Research ArticleTime Delay Effect on Regenerative Chatter inTandem Rolling Mills
Xiaochan Liu Yong Zang Zhiying Gao and Lingqiang Zeng
School of Mechanical Engineering University of Science and Technology Beijing Beijing 100083 China
Correspondence should be addressed to Yong Zang yzangustbeducn
Received 1 September 2015 Revised 8 December 2015 Accepted 17 December 2015
Academic Editor Kumar V Singh
Copyright copy 2016 Xiaochan Liu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The interstand tension coupling effect and strip gauge variation passed on to next stand with time delay are the main causes forregenerative chatter in tandem rolling mills To study the effect of different factors on the stability of tandem rolling mills differentmodels considering different interstand factors were built Through stability analysis of these models by employing the Lyapunovindirectmethod and integral criterionmore detailed and quantitative explanation is put forward to regenerative chattermechanismin rolling To study the time delay effect as a single factor on the stability of tandem rollingmills stability charts of the chatter modelincluding the time delay effect and model neglecting the delay time were compared The results show that the time delay effectreduces the critical velocity of multistand mills slightly in the big picture But it alters the relationship between two adjacent standsby worsening the downstream stand stability To get preferable rolling process parameter configuration for the tandem rollingmillsthe time delay effect in rolling must be involved
1 Introduction
Chatter in high speed rolling mills results in unacceptablegauge variations affects the surface quality of rolled strip andmay damage the mill stand Many studies have been con-ducted to explore the chatter mechanism and put forwardmethods to suppress the vibration [1] The self-excited vibra-tion in rolling is caused by interaction between mill standstructure and rolling process Rolling chatter has three maindifferent types third octave fifth octave and torsional vibra-tion The third octave is the most detrimental [2 3] Thechatter mechanism in tandem rolling mills can be concludedas four reasons namely model matching effect negativedamping effect mode coupling effect and regenerative effectRegenerative effect is themost complexmechanism for insta-bility in tandem rolling mills [4 5]
Regenerative effect in rolling mainly considers the inter-action between adjacent stands in tandem rolling mills Hufirst defined the regenerative chatter in rolling process inanalogy to chatter inmetal cutting process According toHursquosdefinition the regenerative effect in strip rolling refers to thephenomenon that the vibration of a certain stand at a prior
time causes or aggravates vibration of the same stand at thecurrent time through interstand interactions [6] By buildinga dynamic rolling process model and coupling it with standstructure model a multistand regenerative chatter modelwas proposed Simulation and analysis were conducted toreveal the regenerative mechanism The results have shownthat due to the needed time for the strip to travel from onestand to another a tandem mill can become unstable even ifindividual mill stands are stable [5]
Considering the work hardening effect of the rolled pieceand work roll flattening during rolling process a new rollingprocess model was proposed by Zhao and EhmannThroughcoupling it withmill standmodel three differentmodels werebuilt to explore the regenerative mechanism The first one isa single stand chatter model The second one is a two-standchatter model which only considers the interstand tensioncoupling effect [7] The last one is a two-stand chatter modelwhich considers both the interstand tension coupling effectand the strip gauge variation from an upstream stand passedon to downstream stand with time delayThe Routh criterionwas employed to formulate the critical velocity of the first andsecond model Due to the time delay effect the third model
Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 4025650 15 pageshttpdxdoiorg10115520164025650
2 Shock and Vibration
becomes a delay differential system and integral criterion wasemployed to find its critical velocity Comparative analysisof the three models presented a potential explanation to theregenerative mechanism [4]
Based on Zhao and Ehmannrsquos andHursquosmodel more com-plex regenerative chattermodels are proposed in recent yearsKimura et al constructed a five-stand regenerative chattermodel by coupling a five-degree freedom mill stand modeland the dynamic rollingmodel built in [7] togetherThe simu-lation results show that there exists an optimal friction of thefifth stand which relates to the maximum critical velocity ofthe whole tandem rolling mills [8] Instead of static analysisNiroomand et al used wave propagation theory in elasticsolids to formulate the dynamic tensile stress variationbetween two consequent stands and built a new two-standregenerative chatter model The simulation results show thatthe errors of critical velocity and chatter frequency are smallerwith the experimental results when the wave propagationtheory is applied [9] Considering the complex friction con-dition in the rolling gap Heidari et al built a regenerativechatter model using the unsteady lubrication model assum-ing that the rolling gap is in full film regime The effects ofrolling lubricant on the stability of tandem rolling mills werediscussed [10] Based on these regenerative chatter modelsoptimization of multistand rolling process parameters wereconducted to avoid chatter phenomenon using the combina-tion of neural networks and genetic algorithms [11 12]
All these regenerative chatter models have considered thetime delay effect which makes the tandem rolling mills adelay differential system Itmeans thatmore complex stabilitycriterion has to be employed to analyze the delay differen-tial multistand system which results in huge computationHowever the time delay effect on the stability of tandemrolling mills has not been analyzed as an independent factorMore research work has to be done to decide whether theeffect of time delay itself on the stability of tandem rollingmills is worth the great amount of computation Besidesthe relationship between consequent stands is not clearlydescribed though the interaction factors between stands havebeen pointed out The major objective pursed in this paper isto investigate the time delay effect as an independent factorin cold rolling tandem mills and put forward more detailedexplanation to the regenerative chatter mechanism and therelations between consecutive stands
2 Chatter Model
The dynamic rolling gap is shown as in Figure 1 when onlythe vertical roll vibration is considered Set the coordinates 119909and 119910 onto the center line of rolled piece and the work rollcenter line And the intersection point is set as the origin ofcoordinates 120590
119890and 120590
119889are the entry and exit tensile stress
V119890and V119889are the entry and exit strip velocity ℎ
119890and ℎ
119889are
the strip thickness at entry and exit V119903is the work roll linear
velocity ℎ119899is the strip thickness at neutral point 119909
119890 119909119899 and
119909119889are the position of entry point neutral point and exit
point
y
x
e
120590e he hn hd
d
R120601
120590d
xexn
xd
+
Figure 1 Dynamic rolling gap
The strip thickness at any 119909 location in the rolling gap isas follows
ℎ = ℎ119889+
1199092
119877
(1)
Due to the roll vertical vibration the continuity equationis modified as
Vℎ = V119890ℎ119890+ (119909 minus 119909
119890)ℎ119889 (2)
According to (1) and (2) the strip entry velocity V119890 strip
exit velocity V119889 strip entry position 119909
119890 and exit position 119909
119889
can be calculated as follows
V119890=
1
ℎ119890
(V119903ℎ119889+
V119903119909119899
2
119877
+ (119909119890minus 119909119899)ℎ119889)
V119889=
V119890ℎ119890+ (119909119889minus 119909119890)ℎ119889
ℎ119889+ 119909119889
2119877
119909119890= radic119877 (ℎ
119890minus ℎ119889)
119909119889=
119877ℎ119889
ℎ119889
2 (V119890ℎ119890minus 119909119890
ℎ119889)
(3)
Figure 2 illustrates the stresses acting on a vertical slabelement inside the rolling gap Employing the coulombfriction model 120591
119904= 120583119901 and the yield criterion 120590
119909= 119896119891minus 119901
the equilibrium equation for an elemental vertical section ofthe strip in the rolling gap in the 119909 direction is obtained
119889119901
119889119909
∓
2120583119901
ℎ
minus
1
ℎ
119889 (ℎ119896119891)
119889119909
= 0(4)
where the positive sign is for the exit side and the negativesign is for the entry side
Considering the strain hardening effect the deformationresistance of the rolled strip can be expressed as follows
119896119891= 1205900(119860 + ln(
ℎ0
ℎ
))
119899
(5)
Shock and Vibration 3
p p
120591120591
120590x 120590x
h h
120591 120591
p p
120590x +120597120590x120597x
dx 120590x +120597120590x120597x
dx
h +120597h
120597xh +
120597h
120597x
Figure 2 Slab analysis on a volume element of the entry region andexit region
1205900 119860 and 119899 are the material property parameter of the
rolled strip and can be decided by experiments ℎ0is the entry
strip thickness of multistand rolling millsApplying the boundary conditions
1199011003816100381610038161003816119909=119909119890
= 119896119891119890minus 120590119890
1199011003816100381610038161003816119909=119909119889
= 119896119891119889minus 120590119889
(6)
The rolling pressure at entry and exit side can be obtained
119901119890= (119896119891119890minus 120590119890)
ℎ119896119891
ℎ119890119896119891119890
exp(119906(2radic 119877ℎ119889
tanminus1 (119909119890
radic119877ℎ119889
)
minus 2radic
119877
ℎ119889
tanminus1 ( 119909
radic119877ℎ119889
)))
119901119889= (119896119891119889minus 120590119889)
ℎ119896119891
ℎ119889119896119891119889
exp(119906(2radic 119877ℎ119889
tanminus1 ( 119909
radic119877ℎ119889
)
minus 2radic
119877
ℎ119889
tanminus1 (119909119889
radic119877ℎ119889
)))
(7)
Letting 119901119890= 119901119889 the neutral point can be formulated as
follows
119909119899= radic119877ℎ
119889tan
1
4
radicℎ119889
119877
(2radic
119877
ℎ119889
tanminus1 (119909119890
radic119877ℎ119889
)
+ 2radic
119877
ℎ119889
tanminus1 (119909119889
radic119877ℎ119889
)
minus
1
120583
ln(119896119891119889minus 120590119889
119896119891119890minus 120590119890
ℎ119890119896119891119890
ℎ119889119896119891119889
))
(8)
By integrating the rolling stresses the rolling force can bedetermined
119865 = int
119909119890
119909119899
119901119890119889119909 + int
119909119899
119909119889
119901119889119889119909 (9)
M
KC
1
2dhd
dF
Figure 3 Mill stand structure model
The dynamic rolling process model can be obtained byutilizing the first-order Taylor expansion
119889119865 = 1198651119889ℎ119889+ 1198652119889ℎ119889+ 1198653119889120590119890+ 1198654119889120590119889+ 1198655119889ℎ119890
119889V119890= 1198751119889ℎ119889+ 1198752119889ℎ119889+ 1198753119889120590119890+ 1198754119889120590119889+ 1198755119889ℎ119890
119889V119889= 1198761119889ℎ119889+ 1198762119889ℎ119889+ 1198763119889120590119890+ 1198764119889120590119889+ 1198765119889ℎ119890
(10)
119865119895119875119895119876119895 119895 = 1 2 3 4 5 are the partial derivatives of119865 V
119890 V119889
with respect to ℎ119889 ℎ119889 120590119890 120590119889 ℎ119890 respectively The calculation
of 119865119895 119875119895 119876119895 119895 = 1 2 3 4 5 can be found in the Appendix
Considering the work roll flattening effect in rolling theequivalent work roll radius is calculated by employing theHitchcock formula
119877 = 119877119908(1 +
16 (1 minus 1205822
)
120587119864119908
119865
119903
) (11)
where 119877119908is initial work roll radius 120582 is Poissonrsquos ratio for the
work roll material 119865 is rolling pressure 119903 is reduction and119864119908is elasticity modulus of work roll materialChatter in rolling is the result of the interaction between
rolling process and mill stand structure The rolling processmodel and stand structure model are coupled togetherthrough mechanical rolling parameters A one degree offreedom mill stand structure model is shown in Figure 3
By coupling the rolling processmodel and stand structuremodel together the single stand vibrationmodel is as follows
119872119889ℎ119889+ 119862119889
ℎ119889+ 119870119889ℎ
119889= 2119908119889119865
119889119890=
119864
119871119890
119889V119890
119889120590119889= minus
119864
119871119889
119889V119889
(12)
where 119871119890is the distance between the stand and the upstream
stand and 119871119889is the distance between the stand and down-
stream stand
4 Shock and Vibration
Table 1 The initial rolling process parameters for the 1st and 2nd stand
ℎ1198901
(mm) ℎ1198891
ℎ1198902
(mm) ℎ1198892
(mm) 1205901198901
(Mpa) 1205901198891
(Mpa) 1205901198902
(Mpa)045 028 028 019 137 137 1371205901198892
(Mpa) 1199061
1199062
1198771199081
(mm) 1198771199082
(mm) 1205900(Mpa) 119860
98 0018 0015 276 291 8116 8116119899 119871
1198901(m) 119871
1198902(m) 119871
1198892(m) 119864
119908(Gpa) 119864 (Gpa) 119908 (m)
024 45 45 45 210 210 09
The regenerative chatter model for a two-stand rollingmill is as follows
119872119889ℎ1198891+ 119862119889
ℎ1198891+ 119870119889ℎ
1198891= 2119908119889119865
1
1198891198901=
119864
1198711198901
119889V1198901
1198891198891=
119864
1198711198902
(119889V1198902minus 119889V1198891)
119872119889ℎ1198892+ 119862119889
ℎ1198892+ 119870119889ℎ
1198892= 2119908119889119865
2
1198891198892= minus
119864
1198711198892
119889V1198892
(13)
where the second subscripts stand for the number of stands119864 is the elasticity modulus of roll piece material and 119908 is therolled strip width The rolled strip width spread is neglectedSo 119908 is constant
The relationship between the 2nd stand entry strip gaugevariation and the 1st stand exit strip gauge variation is asfollows
119889ℎ1198902(119905) = 119889ℎ
1198891(119905 minus 120591) (14)
where 120591 is the delay time decided by the interstand distanceand the entry velocity of 2nd stand
120591 =
1198711198902
V1198902
(15)
The initial rolling process parameters for the 1st and 2ndstand come from [7] and are shown in Table 1
To explore the effects of interstand coupling factors onregenerative chatter in detail four models were built in thispaperThe first one is a single stand vibrationmodelThe sec-ond one is a two-stand regenerative chattermodel which onlyconsiders the interstand tension coupling effectThe third oneis a two-stand regenerative chatter model which considersthe interstand tension coupling effect and the strip gaugevariation passed on to next stand but neglects the delay timeThe fourth one is a two-stand regenerative chatter modelwhich considers the interstand tension coupling effect and thestrip gauge variation passed on to next stand with time delayeffect
3 Stability Analysis
31 Stability Criterion To study the stabilities of the fourmodels proposed in Section 2 different stability criteria have
to be used to calculate the critical velocity for each modelThe first threemodels are nondelay differential systemsTheircritical velocities can be calculated by employing the Lya-punov indirect method According to the Lyapunov indirectmethod for a linear system = 119860119909 if all the real parts ofeigenvalues of matrix 119860 are negative the system is stable ifnot the system is unstable [13] Based on this method criticalvelocities of the first three models are obtained namely3482ms for the single 2nd stand model 3600ms for thesecond model and 3023ms for the third model Similar tothe single 2nd stand model the critical velocity of the single1st stand is 2478ms
The fourth model is a delay differential system Thestability criterion of time delay system is muchmore complexthan nondelay system The integral criterion is employed toanalyze the fourth model According to the integral criterionif the following inequation is true the delay differentialsystem is asymptotic stable [14]
int
infin
0
119871 (119908) gt
(119873 minus 1)
2
120587 (16)
where
119871 (119908) =
119877 (119908) 1198781015840
(119908) minus 119878 (119908) 1198771015840
(119908)
119877 (119908)2
+ 119878 (119908)2
(17)
119877(119908) and 119878(119908) are the real and imaginary parts of thedeterminant of the systemmatrix respectively119873 is the orderof the system For the fourth model 119873 is 7 The integrationis calculated with the help of MATLAB When the linearvelocity of the 2nd stand is 292ms
int
5000
0
119871 (119908) = 107 (18)
When the linear velocity of the 2nd stand is 293ms
int
5000
0
119871 (119908) = 45 (19)
So the critical velocity of fourth model is about 2925msComparing the critical velocities of the four models it
can be concluded that the interstand tension has just a littleeffect on the critical velocity of the chatter model The stripgauge variation passed on between stands reduces the criticalvelocity dramatically However the time delay effect as anindependent factor on critical velocity is very limited
Shock and Vibration 5
Frequency (Hz)0 200 400
0
1000
2000
Am
plitu
de (P
a)(e)
246
Frequency (Hz)
Am
plitu
de (P
a)
0 200 4000
200
400
(f)
246
0 200 4000
05
1
Frequency (Hz)
Am
plitu
de (m
)
(d)
246
times10minus9
0 005 01
0
1
(a)
minus1
times10minus9dhd2
(m)
t (s)
0 005 01
0
5000
(b)
minus5000
d120590e2
(Pa)
t (s)
0 005 01
0
1000
(c)
minus1000
d120590d2
(Pa)
t (s)
Figure 4 Time and frequency domain of the single 2nd stand when V1199032= 3482ms ((a) and (d)) work roll ((b) and (e)) entry tension ((c)
and (f)) exit tension
32 Simulation and Analysis To study the regenerative chat-ter mechanism and relationship between stands in detailtime and frequency domain simulations of the four modelswere conducted As shown in Figure 4 the single 2nd standvibrates periodically when V
1199032is 3482ms and the frequency
is 246Hz Similarly the single 1st stand vibrates periodicallywhen V
1199031is 2478ms and the frequency is 209Hz as shown
in Figure 5The second model only considers the interstand tension
coupling effect between stands As can been seen in Figure 6the 1st stand takes on periodic oscillation state and thefrequency is 209Hz It is the same with the single 1st standmodel The beat phenomena appear in the 2nd stand and theinterstand tension The frequencies are 209Hz and 251Hz209Hz is the frequency of the 1st stand and 251Hz isapproximate to the frequency of the 2nd stand It is obviousthat the beat phenomena are caused by the interstand tensioncoupling effect between the 1st stand and 2nd stand Besidesthe amplitude of 2nd stand is far less than the 1st stand
It suggests that the vibration in 2nd stand raised by theinterstand tension variation is not violent
The third model considers the interstand tension cou-pling effect and the strip gauge variation passed on to the nextstand but neglects the time delay effect As shown in Figure 7the beat phenomena in the second model disappear Thefrequency of the 1st stand the 2nd stand and the interstandtension are all 212HzThe amplitudes of the 1st and 2nd standare in the same magnitude It indicates that the strip gaugevariation passed on to the next stand makes the 2nd standvibrate more violently
The fourth model considers the interstand tension cou-pling effect between stands the strip gauge variation passedon to the next stand and the time delay effect As shown inFigure 8 the time domain of 1st stand 2nd stand and theinterstand tension are all gourd-shaped which consist withthe test results in [1] The time domain of the 2nd stand lagsbehind the 1st stand by the delay time Due to the time delayeffect the frequency domain of the fourth model is more
6 Shock and Vibration
Frequency (Hz)0 200 400
0
1000
2000
Am
plitu
de (P
a)
(e)
209
Frequency (Hz)
Am
plitu
de (P
a)
0 200 4000
100
200
(f)
209
0 200 4000
05
1
Frequency (Hz)
Am
plitu
de (m
)
(d)
209
times10minus9
0 005 01
0
5000
(b)
minus5000
d120590e1
(Pa)
t (s)
0 005 01
0
1000
(c)
minus1000
d120590d1
(Pa)
t (s)
0 005 01
0
1
(a)
minus1
times10minus9dhd1
(m)
t (s)
Figure 5 Time and frequency domain of the single 1st stand when V1199031= 2478ms ((a) and (d)) work roll ((b) and (e)) entry tension ((c)
and (f)) exit tension
complex than the thirdmodelThemain frequency is 209Hza little smaller than the third model but still in the range ofthe third octave mode chatter
Comparing the above four models a more detailedexplanation for regenerative chattermechanism in rolling canbe presented Consequent stands in tandem rolling mills arecoupled together through rolled piece One of these standsbecomes unstable and oscillates first for example the 119894thstand The 119894th stand vibration gives rise to the interstandtension vibration The interstand vibration makes the 119894 + 1thstand vibrate gently at onceThe amplitude of the 119894+1th standis far smaller than that of the 119894th stand by this time But thestrip gauge variation generated by the 119894th stand is passed onto the 119894+1th stand after delay time It aggravates the vibrationof the 119894 + 1th stand and makes the 119894th and 119894 + 1th standvibrate in the same amplitude The oscillation of the 119894 + 1thstand intensifies vibration of the interstand tension andfinallymakes the 119894th stand vibrate again
It can be concluded from the above analysis that timedelay effect as an independent factor has very limited effects
on both critical velocity and chatter frequency while makingthe stability analysis more complex But it does not mean thattime delay effect can be neglected when modeling the rollingsystem Enough works have been done to compare the firstsecond and the fourth model in [4] To study the time delayeffect as a single factor on multistand rolling system stabilityanalysis comparison of the third model and fourth modelwas done in the next section To simplify the writing modethe third model is called the nondelay system and the fourthmodel is called the delay system
4 Effects of Rolling Process Parameters
41 Effects of Friction Stability of a single stand is very sen-sitive to friction The critical velocity of a single stand growsrapidly with the increase of friction coefficient [2 7] Butthings are different for multistand systems as shown inFigure 9
When 1199061is small enough stability of the 1st stand gets
worse and it becomes more unstable than the 2nd stand
Shock and Vibration 7
Frequency (Hz)
Am
plitu
de (P
a)
0 200 400
(f)
209
251100
200
0
0 200 4000
05
1
Frequency (Hz)
Am
plitu
de (m
)
(d)
209
times10minus9
Frequency (Hz)
Am
plitu
de (m
)
0 200 4000
2
4
(e)
209251
times10minus11
0 005 01
0
1
(a)
minus1
times10minus9dhd1
(m)
t (s)
0 005 01
0
5
(b)
minus5
times10minus11
dhd2
(m)
t (s)
0 005 01
0
500
(c)
minus500
d120590d1
(Pa)
t (s)
Figure 6 The time and frequency domain of the second model V1199032= 3600ms ((a) and (d)) 1st stand work roll ((b) and (e)) 2nd stand
work roll ((c) and (f)) interstand tension
For example as shown in Table 2 when 1199061is 0014 the critical
velocity of single 1st stand is 229ms According to theprinciple of mass conservation the corresponding velocityof 2nd stand is 331ms which is smaller than the single2nd stand critical velocity 348ms The 1st stand becomesunstable before the 2nd standThe critical velocity of the two-stand system is determined by the 1st stand As 119906
1increases
the 1st stand becomes more stable and critical velocity of thetwo-stand system increases sharply until 119906
1reaches a certain
point
Table 2The critical velocities for different stands when 1199061changes
1199061
0014 0018Critical velocity of single 1st stand V
1198881
(ms) 229 248
Corresponding critical velocity ofsingle 2nd stand V
1198882(ms) 331 363
Critical velocity of single 2nd stand V1198882
(ms) 348 348
More unstable stand 1st stand 2nd stand
8 Shock and Vibration
Frequency (Hz)
Am
plitu
de (P
a)
0 200 4000
1000
2000
(f)
212
0 200 4000
05
1
Frequency (Hz)
Am
plitu
de (m
)
(d)
212
times10minus9
Frequency (Hz)
Am
plitu
de (m
)
0 200 4000
05
1
(e)
212
times10minus9
0 005 01
0
2
(b)
minus2
times10minus9
dhd2
(m)
t (s)
0 005 01
0
1
(a)
minus1
times10minus9
dhd1
(m)
t (s)
0 005 01
0
5000
(c)
minus5000
d120590d1
(Pa)
t (s)
Figure 7The time and frequency domain of the third model when V1199032= 3023ms ((a) and (d)) 1st stand work roll ((b) and (e)) 2nd stand
work roll ((c) and (f)) interstand tension
When 1199061is big enough 2nd stand becomesmore unstable
than 1st stand For example as shown in Table 2 when 1199061
is 0018 the critical velocity of the single 1st stand goes upto 248ms and the corresponding velocity of the single 2ndstand rises to 363ms which is bigger than critical velocityof the 2nd stand The 2nd stand becomes unstable first Thecritical velocity of the two-stand system is determined by the2nd stand As 119906
1increases the 1st stand becomesmore stable
but it has little effect on the 2nd stand The stability of 2ndstand is much worse than the 1st stand so the critical velocityof the two-stand system goes down as shown in Figure 9
When 1199061goes up the overall trend of critical velocities
are the same for the delay system and nondelay systemThere exists an optimal 119906
1 which relates to the maximum
critical velocity given that other rolling parameters remainunchanged But the optimal 119906
1values are different for the
delay and nondelay system The optimal 1199061for the delay
system is about 0016 while the optimal 1199061for the nondelay
system is about 002 Critical velocities for the delay systemare smaller than the nondelay system especially when 119906
1is
larger It can be explained by the idea that time delay effect hasa far greater impact on 2nd stand than 1st stand as the strip
Shock and Vibration 9
0 200 4000
50
100
Frequency (Hz)
Am
plitu
de (P
a)
(f)
209
0 200 4000
5
Frequency (Hz)A
mpl
itude
(m)
(e)
209
times10minus11
0 200 4000
5
Am
plitu
de (m
)
(d)
Frequency (Hz)
209
times10minus11
0 05 1
0
1
(a)
minus1
times10minus9dhd1
(m)
t (s)
0 05 1
0
2
(b)
minus2
times10minus9
dhd2
(m)
t (s)
0 05 1
0
2000
(c) t (s)
minus2000
d120590d1
(Pa)
Figure 8 The time and frequency domain of the fourth model when V1199032= 2925ms ((a) and (d)) 1st stand work roll ((b) and (e)) 2nd
stand work roll ((c) and (f)) interstand tension
gauge variation generated in the 1st stand is passed on to the2nd stand with delay time but not in the opposite direction
Figure 10 demonstrates the critical velocities for the delaysystem and nondelay system when the friction coefficient ofthe 2nd stand increases The critical velocity changing trendis much simpler When 119906
2is small enough the 2nd stand is
more unstable and the critical velocity increases as 1199062goes
up When 1199062is big enough the stability of the whole system
becomes better as 1199062goes up The critical velocities for the
delay system and nondelay system are almost the sameWhen1199062is relatively small the critical velocity of the time delay
system is slightly smaller than that of the nondelay systemThe smaller the 119906
2is the more unstable and sensitive to 119906
2
the time delay system is
42 Effects of Interstand Tension The critical velocity for asingle stand decreases as the entry tension and exit tensionincrease But the critical velocity ismore sensitive to the entry
tension due to the fact that the entry zone is much longer thatthe exit zone [2]1205901198891
is the intertension between the 1st and 2nd standnamely the exit tension of the 1st stand and the entry tensionof the 2nd stand The stability of the 1st and 2nd stand bothgoes worse with the increase of 120590
1198891 Therefore the critical
velocities of the delay and nondelay system both fall offas 1205901198891
increases as shown in Figure 11 But the stability ofthe 2nd stand is more sensitive to 120590
1198891 as 120590
1198891is the entry
tension for the 2nd stand As can been seen from Table 3when 120590
1198891is rather small the 1st stand becomes unstable first
and the critical velocity of the system is decided by the 1ststand The critical velocities decrease slowly as 120590
1198891is the exit
tension of the 1st stand But when 1205901198891
is big enough the 2ndstand becomes unstable first and the critical velocity of thetwo-stand system is decided by the 2nd stand The criticalvelocities of the delay system decrease fast with the increaseof 1205901198891 because 120590
1198891is the entry tension of the 2nd stand
10 Shock and Vibration
001 0015 002 0025 00323
24
25
26
27
28
29
Criti
cal s
peed
(ms
)
Delay systemNondelay system
u1
Figure 9 Comparison of stability charts for 1199061
001 0015 002 0025 00322
24
26
28
30
32
34
Criti
cal s
peed
(ms
)
Delay systemNondelay system
u2
Figure 10 Comparison of stability charts for 1199062
When considering the time delay effect the critical velocitygets smaller and more sensitive to 120590
1198891
43 Effects of Interstand Distance Interstand tension vari-ation becomes more gentle with the increase of interstanddistance And so the stability of mill stand becomes better[2] But it is more complex to multistand system 119871
1198902is the
distance between the 1st and 2nd stand If 1198711198902
is changedthe interstand tension variation and the delay time are bothchanged As shown in Figure 12 for time delay system withthe increase of 119871
1198902 the interstand tension variation decreases
and critical velocity of delay system risesMeantime the delaytime becomes larger and critical velocity decreases So thecritical velocities of time delay system form a wavy patternof rising
50 100 150 200 250 30025
26
27
28
29
30
31
32
Criti
cal s
peed
(ms
)
Delay systemNondelay system
120590d1 (MPa)
Figure 11 Comparison of stability charts for 1205901198891
3 35 4 45 5 55 623
24
25
26
27
28
29
30Cr
itica
l spe
ed (m
s)
Delay systemNondelay system
Le2 (m)
Figure 12 Comparison of stability charts for 1198711198902
Table 3 The critical velocities for different stands when 1205901198891
changes
1205901198891
(Mpa) 60 250Critical velocity of single 1st stand V
1198881(ms) 227 225
Corresponding critical velocity of single 2ndstand V
1198882(ms) 326 335
Critical velocity of single 2nd stand V1198882
(ms) 367 331
More unstable stand 1st stand 2nd stand
At first the 2nd stand is more unstable than the 1st oneWith the increase of 119871
1198902 the critical velocities of the 1st
and 2nd stand both rise But the critical velocity of 2ndstand goes up much faster than the 1st stand as 119871
1198902affects
Shock and Vibration 11
1 15 2 25 3 35 420
25
30
35
40
45
Criti
cal s
peed
(ms
)
Delay systemNondelay system
E (GPa)
Figure 13 Comparison of stability charts for 119864
05 1 1526
27
28
29
30
31
32
33
34
Criti
cal s
peed
(ms
)
Delay systemNondelay system
w (m)
Figure 14 Comparison of stability charts for 119908
the entry tension variation of 2nd standThe critical velocitiesof the nondelay and time delay system both rise fast When1198711198902
is bigger than a certain value the 1st stand becomesunstable first The rises of critical velocities for the nondelayand delay system both slow down for 119871
1198902affects the exit
tension variation of 1st stand It can been seen in Table 4 thatwhen 119871
1198902is 35m the 2nd stand goes to unstable first but
when 1198711198902
is 55m chatter occurs in the 1st stand first Butthe instability of the 1st stand occurs when 119871
1198902is quite bigger
than nondelay system for the delay time effect makes the 2ndstand more unstable
44 Effects of Strip Width and Strip Elastic Modulus Thecritical velocity of a single stand system decreases with theincrease of the strip elastic modulus and goes up along with
022 024 026 028 03 032 034 03624
26
28
30
32
34
36
38
40
Criti
cal s
peed
(ms
)
Delay systemNondelay system
he2 (mm)
Figure 15 Comparison of stability charts for ℎ1198902
Table 4 The critical velocities for different stands when 1198711198902
changes
1198711198902
(ms) 35 55Critical velocity of single 1st stand V
1198881
(ms) 2463 2487
Corresponding critical velocity of single2nd stand V
1198882(ms) 3605 3640
Critical velocity of single 2nd stand V1198882
(ms) 3138 3789
More unstable stand 2nd stand 1st stand
the increase of the strip width [2 7] As shown in Figures13 and 14 it shows the same trend for both the time delaysystem and nondelay systemWith the increase of strip elasticmodulus the critical velocities for the time delay system andnondelay system are almost the same The latter is just alittle bigger than the former With the increasing strip widththe critical velocities for the time delay system and nondelaysystem both go up But the critical velocities for the time delaysystem are a littlemore sensitive to the increase of strip width
45 Effects of Reduction Allocation It has been proved thatthe critical velocity for a single stand system decreaseswith the increase of the entry thickness given that the exitthickness stays the same and increase with the rise of the exitthickness given that the entry thickness remains the same[2] But it is more complex for multistand system as shownin Figure 15 Assuming that the entry strip thickness of the1st stand and the exit strip thickness of the 2nd stand remainunaltered the critical velocity would be changed if the entrythickness of the 2nd stand namely ℎ
1198902 is changed
If ℎ1198902
is small enough it means that exit strip thicknessof the 1st stand decreases and entry strip thickness of the 2ndstand increases The 1st stand becomes unstable first But asℎ1198902
rises the 1st stand reduction gets smaller while the 2nd
12 Shock and Vibration
Table 5The critical velocities for different standswhen ℎ1198902changes
ℎ1198902
(mm) 026 029Critical velocity of single 1st stand V
1198881
(ms) 242 252
Corresponding critical velocity of single2nd stand V
1198882(ms) 331 380
Critical velocity of single 2nd stand V1198882
(ms) 368 341
More unstable stand 1st stand 2nd stand
stand reduction gets bigger The 2nd stand becomes moreunstable than the 1st stand As shown in Table 5 when ℎ
1198902
is 026mm the 1st stand is more unstable However whenℎ1198902
increases to 029 the 2nd stand becomes more unstablethan 1st stand There exists an optimal value of ℎ
1198902for the
multistand system which is related to the maximum criticalvelocity as shown in Figure 11
The overall critical velocity changing trend for the timedelay system and nondelay system is the same But theoptimal values of ℎ
1198902are different for the two systems
because the stability of the 2nd stand becomes worse whenconsidering the time delay effect The optimal values of ℎ
1198902
for the two systems are 033mm and 028mm respectivelyObviously the optimal value of ℎ
1198902for the delay system is
more consistent with the actual production It suggests thatby redistributing the reduction of the tandem rolling millswe can improve the stability of the tandem rolling mills
The critical velocity of a single stand changing along withthe increase of the reduction is not linearWhen the reductionis small the critical velocity of the stand is very sensitiveto the reduction changing But as the reduction goes upthe decrease of critical velocity slows down So as shown inFigure 15 when ℎ
1198902is bigger than a certain point the critical
velocities of the two systems both go down slowlyIt has to be clarified that the optimal ℎ
1198902value for the two-
stand system may not work for five stand system There existfour delay times in five-stand tandem rolling mills whichmakes the relationship between stands more complex
5 Conclusions
In this paper the effects of interstand tension variation andstrip variation transportation between adjacent stands withtime delay on tandem rolling mills were studied in detailthrough comparison of different models Stability analy-sis shows that interstand tension couples adjacent standstogether and has just a little effect on the critical velocity of thechatter model The strip gauge variation passed on betweenstands reduces the critical velocity dramatically Howeverthe time delay effect has very limited effects on both thecritical velocity and the frequency But it does not mean thatthe time delay effect can be neglected when modeling therolling system By comparing the critical velocities of thedelay and nondelay system the influences of delay time asa single factor on multistand rolling system were studiedMore detailed and quantitative explanation is put forward to
the relationship of two adjacent stands In the big picturethe critical velocity of the time delay system is just slightlysmaller than the nondelay system But the delay time worsensthe stability of downstream stand and makes the relationshipbetween consecutive stands more complex The time delayeffect changes the optimal process parameters values aimedat obtaining the maximum critical velocity To get preferablerolling process parameter configuration for tandem rollingmills time delay effect must be involved
Appendix
Consider119889119865 = 119865
1119889ℎ119889+ 1198652119889ℎ119889+ 1198653119889120590119890+ 1198654119889120590119889+ 1198655119889ℎ119890
1198651= (
120597119865
120597ℎ119889
)
119904
= int
119909119899
119909119889
(
120597119901119889
120597ℎ119889
)
119904
119889119909 + int
119909119890
119909119899
(
120597119901119890
120597ℎ119889
)
119904
119889119909
+ (119896119891119890minus 120590119890) (
120597119909119890
120597ℎ119889
)
119904
1198652= (
120597119865
120597ℎ119889
)
119904
= int
119909119899
119909119889
(
120597119901119889
120597ℎ119889
)
119904
119889119909 minus (119896119891119889minus 120590119889)(
120597119909119889
120597ℎ119889
)
119904
1198653= (
120597119865
120597120590119890
)
119904
= int
119909119890
119909119899
(
120597119901119890
120597120590119890
)
119904
119889119909
1198654= (
120597119865
120597120590119889
)
119904
= int
119909119899
119909119889
(
120597119901119889
120597120590119889
)
119904
119889119909
1198655= (
120597119865
120597ℎ119890
)
119904
= int
119909119890
119909119899
(
120597119901119890
120597ℎ119890
)
119904
119889119909 + (119896119891119890minus 120590119890) (
120597119909119890
120597ℎ119890
)
119904
119889V119890= 1198751119889ℎ119889+ 1198752119889ℎ119889+ 1198753119889120590119890+ 1198754119889120590119889+ 1198755119889ℎ119890
1198751= (
120597V119890
120597ℎ119889
)
119904
=
V119903
ℎ119890
+
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597ℎ119889
)
119904
1198752= (
120597V119890
120597ℎ119889
)
119904
=
119909119890minus 119909119899
ℎ119890
+
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597ℎ119889
)
119904
1198753= (
120597V119890
120597120590119890
)
119904
=
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597120590119890
)
119904
1198754= (
120597V119890
120597120590119889
)
119904
=
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597120590119889
)
119904
1198755= (
120597V119890
120597ℎ119890
)
119904
= minus
V119903ℎ119899
ℎ119890
2+
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597ℎ119890
)
119904
Shock and Vibration 13
119889V119889= 1198761119889ℎ119889+ 1198762119889ℎ119889+ 1198763119889120590119890+ 1198764119889120590119889+ 1198765119889ℎ119890
1198761= (
120597V119889
120597ℎ119889
)
119904
= minus
V119890ℎ119890
ℎ119889
2+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119889
)
119904
1198762= (
120597V119889
120597ℎ119889
)
119904
=
minus119909119890
ℎ119889
+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119889
)
119904
1198763= (
120597V119889
120597120590119890
)
119904
=
ℎ119890
ℎ119889
(
120597V119890
120597120590119890
)
119904
1198764= (
120597V119889
120597120590119889
)
119904
=
ℎ119890
ℎ119889
(
120597V119890
120597120590119889
)
119904
1198765= (
120597V119889
120597ℎ119890
)
119904
=
V119890
ℎ119889
+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119890
)
119904
(A1)
The intermedia variables are given as follows
(
120597119909119899
120597ℎ119889
)
119904
= (minus
1198921
8119906
+
1
4119906
minus
119909119890
4ℎ119890
) sec (119892)2 + radic 119877
4ℎ119889
sdot tan (119892) + (120597119909119899
120597119909119890
)
119904
(
120597119909119890
120597ℎ119889
)
119904
+ (
120597119909119899
120597119896119891119889
)
119904
sdot (
120597119896119891119889
120597ℎ119889
)
119904
1198921 = log(ℎ119890119896119891119890
ℎ119889119896119891119889
119896119891119889minus 120590119889
119896119891119890minus 120590119890
)
119892 =
1
4
radicℎ119889
119877
(2radic
119877
ℎ119889
tanminus1 (119909119890
radic119877ℎ119889
)
+ 2radic
119877
ℎ119889
tanminus1 (119909119889
radic119877ℎ119889
) minus
1198921
119906
)
(
120597119909119899
120597119909119890
)
119904
=
ℎ119889sec (119892)2
2ℎ119890
(
120597119909119899
120597119896119891119889
)
119904
=
ℎ119889
4119906
(
1
119896119891119889
minus
1
119896119891119889minus 120590119889
) sec (119892)2
(
120597119896119891119889
120597ℎ119889
) = minus
1198991205900
ℎ119889
(119860 + log(ℎ0
ℎ119889
))
119899minus1
(
120597119909119899
120597ℎ119889
)
119904
=
1
2
sec (119892)2 (120597119909119889
120597ℎ119889
)
119904
(
120597119909119899
120597120590119890
)
119904
= minus
ℎ119889sec (119892)2
4119906 (119896119891119890minus 120590119890)
(
120597119909119899
120597120590119889
)
119904
=
ℎ119889sec (119892)2
4119906 (119896119891119889minus 120590119889)
(
120597119909119899
120597ℎ119890
)
119904
= minus
ℎ119889sec (119892)2
4119906ℎ119890
+ (
120597119909119899
120597119909119890
)
119904
(
120597119909119890
120597ℎ119890
)
119904
+ (
120597119909119899
120597119896119891119890
)
119904
(
120597119896119891119890
120597ℎ119890
)
119904
(
120597119909119899
120597119896119891119890
)
119904
=
ℎ119889
4119906
(
1
119896119891119890minus 120590119890
minus
1
119896119891119890
) sec (119892)2
(
120597119896119891119890
120597ℎ119890
)
119904
= minus
1198991205900
ℎ119890
(119860 + log(ℎ0
ℎ119890
))
119899minus1
(
120597119909119890
120597ℎ119889
)
119904
= minusradic
119877
4 (ℎ119890minus ℎ119889)
(
120597119909119890
120597ℎ119890
)
119904
= radic
119877
4 (ℎ119890minus ℎ119889)
(
120597119909119889
120597ℎ119889
)
119904
=
119877ℎ119889
2V119903ℎ119899
(
120597119901119890
120597ℎ119889
)
119904
= (
120597119901119890
120597ℎ
)
119904
+ (
120597119901119890
120597119896119891
)
119904
(
120597119896119891
120597ℎ119889
)
119904
+ (
120597119901119890
120597119867119890
)
119904
sdot (
120597119867119890
120597ℎ119889
)
119904
+ (
120597119901119890
120597119867
)
119904
(
120597119867
120597ℎ119889
)
119904
(
120597119901119890
120597ℎ
)
119904
=
(119896119891119890minus 120590119890) 119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119901119890
120597119896119891
)
119904
=
(119896119891119890minus 120590119890) ℎ119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119896119891
120597ℎ119889
)
119904
= minus
1205900119899
ℎ
(119860 + ln(ℎ0
ℎ
))
119899minus1
(
120597119901119890
120597119867119890
)
119904
=
(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119867119890
120597ℎ119889
)
119904
= minus
119877arctan (119909119890radic119877ℎ
119889)
ℎ119889
2
radic119877ℎ119889
minus
119909119890
ℎ119889ℎ119890
+ (
120597119867119890
120597119909119890
)
119904
(
120597119909119890
120597ℎ119889
)
119904
(
120597119901119890
120597119867
)
119904
= minus
(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119867
120597ℎ119889
)
119904
= minus
119877arctan (119909radic119877ℎ119889)
ℎ119889
2
radic119877ℎ119889
minus
119909
ℎ119889ℎ
(
120597119901119890
120597120590119890
)
119904
= minus
ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
14 Shock and Vibration
(
120597119901e120597ℎe)
119904
= (
120597119901e120597119896119891119890
)
119904
(
120597119896119891119890
120597ℎ119890
)
119904
+ (
120597119901e120597119867119890
)
119904
(
120597119867119890
120597119909119890
)
119904
sdot (
120597119909119890
120597ℎ119890
)
119904
minus
(119896119891119890minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)
ℎ119890
2
119896119891119890
(
120597119901119890
120597119896119891119890
)
119904
=
ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
minus
(119896119891119890 minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
2
(
120597119867119890
120597119909119890
)
119904
=
2
ℎ119890
(
120597119901d120597ℎ119889
)
119904
= (
120597119901119889
120597119896119891119889
)
119904
(
120597119896119891119889
120597ℎ119889
)
119904
+ (
120597119901119889
120597ℎ
)
119904
+ (
120597119901119889
120597119896119891
)
119904
(
120597119896119891
120597ℎ119889
)
119904
+ (
120597119901119889
120597119867
)
119904
(
120597119867
120597ℎ119889
)
119904
minus
(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889
2
119896119891119889
(
120597119901119889
120597119896119891119889
)
119904
=
ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
minus
(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
2
(
120597119901119889
120597ℎ
)
119904
=
(119896119891119889minus 120590119889) 119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119896119891
)
119904
=
(119896119891119889minus 120590119889) ℎ119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119867119889
)
119904
= minus
(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119867
)
119904
=
(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597ℎ119889
)
119904
= (
120597119901119889
120597119867119889
)
119904
(
120597119867119889
120597119909119889
)
119904
(
120597119909119889
120597ℎ119889
)
119904
(
120597119867119889
120597119909119889
)
119904
=
2
ℎ119889
(
120597119901119889
120597120590119889
)
119904
= minus
ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
119867119890= 2radic
119877
ℎ119889
tanminus1 (119909119890
radic119877ℎ119889
)
119867119889= 2radic
119877
ℎ119889
tanminus1 (119909119889
radic119877ℎ119889
)
119867 = 2radic
119877
ℎ119889
tanminus1 ( 119909
radic119877ℎ119889
)
(A2)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors gratefully acknowledge the support of theNational Natural Science Foundation of China no 51175035PhD Programs Foundation of Ministry of Education ofChina no 20100006110024 the Fundamental Research Fundsfor the Central Universities no FRF-BR-14-006A and BeijingHigher EducationYoungElite Teacher Project no YETP0367
References
[1] I S YunW R DWilson and K F Ehmann ldquoReview of chatterstudies in cold rollingrdquo International Journal of Machine Toolsand Manufacture vol 38 no 12 pp 1499ndash1530 1998
[2] J X Zhou TandemMill Vibration Control Metallurgical Indus-try Press Beijing China 1st edition 1998
[3] G Zhiying Z Yong and Z Lingqiang ldquoReview of modellingand theoretical studies on chatter in the rolling millsrdquo Journalof Mechanical Engineering vol 51 no 16 pp 87ndash105 2015
[4] H Zhao and K F Ehmann ldquoStability analysis of chatter intandem rolling millsmdashpart 2 the regenerative effectrdquo Journalof Manufacturing Science and EngineeringmdashTransactions of theASME vol 135 no 3 Article ID 031002 11 pages 2013
[5] P-H Hu and K F Ehmann ldquoRegenerative effect in rollingchatterrdquo Journal ofManufacturing Processes vol 3 no 2 pp 82ndash93 2001
[6] P H Hu Stability and Chatter in Rolling Northwestern Univer-sity Evanston Ill USA 1998
[7] H Y Zhao and K F Ehmann ldquoStability analysis of chatter intandem rolling millsmdashpart 1 single- and multi-stand negativedamping effectrdquo Journal ofManufacturing Science and Engineer-ing vol 135 no 3 Article ID 031001 2013
[8] Y Kimura Y Sodani N Nishiura et al ldquoAnalysis of chaffer intandem cold rolling millsrdquo ISIJ International vol 43 no 1 pp77ndash84 2003
[9] M R Niroomand R M Forouzan andM Salimi ldquoTheoreticaland experimental analysis of chatter in tandem cold rollingmills based on wave propagation theoryrdquo ISIJ International vol55 no 3 pp 637ndash646 2015
[10] A Heidari M R Forouzan and S Akbarzadeh ldquoDevelopmentof a rolling chattermodel considering unsteady lubricationrdquo ISIJInternational vol 54 no 1 pp 165ndash170 2014
[11] A Heidari and M R Forouzan ldquoOptimization of cold rollingprocess parameters in order to increasing rolling speed limitedby chatter vibrationsrdquo Journal of Advanced Research vol 4 no1 pp 27ndash34 2013
Shock and Vibration 15
[12] B BahramiNejadMDehghani and S AMousavi ldquoSimulationof two stands cold rolling mill process using neural networksand genetic algorithms in combination to avoid the chatterphenomenonrdquo Majlesi Journal of Electrical Engineering vol 9no 1 pp 21ndash24 2014
[13] X X Liao Theory Methods and Application of Stability Huaz-hong Science and Engineering University PressWuhan China1999
[14] V B Kolmanovskii and V R Nosov Stability of FunctionalDifferential Equations Academic Press New York NY USA1986
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2 Shock and Vibration
becomes a delay differential system and integral criterion wasemployed to find its critical velocity Comparative analysisof the three models presented a potential explanation to theregenerative mechanism [4]
Based on Zhao and Ehmannrsquos andHursquosmodel more com-plex regenerative chattermodels are proposed in recent yearsKimura et al constructed a five-stand regenerative chattermodel by coupling a five-degree freedom mill stand modeland the dynamic rollingmodel built in [7] togetherThe simu-lation results show that there exists an optimal friction of thefifth stand which relates to the maximum critical velocity ofthe whole tandem rolling mills [8] Instead of static analysisNiroomand et al used wave propagation theory in elasticsolids to formulate the dynamic tensile stress variationbetween two consequent stands and built a new two-standregenerative chatter model The simulation results show thatthe errors of critical velocity and chatter frequency are smallerwith the experimental results when the wave propagationtheory is applied [9] Considering the complex friction con-dition in the rolling gap Heidari et al built a regenerativechatter model using the unsteady lubrication model assum-ing that the rolling gap is in full film regime The effects ofrolling lubricant on the stability of tandem rolling mills werediscussed [10] Based on these regenerative chatter modelsoptimization of multistand rolling process parameters wereconducted to avoid chatter phenomenon using the combina-tion of neural networks and genetic algorithms [11 12]
All these regenerative chatter models have considered thetime delay effect which makes the tandem rolling mills adelay differential system Itmeans thatmore complex stabilitycriterion has to be employed to analyze the delay differen-tial multistand system which results in huge computationHowever the time delay effect on the stability of tandemrolling mills has not been analyzed as an independent factorMore research work has to be done to decide whether theeffect of time delay itself on the stability of tandem rollingmills is worth the great amount of computation Besidesthe relationship between consequent stands is not clearlydescribed though the interaction factors between stands havebeen pointed out The major objective pursed in this paper isto investigate the time delay effect as an independent factorin cold rolling tandem mills and put forward more detailedexplanation to the regenerative chatter mechanism and therelations between consecutive stands
2 Chatter Model
The dynamic rolling gap is shown as in Figure 1 when onlythe vertical roll vibration is considered Set the coordinates 119909and 119910 onto the center line of rolled piece and the work rollcenter line And the intersection point is set as the origin ofcoordinates 120590
119890and 120590
119889are the entry and exit tensile stress
V119890and V119889are the entry and exit strip velocity ℎ
119890and ℎ
119889are
the strip thickness at entry and exit V119903is the work roll linear
velocity ℎ119899is the strip thickness at neutral point 119909
119890 119909119899 and
119909119889are the position of entry point neutral point and exit
point
y
x
e
120590e he hn hd
d
R120601
120590d
xexn
xd
+
Figure 1 Dynamic rolling gap
The strip thickness at any 119909 location in the rolling gap isas follows
ℎ = ℎ119889+
1199092
119877
(1)
Due to the roll vertical vibration the continuity equationis modified as
Vℎ = V119890ℎ119890+ (119909 minus 119909
119890)ℎ119889 (2)
According to (1) and (2) the strip entry velocity V119890 strip
exit velocity V119889 strip entry position 119909
119890 and exit position 119909
119889
can be calculated as follows
V119890=
1
ℎ119890
(V119903ℎ119889+
V119903119909119899
2
119877
+ (119909119890minus 119909119899)ℎ119889)
V119889=
V119890ℎ119890+ (119909119889minus 119909119890)ℎ119889
ℎ119889+ 119909119889
2119877
119909119890= radic119877 (ℎ
119890minus ℎ119889)
119909119889=
119877ℎ119889
ℎ119889
2 (V119890ℎ119890minus 119909119890
ℎ119889)
(3)
Figure 2 illustrates the stresses acting on a vertical slabelement inside the rolling gap Employing the coulombfriction model 120591
119904= 120583119901 and the yield criterion 120590
119909= 119896119891minus 119901
the equilibrium equation for an elemental vertical section ofthe strip in the rolling gap in the 119909 direction is obtained
119889119901
119889119909
∓
2120583119901
ℎ
minus
1
ℎ
119889 (ℎ119896119891)
119889119909
= 0(4)
where the positive sign is for the exit side and the negativesign is for the entry side
Considering the strain hardening effect the deformationresistance of the rolled strip can be expressed as follows
119896119891= 1205900(119860 + ln(
ℎ0
ℎ
))
119899
(5)
Shock and Vibration 3
p p
120591120591
120590x 120590x
h h
120591 120591
p p
120590x +120597120590x120597x
dx 120590x +120597120590x120597x
dx
h +120597h
120597xh +
120597h
120597x
Figure 2 Slab analysis on a volume element of the entry region andexit region
1205900 119860 and 119899 are the material property parameter of the
rolled strip and can be decided by experiments ℎ0is the entry
strip thickness of multistand rolling millsApplying the boundary conditions
1199011003816100381610038161003816119909=119909119890
= 119896119891119890minus 120590119890
1199011003816100381610038161003816119909=119909119889
= 119896119891119889minus 120590119889
(6)
The rolling pressure at entry and exit side can be obtained
119901119890= (119896119891119890minus 120590119890)
ℎ119896119891
ℎ119890119896119891119890
exp(119906(2radic 119877ℎ119889
tanminus1 (119909119890
radic119877ℎ119889
)
minus 2radic
119877
ℎ119889
tanminus1 ( 119909
radic119877ℎ119889
)))
119901119889= (119896119891119889minus 120590119889)
ℎ119896119891
ℎ119889119896119891119889
exp(119906(2radic 119877ℎ119889
tanminus1 ( 119909
radic119877ℎ119889
)
minus 2radic
119877
ℎ119889
tanminus1 (119909119889
radic119877ℎ119889
)))
(7)
Letting 119901119890= 119901119889 the neutral point can be formulated as
follows
119909119899= radic119877ℎ
119889tan
1
4
radicℎ119889
119877
(2radic
119877
ℎ119889
tanminus1 (119909119890
radic119877ℎ119889
)
+ 2radic
119877
ℎ119889
tanminus1 (119909119889
radic119877ℎ119889
)
minus
1
120583
ln(119896119891119889minus 120590119889
119896119891119890minus 120590119890
ℎ119890119896119891119890
ℎ119889119896119891119889
))
(8)
By integrating the rolling stresses the rolling force can bedetermined
119865 = int
119909119890
119909119899
119901119890119889119909 + int
119909119899
119909119889
119901119889119889119909 (9)
M
KC
1
2dhd
dF
Figure 3 Mill stand structure model
The dynamic rolling process model can be obtained byutilizing the first-order Taylor expansion
119889119865 = 1198651119889ℎ119889+ 1198652119889ℎ119889+ 1198653119889120590119890+ 1198654119889120590119889+ 1198655119889ℎ119890
119889V119890= 1198751119889ℎ119889+ 1198752119889ℎ119889+ 1198753119889120590119890+ 1198754119889120590119889+ 1198755119889ℎ119890
119889V119889= 1198761119889ℎ119889+ 1198762119889ℎ119889+ 1198763119889120590119890+ 1198764119889120590119889+ 1198765119889ℎ119890
(10)
119865119895119875119895119876119895 119895 = 1 2 3 4 5 are the partial derivatives of119865 V
119890 V119889
with respect to ℎ119889 ℎ119889 120590119890 120590119889 ℎ119890 respectively The calculation
of 119865119895 119875119895 119876119895 119895 = 1 2 3 4 5 can be found in the Appendix
Considering the work roll flattening effect in rolling theequivalent work roll radius is calculated by employing theHitchcock formula
119877 = 119877119908(1 +
16 (1 minus 1205822
)
120587119864119908
119865
119903
) (11)
where 119877119908is initial work roll radius 120582 is Poissonrsquos ratio for the
work roll material 119865 is rolling pressure 119903 is reduction and119864119908is elasticity modulus of work roll materialChatter in rolling is the result of the interaction between
rolling process and mill stand structure The rolling processmodel and stand structure model are coupled togetherthrough mechanical rolling parameters A one degree offreedom mill stand structure model is shown in Figure 3
By coupling the rolling processmodel and stand structuremodel together the single stand vibrationmodel is as follows
119872119889ℎ119889+ 119862119889
ℎ119889+ 119870119889ℎ
119889= 2119908119889119865
119889119890=
119864
119871119890
119889V119890
119889120590119889= minus
119864
119871119889
119889V119889
(12)
where 119871119890is the distance between the stand and the upstream
stand and 119871119889is the distance between the stand and down-
stream stand
4 Shock and Vibration
Table 1 The initial rolling process parameters for the 1st and 2nd stand
ℎ1198901
(mm) ℎ1198891
ℎ1198902
(mm) ℎ1198892
(mm) 1205901198901
(Mpa) 1205901198891
(Mpa) 1205901198902
(Mpa)045 028 028 019 137 137 1371205901198892
(Mpa) 1199061
1199062
1198771199081
(mm) 1198771199082
(mm) 1205900(Mpa) 119860
98 0018 0015 276 291 8116 8116119899 119871
1198901(m) 119871
1198902(m) 119871
1198892(m) 119864
119908(Gpa) 119864 (Gpa) 119908 (m)
024 45 45 45 210 210 09
The regenerative chatter model for a two-stand rollingmill is as follows
119872119889ℎ1198891+ 119862119889
ℎ1198891+ 119870119889ℎ
1198891= 2119908119889119865
1
1198891198901=
119864
1198711198901
119889V1198901
1198891198891=
119864
1198711198902
(119889V1198902minus 119889V1198891)
119872119889ℎ1198892+ 119862119889
ℎ1198892+ 119870119889ℎ
1198892= 2119908119889119865
2
1198891198892= minus
119864
1198711198892
119889V1198892
(13)
where the second subscripts stand for the number of stands119864 is the elasticity modulus of roll piece material and 119908 is therolled strip width The rolled strip width spread is neglectedSo 119908 is constant
The relationship between the 2nd stand entry strip gaugevariation and the 1st stand exit strip gauge variation is asfollows
119889ℎ1198902(119905) = 119889ℎ
1198891(119905 minus 120591) (14)
where 120591 is the delay time decided by the interstand distanceand the entry velocity of 2nd stand
120591 =
1198711198902
V1198902
(15)
The initial rolling process parameters for the 1st and 2ndstand come from [7] and are shown in Table 1
To explore the effects of interstand coupling factors onregenerative chatter in detail four models were built in thispaperThe first one is a single stand vibrationmodelThe sec-ond one is a two-stand regenerative chattermodel which onlyconsiders the interstand tension coupling effectThe third oneis a two-stand regenerative chatter model which considersthe interstand tension coupling effect and the strip gaugevariation passed on to next stand but neglects the delay timeThe fourth one is a two-stand regenerative chatter modelwhich considers the interstand tension coupling effect and thestrip gauge variation passed on to next stand with time delayeffect
3 Stability Analysis
31 Stability Criterion To study the stabilities of the fourmodels proposed in Section 2 different stability criteria have
to be used to calculate the critical velocity for each modelThe first threemodels are nondelay differential systemsTheircritical velocities can be calculated by employing the Lya-punov indirect method According to the Lyapunov indirectmethod for a linear system = 119860119909 if all the real parts ofeigenvalues of matrix 119860 are negative the system is stable ifnot the system is unstable [13] Based on this method criticalvelocities of the first three models are obtained namely3482ms for the single 2nd stand model 3600ms for thesecond model and 3023ms for the third model Similar tothe single 2nd stand model the critical velocity of the single1st stand is 2478ms
The fourth model is a delay differential system Thestability criterion of time delay system is muchmore complexthan nondelay system The integral criterion is employed toanalyze the fourth model According to the integral criterionif the following inequation is true the delay differentialsystem is asymptotic stable [14]
int
infin
0
119871 (119908) gt
(119873 minus 1)
2
120587 (16)
where
119871 (119908) =
119877 (119908) 1198781015840
(119908) minus 119878 (119908) 1198771015840
(119908)
119877 (119908)2
+ 119878 (119908)2
(17)
119877(119908) and 119878(119908) are the real and imaginary parts of thedeterminant of the systemmatrix respectively119873 is the orderof the system For the fourth model 119873 is 7 The integrationis calculated with the help of MATLAB When the linearvelocity of the 2nd stand is 292ms
int
5000
0
119871 (119908) = 107 (18)
When the linear velocity of the 2nd stand is 293ms
int
5000
0
119871 (119908) = 45 (19)
So the critical velocity of fourth model is about 2925msComparing the critical velocities of the four models it
can be concluded that the interstand tension has just a littleeffect on the critical velocity of the chatter model The stripgauge variation passed on between stands reduces the criticalvelocity dramatically However the time delay effect as anindependent factor on critical velocity is very limited
Shock and Vibration 5
Frequency (Hz)0 200 400
0
1000
2000
Am
plitu
de (P
a)(e)
246
Frequency (Hz)
Am
plitu
de (P
a)
0 200 4000
200
400
(f)
246
0 200 4000
05
1
Frequency (Hz)
Am
plitu
de (m
)
(d)
246
times10minus9
0 005 01
0
1
(a)
minus1
times10minus9dhd2
(m)
t (s)
0 005 01
0
5000
(b)
minus5000
d120590e2
(Pa)
t (s)
0 005 01
0
1000
(c)
minus1000
d120590d2
(Pa)
t (s)
Figure 4 Time and frequency domain of the single 2nd stand when V1199032= 3482ms ((a) and (d)) work roll ((b) and (e)) entry tension ((c)
and (f)) exit tension
32 Simulation and Analysis To study the regenerative chat-ter mechanism and relationship between stands in detailtime and frequency domain simulations of the four modelswere conducted As shown in Figure 4 the single 2nd standvibrates periodically when V
1199032is 3482ms and the frequency
is 246Hz Similarly the single 1st stand vibrates periodicallywhen V
1199031is 2478ms and the frequency is 209Hz as shown
in Figure 5The second model only considers the interstand tension
coupling effect between stands As can been seen in Figure 6the 1st stand takes on periodic oscillation state and thefrequency is 209Hz It is the same with the single 1st standmodel The beat phenomena appear in the 2nd stand and theinterstand tension The frequencies are 209Hz and 251Hz209Hz is the frequency of the 1st stand and 251Hz isapproximate to the frequency of the 2nd stand It is obviousthat the beat phenomena are caused by the interstand tensioncoupling effect between the 1st stand and 2nd stand Besidesthe amplitude of 2nd stand is far less than the 1st stand
It suggests that the vibration in 2nd stand raised by theinterstand tension variation is not violent
The third model considers the interstand tension cou-pling effect and the strip gauge variation passed on to the nextstand but neglects the time delay effect As shown in Figure 7the beat phenomena in the second model disappear Thefrequency of the 1st stand the 2nd stand and the interstandtension are all 212HzThe amplitudes of the 1st and 2nd standare in the same magnitude It indicates that the strip gaugevariation passed on to the next stand makes the 2nd standvibrate more violently
The fourth model considers the interstand tension cou-pling effect between stands the strip gauge variation passedon to the next stand and the time delay effect As shown inFigure 8 the time domain of 1st stand 2nd stand and theinterstand tension are all gourd-shaped which consist withthe test results in [1] The time domain of the 2nd stand lagsbehind the 1st stand by the delay time Due to the time delayeffect the frequency domain of the fourth model is more
6 Shock and Vibration
Frequency (Hz)0 200 400
0
1000
2000
Am
plitu
de (P
a)
(e)
209
Frequency (Hz)
Am
plitu
de (P
a)
0 200 4000
100
200
(f)
209
0 200 4000
05
1
Frequency (Hz)
Am
plitu
de (m
)
(d)
209
times10minus9
0 005 01
0
5000
(b)
minus5000
d120590e1
(Pa)
t (s)
0 005 01
0
1000
(c)
minus1000
d120590d1
(Pa)
t (s)
0 005 01
0
1
(a)
minus1
times10minus9dhd1
(m)
t (s)
Figure 5 Time and frequency domain of the single 1st stand when V1199031= 2478ms ((a) and (d)) work roll ((b) and (e)) entry tension ((c)
and (f)) exit tension
complex than the thirdmodelThemain frequency is 209Hza little smaller than the third model but still in the range ofthe third octave mode chatter
Comparing the above four models a more detailedexplanation for regenerative chattermechanism in rolling canbe presented Consequent stands in tandem rolling mills arecoupled together through rolled piece One of these standsbecomes unstable and oscillates first for example the 119894thstand The 119894th stand vibration gives rise to the interstandtension vibration The interstand vibration makes the 119894 + 1thstand vibrate gently at onceThe amplitude of the 119894+1th standis far smaller than that of the 119894th stand by this time But thestrip gauge variation generated by the 119894th stand is passed onto the 119894+1th stand after delay time It aggravates the vibrationof the 119894 + 1th stand and makes the 119894th and 119894 + 1th standvibrate in the same amplitude The oscillation of the 119894 + 1thstand intensifies vibration of the interstand tension andfinallymakes the 119894th stand vibrate again
It can be concluded from the above analysis that timedelay effect as an independent factor has very limited effects
on both critical velocity and chatter frequency while makingthe stability analysis more complex But it does not mean thattime delay effect can be neglected when modeling the rollingsystem Enough works have been done to compare the firstsecond and the fourth model in [4] To study the time delayeffect as a single factor on multistand rolling system stabilityanalysis comparison of the third model and fourth modelwas done in the next section To simplify the writing modethe third model is called the nondelay system and the fourthmodel is called the delay system
4 Effects of Rolling Process Parameters
41 Effects of Friction Stability of a single stand is very sen-sitive to friction The critical velocity of a single stand growsrapidly with the increase of friction coefficient [2 7] Butthings are different for multistand systems as shown inFigure 9
When 1199061is small enough stability of the 1st stand gets
worse and it becomes more unstable than the 2nd stand
Shock and Vibration 7
Frequency (Hz)
Am
plitu
de (P
a)
0 200 400
(f)
209
251100
200
0
0 200 4000
05
1
Frequency (Hz)
Am
plitu
de (m
)
(d)
209
times10minus9
Frequency (Hz)
Am
plitu
de (m
)
0 200 4000
2
4
(e)
209251
times10minus11
0 005 01
0
1
(a)
minus1
times10minus9dhd1
(m)
t (s)
0 005 01
0
5
(b)
minus5
times10minus11
dhd2
(m)
t (s)
0 005 01
0
500
(c)
minus500
d120590d1
(Pa)
t (s)
Figure 6 The time and frequency domain of the second model V1199032= 3600ms ((a) and (d)) 1st stand work roll ((b) and (e)) 2nd stand
work roll ((c) and (f)) interstand tension
For example as shown in Table 2 when 1199061is 0014 the critical
velocity of single 1st stand is 229ms According to theprinciple of mass conservation the corresponding velocityof 2nd stand is 331ms which is smaller than the single2nd stand critical velocity 348ms The 1st stand becomesunstable before the 2nd standThe critical velocity of the two-stand system is determined by the 1st stand As 119906
1increases
the 1st stand becomes more stable and critical velocity of thetwo-stand system increases sharply until 119906
1reaches a certain
point
Table 2The critical velocities for different stands when 1199061changes
1199061
0014 0018Critical velocity of single 1st stand V
1198881
(ms) 229 248
Corresponding critical velocity ofsingle 2nd stand V
1198882(ms) 331 363
Critical velocity of single 2nd stand V1198882
(ms) 348 348
More unstable stand 1st stand 2nd stand
8 Shock and Vibration
Frequency (Hz)
Am
plitu
de (P
a)
0 200 4000
1000
2000
(f)
212
0 200 4000
05
1
Frequency (Hz)
Am
plitu
de (m
)
(d)
212
times10minus9
Frequency (Hz)
Am
plitu
de (m
)
0 200 4000
05
1
(e)
212
times10minus9
0 005 01
0
2
(b)
minus2
times10minus9
dhd2
(m)
t (s)
0 005 01
0
1
(a)
minus1
times10minus9
dhd1
(m)
t (s)
0 005 01
0
5000
(c)
minus5000
d120590d1
(Pa)
t (s)
Figure 7The time and frequency domain of the third model when V1199032= 3023ms ((a) and (d)) 1st stand work roll ((b) and (e)) 2nd stand
work roll ((c) and (f)) interstand tension
When 1199061is big enough 2nd stand becomesmore unstable
than 1st stand For example as shown in Table 2 when 1199061
is 0018 the critical velocity of the single 1st stand goes upto 248ms and the corresponding velocity of the single 2ndstand rises to 363ms which is bigger than critical velocityof the 2nd stand The 2nd stand becomes unstable first Thecritical velocity of the two-stand system is determined by the2nd stand As 119906
1increases the 1st stand becomesmore stable
but it has little effect on the 2nd stand The stability of 2ndstand is much worse than the 1st stand so the critical velocityof the two-stand system goes down as shown in Figure 9
When 1199061goes up the overall trend of critical velocities
are the same for the delay system and nondelay systemThere exists an optimal 119906
1 which relates to the maximum
critical velocity given that other rolling parameters remainunchanged But the optimal 119906
1values are different for the
delay and nondelay system The optimal 1199061for the delay
system is about 0016 while the optimal 1199061for the nondelay
system is about 002 Critical velocities for the delay systemare smaller than the nondelay system especially when 119906
1is
larger It can be explained by the idea that time delay effect hasa far greater impact on 2nd stand than 1st stand as the strip
Shock and Vibration 9
0 200 4000
50
100
Frequency (Hz)
Am
plitu
de (P
a)
(f)
209
0 200 4000
5
Frequency (Hz)A
mpl
itude
(m)
(e)
209
times10minus11
0 200 4000
5
Am
plitu
de (m
)
(d)
Frequency (Hz)
209
times10minus11
0 05 1
0
1
(a)
minus1
times10minus9dhd1
(m)
t (s)
0 05 1
0
2
(b)
minus2
times10minus9
dhd2
(m)
t (s)
0 05 1
0
2000
(c) t (s)
minus2000
d120590d1
(Pa)
Figure 8 The time and frequency domain of the fourth model when V1199032= 2925ms ((a) and (d)) 1st stand work roll ((b) and (e)) 2nd
stand work roll ((c) and (f)) interstand tension
gauge variation generated in the 1st stand is passed on to the2nd stand with delay time but not in the opposite direction
Figure 10 demonstrates the critical velocities for the delaysystem and nondelay system when the friction coefficient ofthe 2nd stand increases The critical velocity changing trendis much simpler When 119906
2is small enough the 2nd stand is
more unstable and the critical velocity increases as 1199062goes
up When 1199062is big enough the stability of the whole system
becomes better as 1199062goes up The critical velocities for the
delay system and nondelay system are almost the sameWhen1199062is relatively small the critical velocity of the time delay
system is slightly smaller than that of the nondelay systemThe smaller the 119906
2is the more unstable and sensitive to 119906
2
the time delay system is
42 Effects of Interstand Tension The critical velocity for asingle stand decreases as the entry tension and exit tensionincrease But the critical velocity ismore sensitive to the entry
tension due to the fact that the entry zone is much longer thatthe exit zone [2]1205901198891
is the intertension between the 1st and 2nd standnamely the exit tension of the 1st stand and the entry tensionof the 2nd stand The stability of the 1st and 2nd stand bothgoes worse with the increase of 120590
1198891 Therefore the critical
velocities of the delay and nondelay system both fall offas 1205901198891
increases as shown in Figure 11 But the stability ofthe 2nd stand is more sensitive to 120590
1198891 as 120590
1198891is the entry
tension for the 2nd stand As can been seen from Table 3when 120590
1198891is rather small the 1st stand becomes unstable first
and the critical velocity of the system is decided by the 1ststand The critical velocities decrease slowly as 120590
1198891is the exit
tension of the 1st stand But when 1205901198891
is big enough the 2ndstand becomes unstable first and the critical velocity of thetwo-stand system is decided by the 2nd stand The criticalvelocities of the delay system decrease fast with the increaseof 1205901198891 because 120590
1198891is the entry tension of the 2nd stand
10 Shock and Vibration
001 0015 002 0025 00323
24
25
26
27
28
29
Criti
cal s
peed
(ms
)
Delay systemNondelay system
u1
Figure 9 Comparison of stability charts for 1199061
001 0015 002 0025 00322
24
26
28
30
32
34
Criti
cal s
peed
(ms
)
Delay systemNondelay system
u2
Figure 10 Comparison of stability charts for 1199062
When considering the time delay effect the critical velocitygets smaller and more sensitive to 120590
1198891
43 Effects of Interstand Distance Interstand tension vari-ation becomes more gentle with the increase of interstanddistance And so the stability of mill stand becomes better[2] But it is more complex to multistand system 119871
1198902is the
distance between the 1st and 2nd stand If 1198711198902
is changedthe interstand tension variation and the delay time are bothchanged As shown in Figure 12 for time delay system withthe increase of 119871
1198902 the interstand tension variation decreases
and critical velocity of delay system risesMeantime the delaytime becomes larger and critical velocity decreases So thecritical velocities of time delay system form a wavy patternof rising
50 100 150 200 250 30025
26
27
28
29
30
31
32
Criti
cal s
peed
(ms
)
Delay systemNondelay system
120590d1 (MPa)
Figure 11 Comparison of stability charts for 1205901198891
3 35 4 45 5 55 623
24
25
26
27
28
29
30Cr
itica
l spe
ed (m
s)
Delay systemNondelay system
Le2 (m)
Figure 12 Comparison of stability charts for 1198711198902
Table 3 The critical velocities for different stands when 1205901198891
changes
1205901198891
(Mpa) 60 250Critical velocity of single 1st stand V
1198881(ms) 227 225
Corresponding critical velocity of single 2ndstand V
1198882(ms) 326 335
Critical velocity of single 2nd stand V1198882
(ms) 367 331
More unstable stand 1st stand 2nd stand
At first the 2nd stand is more unstable than the 1st oneWith the increase of 119871
1198902 the critical velocities of the 1st
and 2nd stand both rise But the critical velocity of 2ndstand goes up much faster than the 1st stand as 119871
1198902affects
Shock and Vibration 11
1 15 2 25 3 35 420
25
30
35
40
45
Criti
cal s
peed
(ms
)
Delay systemNondelay system
E (GPa)
Figure 13 Comparison of stability charts for 119864
05 1 1526
27
28
29
30
31
32
33
34
Criti
cal s
peed
(ms
)
Delay systemNondelay system
w (m)
Figure 14 Comparison of stability charts for 119908
the entry tension variation of 2nd standThe critical velocitiesof the nondelay and time delay system both rise fast When1198711198902
is bigger than a certain value the 1st stand becomesunstable first The rises of critical velocities for the nondelayand delay system both slow down for 119871
1198902affects the exit
tension variation of 1st stand It can been seen in Table 4 thatwhen 119871
1198902is 35m the 2nd stand goes to unstable first but
when 1198711198902
is 55m chatter occurs in the 1st stand first Butthe instability of the 1st stand occurs when 119871
1198902is quite bigger
than nondelay system for the delay time effect makes the 2ndstand more unstable
44 Effects of Strip Width and Strip Elastic Modulus Thecritical velocity of a single stand system decreases with theincrease of the strip elastic modulus and goes up along with
022 024 026 028 03 032 034 03624
26
28
30
32
34
36
38
40
Criti
cal s
peed
(ms
)
Delay systemNondelay system
he2 (mm)
Figure 15 Comparison of stability charts for ℎ1198902
Table 4 The critical velocities for different stands when 1198711198902
changes
1198711198902
(ms) 35 55Critical velocity of single 1st stand V
1198881
(ms) 2463 2487
Corresponding critical velocity of single2nd stand V
1198882(ms) 3605 3640
Critical velocity of single 2nd stand V1198882
(ms) 3138 3789
More unstable stand 2nd stand 1st stand
the increase of the strip width [2 7] As shown in Figures13 and 14 it shows the same trend for both the time delaysystem and nondelay systemWith the increase of strip elasticmodulus the critical velocities for the time delay system andnondelay system are almost the same The latter is just alittle bigger than the former With the increasing strip widththe critical velocities for the time delay system and nondelaysystem both go up But the critical velocities for the time delaysystem are a littlemore sensitive to the increase of strip width
45 Effects of Reduction Allocation It has been proved thatthe critical velocity for a single stand system decreaseswith the increase of the entry thickness given that the exitthickness stays the same and increase with the rise of the exitthickness given that the entry thickness remains the same[2] But it is more complex for multistand system as shownin Figure 15 Assuming that the entry strip thickness of the1st stand and the exit strip thickness of the 2nd stand remainunaltered the critical velocity would be changed if the entrythickness of the 2nd stand namely ℎ
1198902 is changed
If ℎ1198902
is small enough it means that exit strip thicknessof the 1st stand decreases and entry strip thickness of the 2ndstand increases The 1st stand becomes unstable first But asℎ1198902
rises the 1st stand reduction gets smaller while the 2nd
12 Shock and Vibration
Table 5The critical velocities for different standswhen ℎ1198902changes
ℎ1198902
(mm) 026 029Critical velocity of single 1st stand V
1198881
(ms) 242 252
Corresponding critical velocity of single2nd stand V
1198882(ms) 331 380
Critical velocity of single 2nd stand V1198882
(ms) 368 341
More unstable stand 1st stand 2nd stand
stand reduction gets bigger The 2nd stand becomes moreunstable than the 1st stand As shown in Table 5 when ℎ
1198902
is 026mm the 1st stand is more unstable However whenℎ1198902
increases to 029 the 2nd stand becomes more unstablethan 1st stand There exists an optimal value of ℎ
1198902for the
multistand system which is related to the maximum criticalvelocity as shown in Figure 11
The overall critical velocity changing trend for the timedelay system and nondelay system is the same But theoptimal values of ℎ
1198902are different for the two systems
because the stability of the 2nd stand becomes worse whenconsidering the time delay effect The optimal values of ℎ
1198902
for the two systems are 033mm and 028mm respectivelyObviously the optimal value of ℎ
1198902for the delay system is
more consistent with the actual production It suggests thatby redistributing the reduction of the tandem rolling millswe can improve the stability of the tandem rolling mills
The critical velocity of a single stand changing along withthe increase of the reduction is not linearWhen the reductionis small the critical velocity of the stand is very sensitiveto the reduction changing But as the reduction goes upthe decrease of critical velocity slows down So as shown inFigure 15 when ℎ
1198902is bigger than a certain point the critical
velocities of the two systems both go down slowlyIt has to be clarified that the optimal ℎ
1198902value for the two-
stand system may not work for five stand system There existfour delay times in five-stand tandem rolling mills whichmakes the relationship between stands more complex
5 Conclusions
In this paper the effects of interstand tension variation andstrip variation transportation between adjacent stands withtime delay on tandem rolling mills were studied in detailthrough comparison of different models Stability analy-sis shows that interstand tension couples adjacent standstogether and has just a little effect on the critical velocity of thechatter model The strip gauge variation passed on betweenstands reduces the critical velocity dramatically Howeverthe time delay effect has very limited effects on both thecritical velocity and the frequency But it does not mean thatthe time delay effect can be neglected when modeling therolling system By comparing the critical velocities of thedelay and nondelay system the influences of delay time asa single factor on multistand rolling system were studiedMore detailed and quantitative explanation is put forward to
the relationship of two adjacent stands In the big picturethe critical velocity of the time delay system is just slightlysmaller than the nondelay system But the delay time worsensthe stability of downstream stand and makes the relationshipbetween consecutive stands more complex The time delayeffect changes the optimal process parameters values aimedat obtaining the maximum critical velocity To get preferablerolling process parameter configuration for tandem rollingmills time delay effect must be involved
Appendix
Consider119889119865 = 119865
1119889ℎ119889+ 1198652119889ℎ119889+ 1198653119889120590119890+ 1198654119889120590119889+ 1198655119889ℎ119890
1198651= (
120597119865
120597ℎ119889
)
119904
= int
119909119899
119909119889
(
120597119901119889
120597ℎ119889
)
119904
119889119909 + int
119909119890
119909119899
(
120597119901119890
120597ℎ119889
)
119904
119889119909
+ (119896119891119890minus 120590119890) (
120597119909119890
120597ℎ119889
)
119904
1198652= (
120597119865
120597ℎ119889
)
119904
= int
119909119899
119909119889
(
120597119901119889
120597ℎ119889
)
119904
119889119909 minus (119896119891119889minus 120590119889)(
120597119909119889
120597ℎ119889
)
119904
1198653= (
120597119865
120597120590119890
)
119904
= int
119909119890
119909119899
(
120597119901119890
120597120590119890
)
119904
119889119909
1198654= (
120597119865
120597120590119889
)
119904
= int
119909119899
119909119889
(
120597119901119889
120597120590119889
)
119904
119889119909
1198655= (
120597119865
120597ℎ119890
)
119904
= int
119909119890
119909119899
(
120597119901119890
120597ℎ119890
)
119904
119889119909 + (119896119891119890minus 120590119890) (
120597119909119890
120597ℎ119890
)
119904
119889V119890= 1198751119889ℎ119889+ 1198752119889ℎ119889+ 1198753119889120590119890+ 1198754119889120590119889+ 1198755119889ℎ119890
1198751= (
120597V119890
120597ℎ119889
)
119904
=
V119903
ℎ119890
+
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597ℎ119889
)
119904
1198752= (
120597V119890
120597ℎ119889
)
119904
=
119909119890minus 119909119899
ℎ119890
+
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597ℎ119889
)
119904
1198753= (
120597V119890
120597120590119890
)
119904
=
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597120590119890
)
119904
1198754= (
120597V119890
120597120590119889
)
119904
=
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597120590119889
)
119904
1198755= (
120597V119890
120597ℎ119890
)
119904
= minus
V119903ℎ119899
ℎ119890
2+
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597ℎ119890
)
119904
Shock and Vibration 13
119889V119889= 1198761119889ℎ119889+ 1198762119889ℎ119889+ 1198763119889120590119890+ 1198764119889120590119889+ 1198765119889ℎ119890
1198761= (
120597V119889
120597ℎ119889
)
119904
= minus
V119890ℎ119890
ℎ119889
2+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119889
)
119904
1198762= (
120597V119889
120597ℎ119889
)
119904
=
minus119909119890
ℎ119889
+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119889
)
119904
1198763= (
120597V119889
120597120590119890
)
119904
=
ℎ119890
ℎ119889
(
120597V119890
120597120590119890
)
119904
1198764= (
120597V119889
120597120590119889
)
119904
=
ℎ119890
ℎ119889
(
120597V119890
120597120590119889
)
119904
1198765= (
120597V119889
120597ℎ119890
)
119904
=
V119890
ℎ119889
+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119890
)
119904
(A1)
The intermedia variables are given as follows
(
120597119909119899
120597ℎ119889
)
119904
= (minus
1198921
8119906
+
1
4119906
minus
119909119890
4ℎ119890
) sec (119892)2 + radic 119877
4ℎ119889
sdot tan (119892) + (120597119909119899
120597119909119890
)
119904
(
120597119909119890
120597ℎ119889
)
119904
+ (
120597119909119899
120597119896119891119889
)
119904
sdot (
120597119896119891119889
120597ℎ119889
)
119904
1198921 = log(ℎ119890119896119891119890
ℎ119889119896119891119889
119896119891119889minus 120590119889
119896119891119890minus 120590119890
)
119892 =
1
4
radicℎ119889
119877
(2radic
119877
ℎ119889
tanminus1 (119909119890
radic119877ℎ119889
)
+ 2radic
119877
ℎ119889
tanminus1 (119909119889
radic119877ℎ119889
) minus
1198921
119906
)
(
120597119909119899
120597119909119890
)
119904
=
ℎ119889sec (119892)2
2ℎ119890
(
120597119909119899
120597119896119891119889
)
119904
=
ℎ119889
4119906
(
1
119896119891119889
minus
1
119896119891119889minus 120590119889
) sec (119892)2
(
120597119896119891119889
120597ℎ119889
) = minus
1198991205900
ℎ119889
(119860 + log(ℎ0
ℎ119889
))
119899minus1
(
120597119909119899
120597ℎ119889
)
119904
=
1
2
sec (119892)2 (120597119909119889
120597ℎ119889
)
119904
(
120597119909119899
120597120590119890
)
119904
= minus
ℎ119889sec (119892)2
4119906 (119896119891119890minus 120590119890)
(
120597119909119899
120597120590119889
)
119904
=
ℎ119889sec (119892)2
4119906 (119896119891119889minus 120590119889)
(
120597119909119899
120597ℎ119890
)
119904
= minus
ℎ119889sec (119892)2
4119906ℎ119890
+ (
120597119909119899
120597119909119890
)
119904
(
120597119909119890
120597ℎ119890
)
119904
+ (
120597119909119899
120597119896119891119890
)
119904
(
120597119896119891119890
120597ℎ119890
)
119904
(
120597119909119899
120597119896119891119890
)
119904
=
ℎ119889
4119906
(
1
119896119891119890minus 120590119890
minus
1
119896119891119890
) sec (119892)2
(
120597119896119891119890
120597ℎ119890
)
119904
= minus
1198991205900
ℎ119890
(119860 + log(ℎ0
ℎ119890
))
119899minus1
(
120597119909119890
120597ℎ119889
)
119904
= minusradic
119877
4 (ℎ119890minus ℎ119889)
(
120597119909119890
120597ℎ119890
)
119904
= radic
119877
4 (ℎ119890minus ℎ119889)
(
120597119909119889
120597ℎ119889
)
119904
=
119877ℎ119889
2V119903ℎ119899
(
120597119901119890
120597ℎ119889
)
119904
= (
120597119901119890
120597ℎ
)
119904
+ (
120597119901119890
120597119896119891
)
119904
(
120597119896119891
120597ℎ119889
)
119904
+ (
120597119901119890
120597119867119890
)
119904
sdot (
120597119867119890
120597ℎ119889
)
119904
+ (
120597119901119890
120597119867
)
119904
(
120597119867
120597ℎ119889
)
119904
(
120597119901119890
120597ℎ
)
119904
=
(119896119891119890minus 120590119890) 119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119901119890
120597119896119891
)
119904
=
(119896119891119890minus 120590119890) ℎ119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119896119891
120597ℎ119889
)
119904
= minus
1205900119899
ℎ
(119860 + ln(ℎ0
ℎ
))
119899minus1
(
120597119901119890
120597119867119890
)
119904
=
(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119867119890
120597ℎ119889
)
119904
= minus
119877arctan (119909119890radic119877ℎ
119889)
ℎ119889
2
radic119877ℎ119889
minus
119909119890
ℎ119889ℎ119890
+ (
120597119867119890
120597119909119890
)
119904
(
120597119909119890
120597ℎ119889
)
119904
(
120597119901119890
120597119867
)
119904
= minus
(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119867
120597ℎ119889
)
119904
= minus
119877arctan (119909radic119877ℎ119889)
ℎ119889
2
radic119877ℎ119889
minus
119909
ℎ119889ℎ
(
120597119901119890
120597120590119890
)
119904
= minus
ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
14 Shock and Vibration
(
120597119901e120597ℎe)
119904
= (
120597119901e120597119896119891119890
)
119904
(
120597119896119891119890
120597ℎ119890
)
119904
+ (
120597119901e120597119867119890
)
119904
(
120597119867119890
120597119909119890
)
119904
sdot (
120597119909119890
120597ℎ119890
)
119904
minus
(119896119891119890minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)
ℎ119890
2
119896119891119890
(
120597119901119890
120597119896119891119890
)
119904
=
ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
minus
(119896119891119890 minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
2
(
120597119867119890
120597119909119890
)
119904
=
2
ℎ119890
(
120597119901d120597ℎ119889
)
119904
= (
120597119901119889
120597119896119891119889
)
119904
(
120597119896119891119889
120597ℎ119889
)
119904
+ (
120597119901119889
120597ℎ
)
119904
+ (
120597119901119889
120597119896119891
)
119904
(
120597119896119891
120597ℎ119889
)
119904
+ (
120597119901119889
120597119867
)
119904
(
120597119867
120597ℎ119889
)
119904
minus
(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889
2
119896119891119889
(
120597119901119889
120597119896119891119889
)
119904
=
ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
minus
(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
2
(
120597119901119889
120597ℎ
)
119904
=
(119896119891119889minus 120590119889) 119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119896119891
)
119904
=
(119896119891119889minus 120590119889) ℎ119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119867119889
)
119904
= minus
(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119867
)
119904
=
(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597ℎ119889
)
119904
= (
120597119901119889
120597119867119889
)
119904
(
120597119867119889
120597119909119889
)
119904
(
120597119909119889
120597ℎ119889
)
119904
(
120597119867119889
120597119909119889
)
119904
=
2
ℎ119889
(
120597119901119889
120597120590119889
)
119904
= minus
ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
119867119890= 2radic
119877
ℎ119889
tanminus1 (119909119890
radic119877ℎ119889
)
119867119889= 2radic
119877
ℎ119889
tanminus1 (119909119889
radic119877ℎ119889
)
119867 = 2radic
119877
ℎ119889
tanminus1 ( 119909
radic119877ℎ119889
)
(A2)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors gratefully acknowledge the support of theNational Natural Science Foundation of China no 51175035PhD Programs Foundation of Ministry of Education ofChina no 20100006110024 the Fundamental Research Fundsfor the Central Universities no FRF-BR-14-006A and BeijingHigher EducationYoungElite Teacher Project no YETP0367
References
[1] I S YunW R DWilson and K F Ehmann ldquoReview of chatterstudies in cold rollingrdquo International Journal of Machine Toolsand Manufacture vol 38 no 12 pp 1499ndash1530 1998
[2] J X Zhou TandemMill Vibration Control Metallurgical Indus-try Press Beijing China 1st edition 1998
[3] G Zhiying Z Yong and Z Lingqiang ldquoReview of modellingand theoretical studies on chatter in the rolling millsrdquo Journalof Mechanical Engineering vol 51 no 16 pp 87ndash105 2015
[4] H Zhao and K F Ehmann ldquoStability analysis of chatter intandem rolling millsmdashpart 2 the regenerative effectrdquo Journalof Manufacturing Science and EngineeringmdashTransactions of theASME vol 135 no 3 Article ID 031002 11 pages 2013
[5] P-H Hu and K F Ehmann ldquoRegenerative effect in rollingchatterrdquo Journal ofManufacturing Processes vol 3 no 2 pp 82ndash93 2001
[6] P H Hu Stability and Chatter in Rolling Northwestern Univer-sity Evanston Ill USA 1998
[7] H Y Zhao and K F Ehmann ldquoStability analysis of chatter intandem rolling millsmdashpart 1 single- and multi-stand negativedamping effectrdquo Journal ofManufacturing Science and Engineer-ing vol 135 no 3 Article ID 031001 2013
[8] Y Kimura Y Sodani N Nishiura et al ldquoAnalysis of chaffer intandem cold rolling millsrdquo ISIJ International vol 43 no 1 pp77ndash84 2003
[9] M R Niroomand R M Forouzan andM Salimi ldquoTheoreticaland experimental analysis of chatter in tandem cold rollingmills based on wave propagation theoryrdquo ISIJ International vol55 no 3 pp 637ndash646 2015
[10] A Heidari M R Forouzan and S Akbarzadeh ldquoDevelopmentof a rolling chattermodel considering unsteady lubricationrdquo ISIJInternational vol 54 no 1 pp 165ndash170 2014
[11] A Heidari and M R Forouzan ldquoOptimization of cold rollingprocess parameters in order to increasing rolling speed limitedby chatter vibrationsrdquo Journal of Advanced Research vol 4 no1 pp 27ndash34 2013
Shock and Vibration 15
[12] B BahramiNejadMDehghani and S AMousavi ldquoSimulationof two stands cold rolling mill process using neural networksand genetic algorithms in combination to avoid the chatterphenomenonrdquo Majlesi Journal of Electrical Engineering vol 9no 1 pp 21ndash24 2014
[13] X X Liao Theory Methods and Application of Stability Huaz-hong Science and Engineering University PressWuhan China1999
[14] V B Kolmanovskii and V R Nosov Stability of FunctionalDifferential Equations Academic Press New York NY USA1986
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Shock and Vibration
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International Journal of
Shock and Vibration 3
p p
120591120591
120590x 120590x
h h
120591 120591
p p
120590x +120597120590x120597x
dx 120590x +120597120590x120597x
dx
h +120597h
120597xh +
120597h
120597x
Figure 2 Slab analysis on a volume element of the entry region andexit region
1205900 119860 and 119899 are the material property parameter of the
rolled strip and can be decided by experiments ℎ0is the entry
strip thickness of multistand rolling millsApplying the boundary conditions
1199011003816100381610038161003816119909=119909119890
= 119896119891119890minus 120590119890
1199011003816100381610038161003816119909=119909119889
= 119896119891119889minus 120590119889
(6)
The rolling pressure at entry and exit side can be obtained
119901119890= (119896119891119890minus 120590119890)
ℎ119896119891
ℎ119890119896119891119890
exp(119906(2radic 119877ℎ119889
tanminus1 (119909119890
radic119877ℎ119889
)
minus 2radic
119877
ℎ119889
tanminus1 ( 119909
radic119877ℎ119889
)))
119901119889= (119896119891119889minus 120590119889)
ℎ119896119891
ℎ119889119896119891119889
exp(119906(2radic 119877ℎ119889
tanminus1 ( 119909
radic119877ℎ119889
)
minus 2radic
119877
ℎ119889
tanminus1 (119909119889
radic119877ℎ119889
)))
(7)
Letting 119901119890= 119901119889 the neutral point can be formulated as
follows
119909119899= radic119877ℎ
119889tan
1
4
radicℎ119889
119877
(2radic
119877
ℎ119889
tanminus1 (119909119890
radic119877ℎ119889
)
+ 2radic
119877
ℎ119889
tanminus1 (119909119889
radic119877ℎ119889
)
minus
1
120583
ln(119896119891119889minus 120590119889
119896119891119890minus 120590119890
ℎ119890119896119891119890
ℎ119889119896119891119889
))
(8)
By integrating the rolling stresses the rolling force can bedetermined
119865 = int
119909119890
119909119899
119901119890119889119909 + int
119909119899
119909119889
119901119889119889119909 (9)
M
KC
1
2dhd
dF
Figure 3 Mill stand structure model
The dynamic rolling process model can be obtained byutilizing the first-order Taylor expansion
119889119865 = 1198651119889ℎ119889+ 1198652119889ℎ119889+ 1198653119889120590119890+ 1198654119889120590119889+ 1198655119889ℎ119890
119889V119890= 1198751119889ℎ119889+ 1198752119889ℎ119889+ 1198753119889120590119890+ 1198754119889120590119889+ 1198755119889ℎ119890
119889V119889= 1198761119889ℎ119889+ 1198762119889ℎ119889+ 1198763119889120590119890+ 1198764119889120590119889+ 1198765119889ℎ119890
(10)
119865119895119875119895119876119895 119895 = 1 2 3 4 5 are the partial derivatives of119865 V
119890 V119889
with respect to ℎ119889 ℎ119889 120590119890 120590119889 ℎ119890 respectively The calculation
of 119865119895 119875119895 119876119895 119895 = 1 2 3 4 5 can be found in the Appendix
Considering the work roll flattening effect in rolling theequivalent work roll radius is calculated by employing theHitchcock formula
119877 = 119877119908(1 +
16 (1 minus 1205822
)
120587119864119908
119865
119903
) (11)
where 119877119908is initial work roll radius 120582 is Poissonrsquos ratio for the
work roll material 119865 is rolling pressure 119903 is reduction and119864119908is elasticity modulus of work roll materialChatter in rolling is the result of the interaction between
rolling process and mill stand structure The rolling processmodel and stand structure model are coupled togetherthrough mechanical rolling parameters A one degree offreedom mill stand structure model is shown in Figure 3
By coupling the rolling processmodel and stand structuremodel together the single stand vibrationmodel is as follows
119872119889ℎ119889+ 119862119889
ℎ119889+ 119870119889ℎ
119889= 2119908119889119865
119889119890=
119864
119871119890
119889V119890
119889120590119889= minus
119864
119871119889
119889V119889
(12)
where 119871119890is the distance between the stand and the upstream
stand and 119871119889is the distance between the stand and down-
stream stand
4 Shock and Vibration
Table 1 The initial rolling process parameters for the 1st and 2nd stand
ℎ1198901
(mm) ℎ1198891
ℎ1198902
(mm) ℎ1198892
(mm) 1205901198901
(Mpa) 1205901198891
(Mpa) 1205901198902
(Mpa)045 028 028 019 137 137 1371205901198892
(Mpa) 1199061
1199062
1198771199081
(mm) 1198771199082
(mm) 1205900(Mpa) 119860
98 0018 0015 276 291 8116 8116119899 119871
1198901(m) 119871
1198902(m) 119871
1198892(m) 119864
119908(Gpa) 119864 (Gpa) 119908 (m)
024 45 45 45 210 210 09
The regenerative chatter model for a two-stand rollingmill is as follows
119872119889ℎ1198891+ 119862119889
ℎ1198891+ 119870119889ℎ
1198891= 2119908119889119865
1
1198891198901=
119864
1198711198901
119889V1198901
1198891198891=
119864
1198711198902
(119889V1198902minus 119889V1198891)
119872119889ℎ1198892+ 119862119889
ℎ1198892+ 119870119889ℎ
1198892= 2119908119889119865
2
1198891198892= minus
119864
1198711198892
119889V1198892
(13)
where the second subscripts stand for the number of stands119864 is the elasticity modulus of roll piece material and 119908 is therolled strip width The rolled strip width spread is neglectedSo 119908 is constant
The relationship between the 2nd stand entry strip gaugevariation and the 1st stand exit strip gauge variation is asfollows
119889ℎ1198902(119905) = 119889ℎ
1198891(119905 minus 120591) (14)
where 120591 is the delay time decided by the interstand distanceand the entry velocity of 2nd stand
120591 =
1198711198902
V1198902
(15)
The initial rolling process parameters for the 1st and 2ndstand come from [7] and are shown in Table 1
To explore the effects of interstand coupling factors onregenerative chatter in detail four models were built in thispaperThe first one is a single stand vibrationmodelThe sec-ond one is a two-stand regenerative chattermodel which onlyconsiders the interstand tension coupling effectThe third oneis a two-stand regenerative chatter model which considersthe interstand tension coupling effect and the strip gaugevariation passed on to next stand but neglects the delay timeThe fourth one is a two-stand regenerative chatter modelwhich considers the interstand tension coupling effect and thestrip gauge variation passed on to next stand with time delayeffect
3 Stability Analysis
31 Stability Criterion To study the stabilities of the fourmodels proposed in Section 2 different stability criteria have
to be used to calculate the critical velocity for each modelThe first threemodels are nondelay differential systemsTheircritical velocities can be calculated by employing the Lya-punov indirect method According to the Lyapunov indirectmethod for a linear system = 119860119909 if all the real parts ofeigenvalues of matrix 119860 are negative the system is stable ifnot the system is unstable [13] Based on this method criticalvelocities of the first three models are obtained namely3482ms for the single 2nd stand model 3600ms for thesecond model and 3023ms for the third model Similar tothe single 2nd stand model the critical velocity of the single1st stand is 2478ms
The fourth model is a delay differential system Thestability criterion of time delay system is muchmore complexthan nondelay system The integral criterion is employed toanalyze the fourth model According to the integral criterionif the following inequation is true the delay differentialsystem is asymptotic stable [14]
int
infin
0
119871 (119908) gt
(119873 minus 1)
2
120587 (16)
where
119871 (119908) =
119877 (119908) 1198781015840
(119908) minus 119878 (119908) 1198771015840
(119908)
119877 (119908)2
+ 119878 (119908)2
(17)
119877(119908) and 119878(119908) are the real and imaginary parts of thedeterminant of the systemmatrix respectively119873 is the orderof the system For the fourth model 119873 is 7 The integrationis calculated with the help of MATLAB When the linearvelocity of the 2nd stand is 292ms
int
5000
0
119871 (119908) = 107 (18)
When the linear velocity of the 2nd stand is 293ms
int
5000
0
119871 (119908) = 45 (19)
So the critical velocity of fourth model is about 2925msComparing the critical velocities of the four models it
can be concluded that the interstand tension has just a littleeffect on the critical velocity of the chatter model The stripgauge variation passed on between stands reduces the criticalvelocity dramatically However the time delay effect as anindependent factor on critical velocity is very limited
Shock and Vibration 5
Frequency (Hz)0 200 400
0
1000
2000
Am
plitu
de (P
a)(e)
246
Frequency (Hz)
Am
plitu
de (P
a)
0 200 4000
200
400
(f)
246
0 200 4000
05
1
Frequency (Hz)
Am
plitu
de (m
)
(d)
246
times10minus9
0 005 01
0
1
(a)
minus1
times10minus9dhd2
(m)
t (s)
0 005 01
0
5000
(b)
minus5000
d120590e2
(Pa)
t (s)
0 005 01
0
1000
(c)
minus1000
d120590d2
(Pa)
t (s)
Figure 4 Time and frequency domain of the single 2nd stand when V1199032= 3482ms ((a) and (d)) work roll ((b) and (e)) entry tension ((c)
and (f)) exit tension
32 Simulation and Analysis To study the regenerative chat-ter mechanism and relationship between stands in detailtime and frequency domain simulations of the four modelswere conducted As shown in Figure 4 the single 2nd standvibrates periodically when V
1199032is 3482ms and the frequency
is 246Hz Similarly the single 1st stand vibrates periodicallywhen V
1199031is 2478ms and the frequency is 209Hz as shown
in Figure 5The second model only considers the interstand tension
coupling effect between stands As can been seen in Figure 6the 1st stand takes on periodic oscillation state and thefrequency is 209Hz It is the same with the single 1st standmodel The beat phenomena appear in the 2nd stand and theinterstand tension The frequencies are 209Hz and 251Hz209Hz is the frequency of the 1st stand and 251Hz isapproximate to the frequency of the 2nd stand It is obviousthat the beat phenomena are caused by the interstand tensioncoupling effect between the 1st stand and 2nd stand Besidesthe amplitude of 2nd stand is far less than the 1st stand
It suggests that the vibration in 2nd stand raised by theinterstand tension variation is not violent
The third model considers the interstand tension cou-pling effect and the strip gauge variation passed on to the nextstand but neglects the time delay effect As shown in Figure 7the beat phenomena in the second model disappear Thefrequency of the 1st stand the 2nd stand and the interstandtension are all 212HzThe amplitudes of the 1st and 2nd standare in the same magnitude It indicates that the strip gaugevariation passed on to the next stand makes the 2nd standvibrate more violently
The fourth model considers the interstand tension cou-pling effect between stands the strip gauge variation passedon to the next stand and the time delay effect As shown inFigure 8 the time domain of 1st stand 2nd stand and theinterstand tension are all gourd-shaped which consist withthe test results in [1] The time domain of the 2nd stand lagsbehind the 1st stand by the delay time Due to the time delayeffect the frequency domain of the fourth model is more
6 Shock and Vibration
Frequency (Hz)0 200 400
0
1000
2000
Am
plitu
de (P
a)
(e)
209
Frequency (Hz)
Am
plitu
de (P
a)
0 200 4000
100
200
(f)
209
0 200 4000
05
1
Frequency (Hz)
Am
plitu
de (m
)
(d)
209
times10minus9
0 005 01
0
5000
(b)
minus5000
d120590e1
(Pa)
t (s)
0 005 01
0
1000
(c)
minus1000
d120590d1
(Pa)
t (s)
0 005 01
0
1
(a)
minus1
times10minus9dhd1
(m)
t (s)
Figure 5 Time and frequency domain of the single 1st stand when V1199031= 2478ms ((a) and (d)) work roll ((b) and (e)) entry tension ((c)
and (f)) exit tension
complex than the thirdmodelThemain frequency is 209Hza little smaller than the third model but still in the range ofthe third octave mode chatter
Comparing the above four models a more detailedexplanation for regenerative chattermechanism in rolling canbe presented Consequent stands in tandem rolling mills arecoupled together through rolled piece One of these standsbecomes unstable and oscillates first for example the 119894thstand The 119894th stand vibration gives rise to the interstandtension vibration The interstand vibration makes the 119894 + 1thstand vibrate gently at onceThe amplitude of the 119894+1th standis far smaller than that of the 119894th stand by this time But thestrip gauge variation generated by the 119894th stand is passed onto the 119894+1th stand after delay time It aggravates the vibrationof the 119894 + 1th stand and makes the 119894th and 119894 + 1th standvibrate in the same amplitude The oscillation of the 119894 + 1thstand intensifies vibration of the interstand tension andfinallymakes the 119894th stand vibrate again
It can be concluded from the above analysis that timedelay effect as an independent factor has very limited effects
on both critical velocity and chatter frequency while makingthe stability analysis more complex But it does not mean thattime delay effect can be neglected when modeling the rollingsystem Enough works have been done to compare the firstsecond and the fourth model in [4] To study the time delayeffect as a single factor on multistand rolling system stabilityanalysis comparison of the third model and fourth modelwas done in the next section To simplify the writing modethe third model is called the nondelay system and the fourthmodel is called the delay system
4 Effects of Rolling Process Parameters
41 Effects of Friction Stability of a single stand is very sen-sitive to friction The critical velocity of a single stand growsrapidly with the increase of friction coefficient [2 7] Butthings are different for multistand systems as shown inFigure 9
When 1199061is small enough stability of the 1st stand gets
worse and it becomes more unstable than the 2nd stand
Shock and Vibration 7
Frequency (Hz)
Am
plitu
de (P
a)
0 200 400
(f)
209
251100
200
0
0 200 4000
05
1
Frequency (Hz)
Am
plitu
de (m
)
(d)
209
times10minus9
Frequency (Hz)
Am
plitu
de (m
)
0 200 4000
2
4
(e)
209251
times10minus11
0 005 01
0
1
(a)
minus1
times10minus9dhd1
(m)
t (s)
0 005 01
0
5
(b)
minus5
times10minus11
dhd2
(m)
t (s)
0 005 01
0
500
(c)
minus500
d120590d1
(Pa)
t (s)
Figure 6 The time and frequency domain of the second model V1199032= 3600ms ((a) and (d)) 1st stand work roll ((b) and (e)) 2nd stand
work roll ((c) and (f)) interstand tension
For example as shown in Table 2 when 1199061is 0014 the critical
velocity of single 1st stand is 229ms According to theprinciple of mass conservation the corresponding velocityof 2nd stand is 331ms which is smaller than the single2nd stand critical velocity 348ms The 1st stand becomesunstable before the 2nd standThe critical velocity of the two-stand system is determined by the 1st stand As 119906
1increases
the 1st stand becomes more stable and critical velocity of thetwo-stand system increases sharply until 119906
1reaches a certain
point
Table 2The critical velocities for different stands when 1199061changes
1199061
0014 0018Critical velocity of single 1st stand V
1198881
(ms) 229 248
Corresponding critical velocity ofsingle 2nd stand V
1198882(ms) 331 363
Critical velocity of single 2nd stand V1198882
(ms) 348 348
More unstable stand 1st stand 2nd stand
8 Shock and Vibration
Frequency (Hz)
Am
plitu
de (P
a)
0 200 4000
1000
2000
(f)
212
0 200 4000
05
1
Frequency (Hz)
Am
plitu
de (m
)
(d)
212
times10minus9
Frequency (Hz)
Am
plitu
de (m
)
0 200 4000
05
1
(e)
212
times10minus9
0 005 01
0
2
(b)
minus2
times10minus9
dhd2
(m)
t (s)
0 005 01
0
1
(a)
minus1
times10minus9
dhd1
(m)
t (s)
0 005 01
0
5000
(c)
minus5000
d120590d1
(Pa)
t (s)
Figure 7The time and frequency domain of the third model when V1199032= 3023ms ((a) and (d)) 1st stand work roll ((b) and (e)) 2nd stand
work roll ((c) and (f)) interstand tension
When 1199061is big enough 2nd stand becomesmore unstable
than 1st stand For example as shown in Table 2 when 1199061
is 0018 the critical velocity of the single 1st stand goes upto 248ms and the corresponding velocity of the single 2ndstand rises to 363ms which is bigger than critical velocityof the 2nd stand The 2nd stand becomes unstable first Thecritical velocity of the two-stand system is determined by the2nd stand As 119906
1increases the 1st stand becomesmore stable
but it has little effect on the 2nd stand The stability of 2ndstand is much worse than the 1st stand so the critical velocityof the two-stand system goes down as shown in Figure 9
When 1199061goes up the overall trend of critical velocities
are the same for the delay system and nondelay systemThere exists an optimal 119906
1 which relates to the maximum
critical velocity given that other rolling parameters remainunchanged But the optimal 119906
1values are different for the
delay and nondelay system The optimal 1199061for the delay
system is about 0016 while the optimal 1199061for the nondelay
system is about 002 Critical velocities for the delay systemare smaller than the nondelay system especially when 119906
1is
larger It can be explained by the idea that time delay effect hasa far greater impact on 2nd stand than 1st stand as the strip
Shock and Vibration 9
0 200 4000
50
100
Frequency (Hz)
Am
plitu
de (P
a)
(f)
209
0 200 4000
5
Frequency (Hz)A
mpl
itude
(m)
(e)
209
times10minus11
0 200 4000
5
Am
plitu
de (m
)
(d)
Frequency (Hz)
209
times10minus11
0 05 1
0
1
(a)
minus1
times10minus9dhd1
(m)
t (s)
0 05 1
0
2
(b)
minus2
times10minus9
dhd2
(m)
t (s)
0 05 1
0
2000
(c) t (s)
minus2000
d120590d1
(Pa)
Figure 8 The time and frequency domain of the fourth model when V1199032= 2925ms ((a) and (d)) 1st stand work roll ((b) and (e)) 2nd
stand work roll ((c) and (f)) interstand tension
gauge variation generated in the 1st stand is passed on to the2nd stand with delay time but not in the opposite direction
Figure 10 demonstrates the critical velocities for the delaysystem and nondelay system when the friction coefficient ofthe 2nd stand increases The critical velocity changing trendis much simpler When 119906
2is small enough the 2nd stand is
more unstable and the critical velocity increases as 1199062goes
up When 1199062is big enough the stability of the whole system
becomes better as 1199062goes up The critical velocities for the
delay system and nondelay system are almost the sameWhen1199062is relatively small the critical velocity of the time delay
system is slightly smaller than that of the nondelay systemThe smaller the 119906
2is the more unstable and sensitive to 119906
2
the time delay system is
42 Effects of Interstand Tension The critical velocity for asingle stand decreases as the entry tension and exit tensionincrease But the critical velocity ismore sensitive to the entry
tension due to the fact that the entry zone is much longer thatthe exit zone [2]1205901198891
is the intertension between the 1st and 2nd standnamely the exit tension of the 1st stand and the entry tensionof the 2nd stand The stability of the 1st and 2nd stand bothgoes worse with the increase of 120590
1198891 Therefore the critical
velocities of the delay and nondelay system both fall offas 1205901198891
increases as shown in Figure 11 But the stability ofthe 2nd stand is more sensitive to 120590
1198891 as 120590
1198891is the entry
tension for the 2nd stand As can been seen from Table 3when 120590
1198891is rather small the 1st stand becomes unstable first
and the critical velocity of the system is decided by the 1ststand The critical velocities decrease slowly as 120590
1198891is the exit
tension of the 1st stand But when 1205901198891
is big enough the 2ndstand becomes unstable first and the critical velocity of thetwo-stand system is decided by the 2nd stand The criticalvelocities of the delay system decrease fast with the increaseof 1205901198891 because 120590
1198891is the entry tension of the 2nd stand
10 Shock and Vibration
001 0015 002 0025 00323
24
25
26
27
28
29
Criti
cal s
peed
(ms
)
Delay systemNondelay system
u1
Figure 9 Comparison of stability charts for 1199061
001 0015 002 0025 00322
24
26
28
30
32
34
Criti
cal s
peed
(ms
)
Delay systemNondelay system
u2
Figure 10 Comparison of stability charts for 1199062
When considering the time delay effect the critical velocitygets smaller and more sensitive to 120590
1198891
43 Effects of Interstand Distance Interstand tension vari-ation becomes more gentle with the increase of interstanddistance And so the stability of mill stand becomes better[2] But it is more complex to multistand system 119871
1198902is the
distance between the 1st and 2nd stand If 1198711198902
is changedthe interstand tension variation and the delay time are bothchanged As shown in Figure 12 for time delay system withthe increase of 119871
1198902 the interstand tension variation decreases
and critical velocity of delay system risesMeantime the delaytime becomes larger and critical velocity decreases So thecritical velocities of time delay system form a wavy patternof rising
50 100 150 200 250 30025
26
27
28
29
30
31
32
Criti
cal s
peed
(ms
)
Delay systemNondelay system
120590d1 (MPa)
Figure 11 Comparison of stability charts for 1205901198891
3 35 4 45 5 55 623
24
25
26
27
28
29
30Cr
itica
l spe
ed (m
s)
Delay systemNondelay system
Le2 (m)
Figure 12 Comparison of stability charts for 1198711198902
Table 3 The critical velocities for different stands when 1205901198891
changes
1205901198891
(Mpa) 60 250Critical velocity of single 1st stand V
1198881(ms) 227 225
Corresponding critical velocity of single 2ndstand V
1198882(ms) 326 335
Critical velocity of single 2nd stand V1198882
(ms) 367 331
More unstable stand 1st stand 2nd stand
At first the 2nd stand is more unstable than the 1st oneWith the increase of 119871
1198902 the critical velocities of the 1st
and 2nd stand both rise But the critical velocity of 2ndstand goes up much faster than the 1st stand as 119871
1198902affects
Shock and Vibration 11
1 15 2 25 3 35 420
25
30
35
40
45
Criti
cal s
peed
(ms
)
Delay systemNondelay system
E (GPa)
Figure 13 Comparison of stability charts for 119864
05 1 1526
27
28
29
30
31
32
33
34
Criti
cal s
peed
(ms
)
Delay systemNondelay system
w (m)
Figure 14 Comparison of stability charts for 119908
the entry tension variation of 2nd standThe critical velocitiesof the nondelay and time delay system both rise fast When1198711198902
is bigger than a certain value the 1st stand becomesunstable first The rises of critical velocities for the nondelayand delay system both slow down for 119871
1198902affects the exit
tension variation of 1st stand It can been seen in Table 4 thatwhen 119871
1198902is 35m the 2nd stand goes to unstable first but
when 1198711198902
is 55m chatter occurs in the 1st stand first Butthe instability of the 1st stand occurs when 119871
1198902is quite bigger
than nondelay system for the delay time effect makes the 2ndstand more unstable
44 Effects of Strip Width and Strip Elastic Modulus Thecritical velocity of a single stand system decreases with theincrease of the strip elastic modulus and goes up along with
022 024 026 028 03 032 034 03624
26
28
30
32
34
36
38
40
Criti
cal s
peed
(ms
)
Delay systemNondelay system
he2 (mm)
Figure 15 Comparison of stability charts for ℎ1198902
Table 4 The critical velocities for different stands when 1198711198902
changes
1198711198902
(ms) 35 55Critical velocity of single 1st stand V
1198881
(ms) 2463 2487
Corresponding critical velocity of single2nd stand V
1198882(ms) 3605 3640
Critical velocity of single 2nd stand V1198882
(ms) 3138 3789
More unstable stand 2nd stand 1st stand
the increase of the strip width [2 7] As shown in Figures13 and 14 it shows the same trend for both the time delaysystem and nondelay systemWith the increase of strip elasticmodulus the critical velocities for the time delay system andnondelay system are almost the same The latter is just alittle bigger than the former With the increasing strip widththe critical velocities for the time delay system and nondelaysystem both go up But the critical velocities for the time delaysystem are a littlemore sensitive to the increase of strip width
45 Effects of Reduction Allocation It has been proved thatthe critical velocity for a single stand system decreaseswith the increase of the entry thickness given that the exitthickness stays the same and increase with the rise of the exitthickness given that the entry thickness remains the same[2] But it is more complex for multistand system as shownin Figure 15 Assuming that the entry strip thickness of the1st stand and the exit strip thickness of the 2nd stand remainunaltered the critical velocity would be changed if the entrythickness of the 2nd stand namely ℎ
1198902 is changed
If ℎ1198902
is small enough it means that exit strip thicknessof the 1st stand decreases and entry strip thickness of the 2ndstand increases The 1st stand becomes unstable first But asℎ1198902
rises the 1st stand reduction gets smaller while the 2nd
12 Shock and Vibration
Table 5The critical velocities for different standswhen ℎ1198902changes
ℎ1198902
(mm) 026 029Critical velocity of single 1st stand V
1198881
(ms) 242 252
Corresponding critical velocity of single2nd stand V
1198882(ms) 331 380
Critical velocity of single 2nd stand V1198882
(ms) 368 341
More unstable stand 1st stand 2nd stand
stand reduction gets bigger The 2nd stand becomes moreunstable than the 1st stand As shown in Table 5 when ℎ
1198902
is 026mm the 1st stand is more unstable However whenℎ1198902
increases to 029 the 2nd stand becomes more unstablethan 1st stand There exists an optimal value of ℎ
1198902for the
multistand system which is related to the maximum criticalvelocity as shown in Figure 11
The overall critical velocity changing trend for the timedelay system and nondelay system is the same But theoptimal values of ℎ
1198902are different for the two systems
because the stability of the 2nd stand becomes worse whenconsidering the time delay effect The optimal values of ℎ
1198902
for the two systems are 033mm and 028mm respectivelyObviously the optimal value of ℎ
1198902for the delay system is
more consistent with the actual production It suggests thatby redistributing the reduction of the tandem rolling millswe can improve the stability of the tandem rolling mills
The critical velocity of a single stand changing along withthe increase of the reduction is not linearWhen the reductionis small the critical velocity of the stand is very sensitiveto the reduction changing But as the reduction goes upthe decrease of critical velocity slows down So as shown inFigure 15 when ℎ
1198902is bigger than a certain point the critical
velocities of the two systems both go down slowlyIt has to be clarified that the optimal ℎ
1198902value for the two-
stand system may not work for five stand system There existfour delay times in five-stand tandem rolling mills whichmakes the relationship between stands more complex
5 Conclusions
In this paper the effects of interstand tension variation andstrip variation transportation between adjacent stands withtime delay on tandem rolling mills were studied in detailthrough comparison of different models Stability analy-sis shows that interstand tension couples adjacent standstogether and has just a little effect on the critical velocity of thechatter model The strip gauge variation passed on betweenstands reduces the critical velocity dramatically Howeverthe time delay effect has very limited effects on both thecritical velocity and the frequency But it does not mean thatthe time delay effect can be neglected when modeling therolling system By comparing the critical velocities of thedelay and nondelay system the influences of delay time asa single factor on multistand rolling system were studiedMore detailed and quantitative explanation is put forward to
the relationship of two adjacent stands In the big picturethe critical velocity of the time delay system is just slightlysmaller than the nondelay system But the delay time worsensthe stability of downstream stand and makes the relationshipbetween consecutive stands more complex The time delayeffect changes the optimal process parameters values aimedat obtaining the maximum critical velocity To get preferablerolling process parameter configuration for tandem rollingmills time delay effect must be involved
Appendix
Consider119889119865 = 119865
1119889ℎ119889+ 1198652119889ℎ119889+ 1198653119889120590119890+ 1198654119889120590119889+ 1198655119889ℎ119890
1198651= (
120597119865
120597ℎ119889
)
119904
= int
119909119899
119909119889
(
120597119901119889
120597ℎ119889
)
119904
119889119909 + int
119909119890
119909119899
(
120597119901119890
120597ℎ119889
)
119904
119889119909
+ (119896119891119890minus 120590119890) (
120597119909119890
120597ℎ119889
)
119904
1198652= (
120597119865
120597ℎ119889
)
119904
= int
119909119899
119909119889
(
120597119901119889
120597ℎ119889
)
119904
119889119909 minus (119896119891119889minus 120590119889)(
120597119909119889
120597ℎ119889
)
119904
1198653= (
120597119865
120597120590119890
)
119904
= int
119909119890
119909119899
(
120597119901119890
120597120590119890
)
119904
119889119909
1198654= (
120597119865
120597120590119889
)
119904
= int
119909119899
119909119889
(
120597119901119889
120597120590119889
)
119904
119889119909
1198655= (
120597119865
120597ℎ119890
)
119904
= int
119909119890
119909119899
(
120597119901119890
120597ℎ119890
)
119904
119889119909 + (119896119891119890minus 120590119890) (
120597119909119890
120597ℎ119890
)
119904
119889V119890= 1198751119889ℎ119889+ 1198752119889ℎ119889+ 1198753119889120590119890+ 1198754119889120590119889+ 1198755119889ℎ119890
1198751= (
120597V119890
120597ℎ119889
)
119904
=
V119903
ℎ119890
+
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597ℎ119889
)
119904
1198752= (
120597V119890
120597ℎ119889
)
119904
=
119909119890minus 119909119899
ℎ119890
+
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597ℎ119889
)
119904
1198753= (
120597V119890
120597120590119890
)
119904
=
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597120590119890
)
119904
1198754= (
120597V119890
120597120590119889
)
119904
=
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597120590119889
)
119904
1198755= (
120597V119890
120597ℎ119890
)
119904
= minus
V119903ℎ119899
ℎ119890
2+
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597ℎ119890
)
119904
Shock and Vibration 13
119889V119889= 1198761119889ℎ119889+ 1198762119889ℎ119889+ 1198763119889120590119890+ 1198764119889120590119889+ 1198765119889ℎ119890
1198761= (
120597V119889
120597ℎ119889
)
119904
= minus
V119890ℎ119890
ℎ119889
2+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119889
)
119904
1198762= (
120597V119889
120597ℎ119889
)
119904
=
minus119909119890
ℎ119889
+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119889
)
119904
1198763= (
120597V119889
120597120590119890
)
119904
=
ℎ119890
ℎ119889
(
120597V119890
120597120590119890
)
119904
1198764= (
120597V119889
120597120590119889
)
119904
=
ℎ119890
ℎ119889
(
120597V119890
120597120590119889
)
119904
1198765= (
120597V119889
120597ℎ119890
)
119904
=
V119890
ℎ119889
+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119890
)
119904
(A1)
The intermedia variables are given as follows
(
120597119909119899
120597ℎ119889
)
119904
= (minus
1198921
8119906
+
1
4119906
minus
119909119890
4ℎ119890
) sec (119892)2 + radic 119877
4ℎ119889
sdot tan (119892) + (120597119909119899
120597119909119890
)
119904
(
120597119909119890
120597ℎ119889
)
119904
+ (
120597119909119899
120597119896119891119889
)
119904
sdot (
120597119896119891119889
120597ℎ119889
)
119904
1198921 = log(ℎ119890119896119891119890
ℎ119889119896119891119889
119896119891119889minus 120590119889
119896119891119890minus 120590119890
)
119892 =
1
4
radicℎ119889
119877
(2radic
119877
ℎ119889
tanminus1 (119909119890
radic119877ℎ119889
)
+ 2radic
119877
ℎ119889
tanminus1 (119909119889
radic119877ℎ119889
) minus
1198921
119906
)
(
120597119909119899
120597119909119890
)
119904
=
ℎ119889sec (119892)2
2ℎ119890
(
120597119909119899
120597119896119891119889
)
119904
=
ℎ119889
4119906
(
1
119896119891119889
minus
1
119896119891119889minus 120590119889
) sec (119892)2
(
120597119896119891119889
120597ℎ119889
) = minus
1198991205900
ℎ119889
(119860 + log(ℎ0
ℎ119889
))
119899minus1
(
120597119909119899
120597ℎ119889
)
119904
=
1
2
sec (119892)2 (120597119909119889
120597ℎ119889
)
119904
(
120597119909119899
120597120590119890
)
119904
= minus
ℎ119889sec (119892)2
4119906 (119896119891119890minus 120590119890)
(
120597119909119899
120597120590119889
)
119904
=
ℎ119889sec (119892)2
4119906 (119896119891119889minus 120590119889)
(
120597119909119899
120597ℎ119890
)
119904
= minus
ℎ119889sec (119892)2
4119906ℎ119890
+ (
120597119909119899
120597119909119890
)
119904
(
120597119909119890
120597ℎ119890
)
119904
+ (
120597119909119899
120597119896119891119890
)
119904
(
120597119896119891119890
120597ℎ119890
)
119904
(
120597119909119899
120597119896119891119890
)
119904
=
ℎ119889
4119906
(
1
119896119891119890minus 120590119890
minus
1
119896119891119890
) sec (119892)2
(
120597119896119891119890
120597ℎ119890
)
119904
= minus
1198991205900
ℎ119890
(119860 + log(ℎ0
ℎ119890
))
119899minus1
(
120597119909119890
120597ℎ119889
)
119904
= minusradic
119877
4 (ℎ119890minus ℎ119889)
(
120597119909119890
120597ℎ119890
)
119904
= radic
119877
4 (ℎ119890minus ℎ119889)
(
120597119909119889
120597ℎ119889
)
119904
=
119877ℎ119889
2V119903ℎ119899
(
120597119901119890
120597ℎ119889
)
119904
= (
120597119901119890
120597ℎ
)
119904
+ (
120597119901119890
120597119896119891
)
119904
(
120597119896119891
120597ℎ119889
)
119904
+ (
120597119901119890
120597119867119890
)
119904
sdot (
120597119867119890
120597ℎ119889
)
119904
+ (
120597119901119890
120597119867
)
119904
(
120597119867
120597ℎ119889
)
119904
(
120597119901119890
120597ℎ
)
119904
=
(119896119891119890minus 120590119890) 119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119901119890
120597119896119891
)
119904
=
(119896119891119890minus 120590119890) ℎ119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119896119891
120597ℎ119889
)
119904
= minus
1205900119899
ℎ
(119860 + ln(ℎ0
ℎ
))
119899minus1
(
120597119901119890
120597119867119890
)
119904
=
(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119867119890
120597ℎ119889
)
119904
= minus
119877arctan (119909119890radic119877ℎ
119889)
ℎ119889
2
radic119877ℎ119889
minus
119909119890
ℎ119889ℎ119890
+ (
120597119867119890
120597119909119890
)
119904
(
120597119909119890
120597ℎ119889
)
119904
(
120597119901119890
120597119867
)
119904
= minus
(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119867
120597ℎ119889
)
119904
= minus
119877arctan (119909radic119877ℎ119889)
ℎ119889
2
radic119877ℎ119889
minus
119909
ℎ119889ℎ
(
120597119901119890
120597120590119890
)
119904
= minus
ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
14 Shock and Vibration
(
120597119901e120597ℎe)
119904
= (
120597119901e120597119896119891119890
)
119904
(
120597119896119891119890
120597ℎ119890
)
119904
+ (
120597119901e120597119867119890
)
119904
(
120597119867119890
120597119909119890
)
119904
sdot (
120597119909119890
120597ℎ119890
)
119904
minus
(119896119891119890minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)
ℎ119890
2
119896119891119890
(
120597119901119890
120597119896119891119890
)
119904
=
ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
minus
(119896119891119890 minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
2
(
120597119867119890
120597119909119890
)
119904
=
2
ℎ119890
(
120597119901d120597ℎ119889
)
119904
= (
120597119901119889
120597119896119891119889
)
119904
(
120597119896119891119889
120597ℎ119889
)
119904
+ (
120597119901119889
120597ℎ
)
119904
+ (
120597119901119889
120597119896119891
)
119904
(
120597119896119891
120597ℎ119889
)
119904
+ (
120597119901119889
120597119867
)
119904
(
120597119867
120597ℎ119889
)
119904
minus
(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889
2
119896119891119889
(
120597119901119889
120597119896119891119889
)
119904
=
ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
minus
(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
2
(
120597119901119889
120597ℎ
)
119904
=
(119896119891119889minus 120590119889) 119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119896119891
)
119904
=
(119896119891119889minus 120590119889) ℎ119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119867119889
)
119904
= minus
(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119867
)
119904
=
(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597ℎ119889
)
119904
= (
120597119901119889
120597119867119889
)
119904
(
120597119867119889
120597119909119889
)
119904
(
120597119909119889
120597ℎ119889
)
119904
(
120597119867119889
120597119909119889
)
119904
=
2
ℎ119889
(
120597119901119889
120597120590119889
)
119904
= minus
ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
119867119890= 2radic
119877
ℎ119889
tanminus1 (119909119890
radic119877ℎ119889
)
119867119889= 2radic
119877
ℎ119889
tanminus1 (119909119889
radic119877ℎ119889
)
119867 = 2radic
119877
ℎ119889
tanminus1 ( 119909
radic119877ℎ119889
)
(A2)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors gratefully acknowledge the support of theNational Natural Science Foundation of China no 51175035PhD Programs Foundation of Ministry of Education ofChina no 20100006110024 the Fundamental Research Fundsfor the Central Universities no FRF-BR-14-006A and BeijingHigher EducationYoungElite Teacher Project no YETP0367
References
[1] I S YunW R DWilson and K F Ehmann ldquoReview of chatterstudies in cold rollingrdquo International Journal of Machine Toolsand Manufacture vol 38 no 12 pp 1499ndash1530 1998
[2] J X Zhou TandemMill Vibration Control Metallurgical Indus-try Press Beijing China 1st edition 1998
[3] G Zhiying Z Yong and Z Lingqiang ldquoReview of modellingand theoretical studies on chatter in the rolling millsrdquo Journalof Mechanical Engineering vol 51 no 16 pp 87ndash105 2015
[4] H Zhao and K F Ehmann ldquoStability analysis of chatter intandem rolling millsmdashpart 2 the regenerative effectrdquo Journalof Manufacturing Science and EngineeringmdashTransactions of theASME vol 135 no 3 Article ID 031002 11 pages 2013
[5] P-H Hu and K F Ehmann ldquoRegenerative effect in rollingchatterrdquo Journal ofManufacturing Processes vol 3 no 2 pp 82ndash93 2001
[6] P H Hu Stability and Chatter in Rolling Northwestern Univer-sity Evanston Ill USA 1998
[7] H Y Zhao and K F Ehmann ldquoStability analysis of chatter intandem rolling millsmdashpart 1 single- and multi-stand negativedamping effectrdquo Journal ofManufacturing Science and Engineer-ing vol 135 no 3 Article ID 031001 2013
[8] Y Kimura Y Sodani N Nishiura et al ldquoAnalysis of chaffer intandem cold rolling millsrdquo ISIJ International vol 43 no 1 pp77ndash84 2003
[9] M R Niroomand R M Forouzan andM Salimi ldquoTheoreticaland experimental analysis of chatter in tandem cold rollingmills based on wave propagation theoryrdquo ISIJ International vol55 no 3 pp 637ndash646 2015
[10] A Heidari M R Forouzan and S Akbarzadeh ldquoDevelopmentof a rolling chattermodel considering unsteady lubricationrdquo ISIJInternational vol 54 no 1 pp 165ndash170 2014
[11] A Heidari and M R Forouzan ldquoOptimization of cold rollingprocess parameters in order to increasing rolling speed limitedby chatter vibrationsrdquo Journal of Advanced Research vol 4 no1 pp 27ndash34 2013
Shock and Vibration 15
[12] B BahramiNejadMDehghani and S AMousavi ldquoSimulationof two stands cold rolling mill process using neural networksand genetic algorithms in combination to avoid the chatterphenomenonrdquo Majlesi Journal of Electrical Engineering vol 9no 1 pp 21ndash24 2014
[13] X X Liao Theory Methods and Application of Stability Huaz-hong Science and Engineering University PressWuhan China1999
[14] V B Kolmanovskii and V R Nosov Stability of FunctionalDifferential Equations Academic Press New York NY USA1986
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4 Shock and Vibration
Table 1 The initial rolling process parameters for the 1st and 2nd stand
ℎ1198901
(mm) ℎ1198891
ℎ1198902
(mm) ℎ1198892
(mm) 1205901198901
(Mpa) 1205901198891
(Mpa) 1205901198902
(Mpa)045 028 028 019 137 137 1371205901198892
(Mpa) 1199061
1199062
1198771199081
(mm) 1198771199082
(mm) 1205900(Mpa) 119860
98 0018 0015 276 291 8116 8116119899 119871
1198901(m) 119871
1198902(m) 119871
1198892(m) 119864
119908(Gpa) 119864 (Gpa) 119908 (m)
024 45 45 45 210 210 09
The regenerative chatter model for a two-stand rollingmill is as follows
119872119889ℎ1198891+ 119862119889
ℎ1198891+ 119870119889ℎ
1198891= 2119908119889119865
1
1198891198901=
119864
1198711198901
119889V1198901
1198891198891=
119864
1198711198902
(119889V1198902minus 119889V1198891)
119872119889ℎ1198892+ 119862119889
ℎ1198892+ 119870119889ℎ
1198892= 2119908119889119865
2
1198891198892= minus
119864
1198711198892
119889V1198892
(13)
where the second subscripts stand for the number of stands119864 is the elasticity modulus of roll piece material and 119908 is therolled strip width The rolled strip width spread is neglectedSo 119908 is constant
The relationship between the 2nd stand entry strip gaugevariation and the 1st stand exit strip gauge variation is asfollows
119889ℎ1198902(119905) = 119889ℎ
1198891(119905 minus 120591) (14)
where 120591 is the delay time decided by the interstand distanceand the entry velocity of 2nd stand
120591 =
1198711198902
V1198902
(15)
The initial rolling process parameters for the 1st and 2ndstand come from [7] and are shown in Table 1
To explore the effects of interstand coupling factors onregenerative chatter in detail four models were built in thispaperThe first one is a single stand vibrationmodelThe sec-ond one is a two-stand regenerative chattermodel which onlyconsiders the interstand tension coupling effectThe third oneis a two-stand regenerative chatter model which considersthe interstand tension coupling effect and the strip gaugevariation passed on to next stand but neglects the delay timeThe fourth one is a two-stand regenerative chatter modelwhich considers the interstand tension coupling effect and thestrip gauge variation passed on to next stand with time delayeffect
3 Stability Analysis
31 Stability Criterion To study the stabilities of the fourmodels proposed in Section 2 different stability criteria have
to be used to calculate the critical velocity for each modelThe first threemodels are nondelay differential systemsTheircritical velocities can be calculated by employing the Lya-punov indirect method According to the Lyapunov indirectmethod for a linear system = 119860119909 if all the real parts ofeigenvalues of matrix 119860 are negative the system is stable ifnot the system is unstable [13] Based on this method criticalvelocities of the first three models are obtained namely3482ms for the single 2nd stand model 3600ms for thesecond model and 3023ms for the third model Similar tothe single 2nd stand model the critical velocity of the single1st stand is 2478ms
The fourth model is a delay differential system Thestability criterion of time delay system is muchmore complexthan nondelay system The integral criterion is employed toanalyze the fourth model According to the integral criterionif the following inequation is true the delay differentialsystem is asymptotic stable [14]
int
infin
0
119871 (119908) gt
(119873 minus 1)
2
120587 (16)
where
119871 (119908) =
119877 (119908) 1198781015840
(119908) minus 119878 (119908) 1198771015840
(119908)
119877 (119908)2
+ 119878 (119908)2
(17)
119877(119908) and 119878(119908) are the real and imaginary parts of thedeterminant of the systemmatrix respectively119873 is the orderof the system For the fourth model 119873 is 7 The integrationis calculated with the help of MATLAB When the linearvelocity of the 2nd stand is 292ms
int
5000
0
119871 (119908) = 107 (18)
When the linear velocity of the 2nd stand is 293ms
int
5000
0
119871 (119908) = 45 (19)
So the critical velocity of fourth model is about 2925msComparing the critical velocities of the four models it
can be concluded that the interstand tension has just a littleeffect on the critical velocity of the chatter model The stripgauge variation passed on between stands reduces the criticalvelocity dramatically However the time delay effect as anindependent factor on critical velocity is very limited
Shock and Vibration 5
Frequency (Hz)0 200 400
0
1000
2000
Am
plitu
de (P
a)(e)
246
Frequency (Hz)
Am
plitu
de (P
a)
0 200 4000
200
400
(f)
246
0 200 4000
05
1
Frequency (Hz)
Am
plitu
de (m
)
(d)
246
times10minus9
0 005 01
0
1
(a)
minus1
times10minus9dhd2
(m)
t (s)
0 005 01
0
5000
(b)
minus5000
d120590e2
(Pa)
t (s)
0 005 01
0
1000
(c)
minus1000
d120590d2
(Pa)
t (s)
Figure 4 Time and frequency domain of the single 2nd stand when V1199032= 3482ms ((a) and (d)) work roll ((b) and (e)) entry tension ((c)
and (f)) exit tension
32 Simulation and Analysis To study the regenerative chat-ter mechanism and relationship between stands in detailtime and frequency domain simulations of the four modelswere conducted As shown in Figure 4 the single 2nd standvibrates periodically when V
1199032is 3482ms and the frequency
is 246Hz Similarly the single 1st stand vibrates periodicallywhen V
1199031is 2478ms and the frequency is 209Hz as shown
in Figure 5The second model only considers the interstand tension
coupling effect between stands As can been seen in Figure 6the 1st stand takes on periodic oscillation state and thefrequency is 209Hz It is the same with the single 1st standmodel The beat phenomena appear in the 2nd stand and theinterstand tension The frequencies are 209Hz and 251Hz209Hz is the frequency of the 1st stand and 251Hz isapproximate to the frequency of the 2nd stand It is obviousthat the beat phenomena are caused by the interstand tensioncoupling effect between the 1st stand and 2nd stand Besidesthe amplitude of 2nd stand is far less than the 1st stand
It suggests that the vibration in 2nd stand raised by theinterstand tension variation is not violent
The third model considers the interstand tension cou-pling effect and the strip gauge variation passed on to the nextstand but neglects the time delay effect As shown in Figure 7the beat phenomena in the second model disappear Thefrequency of the 1st stand the 2nd stand and the interstandtension are all 212HzThe amplitudes of the 1st and 2nd standare in the same magnitude It indicates that the strip gaugevariation passed on to the next stand makes the 2nd standvibrate more violently
The fourth model considers the interstand tension cou-pling effect between stands the strip gauge variation passedon to the next stand and the time delay effect As shown inFigure 8 the time domain of 1st stand 2nd stand and theinterstand tension are all gourd-shaped which consist withthe test results in [1] The time domain of the 2nd stand lagsbehind the 1st stand by the delay time Due to the time delayeffect the frequency domain of the fourth model is more
6 Shock and Vibration
Frequency (Hz)0 200 400
0
1000
2000
Am
plitu
de (P
a)
(e)
209
Frequency (Hz)
Am
plitu
de (P
a)
0 200 4000
100
200
(f)
209
0 200 4000
05
1
Frequency (Hz)
Am
plitu
de (m
)
(d)
209
times10minus9
0 005 01
0
5000
(b)
minus5000
d120590e1
(Pa)
t (s)
0 005 01
0
1000
(c)
minus1000
d120590d1
(Pa)
t (s)
0 005 01
0
1
(a)
minus1
times10minus9dhd1
(m)
t (s)
Figure 5 Time and frequency domain of the single 1st stand when V1199031= 2478ms ((a) and (d)) work roll ((b) and (e)) entry tension ((c)
and (f)) exit tension
complex than the thirdmodelThemain frequency is 209Hza little smaller than the third model but still in the range ofthe third octave mode chatter
Comparing the above four models a more detailedexplanation for regenerative chattermechanism in rolling canbe presented Consequent stands in tandem rolling mills arecoupled together through rolled piece One of these standsbecomes unstable and oscillates first for example the 119894thstand The 119894th stand vibration gives rise to the interstandtension vibration The interstand vibration makes the 119894 + 1thstand vibrate gently at onceThe amplitude of the 119894+1th standis far smaller than that of the 119894th stand by this time But thestrip gauge variation generated by the 119894th stand is passed onto the 119894+1th stand after delay time It aggravates the vibrationof the 119894 + 1th stand and makes the 119894th and 119894 + 1th standvibrate in the same amplitude The oscillation of the 119894 + 1thstand intensifies vibration of the interstand tension andfinallymakes the 119894th stand vibrate again
It can be concluded from the above analysis that timedelay effect as an independent factor has very limited effects
on both critical velocity and chatter frequency while makingthe stability analysis more complex But it does not mean thattime delay effect can be neglected when modeling the rollingsystem Enough works have been done to compare the firstsecond and the fourth model in [4] To study the time delayeffect as a single factor on multistand rolling system stabilityanalysis comparison of the third model and fourth modelwas done in the next section To simplify the writing modethe third model is called the nondelay system and the fourthmodel is called the delay system
4 Effects of Rolling Process Parameters
41 Effects of Friction Stability of a single stand is very sen-sitive to friction The critical velocity of a single stand growsrapidly with the increase of friction coefficient [2 7] Butthings are different for multistand systems as shown inFigure 9
When 1199061is small enough stability of the 1st stand gets
worse and it becomes more unstable than the 2nd stand
Shock and Vibration 7
Frequency (Hz)
Am
plitu
de (P
a)
0 200 400
(f)
209
251100
200
0
0 200 4000
05
1
Frequency (Hz)
Am
plitu
de (m
)
(d)
209
times10minus9
Frequency (Hz)
Am
plitu
de (m
)
0 200 4000
2
4
(e)
209251
times10minus11
0 005 01
0
1
(a)
minus1
times10minus9dhd1
(m)
t (s)
0 005 01
0
5
(b)
minus5
times10minus11
dhd2
(m)
t (s)
0 005 01
0
500
(c)
minus500
d120590d1
(Pa)
t (s)
Figure 6 The time and frequency domain of the second model V1199032= 3600ms ((a) and (d)) 1st stand work roll ((b) and (e)) 2nd stand
work roll ((c) and (f)) interstand tension
For example as shown in Table 2 when 1199061is 0014 the critical
velocity of single 1st stand is 229ms According to theprinciple of mass conservation the corresponding velocityof 2nd stand is 331ms which is smaller than the single2nd stand critical velocity 348ms The 1st stand becomesunstable before the 2nd standThe critical velocity of the two-stand system is determined by the 1st stand As 119906
1increases
the 1st stand becomes more stable and critical velocity of thetwo-stand system increases sharply until 119906
1reaches a certain
point
Table 2The critical velocities for different stands when 1199061changes
1199061
0014 0018Critical velocity of single 1st stand V
1198881
(ms) 229 248
Corresponding critical velocity ofsingle 2nd stand V
1198882(ms) 331 363
Critical velocity of single 2nd stand V1198882
(ms) 348 348
More unstable stand 1st stand 2nd stand
8 Shock and Vibration
Frequency (Hz)
Am
plitu
de (P
a)
0 200 4000
1000
2000
(f)
212
0 200 4000
05
1
Frequency (Hz)
Am
plitu
de (m
)
(d)
212
times10minus9
Frequency (Hz)
Am
plitu
de (m
)
0 200 4000
05
1
(e)
212
times10minus9
0 005 01
0
2
(b)
minus2
times10minus9
dhd2
(m)
t (s)
0 005 01
0
1
(a)
minus1
times10minus9
dhd1
(m)
t (s)
0 005 01
0
5000
(c)
minus5000
d120590d1
(Pa)
t (s)
Figure 7The time and frequency domain of the third model when V1199032= 3023ms ((a) and (d)) 1st stand work roll ((b) and (e)) 2nd stand
work roll ((c) and (f)) interstand tension
When 1199061is big enough 2nd stand becomesmore unstable
than 1st stand For example as shown in Table 2 when 1199061
is 0018 the critical velocity of the single 1st stand goes upto 248ms and the corresponding velocity of the single 2ndstand rises to 363ms which is bigger than critical velocityof the 2nd stand The 2nd stand becomes unstable first Thecritical velocity of the two-stand system is determined by the2nd stand As 119906
1increases the 1st stand becomesmore stable
but it has little effect on the 2nd stand The stability of 2ndstand is much worse than the 1st stand so the critical velocityof the two-stand system goes down as shown in Figure 9
When 1199061goes up the overall trend of critical velocities
are the same for the delay system and nondelay systemThere exists an optimal 119906
1 which relates to the maximum
critical velocity given that other rolling parameters remainunchanged But the optimal 119906
1values are different for the
delay and nondelay system The optimal 1199061for the delay
system is about 0016 while the optimal 1199061for the nondelay
system is about 002 Critical velocities for the delay systemare smaller than the nondelay system especially when 119906
1is
larger It can be explained by the idea that time delay effect hasa far greater impact on 2nd stand than 1st stand as the strip
Shock and Vibration 9
0 200 4000
50
100
Frequency (Hz)
Am
plitu
de (P
a)
(f)
209
0 200 4000
5
Frequency (Hz)A
mpl
itude
(m)
(e)
209
times10minus11
0 200 4000
5
Am
plitu
de (m
)
(d)
Frequency (Hz)
209
times10minus11
0 05 1
0
1
(a)
minus1
times10minus9dhd1
(m)
t (s)
0 05 1
0
2
(b)
minus2
times10minus9
dhd2
(m)
t (s)
0 05 1
0
2000
(c) t (s)
minus2000
d120590d1
(Pa)
Figure 8 The time and frequency domain of the fourth model when V1199032= 2925ms ((a) and (d)) 1st stand work roll ((b) and (e)) 2nd
stand work roll ((c) and (f)) interstand tension
gauge variation generated in the 1st stand is passed on to the2nd stand with delay time but not in the opposite direction
Figure 10 demonstrates the critical velocities for the delaysystem and nondelay system when the friction coefficient ofthe 2nd stand increases The critical velocity changing trendis much simpler When 119906
2is small enough the 2nd stand is
more unstable and the critical velocity increases as 1199062goes
up When 1199062is big enough the stability of the whole system
becomes better as 1199062goes up The critical velocities for the
delay system and nondelay system are almost the sameWhen1199062is relatively small the critical velocity of the time delay
system is slightly smaller than that of the nondelay systemThe smaller the 119906
2is the more unstable and sensitive to 119906
2
the time delay system is
42 Effects of Interstand Tension The critical velocity for asingle stand decreases as the entry tension and exit tensionincrease But the critical velocity ismore sensitive to the entry
tension due to the fact that the entry zone is much longer thatthe exit zone [2]1205901198891
is the intertension between the 1st and 2nd standnamely the exit tension of the 1st stand and the entry tensionof the 2nd stand The stability of the 1st and 2nd stand bothgoes worse with the increase of 120590
1198891 Therefore the critical
velocities of the delay and nondelay system both fall offas 1205901198891
increases as shown in Figure 11 But the stability ofthe 2nd stand is more sensitive to 120590
1198891 as 120590
1198891is the entry
tension for the 2nd stand As can been seen from Table 3when 120590
1198891is rather small the 1st stand becomes unstable first
and the critical velocity of the system is decided by the 1ststand The critical velocities decrease slowly as 120590
1198891is the exit
tension of the 1st stand But when 1205901198891
is big enough the 2ndstand becomes unstable first and the critical velocity of thetwo-stand system is decided by the 2nd stand The criticalvelocities of the delay system decrease fast with the increaseof 1205901198891 because 120590
1198891is the entry tension of the 2nd stand
10 Shock and Vibration
001 0015 002 0025 00323
24
25
26
27
28
29
Criti
cal s
peed
(ms
)
Delay systemNondelay system
u1
Figure 9 Comparison of stability charts for 1199061
001 0015 002 0025 00322
24
26
28
30
32
34
Criti
cal s
peed
(ms
)
Delay systemNondelay system
u2
Figure 10 Comparison of stability charts for 1199062
When considering the time delay effect the critical velocitygets smaller and more sensitive to 120590
1198891
43 Effects of Interstand Distance Interstand tension vari-ation becomes more gentle with the increase of interstanddistance And so the stability of mill stand becomes better[2] But it is more complex to multistand system 119871
1198902is the
distance between the 1st and 2nd stand If 1198711198902
is changedthe interstand tension variation and the delay time are bothchanged As shown in Figure 12 for time delay system withthe increase of 119871
1198902 the interstand tension variation decreases
and critical velocity of delay system risesMeantime the delaytime becomes larger and critical velocity decreases So thecritical velocities of time delay system form a wavy patternof rising
50 100 150 200 250 30025
26
27
28
29
30
31
32
Criti
cal s
peed
(ms
)
Delay systemNondelay system
120590d1 (MPa)
Figure 11 Comparison of stability charts for 1205901198891
3 35 4 45 5 55 623
24
25
26
27
28
29
30Cr
itica
l spe
ed (m
s)
Delay systemNondelay system
Le2 (m)
Figure 12 Comparison of stability charts for 1198711198902
Table 3 The critical velocities for different stands when 1205901198891
changes
1205901198891
(Mpa) 60 250Critical velocity of single 1st stand V
1198881(ms) 227 225
Corresponding critical velocity of single 2ndstand V
1198882(ms) 326 335
Critical velocity of single 2nd stand V1198882
(ms) 367 331
More unstable stand 1st stand 2nd stand
At first the 2nd stand is more unstable than the 1st oneWith the increase of 119871
1198902 the critical velocities of the 1st
and 2nd stand both rise But the critical velocity of 2ndstand goes up much faster than the 1st stand as 119871
1198902affects
Shock and Vibration 11
1 15 2 25 3 35 420
25
30
35
40
45
Criti
cal s
peed
(ms
)
Delay systemNondelay system
E (GPa)
Figure 13 Comparison of stability charts for 119864
05 1 1526
27
28
29
30
31
32
33
34
Criti
cal s
peed
(ms
)
Delay systemNondelay system
w (m)
Figure 14 Comparison of stability charts for 119908
the entry tension variation of 2nd standThe critical velocitiesof the nondelay and time delay system both rise fast When1198711198902
is bigger than a certain value the 1st stand becomesunstable first The rises of critical velocities for the nondelayand delay system both slow down for 119871
1198902affects the exit
tension variation of 1st stand It can been seen in Table 4 thatwhen 119871
1198902is 35m the 2nd stand goes to unstable first but
when 1198711198902
is 55m chatter occurs in the 1st stand first Butthe instability of the 1st stand occurs when 119871
1198902is quite bigger
than nondelay system for the delay time effect makes the 2ndstand more unstable
44 Effects of Strip Width and Strip Elastic Modulus Thecritical velocity of a single stand system decreases with theincrease of the strip elastic modulus and goes up along with
022 024 026 028 03 032 034 03624
26
28
30
32
34
36
38
40
Criti
cal s
peed
(ms
)
Delay systemNondelay system
he2 (mm)
Figure 15 Comparison of stability charts for ℎ1198902
Table 4 The critical velocities for different stands when 1198711198902
changes
1198711198902
(ms) 35 55Critical velocity of single 1st stand V
1198881
(ms) 2463 2487
Corresponding critical velocity of single2nd stand V
1198882(ms) 3605 3640
Critical velocity of single 2nd stand V1198882
(ms) 3138 3789
More unstable stand 2nd stand 1st stand
the increase of the strip width [2 7] As shown in Figures13 and 14 it shows the same trend for both the time delaysystem and nondelay systemWith the increase of strip elasticmodulus the critical velocities for the time delay system andnondelay system are almost the same The latter is just alittle bigger than the former With the increasing strip widththe critical velocities for the time delay system and nondelaysystem both go up But the critical velocities for the time delaysystem are a littlemore sensitive to the increase of strip width
45 Effects of Reduction Allocation It has been proved thatthe critical velocity for a single stand system decreaseswith the increase of the entry thickness given that the exitthickness stays the same and increase with the rise of the exitthickness given that the entry thickness remains the same[2] But it is more complex for multistand system as shownin Figure 15 Assuming that the entry strip thickness of the1st stand and the exit strip thickness of the 2nd stand remainunaltered the critical velocity would be changed if the entrythickness of the 2nd stand namely ℎ
1198902 is changed
If ℎ1198902
is small enough it means that exit strip thicknessof the 1st stand decreases and entry strip thickness of the 2ndstand increases The 1st stand becomes unstable first But asℎ1198902
rises the 1st stand reduction gets smaller while the 2nd
12 Shock and Vibration
Table 5The critical velocities for different standswhen ℎ1198902changes
ℎ1198902
(mm) 026 029Critical velocity of single 1st stand V
1198881
(ms) 242 252
Corresponding critical velocity of single2nd stand V
1198882(ms) 331 380
Critical velocity of single 2nd stand V1198882
(ms) 368 341
More unstable stand 1st stand 2nd stand
stand reduction gets bigger The 2nd stand becomes moreunstable than the 1st stand As shown in Table 5 when ℎ
1198902
is 026mm the 1st stand is more unstable However whenℎ1198902
increases to 029 the 2nd stand becomes more unstablethan 1st stand There exists an optimal value of ℎ
1198902for the
multistand system which is related to the maximum criticalvelocity as shown in Figure 11
The overall critical velocity changing trend for the timedelay system and nondelay system is the same But theoptimal values of ℎ
1198902are different for the two systems
because the stability of the 2nd stand becomes worse whenconsidering the time delay effect The optimal values of ℎ
1198902
for the two systems are 033mm and 028mm respectivelyObviously the optimal value of ℎ
1198902for the delay system is
more consistent with the actual production It suggests thatby redistributing the reduction of the tandem rolling millswe can improve the stability of the tandem rolling mills
The critical velocity of a single stand changing along withthe increase of the reduction is not linearWhen the reductionis small the critical velocity of the stand is very sensitiveto the reduction changing But as the reduction goes upthe decrease of critical velocity slows down So as shown inFigure 15 when ℎ
1198902is bigger than a certain point the critical
velocities of the two systems both go down slowlyIt has to be clarified that the optimal ℎ
1198902value for the two-
stand system may not work for five stand system There existfour delay times in five-stand tandem rolling mills whichmakes the relationship between stands more complex
5 Conclusions
In this paper the effects of interstand tension variation andstrip variation transportation between adjacent stands withtime delay on tandem rolling mills were studied in detailthrough comparison of different models Stability analy-sis shows that interstand tension couples adjacent standstogether and has just a little effect on the critical velocity of thechatter model The strip gauge variation passed on betweenstands reduces the critical velocity dramatically Howeverthe time delay effect has very limited effects on both thecritical velocity and the frequency But it does not mean thatthe time delay effect can be neglected when modeling therolling system By comparing the critical velocities of thedelay and nondelay system the influences of delay time asa single factor on multistand rolling system were studiedMore detailed and quantitative explanation is put forward to
the relationship of two adjacent stands In the big picturethe critical velocity of the time delay system is just slightlysmaller than the nondelay system But the delay time worsensthe stability of downstream stand and makes the relationshipbetween consecutive stands more complex The time delayeffect changes the optimal process parameters values aimedat obtaining the maximum critical velocity To get preferablerolling process parameter configuration for tandem rollingmills time delay effect must be involved
Appendix
Consider119889119865 = 119865
1119889ℎ119889+ 1198652119889ℎ119889+ 1198653119889120590119890+ 1198654119889120590119889+ 1198655119889ℎ119890
1198651= (
120597119865
120597ℎ119889
)
119904
= int
119909119899
119909119889
(
120597119901119889
120597ℎ119889
)
119904
119889119909 + int
119909119890
119909119899
(
120597119901119890
120597ℎ119889
)
119904
119889119909
+ (119896119891119890minus 120590119890) (
120597119909119890
120597ℎ119889
)
119904
1198652= (
120597119865
120597ℎ119889
)
119904
= int
119909119899
119909119889
(
120597119901119889
120597ℎ119889
)
119904
119889119909 minus (119896119891119889minus 120590119889)(
120597119909119889
120597ℎ119889
)
119904
1198653= (
120597119865
120597120590119890
)
119904
= int
119909119890
119909119899
(
120597119901119890
120597120590119890
)
119904
119889119909
1198654= (
120597119865
120597120590119889
)
119904
= int
119909119899
119909119889
(
120597119901119889
120597120590119889
)
119904
119889119909
1198655= (
120597119865
120597ℎ119890
)
119904
= int
119909119890
119909119899
(
120597119901119890
120597ℎ119890
)
119904
119889119909 + (119896119891119890minus 120590119890) (
120597119909119890
120597ℎ119890
)
119904
119889V119890= 1198751119889ℎ119889+ 1198752119889ℎ119889+ 1198753119889120590119890+ 1198754119889120590119889+ 1198755119889ℎ119890
1198751= (
120597V119890
120597ℎ119889
)
119904
=
V119903
ℎ119890
+
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597ℎ119889
)
119904
1198752= (
120597V119890
120597ℎ119889
)
119904
=
119909119890minus 119909119899
ℎ119890
+
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597ℎ119889
)
119904
1198753= (
120597V119890
120597120590119890
)
119904
=
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597120590119890
)
119904
1198754= (
120597V119890
120597120590119889
)
119904
=
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597120590119889
)
119904
1198755= (
120597V119890
120597ℎ119890
)
119904
= minus
V119903ℎ119899
ℎ119890
2+
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597ℎ119890
)
119904
Shock and Vibration 13
119889V119889= 1198761119889ℎ119889+ 1198762119889ℎ119889+ 1198763119889120590119890+ 1198764119889120590119889+ 1198765119889ℎ119890
1198761= (
120597V119889
120597ℎ119889
)
119904
= minus
V119890ℎ119890
ℎ119889
2+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119889
)
119904
1198762= (
120597V119889
120597ℎ119889
)
119904
=
minus119909119890
ℎ119889
+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119889
)
119904
1198763= (
120597V119889
120597120590119890
)
119904
=
ℎ119890
ℎ119889
(
120597V119890
120597120590119890
)
119904
1198764= (
120597V119889
120597120590119889
)
119904
=
ℎ119890
ℎ119889
(
120597V119890
120597120590119889
)
119904
1198765= (
120597V119889
120597ℎ119890
)
119904
=
V119890
ℎ119889
+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119890
)
119904
(A1)
The intermedia variables are given as follows
(
120597119909119899
120597ℎ119889
)
119904
= (minus
1198921
8119906
+
1
4119906
minus
119909119890
4ℎ119890
) sec (119892)2 + radic 119877
4ℎ119889
sdot tan (119892) + (120597119909119899
120597119909119890
)
119904
(
120597119909119890
120597ℎ119889
)
119904
+ (
120597119909119899
120597119896119891119889
)
119904
sdot (
120597119896119891119889
120597ℎ119889
)
119904
1198921 = log(ℎ119890119896119891119890
ℎ119889119896119891119889
119896119891119889minus 120590119889
119896119891119890minus 120590119890
)
119892 =
1
4
radicℎ119889
119877
(2radic
119877
ℎ119889
tanminus1 (119909119890
radic119877ℎ119889
)
+ 2radic
119877
ℎ119889
tanminus1 (119909119889
radic119877ℎ119889
) minus
1198921
119906
)
(
120597119909119899
120597119909119890
)
119904
=
ℎ119889sec (119892)2
2ℎ119890
(
120597119909119899
120597119896119891119889
)
119904
=
ℎ119889
4119906
(
1
119896119891119889
minus
1
119896119891119889minus 120590119889
) sec (119892)2
(
120597119896119891119889
120597ℎ119889
) = minus
1198991205900
ℎ119889
(119860 + log(ℎ0
ℎ119889
))
119899minus1
(
120597119909119899
120597ℎ119889
)
119904
=
1
2
sec (119892)2 (120597119909119889
120597ℎ119889
)
119904
(
120597119909119899
120597120590119890
)
119904
= minus
ℎ119889sec (119892)2
4119906 (119896119891119890minus 120590119890)
(
120597119909119899
120597120590119889
)
119904
=
ℎ119889sec (119892)2
4119906 (119896119891119889minus 120590119889)
(
120597119909119899
120597ℎ119890
)
119904
= minus
ℎ119889sec (119892)2
4119906ℎ119890
+ (
120597119909119899
120597119909119890
)
119904
(
120597119909119890
120597ℎ119890
)
119904
+ (
120597119909119899
120597119896119891119890
)
119904
(
120597119896119891119890
120597ℎ119890
)
119904
(
120597119909119899
120597119896119891119890
)
119904
=
ℎ119889
4119906
(
1
119896119891119890minus 120590119890
minus
1
119896119891119890
) sec (119892)2
(
120597119896119891119890
120597ℎ119890
)
119904
= minus
1198991205900
ℎ119890
(119860 + log(ℎ0
ℎ119890
))
119899minus1
(
120597119909119890
120597ℎ119889
)
119904
= minusradic
119877
4 (ℎ119890minus ℎ119889)
(
120597119909119890
120597ℎ119890
)
119904
= radic
119877
4 (ℎ119890minus ℎ119889)
(
120597119909119889
120597ℎ119889
)
119904
=
119877ℎ119889
2V119903ℎ119899
(
120597119901119890
120597ℎ119889
)
119904
= (
120597119901119890
120597ℎ
)
119904
+ (
120597119901119890
120597119896119891
)
119904
(
120597119896119891
120597ℎ119889
)
119904
+ (
120597119901119890
120597119867119890
)
119904
sdot (
120597119867119890
120597ℎ119889
)
119904
+ (
120597119901119890
120597119867
)
119904
(
120597119867
120597ℎ119889
)
119904
(
120597119901119890
120597ℎ
)
119904
=
(119896119891119890minus 120590119890) 119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119901119890
120597119896119891
)
119904
=
(119896119891119890minus 120590119890) ℎ119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119896119891
120597ℎ119889
)
119904
= minus
1205900119899
ℎ
(119860 + ln(ℎ0
ℎ
))
119899minus1
(
120597119901119890
120597119867119890
)
119904
=
(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119867119890
120597ℎ119889
)
119904
= minus
119877arctan (119909119890radic119877ℎ
119889)
ℎ119889
2
radic119877ℎ119889
minus
119909119890
ℎ119889ℎ119890
+ (
120597119867119890
120597119909119890
)
119904
(
120597119909119890
120597ℎ119889
)
119904
(
120597119901119890
120597119867
)
119904
= minus
(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119867
120597ℎ119889
)
119904
= minus
119877arctan (119909radic119877ℎ119889)
ℎ119889
2
radic119877ℎ119889
minus
119909
ℎ119889ℎ
(
120597119901119890
120597120590119890
)
119904
= minus
ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
14 Shock and Vibration
(
120597119901e120597ℎe)
119904
= (
120597119901e120597119896119891119890
)
119904
(
120597119896119891119890
120597ℎ119890
)
119904
+ (
120597119901e120597119867119890
)
119904
(
120597119867119890
120597119909119890
)
119904
sdot (
120597119909119890
120597ℎ119890
)
119904
minus
(119896119891119890minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)
ℎ119890
2
119896119891119890
(
120597119901119890
120597119896119891119890
)
119904
=
ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
minus
(119896119891119890 minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
2
(
120597119867119890
120597119909119890
)
119904
=
2
ℎ119890
(
120597119901d120597ℎ119889
)
119904
= (
120597119901119889
120597119896119891119889
)
119904
(
120597119896119891119889
120597ℎ119889
)
119904
+ (
120597119901119889
120597ℎ
)
119904
+ (
120597119901119889
120597119896119891
)
119904
(
120597119896119891
120597ℎ119889
)
119904
+ (
120597119901119889
120597119867
)
119904
(
120597119867
120597ℎ119889
)
119904
minus
(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889
2
119896119891119889
(
120597119901119889
120597119896119891119889
)
119904
=
ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
minus
(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
2
(
120597119901119889
120597ℎ
)
119904
=
(119896119891119889minus 120590119889) 119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119896119891
)
119904
=
(119896119891119889minus 120590119889) ℎ119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119867119889
)
119904
= minus
(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119867
)
119904
=
(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597ℎ119889
)
119904
= (
120597119901119889
120597119867119889
)
119904
(
120597119867119889
120597119909119889
)
119904
(
120597119909119889
120597ℎ119889
)
119904
(
120597119867119889
120597119909119889
)
119904
=
2
ℎ119889
(
120597119901119889
120597120590119889
)
119904
= minus
ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
119867119890= 2radic
119877
ℎ119889
tanminus1 (119909119890
radic119877ℎ119889
)
119867119889= 2radic
119877
ℎ119889
tanminus1 (119909119889
radic119877ℎ119889
)
119867 = 2radic
119877
ℎ119889
tanminus1 ( 119909
radic119877ℎ119889
)
(A2)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors gratefully acknowledge the support of theNational Natural Science Foundation of China no 51175035PhD Programs Foundation of Ministry of Education ofChina no 20100006110024 the Fundamental Research Fundsfor the Central Universities no FRF-BR-14-006A and BeijingHigher EducationYoungElite Teacher Project no YETP0367
References
[1] I S YunW R DWilson and K F Ehmann ldquoReview of chatterstudies in cold rollingrdquo International Journal of Machine Toolsand Manufacture vol 38 no 12 pp 1499ndash1530 1998
[2] J X Zhou TandemMill Vibration Control Metallurgical Indus-try Press Beijing China 1st edition 1998
[3] G Zhiying Z Yong and Z Lingqiang ldquoReview of modellingand theoretical studies on chatter in the rolling millsrdquo Journalof Mechanical Engineering vol 51 no 16 pp 87ndash105 2015
[4] H Zhao and K F Ehmann ldquoStability analysis of chatter intandem rolling millsmdashpart 2 the regenerative effectrdquo Journalof Manufacturing Science and EngineeringmdashTransactions of theASME vol 135 no 3 Article ID 031002 11 pages 2013
[5] P-H Hu and K F Ehmann ldquoRegenerative effect in rollingchatterrdquo Journal ofManufacturing Processes vol 3 no 2 pp 82ndash93 2001
[6] P H Hu Stability and Chatter in Rolling Northwestern Univer-sity Evanston Ill USA 1998
[7] H Y Zhao and K F Ehmann ldquoStability analysis of chatter intandem rolling millsmdashpart 1 single- and multi-stand negativedamping effectrdquo Journal ofManufacturing Science and Engineer-ing vol 135 no 3 Article ID 031001 2013
[8] Y Kimura Y Sodani N Nishiura et al ldquoAnalysis of chaffer intandem cold rolling millsrdquo ISIJ International vol 43 no 1 pp77ndash84 2003
[9] M R Niroomand R M Forouzan andM Salimi ldquoTheoreticaland experimental analysis of chatter in tandem cold rollingmills based on wave propagation theoryrdquo ISIJ International vol55 no 3 pp 637ndash646 2015
[10] A Heidari M R Forouzan and S Akbarzadeh ldquoDevelopmentof a rolling chattermodel considering unsteady lubricationrdquo ISIJInternational vol 54 no 1 pp 165ndash170 2014
[11] A Heidari and M R Forouzan ldquoOptimization of cold rollingprocess parameters in order to increasing rolling speed limitedby chatter vibrationsrdquo Journal of Advanced Research vol 4 no1 pp 27ndash34 2013
Shock and Vibration 15
[12] B BahramiNejadMDehghani and S AMousavi ldquoSimulationof two stands cold rolling mill process using neural networksand genetic algorithms in combination to avoid the chatterphenomenonrdquo Majlesi Journal of Electrical Engineering vol 9no 1 pp 21ndash24 2014
[13] X X Liao Theory Methods and Application of Stability Huaz-hong Science and Engineering University PressWuhan China1999
[14] V B Kolmanovskii and V R Nosov Stability of FunctionalDifferential Equations Academic Press New York NY USA1986
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VLSI Design
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Chemical EngineeringInternational Journal of Antennas and
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International Journal of
Shock and Vibration 5
Frequency (Hz)0 200 400
0
1000
2000
Am
plitu
de (P
a)(e)
246
Frequency (Hz)
Am
plitu
de (P
a)
0 200 4000
200
400
(f)
246
0 200 4000
05
1
Frequency (Hz)
Am
plitu
de (m
)
(d)
246
times10minus9
0 005 01
0
1
(a)
minus1
times10minus9dhd2
(m)
t (s)
0 005 01
0
5000
(b)
minus5000
d120590e2
(Pa)
t (s)
0 005 01
0
1000
(c)
minus1000
d120590d2
(Pa)
t (s)
Figure 4 Time and frequency domain of the single 2nd stand when V1199032= 3482ms ((a) and (d)) work roll ((b) and (e)) entry tension ((c)
and (f)) exit tension
32 Simulation and Analysis To study the regenerative chat-ter mechanism and relationship between stands in detailtime and frequency domain simulations of the four modelswere conducted As shown in Figure 4 the single 2nd standvibrates periodically when V
1199032is 3482ms and the frequency
is 246Hz Similarly the single 1st stand vibrates periodicallywhen V
1199031is 2478ms and the frequency is 209Hz as shown
in Figure 5The second model only considers the interstand tension
coupling effect between stands As can been seen in Figure 6the 1st stand takes on periodic oscillation state and thefrequency is 209Hz It is the same with the single 1st standmodel The beat phenomena appear in the 2nd stand and theinterstand tension The frequencies are 209Hz and 251Hz209Hz is the frequency of the 1st stand and 251Hz isapproximate to the frequency of the 2nd stand It is obviousthat the beat phenomena are caused by the interstand tensioncoupling effect between the 1st stand and 2nd stand Besidesthe amplitude of 2nd stand is far less than the 1st stand
It suggests that the vibration in 2nd stand raised by theinterstand tension variation is not violent
The third model considers the interstand tension cou-pling effect and the strip gauge variation passed on to the nextstand but neglects the time delay effect As shown in Figure 7the beat phenomena in the second model disappear Thefrequency of the 1st stand the 2nd stand and the interstandtension are all 212HzThe amplitudes of the 1st and 2nd standare in the same magnitude It indicates that the strip gaugevariation passed on to the next stand makes the 2nd standvibrate more violently
The fourth model considers the interstand tension cou-pling effect between stands the strip gauge variation passedon to the next stand and the time delay effect As shown inFigure 8 the time domain of 1st stand 2nd stand and theinterstand tension are all gourd-shaped which consist withthe test results in [1] The time domain of the 2nd stand lagsbehind the 1st stand by the delay time Due to the time delayeffect the frequency domain of the fourth model is more
6 Shock and Vibration
Frequency (Hz)0 200 400
0
1000
2000
Am
plitu
de (P
a)
(e)
209
Frequency (Hz)
Am
plitu
de (P
a)
0 200 4000
100
200
(f)
209
0 200 4000
05
1
Frequency (Hz)
Am
plitu
de (m
)
(d)
209
times10minus9
0 005 01
0
5000
(b)
minus5000
d120590e1
(Pa)
t (s)
0 005 01
0
1000
(c)
minus1000
d120590d1
(Pa)
t (s)
0 005 01
0
1
(a)
minus1
times10minus9dhd1
(m)
t (s)
Figure 5 Time and frequency domain of the single 1st stand when V1199031= 2478ms ((a) and (d)) work roll ((b) and (e)) entry tension ((c)
and (f)) exit tension
complex than the thirdmodelThemain frequency is 209Hza little smaller than the third model but still in the range ofthe third octave mode chatter
Comparing the above four models a more detailedexplanation for regenerative chattermechanism in rolling canbe presented Consequent stands in tandem rolling mills arecoupled together through rolled piece One of these standsbecomes unstable and oscillates first for example the 119894thstand The 119894th stand vibration gives rise to the interstandtension vibration The interstand vibration makes the 119894 + 1thstand vibrate gently at onceThe amplitude of the 119894+1th standis far smaller than that of the 119894th stand by this time But thestrip gauge variation generated by the 119894th stand is passed onto the 119894+1th stand after delay time It aggravates the vibrationof the 119894 + 1th stand and makes the 119894th and 119894 + 1th standvibrate in the same amplitude The oscillation of the 119894 + 1thstand intensifies vibration of the interstand tension andfinallymakes the 119894th stand vibrate again
It can be concluded from the above analysis that timedelay effect as an independent factor has very limited effects
on both critical velocity and chatter frequency while makingthe stability analysis more complex But it does not mean thattime delay effect can be neglected when modeling the rollingsystem Enough works have been done to compare the firstsecond and the fourth model in [4] To study the time delayeffect as a single factor on multistand rolling system stabilityanalysis comparison of the third model and fourth modelwas done in the next section To simplify the writing modethe third model is called the nondelay system and the fourthmodel is called the delay system
4 Effects of Rolling Process Parameters
41 Effects of Friction Stability of a single stand is very sen-sitive to friction The critical velocity of a single stand growsrapidly with the increase of friction coefficient [2 7] Butthings are different for multistand systems as shown inFigure 9
When 1199061is small enough stability of the 1st stand gets
worse and it becomes more unstable than the 2nd stand
Shock and Vibration 7
Frequency (Hz)
Am
plitu
de (P
a)
0 200 400
(f)
209
251100
200
0
0 200 4000
05
1
Frequency (Hz)
Am
plitu
de (m
)
(d)
209
times10minus9
Frequency (Hz)
Am
plitu
de (m
)
0 200 4000
2
4
(e)
209251
times10minus11
0 005 01
0
1
(a)
minus1
times10minus9dhd1
(m)
t (s)
0 005 01
0
5
(b)
minus5
times10minus11
dhd2
(m)
t (s)
0 005 01
0
500
(c)
minus500
d120590d1
(Pa)
t (s)
Figure 6 The time and frequency domain of the second model V1199032= 3600ms ((a) and (d)) 1st stand work roll ((b) and (e)) 2nd stand
work roll ((c) and (f)) interstand tension
For example as shown in Table 2 when 1199061is 0014 the critical
velocity of single 1st stand is 229ms According to theprinciple of mass conservation the corresponding velocityof 2nd stand is 331ms which is smaller than the single2nd stand critical velocity 348ms The 1st stand becomesunstable before the 2nd standThe critical velocity of the two-stand system is determined by the 1st stand As 119906
1increases
the 1st stand becomes more stable and critical velocity of thetwo-stand system increases sharply until 119906
1reaches a certain
point
Table 2The critical velocities for different stands when 1199061changes
1199061
0014 0018Critical velocity of single 1st stand V
1198881
(ms) 229 248
Corresponding critical velocity ofsingle 2nd stand V
1198882(ms) 331 363
Critical velocity of single 2nd stand V1198882
(ms) 348 348
More unstable stand 1st stand 2nd stand
8 Shock and Vibration
Frequency (Hz)
Am
plitu
de (P
a)
0 200 4000
1000
2000
(f)
212
0 200 4000
05
1
Frequency (Hz)
Am
plitu
de (m
)
(d)
212
times10minus9
Frequency (Hz)
Am
plitu
de (m
)
0 200 4000
05
1
(e)
212
times10minus9
0 005 01
0
2
(b)
minus2
times10minus9
dhd2
(m)
t (s)
0 005 01
0
1
(a)
minus1
times10minus9
dhd1
(m)
t (s)
0 005 01
0
5000
(c)
minus5000
d120590d1
(Pa)
t (s)
Figure 7The time and frequency domain of the third model when V1199032= 3023ms ((a) and (d)) 1st stand work roll ((b) and (e)) 2nd stand
work roll ((c) and (f)) interstand tension
When 1199061is big enough 2nd stand becomesmore unstable
than 1st stand For example as shown in Table 2 when 1199061
is 0018 the critical velocity of the single 1st stand goes upto 248ms and the corresponding velocity of the single 2ndstand rises to 363ms which is bigger than critical velocityof the 2nd stand The 2nd stand becomes unstable first Thecritical velocity of the two-stand system is determined by the2nd stand As 119906
1increases the 1st stand becomesmore stable
but it has little effect on the 2nd stand The stability of 2ndstand is much worse than the 1st stand so the critical velocityof the two-stand system goes down as shown in Figure 9
When 1199061goes up the overall trend of critical velocities
are the same for the delay system and nondelay systemThere exists an optimal 119906
1 which relates to the maximum
critical velocity given that other rolling parameters remainunchanged But the optimal 119906
1values are different for the
delay and nondelay system The optimal 1199061for the delay
system is about 0016 while the optimal 1199061for the nondelay
system is about 002 Critical velocities for the delay systemare smaller than the nondelay system especially when 119906
1is
larger It can be explained by the idea that time delay effect hasa far greater impact on 2nd stand than 1st stand as the strip
Shock and Vibration 9
0 200 4000
50
100
Frequency (Hz)
Am
plitu
de (P
a)
(f)
209
0 200 4000
5
Frequency (Hz)A
mpl
itude
(m)
(e)
209
times10minus11
0 200 4000
5
Am
plitu
de (m
)
(d)
Frequency (Hz)
209
times10minus11
0 05 1
0
1
(a)
minus1
times10minus9dhd1
(m)
t (s)
0 05 1
0
2
(b)
minus2
times10minus9
dhd2
(m)
t (s)
0 05 1
0
2000
(c) t (s)
minus2000
d120590d1
(Pa)
Figure 8 The time and frequency domain of the fourth model when V1199032= 2925ms ((a) and (d)) 1st stand work roll ((b) and (e)) 2nd
stand work roll ((c) and (f)) interstand tension
gauge variation generated in the 1st stand is passed on to the2nd stand with delay time but not in the opposite direction
Figure 10 demonstrates the critical velocities for the delaysystem and nondelay system when the friction coefficient ofthe 2nd stand increases The critical velocity changing trendis much simpler When 119906
2is small enough the 2nd stand is
more unstable and the critical velocity increases as 1199062goes
up When 1199062is big enough the stability of the whole system
becomes better as 1199062goes up The critical velocities for the
delay system and nondelay system are almost the sameWhen1199062is relatively small the critical velocity of the time delay
system is slightly smaller than that of the nondelay systemThe smaller the 119906
2is the more unstable and sensitive to 119906
2
the time delay system is
42 Effects of Interstand Tension The critical velocity for asingle stand decreases as the entry tension and exit tensionincrease But the critical velocity ismore sensitive to the entry
tension due to the fact that the entry zone is much longer thatthe exit zone [2]1205901198891
is the intertension between the 1st and 2nd standnamely the exit tension of the 1st stand and the entry tensionof the 2nd stand The stability of the 1st and 2nd stand bothgoes worse with the increase of 120590
1198891 Therefore the critical
velocities of the delay and nondelay system both fall offas 1205901198891
increases as shown in Figure 11 But the stability ofthe 2nd stand is more sensitive to 120590
1198891 as 120590
1198891is the entry
tension for the 2nd stand As can been seen from Table 3when 120590
1198891is rather small the 1st stand becomes unstable first
and the critical velocity of the system is decided by the 1ststand The critical velocities decrease slowly as 120590
1198891is the exit
tension of the 1st stand But when 1205901198891
is big enough the 2ndstand becomes unstable first and the critical velocity of thetwo-stand system is decided by the 2nd stand The criticalvelocities of the delay system decrease fast with the increaseof 1205901198891 because 120590
1198891is the entry tension of the 2nd stand
10 Shock and Vibration
001 0015 002 0025 00323
24
25
26
27
28
29
Criti
cal s
peed
(ms
)
Delay systemNondelay system
u1
Figure 9 Comparison of stability charts for 1199061
001 0015 002 0025 00322
24
26
28
30
32
34
Criti
cal s
peed
(ms
)
Delay systemNondelay system
u2
Figure 10 Comparison of stability charts for 1199062
When considering the time delay effect the critical velocitygets smaller and more sensitive to 120590
1198891
43 Effects of Interstand Distance Interstand tension vari-ation becomes more gentle with the increase of interstanddistance And so the stability of mill stand becomes better[2] But it is more complex to multistand system 119871
1198902is the
distance between the 1st and 2nd stand If 1198711198902
is changedthe interstand tension variation and the delay time are bothchanged As shown in Figure 12 for time delay system withthe increase of 119871
1198902 the interstand tension variation decreases
and critical velocity of delay system risesMeantime the delaytime becomes larger and critical velocity decreases So thecritical velocities of time delay system form a wavy patternof rising
50 100 150 200 250 30025
26
27
28
29
30
31
32
Criti
cal s
peed
(ms
)
Delay systemNondelay system
120590d1 (MPa)
Figure 11 Comparison of stability charts for 1205901198891
3 35 4 45 5 55 623
24
25
26
27
28
29
30Cr
itica
l spe
ed (m
s)
Delay systemNondelay system
Le2 (m)
Figure 12 Comparison of stability charts for 1198711198902
Table 3 The critical velocities for different stands when 1205901198891
changes
1205901198891
(Mpa) 60 250Critical velocity of single 1st stand V
1198881(ms) 227 225
Corresponding critical velocity of single 2ndstand V
1198882(ms) 326 335
Critical velocity of single 2nd stand V1198882
(ms) 367 331
More unstable stand 1st stand 2nd stand
At first the 2nd stand is more unstable than the 1st oneWith the increase of 119871
1198902 the critical velocities of the 1st
and 2nd stand both rise But the critical velocity of 2ndstand goes up much faster than the 1st stand as 119871
1198902affects
Shock and Vibration 11
1 15 2 25 3 35 420
25
30
35
40
45
Criti
cal s
peed
(ms
)
Delay systemNondelay system
E (GPa)
Figure 13 Comparison of stability charts for 119864
05 1 1526
27
28
29
30
31
32
33
34
Criti
cal s
peed
(ms
)
Delay systemNondelay system
w (m)
Figure 14 Comparison of stability charts for 119908
the entry tension variation of 2nd standThe critical velocitiesof the nondelay and time delay system both rise fast When1198711198902
is bigger than a certain value the 1st stand becomesunstable first The rises of critical velocities for the nondelayand delay system both slow down for 119871
1198902affects the exit
tension variation of 1st stand It can been seen in Table 4 thatwhen 119871
1198902is 35m the 2nd stand goes to unstable first but
when 1198711198902
is 55m chatter occurs in the 1st stand first Butthe instability of the 1st stand occurs when 119871
1198902is quite bigger
than nondelay system for the delay time effect makes the 2ndstand more unstable
44 Effects of Strip Width and Strip Elastic Modulus Thecritical velocity of a single stand system decreases with theincrease of the strip elastic modulus and goes up along with
022 024 026 028 03 032 034 03624
26
28
30
32
34
36
38
40
Criti
cal s
peed
(ms
)
Delay systemNondelay system
he2 (mm)
Figure 15 Comparison of stability charts for ℎ1198902
Table 4 The critical velocities for different stands when 1198711198902
changes
1198711198902
(ms) 35 55Critical velocity of single 1st stand V
1198881
(ms) 2463 2487
Corresponding critical velocity of single2nd stand V
1198882(ms) 3605 3640
Critical velocity of single 2nd stand V1198882
(ms) 3138 3789
More unstable stand 2nd stand 1st stand
the increase of the strip width [2 7] As shown in Figures13 and 14 it shows the same trend for both the time delaysystem and nondelay systemWith the increase of strip elasticmodulus the critical velocities for the time delay system andnondelay system are almost the same The latter is just alittle bigger than the former With the increasing strip widththe critical velocities for the time delay system and nondelaysystem both go up But the critical velocities for the time delaysystem are a littlemore sensitive to the increase of strip width
45 Effects of Reduction Allocation It has been proved thatthe critical velocity for a single stand system decreaseswith the increase of the entry thickness given that the exitthickness stays the same and increase with the rise of the exitthickness given that the entry thickness remains the same[2] But it is more complex for multistand system as shownin Figure 15 Assuming that the entry strip thickness of the1st stand and the exit strip thickness of the 2nd stand remainunaltered the critical velocity would be changed if the entrythickness of the 2nd stand namely ℎ
1198902 is changed
If ℎ1198902
is small enough it means that exit strip thicknessof the 1st stand decreases and entry strip thickness of the 2ndstand increases The 1st stand becomes unstable first But asℎ1198902
rises the 1st stand reduction gets smaller while the 2nd
12 Shock and Vibration
Table 5The critical velocities for different standswhen ℎ1198902changes
ℎ1198902
(mm) 026 029Critical velocity of single 1st stand V
1198881
(ms) 242 252
Corresponding critical velocity of single2nd stand V
1198882(ms) 331 380
Critical velocity of single 2nd stand V1198882
(ms) 368 341
More unstable stand 1st stand 2nd stand
stand reduction gets bigger The 2nd stand becomes moreunstable than the 1st stand As shown in Table 5 when ℎ
1198902
is 026mm the 1st stand is more unstable However whenℎ1198902
increases to 029 the 2nd stand becomes more unstablethan 1st stand There exists an optimal value of ℎ
1198902for the
multistand system which is related to the maximum criticalvelocity as shown in Figure 11
The overall critical velocity changing trend for the timedelay system and nondelay system is the same But theoptimal values of ℎ
1198902are different for the two systems
because the stability of the 2nd stand becomes worse whenconsidering the time delay effect The optimal values of ℎ
1198902
for the two systems are 033mm and 028mm respectivelyObviously the optimal value of ℎ
1198902for the delay system is
more consistent with the actual production It suggests thatby redistributing the reduction of the tandem rolling millswe can improve the stability of the tandem rolling mills
The critical velocity of a single stand changing along withthe increase of the reduction is not linearWhen the reductionis small the critical velocity of the stand is very sensitiveto the reduction changing But as the reduction goes upthe decrease of critical velocity slows down So as shown inFigure 15 when ℎ
1198902is bigger than a certain point the critical
velocities of the two systems both go down slowlyIt has to be clarified that the optimal ℎ
1198902value for the two-
stand system may not work for five stand system There existfour delay times in five-stand tandem rolling mills whichmakes the relationship between stands more complex
5 Conclusions
In this paper the effects of interstand tension variation andstrip variation transportation between adjacent stands withtime delay on tandem rolling mills were studied in detailthrough comparison of different models Stability analy-sis shows that interstand tension couples adjacent standstogether and has just a little effect on the critical velocity of thechatter model The strip gauge variation passed on betweenstands reduces the critical velocity dramatically Howeverthe time delay effect has very limited effects on both thecritical velocity and the frequency But it does not mean thatthe time delay effect can be neglected when modeling therolling system By comparing the critical velocities of thedelay and nondelay system the influences of delay time asa single factor on multistand rolling system were studiedMore detailed and quantitative explanation is put forward to
the relationship of two adjacent stands In the big picturethe critical velocity of the time delay system is just slightlysmaller than the nondelay system But the delay time worsensthe stability of downstream stand and makes the relationshipbetween consecutive stands more complex The time delayeffect changes the optimal process parameters values aimedat obtaining the maximum critical velocity To get preferablerolling process parameter configuration for tandem rollingmills time delay effect must be involved
Appendix
Consider119889119865 = 119865
1119889ℎ119889+ 1198652119889ℎ119889+ 1198653119889120590119890+ 1198654119889120590119889+ 1198655119889ℎ119890
1198651= (
120597119865
120597ℎ119889
)
119904
= int
119909119899
119909119889
(
120597119901119889
120597ℎ119889
)
119904
119889119909 + int
119909119890
119909119899
(
120597119901119890
120597ℎ119889
)
119904
119889119909
+ (119896119891119890minus 120590119890) (
120597119909119890
120597ℎ119889
)
119904
1198652= (
120597119865
120597ℎ119889
)
119904
= int
119909119899
119909119889
(
120597119901119889
120597ℎ119889
)
119904
119889119909 minus (119896119891119889minus 120590119889)(
120597119909119889
120597ℎ119889
)
119904
1198653= (
120597119865
120597120590119890
)
119904
= int
119909119890
119909119899
(
120597119901119890
120597120590119890
)
119904
119889119909
1198654= (
120597119865
120597120590119889
)
119904
= int
119909119899
119909119889
(
120597119901119889
120597120590119889
)
119904
119889119909
1198655= (
120597119865
120597ℎ119890
)
119904
= int
119909119890
119909119899
(
120597119901119890
120597ℎ119890
)
119904
119889119909 + (119896119891119890minus 120590119890) (
120597119909119890
120597ℎ119890
)
119904
119889V119890= 1198751119889ℎ119889+ 1198752119889ℎ119889+ 1198753119889120590119890+ 1198754119889120590119889+ 1198755119889ℎ119890
1198751= (
120597V119890
120597ℎ119889
)
119904
=
V119903
ℎ119890
+
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597ℎ119889
)
119904
1198752= (
120597V119890
120597ℎ119889
)
119904
=
119909119890minus 119909119899
ℎ119890
+
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597ℎ119889
)
119904
1198753= (
120597V119890
120597120590119890
)
119904
=
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597120590119890
)
119904
1198754= (
120597V119890
120597120590119889
)
119904
=
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597120590119889
)
119904
1198755= (
120597V119890
120597ℎ119890
)
119904
= minus
V119903ℎ119899
ℎ119890
2+
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597ℎ119890
)
119904
Shock and Vibration 13
119889V119889= 1198761119889ℎ119889+ 1198762119889ℎ119889+ 1198763119889120590119890+ 1198764119889120590119889+ 1198765119889ℎ119890
1198761= (
120597V119889
120597ℎ119889
)
119904
= minus
V119890ℎ119890
ℎ119889
2+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119889
)
119904
1198762= (
120597V119889
120597ℎ119889
)
119904
=
minus119909119890
ℎ119889
+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119889
)
119904
1198763= (
120597V119889
120597120590119890
)
119904
=
ℎ119890
ℎ119889
(
120597V119890
120597120590119890
)
119904
1198764= (
120597V119889
120597120590119889
)
119904
=
ℎ119890
ℎ119889
(
120597V119890
120597120590119889
)
119904
1198765= (
120597V119889
120597ℎ119890
)
119904
=
V119890
ℎ119889
+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119890
)
119904
(A1)
The intermedia variables are given as follows
(
120597119909119899
120597ℎ119889
)
119904
= (minus
1198921
8119906
+
1
4119906
minus
119909119890
4ℎ119890
) sec (119892)2 + radic 119877
4ℎ119889
sdot tan (119892) + (120597119909119899
120597119909119890
)
119904
(
120597119909119890
120597ℎ119889
)
119904
+ (
120597119909119899
120597119896119891119889
)
119904
sdot (
120597119896119891119889
120597ℎ119889
)
119904
1198921 = log(ℎ119890119896119891119890
ℎ119889119896119891119889
119896119891119889minus 120590119889
119896119891119890minus 120590119890
)
119892 =
1
4
radicℎ119889
119877
(2radic
119877
ℎ119889
tanminus1 (119909119890
radic119877ℎ119889
)
+ 2radic
119877
ℎ119889
tanminus1 (119909119889
radic119877ℎ119889
) minus
1198921
119906
)
(
120597119909119899
120597119909119890
)
119904
=
ℎ119889sec (119892)2
2ℎ119890
(
120597119909119899
120597119896119891119889
)
119904
=
ℎ119889
4119906
(
1
119896119891119889
minus
1
119896119891119889minus 120590119889
) sec (119892)2
(
120597119896119891119889
120597ℎ119889
) = minus
1198991205900
ℎ119889
(119860 + log(ℎ0
ℎ119889
))
119899minus1
(
120597119909119899
120597ℎ119889
)
119904
=
1
2
sec (119892)2 (120597119909119889
120597ℎ119889
)
119904
(
120597119909119899
120597120590119890
)
119904
= minus
ℎ119889sec (119892)2
4119906 (119896119891119890minus 120590119890)
(
120597119909119899
120597120590119889
)
119904
=
ℎ119889sec (119892)2
4119906 (119896119891119889minus 120590119889)
(
120597119909119899
120597ℎ119890
)
119904
= minus
ℎ119889sec (119892)2
4119906ℎ119890
+ (
120597119909119899
120597119909119890
)
119904
(
120597119909119890
120597ℎ119890
)
119904
+ (
120597119909119899
120597119896119891119890
)
119904
(
120597119896119891119890
120597ℎ119890
)
119904
(
120597119909119899
120597119896119891119890
)
119904
=
ℎ119889
4119906
(
1
119896119891119890minus 120590119890
minus
1
119896119891119890
) sec (119892)2
(
120597119896119891119890
120597ℎ119890
)
119904
= minus
1198991205900
ℎ119890
(119860 + log(ℎ0
ℎ119890
))
119899minus1
(
120597119909119890
120597ℎ119889
)
119904
= minusradic
119877
4 (ℎ119890minus ℎ119889)
(
120597119909119890
120597ℎ119890
)
119904
= radic
119877
4 (ℎ119890minus ℎ119889)
(
120597119909119889
120597ℎ119889
)
119904
=
119877ℎ119889
2V119903ℎ119899
(
120597119901119890
120597ℎ119889
)
119904
= (
120597119901119890
120597ℎ
)
119904
+ (
120597119901119890
120597119896119891
)
119904
(
120597119896119891
120597ℎ119889
)
119904
+ (
120597119901119890
120597119867119890
)
119904
sdot (
120597119867119890
120597ℎ119889
)
119904
+ (
120597119901119890
120597119867
)
119904
(
120597119867
120597ℎ119889
)
119904
(
120597119901119890
120597ℎ
)
119904
=
(119896119891119890minus 120590119890) 119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119901119890
120597119896119891
)
119904
=
(119896119891119890minus 120590119890) ℎ119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119896119891
120597ℎ119889
)
119904
= minus
1205900119899
ℎ
(119860 + ln(ℎ0
ℎ
))
119899minus1
(
120597119901119890
120597119867119890
)
119904
=
(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119867119890
120597ℎ119889
)
119904
= minus
119877arctan (119909119890radic119877ℎ
119889)
ℎ119889
2
radic119877ℎ119889
minus
119909119890
ℎ119889ℎ119890
+ (
120597119867119890
120597119909119890
)
119904
(
120597119909119890
120597ℎ119889
)
119904
(
120597119901119890
120597119867
)
119904
= minus
(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119867
120597ℎ119889
)
119904
= minus
119877arctan (119909radic119877ℎ119889)
ℎ119889
2
radic119877ℎ119889
minus
119909
ℎ119889ℎ
(
120597119901119890
120597120590119890
)
119904
= minus
ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
14 Shock and Vibration
(
120597119901e120597ℎe)
119904
= (
120597119901e120597119896119891119890
)
119904
(
120597119896119891119890
120597ℎ119890
)
119904
+ (
120597119901e120597119867119890
)
119904
(
120597119867119890
120597119909119890
)
119904
sdot (
120597119909119890
120597ℎ119890
)
119904
minus
(119896119891119890minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)
ℎ119890
2
119896119891119890
(
120597119901119890
120597119896119891119890
)
119904
=
ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
minus
(119896119891119890 minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
2
(
120597119867119890
120597119909119890
)
119904
=
2
ℎ119890
(
120597119901d120597ℎ119889
)
119904
= (
120597119901119889
120597119896119891119889
)
119904
(
120597119896119891119889
120597ℎ119889
)
119904
+ (
120597119901119889
120597ℎ
)
119904
+ (
120597119901119889
120597119896119891
)
119904
(
120597119896119891
120597ℎ119889
)
119904
+ (
120597119901119889
120597119867
)
119904
(
120597119867
120597ℎ119889
)
119904
minus
(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889
2
119896119891119889
(
120597119901119889
120597119896119891119889
)
119904
=
ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
minus
(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
2
(
120597119901119889
120597ℎ
)
119904
=
(119896119891119889minus 120590119889) 119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119896119891
)
119904
=
(119896119891119889minus 120590119889) ℎ119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119867119889
)
119904
= minus
(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119867
)
119904
=
(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597ℎ119889
)
119904
= (
120597119901119889
120597119867119889
)
119904
(
120597119867119889
120597119909119889
)
119904
(
120597119909119889
120597ℎ119889
)
119904
(
120597119867119889
120597119909119889
)
119904
=
2
ℎ119889
(
120597119901119889
120597120590119889
)
119904
= minus
ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
119867119890= 2radic
119877
ℎ119889
tanminus1 (119909119890
radic119877ℎ119889
)
119867119889= 2radic
119877
ℎ119889
tanminus1 (119909119889
radic119877ℎ119889
)
119867 = 2radic
119877
ℎ119889
tanminus1 ( 119909
radic119877ℎ119889
)
(A2)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors gratefully acknowledge the support of theNational Natural Science Foundation of China no 51175035PhD Programs Foundation of Ministry of Education ofChina no 20100006110024 the Fundamental Research Fundsfor the Central Universities no FRF-BR-14-006A and BeijingHigher EducationYoungElite Teacher Project no YETP0367
References
[1] I S YunW R DWilson and K F Ehmann ldquoReview of chatterstudies in cold rollingrdquo International Journal of Machine Toolsand Manufacture vol 38 no 12 pp 1499ndash1530 1998
[2] J X Zhou TandemMill Vibration Control Metallurgical Indus-try Press Beijing China 1st edition 1998
[3] G Zhiying Z Yong and Z Lingqiang ldquoReview of modellingand theoretical studies on chatter in the rolling millsrdquo Journalof Mechanical Engineering vol 51 no 16 pp 87ndash105 2015
[4] H Zhao and K F Ehmann ldquoStability analysis of chatter intandem rolling millsmdashpart 2 the regenerative effectrdquo Journalof Manufacturing Science and EngineeringmdashTransactions of theASME vol 135 no 3 Article ID 031002 11 pages 2013
[5] P-H Hu and K F Ehmann ldquoRegenerative effect in rollingchatterrdquo Journal ofManufacturing Processes vol 3 no 2 pp 82ndash93 2001
[6] P H Hu Stability and Chatter in Rolling Northwestern Univer-sity Evanston Ill USA 1998
[7] H Y Zhao and K F Ehmann ldquoStability analysis of chatter intandem rolling millsmdashpart 1 single- and multi-stand negativedamping effectrdquo Journal ofManufacturing Science and Engineer-ing vol 135 no 3 Article ID 031001 2013
[8] Y Kimura Y Sodani N Nishiura et al ldquoAnalysis of chaffer intandem cold rolling millsrdquo ISIJ International vol 43 no 1 pp77ndash84 2003
[9] M R Niroomand R M Forouzan andM Salimi ldquoTheoreticaland experimental analysis of chatter in tandem cold rollingmills based on wave propagation theoryrdquo ISIJ International vol55 no 3 pp 637ndash646 2015
[10] A Heidari M R Forouzan and S Akbarzadeh ldquoDevelopmentof a rolling chattermodel considering unsteady lubricationrdquo ISIJInternational vol 54 no 1 pp 165ndash170 2014
[11] A Heidari and M R Forouzan ldquoOptimization of cold rollingprocess parameters in order to increasing rolling speed limitedby chatter vibrationsrdquo Journal of Advanced Research vol 4 no1 pp 27ndash34 2013
Shock and Vibration 15
[12] B BahramiNejadMDehghani and S AMousavi ldquoSimulationof two stands cold rolling mill process using neural networksand genetic algorithms in combination to avoid the chatterphenomenonrdquo Majlesi Journal of Electrical Engineering vol 9no 1 pp 21ndash24 2014
[13] X X Liao Theory Methods and Application of Stability Huaz-hong Science and Engineering University PressWuhan China1999
[14] V B Kolmanovskii and V R Nosov Stability of FunctionalDifferential Equations Academic Press New York NY USA1986
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Shock and Vibration
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International Journal of
6 Shock and Vibration
Frequency (Hz)0 200 400
0
1000
2000
Am
plitu
de (P
a)
(e)
209
Frequency (Hz)
Am
plitu
de (P
a)
0 200 4000
100
200
(f)
209
0 200 4000
05
1
Frequency (Hz)
Am
plitu
de (m
)
(d)
209
times10minus9
0 005 01
0
5000
(b)
minus5000
d120590e1
(Pa)
t (s)
0 005 01
0
1000
(c)
minus1000
d120590d1
(Pa)
t (s)
0 005 01
0
1
(a)
minus1
times10minus9dhd1
(m)
t (s)
Figure 5 Time and frequency domain of the single 1st stand when V1199031= 2478ms ((a) and (d)) work roll ((b) and (e)) entry tension ((c)
and (f)) exit tension
complex than the thirdmodelThemain frequency is 209Hza little smaller than the third model but still in the range ofthe third octave mode chatter
Comparing the above four models a more detailedexplanation for regenerative chattermechanism in rolling canbe presented Consequent stands in tandem rolling mills arecoupled together through rolled piece One of these standsbecomes unstable and oscillates first for example the 119894thstand The 119894th stand vibration gives rise to the interstandtension vibration The interstand vibration makes the 119894 + 1thstand vibrate gently at onceThe amplitude of the 119894+1th standis far smaller than that of the 119894th stand by this time But thestrip gauge variation generated by the 119894th stand is passed onto the 119894+1th stand after delay time It aggravates the vibrationof the 119894 + 1th stand and makes the 119894th and 119894 + 1th standvibrate in the same amplitude The oscillation of the 119894 + 1thstand intensifies vibration of the interstand tension andfinallymakes the 119894th stand vibrate again
It can be concluded from the above analysis that timedelay effect as an independent factor has very limited effects
on both critical velocity and chatter frequency while makingthe stability analysis more complex But it does not mean thattime delay effect can be neglected when modeling the rollingsystem Enough works have been done to compare the firstsecond and the fourth model in [4] To study the time delayeffect as a single factor on multistand rolling system stabilityanalysis comparison of the third model and fourth modelwas done in the next section To simplify the writing modethe third model is called the nondelay system and the fourthmodel is called the delay system
4 Effects of Rolling Process Parameters
41 Effects of Friction Stability of a single stand is very sen-sitive to friction The critical velocity of a single stand growsrapidly with the increase of friction coefficient [2 7] Butthings are different for multistand systems as shown inFigure 9
When 1199061is small enough stability of the 1st stand gets
worse and it becomes more unstable than the 2nd stand
Shock and Vibration 7
Frequency (Hz)
Am
plitu
de (P
a)
0 200 400
(f)
209
251100
200
0
0 200 4000
05
1
Frequency (Hz)
Am
plitu
de (m
)
(d)
209
times10minus9
Frequency (Hz)
Am
plitu
de (m
)
0 200 4000
2
4
(e)
209251
times10minus11
0 005 01
0
1
(a)
minus1
times10minus9dhd1
(m)
t (s)
0 005 01
0
5
(b)
minus5
times10minus11
dhd2
(m)
t (s)
0 005 01
0
500
(c)
minus500
d120590d1
(Pa)
t (s)
Figure 6 The time and frequency domain of the second model V1199032= 3600ms ((a) and (d)) 1st stand work roll ((b) and (e)) 2nd stand
work roll ((c) and (f)) interstand tension
For example as shown in Table 2 when 1199061is 0014 the critical
velocity of single 1st stand is 229ms According to theprinciple of mass conservation the corresponding velocityof 2nd stand is 331ms which is smaller than the single2nd stand critical velocity 348ms The 1st stand becomesunstable before the 2nd standThe critical velocity of the two-stand system is determined by the 1st stand As 119906
1increases
the 1st stand becomes more stable and critical velocity of thetwo-stand system increases sharply until 119906
1reaches a certain
point
Table 2The critical velocities for different stands when 1199061changes
1199061
0014 0018Critical velocity of single 1st stand V
1198881
(ms) 229 248
Corresponding critical velocity ofsingle 2nd stand V
1198882(ms) 331 363
Critical velocity of single 2nd stand V1198882
(ms) 348 348
More unstable stand 1st stand 2nd stand
8 Shock and Vibration
Frequency (Hz)
Am
plitu
de (P
a)
0 200 4000
1000
2000
(f)
212
0 200 4000
05
1
Frequency (Hz)
Am
plitu
de (m
)
(d)
212
times10minus9
Frequency (Hz)
Am
plitu
de (m
)
0 200 4000
05
1
(e)
212
times10minus9
0 005 01
0
2
(b)
minus2
times10minus9
dhd2
(m)
t (s)
0 005 01
0
1
(a)
minus1
times10minus9
dhd1
(m)
t (s)
0 005 01
0
5000
(c)
minus5000
d120590d1
(Pa)
t (s)
Figure 7The time and frequency domain of the third model when V1199032= 3023ms ((a) and (d)) 1st stand work roll ((b) and (e)) 2nd stand
work roll ((c) and (f)) interstand tension
When 1199061is big enough 2nd stand becomesmore unstable
than 1st stand For example as shown in Table 2 when 1199061
is 0018 the critical velocity of the single 1st stand goes upto 248ms and the corresponding velocity of the single 2ndstand rises to 363ms which is bigger than critical velocityof the 2nd stand The 2nd stand becomes unstable first Thecritical velocity of the two-stand system is determined by the2nd stand As 119906
1increases the 1st stand becomesmore stable
but it has little effect on the 2nd stand The stability of 2ndstand is much worse than the 1st stand so the critical velocityof the two-stand system goes down as shown in Figure 9
When 1199061goes up the overall trend of critical velocities
are the same for the delay system and nondelay systemThere exists an optimal 119906
1 which relates to the maximum
critical velocity given that other rolling parameters remainunchanged But the optimal 119906
1values are different for the
delay and nondelay system The optimal 1199061for the delay
system is about 0016 while the optimal 1199061for the nondelay
system is about 002 Critical velocities for the delay systemare smaller than the nondelay system especially when 119906
1is
larger It can be explained by the idea that time delay effect hasa far greater impact on 2nd stand than 1st stand as the strip
Shock and Vibration 9
0 200 4000
50
100
Frequency (Hz)
Am
plitu
de (P
a)
(f)
209
0 200 4000
5
Frequency (Hz)A
mpl
itude
(m)
(e)
209
times10minus11
0 200 4000
5
Am
plitu
de (m
)
(d)
Frequency (Hz)
209
times10minus11
0 05 1
0
1
(a)
minus1
times10minus9dhd1
(m)
t (s)
0 05 1
0
2
(b)
minus2
times10minus9
dhd2
(m)
t (s)
0 05 1
0
2000
(c) t (s)
minus2000
d120590d1
(Pa)
Figure 8 The time and frequency domain of the fourth model when V1199032= 2925ms ((a) and (d)) 1st stand work roll ((b) and (e)) 2nd
stand work roll ((c) and (f)) interstand tension
gauge variation generated in the 1st stand is passed on to the2nd stand with delay time but not in the opposite direction
Figure 10 demonstrates the critical velocities for the delaysystem and nondelay system when the friction coefficient ofthe 2nd stand increases The critical velocity changing trendis much simpler When 119906
2is small enough the 2nd stand is
more unstable and the critical velocity increases as 1199062goes
up When 1199062is big enough the stability of the whole system
becomes better as 1199062goes up The critical velocities for the
delay system and nondelay system are almost the sameWhen1199062is relatively small the critical velocity of the time delay
system is slightly smaller than that of the nondelay systemThe smaller the 119906
2is the more unstable and sensitive to 119906
2
the time delay system is
42 Effects of Interstand Tension The critical velocity for asingle stand decreases as the entry tension and exit tensionincrease But the critical velocity ismore sensitive to the entry
tension due to the fact that the entry zone is much longer thatthe exit zone [2]1205901198891
is the intertension between the 1st and 2nd standnamely the exit tension of the 1st stand and the entry tensionof the 2nd stand The stability of the 1st and 2nd stand bothgoes worse with the increase of 120590
1198891 Therefore the critical
velocities of the delay and nondelay system both fall offas 1205901198891
increases as shown in Figure 11 But the stability ofthe 2nd stand is more sensitive to 120590
1198891 as 120590
1198891is the entry
tension for the 2nd stand As can been seen from Table 3when 120590
1198891is rather small the 1st stand becomes unstable first
and the critical velocity of the system is decided by the 1ststand The critical velocities decrease slowly as 120590
1198891is the exit
tension of the 1st stand But when 1205901198891
is big enough the 2ndstand becomes unstable first and the critical velocity of thetwo-stand system is decided by the 2nd stand The criticalvelocities of the delay system decrease fast with the increaseof 1205901198891 because 120590
1198891is the entry tension of the 2nd stand
10 Shock and Vibration
001 0015 002 0025 00323
24
25
26
27
28
29
Criti
cal s
peed
(ms
)
Delay systemNondelay system
u1
Figure 9 Comparison of stability charts for 1199061
001 0015 002 0025 00322
24
26
28
30
32
34
Criti
cal s
peed
(ms
)
Delay systemNondelay system
u2
Figure 10 Comparison of stability charts for 1199062
When considering the time delay effect the critical velocitygets smaller and more sensitive to 120590
1198891
43 Effects of Interstand Distance Interstand tension vari-ation becomes more gentle with the increase of interstanddistance And so the stability of mill stand becomes better[2] But it is more complex to multistand system 119871
1198902is the
distance between the 1st and 2nd stand If 1198711198902
is changedthe interstand tension variation and the delay time are bothchanged As shown in Figure 12 for time delay system withthe increase of 119871
1198902 the interstand tension variation decreases
and critical velocity of delay system risesMeantime the delaytime becomes larger and critical velocity decreases So thecritical velocities of time delay system form a wavy patternof rising
50 100 150 200 250 30025
26
27
28
29
30
31
32
Criti
cal s
peed
(ms
)
Delay systemNondelay system
120590d1 (MPa)
Figure 11 Comparison of stability charts for 1205901198891
3 35 4 45 5 55 623
24
25
26
27
28
29
30Cr
itica
l spe
ed (m
s)
Delay systemNondelay system
Le2 (m)
Figure 12 Comparison of stability charts for 1198711198902
Table 3 The critical velocities for different stands when 1205901198891
changes
1205901198891
(Mpa) 60 250Critical velocity of single 1st stand V
1198881(ms) 227 225
Corresponding critical velocity of single 2ndstand V
1198882(ms) 326 335
Critical velocity of single 2nd stand V1198882
(ms) 367 331
More unstable stand 1st stand 2nd stand
At first the 2nd stand is more unstable than the 1st oneWith the increase of 119871
1198902 the critical velocities of the 1st
and 2nd stand both rise But the critical velocity of 2ndstand goes up much faster than the 1st stand as 119871
1198902affects
Shock and Vibration 11
1 15 2 25 3 35 420
25
30
35
40
45
Criti
cal s
peed
(ms
)
Delay systemNondelay system
E (GPa)
Figure 13 Comparison of stability charts for 119864
05 1 1526
27
28
29
30
31
32
33
34
Criti
cal s
peed
(ms
)
Delay systemNondelay system
w (m)
Figure 14 Comparison of stability charts for 119908
the entry tension variation of 2nd standThe critical velocitiesof the nondelay and time delay system both rise fast When1198711198902
is bigger than a certain value the 1st stand becomesunstable first The rises of critical velocities for the nondelayand delay system both slow down for 119871
1198902affects the exit
tension variation of 1st stand It can been seen in Table 4 thatwhen 119871
1198902is 35m the 2nd stand goes to unstable first but
when 1198711198902
is 55m chatter occurs in the 1st stand first Butthe instability of the 1st stand occurs when 119871
1198902is quite bigger
than nondelay system for the delay time effect makes the 2ndstand more unstable
44 Effects of Strip Width and Strip Elastic Modulus Thecritical velocity of a single stand system decreases with theincrease of the strip elastic modulus and goes up along with
022 024 026 028 03 032 034 03624
26
28
30
32
34
36
38
40
Criti
cal s
peed
(ms
)
Delay systemNondelay system
he2 (mm)
Figure 15 Comparison of stability charts for ℎ1198902
Table 4 The critical velocities for different stands when 1198711198902
changes
1198711198902
(ms) 35 55Critical velocity of single 1st stand V
1198881
(ms) 2463 2487
Corresponding critical velocity of single2nd stand V
1198882(ms) 3605 3640
Critical velocity of single 2nd stand V1198882
(ms) 3138 3789
More unstable stand 2nd stand 1st stand
the increase of the strip width [2 7] As shown in Figures13 and 14 it shows the same trend for both the time delaysystem and nondelay systemWith the increase of strip elasticmodulus the critical velocities for the time delay system andnondelay system are almost the same The latter is just alittle bigger than the former With the increasing strip widththe critical velocities for the time delay system and nondelaysystem both go up But the critical velocities for the time delaysystem are a littlemore sensitive to the increase of strip width
45 Effects of Reduction Allocation It has been proved thatthe critical velocity for a single stand system decreaseswith the increase of the entry thickness given that the exitthickness stays the same and increase with the rise of the exitthickness given that the entry thickness remains the same[2] But it is more complex for multistand system as shownin Figure 15 Assuming that the entry strip thickness of the1st stand and the exit strip thickness of the 2nd stand remainunaltered the critical velocity would be changed if the entrythickness of the 2nd stand namely ℎ
1198902 is changed
If ℎ1198902
is small enough it means that exit strip thicknessof the 1st stand decreases and entry strip thickness of the 2ndstand increases The 1st stand becomes unstable first But asℎ1198902
rises the 1st stand reduction gets smaller while the 2nd
12 Shock and Vibration
Table 5The critical velocities for different standswhen ℎ1198902changes
ℎ1198902
(mm) 026 029Critical velocity of single 1st stand V
1198881
(ms) 242 252
Corresponding critical velocity of single2nd stand V
1198882(ms) 331 380
Critical velocity of single 2nd stand V1198882
(ms) 368 341
More unstable stand 1st stand 2nd stand
stand reduction gets bigger The 2nd stand becomes moreunstable than the 1st stand As shown in Table 5 when ℎ
1198902
is 026mm the 1st stand is more unstable However whenℎ1198902
increases to 029 the 2nd stand becomes more unstablethan 1st stand There exists an optimal value of ℎ
1198902for the
multistand system which is related to the maximum criticalvelocity as shown in Figure 11
The overall critical velocity changing trend for the timedelay system and nondelay system is the same But theoptimal values of ℎ
1198902are different for the two systems
because the stability of the 2nd stand becomes worse whenconsidering the time delay effect The optimal values of ℎ
1198902
for the two systems are 033mm and 028mm respectivelyObviously the optimal value of ℎ
1198902for the delay system is
more consistent with the actual production It suggests thatby redistributing the reduction of the tandem rolling millswe can improve the stability of the tandem rolling mills
The critical velocity of a single stand changing along withthe increase of the reduction is not linearWhen the reductionis small the critical velocity of the stand is very sensitiveto the reduction changing But as the reduction goes upthe decrease of critical velocity slows down So as shown inFigure 15 when ℎ
1198902is bigger than a certain point the critical
velocities of the two systems both go down slowlyIt has to be clarified that the optimal ℎ
1198902value for the two-
stand system may not work for five stand system There existfour delay times in five-stand tandem rolling mills whichmakes the relationship between stands more complex
5 Conclusions
In this paper the effects of interstand tension variation andstrip variation transportation between adjacent stands withtime delay on tandem rolling mills were studied in detailthrough comparison of different models Stability analy-sis shows that interstand tension couples adjacent standstogether and has just a little effect on the critical velocity of thechatter model The strip gauge variation passed on betweenstands reduces the critical velocity dramatically Howeverthe time delay effect has very limited effects on both thecritical velocity and the frequency But it does not mean thatthe time delay effect can be neglected when modeling therolling system By comparing the critical velocities of thedelay and nondelay system the influences of delay time asa single factor on multistand rolling system were studiedMore detailed and quantitative explanation is put forward to
the relationship of two adjacent stands In the big picturethe critical velocity of the time delay system is just slightlysmaller than the nondelay system But the delay time worsensthe stability of downstream stand and makes the relationshipbetween consecutive stands more complex The time delayeffect changes the optimal process parameters values aimedat obtaining the maximum critical velocity To get preferablerolling process parameter configuration for tandem rollingmills time delay effect must be involved
Appendix
Consider119889119865 = 119865
1119889ℎ119889+ 1198652119889ℎ119889+ 1198653119889120590119890+ 1198654119889120590119889+ 1198655119889ℎ119890
1198651= (
120597119865
120597ℎ119889
)
119904
= int
119909119899
119909119889
(
120597119901119889
120597ℎ119889
)
119904
119889119909 + int
119909119890
119909119899
(
120597119901119890
120597ℎ119889
)
119904
119889119909
+ (119896119891119890minus 120590119890) (
120597119909119890
120597ℎ119889
)
119904
1198652= (
120597119865
120597ℎ119889
)
119904
= int
119909119899
119909119889
(
120597119901119889
120597ℎ119889
)
119904
119889119909 minus (119896119891119889minus 120590119889)(
120597119909119889
120597ℎ119889
)
119904
1198653= (
120597119865
120597120590119890
)
119904
= int
119909119890
119909119899
(
120597119901119890
120597120590119890
)
119904
119889119909
1198654= (
120597119865
120597120590119889
)
119904
= int
119909119899
119909119889
(
120597119901119889
120597120590119889
)
119904
119889119909
1198655= (
120597119865
120597ℎ119890
)
119904
= int
119909119890
119909119899
(
120597119901119890
120597ℎ119890
)
119904
119889119909 + (119896119891119890minus 120590119890) (
120597119909119890
120597ℎ119890
)
119904
119889V119890= 1198751119889ℎ119889+ 1198752119889ℎ119889+ 1198753119889120590119890+ 1198754119889120590119889+ 1198755119889ℎ119890
1198751= (
120597V119890
120597ℎ119889
)
119904
=
V119903
ℎ119890
+
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597ℎ119889
)
119904
1198752= (
120597V119890
120597ℎ119889
)
119904
=
119909119890minus 119909119899
ℎ119890
+
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597ℎ119889
)
119904
1198753= (
120597V119890
120597120590119890
)
119904
=
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597120590119890
)
119904
1198754= (
120597V119890
120597120590119889
)
119904
=
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597120590119889
)
119904
1198755= (
120597V119890
120597ℎ119890
)
119904
= minus
V119903ℎ119899
ℎ119890
2+
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597ℎ119890
)
119904
Shock and Vibration 13
119889V119889= 1198761119889ℎ119889+ 1198762119889ℎ119889+ 1198763119889120590119890+ 1198764119889120590119889+ 1198765119889ℎ119890
1198761= (
120597V119889
120597ℎ119889
)
119904
= minus
V119890ℎ119890
ℎ119889
2+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119889
)
119904
1198762= (
120597V119889
120597ℎ119889
)
119904
=
minus119909119890
ℎ119889
+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119889
)
119904
1198763= (
120597V119889
120597120590119890
)
119904
=
ℎ119890
ℎ119889
(
120597V119890
120597120590119890
)
119904
1198764= (
120597V119889
120597120590119889
)
119904
=
ℎ119890
ℎ119889
(
120597V119890
120597120590119889
)
119904
1198765= (
120597V119889
120597ℎ119890
)
119904
=
V119890
ℎ119889
+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119890
)
119904
(A1)
The intermedia variables are given as follows
(
120597119909119899
120597ℎ119889
)
119904
= (minus
1198921
8119906
+
1
4119906
minus
119909119890
4ℎ119890
) sec (119892)2 + radic 119877
4ℎ119889
sdot tan (119892) + (120597119909119899
120597119909119890
)
119904
(
120597119909119890
120597ℎ119889
)
119904
+ (
120597119909119899
120597119896119891119889
)
119904
sdot (
120597119896119891119889
120597ℎ119889
)
119904
1198921 = log(ℎ119890119896119891119890
ℎ119889119896119891119889
119896119891119889minus 120590119889
119896119891119890minus 120590119890
)
119892 =
1
4
radicℎ119889
119877
(2radic
119877
ℎ119889
tanminus1 (119909119890
radic119877ℎ119889
)
+ 2radic
119877
ℎ119889
tanminus1 (119909119889
radic119877ℎ119889
) minus
1198921
119906
)
(
120597119909119899
120597119909119890
)
119904
=
ℎ119889sec (119892)2
2ℎ119890
(
120597119909119899
120597119896119891119889
)
119904
=
ℎ119889
4119906
(
1
119896119891119889
minus
1
119896119891119889minus 120590119889
) sec (119892)2
(
120597119896119891119889
120597ℎ119889
) = minus
1198991205900
ℎ119889
(119860 + log(ℎ0
ℎ119889
))
119899minus1
(
120597119909119899
120597ℎ119889
)
119904
=
1
2
sec (119892)2 (120597119909119889
120597ℎ119889
)
119904
(
120597119909119899
120597120590119890
)
119904
= minus
ℎ119889sec (119892)2
4119906 (119896119891119890minus 120590119890)
(
120597119909119899
120597120590119889
)
119904
=
ℎ119889sec (119892)2
4119906 (119896119891119889minus 120590119889)
(
120597119909119899
120597ℎ119890
)
119904
= minus
ℎ119889sec (119892)2
4119906ℎ119890
+ (
120597119909119899
120597119909119890
)
119904
(
120597119909119890
120597ℎ119890
)
119904
+ (
120597119909119899
120597119896119891119890
)
119904
(
120597119896119891119890
120597ℎ119890
)
119904
(
120597119909119899
120597119896119891119890
)
119904
=
ℎ119889
4119906
(
1
119896119891119890minus 120590119890
minus
1
119896119891119890
) sec (119892)2
(
120597119896119891119890
120597ℎ119890
)
119904
= minus
1198991205900
ℎ119890
(119860 + log(ℎ0
ℎ119890
))
119899minus1
(
120597119909119890
120597ℎ119889
)
119904
= minusradic
119877
4 (ℎ119890minus ℎ119889)
(
120597119909119890
120597ℎ119890
)
119904
= radic
119877
4 (ℎ119890minus ℎ119889)
(
120597119909119889
120597ℎ119889
)
119904
=
119877ℎ119889
2V119903ℎ119899
(
120597119901119890
120597ℎ119889
)
119904
= (
120597119901119890
120597ℎ
)
119904
+ (
120597119901119890
120597119896119891
)
119904
(
120597119896119891
120597ℎ119889
)
119904
+ (
120597119901119890
120597119867119890
)
119904
sdot (
120597119867119890
120597ℎ119889
)
119904
+ (
120597119901119890
120597119867
)
119904
(
120597119867
120597ℎ119889
)
119904
(
120597119901119890
120597ℎ
)
119904
=
(119896119891119890minus 120590119890) 119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119901119890
120597119896119891
)
119904
=
(119896119891119890minus 120590119890) ℎ119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119896119891
120597ℎ119889
)
119904
= minus
1205900119899
ℎ
(119860 + ln(ℎ0
ℎ
))
119899minus1
(
120597119901119890
120597119867119890
)
119904
=
(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119867119890
120597ℎ119889
)
119904
= minus
119877arctan (119909119890radic119877ℎ
119889)
ℎ119889
2
radic119877ℎ119889
minus
119909119890
ℎ119889ℎ119890
+ (
120597119867119890
120597119909119890
)
119904
(
120597119909119890
120597ℎ119889
)
119904
(
120597119901119890
120597119867
)
119904
= minus
(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119867
120597ℎ119889
)
119904
= minus
119877arctan (119909radic119877ℎ119889)
ℎ119889
2
radic119877ℎ119889
minus
119909
ℎ119889ℎ
(
120597119901119890
120597120590119890
)
119904
= minus
ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
14 Shock and Vibration
(
120597119901e120597ℎe)
119904
= (
120597119901e120597119896119891119890
)
119904
(
120597119896119891119890
120597ℎ119890
)
119904
+ (
120597119901e120597119867119890
)
119904
(
120597119867119890
120597119909119890
)
119904
sdot (
120597119909119890
120597ℎ119890
)
119904
minus
(119896119891119890minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)
ℎ119890
2
119896119891119890
(
120597119901119890
120597119896119891119890
)
119904
=
ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
minus
(119896119891119890 minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
2
(
120597119867119890
120597119909119890
)
119904
=
2
ℎ119890
(
120597119901d120597ℎ119889
)
119904
= (
120597119901119889
120597119896119891119889
)
119904
(
120597119896119891119889
120597ℎ119889
)
119904
+ (
120597119901119889
120597ℎ
)
119904
+ (
120597119901119889
120597119896119891
)
119904
(
120597119896119891
120597ℎ119889
)
119904
+ (
120597119901119889
120597119867
)
119904
(
120597119867
120597ℎ119889
)
119904
minus
(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889
2
119896119891119889
(
120597119901119889
120597119896119891119889
)
119904
=
ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
minus
(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
2
(
120597119901119889
120597ℎ
)
119904
=
(119896119891119889minus 120590119889) 119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119896119891
)
119904
=
(119896119891119889minus 120590119889) ℎ119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119867119889
)
119904
= minus
(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119867
)
119904
=
(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597ℎ119889
)
119904
= (
120597119901119889
120597119867119889
)
119904
(
120597119867119889
120597119909119889
)
119904
(
120597119909119889
120597ℎ119889
)
119904
(
120597119867119889
120597119909119889
)
119904
=
2
ℎ119889
(
120597119901119889
120597120590119889
)
119904
= minus
ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
119867119890= 2radic
119877
ℎ119889
tanminus1 (119909119890
radic119877ℎ119889
)
119867119889= 2radic
119877
ℎ119889
tanminus1 (119909119889
radic119877ℎ119889
)
119867 = 2radic
119877
ℎ119889
tanminus1 ( 119909
radic119877ℎ119889
)
(A2)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors gratefully acknowledge the support of theNational Natural Science Foundation of China no 51175035PhD Programs Foundation of Ministry of Education ofChina no 20100006110024 the Fundamental Research Fundsfor the Central Universities no FRF-BR-14-006A and BeijingHigher EducationYoungElite Teacher Project no YETP0367
References
[1] I S YunW R DWilson and K F Ehmann ldquoReview of chatterstudies in cold rollingrdquo International Journal of Machine Toolsand Manufacture vol 38 no 12 pp 1499ndash1530 1998
[2] J X Zhou TandemMill Vibration Control Metallurgical Indus-try Press Beijing China 1st edition 1998
[3] G Zhiying Z Yong and Z Lingqiang ldquoReview of modellingand theoretical studies on chatter in the rolling millsrdquo Journalof Mechanical Engineering vol 51 no 16 pp 87ndash105 2015
[4] H Zhao and K F Ehmann ldquoStability analysis of chatter intandem rolling millsmdashpart 2 the regenerative effectrdquo Journalof Manufacturing Science and EngineeringmdashTransactions of theASME vol 135 no 3 Article ID 031002 11 pages 2013
[5] P-H Hu and K F Ehmann ldquoRegenerative effect in rollingchatterrdquo Journal ofManufacturing Processes vol 3 no 2 pp 82ndash93 2001
[6] P H Hu Stability and Chatter in Rolling Northwestern Univer-sity Evanston Ill USA 1998
[7] H Y Zhao and K F Ehmann ldquoStability analysis of chatter intandem rolling millsmdashpart 1 single- and multi-stand negativedamping effectrdquo Journal ofManufacturing Science and Engineer-ing vol 135 no 3 Article ID 031001 2013
[8] Y Kimura Y Sodani N Nishiura et al ldquoAnalysis of chaffer intandem cold rolling millsrdquo ISIJ International vol 43 no 1 pp77ndash84 2003
[9] M R Niroomand R M Forouzan andM Salimi ldquoTheoreticaland experimental analysis of chatter in tandem cold rollingmills based on wave propagation theoryrdquo ISIJ International vol55 no 3 pp 637ndash646 2015
[10] A Heidari M R Forouzan and S Akbarzadeh ldquoDevelopmentof a rolling chattermodel considering unsteady lubricationrdquo ISIJInternational vol 54 no 1 pp 165ndash170 2014
[11] A Heidari and M R Forouzan ldquoOptimization of cold rollingprocess parameters in order to increasing rolling speed limitedby chatter vibrationsrdquo Journal of Advanced Research vol 4 no1 pp 27ndash34 2013
Shock and Vibration 15
[12] B BahramiNejadMDehghani and S AMousavi ldquoSimulationof two stands cold rolling mill process using neural networksand genetic algorithms in combination to avoid the chatterphenomenonrdquo Majlesi Journal of Electrical Engineering vol 9no 1 pp 21ndash24 2014
[13] X X Liao Theory Methods and Application of Stability Huaz-hong Science and Engineering University PressWuhan China1999
[14] V B Kolmanovskii and V R Nosov Stability of FunctionalDifferential Equations Academic Press New York NY USA1986
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Shock and Vibration
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International Journal of
Shock and Vibration 7
Frequency (Hz)
Am
plitu
de (P
a)
0 200 400
(f)
209
251100
200
0
0 200 4000
05
1
Frequency (Hz)
Am
plitu
de (m
)
(d)
209
times10minus9
Frequency (Hz)
Am
plitu
de (m
)
0 200 4000
2
4
(e)
209251
times10minus11
0 005 01
0
1
(a)
minus1
times10minus9dhd1
(m)
t (s)
0 005 01
0
5
(b)
minus5
times10minus11
dhd2
(m)
t (s)
0 005 01
0
500
(c)
minus500
d120590d1
(Pa)
t (s)
Figure 6 The time and frequency domain of the second model V1199032= 3600ms ((a) and (d)) 1st stand work roll ((b) and (e)) 2nd stand
work roll ((c) and (f)) interstand tension
For example as shown in Table 2 when 1199061is 0014 the critical
velocity of single 1st stand is 229ms According to theprinciple of mass conservation the corresponding velocityof 2nd stand is 331ms which is smaller than the single2nd stand critical velocity 348ms The 1st stand becomesunstable before the 2nd standThe critical velocity of the two-stand system is determined by the 1st stand As 119906
1increases
the 1st stand becomes more stable and critical velocity of thetwo-stand system increases sharply until 119906
1reaches a certain
point
Table 2The critical velocities for different stands when 1199061changes
1199061
0014 0018Critical velocity of single 1st stand V
1198881
(ms) 229 248
Corresponding critical velocity ofsingle 2nd stand V
1198882(ms) 331 363
Critical velocity of single 2nd stand V1198882
(ms) 348 348
More unstable stand 1st stand 2nd stand
8 Shock and Vibration
Frequency (Hz)
Am
plitu
de (P
a)
0 200 4000
1000
2000
(f)
212
0 200 4000
05
1
Frequency (Hz)
Am
plitu
de (m
)
(d)
212
times10minus9
Frequency (Hz)
Am
plitu
de (m
)
0 200 4000
05
1
(e)
212
times10minus9
0 005 01
0
2
(b)
minus2
times10minus9
dhd2
(m)
t (s)
0 005 01
0
1
(a)
minus1
times10minus9
dhd1
(m)
t (s)
0 005 01
0
5000
(c)
minus5000
d120590d1
(Pa)
t (s)
Figure 7The time and frequency domain of the third model when V1199032= 3023ms ((a) and (d)) 1st stand work roll ((b) and (e)) 2nd stand
work roll ((c) and (f)) interstand tension
When 1199061is big enough 2nd stand becomesmore unstable
than 1st stand For example as shown in Table 2 when 1199061
is 0018 the critical velocity of the single 1st stand goes upto 248ms and the corresponding velocity of the single 2ndstand rises to 363ms which is bigger than critical velocityof the 2nd stand The 2nd stand becomes unstable first Thecritical velocity of the two-stand system is determined by the2nd stand As 119906
1increases the 1st stand becomesmore stable
but it has little effect on the 2nd stand The stability of 2ndstand is much worse than the 1st stand so the critical velocityof the two-stand system goes down as shown in Figure 9
When 1199061goes up the overall trend of critical velocities
are the same for the delay system and nondelay systemThere exists an optimal 119906
1 which relates to the maximum
critical velocity given that other rolling parameters remainunchanged But the optimal 119906
1values are different for the
delay and nondelay system The optimal 1199061for the delay
system is about 0016 while the optimal 1199061for the nondelay
system is about 002 Critical velocities for the delay systemare smaller than the nondelay system especially when 119906
1is
larger It can be explained by the idea that time delay effect hasa far greater impact on 2nd stand than 1st stand as the strip
Shock and Vibration 9
0 200 4000
50
100
Frequency (Hz)
Am
plitu
de (P
a)
(f)
209
0 200 4000
5
Frequency (Hz)A
mpl
itude
(m)
(e)
209
times10minus11
0 200 4000
5
Am
plitu
de (m
)
(d)
Frequency (Hz)
209
times10minus11
0 05 1
0
1
(a)
minus1
times10minus9dhd1
(m)
t (s)
0 05 1
0
2
(b)
minus2
times10minus9
dhd2
(m)
t (s)
0 05 1
0
2000
(c) t (s)
minus2000
d120590d1
(Pa)
Figure 8 The time and frequency domain of the fourth model when V1199032= 2925ms ((a) and (d)) 1st stand work roll ((b) and (e)) 2nd
stand work roll ((c) and (f)) interstand tension
gauge variation generated in the 1st stand is passed on to the2nd stand with delay time but not in the opposite direction
Figure 10 demonstrates the critical velocities for the delaysystem and nondelay system when the friction coefficient ofthe 2nd stand increases The critical velocity changing trendis much simpler When 119906
2is small enough the 2nd stand is
more unstable and the critical velocity increases as 1199062goes
up When 1199062is big enough the stability of the whole system
becomes better as 1199062goes up The critical velocities for the
delay system and nondelay system are almost the sameWhen1199062is relatively small the critical velocity of the time delay
system is slightly smaller than that of the nondelay systemThe smaller the 119906
2is the more unstable and sensitive to 119906
2
the time delay system is
42 Effects of Interstand Tension The critical velocity for asingle stand decreases as the entry tension and exit tensionincrease But the critical velocity ismore sensitive to the entry
tension due to the fact that the entry zone is much longer thatthe exit zone [2]1205901198891
is the intertension between the 1st and 2nd standnamely the exit tension of the 1st stand and the entry tensionof the 2nd stand The stability of the 1st and 2nd stand bothgoes worse with the increase of 120590
1198891 Therefore the critical
velocities of the delay and nondelay system both fall offas 1205901198891
increases as shown in Figure 11 But the stability ofthe 2nd stand is more sensitive to 120590
1198891 as 120590
1198891is the entry
tension for the 2nd stand As can been seen from Table 3when 120590
1198891is rather small the 1st stand becomes unstable first
and the critical velocity of the system is decided by the 1ststand The critical velocities decrease slowly as 120590
1198891is the exit
tension of the 1st stand But when 1205901198891
is big enough the 2ndstand becomes unstable first and the critical velocity of thetwo-stand system is decided by the 2nd stand The criticalvelocities of the delay system decrease fast with the increaseof 1205901198891 because 120590
1198891is the entry tension of the 2nd stand
10 Shock and Vibration
001 0015 002 0025 00323
24
25
26
27
28
29
Criti
cal s
peed
(ms
)
Delay systemNondelay system
u1
Figure 9 Comparison of stability charts for 1199061
001 0015 002 0025 00322
24
26
28
30
32
34
Criti
cal s
peed
(ms
)
Delay systemNondelay system
u2
Figure 10 Comparison of stability charts for 1199062
When considering the time delay effect the critical velocitygets smaller and more sensitive to 120590
1198891
43 Effects of Interstand Distance Interstand tension vari-ation becomes more gentle with the increase of interstanddistance And so the stability of mill stand becomes better[2] But it is more complex to multistand system 119871
1198902is the
distance between the 1st and 2nd stand If 1198711198902
is changedthe interstand tension variation and the delay time are bothchanged As shown in Figure 12 for time delay system withthe increase of 119871
1198902 the interstand tension variation decreases
and critical velocity of delay system risesMeantime the delaytime becomes larger and critical velocity decreases So thecritical velocities of time delay system form a wavy patternof rising
50 100 150 200 250 30025
26
27
28
29
30
31
32
Criti
cal s
peed
(ms
)
Delay systemNondelay system
120590d1 (MPa)
Figure 11 Comparison of stability charts for 1205901198891
3 35 4 45 5 55 623
24
25
26
27
28
29
30Cr
itica
l spe
ed (m
s)
Delay systemNondelay system
Le2 (m)
Figure 12 Comparison of stability charts for 1198711198902
Table 3 The critical velocities for different stands when 1205901198891
changes
1205901198891
(Mpa) 60 250Critical velocity of single 1st stand V
1198881(ms) 227 225
Corresponding critical velocity of single 2ndstand V
1198882(ms) 326 335
Critical velocity of single 2nd stand V1198882
(ms) 367 331
More unstable stand 1st stand 2nd stand
At first the 2nd stand is more unstable than the 1st oneWith the increase of 119871
1198902 the critical velocities of the 1st
and 2nd stand both rise But the critical velocity of 2ndstand goes up much faster than the 1st stand as 119871
1198902affects
Shock and Vibration 11
1 15 2 25 3 35 420
25
30
35
40
45
Criti
cal s
peed
(ms
)
Delay systemNondelay system
E (GPa)
Figure 13 Comparison of stability charts for 119864
05 1 1526
27
28
29
30
31
32
33
34
Criti
cal s
peed
(ms
)
Delay systemNondelay system
w (m)
Figure 14 Comparison of stability charts for 119908
the entry tension variation of 2nd standThe critical velocitiesof the nondelay and time delay system both rise fast When1198711198902
is bigger than a certain value the 1st stand becomesunstable first The rises of critical velocities for the nondelayand delay system both slow down for 119871
1198902affects the exit
tension variation of 1st stand It can been seen in Table 4 thatwhen 119871
1198902is 35m the 2nd stand goes to unstable first but
when 1198711198902
is 55m chatter occurs in the 1st stand first Butthe instability of the 1st stand occurs when 119871
1198902is quite bigger
than nondelay system for the delay time effect makes the 2ndstand more unstable
44 Effects of Strip Width and Strip Elastic Modulus Thecritical velocity of a single stand system decreases with theincrease of the strip elastic modulus and goes up along with
022 024 026 028 03 032 034 03624
26
28
30
32
34
36
38
40
Criti
cal s
peed
(ms
)
Delay systemNondelay system
he2 (mm)
Figure 15 Comparison of stability charts for ℎ1198902
Table 4 The critical velocities for different stands when 1198711198902
changes
1198711198902
(ms) 35 55Critical velocity of single 1st stand V
1198881
(ms) 2463 2487
Corresponding critical velocity of single2nd stand V
1198882(ms) 3605 3640
Critical velocity of single 2nd stand V1198882
(ms) 3138 3789
More unstable stand 2nd stand 1st stand
the increase of the strip width [2 7] As shown in Figures13 and 14 it shows the same trend for both the time delaysystem and nondelay systemWith the increase of strip elasticmodulus the critical velocities for the time delay system andnondelay system are almost the same The latter is just alittle bigger than the former With the increasing strip widththe critical velocities for the time delay system and nondelaysystem both go up But the critical velocities for the time delaysystem are a littlemore sensitive to the increase of strip width
45 Effects of Reduction Allocation It has been proved thatthe critical velocity for a single stand system decreaseswith the increase of the entry thickness given that the exitthickness stays the same and increase with the rise of the exitthickness given that the entry thickness remains the same[2] But it is more complex for multistand system as shownin Figure 15 Assuming that the entry strip thickness of the1st stand and the exit strip thickness of the 2nd stand remainunaltered the critical velocity would be changed if the entrythickness of the 2nd stand namely ℎ
1198902 is changed
If ℎ1198902
is small enough it means that exit strip thicknessof the 1st stand decreases and entry strip thickness of the 2ndstand increases The 1st stand becomes unstable first But asℎ1198902
rises the 1st stand reduction gets smaller while the 2nd
12 Shock and Vibration
Table 5The critical velocities for different standswhen ℎ1198902changes
ℎ1198902
(mm) 026 029Critical velocity of single 1st stand V
1198881
(ms) 242 252
Corresponding critical velocity of single2nd stand V
1198882(ms) 331 380
Critical velocity of single 2nd stand V1198882
(ms) 368 341
More unstable stand 1st stand 2nd stand
stand reduction gets bigger The 2nd stand becomes moreunstable than the 1st stand As shown in Table 5 when ℎ
1198902
is 026mm the 1st stand is more unstable However whenℎ1198902
increases to 029 the 2nd stand becomes more unstablethan 1st stand There exists an optimal value of ℎ
1198902for the
multistand system which is related to the maximum criticalvelocity as shown in Figure 11
The overall critical velocity changing trend for the timedelay system and nondelay system is the same But theoptimal values of ℎ
1198902are different for the two systems
because the stability of the 2nd stand becomes worse whenconsidering the time delay effect The optimal values of ℎ
1198902
for the two systems are 033mm and 028mm respectivelyObviously the optimal value of ℎ
1198902for the delay system is
more consistent with the actual production It suggests thatby redistributing the reduction of the tandem rolling millswe can improve the stability of the tandem rolling mills
The critical velocity of a single stand changing along withthe increase of the reduction is not linearWhen the reductionis small the critical velocity of the stand is very sensitiveto the reduction changing But as the reduction goes upthe decrease of critical velocity slows down So as shown inFigure 15 when ℎ
1198902is bigger than a certain point the critical
velocities of the two systems both go down slowlyIt has to be clarified that the optimal ℎ
1198902value for the two-
stand system may not work for five stand system There existfour delay times in five-stand tandem rolling mills whichmakes the relationship between stands more complex
5 Conclusions
In this paper the effects of interstand tension variation andstrip variation transportation between adjacent stands withtime delay on tandem rolling mills were studied in detailthrough comparison of different models Stability analy-sis shows that interstand tension couples adjacent standstogether and has just a little effect on the critical velocity of thechatter model The strip gauge variation passed on betweenstands reduces the critical velocity dramatically Howeverthe time delay effect has very limited effects on both thecritical velocity and the frequency But it does not mean thatthe time delay effect can be neglected when modeling therolling system By comparing the critical velocities of thedelay and nondelay system the influences of delay time asa single factor on multistand rolling system were studiedMore detailed and quantitative explanation is put forward to
the relationship of two adjacent stands In the big picturethe critical velocity of the time delay system is just slightlysmaller than the nondelay system But the delay time worsensthe stability of downstream stand and makes the relationshipbetween consecutive stands more complex The time delayeffect changes the optimal process parameters values aimedat obtaining the maximum critical velocity To get preferablerolling process parameter configuration for tandem rollingmills time delay effect must be involved
Appendix
Consider119889119865 = 119865
1119889ℎ119889+ 1198652119889ℎ119889+ 1198653119889120590119890+ 1198654119889120590119889+ 1198655119889ℎ119890
1198651= (
120597119865
120597ℎ119889
)
119904
= int
119909119899
119909119889
(
120597119901119889
120597ℎ119889
)
119904
119889119909 + int
119909119890
119909119899
(
120597119901119890
120597ℎ119889
)
119904
119889119909
+ (119896119891119890minus 120590119890) (
120597119909119890
120597ℎ119889
)
119904
1198652= (
120597119865
120597ℎ119889
)
119904
= int
119909119899
119909119889
(
120597119901119889
120597ℎ119889
)
119904
119889119909 minus (119896119891119889minus 120590119889)(
120597119909119889
120597ℎ119889
)
119904
1198653= (
120597119865
120597120590119890
)
119904
= int
119909119890
119909119899
(
120597119901119890
120597120590119890
)
119904
119889119909
1198654= (
120597119865
120597120590119889
)
119904
= int
119909119899
119909119889
(
120597119901119889
120597120590119889
)
119904
119889119909
1198655= (
120597119865
120597ℎ119890
)
119904
= int
119909119890
119909119899
(
120597119901119890
120597ℎ119890
)
119904
119889119909 + (119896119891119890minus 120590119890) (
120597119909119890
120597ℎ119890
)
119904
119889V119890= 1198751119889ℎ119889+ 1198752119889ℎ119889+ 1198753119889120590119890+ 1198754119889120590119889+ 1198755119889ℎ119890
1198751= (
120597V119890
120597ℎ119889
)
119904
=
V119903
ℎ119890
+
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597ℎ119889
)
119904
1198752= (
120597V119890
120597ℎ119889
)
119904
=
119909119890minus 119909119899
ℎ119890
+
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597ℎ119889
)
119904
1198753= (
120597V119890
120597120590119890
)
119904
=
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597120590119890
)
119904
1198754= (
120597V119890
120597120590119889
)
119904
=
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597120590119889
)
119904
1198755= (
120597V119890
120597ℎ119890
)
119904
= minus
V119903ℎ119899
ℎ119890
2+
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597ℎ119890
)
119904
Shock and Vibration 13
119889V119889= 1198761119889ℎ119889+ 1198762119889ℎ119889+ 1198763119889120590119890+ 1198764119889120590119889+ 1198765119889ℎ119890
1198761= (
120597V119889
120597ℎ119889
)
119904
= minus
V119890ℎ119890
ℎ119889
2+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119889
)
119904
1198762= (
120597V119889
120597ℎ119889
)
119904
=
minus119909119890
ℎ119889
+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119889
)
119904
1198763= (
120597V119889
120597120590119890
)
119904
=
ℎ119890
ℎ119889
(
120597V119890
120597120590119890
)
119904
1198764= (
120597V119889
120597120590119889
)
119904
=
ℎ119890
ℎ119889
(
120597V119890
120597120590119889
)
119904
1198765= (
120597V119889
120597ℎ119890
)
119904
=
V119890
ℎ119889
+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119890
)
119904
(A1)
The intermedia variables are given as follows
(
120597119909119899
120597ℎ119889
)
119904
= (minus
1198921
8119906
+
1
4119906
minus
119909119890
4ℎ119890
) sec (119892)2 + radic 119877
4ℎ119889
sdot tan (119892) + (120597119909119899
120597119909119890
)
119904
(
120597119909119890
120597ℎ119889
)
119904
+ (
120597119909119899
120597119896119891119889
)
119904
sdot (
120597119896119891119889
120597ℎ119889
)
119904
1198921 = log(ℎ119890119896119891119890
ℎ119889119896119891119889
119896119891119889minus 120590119889
119896119891119890minus 120590119890
)
119892 =
1
4
radicℎ119889
119877
(2radic
119877
ℎ119889
tanminus1 (119909119890
radic119877ℎ119889
)
+ 2radic
119877
ℎ119889
tanminus1 (119909119889
radic119877ℎ119889
) minus
1198921
119906
)
(
120597119909119899
120597119909119890
)
119904
=
ℎ119889sec (119892)2
2ℎ119890
(
120597119909119899
120597119896119891119889
)
119904
=
ℎ119889
4119906
(
1
119896119891119889
minus
1
119896119891119889minus 120590119889
) sec (119892)2
(
120597119896119891119889
120597ℎ119889
) = minus
1198991205900
ℎ119889
(119860 + log(ℎ0
ℎ119889
))
119899minus1
(
120597119909119899
120597ℎ119889
)
119904
=
1
2
sec (119892)2 (120597119909119889
120597ℎ119889
)
119904
(
120597119909119899
120597120590119890
)
119904
= minus
ℎ119889sec (119892)2
4119906 (119896119891119890minus 120590119890)
(
120597119909119899
120597120590119889
)
119904
=
ℎ119889sec (119892)2
4119906 (119896119891119889minus 120590119889)
(
120597119909119899
120597ℎ119890
)
119904
= minus
ℎ119889sec (119892)2
4119906ℎ119890
+ (
120597119909119899
120597119909119890
)
119904
(
120597119909119890
120597ℎ119890
)
119904
+ (
120597119909119899
120597119896119891119890
)
119904
(
120597119896119891119890
120597ℎ119890
)
119904
(
120597119909119899
120597119896119891119890
)
119904
=
ℎ119889
4119906
(
1
119896119891119890minus 120590119890
minus
1
119896119891119890
) sec (119892)2
(
120597119896119891119890
120597ℎ119890
)
119904
= minus
1198991205900
ℎ119890
(119860 + log(ℎ0
ℎ119890
))
119899minus1
(
120597119909119890
120597ℎ119889
)
119904
= minusradic
119877
4 (ℎ119890minus ℎ119889)
(
120597119909119890
120597ℎ119890
)
119904
= radic
119877
4 (ℎ119890minus ℎ119889)
(
120597119909119889
120597ℎ119889
)
119904
=
119877ℎ119889
2V119903ℎ119899
(
120597119901119890
120597ℎ119889
)
119904
= (
120597119901119890
120597ℎ
)
119904
+ (
120597119901119890
120597119896119891
)
119904
(
120597119896119891
120597ℎ119889
)
119904
+ (
120597119901119890
120597119867119890
)
119904
sdot (
120597119867119890
120597ℎ119889
)
119904
+ (
120597119901119890
120597119867
)
119904
(
120597119867
120597ℎ119889
)
119904
(
120597119901119890
120597ℎ
)
119904
=
(119896119891119890minus 120590119890) 119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119901119890
120597119896119891
)
119904
=
(119896119891119890minus 120590119890) ℎ119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119896119891
120597ℎ119889
)
119904
= minus
1205900119899
ℎ
(119860 + ln(ℎ0
ℎ
))
119899minus1
(
120597119901119890
120597119867119890
)
119904
=
(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119867119890
120597ℎ119889
)
119904
= minus
119877arctan (119909119890radic119877ℎ
119889)
ℎ119889
2
radic119877ℎ119889
minus
119909119890
ℎ119889ℎ119890
+ (
120597119867119890
120597119909119890
)
119904
(
120597119909119890
120597ℎ119889
)
119904
(
120597119901119890
120597119867
)
119904
= minus
(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119867
120597ℎ119889
)
119904
= minus
119877arctan (119909radic119877ℎ119889)
ℎ119889
2
radic119877ℎ119889
minus
119909
ℎ119889ℎ
(
120597119901119890
120597120590119890
)
119904
= minus
ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
14 Shock and Vibration
(
120597119901e120597ℎe)
119904
= (
120597119901e120597119896119891119890
)
119904
(
120597119896119891119890
120597ℎ119890
)
119904
+ (
120597119901e120597119867119890
)
119904
(
120597119867119890
120597119909119890
)
119904
sdot (
120597119909119890
120597ℎ119890
)
119904
minus
(119896119891119890minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)
ℎ119890
2
119896119891119890
(
120597119901119890
120597119896119891119890
)
119904
=
ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
minus
(119896119891119890 minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
2
(
120597119867119890
120597119909119890
)
119904
=
2
ℎ119890
(
120597119901d120597ℎ119889
)
119904
= (
120597119901119889
120597119896119891119889
)
119904
(
120597119896119891119889
120597ℎ119889
)
119904
+ (
120597119901119889
120597ℎ
)
119904
+ (
120597119901119889
120597119896119891
)
119904
(
120597119896119891
120597ℎ119889
)
119904
+ (
120597119901119889
120597119867
)
119904
(
120597119867
120597ℎ119889
)
119904
minus
(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889
2
119896119891119889
(
120597119901119889
120597119896119891119889
)
119904
=
ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
minus
(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
2
(
120597119901119889
120597ℎ
)
119904
=
(119896119891119889minus 120590119889) 119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119896119891
)
119904
=
(119896119891119889minus 120590119889) ℎ119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119867119889
)
119904
= minus
(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119867
)
119904
=
(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597ℎ119889
)
119904
= (
120597119901119889
120597119867119889
)
119904
(
120597119867119889
120597119909119889
)
119904
(
120597119909119889
120597ℎ119889
)
119904
(
120597119867119889
120597119909119889
)
119904
=
2
ℎ119889
(
120597119901119889
120597120590119889
)
119904
= minus
ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
119867119890= 2radic
119877
ℎ119889
tanminus1 (119909119890
radic119877ℎ119889
)
119867119889= 2radic
119877
ℎ119889
tanminus1 (119909119889
radic119877ℎ119889
)
119867 = 2radic
119877
ℎ119889
tanminus1 ( 119909
radic119877ℎ119889
)
(A2)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors gratefully acknowledge the support of theNational Natural Science Foundation of China no 51175035PhD Programs Foundation of Ministry of Education ofChina no 20100006110024 the Fundamental Research Fundsfor the Central Universities no FRF-BR-14-006A and BeijingHigher EducationYoungElite Teacher Project no YETP0367
References
[1] I S YunW R DWilson and K F Ehmann ldquoReview of chatterstudies in cold rollingrdquo International Journal of Machine Toolsand Manufacture vol 38 no 12 pp 1499ndash1530 1998
[2] J X Zhou TandemMill Vibration Control Metallurgical Indus-try Press Beijing China 1st edition 1998
[3] G Zhiying Z Yong and Z Lingqiang ldquoReview of modellingand theoretical studies on chatter in the rolling millsrdquo Journalof Mechanical Engineering vol 51 no 16 pp 87ndash105 2015
[4] H Zhao and K F Ehmann ldquoStability analysis of chatter intandem rolling millsmdashpart 2 the regenerative effectrdquo Journalof Manufacturing Science and EngineeringmdashTransactions of theASME vol 135 no 3 Article ID 031002 11 pages 2013
[5] P-H Hu and K F Ehmann ldquoRegenerative effect in rollingchatterrdquo Journal ofManufacturing Processes vol 3 no 2 pp 82ndash93 2001
[6] P H Hu Stability and Chatter in Rolling Northwestern Univer-sity Evanston Ill USA 1998
[7] H Y Zhao and K F Ehmann ldquoStability analysis of chatter intandem rolling millsmdashpart 1 single- and multi-stand negativedamping effectrdquo Journal ofManufacturing Science and Engineer-ing vol 135 no 3 Article ID 031001 2013
[8] Y Kimura Y Sodani N Nishiura et al ldquoAnalysis of chaffer intandem cold rolling millsrdquo ISIJ International vol 43 no 1 pp77ndash84 2003
[9] M R Niroomand R M Forouzan andM Salimi ldquoTheoreticaland experimental analysis of chatter in tandem cold rollingmills based on wave propagation theoryrdquo ISIJ International vol55 no 3 pp 637ndash646 2015
[10] A Heidari M R Forouzan and S Akbarzadeh ldquoDevelopmentof a rolling chattermodel considering unsteady lubricationrdquo ISIJInternational vol 54 no 1 pp 165ndash170 2014
[11] A Heidari and M R Forouzan ldquoOptimization of cold rollingprocess parameters in order to increasing rolling speed limitedby chatter vibrationsrdquo Journal of Advanced Research vol 4 no1 pp 27ndash34 2013
Shock and Vibration 15
[12] B BahramiNejadMDehghani and S AMousavi ldquoSimulationof two stands cold rolling mill process using neural networksand genetic algorithms in combination to avoid the chatterphenomenonrdquo Majlesi Journal of Electrical Engineering vol 9no 1 pp 21ndash24 2014
[13] X X Liao Theory Methods and Application of Stability Huaz-hong Science and Engineering University PressWuhan China1999
[14] V B Kolmanovskii and V R Nosov Stability of FunctionalDifferential Equations Academic Press New York NY USA1986
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Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Chemical EngineeringInternational Journal of Antennas and
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International Journal of
8 Shock and Vibration
Frequency (Hz)
Am
plitu
de (P
a)
0 200 4000
1000
2000
(f)
212
0 200 4000
05
1
Frequency (Hz)
Am
plitu
de (m
)
(d)
212
times10minus9
Frequency (Hz)
Am
plitu
de (m
)
0 200 4000
05
1
(e)
212
times10minus9
0 005 01
0
2
(b)
minus2
times10minus9
dhd2
(m)
t (s)
0 005 01
0
1
(a)
minus1
times10minus9
dhd1
(m)
t (s)
0 005 01
0
5000
(c)
minus5000
d120590d1
(Pa)
t (s)
Figure 7The time and frequency domain of the third model when V1199032= 3023ms ((a) and (d)) 1st stand work roll ((b) and (e)) 2nd stand
work roll ((c) and (f)) interstand tension
When 1199061is big enough 2nd stand becomesmore unstable
than 1st stand For example as shown in Table 2 when 1199061
is 0018 the critical velocity of the single 1st stand goes upto 248ms and the corresponding velocity of the single 2ndstand rises to 363ms which is bigger than critical velocityof the 2nd stand The 2nd stand becomes unstable first Thecritical velocity of the two-stand system is determined by the2nd stand As 119906
1increases the 1st stand becomesmore stable
but it has little effect on the 2nd stand The stability of 2ndstand is much worse than the 1st stand so the critical velocityof the two-stand system goes down as shown in Figure 9
When 1199061goes up the overall trend of critical velocities
are the same for the delay system and nondelay systemThere exists an optimal 119906
1 which relates to the maximum
critical velocity given that other rolling parameters remainunchanged But the optimal 119906
1values are different for the
delay and nondelay system The optimal 1199061for the delay
system is about 0016 while the optimal 1199061for the nondelay
system is about 002 Critical velocities for the delay systemare smaller than the nondelay system especially when 119906
1is
larger It can be explained by the idea that time delay effect hasa far greater impact on 2nd stand than 1st stand as the strip
Shock and Vibration 9
0 200 4000
50
100
Frequency (Hz)
Am
plitu
de (P
a)
(f)
209
0 200 4000
5
Frequency (Hz)A
mpl
itude
(m)
(e)
209
times10minus11
0 200 4000
5
Am
plitu
de (m
)
(d)
Frequency (Hz)
209
times10minus11
0 05 1
0
1
(a)
minus1
times10minus9dhd1
(m)
t (s)
0 05 1
0
2
(b)
minus2
times10minus9
dhd2
(m)
t (s)
0 05 1
0
2000
(c) t (s)
minus2000
d120590d1
(Pa)
Figure 8 The time and frequency domain of the fourth model when V1199032= 2925ms ((a) and (d)) 1st stand work roll ((b) and (e)) 2nd
stand work roll ((c) and (f)) interstand tension
gauge variation generated in the 1st stand is passed on to the2nd stand with delay time but not in the opposite direction
Figure 10 demonstrates the critical velocities for the delaysystem and nondelay system when the friction coefficient ofthe 2nd stand increases The critical velocity changing trendis much simpler When 119906
2is small enough the 2nd stand is
more unstable and the critical velocity increases as 1199062goes
up When 1199062is big enough the stability of the whole system
becomes better as 1199062goes up The critical velocities for the
delay system and nondelay system are almost the sameWhen1199062is relatively small the critical velocity of the time delay
system is slightly smaller than that of the nondelay systemThe smaller the 119906
2is the more unstable and sensitive to 119906
2
the time delay system is
42 Effects of Interstand Tension The critical velocity for asingle stand decreases as the entry tension and exit tensionincrease But the critical velocity ismore sensitive to the entry
tension due to the fact that the entry zone is much longer thatthe exit zone [2]1205901198891
is the intertension between the 1st and 2nd standnamely the exit tension of the 1st stand and the entry tensionof the 2nd stand The stability of the 1st and 2nd stand bothgoes worse with the increase of 120590
1198891 Therefore the critical
velocities of the delay and nondelay system both fall offas 1205901198891
increases as shown in Figure 11 But the stability ofthe 2nd stand is more sensitive to 120590
1198891 as 120590
1198891is the entry
tension for the 2nd stand As can been seen from Table 3when 120590
1198891is rather small the 1st stand becomes unstable first
and the critical velocity of the system is decided by the 1ststand The critical velocities decrease slowly as 120590
1198891is the exit
tension of the 1st stand But when 1205901198891
is big enough the 2ndstand becomes unstable first and the critical velocity of thetwo-stand system is decided by the 2nd stand The criticalvelocities of the delay system decrease fast with the increaseof 1205901198891 because 120590
1198891is the entry tension of the 2nd stand
10 Shock and Vibration
001 0015 002 0025 00323
24
25
26
27
28
29
Criti
cal s
peed
(ms
)
Delay systemNondelay system
u1
Figure 9 Comparison of stability charts for 1199061
001 0015 002 0025 00322
24
26
28
30
32
34
Criti
cal s
peed
(ms
)
Delay systemNondelay system
u2
Figure 10 Comparison of stability charts for 1199062
When considering the time delay effect the critical velocitygets smaller and more sensitive to 120590
1198891
43 Effects of Interstand Distance Interstand tension vari-ation becomes more gentle with the increase of interstanddistance And so the stability of mill stand becomes better[2] But it is more complex to multistand system 119871
1198902is the
distance between the 1st and 2nd stand If 1198711198902
is changedthe interstand tension variation and the delay time are bothchanged As shown in Figure 12 for time delay system withthe increase of 119871
1198902 the interstand tension variation decreases
and critical velocity of delay system risesMeantime the delaytime becomes larger and critical velocity decreases So thecritical velocities of time delay system form a wavy patternof rising
50 100 150 200 250 30025
26
27
28
29
30
31
32
Criti
cal s
peed
(ms
)
Delay systemNondelay system
120590d1 (MPa)
Figure 11 Comparison of stability charts for 1205901198891
3 35 4 45 5 55 623
24
25
26
27
28
29
30Cr
itica
l spe
ed (m
s)
Delay systemNondelay system
Le2 (m)
Figure 12 Comparison of stability charts for 1198711198902
Table 3 The critical velocities for different stands when 1205901198891
changes
1205901198891
(Mpa) 60 250Critical velocity of single 1st stand V
1198881(ms) 227 225
Corresponding critical velocity of single 2ndstand V
1198882(ms) 326 335
Critical velocity of single 2nd stand V1198882
(ms) 367 331
More unstable stand 1st stand 2nd stand
At first the 2nd stand is more unstable than the 1st oneWith the increase of 119871
1198902 the critical velocities of the 1st
and 2nd stand both rise But the critical velocity of 2ndstand goes up much faster than the 1st stand as 119871
1198902affects
Shock and Vibration 11
1 15 2 25 3 35 420
25
30
35
40
45
Criti
cal s
peed
(ms
)
Delay systemNondelay system
E (GPa)
Figure 13 Comparison of stability charts for 119864
05 1 1526
27
28
29
30
31
32
33
34
Criti
cal s
peed
(ms
)
Delay systemNondelay system
w (m)
Figure 14 Comparison of stability charts for 119908
the entry tension variation of 2nd standThe critical velocitiesof the nondelay and time delay system both rise fast When1198711198902
is bigger than a certain value the 1st stand becomesunstable first The rises of critical velocities for the nondelayand delay system both slow down for 119871
1198902affects the exit
tension variation of 1st stand It can been seen in Table 4 thatwhen 119871
1198902is 35m the 2nd stand goes to unstable first but
when 1198711198902
is 55m chatter occurs in the 1st stand first Butthe instability of the 1st stand occurs when 119871
1198902is quite bigger
than nondelay system for the delay time effect makes the 2ndstand more unstable
44 Effects of Strip Width and Strip Elastic Modulus Thecritical velocity of a single stand system decreases with theincrease of the strip elastic modulus and goes up along with
022 024 026 028 03 032 034 03624
26
28
30
32
34
36
38
40
Criti
cal s
peed
(ms
)
Delay systemNondelay system
he2 (mm)
Figure 15 Comparison of stability charts for ℎ1198902
Table 4 The critical velocities for different stands when 1198711198902
changes
1198711198902
(ms) 35 55Critical velocity of single 1st stand V
1198881
(ms) 2463 2487
Corresponding critical velocity of single2nd stand V
1198882(ms) 3605 3640
Critical velocity of single 2nd stand V1198882
(ms) 3138 3789
More unstable stand 2nd stand 1st stand
the increase of the strip width [2 7] As shown in Figures13 and 14 it shows the same trend for both the time delaysystem and nondelay systemWith the increase of strip elasticmodulus the critical velocities for the time delay system andnondelay system are almost the same The latter is just alittle bigger than the former With the increasing strip widththe critical velocities for the time delay system and nondelaysystem both go up But the critical velocities for the time delaysystem are a littlemore sensitive to the increase of strip width
45 Effects of Reduction Allocation It has been proved thatthe critical velocity for a single stand system decreaseswith the increase of the entry thickness given that the exitthickness stays the same and increase with the rise of the exitthickness given that the entry thickness remains the same[2] But it is more complex for multistand system as shownin Figure 15 Assuming that the entry strip thickness of the1st stand and the exit strip thickness of the 2nd stand remainunaltered the critical velocity would be changed if the entrythickness of the 2nd stand namely ℎ
1198902 is changed
If ℎ1198902
is small enough it means that exit strip thicknessof the 1st stand decreases and entry strip thickness of the 2ndstand increases The 1st stand becomes unstable first But asℎ1198902
rises the 1st stand reduction gets smaller while the 2nd
12 Shock and Vibration
Table 5The critical velocities for different standswhen ℎ1198902changes
ℎ1198902
(mm) 026 029Critical velocity of single 1st stand V
1198881
(ms) 242 252
Corresponding critical velocity of single2nd stand V
1198882(ms) 331 380
Critical velocity of single 2nd stand V1198882
(ms) 368 341
More unstable stand 1st stand 2nd stand
stand reduction gets bigger The 2nd stand becomes moreunstable than the 1st stand As shown in Table 5 when ℎ
1198902
is 026mm the 1st stand is more unstable However whenℎ1198902
increases to 029 the 2nd stand becomes more unstablethan 1st stand There exists an optimal value of ℎ
1198902for the
multistand system which is related to the maximum criticalvelocity as shown in Figure 11
The overall critical velocity changing trend for the timedelay system and nondelay system is the same But theoptimal values of ℎ
1198902are different for the two systems
because the stability of the 2nd stand becomes worse whenconsidering the time delay effect The optimal values of ℎ
1198902
for the two systems are 033mm and 028mm respectivelyObviously the optimal value of ℎ
1198902for the delay system is
more consistent with the actual production It suggests thatby redistributing the reduction of the tandem rolling millswe can improve the stability of the tandem rolling mills
The critical velocity of a single stand changing along withthe increase of the reduction is not linearWhen the reductionis small the critical velocity of the stand is very sensitiveto the reduction changing But as the reduction goes upthe decrease of critical velocity slows down So as shown inFigure 15 when ℎ
1198902is bigger than a certain point the critical
velocities of the two systems both go down slowlyIt has to be clarified that the optimal ℎ
1198902value for the two-
stand system may not work for five stand system There existfour delay times in five-stand tandem rolling mills whichmakes the relationship between stands more complex
5 Conclusions
In this paper the effects of interstand tension variation andstrip variation transportation between adjacent stands withtime delay on tandem rolling mills were studied in detailthrough comparison of different models Stability analy-sis shows that interstand tension couples adjacent standstogether and has just a little effect on the critical velocity of thechatter model The strip gauge variation passed on betweenstands reduces the critical velocity dramatically Howeverthe time delay effect has very limited effects on both thecritical velocity and the frequency But it does not mean thatthe time delay effect can be neglected when modeling therolling system By comparing the critical velocities of thedelay and nondelay system the influences of delay time asa single factor on multistand rolling system were studiedMore detailed and quantitative explanation is put forward to
the relationship of two adjacent stands In the big picturethe critical velocity of the time delay system is just slightlysmaller than the nondelay system But the delay time worsensthe stability of downstream stand and makes the relationshipbetween consecutive stands more complex The time delayeffect changes the optimal process parameters values aimedat obtaining the maximum critical velocity To get preferablerolling process parameter configuration for tandem rollingmills time delay effect must be involved
Appendix
Consider119889119865 = 119865
1119889ℎ119889+ 1198652119889ℎ119889+ 1198653119889120590119890+ 1198654119889120590119889+ 1198655119889ℎ119890
1198651= (
120597119865
120597ℎ119889
)
119904
= int
119909119899
119909119889
(
120597119901119889
120597ℎ119889
)
119904
119889119909 + int
119909119890
119909119899
(
120597119901119890
120597ℎ119889
)
119904
119889119909
+ (119896119891119890minus 120590119890) (
120597119909119890
120597ℎ119889
)
119904
1198652= (
120597119865
120597ℎ119889
)
119904
= int
119909119899
119909119889
(
120597119901119889
120597ℎ119889
)
119904
119889119909 minus (119896119891119889minus 120590119889)(
120597119909119889
120597ℎ119889
)
119904
1198653= (
120597119865
120597120590119890
)
119904
= int
119909119890
119909119899
(
120597119901119890
120597120590119890
)
119904
119889119909
1198654= (
120597119865
120597120590119889
)
119904
= int
119909119899
119909119889
(
120597119901119889
120597120590119889
)
119904
119889119909
1198655= (
120597119865
120597ℎ119890
)
119904
= int
119909119890
119909119899
(
120597119901119890
120597ℎ119890
)
119904
119889119909 + (119896119891119890minus 120590119890) (
120597119909119890
120597ℎ119890
)
119904
119889V119890= 1198751119889ℎ119889+ 1198752119889ℎ119889+ 1198753119889120590119890+ 1198754119889120590119889+ 1198755119889ℎ119890
1198751= (
120597V119890
120597ℎ119889
)
119904
=
V119903
ℎ119890
+
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597ℎ119889
)
119904
1198752= (
120597V119890
120597ℎ119889
)
119904
=
119909119890minus 119909119899
ℎ119890
+
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597ℎ119889
)
119904
1198753= (
120597V119890
120597120590119890
)
119904
=
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597120590119890
)
119904
1198754= (
120597V119890
120597120590119889
)
119904
=
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597120590119889
)
119904
1198755= (
120597V119890
120597ℎ119890
)
119904
= minus
V119903ℎ119899
ℎ119890
2+
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597ℎ119890
)
119904
Shock and Vibration 13
119889V119889= 1198761119889ℎ119889+ 1198762119889ℎ119889+ 1198763119889120590119890+ 1198764119889120590119889+ 1198765119889ℎ119890
1198761= (
120597V119889
120597ℎ119889
)
119904
= minus
V119890ℎ119890
ℎ119889
2+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119889
)
119904
1198762= (
120597V119889
120597ℎ119889
)
119904
=
minus119909119890
ℎ119889
+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119889
)
119904
1198763= (
120597V119889
120597120590119890
)
119904
=
ℎ119890
ℎ119889
(
120597V119890
120597120590119890
)
119904
1198764= (
120597V119889
120597120590119889
)
119904
=
ℎ119890
ℎ119889
(
120597V119890
120597120590119889
)
119904
1198765= (
120597V119889
120597ℎ119890
)
119904
=
V119890
ℎ119889
+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119890
)
119904
(A1)
The intermedia variables are given as follows
(
120597119909119899
120597ℎ119889
)
119904
= (minus
1198921
8119906
+
1
4119906
minus
119909119890
4ℎ119890
) sec (119892)2 + radic 119877
4ℎ119889
sdot tan (119892) + (120597119909119899
120597119909119890
)
119904
(
120597119909119890
120597ℎ119889
)
119904
+ (
120597119909119899
120597119896119891119889
)
119904
sdot (
120597119896119891119889
120597ℎ119889
)
119904
1198921 = log(ℎ119890119896119891119890
ℎ119889119896119891119889
119896119891119889minus 120590119889
119896119891119890minus 120590119890
)
119892 =
1
4
radicℎ119889
119877
(2radic
119877
ℎ119889
tanminus1 (119909119890
radic119877ℎ119889
)
+ 2radic
119877
ℎ119889
tanminus1 (119909119889
radic119877ℎ119889
) minus
1198921
119906
)
(
120597119909119899
120597119909119890
)
119904
=
ℎ119889sec (119892)2
2ℎ119890
(
120597119909119899
120597119896119891119889
)
119904
=
ℎ119889
4119906
(
1
119896119891119889
minus
1
119896119891119889minus 120590119889
) sec (119892)2
(
120597119896119891119889
120597ℎ119889
) = minus
1198991205900
ℎ119889
(119860 + log(ℎ0
ℎ119889
))
119899minus1
(
120597119909119899
120597ℎ119889
)
119904
=
1
2
sec (119892)2 (120597119909119889
120597ℎ119889
)
119904
(
120597119909119899
120597120590119890
)
119904
= minus
ℎ119889sec (119892)2
4119906 (119896119891119890minus 120590119890)
(
120597119909119899
120597120590119889
)
119904
=
ℎ119889sec (119892)2
4119906 (119896119891119889minus 120590119889)
(
120597119909119899
120597ℎ119890
)
119904
= minus
ℎ119889sec (119892)2
4119906ℎ119890
+ (
120597119909119899
120597119909119890
)
119904
(
120597119909119890
120597ℎ119890
)
119904
+ (
120597119909119899
120597119896119891119890
)
119904
(
120597119896119891119890
120597ℎ119890
)
119904
(
120597119909119899
120597119896119891119890
)
119904
=
ℎ119889
4119906
(
1
119896119891119890minus 120590119890
minus
1
119896119891119890
) sec (119892)2
(
120597119896119891119890
120597ℎ119890
)
119904
= minus
1198991205900
ℎ119890
(119860 + log(ℎ0
ℎ119890
))
119899minus1
(
120597119909119890
120597ℎ119889
)
119904
= minusradic
119877
4 (ℎ119890minus ℎ119889)
(
120597119909119890
120597ℎ119890
)
119904
= radic
119877
4 (ℎ119890minus ℎ119889)
(
120597119909119889
120597ℎ119889
)
119904
=
119877ℎ119889
2V119903ℎ119899
(
120597119901119890
120597ℎ119889
)
119904
= (
120597119901119890
120597ℎ
)
119904
+ (
120597119901119890
120597119896119891
)
119904
(
120597119896119891
120597ℎ119889
)
119904
+ (
120597119901119890
120597119867119890
)
119904
sdot (
120597119867119890
120597ℎ119889
)
119904
+ (
120597119901119890
120597119867
)
119904
(
120597119867
120597ℎ119889
)
119904
(
120597119901119890
120597ℎ
)
119904
=
(119896119891119890minus 120590119890) 119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119901119890
120597119896119891
)
119904
=
(119896119891119890minus 120590119890) ℎ119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119896119891
120597ℎ119889
)
119904
= minus
1205900119899
ℎ
(119860 + ln(ℎ0
ℎ
))
119899minus1
(
120597119901119890
120597119867119890
)
119904
=
(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119867119890
120597ℎ119889
)
119904
= minus
119877arctan (119909119890radic119877ℎ
119889)
ℎ119889
2
radic119877ℎ119889
minus
119909119890
ℎ119889ℎ119890
+ (
120597119867119890
120597119909119890
)
119904
(
120597119909119890
120597ℎ119889
)
119904
(
120597119901119890
120597119867
)
119904
= minus
(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119867
120597ℎ119889
)
119904
= minus
119877arctan (119909radic119877ℎ119889)
ℎ119889
2
radic119877ℎ119889
minus
119909
ℎ119889ℎ
(
120597119901119890
120597120590119890
)
119904
= minus
ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
14 Shock and Vibration
(
120597119901e120597ℎe)
119904
= (
120597119901e120597119896119891119890
)
119904
(
120597119896119891119890
120597ℎ119890
)
119904
+ (
120597119901e120597119867119890
)
119904
(
120597119867119890
120597119909119890
)
119904
sdot (
120597119909119890
120597ℎ119890
)
119904
minus
(119896119891119890minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)
ℎ119890
2
119896119891119890
(
120597119901119890
120597119896119891119890
)
119904
=
ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
minus
(119896119891119890 minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
2
(
120597119867119890
120597119909119890
)
119904
=
2
ℎ119890
(
120597119901d120597ℎ119889
)
119904
= (
120597119901119889
120597119896119891119889
)
119904
(
120597119896119891119889
120597ℎ119889
)
119904
+ (
120597119901119889
120597ℎ
)
119904
+ (
120597119901119889
120597119896119891
)
119904
(
120597119896119891
120597ℎ119889
)
119904
+ (
120597119901119889
120597119867
)
119904
(
120597119867
120597ℎ119889
)
119904
minus
(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889
2
119896119891119889
(
120597119901119889
120597119896119891119889
)
119904
=
ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
minus
(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
2
(
120597119901119889
120597ℎ
)
119904
=
(119896119891119889minus 120590119889) 119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119896119891
)
119904
=
(119896119891119889minus 120590119889) ℎ119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119867119889
)
119904
= minus
(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119867
)
119904
=
(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597ℎ119889
)
119904
= (
120597119901119889
120597119867119889
)
119904
(
120597119867119889
120597119909119889
)
119904
(
120597119909119889
120597ℎ119889
)
119904
(
120597119867119889
120597119909119889
)
119904
=
2
ℎ119889
(
120597119901119889
120597120590119889
)
119904
= minus
ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
119867119890= 2radic
119877
ℎ119889
tanminus1 (119909119890
radic119877ℎ119889
)
119867119889= 2radic
119877
ℎ119889
tanminus1 (119909119889
radic119877ℎ119889
)
119867 = 2radic
119877
ℎ119889
tanminus1 ( 119909
radic119877ℎ119889
)
(A2)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors gratefully acknowledge the support of theNational Natural Science Foundation of China no 51175035PhD Programs Foundation of Ministry of Education ofChina no 20100006110024 the Fundamental Research Fundsfor the Central Universities no FRF-BR-14-006A and BeijingHigher EducationYoungElite Teacher Project no YETP0367
References
[1] I S YunW R DWilson and K F Ehmann ldquoReview of chatterstudies in cold rollingrdquo International Journal of Machine Toolsand Manufacture vol 38 no 12 pp 1499ndash1530 1998
[2] J X Zhou TandemMill Vibration Control Metallurgical Indus-try Press Beijing China 1st edition 1998
[3] G Zhiying Z Yong and Z Lingqiang ldquoReview of modellingand theoretical studies on chatter in the rolling millsrdquo Journalof Mechanical Engineering vol 51 no 16 pp 87ndash105 2015
[4] H Zhao and K F Ehmann ldquoStability analysis of chatter intandem rolling millsmdashpart 2 the regenerative effectrdquo Journalof Manufacturing Science and EngineeringmdashTransactions of theASME vol 135 no 3 Article ID 031002 11 pages 2013
[5] P-H Hu and K F Ehmann ldquoRegenerative effect in rollingchatterrdquo Journal ofManufacturing Processes vol 3 no 2 pp 82ndash93 2001
[6] P H Hu Stability and Chatter in Rolling Northwestern Univer-sity Evanston Ill USA 1998
[7] H Y Zhao and K F Ehmann ldquoStability analysis of chatter intandem rolling millsmdashpart 1 single- and multi-stand negativedamping effectrdquo Journal ofManufacturing Science and Engineer-ing vol 135 no 3 Article ID 031001 2013
[8] Y Kimura Y Sodani N Nishiura et al ldquoAnalysis of chaffer intandem cold rolling millsrdquo ISIJ International vol 43 no 1 pp77ndash84 2003
[9] M R Niroomand R M Forouzan andM Salimi ldquoTheoreticaland experimental analysis of chatter in tandem cold rollingmills based on wave propagation theoryrdquo ISIJ International vol55 no 3 pp 637ndash646 2015
[10] A Heidari M R Forouzan and S Akbarzadeh ldquoDevelopmentof a rolling chattermodel considering unsteady lubricationrdquo ISIJInternational vol 54 no 1 pp 165ndash170 2014
[11] A Heidari and M R Forouzan ldquoOptimization of cold rollingprocess parameters in order to increasing rolling speed limitedby chatter vibrationsrdquo Journal of Advanced Research vol 4 no1 pp 27ndash34 2013
Shock and Vibration 15
[12] B BahramiNejadMDehghani and S AMousavi ldquoSimulationof two stands cold rolling mill process using neural networksand genetic algorithms in combination to avoid the chatterphenomenonrdquo Majlesi Journal of Electrical Engineering vol 9no 1 pp 21ndash24 2014
[13] X X Liao Theory Methods and Application of Stability Huaz-hong Science and Engineering University PressWuhan China1999
[14] V B Kolmanovskii and V R Nosov Stability of FunctionalDifferential Equations Academic Press New York NY USA1986
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VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
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Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
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Volume 2014
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Chemical EngineeringInternational Journal of Antennas and
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International Journal of
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International Journal of
Shock and Vibration 9
0 200 4000
50
100
Frequency (Hz)
Am
plitu
de (P
a)
(f)
209
0 200 4000
5
Frequency (Hz)A
mpl
itude
(m)
(e)
209
times10minus11
0 200 4000
5
Am
plitu
de (m
)
(d)
Frequency (Hz)
209
times10minus11
0 05 1
0
1
(a)
minus1
times10minus9dhd1
(m)
t (s)
0 05 1
0
2
(b)
minus2
times10minus9
dhd2
(m)
t (s)
0 05 1
0
2000
(c) t (s)
minus2000
d120590d1
(Pa)
Figure 8 The time and frequency domain of the fourth model when V1199032= 2925ms ((a) and (d)) 1st stand work roll ((b) and (e)) 2nd
stand work roll ((c) and (f)) interstand tension
gauge variation generated in the 1st stand is passed on to the2nd stand with delay time but not in the opposite direction
Figure 10 demonstrates the critical velocities for the delaysystem and nondelay system when the friction coefficient ofthe 2nd stand increases The critical velocity changing trendis much simpler When 119906
2is small enough the 2nd stand is
more unstable and the critical velocity increases as 1199062goes
up When 1199062is big enough the stability of the whole system
becomes better as 1199062goes up The critical velocities for the
delay system and nondelay system are almost the sameWhen1199062is relatively small the critical velocity of the time delay
system is slightly smaller than that of the nondelay systemThe smaller the 119906
2is the more unstable and sensitive to 119906
2
the time delay system is
42 Effects of Interstand Tension The critical velocity for asingle stand decreases as the entry tension and exit tensionincrease But the critical velocity ismore sensitive to the entry
tension due to the fact that the entry zone is much longer thatthe exit zone [2]1205901198891
is the intertension between the 1st and 2nd standnamely the exit tension of the 1st stand and the entry tensionof the 2nd stand The stability of the 1st and 2nd stand bothgoes worse with the increase of 120590
1198891 Therefore the critical
velocities of the delay and nondelay system both fall offas 1205901198891
increases as shown in Figure 11 But the stability ofthe 2nd stand is more sensitive to 120590
1198891 as 120590
1198891is the entry
tension for the 2nd stand As can been seen from Table 3when 120590
1198891is rather small the 1st stand becomes unstable first
and the critical velocity of the system is decided by the 1ststand The critical velocities decrease slowly as 120590
1198891is the exit
tension of the 1st stand But when 1205901198891
is big enough the 2ndstand becomes unstable first and the critical velocity of thetwo-stand system is decided by the 2nd stand The criticalvelocities of the delay system decrease fast with the increaseof 1205901198891 because 120590
1198891is the entry tension of the 2nd stand
10 Shock and Vibration
001 0015 002 0025 00323
24
25
26
27
28
29
Criti
cal s
peed
(ms
)
Delay systemNondelay system
u1
Figure 9 Comparison of stability charts for 1199061
001 0015 002 0025 00322
24
26
28
30
32
34
Criti
cal s
peed
(ms
)
Delay systemNondelay system
u2
Figure 10 Comparison of stability charts for 1199062
When considering the time delay effect the critical velocitygets smaller and more sensitive to 120590
1198891
43 Effects of Interstand Distance Interstand tension vari-ation becomes more gentle with the increase of interstanddistance And so the stability of mill stand becomes better[2] But it is more complex to multistand system 119871
1198902is the
distance between the 1st and 2nd stand If 1198711198902
is changedthe interstand tension variation and the delay time are bothchanged As shown in Figure 12 for time delay system withthe increase of 119871
1198902 the interstand tension variation decreases
and critical velocity of delay system risesMeantime the delaytime becomes larger and critical velocity decreases So thecritical velocities of time delay system form a wavy patternof rising
50 100 150 200 250 30025
26
27
28
29
30
31
32
Criti
cal s
peed
(ms
)
Delay systemNondelay system
120590d1 (MPa)
Figure 11 Comparison of stability charts for 1205901198891
3 35 4 45 5 55 623
24
25
26
27
28
29
30Cr
itica
l spe
ed (m
s)
Delay systemNondelay system
Le2 (m)
Figure 12 Comparison of stability charts for 1198711198902
Table 3 The critical velocities for different stands when 1205901198891
changes
1205901198891
(Mpa) 60 250Critical velocity of single 1st stand V
1198881(ms) 227 225
Corresponding critical velocity of single 2ndstand V
1198882(ms) 326 335
Critical velocity of single 2nd stand V1198882
(ms) 367 331
More unstable stand 1st stand 2nd stand
At first the 2nd stand is more unstable than the 1st oneWith the increase of 119871
1198902 the critical velocities of the 1st
and 2nd stand both rise But the critical velocity of 2ndstand goes up much faster than the 1st stand as 119871
1198902affects
Shock and Vibration 11
1 15 2 25 3 35 420
25
30
35
40
45
Criti
cal s
peed
(ms
)
Delay systemNondelay system
E (GPa)
Figure 13 Comparison of stability charts for 119864
05 1 1526
27
28
29
30
31
32
33
34
Criti
cal s
peed
(ms
)
Delay systemNondelay system
w (m)
Figure 14 Comparison of stability charts for 119908
the entry tension variation of 2nd standThe critical velocitiesof the nondelay and time delay system both rise fast When1198711198902
is bigger than a certain value the 1st stand becomesunstable first The rises of critical velocities for the nondelayand delay system both slow down for 119871
1198902affects the exit
tension variation of 1st stand It can been seen in Table 4 thatwhen 119871
1198902is 35m the 2nd stand goes to unstable first but
when 1198711198902
is 55m chatter occurs in the 1st stand first Butthe instability of the 1st stand occurs when 119871
1198902is quite bigger
than nondelay system for the delay time effect makes the 2ndstand more unstable
44 Effects of Strip Width and Strip Elastic Modulus Thecritical velocity of a single stand system decreases with theincrease of the strip elastic modulus and goes up along with
022 024 026 028 03 032 034 03624
26
28
30
32
34
36
38
40
Criti
cal s
peed
(ms
)
Delay systemNondelay system
he2 (mm)
Figure 15 Comparison of stability charts for ℎ1198902
Table 4 The critical velocities for different stands when 1198711198902
changes
1198711198902
(ms) 35 55Critical velocity of single 1st stand V
1198881
(ms) 2463 2487
Corresponding critical velocity of single2nd stand V
1198882(ms) 3605 3640
Critical velocity of single 2nd stand V1198882
(ms) 3138 3789
More unstable stand 2nd stand 1st stand
the increase of the strip width [2 7] As shown in Figures13 and 14 it shows the same trend for both the time delaysystem and nondelay systemWith the increase of strip elasticmodulus the critical velocities for the time delay system andnondelay system are almost the same The latter is just alittle bigger than the former With the increasing strip widththe critical velocities for the time delay system and nondelaysystem both go up But the critical velocities for the time delaysystem are a littlemore sensitive to the increase of strip width
45 Effects of Reduction Allocation It has been proved thatthe critical velocity for a single stand system decreaseswith the increase of the entry thickness given that the exitthickness stays the same and increase with the rise of the exitthickness given that the entry thickness remains the same[2] But it is more complex for multistand system as shownin Figure 15 Assuming that the entry strip thickness of the1st stand and the exit strip thickness of the 2nd stand remainunaltered the critical velocity would be changed if the entrythickness of the 2nd stand namely ℎ
1198902 is changed
If ℎ1198902
is small enough it means that exit strip thicknessof the 1st stand decreases and entry strip thickness of the 2ndstand increases The 1st stand becomes unstable first But asℎ1198902
rises the 1st stand reduction gets smaller while the 2nd
12 Shock and Vibration
Table 5The critical velocities for different standswhen ℎ1198902changes
ℎ1198902
(mm) 026 029Critical velocity of single 1st stand V
1198881
(ms) 242 252
Corresponding critical velocity of single2nd stand V
1198882(ms) 331 380
Critical velocity of single 2nd stand V1198882
(ms) 368 341
More unstable stand 1st stand 2nd stand
stand reduction gets bigger The 2nd stand becomes moreunstable than the 1st stand As shown in Table 5 when ℎ
1198902
is 026mm the 1st stand is more unstable However whenℎ1198902
increases to 029 the 2nd stand becomes more unstablethan 1st stand There exists an optimal value of ℎ
1198902for the
multistand system which is related to the maximum criticalvelocity as shown in Figure 11
The overall critical velocity changing trend for the timedelay system and nondelay system is the same But theoptimal values of ℎ
1198902are different for the two systems
because the stability of the 2nd stand becomes worse whenconsidering the time delay effect The optimal values of ℎ
1198902
for the two systems are 033mm and 028mm respectivelyObviously the optimal value of ℎ
1198902for the delay system is
more consistent with the actual production It suggests thatby redistributing the reduction of the tandem rolling millswe can improve the stability of the tandem rolling mills
The critical velocity of a single stand changing along withthe increase of the reduction is not linearWhen the reductionis small the critical velocity of the stand is very sensitiveto the reduction changing But as the reduction goes upthe decrease of critical velocity slows down So as shown inFigure 15 when ℎ
1198902is bigger than a certain point the critical
velocities of the two systems both go down slowlyIt has to be clarified that the optimal ℎ
1198902value for the two-
stand system may not work for five stand system There existfour delay times in five-stand tandem rolling mills whichmakes the relationship between stands more complex
5 Conclusions
In this paper the effects of interstand tension variation andstrip variation transportation between adjacent stands withtime delay on tandem rolling mills were studied in detailthrough comparison of different models Stability analy-sis shows that interstand tension couples adjacent standstogether and has just a little effect on the critical velocity of thechatter model The strip gauge variation passed on betweenstands reduces the critical velocity dramatically Howeverthe time delay effect has very limited effects on both thecritical velocity and the frequency But it does not mean thatthe time delay effect can be neglected when modeling therolling system By comparing the critical velocities of thedelay and nondelay system the influences of delay time asa single factor on multistand rolling system were studiedMore detailed and quantitative explanation is put forward to
the relationship of two adjacent stands In the big picturethe critical velocity of the time delay system is just slightlysmaller than the nondelay system But the delay time worsensthe stability of downstream stand and makes the relationshipbetween consecutive stands more complex The time delayeffect changes the optimal process parameters values aimedat obtaining the maximum critical velocity To get preferablerolling process parameter configuration for tandem rollingmills time delay effect must be involved
Appendix
Consider119889119865 = 119865
1119889ℎ119889+ 1198652119889ℎ119889+ 1198653119889120590119890+ 1198654119889120590119889+ 1198655119889ℎ119890
1198651= (
120597119865
120597ℎ119889
)
119904
= int
119909119899
119909119889
(
120597119901119889
120597ℎ119889
)
119904
119889119909 + int
119909119890
119909119899
(
120597119901119890
120597ℎ119889
)
119904
119889119909
+ (119896119891119890minus 120590119890) (
120597119909119890
120597ℎ119889
)
119904
1198652= (
120597119865
120597ℎ119889
)
119904
= int
119909119899
119909119889
(
120597119901119889
120597ℎ119889
)
119904
119889119909 minus (119896119891119889minus 120590119889)(
120597119909119889
120597ℎ119889
)
119904
1198653= (
120597119865
120597120590119890
)
119904
= int
119909119890
119909119899
(
120597119901119890
120597120590119890
)
119904
119889119909
1198654= (
120597119865
120597120590119889
)
119904
= int
119909119899
119909119889
(
120597119901119889
120597120590119889
)
119904
119889119909
1198655= (
120597119865
120597ℎ119890
)
119904
= int
119909119890
119909119899
(
120597119901119890
120597ℎ119890
)
119904
119889119909 + (119896119891119890minus 120590119890) (
120597119909119890
120597ℎ119890
)
119904
119889V119890= 1198751119889ℎ119889+ 1198752119889ℎ119889+ 1198753119889120590119890+ 1198754119889120590119889+ 1198755119889ℎ119890
1198751= (
120597V119890
120597ℎ119889
)
119904
=
V119903
ℎ119890
+
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597ℎ119889
)
119904
1198752= (
120597V119890
120597ℎ119889
)
119904
=
119909119890minus 119909119899
ℎ119890
+
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597ℎ119889
)
119904
1198753= (
120597V119890
120597120590119890
)
119904
=
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597120590119890
)
119904
1198754= (
120597V119890
120597120590119889
)
119904
=
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597120590119889
)
119904
1198755= (
120597V119890
120597ℎ119890
)
119904
= minus
V119903ℎ119899
ℎ119890
2+
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597ℎ119890
)
119904
Shock and Vibration 13
119889V119889= 1198761119889ℎ119889+ 1198762119889ℎ119889+ 1198763119889120590119890+ 1198764119889120590119889+ 1198765119889ℎ119890
1198761= (
120597V119889
120597ℎ119889
)
119904
= minus
V119890ℎ119890
ℎ119889
2+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119889
)
119904
1198762= (
120597V119889
120597ℎ119889
)
119904
=
minus119909119890
ℎ119889
+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119889
)
119904
1198763= (
120597V119889
120597120590119890
)
119904
=
ℎ119890
ℎ119889
(
120597V119890
120597120590119890
)
119904
1198764= (
120597V119889
120597120590119889
)
119904
=
ℎ119890
ℎ119889
(
120597V119890
120597120590119889
)
119904
1198765= (
120597V119889
120597ℎ119890
)
119904
=
V119890
ℎ119889
+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119890
)
119904
(A1)
The intermedia variables are given as follows
(
120597119909119899
120597ℎ119889
)
119904
= (minus
1198921
8119906
+
1
4119906
minus
119909119890
4ℎ119890
) sec (119892)2 + radic 119877
4ℎ119889
sdot tan (119892) + (120597119909119899
120597119909119890
)
119904
(
120597119909119890
120597ℎ119889
)
119904
+ (
120597119909119899
120597119896119891119889
)
119904
sdot (
120597119896119891119889
120597ℎ119889
)
119904
1198921 = log(ℎ119890119896119891119890
ℎ119889119896119891119889
119896119891119889minus 120590119889
119896119891119890minus 120590119890
)
119892 =
1
4
radicℎ119889
119877
(2radic
119877
ℎ119889
tanminus1 (119909119890
radic119877ℎ119889
)
+ 2radic
119877
ℎ119889
tanminus1 (119909119889
radic119877ℎ119889
) minus
1198921
119906
)
(
120597119909119899
120597119909119890
)
119904
=
ℎ119889sec (119892)2
2ℎ119890
(
120597119909119899
120597119896119891119889
)
119904
=
ℎ119889
4119906
(
1
119896119891119889
minus
1
119896119891119889minus 120590119889
) sec (119892)2
(
120597119896119891119889
120597ℎ119889
) = minus
1198991205900
ℎ119889
(119860 + log(ℎ0
ℎ119889
))
119899minus1
(
120597119909119899
120597ℎ119889
)
119904
=
1
2
sec (119892)2 (120597119909119889
120597ℎ119889
)
119904
(
120597119909119899
120597120590119890
)
119904
= minus
ℎ119889sec (119892)2
4119906 (119896119891119890minus 120590119890)
(
120597119909119899
120597120590119889
)
119904
=
ℎ119889sec (119892)2
4119906 (119896119891119889minus 120590119889)
(
120597119909119899
120597ℎ119890
)
119904
= minus
ℎ119889sec (119892)2
4119906ℎ119890
+ (
120597119909119899
120597119909119890
)
119904
(
120597119909119890
120597ℎ119890
)
119904
+ (
120597119909119899
120597119896119891119890
)
119904
(
120597119896119891119890
120597ℎ119890
)
119904
(
120597119909119899
120597119896119891119890
)
119904
=
ℎ119889
4119906
(
1
119896119891119890minus 120590119890
minus
1
119896119891119890
) sec (119892)2
(
120597119896119891119890
120597ℎ119890
)
119904
= minus
1198991205900
ℎ119890
(119860 + log(ℎ0
ℎ119890
))
119899minus1
(
120597119909119890
120597ℎ119889
)
119904
= minusradic
119877
4 (ℎ119890minus ℎ119889)
(
120597119909119890
120597ℎ119890
)
119904
= radic
119877
4 (ℎ119890minus ℎ119889)
(
120597119909119889
120597ℎ119889
)
119904
=
119877ℎ119889
2V119903ℎ119899
(
120597119901119890
120597ℎ119889
)
119904
= (
120597119901119890
120597ℎ
)
119904
+ (
120597119901119890
120597119896119891
)
119904
(
120597119896119891
120597ℎ119889
)
119904
+ (
120597119901119890
120597119867119890
)
119904
sdot (
120597119867119890
120597ℎ119889
)
119904
+ (
120597119901119890
120597119867
)
119904
(
120597119867
120597ℎ119889
)
119904
(
120597119901119890
120597ℎ
)
119904
=
(119896119891119890minus 120590119890) 119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119901119890
120597119896119891
)
119904
=
(119896119891119890minus 120590119890) ℎ119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119896119891
120597ℎ119889
)
119904
= minus
1205900119899
ℎ
(119860 + ln(ℎ0
ℎ
))
119899minus1
(
120597119901119890
120597119867119890
)
119904
=
(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119867119890
120597ℎ119889
)
119904
= minus
119877arctan (119909119890radic119877ℎ
119889)
ℎ119889
2
radic119877ℎ119889
minus
119909119890
ℎ119889ℎ119890
+ (
120597119867119890
120597119909119890
)
119904
(
120597119909119890
120597ℎ119889
)
119904
(
120597119901119890
120597119867
)
119904
= minus
(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119867
120597ℎ119889
)
119904
= minus
119877arctan (119909radic119877ℎ119889)
ℎ119889
2
radic119877ℎ119889
minus
119909
ℎ119889ℎ
(
120597119901119890
120597120590119890
)
119904
= minus
ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
14 Shock and Vibration
(
120597119901e120597ℎe)
119904
= (
120597119901e120597119896119891119890
)
119904
(
120597119896119891119890
120597ℎ119890
)
119904
+ (
120597119901e120597119867119890
)
119904
(
120597119867119890
120597119909119890
)
119904
sdot (
120597119909119890
120597ℎ119890
)
119904
minus
(119896119891119890minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)
ℎ119890
2
119896119891119890
(
120597119901119890
120597119896119891119890
)
119904
=
ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
minus
(119896119891119890 minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
2
(
120597119867119890
120597119909119890
)
119904
=
2
ℎ119890
(
120597119901d120597ℎ119889
)
119904
= (
120597119901119889
120597119896119891119889
)
119904
(
120597119896119891119889
120597ℎ119889
)
119904
+ (
120597119901119889
120597ℎ
)
119904
+ (
120597119901119889
120597119896119891
)
119904
(
120597119896119891
120597ℎ119889
)
119904
+ (
120597119901119889
120597119867
)
119904
(
120597119867
120597ℎ119889
)
119904
minus
(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889
2
119896119891119889
(
120597119901119889
120597119896119891119889
)
119904
=
ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
minus
(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
2
(
120597119901119889
120597ℎ
)
119904
=
(119896119891119889minus 120590119889) 119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119896119891
)
119904
=
(119896119891119889minus 120590119889) ℎ119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119867119889
)
119904
= minus
(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119867
)
119904
=
(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597ℎ119889
)
119904
= (
120597119901119889
120597119867119889
)
119904
(
120597119867119889
120597119909119889
)
119904
(
120597119909119889
120597ℎ119889
)
119904
(
120597119867119889
120597119909119889
)
119904
=
2
ℎ119889
(
120597119901119889
120597120590119889
)
119904
= minus
ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
119867119890= 2radic
119877
ℎ119889
tanminus1 (119909119890
radic119877ℎ119889
)
119867119889= 2radic
119877
ℎ119889
tanminus1 (119909119889
radic119877ℎ119889
)
119867 = 2radic
119877
ℎ119889
tanminus1 ( 119909
radic119877ℎ119889
)
(A2)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors gratefully acknowledge the support of theNational Natural Science Foundation of China no 51175035PhD Programs Foundation of Ministry of Education ofChina no 20100006110024 the Fundamental Research Fundsfor the Central Universities no FRF-BR-14-006A and BeijingHigher EducationYoungElite Teacher Project no YETP0367
References
[1] I S YunW R DWilson and K F Ehmann ldquoReview of chatterstudies in cold rollingrdquo International Journal of Machine Toolsand Manufacture vol 38 no 12 pp 1499ndash1530 1998
[2] J X Zhou TandemMill Vibration Control Metallurgical Indus-try Press Beijing China 1st edition 1998
[3] G Zhiying Z Yong and Z Lingqiang ldquoReview of modellingand theoretical studies on chatter in the rolling millsrdquo Journalof Mechanical Engineering vol 51 no 16 pp 87ndash105 2015
[4] H Zhao and K F Ehmann ldquoStability analysis of chatter intandem rolling millsmdashpart 2 the regenerative effectrdquo Journalof Manufacturing Science and EngineeringmdashTransactions of theASME vol 135 no 3 Article ID 031002 11 pages 2013
[5] P-H Hu and K F Ehmann ldquoRegenerative effect in rollingchatterrdquo Journal ofManufacturing Processes vol 3 no 2 pp 82ndash93 2001
[6] P H Hu Stability and Chatter in Rolling Northwestern Univer-sity Evanston Ill USA 1998
[7] H Y Zhao and K F Ehmann ldquoStability analysis of chatter intandem rolling millsmdashpart 1 single- and multi-stand negativedamping effectrdquo Journal ofManufacturing Science and Engineer-ing vol 135 no 3 Article ID 031001 2013
[8] Y Kimura Y Sodani N Nishiura et al ldquoAnalysis of chaffer intandem cold rolling millsrdquo ISIJ International vol 43 no 1 pp77ndash84 2003
[9] M R Niroomand R M Forouzan andM Salimi ldquoTheoreticaland experimental analysis of chatter in tandem cold rollingmills based on wave propagation theoryrdquo ISIJ International vol55 no 3 pp 637ndash646 2015
[10] A Heidari M R Forouzan and S Akbarzadeh ldquoDevelopmentof a rolling chattermodel considering unsteady lubricationrdquo ISIJInternational vol 54 no 1 pp 165ndash170 2014
[11] A Heidari and M R Forouzan ldquoOptimization of cold rollingprocess parameters in order to increasing rolling speed limitedby chatter vibrationsrdquo Journal of Advanced Research vol 4 no1 pp 27ndash34 2013
Shock and Vibration 15
[12] B BahramiNejadMDehghani and S AMousavi ldquoSimulationof two stands cold rolling mill process using neural networksand genetic algorithms in combination to avoid the chatterphenomenonrdquo Majlesi Journal of Electrical Engineering vol 9no 1 pp 21ndash24 2014
[13] X X Liao Theory Methods and Application of Stability Huaz-hong Science and Engineering University PressWuhan China1999
[14] V B Kolmanovskii and V R Nosov Stability of FunctionalDifferential Equations Academic Press New York NY USA1986
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Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
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Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
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Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Chemical EngineeringInternational Journal of Antennas and
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International Journal of
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DistributedSensor Networks
International Journal of
10 Shock and Vibration
001 0015 002 0025 00323
24
25
26
27
28
29
Criti
cal s
peed
(ms
)
Delay systemNondelay system
u1
Figure 9 Comparison of stability charts for 1199061
001 0015 002 0025 00322
24
26
28
30
32
34
Criti
cal s
peed
(ms
)
Delay systemNondelay system
u2
Figure 10 Comparison of stability charts for 1199062
When considering the time delay effect the critical velocitygets smaller and more sensitive to 120590
1198891
43 Effects of Interstand Distance Interstand tension vari-ation becomes more gentle with the increase of interstanddistance And so the stability of mill stand becomes better[2] But it is more complex to multistand system 119871
1198902is the
distance between the 1st and 2nd stand If 1198711198902
is changedthe interstand tension variation and the delay time are bothchanged As shown in Figure 12 for time delay system withthe increase of 119871
1198902 the interstand tension variation decreases
and critical velocity of delay system risesMeantime the delaytime becomes larger and critical velocity decreases So thecritical velocities of time delay system form a wavy patternof rising
50 100 150 200 250 30025
26
27
28
29
30
31
32
Criti
cal s
peed
(ms
)
Delay systemNondelay system
120590d1 (MPa)
Figure 11 Comparison of stability charts for 1205901198891
3 35 4 45 5 55 623
24
25
26
27
28
29
30Cr
itica
l spe
ed (m
s)
Delay systemNondelay system
Le2 (m)
Figure 12 Comparison of stability charts for 1198711198902
Table 3 The critical velocities for different stands when 1205901198891
changes
1205901198891
(Mpa) 60 250Critical velocity of single 1st stand V
1198881(ms) 227 225
Corresponding critical velocity of single 2ndstand V
1198882(ms) 326 335
Critical velocity of single 2nd stand V1198882
(ms) 367 331
More unstable stand 1st stand 2nd stand
At first the 2nd stand is more unstable than the 1st oneWith the increase of 119871
1198902 the critical velocities of the 1st
and 2nd stand both rise But the critical velocity of 2ndstand goes up much faster than the 1st stand as 119871
1198902affects
Shock and Vibration 11
1 15 2 25 3 35 420
25
30
35
40
45
Criti
cal s
peed
(ms
)
Delay systemNondelay system
E (GPa)
Figure 13 Comparison of stability charts for 119864
05 1 1526
27
28
29
30
31
32
33
34
Criti
cal s
peed
(ms
)
Delay systemNondelay system
w (m)
Figure 14 Comparison of stability charts for 119908
the entry tension variation of 2nd standThe critical velocitiesof the nondelay and time delay system both rise fast When1198711198902
is bigger than a certain value the 1st stand becomesunstable first The rises of critical velocities for the nondelayand delay system both slow down for 119871
1198902affects the exit
tension variation of 1st stand It can been seen in Table 4 thatwhen 119871
1198902is 35m the 2nd stand goes to unstable first but
when 1198711198902
is 55m chatter occurs in the 1st stand first Butthe instability of the 1st stand occurs when 119871
1198902is quite bigger
than nondelay system for the delay time effect makes the 2ndstand more unstable
44 Effects of Strip Width and Strip Elastic Modulus Thecritical velocity of a single stand system decreases with theincrease of the strip elastic modulus and goes up along with
022 024 026 028 03 032 034 03624
26
28
30
32
34
36
38
40
Criti
cal s
peed
(ms
)
Delay systemNondelay system
he2 (mm)
Figure 15 Comparison of stability charts for ℎ1198902
Table 4 The critical velocities for different stands when 1198711198902
changes
1198711198902
(ms) 35 55Critical velocity of single 1st stand V
1198881
(ms) 2463 2487
Corresponding critical velocity of single2nd stand V
1198882(ms) 3605 3640
Critical velocity of single 2nd stand V1198882
(ms) 3138 3789
More unstable stand 2nd stand 1st stand
the increase of the strip width [2 7] As shown in Figures13 and 14 it shows the same trend for both the time delaysystem and nondelay systemWith the increase of strip elasticmodulus the critical velocities for the time delay system andnondelay system are almost the same The latter is just alittle bigger than the former With the increasing strip widththe critical velocities for the time delay system and nondelaysystem both go up But the critical velocities for the time delaysystem are a littlemore sensitive to the increase of strip width
45 Effects of Reduction Allocation It has been proved thatthe critical velocity for a single stand system decreaseswith the increase of the entry thickness given that the exitthickness stays the same and increase with the rise of the exitthickness given that the entry thickness remains the same[2] But it is more complex for multistand system as shownin Figure 15 Assuming that the entry strip thickness of the1st stand and the exit strip thickness of the 2nd stand remainunaltered the critical velocity would be changed if the entrythickness of the 2nd stand namely ℎ
1198902 is changed
If ℎ1198902
is small enough it means that exit strip thicknessof the 1st stand decreases and entry strip thickness of the 2ndstand increases The 1st stand becomes unstable first But asℎ1198902
rises the 1st stand reduction gets smaller while the 2nd
12 Shock and Vibration
Table 5The critical velocities for different standswhen ℎ1198902changes
ℎ1198902
(mm) 026 029Critical velocity of single 1st stand V
1198881
(ms) 242 252
Corresponding critical velocity of single2nd stand V
1198882(ms) 331 380
Critical velocity of single 2nd stand V1198882
(ms) 368 341
More unstable stand 1st stand 2nd stand
stand reduction gets bigger The 2nd stand becomes moreunstable than the 1st stand As shown in Table 5 when ℎ
1198902
is 026mm the 1st stand is more unstable However whenℎ1198902
increases to 029 the 2nd stand becomes more unstablethan 1st stand There exists an optimal value of ℎ
1198902for the
multistand system which is related to the maximum criticalvelocity as shown in Figure 11
The overall critical velocity changing trend for the timedelay system and nondelay system is the same But theoptimal values of ℎ
1198902are different for the two systems
because the stability of the 2nd stand becomes worse whenconsidering the time delay effect The optimal values of ℎ
1198902
for the two systems are 033mm and 028mm respectivelyObviously the optimal value of ℎ
1198902for the delay system is
more consistent with the actual production It suggests thatby redistributing the reduction of the tandem rolling millswe can improve the stability of the tandem rolling mills
The critical velocity of a single stand changing along withthe increase of the reduction is not linearWhen the reductionis small the critical velocity of the stand is very sensitiveto the reduction changing But as the reduction goes upthe decrease of critical velocity slows down So as shown inFigure 15 when ℎ
1198902is bigger than a certain point the critical
velocities of the two systems both go down slowlyIt has to be clarified that the optimal ℎ
1198902value for the two-
stand system may not work for five stand system There existfour delay times in five-stand tandem rolling mills whichmakes the relationship between stands more complex
5 Conclusions
In this paper the effects of interstand tension variation andstrip variation transportation between adjacent stands withtime delay on tandem rolling mills were studied in detailthrough comparison of different models Stability analy-sis shows that interstand tension couples adjacent standstogether and has just a little effect on the critical velocity of thechatter model The strip gauge variation passed on betweenstands reduces the critical velocity dramatically Howeverthe time delay effect has very limited effects on both thecritical velocity and the frequency But it does not mean thatthe time delay effect can be neglected when modeling therolling system By comparing the critical velocities of thedelay and nondelay system the influences of delay time asa single factor on multistand rolling system were studiedMore detailed and quantitative explanation is put forward to
the relationship of two adjacent stands In the big picturethe critical velocity of the time delay system is just slightlysmaller than the nondelay system But the delay time worsensthe stability of downstream stand and makes the relationshipbetween consecutive stands more complex The time delayeffect changes the optimal process parameters values aimedat obtaining the maximum critical velocity To get preferablerolling process parameter configuration for tandem rollingmills time delay effect must be involved
Appendix
Consider119889119865 = 119865
1119889ℎ119889+ 1198652119889ℎ119889+ 1198653119889120590119890+ 1198654119889120590119889+ 1198655119889ℎ119890
1198651= (
120597119865
120597ℎ119889
)
119904
= int
119909119899
119909119889
(
120597119901119889
120597ℎ119889
)
119904
119889119909 + int
119909119890
119909119899
(
120597119901119890
120597ℎ119889
)
119904
119889119909
+ (119896119891119890minus 120590119890) (
120597119909119890
120597ℎ119889
)
119904
1198652= (
120597119865
120597ℎ119889
)
119904
= int
119909119899
119909119889
(
120597119901119889
120597ℎ119889
)
119904
119889119909 minus (119896119891119889minus 120590119889)(
120597119909119889
120597ℎ119889
)
119904
1198653= (
120597119865
120597120590119890
)
119904
= int
119909119890
119909119899
(
120597119901119890
120597120590119890
)
119904
119889119909
1198654= (
120597119865
120597120590119889
)
119904
= int
119909119899
119909119889
(
120597119901119889
120597120590119889
)
119904
119889119909
1198655= (
120597119865
120597ℎ119890
)
119904
= int
119909119890
119909119899
(
120597119901119890
120597ℎ119890
)
119904
119889119909 + (119896119891119890minus 120590119890) (
120597119909119890
120597ℎ119890
)
119904
119889V119890= 1198751119889ℎ119889+ 1198752119889ℎ119889+ 1198753119889120590119890+ 1198754119889120590119889+ 1198755119889ℎ119890
1198751= (
120597V119890
120597ℎ119889
)
119904
=
V119903
ℎ119890
+
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597ℎ119889
)
119904
1198752= (
120597V119890
120597ℎ119889
)
119904
=
119909119890minus 119909119899
ℎ119890
+
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597ℎ119889
)
119904
1198753= (
120597V119890
120597120590119890
)
119904
=
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597120590119890
)
119904
1198754= (
120597V119890
120597120590119889
)
119904
=
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597120590119889
)
119904
1198755= (
120597V119890
120597ℎ119890
)
119904
= minus
V119903ℎ119899
ℎ119890
2+
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597ℎ119890
)
119904
Shock and Vibration 13
119889V119889= 1198761119889ℎ119889+ 1198762119889ℎ119889+ 1198763119889120590119890+ 1198764119889120590119889+ 1198765119889ℎ119890
1198761= (
120597V119889
120597ℎ119889
)
119904
= minus
V119890ℎ119890
ℎ119889
2+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119889
)
119904
1198762= (
120597V119889
120597ℎ119889
)
119904
=
minus119909119890
ℎ119889
+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119889
)
119904
1198763= (
120597V119889
120597120590119890
)
119904
=
ℎ119890
ℎ119889
(
120597V119890
120597120590119890
)
119904
1198764= (
120597V119889
120597120590119889
)
119904
=
ℎ119890
ℎ119889
(
120597V119890
120597120590119889
)
119904
1198765= (
120597V119889
120597ℎ119890
)
119904
=
V119890
ℎ119889
+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119890
)
119904
(A1)
The intermedia variables are given as follows
(
120597119909119899
120597ℎ119889
)
119904
= (minus
1198921
8119906
+
1
4119906
minus
119909119890
4ℎ119890
) sec (119892)2 + radic 119877
4ℎ119889
sdot tan (119892) + (120597119909119899
120597119909119890
)
119904
(
120597119909119890
120597ℎ119889
)
119904
+ (
120597119909119899
120597119896119891119889
)
119904
sdot (
120597119896119891119889
120597ℎ119889
)
119904
1198921 = log(ℎ119890119896119891119890
ℎ119889119896119891119889
119896119891119889minus 120590119889
119896119891119890minus 120590119890
)
119892 =
1
4
radicℎ119889
119877
(2radic
119877
ℎ119889
tanminus1 (119909119890
radic119877ℎ119889
)
+ 2radic
119877
ℎ119889
tanminus1 (119909119889
radic119877ℎ119889
) minus
1198921
119906
)
(
120597119909119899
120597119909119890
)
119904
=
ℎ119889sec (119892)2
2ℎ119890
(
120597119909119899
120597119896119891119889
)
119904
=
ℎ119889
4119906
(
1
119896119891119889
minus
1
119896119891119889minus 120590119889
) sec (119892)2
(
120597119896119891119889
120597ℎ119889
) = minus
1198991205900
ℎ119889
(119860 + log(ℎ0
ℎ119889
))
119899minus1
(
120597119909119899
120597ℎ119889
)
119904
=
1
2
sec (119892)2 (120597119909119889
120597ℎ119889
)
119904
(
120597119909119899
120597120590119890
)
119904
= minus
ℎ119889sec (119892)2
4119906 (119896119891119890minus 120590119890)
(
120597119909119899
120597120590119889
)
119904
=
ℎ119889sec (119892)2
4119906 (119896119891119889minus 120590119889)
(
120597119909119899
120597ℎ119890
)
119904
= minus
ℎ119889sec (119892)2
4119906ℎ119890
+ (
120597119909119899
120597119909119890
)
119904
(
120597119909119890
120597ℎ119890
)
119904
+ (
120597119909119899
120597119896119891119890
)
119904
(
120597119896119891119890
120597ℎ119890
)
119904
(
120597119909119899
120597119896119891119890
)
119904
=
ℎ119889
4119906
(
1
119896119891119890minus 120590119890
minus
1
119896119891119890
) sec (119892)2
(
120597119896119891119890
120597ℎ119890
)
119904
= minus
1198991205900
ℎ119890
(119860 + log(ℎ0
ℎ119890
))
119899minus1
(
120597119909119890
120597ℎ119889
)
119904
= minusradic
119877
4 (ℎ119890minus ℎ119889)
(
120597119909119890
120597ℎ119890
)
119904
= radic
119877
4 (ℎ119890minus ℎ119889)
(
120597119909119889
120597ℎ119889
)
119904
=
119877ℎ119889
2V119903ℎ119899
(
120597119901119890
120597ℎ119889
)
119904
= (
120597119901119890
120597ℎ
)
119904
+ (
120597119901119890
120597119896119891
)
119904
(
120597119896119891
120597ℎ119889
)
119904
+ (
120597119901119890
120597119867119890
)
119904
sdot (
120597119867119890
120597ℎ119889
)
119904
+ (
120597119901119890
120597119867
)
119904
(
120597119867
120597ℎ119889
)
119904
(
120597119901119890
120597ℎ
)
119904
=
(119896119891119890minus 120590119890) 119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119901119890
120597119896119891
)
119904
=
(119896119891119890minus 120590119890) ℎ119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119896119891
120597ℎ119889
)
119904
= minus
1205900119899
ℎ
(119860 + ln(ℎ0
ℎ
))
119899minus1
(
120597119901119890
120597119867119890
)
119904
=
(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119867119890
120597ℎ119889
)
119904
= minus
119877arctan (119909119890radic119877ℎ
119889)
ℎ119889
2
radic119877ℎ119889
minus
119909119890
ℎ119889ℎ119890
+ (
120597119867119890
120597119909119890
)
119904
(
120597119909119890
120597ℎ119889
)
119904
(
120597119901119890
120597119867
)
119904
= minus
(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119867
120597ℎ119889
)
119904
= minus
119877arctan (119909radic119877ℎ119889)
ℎ119889
2
radic119877ℎ119889
minus
119909
ℎ119889ℎ
(
120597119901119890
120597120590119890
)
119904
= minus
ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
14 Shock and Vibration
(
120597119901e120597ℎe)
119904
= (
120597119901e120597119896119891119890
)
119904
(
120597119896119891119890
120597ℎ119890
)
119904
+ (
120597119901e120597119867119890
)
119904
(
120597119867119890
120597119909119890
)
119904
sdot (
120597119909119890
120597ℎ119890
)
119904
minus
(119896119891119890minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)
ℎ119890
2
119896119891119890
(
120597119901119890
120597119896119891119890
)
119904
=
ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
minus
(119896119891119890 minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
2
(
120597119867119890
120597119909119890
)
119904
=
2
ℎ119890
(
120597119901d120597ℎ119889
)
119904
= (
120597119901119889
120597119896119891119889
)
119904
(
120597119896119891119889
120597ℎ119889
)
119904
+ (
120597119901119889
120597ℎ
)
119904
+ (
120597119901119889
120597119896119891
)
119904
(
120597119896119891
120597ℎ119889
)
119904
+ (
120597119901119889
120597119867
)
119904
(
120597119867
120597ℎ119889
)
119904
minus
(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889
2
119896119891119889
(
120597119901119889
120597119896119891119889
)
119904
=
ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
minus
(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
2
(
120597119901119889
120597ℎ
)
119904
=
(119896119891119889minus 120590119889) 119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119896119891
)
119904
=
(119896119891119889minus 120590119889) ℎ119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119867119889
)
119904
= minus
(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119867
)
119904
=
(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597ℎ119889
)
119904
= (
120597119901119889
120597119867119889
)
119904
(
120597119867119889
120597119909119889
)
119904
(
120597119909119889
120597ℎ119889
)
119904
(
120597119867119889
120597119909119889
)
119904
=
2
ℎ119889
(
120597119901119889
120597120590119889
)
119904
= minus
ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
119867119890= 2radic
119877
ℎ119889
tanminus1 (119909119890
radic119877ℎ119889
)
119867119889= 2radic
119877
ℎ119889
tanminus1 (119909119889
radic119877ℎ119889
)
119867 = 2radic
119877
ℎ119889
tanminus1 ( 119909
radic119877ℎ119889
)
(A2)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors gratefully acknowledge the support of theNational Natural Science Foundation of China no 51175035PhD Programs Foundation of Ministry of Education ofChina no 20100006110024 the Fundamental Research Fundsfor the Central Universities no FRF-BR-14-006A and BeijingHigher EducationYoungElite Teacher Project no YETP0367
References
[1] I S YunW R DWilson and K F Ehmann ldquoReview of chatterstudies in cold rollingrdquo International Journal of Machine Toolsand Manufacture vol 38 no 12 pp 1499ndash1530 1998
[2] J X Zhou TandemMill Vibration Control Metallurgical Indus-try Press Beijing China 1st edition 1998
[3] G Zhiying Z Yong and Z Lingqiang ldquoReview of modellingand theoretical studies on chatter in the rolling millsrdquo Journalof Mechanical Engineering vol 51 no 16 pp 87ndash105 2015
[4] H Zhao and K F Ehmann ldquoStability analysis of chatter intandem rolling millsmdashpart 2 the regenerative effectrdquo Journalof Manufacturing Science and EngineeringmdashTransactions of theASME vol 135 no 3 Article ID 031002 11 pages 2013
[5] P-H Hu and K F Ehmann ldquoRegenerative effect in rollingchatterrdquo Journal ofManufacturing Processes vol 3 no 2 pp 82ndash93 2001
[6] P H Hu Stability and Chatter in Rolling Northwestern Univer-sity Evanston Ill USA 1998
[7] H Y Zhao and K F Ehmann ldquoStability analysis of chatter intandem rolling millsmdashpart 1 single- and multi-stand negativedamping effectrdquo Journal ofManufacturing Science and Engineer-ing vol 135 no 3 Article ID 031001 2013
[8] Y Kimura Y Sodani N Nishiura et al ldquoAnalysis of chaffer intandem cold rolling millsrdquo ISIJ International vol 43 no 1 pp77ndash84 2003
[9] M R Niroomand R M Forouzan andM Salimi ldquoTheoreticaland experimental analysis of chatter in tandem cold rollingmills based on wave propagation theoryrdquo ISIJ International vol55 no 3 pp 637ndash646 2015
[10] A Heidari M R Forouzan and S Akbarzadeh ldquoDevelopmentof a rolling chattermodel considering unsteady lubricationrdquo ISIJInternational vol 54 no 1 pp 165ndash170 2014
[11] A Heidari and M R Forouzan ldquoOptimization of cold rollingprocess parameters in order to increasing rolling speed limitedby chatter vibrationsrdquo Journal of Advanced Research vol 4 no1 pp 27ndash34 2013
Shock and Vibration 15
[12] B BahramiNejadMDehghani and S AMousavi ldquoSimulationof two stands cold rolling mill process using neural networksand genetic algorithms in combination to avoid the chatterphenomenonrdquo Majlesi Journal of Electrical Engineering vol 9no 1 pp 21ndash24 2014
[13] X X Liao Theory Methods and Application of Stability Huaz-hong Science and Engineering University PressWuhan China1999
[14] V B Kolmanovskii and V R Nosov Stability of FunctionalDifferential Equations Academic Press New York NY USA1986
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Shock and Vibration
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Civil EngineeringAdvances in
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International Journal of
Shock and Vibration 11
1 15 2 25 3 35 420
25
30
35
40
45
Criti
cal s
peed
(ms
)
Delay systemNondelay system
E (GPa)
Figure 13 Comparison of stability charts for 119864
05 1 1526
27
28
29
30
31
32
33
34
Criti
cal s
peed
(ms
)
Delay systemNondelay system
w (m)
Figure 14 Comparison of stability charts for 119908
the entry tension variation of 2nd standThe critical velocitiesof the nondelay and time delay system both rise fast When1198711198902
is bigger than a certain value the 1st stand becomesunstable first The rises of critical velocities for the nondelayand delay system both slow down for 119871
1198902affects the exit
tension variation of 1st stand It can been seen in Table 4 thatwhen 119871
1198902is 35m the 2nd stand goes to unstable first but
when 1198711198902
is 55m chatter occurs in the 1st stand first Butthe instability of the 1st stand occurs when 119871
1198902is quite bigger
than nondelay system for the delay time effect makes the 2ndstand more unstable
44 Effects of Strip Width and Strip Elastic Modulus Thecritical velocity of a single stand system decreases with theincrease of the strip elastic modulus and goes up along with
022 024 026 028 03 032 034 03624
26
28
30
32
34
36
38
40
Criti
cal s
peed
(ms
)
Delay systemNondelay system
he2 (mm)
Figure 15 Comparison of stability charts for ℎ1198902
Table 4 The critical velocities for different stands when 1198711198902
changes
1198711198902
(ms) 35 55Critical velocity of single 1st stand V
1198881
(ms) 2463 2487
Corresponding critical velocity of single2nd stand V
1198882(ms) 3605 3640
Critical velocity of single 2nd stand V1198882
(ms) 3138 3789
More unstable stand 2nd stand 1st stand
the increase of the strip width [2 7] As shown in Figures13 and 14 it shows the same trend for both the time delaysystem and nondelay systemWith the increase of strip elasticmodulus the critical velocities for the time delay system andnondelay system are almost the same The latter is just alittle bigger than the former With the increasing strip widththe critical velocities for the time delay system and nondelaysystem both go up But the critical velocities for the time delaysystem are a littlemore sensitive to the increase of strip width
45 Effects of Reduction Allocation It has been proved thatthe critical velocity for a single stand system decreaseswith the increase of the entry thickness given that the exitthickness stays the same and increase with the rise of the exitthickness given that the entry thickness remains the same[2] But it is more complex for multistand system as shownin Figure 15 Assuming that the entry strip thickness of the1st stand and the exit strip thickness of the 2nd stand remainunaltered the critical velocity would be changed if the entrythickness of the 2nd stand namely ℎ
1198902 is changed
If ℎ1198902
is small enough it means that exit strip thicknessof the 1st stand decreases and entry strip thickness of the 2ndstand increases The 1st stand becomes unstable first But asℎ1198902
rises the 1st stand reduction gets smaller while the 2nd
12 Shock and Vibration
Table 5The critical velocities for different standswhen ℎ1198902changes
ℎ1198902
(mm) 026 029Critical velocity of single 1st stand V
1198881
(ms) 242 252
Corresponding critical velocity of single2nd stand V
1198882(ms) 331 380
Critical velocity of single 2nd stand V1198882
(ms) 368 341
More unstable stand 1st stand 2nd stand
stand reduction gets bigger The 2nd stand becomes moreunstable than the 1st stand As shown in Table 5 when ℎ
1198902
is 026mm the 1st stand is more unstable However whenℎ1198902
increases to 029 the 2nd stand becomes more unstablethan 1st stand There exists an optimal value of ℎ
1198902for the
multistand system which is related to the maximum criticalvelocity as shown in Figure 11
The overall critical velocity changing trend for the timedelay system and nondelay system is the same But theoptimal values of ℎ
1198902are different for the two systems
because the stability of the 2nd stand becomes worse whenconsidering the time delay effect The optimal values of ℎ
1198902
for the two systems are 033mm and 028mm respectivelyObviously the optimal value of ℎ
1198902for the delay system is
more consistent with the actual production It suggests thatby redistributing the reduction of the tandem rolling millswe can improve the stability of the tandem rolling mills
The critical velocity of a single stand changing along withthe increase of the reduction is not linearWhen the reductionis small the critical velocity of the stand is very sensitiveto the reduction changing But as the reduction goes upthe decrease of critical velocity slows down So as shown inFigure 15 when ℎ
1198902is bigger than a certain point the critical
velocities of the two systems both go down slowlyIt has to be clarified that the optimal ℎ
1198902value for the two-
stand system may not work for five stand system There existfour delay times in five-stand tandem rolling mills whichmakes the relationship between stands more complex
5 Conclusions
In this paper the effects of interstand tension variation andstrip variation transportation between adjacent stands withtime delay on tandem rolling mills were studied in detailthrough comparison of different models Stability analy-sis shows that interstand tension couples adjacent standstogether and has just a little effect on the critical velocity of thechatter model The strip gauge variation passed on betweenstands reduces the critical velocity dramatically Howeverthe time delay effect has very limited effects on both thecritical velocity and the frequency But it does not mean thatthe time delay effect can be neglected when modeling therolling system By comparing the critical velocities of thedelay and nondelay system the influences of delay time asa single factor on multistand rolling system were studiedMore detailed and quantitative explanation is put forward to
the relationship of two adjacent stands In the big picturethe critical velocity of the time delay system is just slightlysmaller than the nondelay system But the delay time worsensthe stability of downstream stand and makes the relationshipbetween consecutive stands more complex The time delayeffect changes the optimal process parameters values aimedat obtaining the maximum critical velocity To get preferablerolling process parameter configuration for tandem rollingmills time delay effect must be involved
Appendix
Consider119889119865 = 119865
1119889ℎ119889+ 1198652119889ℎ119889+ 1198653119889120590119890+ 1198654119889120590119889+ 1198655119889ℎ119890
1198651= (
120597119865
120597ℎ119889
)
119904
= int
119909119899
119909119889
(
120597119901119889
120597ℎ119889
)
119904
119889119909 + int
119909119890
119909119899
(
120597119901119890
120597ℎ119889
)
119904
119889119909
+ (119896119891119890minus 120590119890) (
120597119909119890
120597ℎ119889
)
119904
1198652= (
120597119865
120597ℎ119889
)
119904
= int
119909119899
119909119889
(
120597119901119889
120597ℎ119889
)
119904
119889119909 minus (119896119891119889minus 120590119889)(
120597119909119889
120597ℎ119889
)
119904
1198653= (
120597119865
120597120590119890
)
119904
= int
119909119890
119909119899
(
120597119901119890
120597120590119890
)
119904
119889119909
1198654= (
120597119865
120597120590119889
)
119904
= int
119909119899
119909119889
(
120597119901119889
120597120590119889
)
119904
119889119909
1198655= (
120597119865
120597ℎ119890
)
119904
= int
119909119890
119909119899
(
120597119901119890
120597ℎ119890
)
119904
119889119909 + (119896119891119890minus 120590119890) (
120597119909119890
120597ℎ119890
)
119904
119889V119890= 1198751119889ℎ119889+ 1198752119889ℎ119889+ 1198753119889120590119890+ 1198754119889120590119889+ 1198755119889ℎ119890
1198751= (
120597V119890
120597ℎ119889
)
119904
=
V119903
ℎ119890
+
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597ℎ119889
)
119904
1198752= (
120597V119890
120597ℎ119889
)
119904
=
119909119890minus 119909119899
ℎ119890
+
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597ℎ119889
)
119904
1198753= (
120597V119890
120597120590119890
)
119904
=
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597120590119890
)
119904
1198754= (
120597V119890
120597120590119889
)
119904
=
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597120590119889
)
119904
1198755= (
120597V119890
120597ℎ119890
)
119904
= minus
V119903ℎ119899
ℎ119890
2+
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597ℎ119890
)
119904
Shock and Vibration 13
119889V119889= 1198761119889ℎ119889+ 1198762119889ℎ119889+ 1198763119889120590119890+ 1198764119889120590119889+ 1198765119889ℎ119890
1198761= (
120597V119889
120597ℎ119889
)
119904
= minus
V119890ℎ119890
ℎ119889
2+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119889
)
119904
1198762= (
120597V119889
120597ℎ119889
)
119904
=
minus119909119890
ℎ119889
+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119889
)
119904
1198763= (
120597V119889
120597120590119890
)
119904
=
ℎ119890
ℎ119889
(
120597V119890
120597120590119890
)
119904
1198764= (
120597V119889
120597120590119889
)
119904
=
ℎ119890
ℎ119889
(
120597V119890
120597120590119889
)
119904
1198765= (
120597V119889
120597ℎ119890
)
119904
=
V119890
ℎ119889
+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119890
)
119904
(A1)
The intermedia variables are given as follows
(
120597119909119899
120597ℎ119889
)
119904
= (minus
1198921
8119906
+
1
4119906
minus
119909119890
4ℎ119890
) sec (119892)2 + radic 119877
4ℎ119889
sdot tan (119892) + (120597119909119899
120597119909119890
)
119904
(
120597119909119890
120597ℎ119889
)
119904
+ (
120597119909119899
120597119896119891119889
)
119904
sdot (
120597119896119891119889
120597ℎ119889
)
119904
1198921 = log(ℎ119890119896119891119890
ℎ119889119896119891119889
119896119891119889minus 120590119889
119896119891119890minus 120590119890
)
119892 =
1
4
radicℎ119889
119877
(2radic
119877
ℎ119889
tanminus1 (119909119890
radic119877ℎ119889
)
+ 2radic
119877
ℎ119889
tanminus1 (119909119889
radic119877ℎ119889
) minus
1198921
119906
)
(
120597119909119899
120597119909119890
)
119904
=
ℎ119889sec (119892)2
2ℎ119890
(
120597119909119899
120597119896119891119889
)
119904
=
ℎ119889
4119906
(
1
119896119891119889
minus
1
119896119891119889minus 120590119889
) sec (119892)2
(
120597119896119891119889
120597ℎ119889
) = minus
1198991205900
ℎ119889
(119860 + log(ℎ0
ℎ119889
))
119899minus1
(
120597119909119899
120597ℎ119889
)
119904
=
1
2
sec (119892)2 (120597119909119889
120597ℎ119889
)
119904
(
120597119909119899
120597120590119890
)
119904
= minus
ℎ119889sec (119892)2
4119906 (119896119891119890minus 120590119890)
(
120597119909119899
120597120590119889
)
119904
=
ℎ119889sec (119892)2
4119906 (119896119891119889minus 120590119889)
(
120597119909119899
120597ℎ119890
)
119904
= minus
ℎ119889sec (119892)2
4119906ℎ119890
+ (
120597119909119899
120597119909119890
)
119904
(
120597119909119890
120597ℎ119890
)
119904
+ (
120597119909119899
120597119896119891119890
)
119904
(
120597119896119891119890
120597ℎ119890
)
119904
(
120597119909119899
120597119896119891119890
)
119904
=
ℎ119889
4119906
(
1
119896119891119890minus 120590119890
minus
1
119896119891119890
) sec (119892)2
(
120597119896119891119890
120597ℎ119890
)
119904
= minus
1198991205900
ℎ119890
(119860 + log(ℎ0
ℎ119890
))
119899minus1
(
120597119909119890
120597ℎ119889
)
119904
= minusradic
119877
4 (ℎ119890minus ℎ119889)
(
120597119909119890
120597ℎ119890
)
119904
= radic
119877
4 (ℎ119890minus ℎ119889)
(
120597119909119889
120597ℎ119889
)
119904
=
119877ℎ119889
2V119903ℎ119899
(
120597119901119890
120597ℎ119889
)
119904
= (
120597119901119890
120597ℎ
)
119904
+ (
120597119901119890
120597119896119891
)
119904
(
120597119896119891
120597ℎ119889
)
119904
+ (
120597119901119890
120597119867119890
)
119904
sdot (
120597119867119890
120597ℎ119889
)
119904
+ (
120597119901119890
120597119867
)
119904
(
120597119867
120597ℎ119889
)
119904
(
120597119901119890
120597ℎ
)
119904
=
(119896119891119890minus 120590119890) 119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119901119890
120597119896119891
)
119904
=
(119896119891119890minus 120590119890) ℎ119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119896119891
120597ℎ119889
)
119904
= minus
1205900119899
ℎ
(119860 + ln(ℎ0
ℎ
))
119899minus1
(
120597119901119890
120597119867119890
)
119904
=
(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119867119890
120597ℎ119889
)
119904
= minus
119877arctan (119909119890radic119877ℎ
119889)
ℎ119889
2
radic119877ℎ119889
minus
119909119890
ℎ119889ℎ119890
+ (
120597119867119890
120597119909119890
)
119904
(
120597119909119890
120597ℎ119889
)
119904
(
120597119901119890
120597119867
)
119904
= minus
(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119867
120597ℎ119889
)
119904
= minus
119877arctan (119909radic119877ℎ119889)
ℎ119889
2
radic119877ℎ119889
minus
119909
ℎ119889ℎ
(
120597119901119890
120597120590119890
)
119904
= minus
ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
14 Shock and Vibration
(
120597119901e120597ℎe)
119904
= (
120597119901e120597119896119891119890
)
119904
(
120597119896119891119890
120597ℎ119890
)
119904
+ (
120597119901e120597119867119890
)
119904
(
120597119867119890
120597119909119890
)
119904
sdot (
120597119909119890
120597ℎ119890
)
119904
minus
(119896119891119890minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)
ℎ119890
2
119896119891119890
(
120597119901119890
120597119896119891119890
)
119904
=
ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
minus
(119896119891119890 minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
2
(
120597119867119890
120597119909119890
)
119904
=
2
ℎ119890
(
120597119901d120597ℎ119889
)
119904
= (
120597119901119889
120597119896119891119889
)
119904
(
120597119896119891119889
120597ℎ119889
)
119904
+ (
120597119901119889
120597ℎ
)
119904
+ (
120597119901119889
120597119896119891
)
119904
(
120597119896119891
120597ℎ119889
)
119904
+ (
120597119901119889
120597119867
)
119904
(
120597119867
120597ℎ119889
)
119904
minus
(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889
2
119896119891119889
(
120597119901119889
120597119896119891119889
)
119904
=
ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
minus
(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
2
(
120597119901119889
120597ℎ
)
119904
=
(119896119891119889minus 120590119889) 119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119896119891
)
119904
=
(119896119891119889minus 120590119889) ℎ119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119867119889
)
119904
= minus
(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119867
)
119904
=
(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597ℎ119889
)
119904
= (
120597119901119889
120597119867119889
)
119904
(
120597119867119889
120597119909119889
)
119904
(
120597119909119889
120597ℎ119889
)
119904
(
120597119867119889
120597119909119889
)
119904
=
2
ℎ119889
(
120597119901119889
120597120590119889
)
119904
= minus
ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
119867119890= 2radic
119877
ℎ119889
tanminus1 (119909119890
radic119877ℎ119889
)
119867119889= 2radic
119877
ℎ119889
tanminus1 (119909119889
radic119877ℎ119889
)
119867 = 2radic
119877
ℎ119889
tanminus1 ( 119909
radic119877ℎ119889
)
(A2)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors gratefully acknowledge the support of theNational Natural Science Foundation of China no 51175035PhD Programs Foundation of Ministry of Education ofChina no 20100006110024 the Fundamental Research Fundsfor the Central Universities no FRF-BR-14-006A and BeijingHigher EducationYoungElite Teacher Project no YETP0367
References
[1] I S YunW R DWilson and K F Ehmann ldquoReview of chatterstudies in cold rollingrdquo International Journal of Machine Toolsand Manufacture vol 38 no 12 pp 1499ndash1530 1998
[2] J X Zhou TandemMill Vibration Control Metallurgical Indus-try Press Beijing China 1st edition 1998
[3] G Zhiying Z Yong and Z Lingqiang ldquoReview of modellingand theoretical studies on chatter in the rolling millsrdquo Journalof Mechanical Engineering vol 51 no 16 pp 87ndash105 2015
[4] H Zhao and K F Ehmann ldquoStability analysis of chatter intandem rolling millsmdashpart 2 the regenerative effectrdquo Journalof Manufacturing Science and EngineeringmdashTransactions of theASME vol 135 no 3 Article ID 031002 11 pages 2013
[5] P-H Hu and K F Ehmann ldquoRegenerative effect in rollingchatterrdquo Journal ofManufacturing Processes vol 3 no 2 pp 82ndash93 2001
[6] P H Hu Stability and Chatter in Rolling Northwestern Univer-sity Evanston Ill USA 1998
[7] H Y Zhao and K F Ehmann ldquoStability analysis of chatter intandem rolling millsmdashpart 1 single- and multi-stand negativedamping effectrdquo Journal ofManufacturing Science and Engineer-ing vol 135 no 3 Article ID 031001 2013
[8] Y Kimura Y Sodani N Nishiura et al ldquoAnalysis of chaffer intandem cold rolling millsrdquo ISIJ International vol 43 no 1 pp77ndash84 2003
[9] M R Niroomand R M Forouzan andM Salimi ldquoTheoreticaland experimental analysis of chatter in tandem cold rollingmills based on wave propagation theoryrdquo ISIJ International vol55 no 3 pp 637ndash646 2015
[10] A Heidari M R Forouzan and S Akbarzadeh ldquoDevelopmentof a rolling chattermodel considering unsteady lubricationrdquo ISIJInternational vol 54 no 1 pp 165ndash170 2014
[11] A Heidari and M R Forouzan ldquoOptimization of cold rollingprocess parameters in order to increasing rolling speed limitedby chatter vibrationsrdquo Journal of Advanced Research vol 4 no1 pp 27ndash34 2013
Shock and Vibration 15
[12] B BahramiNejadMDehghani and S AMousavi ldquoSimulationof two stands cold rolling mill process using neural networksand genetic algorithms in combination to avoid the chatterphenomenonrdquo Majlesi Journal of Electrical Engineering vol 9no 1 pp 21ndash24 2014
[13] X X Liao Theory Methods and Application of Stability Huaz-hong Science and Engineering University PressWuhan China1999
[14] V B Kolmanovskii and V R Nosov Stability of FunctionalDifferential Equations Academic Press New York NY USA1986
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International Journal of
12 Shock and Vibration
Table 5The critical velocities for different standswhen ℎ1198902changes
ℎ1198902
(mm) 026 029Critical velocity of single 1st stand V
1198881
(ms) 242 252
Corresponding critical velocity of single2nd stand V
1198882(ms) 331 380
Critical velocity of single 2nd stand V1198882
(ms) 368 341
More unstable stand 1st stand 2nd stand
stand reduction gets bigger The 2nd stand becomes moreunstable than the 1st stand As shown in Table 5 when ℎ
1198902
is 026mm the 1st stand is more unstable However whenℎ1198902
increases to 029 the 2nd stand becomes more unstablethan 1st stand There exists an optimal value of ℎ
1198902for the
multistand system which is related to the maximum criticalvelocity as shown in Figure 11
The overall critical velocity changing trend for the timedelay system and nondelay system is the same But theoptimal values of ℎ
1198902are different for the two systems
because the stability of the 2nd stand becomes worse whenconsidering the time delay effect The optimal values of ℎ
1198902
for the two systems are 033mm and 028mm respectivelyObviously the optimal value of ℎ
1198902for the delay system is
more consistent with the actual production It suggests thatby redistributing the reduction of the tandem rolling millswe can improve the stability of the tandem rolling mills
The critical velocity of a single stand changing along withthe increase of the reduction is not linearWhen the reductionis small the critical velocity of the stand is very sensitiveto the reduction changing But as the reduction goes upthe decrease of critical velocity slows down So as shown inFigure 15 when ℎ
1198902is bigger than a certain point the critical
velocities of the two systems both go down slowlyIt has to be clarified that the optimal ℎ
1198902value for the two-
stand system may not work for five stand system There existfour delay times in five-stand tandem rolling mills whichmakes the relationship between stands more complex
5 Conclusions
In this paper the effects of interstand tension variation andstrip variation transportation between adjacent stands withtime delay on tandem rolling mills were studied in detailthrough comparison of different models Stability analy-sis shows that interstand tension couples adjacent standstogether and has just a little effect on the critical velocity of thechatter model The strip gauge variation passed on betweenstands reduces the critical velocity dramatically Howeverthe time delay effect has very limited effects on both thecritical velocity and the frequency But it does not mean thatthe time delay effect can be neglected when modeling therolling system By comparing the critical velocities of thedelay and nondelay system the influences of delay time asa single factor on multistand rolling system were studiedMore detailed and quantitative explanation is put forward to
the relationship of two adjacent stands In the big picturethe critical velocity of the time delay system is just slightlysmaller than the nondelay system But the delay time worsensthe stability of downstream stand and makes the relationshipbetween consecutive stands more complex The time delayeffect changes the optimal process parameters values aimedat obtaining the maximum critical velocity To get preferablerolling process parameter configuration for tandem rollingmills time delay effect must be involved
Appendix
Consider119889119865 = 119865
1119889ℎ119889+ 1198652119889ℎ119889+ 1198653119889120590119890+ 1198654119889120590119889+ 1198655119889ℎ119890
1198651= (
120597119865
120597ℎ119889
)
119904
= int
119909119899
119909119889
(
120597119901119889
120597ℎ119889
)
119904
119889119909 + int
119909119890
119909119899
(
120597119901119890
120597ℎ119889
)
119904
119889119909
+ (119896119891119890minus 120590119890) (
120597119909119890
120597ℎ119889
)
119904
1198652= (
120597119865
120597ℎ119889
)
119904
= int
119909119899
119909119889
(
120597119901119889
120597ℎ119889
)
119904
119889119909 minus (119896119891119889minus 120590119889)(
120597119909119889
120597ℎ119889
)
119904
1198653= (
120597119865
120597120590119890
)
119904
= int
119909119890
119909119899
(
120597119901119890
120597120590119890
)
119904
119889119909
1198654= (
120597119865
120597120590119889
)
119904
= int
119909119899
119909119889
(
120597119901119889
120597120590119889
)
119904
119889119909
1198655= (
120597119865
120597ℎ119890
)
119904
= int
119909119890
119909119899
(
120597119901119890
120597ℎ119890
)
119904
119889119909 + (119896119891119890minus 120590119890) (
120597119909119890
120597ℎ119890
)
119904
119889V119890= 1198751119889ℎ119889+ 1198752119889ℎ119889+ 1198753119889120590119890+ 1198754119889120590119889+ 1198755119889ℎ119890
1198751= (
120597V119890
120597ℎ119889
)
119904
=
V119903
ℎ119890
+
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597ℎ119889
)
119904
1198752= (
120597V119890
120597ℎ119889
)
119904
=
119909119890minus 119909119899
ℎ119890
+
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597ℎ119889
)
119904
1198753= (
120597V119890
120597120590119890
)
119904
=
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597120590119890
)
119904
1198754= (
120597V119890
120597120590119889
)
119904
=
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597120590119889
)
119904
1198755= (
120597V119890
120597ℎ119890
)
119904
= minus
V119903ℎ119899
ℎ119890
2+
2V119903119909119899
ℎ119890119877
(
120597119909119899
120597ℎ119890
)
119904
Shock and Vibration 13
119889V119889= 1198761119889ℎ119889+ 1198762119889ℎ119889+ 1198763119889120590119890+ 1198764119889120590119889+ 1198765119889ℎ119890
1198761= (
120597V119889
120597ℎ119889
)
119904
= minus
V119890ℎ119890
ℎ119889
2+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119889
)
119904
1198762= (
120597V119889
120597ℎ119889
)
119904
=
minus119909119890
ℎ119889
+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119889
)
119904
1198763= (
120597V119889
120597120590119890
)
119904
=
ℎ119890
ℎ119889
(
120597V119890
120597120590119890
)
119904
1198764= (
120597V119889
120597120590119889
)
119904
=
ℎ119890
ℎ119889
(
120597V119890
120597120590119889
)
119904
1198765= (
120597V119889
120597ℎ119890
)
119904
=
V119890
ℎ119889
+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119890
)
119904
(A1)
The intermedia variables are given as follows
(
120597119909119899
120597ℎ119889
)
119904
= (minus
1198921
8119906
+
1
4119906
minus
119909119890
4ℎ119890
) sec (119892)2 + radic 119877
4ℎ119889
sdot tan (119892) + (120597119909119899
120597119909119890
)
119904
(
120597119909119890
120597ℎ119889
)
119904
+ (
120597119909119899
120597119896119891119889
)
119904
sdot (
120597119896119891119889
120597ℎ119889
)
119904
1198921 = log(ℎ119890119896119891119890
ℎ119889119896119891119889
119896119891119889minus 120590119889
119896119891119890minus 120590119890
)
119892 =
1
4
radicℎ119889
119877
(2radic
119877
ℎ119889
tanminus1 (119909119890
radic119877ℎ119889
)
+ 2radic
119877
ℎ119889
tanminus1 (119909119889
radic119877ℎ119889
) minus
1198921
119906
)
(
120597119909119899
120597119909119890
)
119904
=
ℎ119889sec (119892)2
2ℎ119890
(
120597119909119899
120597119896119891119889
)
119904
=
ℎ119889
4119906
(
1
119896119891119889
minus
1
119896119891119889minus 120590119889
) sec (119892)2
(
120597119896119891119889
120597ℎ119889
) = minus
1198991205900
ℎ119889
(119860 + log(ℎ0
ℎ119889
))
119899minus1
(
120597119909119899
120597ℎ119889
)
119904
=
1
2
sec (119892)2 (120597119909119889
120597ℎ119889
)
119904
(
120597119909119899
120597120590119890
)
119904
= minus
ℎ119889sec (119892)2
4119906 (119896119891119890minus 120590119890)
(
120597119909119899
120597120590119889
)
119904
=
ℎ119889sec (119892)2
4119906 (119896119891119889minus 120590119889)
(
120597119909119899
120597ℎ119890
)
119904
= minus
ℎ119889sec (119892)2
4119906ℎ119890
+ (
120597119909119899
120597119909119890
)
119904
(
120597119909119890
120597ℎ119890
)
119904
+ (
120597119909119899
120597119896119891119890
)
119904
(
120597119896119891119890
120597ℎ119890
)
119904
(
120597119909119899
120597119896119891119890
)
119904
=
ℎ119889
4119906
(
1
119896119891119890minus 120590119890
minus
1
119896119891119890
) sec (119892)2
(
120597119896119891119890
120597ℎ119890
)
119904
= minus
1198991205900
ℎ119890
(119860 + log(ℎ0
ℎ119890
))
119899minus1
(
120597119909119890
120597ℎ119889
)
119904
= minusradic
119877
4 (ℎ119890minus ℎ119889)
(
120597119909119890
120597ℎ119890
)
119904
= radic
119877
4 (ℎ119890minus ℎ119889)
(
120597119909119889
120597ℎ119889
)
119904
=
119877ℎ119889
2V119903ℎ119899
(
120597119901119890
120597ℎ119889
)
119904
= (
120597119901119890
120597ℎ
)
119904
+ (
120597119901119890
120597119896119891
)
119904
(
120597119896119891
120597ℎ119889
)
119904
+ (
120597119901119890
120597119867119890
)
119904
sdot (
120597119867119890
120597ℎ119889
)
119904
+ (
120597119901119890
120597119867
)
119904
(
120597119867
120597ℎ119889
)
119904
(
120597119901119890
120597ℎ
)
119904
=
(119896119891119890minus 120590119890) 119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119901119890
120597119896119891
)
119904
=
(119896119891119890minus 120590119890) ℎ119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119896119891
120597ℎ119889
)
119904
= minus
1205900119899
ℎ
(119860 + ln(ℎ0
ℎ
))
119899minus1
(
120597119901119890
120597119867119890
)
119904
=
(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119867119890
120597ℎ119889
)
119904
= minus
119877arctan (119909119890radic119877ℎ
119889)
ℎ119889
2
radic119877ℎ119889
minus
119909119890
ℎ119889ℎ119890
+ (
120597119867119890
120597119909119890
)
119904
(
120597119909119890
120597ℎ119889
)
119904
(
120597119901119890
120597119867
)
119904
= minus
(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119867
120597ℎ119889
)
119904
= minus
119877arctan (119909radic119877ℎ119889)
ℎ119889
2
radic119877ℎ119889
minus
119909
ℎ119889ℎ
(
120597119901119890
120597120590119890
)
119904
= minus
ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
14 Shock and Vibration
(
120597119901e120597ℎe)
119904
= (
120597119901e120597119896119891119890
)
119904
(
120597119896119891119890
120597ℎ119890
)
119904
+ (
120597119901e120597119867119890
)
119904
(
120597119867119890
120597119909119890
)
119904
sdot (
120597119909119890
120597ℎ119890
)
119904
minus
(119896119891119890minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)
ℎ119890
2
119896119891119890
(
120597119901119890
120597119896119891119890
)
119904
=
ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
minus
(119896119891119890 minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
2
(
120597119867119890
120597119909119890
)
119904
=
2
ℎ119890
(
120597119901d120597ℎ119889
)
119904
= (
120597119901119889
120597119896119891119889
)
119904
(
120597119896119891119889
120597ℎ119889
)
119904
+ (
120597119901119889
120597ℎ
)
119904
+ (
120597119901119889
120597119896119891
)
119904
(
120597119896119891
120597ℎ119889
)
119904
+ (
120597119901119889
120597119867
)
119904
(
120597119867
120597ℎ119889
)
119904
minus
(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889
2
119896119891119889
(
120597119901119889
120597119896119891119889
)
119904
=
ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
minus
(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
2
(
120597119901119889
120597ℎ
)
119904
=
(119896119891119889minus 120590119889) 119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119896119891
)
119904
=
(119896119891119889minus 120590119889) ℎ119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119867119889
)
119904
= minus
(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119867
)
119904
=
(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597ℎ119889
)
119904
= (
120597119901119889
120597119867119889
)
119904
(
120597119867119889
120597119909119889
)
119904
(
120597119909119889
120597ℎ119889
)
119904
(
120597119867119889
120597119909119889
)
119904
=
2
ℎ119889
(
120597119901119889
120597120590119889
)
119904
= minus
ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
119867119890= 2radic
119877
ℎ119889
tanminus1 (119909119890
radic119877ℎ119889
)
119867119889= 2radic
119877
ℎ119889
tanminus1 (119909119889
radic119877ℎ119889
)
119867 = 2radic
119877
ℎ119889
tanminus1 ( 119909
radic119877ℎ119889
)
(A2)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors gratefully acknowledge the support of theNational Natural Science Foundation of China no 51175035PhD Programs Foundation of Ministry of Education ofChina no 20100006110024 the Fundamental Research Fundsfor the Central Universities no FRF-BR-14-006A and BeijingHigher EducationYoungElite Teacher Project no YETP0367
References
[1] I S YunW R DWilson and K F Ehmann ldquoReview of chatterstudies in cold rollingrdquo International Journal of Machine Toolsand Manufacture vol 38 no 12 pp 1499ndash1530 1998
[2] J X Zhou TandemMill Vibration Control Metallurgical Indus-try Press Beijing China 1st edition 1998
[3] G Zhiying Z Yong and Z Lingqiang ldquoReview of modellingand theoretical studies on chatter in the rolling millsrdquo Journalof Mechanical Engineering vol 51 no 16 pp 87ndash105 2015
[4] H Zhao and K F Ehmann ldquoStability analysis of chatter intandem rolling millsmdashpart 2 the regenerative effectrdquo Journalof Manufacturing Science and EngineeringmdashTransactions of theASME vol 135 no 3 Article ID 031002 11 pages 2013
[5] P-H Hu and K F Ehmann ldquoRegenerative effect in rollingchatterrdquo Journal ofManufacturing Processes vol 3 no 2 pp 82ndash93 2001
[6] P H Hu Stability and Chatter in Rolling Northwestern Univer-sity Evanston Ill USA 1998
[7] H Y Zhao and K F Ehmann ldquoStability analysis of chatter intandem rolling millsmdashpart 1 single- and multi-stand negativedamping effectrdquo Journal ofManufacturing Science and Engineer-ing vol 135 no 3 Article ID 031001 2013
[8] Y Kimura Y Sodani N Nishiura et al ldquoAnalysis of chaffer intandem cold rolling millsrdquo ISIJ International vol 43 no 1 pp77ndash84 2003
[9] M R Niroomand R M Forouzan andM Salimi ldquoTheoreticaland experimental analysis of chatter in tandem cold rollingmills based on wave propagation theoryrdquo ISIJ International vol55 no 3 pp 637ndash646 2015
[10] A Heidari M R Forouzan and S Akbarzadeh ldquoDevelopmentof a rolling chattermodel considering unsteady lubricationrdquo ISIJInternational vol 54 no 1 pp 165ndash170 2014
[11] A Heidari and M R Forouzan ldquoOptimization of cold rollingprocess parameters in order to increasing rolling speed limitedby chatter vibrationsrdquo Journal of Advanced Research vol 4 no1 pp 27ndash34 2013
Shock and Vibration 15
[12] B BahramiNejadMDehghani and S AMousavi ldquoSimulationof two stands cold rolling mill process using neural networksand genetic algorithms in combination to avoid the chatterphenomenonrdquo Majlesi Journal of Electrical Engineering vol 9no 1 pp 21ndash24 2014
[13] X X Liao Theory Methods and Application of Stability Huaz-hong Science and Engineering University PressWuhan China1999
[14] V B Kolmanovskii and V R Nosov Stability of FunctionalDifferential Equations Academic Press New York NY USA1986
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Shock and Vibration
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International Journal of
Shock and Vibration 13
119889V119889= 1198761119889ℎ119889+ 1198762119889ℎ119889+ 1198763119889120590119890+ 1198764119889120590119889+ 1198765119889ℎ119890
1198761= (
120597V119889
120597ℎ119889
)
119904
= minus
V119890ℎ119890
ℎ119889
2+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119889
)
119904
1198762= (
120597V119889
120597ℎ119889
)
119904
=
minus119909119890
ℎ119889
+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119889
)
119904
1198763= (
120597V119889
120597120590119890
)
119904
=
ℎ119890
ℎ119889
(
120597V119890
120597120590119890
)
119904
1198764= (
120597V119889
120597120590119889
)
119904
=
ℎ119890
ℎ119889
(
120597V119890
120597120590119889
)
119904
1198765= (
120597V119889
120597ℎ119890
)
119904
=
V119890
ℎ119889
+
ℎ119890
ℎ119889
(
120597V119890
120597ℎ119890
)
119904
(A1)
The intermedia variables are given as follows
(
120597119909119899
120597ℎ119889
)
119904
= (minus
1198921
8119906
+
1
4119906
minus
119909119890
4ℎ119890
) sec (119892)2 + radic 119877
4ℎ119889
sdot tan (119892) + (120597119909119899
120597119909119890
)
119904
(
120597119909119890
120597ℎ119889
)
119904
+ (
120597119909119899
120597119896119891119889
)
119904
sdot (
120597119896119891119889
120597ℎ119889
)
119904
1198921 = log(ℎ119890119896119891119890
ℎ119889119896119891119889
119896119891119889minus 120590119889
119896119891119890minus 120590119890
)
119892 =
1
4
radicℎ119889
119877
(2radic
119877
ℎ119889
tanminus1 (119909119890
radic119877ℎ119889
)
+ 2radic
119877
ℎ119889
tanminus1 (119909119889
radic119877ℎ119889
) minus
1198921
119906
)
(
120597119909119899
120597119909119890
)
119904
=
ℎ119889sec (119892)2
2ℎ119890
(
120597119909119899
120597119896119891119889
)
119904
=
ℎ119889
4119906
(
1
119896119891119889
minus
1
119896119891119889minus 120590119889
) sec (119892)2
(
120597119896119891119889
120597ℎ119889
) = minus
1198991205900
ℎ119889
(119860 + log(ℎ0
ℎ119889
))
119899minus1
(
120597119909119899
120597ℎ119889
)
119904
=
1
2
sec (119892)2 (120597119909119889
120597ℎ119889
)
119904
(
120597119909119899
120597120590119890
)
119904
= minus
ℎ119889sec (119892)2
4119906 (119896119891119890minus 120590119890)
(
120597119909119899
120597120590119889
)
119904
=
ℎ119889sec (119892)2
4119906 (119896119891119889minus 120590119889)
(
120597119909119899
120597ℎ119890
)
119904
= minus
ℎ119889sec (119892)2
4119906ℎ119890
+ (
120597119909119899
120597119909119890
)
119904
(
120597119909119890
120597ℎ119890
)
119904
+ (
120597119909119899
120597119896119891119890
)
119904
(
120597119896119891119890
120597ℎ119890
)
119904
(
120597119909119899
120597119896119891119890
)
119904
=
ℎ119889
4119906
(
1
119896119891119890minus 120590119890
minus
1
119896119891119890
) sec (119892)2
(
120597119896119891119890
120597ℎ119890
)
119904
= minus
1198991205900
ℎ119890
(119860 + log(ℎ0
ℎ119890
))
119899minus1
(
120597119909119890
120597ℎ119889
)
119904
= minusradic
119877
4 (ℎ119890minus ℎ119889)
(
120597119909119890
120597ℎ119890
)
119904
= radic
119877
4 (ℎ119890minus ℎ119889)
(
120597119909119889
120597ℎ119889
)
119904
=
119877ℎ119889
2V119903ℎ119899
(
120597119901119890
120597ℎ119889
)
119904
= (
120597119901119890
120597ℎ
)
119904
+ (
120597119901119890
120597119896119891
)
119904
(
120597119896119891
120597ℎ119889
)
119904
+ (
120597119901119890
120597119867119890
)
119904
sdot (
120597119867119890
120597ℎ119889
)
119904
+ (
120597119901119890
120597119867
)
119904
(
120597119867
120597ℎ119889
)
119904
(
120597119901119890
120597ℎ
)
119904
=
(119896119891119890minus 120590119890) 119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119901119890
120597119896119891
)
119904
=
(119896119891119890minus 120590119890) ℎ119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119896119891
120597ℎ119889
)
119904
= minus
1205900119899
ℎ
(119860 + ln(ℎ0
ℎ
))
119899minus1
(
120597119901119890
120597119867119890
)
119904
=
(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119867119890
120597ℎ119889
)
119904
= minus
119877arctan (119909119890radic119877ℎ
119889)
ℎ119889
2
radic119877ℎ119889
minus
119909119890
ℎ119889ℎ119890
+ (
120597119867119890
120597119909119890
)
119904
(
120597119909119890
120597ℎ119889
)
119904
(
120597119901119890
120597119867
)
119904
= minus
(119896119891119890minus 120590119890) ℎ119896119891119906119890119906(119867119890minus119867)
ℎ119890119896119891119890
(
120597119867
120597ℎ119889
)
119904
= minus
119877arctan (119909radic119877ℎ119889)
ℎ119889
2
radic119877ℎ119889
minus
119909
ℎ119889ℎ
(
120597119901119890
120597120590119890
)
119904
= minus
ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
14 Shock and Vibration
(
120597119901e120597ℎe)
119904
= (
120597119901e120597119896119891119890
)
119904
(
120597119896119891119890
120597ℎ119890
)
119904
+ (
120597119901e120597119867119890
)
119904
(
120597119867119890
120597119909119890
)
119904
sdot (
120597119909119890
120597ℎ119890
)
119904
minus
(119896119891119890minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)
ℎ119890
2
119896119891119890
(
120597119901119890
120597119896119891119890
)
119904
=
ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
minus
(119896119891119890 minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
2
(
120597119867119890
120597119909119890
)
119904
=
2
ℎ119890
(
120597119901d120597ℎ119889
)
119904
= (
120597119901119889
120597119896119891119889
)
119904
(
120597119896119891119889
120597ℎ119889
)
119904
+ (
120597119901119889
120597ℎ
)
119904
+ (
120597119901119889
120597119896119891
)
119904
(
120597119896119891
120597ℎ119889
)
119904
+ (
120597119901119889
120597119867
)
119904
(
120597119867
120597ℎ119889
)
119904
minus
(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889
2
119896119891119889
(
120597119901119889
120597119896119891119889
)
119904
=
ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
minus
(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
2
(
120597119901119889
120597ℎ
)
119904
=
(119896119891119889minus 120590119889) 119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119896119891
)
119904
=
(119896119891119889minus 120590119889) ℎ119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119867119889
)
119904
= minus
(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119867
)
119904
=
(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597ℎ119889
)
119904
= (
120597119901119889
120597119867119889
)
119904
(
120597119867119889
120597119909119889
)
119904
(
120597119909119889
120597ℎ119889
)
119904
(
120597119867119889
120597119909119889
)
119904
=
2
ℎ119889
(
120597119901119889
120597120590119889
)
119904
= minus
ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
119867119890= 2radic
119877
ℎ119889
tanminus1 (119909119890
radic119877ℎ119889
)
119867119889= 2radic
119877
ℎ119889
tanminus1 (119909119889
radic119877ℎ119889
)
119867 = 2radic
119877
ℎ119889
tanminus1 ( 119909
radic119877ℎ119889
)
(A2)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors gratefully acknowledge the support of theNational Natural Science Foundation of China no 51175035PhD Programs Foundation of Ministry of Education ofChina no 20100006110024 the Fundamental Research Fundsfor the Central Universities no FRF-BR-14-006A and BeijingHigher EducationYoungElite Teacher Project no YETP0367
References
[1] I S YunW R DWilson and K F Ehmann ldquoReview of chatterstudies in cold rollingrdquo International Journal of Machine Toolsand Manufacture vol 38 no 12 pp 1499ndash1530 1998
[2] J X Zhou TandemMill Vibration Control Metallurgical Indus-try Press Beijing China 1st edition 1998
[3] G Zhiying Z Yong and Z Lingqiang ldquoReview of modellingand theoretical studies on chatter in the rolling millsrdquo Journalof Mechanical Engineering vol 51 no 16 pp 87ndash105 2015
[4] H Zhao and K F Ehmann ldquoStability analysis of chatter intandem rolling millsmdashpart 2 the regenerative effectrdquo Journalof Manufacturing Science and EngineeringmdashTransactions of theASME vol 135 no 3 Article ID 031002 11 pages 2013
[5] P-H Hu and K F Ehmann ldquoRegenerative effect in rollingchatterrdquo Journal ofManufacturing Processes vol 3 no 2 pp 82ndash93 2001
[6] P H Hu Stability and Chatter in Rolling Northwestern Univer-sity Evanston Ill USA 1998
[7] H Y Zhao and K F Ehmann ldquoStability analysis of chatter intandem rolling millsmdashpart 1 single- and multi-stand negativedamping effectrdquo Journal ofManufacturing Science and Engineer-ing vol 135 no 3 Article ID 031001 2013
[8] Y Kimura Y Sodani N Nishiura et al ldquoAnalysis of chaffer intandem cold rolling millsrdquo ISIJ International vol 43 no 1 pp77ndash84 2003
[9] M R Niroomand R M Forouzan andM Salimi ldquoTheoreticaland experimental analysis of chatter in tandem cold rollingmills based on wave propagation theoryrdquo ISIJ International vol55 no 3 pp 637ndash646 2015
[10] A Heidari M R Forouzan and S Akbarzadeh ldquoDevelopmentof a rolling chattermodel considering unsteady lubricationrdquo ISIJInternational vol 54 no 1 pp 165ndash170 2014
[11] A Heidari and M R Forouzan ldquoOptimization of cold rollingprocess parameters in order to increasing rolling speed limitedby chatter vibrationsrdquo Journal of Advanced Research vol 4 no1 pp 27ndash34 2013
Shock and Vibration 15
[12] B BahramiNejadMDehghani and S AMousavi ldquoSimulationof two stands cold rolling mill process using neural networksand genetic algorithms in combination to avoid the chatterphenomenonrdquo Majlesi Journal of Electrical Engineering vol 9no 1 pp 21ndash24 2014
[13] X X Liao Theory Methods and Application of Stability Huaz-hong Science and Engineering University PressWuhan China1999
[14] V B Kolmanovskii and V R Nosov Stability of FunctionalDifferential Equations Academic Press New York NY USA1986
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
14 Shock and Vibration
(
120597119901e120597ℎe)
119904
= (
120597119901e120597119896119891119890
)
119904
(
120597119896119891119890
120597ℎ119890
)
119904
+ (
120597119901e120597119867119890
)
119904
(
120597119867119890
120597119909119890
)
119904
sdot (
120597119909119890
120597ℎ119890
)
119904
minus
(119896119891119890minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)
ℎ119890
2
119896119891119890
(
120597119901119890
120597119896119891119890
)
119904
=
ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
minus
(119896119891119890 minus 120590119890) ℎ119896119891119890119906(119867119890minus119867)
ℎ119890119896119891119890
2
(
120597119867119890
120597119909119890
)
119904
=
2
ℎ119890
(
120597119901d120597ℎ119889
)
119904
= (
120597119901119889
120597119896119891119889
)
119904
(
120597119896119891119889
120597ℎ119889
)
119904
+ (
120597119901119889
120597ℎ
)
119904
+ (
120597119901119889
120597119896119891
)
119904
(
120597119896119891
120597ℎ119889
)
119904
+ (
120597119901119889
120597119867
)
119904
(
120597119867
120597ℎ119889
)
119904
minus
(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889
2
119896119891119889
(
120597119901119889
120597119896119891119889
)
119904
=
ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
minus
(119896119891119889minus 120590119889) ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
2
(
120597119901119889
120597ℎ
)
119904
=
(119896119891119889minus 120590119889) 119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119896119891
)
119904
=
(119896119891119889minus 120590119889) ℎ119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119867119889
)
119904
= minus
(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597119867
)
119904
=
(119896119891119889minus 120590119889) ℎ119896119891119906119890119906(119867minus119867
119889)
ℎ119889119896119891119889
(
120597119901119889
120597ℎ119889
)
119904
= (
120597119901119889
120597119867119889
)
119904
(
120597119867119889
120597119909119889
)
119904
(
120597119909119889
120597ℎ119889
)
119904
(
120597119867119889
120597119909119889
)
119904
=
2
ℎ119889
(
120597119901119889
120597120590119889
)
119904
= minus
ℎ119896119891119890119906(119867minus119867
119889)
ℎ119889119896119891119889
119867119890= 2radic
119877
ℎ119889
tanminus1 (119909119890
radic119877ℎ119889
)
119867119889= 2radic
119877
ℎ119889
tanminus1 (119909119889
radic119877ℎ119889
)
119867 = 2radic
119877
ℎ119889
tanminus1 ( 119909
radic119877ℎ119889
)
(A2)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors gratefully acknowledge the support of theNational Natural Science Foundation of China no 51175035PhD Programs Foundation of Ministry of Education ofChina no 20100006110024 the Fundamental Research Fundsfor the Central Universities no FRF-BR-14-006A and BeijingHigher EducationYoungElite Teacher Project no YETP0367
References
[1] I S YunW R DWilson and K F Ehmann ldquoReview of chatterstudies in cold rollingrdquo International Journal of Machine Toolsand Manufacture vol 38 no 12 pp 1499ndash1530 1998
[2] J X Zhou TandemMill Vibration Control Metallurgical Indus-try Press Beijing China 1st edition 1998
[3] G Zhiying Z Yong and Z Lingqiang ldquoReview of modellingand theoretical studies on chatter in the rolling millsrdquo Journalof Mechanical Engineering vol 51 no 16 pp 87ndash105 2015
[4] H Zhao and K F Ehmann ldquoStability analysis of chatter intandem rolling millsmdashpart 2 the regenerative effectrdquo Journalof Manufacturing Science and EngineeringmdashTransactions of theASME vol 135 no 3 Article ID 031002 11 pages 2013
[5] P-H Hu and K F Ehmann ldquoRegenerative effect in rollingchatterrdquo Journal ofManufacturing Processes vol 3 no 2 pp 82ndash93 2001
[6] P H Hu Stability and Chatter in Rolling Northwestern Univer-sity Evanston Ill USA 1998
[7] H Y Zhao and K F Ehmann ldquoStability analysis of chatter intandem rolling millsmdashpart 1 single- and multi-stand negativedamping effectrdquo Journal ofManufacturing Science and Engineer-ing vol 135 no 3 Article ID 031001 2013
[8] Y Kimura Y Sodani N Nishiura et al ldquoAnalysis of chaffer intandem cold rolling millsrdquo ISIJ International vol 43 no 1 pp77ndash84 2003
[9] M R Niroomand R M Forouzan andM Salimi ldquoTheoreticaland experimental analysis of chatter in tandem cold rollingmills based on wave propagation theoryrdquo ISIJ International vol55 no 3 pp 637ndash646 2015
[10] A Heidari M R Forouzan and S Akbarzadeh ldquoDevelopmentof a rolling chattermodel considering unsteady lubricationrdquo ISIJInternational vol 54 no 1 pp 165ndash170 2014
[11] A Heidari and M R Forouzan ldquoOptimization of cold rollingprocess parameters in order to increasing rolling speed limitedby chatter vibrationsrdquo Journal of Advanced Research vol 4 no1 pp 27ndash34 2013
Shock and Vibration 15
[12] B BahramiNejadMDehghani and S AMousavi ldquoSimulationof two stands cold rolling mill process using neural networksand genetic algorithms in combination to avoid the chatterphenomenonrdquo Majlesi Journal of Electrical Engineering vol 9no 1 pp 21ndash24 2014
[13] X X Liao Theory Methods and Application of Stability Huaz-hong Science and Engineering University PressWuhan China1999
[14] V B Kolmanovskii and V R Nosov Stability of FunctionalDifferential Equations Academic Press New York NY USA1986
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Shock and Vibration 15
[12] B BahramiNejadMDehghani and S AMousavi ldquoSimulationof two stands cold rolling mill process using neural networksand genetic algorithms in combination to avoid the chatterphenomenonrdquo Majlesi Journal of Electrical Engineering vol 9no 1 pp 21ndash24 2014
[13] X X Liao Theory Methods and Application of Stability Huaz-hong Science and Engineering University PressWuhan China1999
[14] V B Kolmanovskii and V R Nosov Stability of FunctionalDifferential Equations Academic Press New York NY USA1986
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
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