couplingmodeling and stability analysis of cold rolling mill vibra-tionrdquo Shock and Vibration vol 2016 Article ID 234738626 pages 2016

14 Shock and Vibration

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

(3) Based on the coupled mill chatter model the millsystemrsquos stable limited rolling speed stable range ofreduction and front tension are investigatedWithout changing the rolling speed the most ef-fective way to eliminate mill chatter is to change thelubrication conditions and adjust the reductionschedule And the effect of tension on the vibration isquite limited )ese methods can be used to improvethe chatter phenomena in rolling mill under highrolling speed )e proposed vibration model canalso be used to design an optimum rolling schedulein the UCM tandem cold rolling mill and to de-termine the appearance of mill chatter under certainrolling conditions

Data Availability

)e rolling conditions data used to support the findings ofthis study are included within the article

Conflicts of Interest

)e authors declare that they have no conflicts of interestregarding the publication of this manuscript

Acknowledgments

)is work was supported by the National Natural ScienceFoundation of China (51774084 and 51634002) the NationalKey RampD Program of China (2017YFB0304100) the Fun-damental Research Funds for the Central Universities(N170708020) and the Open Research Fund from the StateKey Laboratory of Rolling and Automation NortheasternUniversity (2017RALKFKT006)

References

[1] I S Yun W R D Wilson and K F Ehmann ldquoReview ofchatter studies in cold rollingrdquo International Journal ofMachine Tools and Manufacture vol 38 no 12pp 1499ndash1530 1998

[2] I-S Yun W R D Wilson and K F Ehmann ldquoChatter in thestrip rolling process Part 1 dynamic model of rollingrdquoJournal of Manufacturing Science and Engineering vol 120no 2 pp 330ndash336 1998

[3] J Tlusty G Chandra S Critchley and D Paton ldquoChatter incold rollingrdquo CIRP Annals vol 31 no 1 pp 195ndash199 1982

[4] N M Reza F M Reza M Salimi and H Shojaei ldquoExper-imental investigations and ALE finite element methodanalysis of chatter in cold strip rollingrdquo ISIJ Internationalvol 52 no 12 pp 2245ndash2253 2012

[5] M R Niroomand M R Forouzan M Salimi and H ShojaeildquoExperimental investigations and ALE finite element methodanalysis of chatter in cold strip rollingrdquo ISIJ Internationalvol 52 no 12 pp 2245ndash2253 2012

[6] M R Niroomand M R Forouzan and M Salimi ldquo)eo-retical and experimental analysis of chatter in tandem coldrolling mills based on wave propagation theoryrdquo ISIJ In-ternational vol 55 no 3 pp 637ndash646 2015

[7] S Kapil P Eberhard and S K Dwivedy ldquoNonlinear dynamicanalysis of a parametrically excited cold rolling millrdquo Journal

of Manufacturing Science and Engineering vol 136 no 4p 041012 2014

[8] Y Kimura Y Sodani N Nishiura N Ikeuchi and Y MiharaldquoAnalysis of chaffer in tandem cold rolling millsrdquo ISIJ In-ternational vol 43 no 1 pp 77ndash84 2003

[9] N Fujita Y Kimura K Kobayashi et al ldquoDynamic control oflubrication characteristics in high speed tandem cold rollingrdquoJournal of Materials Processing Technology vol 229pp 407ndash416 2016

[10] A Heidari and M R Forouzan ldquoOptimization of cold rollingprocess parameters in order to increasing rolling speedlimited by chatter vibrationsrdquo Journal of Advanced Researchvol 4 no 1 pp 27ndash34 2013

[11] A Heidari M R Forouzan and S Akbarzadeh ldquoEffect offriction on tandem cold rolling mills chatteringrdquo ISIJ In-ternational vol 54 no 10 pp 2349ndash2356 2014

[12] A Heidari M R Forouzan and S Akbarzadeh ldquoDevelop-ment of a rolling chatter model considering unsteady lubri-cationrdquo ISIJ International vol 54 no 1 pp 165ndash170 2014

[13] A Heidari M R Forouzan and M R Niroomand ldquoDe-velopment and evaluation of friction models for chattersimulation in cold strip rollingrdquo International Journal ofAdvanced Manufacturing Technology vol 96 no 4 pp 1ndash212018

[14] Y Kim C-W Kim S Lee and H Park ldquoDynamic modelingand numerical analysis of a cold rolling millrdquo InternationalJournal of Precision Engineering and Manufacturing vol 14no 3 pp 407ndash413 2013

[15] P V Krot ldquoNonlinear vibrations and backlashes diagnosticsin the rolling mills drive trainsrdquo in Proceedings of the 6thEUROMECH Nonlinear Dynamics Conference Saint Peters-burg Russia July 2008

[16] S Liu B Sun S Zhao J Li and W Zhang ldquoStronglynonlinear dynamics of torsional vibration for rolling millrsquoselectromechanical coupling systemrdquo Steel Research In-ternational vol 86 no 9 pp 984ndash992 2015

[17] S Liu J Wang J Liu and Y Li ldquoNonlinear parametricallyexcited vibration and active control of gear pair system withtime-varying characteristicrdquo Chinese Physics B vol 24 no 10pp 303ndash311 2015

[18] B D Hassard N D Kazarinoff and Y H Wan eory andApplications of Hopf Bifurcation Cambridge University PressCambridge UK 1981

[19] B H K Lee S J Price and Y SWong ldquoNonlinear aeroelasticanalysis of airfoils bifurcation and chaosrdquo Progress inAerospace Sciences vol 35 no 3 pp 205ndash334 1999

[20] K Nishimura T Ikeda and Y Harata ldquoLocalization phe-nomena in torsional rotating shaft systems with multiplecentrifugal pendulum vibration absorbersrdquo Nonlinear Dy-namics vol 83 no 3 pp 1705ndash1726 2015

[21] T Zhang and H Dai ldquoBifurcation analysis of high-speedrailway wheel-setrdquo Nonlinear Dynamics vol 83 no 3pp 1511ndash1528 2015

[22] Z Y Gao Y Zang and T Han ldquoHopf bifurcation andstability analysis on the mill drive systemrdquo Applied Mechanicsamp Materials vol 121-126 pp 1514ndash1520 2012

[23] L Zeng Y Zang and Z Gao ldquoHopf bifurcation control forrolling mill multiple-mode-coupling vibration under non-linear frictionrdquo Journal of Vibration and Acoustics vol 139no 6 article 061015 2017

[24] S Liu Z Wang J Wang and H Li ldquoSliding bifurcationresearch of a horizontal-torsional coupled main drive systemof rolling millrdquo Nonlinear Dynamics vol 83 pp 441ndash4552015

Shock and Vibration 13

[25] Y Peng M Zhang J-L Sun and Y Zhang ldquoExperimentaland numerical investigation on the roll system swing vibra-tion characteristics of a hot rolling millrdquo ISIJ Internationalvol 57 no 9 pp 1567ndash1576 2017

[26] J Pittner and M A Simaan ldquoOptimal control of continuoustandem cold metal rollingrdquo in Proceedings of 2008 AmericanControl Conference Seattle WA USA June 2008

[27] R B Sims and D F S Arthur ldquoSpeed dependent variables incold strip rollingrdquo Journal of Iron and Steel Institute vol 172no 3 pp 285ndash295 1952

[28] S Z Chen D H Zhang J Sun J S Wang and J SongldquoOnline calculation model of rolling force for cold rolling millbased on numerical integrationrdquo in Proceedings of the 24thChinese Control and Decision Conference (CCDC) TaiyuanChina May 2012

[29] D R Bland and H Ford ldquo)e calculation of roll force andtorque in cold strip rolling with tensionsrdquo Proceedings of theInstitution of Mechanical Engineers vol 159 no 1 pp 144ndash163 2006

[30] R Hill ldquoRelations between roll-force torque and the appliedtensions in strip-rollingrdquo Proceedings of the Institution ofMechanical Engineers vol 163 no 1 pp 135ndash140 2006

[31] R E Johnson and Q Qi ldquoChatter dynamics in sheet rollingrdquoInternational Journal of Mechanical Sciences vol 36 no 7pp 617ndash630 1994

[32] Z Gao L Bai and Q Li ldquoResearch on critical rolling speed ofself-excited vibration in the tandem rolling process of thinstriprdquo Journal of Mechanical Engineering vol 53 no 12pp 118ndash132 2017

[33] R C Dorf and R H Bishop Modern Control SystemsUniversity of California Davis CA USA 12th edition 2010

[34] W M Liu ldquoCriterion of hopf bifurcations without usingeigenvaluesrdquo Journal of Mathematical Analysis and Appli-cations vol 182 no 1 pp 250ndash256 1994

[35] L Zeng Y Zang and Z Gao ldquoMultiple-modal-couplingmodeling and stability analysis of cold rolling mill vibra-tionrdquo Shock and Vibration vol 2016 Article ID 234738626 pages 2016

14 Shock and Vibration

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

[25] Y Peng M Zhang J-L Sun and Y Zhang ldquoExperimentaland numerical investigation on the roll system swing vibra-tion characteristics of a hot rolling millrdquo ISIJ Internationalvol 57 no 9 pp 1567ndash1576 2017

[26] J Pittner and M A Simaan ldquoOptimal control of continuoustandem cold metal rollingrdquo in Proceedings of 2008 AmericanControl Conference Seattle WA USA June 2008

[27] R B Sims and D F S Arthur ldquoSpeed dependent variables incold strip rollingrdquo Journal of Iron and Steel Institute vol 172no 3 pp 285ndash295 1952

[28] S Z Chen D H Zhang J Sun J S Wang and J SongldquoOnline calculation model of rolling force for cold rolling millbased on numerical integrationrdquo in Proceedings of the 24thChinese Control and Decision Conference (CCDC) TaiyuanChina May 2012

[29] D R Bland and H Ford ldquo)e calculation of roll force andtorque in cold strip rolling with tensionsrdquo Proceedings of theInstitution of Mechanical Engineers vol 159 no 1 pp 144ndash163 2006

[30] R Hill ldquoRelations between roll-force torque and the appliedtensions in strip-rollingrdquo Proceedings of the Institution ofMechanical Engineers vol 163 no 1 pp 135ndash140 2006

[31] R E Johnson and Q Qi ldquoChatter dynamics in sheet rollingrdquoInternational Journal of Mechanical Sciences vol 36 no 7pp 617ndash630 1994

[32] Z Gao L Bai and Q Li ldquoResearch on critical rolling speed ofself-excited vibration in the tandem rolling process of thinstriprdquo Journal of Mechanical Engineering vol 53 no 12pp 118ndash132 2017

[33] R C Dorf and R H Bishop Modern Control SystemsUniversity of California Davis CA USA 12th edition 2010

[34] W M Liu ldquoCriterion of hopf bifurcations without usingeigenvaluesrdquo Journal of Mathematical Analysis and Appli-cations vol 182 no 1 pp 250ndash256 1994

[35] L Zeng Y Zang and Z Gao ldquoMultiple-modal-couplingmodeling and stability analysis of cold rolling mill vibra-tionrdquo Shock and Vibration vol 2016 Article ID 234738626 pages 2016

14 Shock and Vibration

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom