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A Model Coupling Method for Shape Prediction
Available online at www.sciencedirect.com
ScienceDirect JOURNAL OF IRON AND STEEL RESEARCH, INTERNATIONAL. 2012, 19(2): 22-27
A Model Coupling Method for Shape Prediction WANG Dong-cheng,
LID Hong-min
(State Key Laboratory of Metastable Materials Science and Technology, National Engineering Research Center for Equipment and Technology of Cold Rolling Strip, Yanshan University, Qinhuangdao 066004, Hebei , China) Abstract: The shape of strip is calculated by iterative method which combines strip plastic deformation model with rolls elastic deformation model through their calculation results, which can be called results coupling method. Because the shape and rolling force distribution are very sensitive to strip thickness transverse distribution's variation, the iterative course is rather unstable and sometimes convergence cannot be achieved. In addition, the calculating speed of results coupling method is low, which restricts its usable range. To solve the problem, a new model coupling method is developed, which takes the force distribution between rolls, rolling force distribution and strip's exit transverse displacement distribution as basic unknowns, and integrates strip plastic deformation model and rolls elastic deformation model as a unified linear equations through their internal relation, so the iterative calculation between the strip plastic deformation model and rolls elastic deformation model can be avoided. To prove the effectiveness of the model coupling method, two examples are calculated by results coupling method and model coupling method respectively. The results of front tension stress, back tension stress, strip's exit gauge, the force between rolls and rolling force distribution calculated by model coupling method coincide very well with results coupling method. However the calculation course of model coupling method is more steady than results coupling method, and its calculating speed is about ten times as much as the maximal speed of results coupling method, which validates its practicability and reliability. Key words: shape prediction; results coupling method; model coupling method; strip plastic deformation; rolls elastic deformation
The shape is an important quality index of strip, while the shape control is a key and high level technology. The domestic level of shape control is relatively backward than the international advanced level, and the bottleneck restricting improvement of shape control technology is that the research on shape theory and mathematical models falls behind the technology's development. The whole shape control models system includes eight aspects'{", 1) strip plastic deformation model; 2) rolls elastic deformation model; 3) strip temperature field and rolls thermal deformation model; 4) rolls wear model; 5) shape discrimination model; 6) shape deviation pattern recognition model; 7) shape control object model; 8) shape feedback control model. If the eight aspects could be integrated in whole, many specific shape control models and technologies can be developed, such as shape prediction model, shape
pre-set model, roll shape optimization technology and so on. The core of shape prediction model is strip plastic deformation model and rolls elastic deformation model, which are usually integrated by results coupling method[2-8]. That means the two models are calculated independently, and the link between them is their calculation results. Convergence can be achieved only by introducing a relaxation factor. The substantial computation cost and the fact that the relaxation factor requires adjustment for different mill schedules represent the main trouble to on-line application. The basic reason of the trouble is that the transverse distribution of tension stress (shape) and rolling force distribution are very sensitive to the transverse distribution of strip's exit gauge, especially when the strip is very thin. In this case, minute change of the transverse distribution of strip's
Foundation Item: Item Sponsored by National Science and Technology Support Plan of China (2009AA04Z143); Science and Technology Support Plan of Hebei Province of China (l0212101D); Important Natural Science Foundation of Hebei Province of China (E2006001038) Biography: WANG Dong-cheng(l981-) , Male, Doctor; E-mail: [email protected]; Received Date: February 28, 2011
Issue 2
exit gauge may cause considerable change of tension stress and rolling force. and the rolling force even becomes negative (that is impossible in practice). at which time no convergence will appear. To solve this problem. a new model coupling method is put forward. The model coupling method integrates strip plastic deformation model and rolls elastic deformation model as one set of linear equations through their internal relation. which avoids the unstable phenomenon of results coupling method. At the same time the calculating speed is enhanced remarkably. So it is more applicable and reliable than results coupling method.
1. 1
Strip element variation method As shown in Fig. 1. the strip's width B is divided into n (that is an odd) independent strip elements in the deformation zone's length l, The transverse coordinate of strip element's line is expressed as Yi' and the exit transverse displacement of strip element's line is expressed as u., Strip element's width is Si=Yi-Yi-l. yo
hOi t; h -L
i; and I o are strip's
o
o
I
000
n-I
2
I
n+I
2
n+3
2
y. Y Exit
Y.-l
000
n
I
Entry x
B/2
Fig. 1
B/2
Strip element partition in deformation zone
According to the theory of minimum energy. variational calculation is applied on every strip element and the whole deformation zone. Finally n 1 linear equations can be got and shown in Eqn. (1):
+
(al-eb)UO+{3IUI +ebun=-ks+cI {3jUj-1 + (aj
+aj+1
Cj+I-Cj
E
). eb = (l-v 2 )B' h o•
I-v
h.,
entry mean gauge. exit mean gauge and entry mean length; hOi. h u and lOi are every strip element's entry gauge. exit gauge and entry length; E is strip's elastic modulus; and v is strip's Poisson's ratio. When the displacements u., - u; of every strip element's line are got. the transverse distribution of front tension stress. back tension stress and rolling force can be calculated by Eqn. (2) to Eqn. (4).
=_
111
+.--£2 [1 +h li _hOi _loi +u I-v hI t, t, ' l
P» =wowTll1 s
_
un -UO]
B
(i=l-n)
(2)
(i=l-n)
(3)
(i=l-n)
(4)
where. I1li' 110i and Pu are the front tension stress. back tension stress and rolling force of every strip element; I1 s is strip's deformation resistance; Wo is the influence coefficient of every factor except tension action on rolling force; wT is the influence coefficient of tension action on rolling force; wT = 1 0• 311Ii 0.7110i+ . f ront an db ac k ---=-'-------=-=-. 0'1 an d 0'0 are mean
I1 s tension stress; U,' =csch(Kisi)K,[ -ch(Kis'/ 2 ) u i- 1+ ch(K is ,/2 ) u ,J ; and K, is the parameter corresponding to the strip's exit gauge distribution. 1. 2
y~oy~
Yl
.--£2 (1 + h- li -
C,
Basic Mathematic Equations
Through analyzing the characteristics of every model. strip element variation method'f' and influence function method[2,8] are adopted as strip plastic deformation model and rolls elastic deformation model respectively. To display the model coupling method in detail. the basic mathematic equations of the two methods need to be given.
=
strip's exit gauge distribution;
111,
1
• 23 •
A Model Coupling Method for Shape Prediction
)Uj +~j+1 Uj+1 = [J=I-(n-1)]
1ebUO +{3nUn-1 + (an -eb)Un=k s -Cn
Influence function method Fig. 2 shows the force diagram of 4-high roll milL As shown in Fig. 3. during the length L b of backup roll. rolls are divided into m sections (The sections of strip must be coincided with strip element variation method. and be numbered alone. and its number range is 1 - n ) , the forces action on strip and rolls are divided into the same sections. while the width of every section is boYi. the midpoint coordinate of every section is y., Taking the left pressure cylinder as the origin of coordinates. every section's deflection fbi of backup roll's axis can be expressed as fbi
(1)
where. a, and {3i are parameters corresponding to the
=
m
~ abij boYjqj
(i= I-m)
(5)
j~1
where. abij roll caused Every axis can be
is the bending influence function of backup by the force between rolls. section's displacement fWi of work roll's expressed as
Vol. 19
Journal of Iron and Steel Research, International
• 24 •
z
Lb
I
F;
I I
c
-E
.... ........... q(y)
....
~_
- -p,(y) - -
.,.....,.
--.
I
i~l m
i~l
1=1
I I
C
_~
~ - -q(y) -
Fw
n
1k qiYi!J.Yi =k PUYi!J.Yi + FwL,
F's
I
0
m
kqi!J.Yi= kPli!J.Yi+ 2 Fw
i;
t=1
Combining Eqn. (9) with Eqn, (10), m+2 linear equations can be got, while the number of unknowns qi(i=l-m), C 1 and C z is m+2 too, so it can be solved. But the linear equation's coefficient matrix contains Ywbi. It is the function of qi' so an iteration calculation is necessary. The calculation flow is shown as Fig. 4.
y
3Fw
I
B
(10)
n
Lw Lfw pt(y)-Rolling force action on unit width;
rolls action on unit width; pressure cylinder; cylinder;
F;,
q(y)-Force between
F~ -Counterforce
Assuming uniform force distribution between rolls
of left and right
L, - Distance between left and right pressure
L w' L b - Length of work roll and backup roll;
Fw-Bending roll force;
Lrw-Distance between left and right
bending roll cylinder.
Fig. 2
Solving linear equations and getting new force distribution between rolls
Force diagram of 4-high roll mill
No
Fig. 4
Fig. 3
Section partition diagram of 4-high roll mill
t : = f~i + k awij !J.Yj(Plj -qj) +aFwiF m
w
j~l
(i=l-m)
(6)
where, awij is the bending influence function of work roll caused by the rolling force and the force between rolls; aFwi is the bending influence function of work roll caused by the bending roll force; and f~i is rigid displacement of work roll's axis, which can be expressed as: 2 1 f = CI + (C L- C ) K we
(
w
Y,
-
C)
(i =
1- m) (7)
where, C 1 and C2 are rigid displacement of work roll axis's left and right end; and C= CL, - L w ) /2. The elastic squashed quantity between work roll and backup roll can be expressed as ()wbi=Ywbiqi
U=l-m)
(8)
where Ywbi is elastic squashed influence function between rolls, which is the function of qi. The deformation compatibility equations between rolls can be expressed as
i-. =
fbi
+ ()wbi +
!J.Di
U= I-m)
(9)
where !J.Di is the no load clearance caused by roll shape. Force and torque balance equations are
Iteration flow for calculating force between rolls
When the iteration calculation is over, strip's exit gauge distribution can be expressed as hi -
()wi
= k
j~l
YwijPlj' Ywij is elastic squashed coefficient of
rolling force action on work roll, !J.Dwi is no load clearance between work roll and strip caused by roll shape.
2
Model Coupling Method
As mentioned above, the calculation course of results coupling method is slow and unstable, which may influence its engineering application. Considering the calculation courses of strip element variation method and influence function method can all attribute to linear equations, if integrating and solving them at the same time, the iteration calculation between the strip plastic deformation model and rolls elastic deformation model can be avoided. But for the problem, not only the force between rolls and strip's exit transverse displacement but also the strip's exit gauge distribution, rolling force distribution, front tension stress and back tension stress
A Model Coupling Method for Shape Prediction
Issue 2
distribution are unknown, so the linear equations cannot be integrated easily. Through in-depth analysis and research, the force between rolls, rolling force distribution and strip's exit transverse displacement are taken as basic unknowns. Except for Eqn. 0), Eqn. (9) and Eqn. (0), Eqn. (1) is taken as other linear equations. Now the number of unknowns is m+2n+3, while the number of linear equations is m+2n+3 too, so it can be solved. In the m 2 linear equations of Eqn. (9) and Eqn. (0), the only unknowns are the force between rolls and rolling force distribution, and can be used directly. But for Eqn. (1), the constant matrix of linear equations contains unknowns c, (because h li are unknowns) which must be expressed by basic unknowns including the force between rolls, rolling force distribution and strip's exit transverse displacement. Substitution of Eqn. (2) and Eqn. (3) into Eqn. (4), and the relational expression between rolling force distribution and strip's exit transverse displacement with h li can be derived. However the back tension stress is nonlinear to strip's exit transverse displacement, which makes a trouble. A lot of previous studies[2] show a rule that the distribution shape of back tension stress is very similar to front tension stress, so Eqn. (3) can be substituted approximatively by linear Eqn. (2) as follows
(UI- eai)UO+(j11 +eal)uI +Pll/(W°l) = -k s +a:q
(6)
Plj+1 =0
=_ +~[ 1 +':...Ii _':...Oi _~Oi +ul _Un-UO] 2 ao
I-v
hi
ho
lo
B
t:
(i=l-n)
(2)
Substitution of Eqn. (2) and Eqn. (12) into Eqn. (4), Eqn. (3) can be derived
[j = 1- (n-l) ]
«:
+
ao,
• 25 •
(j1n +ean)un-I +(u n-ean)u n- Pin /(wOl) =
-a:
q h, where, eai =EA,/(I-v2).
To solve the problem, Eqn. (11) has to be introduced as other n linear equations. Substitution of Eqn. (14) and Eqn, (6) into Eqn. (11), new linear equations are got as m 2(Yi- Lw- C) 2(C-Yi) 2~Uwij~Yjqj+ L CI L C2-
+
w
)=1
(7)
Combining the Eqn. (9), Eqn. (10), Eqn. (16) with Eqn, (7), the linear equations can be solved. But we need to pay attention to the linear equations' coefficient matrix, which contains Ywbi' a, and fJ; that are the function of qi and h l i , so an iteration calculation is necessary, the calculation flow is shown as Fig. 5.
- 0l eq_ walE [-A +A _Un-Uo]_ Pli -was I-v2 iUi-1 .u, B
Giving basic process parameters
wOl~hli_ wOl~ [1_hoi_loi] (3) I-v hi I-v s, t; where, Ai = csch (Kis i) Kich (K isj2), a: q = as -
Assuming uniform strip's exit gauge distribution
uniform force distribution between rolls
o. ts, - o. 3(i1.
Replacing strip's exit gauge distribution and force distribution between rolls
From Eqn, (13), the relational expression between rolling force distribution and strip's exit transverse displacement with h-u and Ci can be expressed as
__ 0-v2)h l
h li-
walE
Pli+
0-v 2)hl E
eq
-
as +AihIUi-1
hi lOi]A,hIUi+ B(Un-UO)- [hOi I-To hi
ho
w
Solving linear equations and getting new strip's exit gauge distribution and new force distribution between rolls I L-":":':":":"';:'=':":==--....I
_
Convergence
No
Yes End
(4)
k + aseq+~[A c, - _ wOl I-v2 iUi-1 -A .u, +Un-Uo] B (5)
Substitution of Eqn. (5) into Eqn. 0), new linear equations are got as
Fig. 5
3
Calculation flow of model coupling method
Results Analysis To prove the effectiveness of the model coupling
• 26 •
Journal of Iron and Steel Research, International
method put forward by this paper, two examples are calculated using a 1220 mm four-high cold rolling mill's parameters. Work roll's diameter is 500 mm , backup roll's diameter is 1250 mm, work roll's length is 1220 mm, and backup roll's length is 1092 mm. The process parameters of two examples are listed in Table 1. The two examples are calculated by results Table 1 Example
1 2
coupling method and model coupling method respectively. The comparison of calculation time and stability between the two methods are listed in Table 2, while the results of front tension stress, back tension stress, strip's exit gauge, the force between rolls and rolling force distribution are shown in Fig. 6 and Fig. 7.
Process parameters of two examples
Width/
Entry
Exit
Entry
Exit
mm
gauge/mm
gauge/mm
tension/kN
tension/kN
Deformation resistance/MPa
force/kN
900 900
1. 00
0.60 0.22
100 50
100 50
500 800
200 320
0.3
Table 2
11=0.02
1
No convergence
1765
3468
2
No convergence
No convergence
No convergence
•
-,,,,o.._~._ ..,..,•.,....,
_.~..-..... 4'-....
220
180
•,
80 .........' - - _ . L -_ _.J..--'-'--'-'-...1-_ _- ' - _............J -500 -300 -100 0 100 300 500 l' .....
~ 0.585 0.575 0.565 '--_----J'--_----J'---'-----'_ _----'_ _--I -500 -300 -100 0 100 300 500 5050 , . . . . - - - - - - - - - - - - - - - , --- Results 5000 Model / ~ 4950 . 4900 Cl
i
/ .. L_--'-_....::::=:J:::=====::.::::._-'-_---.J /'
_'"' 4800 -600 -400 -200 400 600 0 200 5800 ...........- - - - - - - - - - - - - , . . - , (d) If/'r'···'·~- ......., .. "
\\ \
/ -
Results • Model
I
,I i
5 200 '---''''''--'-_ _---l_--'-_-'-_ _.......- ''''''------' -500 -300 -100 0 100 500 300 Widthlmm (a) Transverse distribution of front and back tension stress; (b) Transverse distribution of strip's exit gauge; (c) Transverse distribution of the force between rolls; (d) Transverse distribution of rolling force.
Fig. 6
!
.........
- Model-front
•••.. Model-back ••• -.".'............... ~ .................... .-
....
0.226 (b)
§ 0.222
-... 0.595
& 5400
/t
Calculation results of example 1
._.
140 '--_----''--_----'_-'---I._ _-'-'-_ _....J -500 -300 -100 0 100 300 500
0.615 (b)
§ 0.605
~
172 179
260 - Results-front - Model-front r.:::::.!'esults-back • Model-b~)s.•",
120
S 5600
6953 7205
300,....------::---:-:--------,
160
.~
coupling method/ms
0
culated by model coupling method coincide very well with results coupling method, which indicates the model coupling method is precise. From Table 2, if
200
.~
Calculation time of model
11=0.005
0
In Fig. 6 and Fig. 7, the results of front tension stress, back tension stress, strip's exit gauge, the force between rolls and rolling force distribution cal-
"0
Bending
Calculation time and stability
Calculation time of results coupling method/ms
Example
Z ~ 4850
Vol. 19
•-
--
.....-... --•. <,.•• ' "
"aJ 0.218
1,
~ 0.214
-Results \ 0.210 - - Model \ 0.206 L.....J ----'_ _--I._-'----'-_ _--'-_......I.......I -500 -300 -100 0 100 300 500 6000 , . . . . . . , . - - - - - - - - - - - - - - , . . . . , Results -. Model :-5700 Cl
o
'§
Z 5400
i
L_.L.-_~::::::::::t:::::::::::::~_
5 100 ___1.._ __.J -600 -400 -200 o 200 400 600 ~ 6050 , - - - - - - - - - - - - - - ,
.~
S 5950 ~ 5850 o r:r.. 5750 5650 '--_""'-'_ _--1._-'----'-_ _---'-""'-_--' -300 -500 300 -100 0 100 500 Width/mm
(a) Transverse distribution of front and back tension stress; (b) Transverse distribution of strip's exit gauge; (c) Transverse distribution of force between rolls; (d) Transverse distribution of rolling force.
Fig. 7
Calculation results of example 2
Issue 2
A Model Coupling Method for Shape Prediction
results coupling method is used for example 1, the calculation time decreases remarkably along with increasing relaxation factor. When the value of relaxation factor is about O. 02, the calculation speed achieves maximum and the calculation time is 1765 ms , while the value of relaxation factor is O. 03, there is no convergence. If model coupling method is used for example 1, the calculation time is only 172 ms , and its calculation speed is ten times as much as results coupling method's maximum speed. If results coupling method is used for example 2, when the value of relaxation factor is bigger than 0.01, there is no convergence, while the value of relaxation factor is O. 005, the calculation time is 7205 ms, If model coupling method is used for example 2, the calculation time is only 179 ms , and its calculation speed is forty times as much as results coupling method's maximum speed. Through above analysis, a conclusion can be obtained that is when results coupling method is used for shape prediction, the best choice is decreasing the relaxation factor's value when strip gauge becomes smaller, or else no convergence will appear. On the contrary the model coupling method's calculation course is very stable and need not to change parameters, namely the model coupling method has better universality. In addition, even when the most appropriate relaxation factor's value is adopted, the calculation speed of model coupling method is about ten times as much as results coupling method. For five stands tandem rolling mill, if results coupling method is used for whole mill's shape prediction, its calculation time is above 10000 rns , while the calculation time of model coupling method is only 1200 rns , so the computation efficiency can be improved significantly.
4
Conclusions
1) A model coupling method is put forward which combines strip plastic deformation model with rolls elastic deformation model through their basic
• 27 •
equations, and the defect of general results coupling method can be avoided. 2) The results of front tension stress, back tension stress, strip's exit gauge, the force between rolls and rolling force distribution calculated by model coupling method coincide very well with results coupling method, which indicates the model coupling method is precise. 3) The calculation speed of model coupling method is at least ten times as much as results coupling method. The computation efficiency can be improved significantly by employing model coupling method. References: [lJ
LIU Hong-min, DING Kai-rong , LI Xing-dong , et al. Theoretical Computational Method of Shape Standard Curve [JJ. Chinese Journal of Mechanical Engineering, 2008, 44(8): 137 (in Chinese).
[2J
LIU Hong-min. Three-Dimensional Rolling Theory and Its Ap-
[3J
LIAN Iia-chuang , LIU Hong-min. Gauge Control and Shape Control of Strip [M]. Beijing: Arms Industry Press, 1996: 65 (in Chinese). YANG Yi-ting , ZHANG Qing-dong, WANG W en-guang , et
plication [MJ. Beijing: Science Press, 1999: 128 (in Chinese).
[ 4J
[5J
[6J
[7J
al. Research on An Integrative Simulating Model for Strip Shape Prediction during Cold Rolling [J]. Steel Rolling, 2007, 24(5): 13 (in Chinese). BAI Iing-Ian , ZHANG Rui, LIU Hong, et al. Calculation of Rolling Pressure Distribution in Strip Cold Rolling [JJ. Journal of Shenyang Institute of Aeronautical Engineering, 2004, 21 (4): 26 (in Chinese). LIU Yu-li, HU Xi-zeng, ZHAO Yong-he, Calculation of Contact Pressure Distribution Between Rolls [JJ. Journal of Northeast Institute of Heavy Machinery, 1989, 13(4): 39 (in Chinese). LIU Xue-Ieng , WANG Ling-yun. Analysis of Elastic Deformation of Rollers System in Rolling Mill Based on Influential Function and Prediction of Plate Shape [J]. Journal of Chongqing University (Natural Science Edition), 2000, 23(6): 87 (in Chinese).
[8J [9J
WANG Guo-dong. Shape Control and Shape Theary [MJ. Beijing: Metallurgical Industry Press, 1986: 289 (in Chinese). ZHENG Zhen-zhong , PENG yan, LIU Hong-min. A New Strip Element Variation Method for Analysing Lateral Flow of Metal and Transverse Distribution of Front Tension Stress of Cold Rolled Strip [J]. Journal of Iron and Steel Research, 1999, 11 (5): 21 (in Chinese).
ScienceDirect JOURNAL OF IRON AND STEEL RESEARCH, INTERNATIONAL. 2012, 19(2): 22-27
A Model Coupling Method for Shape Prediction WANG Dong-cheng,
LID Hong-min
(State Key Laboratory of Metastable Materials Science and Technology, National Engineering Research Center for Equipment and Technology of Cold Rolling Strip, Yanshan University, Qinhuangdao 066004, Hebei , China) Abstract: The shape of strip is calculated by iterative method which combines strip plastic deformation model with rolls elastic deformation model through their calculation results, which can be called results coupling method. Because the shape and rolling force distribution are very sensitive to strip thickness transverse distribution's variation, the iterative course is rather unstable and sometimes convergence cannot be achieved. In addition, the calculating speed of results coupling method is low, which restricts its usable range. To solve the problem, a new model coupling method is developed, which takes the force distribution between rolls, rolling force distribution and strip's exit transverse displacement distribution as basic unknowns, and integrates strip plastic deformation model and rolls elastic deformation model as a unified linear equations through their internal relation, so the iterative calculation between the strip plastic deformation model and rolls elastic deformation model can be avoided. To prove the effectiveness of the model coupling method, two examples are calculated by results coupling method and model coupling method respectively. The results of front tension stress, back tension stress, strip's exit gauge, the force between rolls and rolling force distribution calculated by model coupling method coincide very well with results coupling method. However the calculation course of model coupling method is more steady than results coupling method, and its calculating speed is about ten times as much as the maximal speed of results coupling method, which validates its practicability and reliability. Key words: shape prediction; results coupling method; model coupling method; strip plastic deformation; rolls elastic deformation
The shape is an important quality index of strip, while the shape control is a key and high level technology. The domestic level of shape control is relatively backward than the international advanced level, and the bottleneck restricting improvement of shape control technology is that the research on shape theory and mathematical models falls behind the technology's development. The whole shape control models system includes eight aspects'{", 1) strip plastic deformation model; 2) rolls elastic deformation model; 3) strip temperature field and rolls thermal deformation model; 4) rolls wear model; 5) shape discrimination model; 6) shape deviation pattern recognition model; 7) shape control object model; 8) shape feedback control model. If the eight aspects could be integrated in whole, many specific shape control models and technologies can be developed, such as shape prediction model, shape
pre-set model, roll shape optimization technology and so on. The core of shape prediction model is strip plastic deformation model and rolls elastic deformation model, which are usually integrated by results coupling method[2-8]. That means the two models are calculated independently, and the link between them is their calculation results. Convergence can be achieved only by introducing a relaxation factor. The substantial computation cost and the fact that the relaxation factor requires adjustment for different mill schedules represent the main trouble to on-line application. The basic reason of the trouble is that the transverse distribution of tension stress (shape) and rolling force distribution are very sensitive to the transverse distribution of strip's exit gauge, especially when the strip is very thin. In this case, minute change of the transverse distribution of strip's
Foundation Item: Item Sponsored by National Science and Technology Support Plan of China (2009AA04Z143); Science and Technology Support Plan of Hebei Province of China (l0212101D); Important Natural Science Foundation of Hebei Province of China (E2006001038) Biography: WANG Dong-cheng(l981-) , Male, Doctor; E-mail: [email protected]; Received Date: February 28, 2011
Issue 2
exit gauge may cause considerable change of tension stress and rolling force. and the rolling force even becomes negative (that is impossible in practice). at which time no convergence will appear. To solve this problem. a new model coupling method is put forward. The model coupling method integrates strip plastic deformation model and rolls elastic deformation model as one set of linear equations through their internal relation. which avoids the unstable phenomenon of results coupling method. At the same time the calculating speed is enhanced remarkably. So it is more applicable and reliable than results coupling method.
1. 1
Strip element variation method As shown in Fig. 1. the strip's width B is divided into n (that is an odd) independent strip elements in the deformation zone's length l, The transverse coordinate of strip element's line is expressed as Yi' and the exit transverse displacement of strip element's line is expressed as u., Strip element's width is Si=Yi-Yi-l. yo
hOi t; h -L
i; and I o are strip's
o
o
I
000
n-I
2
I
n+I
2
n+3
2
y. Y Exit
Y.-l
000
n
I
Entry x
B/2
Fig. 1
B/2
Strip element partition in deformation zone
According to the theory of minimum energy. variational calculation is applied on every strip element and the whole deformation zone. Finally n 1 linear equations can be got and shown in Eqn. (1):
+
(al-eb)UO+{3IUI +ebun=-ks+cI {3jUj-1 + (aj
+aj+1
Cj+I-Cj
E
). eb = (l-v 2 )B' h o•
I-v
h.,
entry mean gauge. exit mean gauge and entry mean length; hOi. h u and lOi are every strip element's entry gauge. exit gauge and entry length; E is strip's elastic modulus; and v is strip's Poisson's ratio. When the displacements u., - u; of every strip element's line are got. the transverse distribution of front tension stress. back tension stress and rolling force can be calculated by Eqn. (2) to Eqn. (4).
=_
111
+.--£2 [1 +h li _hOi _loi +u I-v hI t, t, ' l
P» =wowTll1 s
_
un -UO]
B
(i=l-n)
(2)
(i=l-n)
(3)
(i=l-n)
(4)
where. I1li' 110i and Pu are the front tension stress. back tension stress and rolling force of every strip element; I1 s is strip's deformation resistance; Wo is the influence coefficient of every factor except tension action on rolling force; wT is the influence coefficient of tension action on rolling force; wT = 1 0• 311Ii 0.7110i+ . f ront an db ac k ---=-'-------=-=-. 0'1 an d 0'0 are mean
I1 s tension stress; U,' =csch(Kisi)K,[ -ch(Kis'/ 2 ) u i- 1+ ch(K is ,/2 ) u ,J ; and K, is the parameter corresponding to the strip's exit gauge distribution. 1. 2
y~oy~
Yl
.--£2 (1 + h- li -
C,
Basic Mathematic Equations
Through analyzing the characteristics of every model. strip element variation method'f' and influence function method[2,8] are adopted as strip plastic deformation model and rolls elastic deformation model respectively. To display the model coupling method in detail. the basic mathematic equations of the two methods need to be given.
=
strip's exit gauge distribution;
111,
1
• 23 •
A Model Coupling Method for Shape Prediction
)Uj +~j+1 Uj+1 = [J=I-(n-1)]
1ebUO +{3nUn-1 + (an -eb)Un=k s -Cn
Influence function method Fig. 2 shows the force diagram of 4-high roll milL As shown in Fig. 3. during the length L b of backup roll. rolls are divided into m sections (The sections of strip must be coincided with strip element variation method. and be numbered alone. and its number range is 1 - n ) , the forces action on strip and rolls are divided into the same sections. while the width of every section is boYi. the midpoint coordinate of every section is y., Taking the left pressure cylinder as the origin of coordinates. every section's deflection fbi of backup roll's axis can be expressed as fbi
(1)
where. a, and {3i are parameters corresponding to the
=
m
~ abij boYjqj
(i= I-m)
(5)
j~1
where. abij roll caused Every axis can be
is the bending influence function of backup by the force between rolls. section's displacement fWi of work roll's expressed as
Vol. 19
Journal of Iron and Steel Research, International
• 24 •
z
Lb
I
F;
I I
c
-E
.... ........... q(y)
....
~_
- -p,(y) - -
.,.....,.
--.
I
i~l m
i~l
1=1
I I
C
_~
~ - -q(y) -
Fw
n
1k qiYi!J.Yi =k PUYi!J.Yi + FwL,
F's
I
0
m
kqi!J.Yi= kPli!J.Yi+ 2 Fw
i;
t=1
Combining Eqn. (9) with Eqn, (10), m+2 linear equations can be got, while the number of unknowns qi(i=l-m), C 1 and C z is m+2 too, so it can be solved. But the linear equation's coefficient matrix contains Ywbi. It is the function of qi' so an iteration calculation is necessary. The calculation flow is shown as Fig. 4.
y
3Fw
I
B
(10)
n
Lw Lfw pt(y)-Rolling force action on unit width;
rolls action on unit width; pressure cylinder; cylinder;
F;,
q(y)-Force between
F~ -Counterforce
Assuming uniform force distribution between rolls
of left and right
L, - Distance between left and right pressure
L w' L b - Length of work roll and backup roll;
Fw-Bending roll force;
Lrw-Distance between left and right
bending roll cylinder.
Fig. 2
Solving linear equations and getting new force distribution between rolls
Force diagram of 4-high roll mill
No
Fig. 4
Fig. 3
Section partition diagram of 4-high roll mill
t : = f~i + k awij !J.Yj(Plj -qj) +aFwiF m
w
j~l
(i=l-m)
(6)
where, awij is the bending influence function of work roll caused by the rolling force and the force between rolls; aFwi is the bending influence function of work roll caused by the bending roll force; and f~i is rigid displacement of work roll's axis, which can be expressed as: 2 1 f = CI + (C L- C ) K we
(
w
Y,
-
C)
(i =
1- m) (7)
where, C 1 and C2 are rigid displacement of work roll axis's left and right end; and C= CL, - L w ) /2. The elastic squashed quantity between work roll and backup roll can be expressed as ()wbi=Ywbiqi
U=l-m)
(8)
where Ywbi is elastic squashed influence function between rolls, which is the function of qi. The deformation compatibility equations between rolls can be expressed as
i-. =
fbi
+ ()wbi +
!J.Di
U= I-m)
(9)
where !J.Di is the no load clearance caused by roll shape. Force and torque balance equations are
Iteration flow for calculating force between rolls
When the iteration calculation is over, strip's exit gauge distribution can be expressed as hi -
()wi
= k
j~l
YwijPlj' Ywij is elastic squashed coefficient of
rolling force action on work roll, !J.Dwi is no load clearance between work roll and strip caused by roll shape.
2
Model Coupling Method
As mentioned above, the calculation course of results coupling method is slow and unstable, which may influence its engineering application. Considering the calculation courses of strip element variation method and influence function method can all attribute to linear equations, if integrating and solving them at the same time, the iteration calculation between the strip plastic deformation model and rolls elastic deformation model can be avoided. But for the problem, not only the force between rolls and strip's exit transverse displacement but also the strip's exit gauge distribution, rolling force distribution, front tension stress and back tension stress
A Model Coupling Method for Shape Prediction
Issue 2
distribution are unknown, so the linear equations cannot be integrated easily. Through in-depth analysis and research, the force between rolls, rolling force distribution and strip's exit transverse displacement are taken as basic unknowns. Except for Eqn. 0), Eqn. (9) and Eqn. (0), Eqn. (1) is taken as other linear equations. Now the number of unknowns is m+2n+3, while the number of linear equations is m+2n+3 too, so it can be solved. In the m 2 linear equations of Eqn. (9) and Eqn. (0), the only unknowns are the force between rolls and rolling force distribution, and can be used directly. But for Eqn. (1), the constant matrix of linear equations contains unknowns c, (because h li are unknowns) which must be expressed by basic unknowns including the force between rolls, rolling force distribution and strip's exit transverse displacement. Substitution of Eqn. (2) and Eqn. (3) into Eqn. (4), and the relational expression between rolling force distribution and strip's exit transverse displacement with h li can be derived. However the back tension stress is nonlinear to strip's exit transverse displacement, which makes a trouble. A lot of previous studies[2] show a rule that the distribution shape of back tension stress is very similar to front tension stress, so Eqn. (3) can be substituted approximatively by linear Eqn. (2) as follows
(UI- eai)UO+(j11 +eal)uI +Pll/(W°l) = -k s +a:q
(6)
Plj+1 =0
=_ +~[ 1 +':...Ii _':...Oi _~Oi +ul _Un-UO] 2 ao
I-v
hi
ho
lo
B
t:
(i=l-n)
(2)
Substitution of Eqn. (2) and Eqn. (12) into Eqn. (4), Eqn. (3) can be derived
[j = 1- (n-l) ]
«:
+
ao,
• 25 •
(j1n +ean)un-I +(u n-ean)u n- Pin /(wOl) =
-a:
q h, where, eai =EA,/(I-v2).
To solve the problem, Eqn. (11) has to be introduced as other n linear equations. Substitution of Eqn. (14) and Eqn, (6) into Eqn. (11), new linear equations are got as m 2(Yi- Lw- C) 2(C-Yi) 2~Uwij~Yjqj+ L CI L C2-
+
w
)=1
(7)
Combining the Eqn. (9), Eqn. (10), Eqn. (16) with Eqn, (7), the linear equations can be solved. But we need to pay attention to the linear equations' coefficient matrix, which contains Ywbi' a, and fJ; that are the function of qi and h l i , so an iteration calculation is necessary, the calculation flow is shown as Fig. 5.
- 0l eq_ walE [-A +A _Un-Uo]_ Pli -was I-v2 iUi-1 .u, B
Giving basic process parameters
wOl~hli_ wOl~ [1_hoi_loi] (3) I-v hi I-v s, t; where, Ai = csch (Kis i) Kich (K isj2), a: q = as -
Assuming uniform strip's exit gauge distribution
uniform force distribution between rolls
o. ts, - o. 3(i1.
Replacing strip's exit gauge distribution and force distribution between rolls
From Eqn, (13), the relational expression between rolling force distribution and strip's exit transverse displacement with h-u and Ci can be expressed as
__ 0-v2)h l
h li-
walE
Pli+
0-v 2)hl E
eq
-
as +AihIUi-1
hi lOi]A,hIUi+ B(Un-UO)- [hOi I-To hi
ho
w
Solving linear equations and getting new strip's exit gauge distribution and new force distribution between rolls I L-":":':":":"';:'=':":==--....I
_
Convergence
No
Yes End
(4)
k + aseq+~[A c, - _ wOl I-v2 iUi-1 -A .u, +Un-Uo] B (5)
Substitution of Eqn. (5) into Eqn. 0), new linear equations are got as
Fig. 5
3
Calculation flow of model coupling method
Results Analysis To prove the effectiveness of the model coupling
• 26 •
Journal of Iron and Steel Research, International
method put forward by this paper, two examples are calculated using a 1220 mm four-high cold rolling mill's parameters. Work roll's diameter is 500 mm , backup roll's diameter is 1250 mm, work roll's length is 1220 mm, and backup roll's length is 1092 mm. The process parameters of two examples are listed in Table 1. The two examples are calculated by results Table 1 Example
1 2
coupling method and model coupling method respectively. The comparison of calculation time and stability between the two methods are listed in Table 2, while the results of front tension stress, back tension stress, strip's exit gauge, the force between rolls and rolling force distribution are shown in Fig. 6 and Fig. 7.
Process parameters of two examples
Width/
Entry
Exit
Entry
Exit
mm
gauge/mm
gauge/mm
tension/kN
tension/kN
Deformation resistance/MPa
force/kN
900 900
1. 00
0.60 0.22
100 50
100 50
500 800
200 320
0.3
Table 2
11=0.02
1
No convergence
1765
3468
2
No convergence
No convergence
No convergence
•
-,,,,o.._~._ ..,..,•.,....,
_.~..-..... 4'-....
220
180
•,
80 .........' - - _ . L -_ _.J..--'-'--'-'-...1-_ _- ' - _............J -500 -300 -100 0 100 300 500 l' .....
~ 0.585 0.575 0.565 '--_----J'--_----J'---'-----'_ _----'_ _--I -500 -300 -100 0 100 300 500 5050 , . . . . - - - - - - - - - - - - - - - , --- Results 5000 Model / ~ 4950 . 4900 Cl
i
/ .. L_--'-_....::::=:J:::=====::.::::._-'-_---.J /'
_'"' 4800 -600 -400 -200 400 600 0 200 5800 ...........- - - - - - - - - - - - - , . . - , (d) If/'r'···'·~- ......., .. "
\\ \
/ -
Results • Model
I
,I i
5 200 '---''''''--'-_ _---l_--'-_-'-_ _.......- ''''''------' -500 -300 -100 0 100 500 300 Widthlmm (a) Transverse distribution of front and back tension stress; (b) Transverse distribution of strip's exit gauge; (c) Transverse distribution of the force between rolls; (d) Transverse distribution of rolling force.
Fig. 6
!
.........
- Model-front
•••.. Model-back ••• -.".'............... ~ .................... .-
....
0.226 (b)
§ 0.222
-... 0.595
& 5400
/t
Calculation results of example 1
._.
140 '--_----''--_----'_-'---I._ _-'-'-_ _....J -500 -300 -100 0 100 300 500
0.615 (b)
§ 0.605
~
172 179
260 - Results-front - Model-front r.:::::.!'esults-back • Model-b~)s.•",
120
S 5600
6953 7205
300,....------::---:-:--------,
160
.~
coupling method/ms
0
culated by model coupling method coincide very well with results coupling method, which indicates the model coupling method is precise. From Table 2, if
200
.~
Calculation time of model
11=0.005
0
In Fig. 6 and Fig. 7, the results of front tension stress, back tension stress, strip's exit gauge, the force between rolls and rolling force distribution cal-
"0
Bending
Calculation time and stability
Calculation time of results coupling method/ms
Example
Z ~ 4850
Vol. 19
•-
--
.....-... --•. <,.•• ' "
"aJ 0.218
1,
~ 0.214
-Results \ 0.210 - - Model \ 0.206 L.....J ----'_ _--I._-'----'-_ _--'-_......I.......I -500 -300 -100 0 100 300 500 6000 , . . . . . . , . - - - - - - - - - - - - - - , . . . . , Results -. Model :-5700 Cl
o
'§
Z 5400
i
L_.L.-_~::::::::::t:::::::::::::~_
5 100 ___1.._ __.J -600 -400 -200 o 200 400 600 ~ 6050 , - - - - - - - - - - - - - - ,
.~
S 5950 ~ 5850 o r:r.. 5750 5650 '--_""'-'_ _--1._-'----'-_ _---'-""'-_--' -300 -500 300 -100 0 100 500 Width/mm
(a) Transverse distribution of front and back tension stress; (b) Transverse distribution of strip's exit gauge; (c) Transverse distribution of force between rolls; (d) Transverse distribution of rolling force.
Fig. 7
Calculation results of example 2
Issue 2
A Model Coupling Method for Shape Prediction
results coupling method is used for example 1, the calculation time decreases remarkably along with increasing relaxation factor. When the value of relaxation factor is about O. 02, the calculation speed achieves maximum and the calculation time is 1765 ms , while the value of relaxation factor is O. 03, there is no convergence. If model coupling method is used for example 1, the calculation time is only 172 ms , and its calculation speed is ten times as much as results coupling method's maximum speed. If results coupling method is used for example 2, when the value of relaxation factor is bigger than 0.01, there is no convergence, while the value of relaxation factor is O. 005, the calculation time is 7205 ms, If model coupling method is used for example 2, the calculation time is only 179 ms , and its calculation speed is forty times as much as results coupling method's maximum speed. Through above analysis, a conclusion can be obtained that is when results coupling method is used for shape prediction, the best choice is decreasing the relaxation factor's value when strip gauge becomes smaller, or else no convergence will appear. On the contrary the model coupling method's calculation course is very stable and need not to change parameters, namely the model coupling method has better universality. In addition, even when the most appropriate relaxation factor's value is adopted, the calculation speed of model coupling method is about ten times as much as results coupling method. For five stands tandem rolling mill, if results coupling method is used for whole mill's shape prediction, its calculation time is above 10000 rns , while the calculation time of model coupling method is only 1200 rns , so the computation efficiency can be improved significantly.
4
Conclusions
1) A model coupling method is put forward which combines strip plastic deformation model with rolls elastic deformation model through their basic
• 27 •
equations, and the defect of general results coupling method can be avoided. 2) The results of front tension stress, back tension stress, strip's exit gauge, the force between rolls and rolling force distribution calculated by model coupling method coincide very well with results coupling method, which indicates the model coupling method is precise. 3) The calculation speed of model coupling method is at least ten times as much as results coupling method. The computation efficiency can be improved significantly by employing model coupling method. References: [lJ
LIU Hong-min, DING Kai-rong , LI Xing-dong , et al. Theoretical Computational Method of Shape Standard Curve [JJ. Chinese Journal of Mechanical Engineering, 2008, 44(8): 137 (in Chinese).
[2J
LIU Hong-min. Three-Dimensional Rolling Theory and Its Ap-
[3J
LIAN Iia-chuang , LIU Hong-min. Gauge Control and Shape Control of Strip [M]. Beijing: Arms Industry Press, 1996: 65 (in Chinese). YANG Yi-ting , ZHANG Qing-dong, WANG W en-guang , et
plication [MJ. Beijing: Science Press, 1999: 128 (in Chinese).
[ 4J
[5J
[6J
[7J
al. Research on An Integrative Simulating Model for Strip Shape Prediction during Cold Rolling [J]. Steel Rolling, 2007, 24(5): 13 (in Chinese). BAI Iing-Ian , ZHANG Rui, LIU Hong, et al. Calculation of Rolling Pressure Distribution in Strip Cold Rolling [JJ. Journal of Shenyang Institute of Aeronautical Engineering, 2004, 21 (4): 26 (in Chinese). LIU Yu-li, HU Xi-zeng, ZHAO Yong-he, Calculation of Contact Pressure Distribution Between Rolls [JJ. Journal of Northeast Institute of Heavy Machinery, 1989, 13(4): 39 (in Chinese). LIU Xue-Ieng , WANG Ling-yun. Analysis of Elastic Deformation of Rollers System in Rolling Mill Based on Influential Function and Prediction of Plate Shape [J]. Journal of Chongqing University (Natural Science Edition), 2000, 23(6): 87 (in Chinese).
[8J [9J
WANG Guo-dong. Shape Control and Shape Theary [MJ. Beijing: Metallurgical Industry Press, 1986: 289 (in Chinese). ZHENG Zhen-zhong , PENG yan, LIU Hong-min. A New Strip Element Variation Method for Analysing Lateral Flow of Metal and Transverse Distribution of Front Tension Stress of Cold Rolled Strip [J]. Journal of Iron and Steel Research, 1999, 11 (5): 21 (in Chinese).