Post on 02-Sep-2020
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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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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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
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
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