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Compensation Model for Shape Measuring of Cold Strip Rolling
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ScienceDirect JOURNAL OF IRON AND STEEL RESEARCH, INTERNATIONAL. 2010, 17(6): 21-26
Compensation Model for Shape Measuring of Cold Strip Rolling YU Bing-qiang,
SUN Ya-bo,
LIU Hong-min,
YOU Lei,
PENG Yan
(Engineering Research Center of Rolling Equipment and Complete Technology of Ministry of Education, Yanshan University, Qinhuangdao 066004, Hebei, China) Abstract: Some unavoidable factors in the process of cold strip shape measurement interfere with the shape meter, so the shape measuring results cannot reflect the true shape of the strip and the measuring precision is low. The influences of the measuring error of the strip edges, the transverse temperature difference of the strip, the deflection of shape detection roller, and the shape of the strip coil on the shape measuring results were analyzed in detail, and the corresponding compensation models were established. The simulation calculation and analysis were carried out on a cold strip mill, and a number of disciplinarian cognitions were obtained. Key words: cold strip rolling; shape measuring; compensation model
During cold strip rolling, the additional stresses are generated in the rolled strip owing to the existence of some interferers, so the online shape is changed, and the shape measuring results cannot accurately reflect the true shape of the strip. After the strip with good online shape is offline, the shape is becoming bad, so the precision of shape measuring should be ensured through compensation for these interferers. The experiential models for compensating various additional stresses are given in Ref. [1 J, and their precision depends on the compensation coefficients and is difficult to be ensured. In this study, the influences of the measuring error of the strip edges, the transverse temperature difference of the strip, the deflection of shape detection roller, and the shape of the strip coil on the shape measuring results were analyzed in detail, and corresponding compensation models were established. Finally, the simulation calculation and analysis were carried out on a 1050 HC cold mill of Hengshui sheet Inc of Handan Iron and Steel Co as an example, and a number of disciplinarian cogrutions were obtained.
1
Compensation Models for Shape Measuring
1. 1 Compensation model for measuring error of strip edges When the width of the strip is not equal to the effective measuring width of the shape detection roller, a portion of the limbic measuring unit of the shape detection roller is covered by the strip edges. The tensile stress of the strip calculated according to the conventional arithmetic is less than the true tensile stress of the strip, and the compensation should be done on the limbic shape measuring results. As shown in Fig. 1, the stress of the strip by which the measuring unit of the shape detection roller is covered is assumed to be T;, and it can be obtained according to the pressure acting on the measuring unit of the shape detection roller by the strip, P;, as follows:
T;=~ 2sin
(1)
2
e
where is the rounding angle, that is, the size of the shape detection roller circle covered by the strip.
Foundation Item: Item Sponsored by National Science and Technology Support Plan of China (2007BAF02BI0): Provincial Natural Science Foundation of Hebei of China (E2006001038) Biography: YU Bing-qiang0963-), Male, Doctor; E-mail: ybingq@ysu. edu, en; Received Date: August 21, 2009
Vol. 17
Journal of Iron and Steel Research. International
• 22 •
strip; E is the elastic modulus of the strip; and t is the average value of t(y) across the strip width B.
Fig. 1
Mechanical model of measuring unit of shape detection roller
When the measuring units of the shape detection roller are completely covered by the strip, the tensile stress of the strip (1m (i) is (1m
Ti ( l") = bh
l•
= 2 , 3 ,"', n 1 r:
(2)
where, b is the width of the measuring unit of the shape detection roller; and h is the thickness of the strip. For. the strip edges, the limbic measuring units of the shape detection roller are not always covered by the strip completely. Thus, the tensile stresses of the strip edges should be calculated according to the actual width of measuring unit of the shape detection roller covered by the strip edges, there is
(1m(i)=b~h
i=l, n
(3)
where b, is the actual width of the limbic measuring unit of the shape detection roller covered by the strip edges. 1. 2 Compensation model for transverse temperature difference of strip During cold strip rolling, the transverse distribution of the thermal elongation of strip is uneven because the temperature difference exists along the strip width direction, so the thermal stress is generated. The compensation of the transverse temperature difference of the strip should be done on the shape measuring results[Z-3]. The transverse distribution of the measured or calculated strip temperature t (y) can be expressed as the quartic polynomial t(y) = to+tzyZ + t4y4 (4) where to' tz' t 4 are fitting coefficients. The compensation quantity caused by the transverse temperature difference of the strip, i. e. , the thermal stress that should be subtracted from the shape measuring results, (1,(y) , can be obtained (1,(y)=a, • E· [t-t(y)] (5) where, a, is the linear expansion coefficient of the
1.3 Compensation model for deflection of shape detection roller and shape of strip coil During shape measuring, the shape detection roller is pressed by the strip. The deflection of shape detection roller occurs, and the upper surface of the shape detection roller shows the concave state. Then, the transverse distribution of the elongation of the strip is changed, and the additional stress is generated. At the same time, the transverse distribution of the tensile stress of the strip is changed. Thus, the transverse distribution of pressure measured by shape meter is not transformed from the forward tension which reflects the true shape, and the measuring error is formed. Thus, the compensation of the shape detection roller deflection should be done on the shape measuring results. After the rolled strip with crown is coiled, the crown of the strip coil is formed, and the outside radiuses of the strip coil are not equal along the strip width direction. Thus, the transverse distribution of the elongation of the strip is changed, and the additional stress is generated. In the meanwhile, the transverse distribution of the tensile stress of the strip is changed, and the measuring error is formed. The compensation of the shape of the strip coil should also be done on the shape measuring results. After the tensile stresses are measured by the shape meter, its transverse distribution (1m (y) IS expressed (y) =bo+bzYz +b4y4 (6) where bo' bz' and b, are fitting coefficients. 1. 3. 1 Calculation of shape detection roller deflection The shape detection roller deflection is calculated with the influence coefficient method[4-5]. The mechanical model is shown in Fig. 2. The transverse distribution of the unit width con(1m
Ls
c
y
Fig. 2
Mechanical model of shape detection roller
Issue 6
• 23 •
Compensation Model for Shape Measuring of Cold Strip Rolling
tact pressure acting on the shape detection roller q(y) is obtained as follows:
q(y)=20'm(y)h(y)sin[
~]
(7)
where h(y) is the transverse distribution of the strip thickness. It is assumed that n units (measuring units) of the shape detection roller are covered by the strip. And q(y) and the strip are also divided into the same units. The deflection of the ith unit of the shape detection roller, i. e. , the deflection in the point Yi caused by all the unit width contact pressures qj (j = 1, 2, "', n) can be obtained
. . L>
fi= ~aijbqj
(i=2,
(8)
, n-1)
j=l
~aijbsqj
(i=1, n)
(9)
j=l
where aij is the bending influence coefficient of the shape detection roller. 1.3.2 Calculation of outside radius of strip coil Because the axial stress of the strip coil is very small, which can be divided into n small strip coils along the axial direction according to the measuring units of the shape detection roller, and the iteration of layer by layer[6-7] is taken to calculate the outside radius of the strip coil. During coiling, the radial displacement of the ith small strip coil u, can be expressed as
of the ith small strip coil Ui,k.j can be obtained _ri,k,j[ + ( l - ) p Ui,k,j - -E O'Oi u i,k,j
+ r..«, Pi,k+l,j-Pi,k,j] h ,.«, tlk,)
2, "', j-1) (2) where ri,k,j' hi,k,j are the coiling radius and the strip thickness of the kth layer strip of the ith small strip coil, respectively, when coiling the jth layer strip; Pi.s.,» Pi,k+l,j are the radial pressure of the inside and outside surfaces of the kth layer strip of the ith small strip coil, respectively, when coiling the jth layer strip. Therefore, the radial displacement increment of the kth layer strip of the ith small strip coil for a period of coiling time from the (j -1) th layer strip to the jth layer strip can be expressed as (k=l,
tiUi,k,j
(3)
Eqn. (11) and Eqn. (12) are combined with Eqn, (3), and Pi,k,j can be obtained from Eqn. (4). 1 Pi,k,j
=
2
ri,k,j - _ -n.«: -( 1 -u ) E Ehi,k,j
+
ri,k,j [
[tiUi,k,j +Ui,k,j-l -
Pi,k+l,j
1]
(4)
E O'Oi rr,k,j h.z,ktj. It can be seen that r..s.; and Pi,k,j can be tained, synchronously. Then, when coiling the layer strip, the outside radius of the ith strip R Dj can be expressed as RDj(i)=ri,j,j (i=1, 2, "', n; j=l, 2, "',
r, [ O'Oi +(1-) + r, dPi,j] u, -_ E U Pi.s dri
m)
(5)
(i =
1, 2, "', n; j = 1, 2, "', m) (0) where, O'Oi is the coiling tensile stress of the ith small strip coil; Pi.} is the inner radial pressure of the j th layer strip of the ith small strip coil; u is the Poisson coefficient of strip; r, is the coiling radius of the ith small strip coil; and m is the number of layers of the strip coil. For calculation, dr, is set to be equal to hi' where hi is the strip thickness of the ith small strip coil, and dpi,j is the radial pressure difference between the inside and outside surfaces of the jth layer strip of the ith small strip coil. When coiling the (j -1) th layer strip, the radial displacement of the kth layer strip of the ith small strip coil Ui,k,j-l can be expressed as
1.3. 3 Calculation of compensation quantity caused by deflection of shape detection roller and shape of strip coil The additional stress caused by the deflection of shape detection roller and the shape of the strip coil is acting on a section of the strip from the mill exit to the coiler and the outermost layer strip of the strip coil. It is assumed that the strip is always contacting with the upper surface of the shape detection roller under O'm during coiling, the longitudinal profile figure of the ith unit is shown in Fig. 3. After the shape detection roller is bended , the Coller
Mill exit
Shape detection roller 02
01
-r
Ui,k,j-l -_ri,k,j-l[ O'Oi + ( l - u) p i,k,j-l + (k=l,
objth coil
2, "',
j-1)
(1)
In the same way, when coiling the jth layer strip, the radial displacement of the kth layer strip
Fig, 3
Longitudinal profile of the ith unit
• 24 •
Vol. 17
Journal of Iron and Steel Research, International (j(i)=(jm(i)-(jt(i)-(jsj(i)
geometric relationships in Fig. 3 are as follows:
I; COSAo a=arctan -=arctan +1 ._, a2 a20 S1I1I\o
(16)
_ [R+ R Oj (i) ]sina j3- arccos
(17)
y= ~
(18)
e
eo -
(j=1, 2,
«
m)
(26)
i
I ' iCOSAO
eo -
-(j3-a)
S= arctan
liCOSAo
a 10 -
I' _, i S1I1I\o
(19)
B=y-S
(20)
where, eo, ajO' and a20 are the center distances before the shape detection roller is bended; e , aj' and a2 are the center distances after the shape detection roller is bended; and R is the outside radius of the shape detection roller. When coiling the jth layer strip, for the ith unit, the length of the strip from the mill exit to the coiler and the outermost layer strip of the strip coil lj (i) is expressed as [ R+Ro/I) . ] tanj3+alO - l;sinAo S cos RtanS+2n:R oj (i)
.
lj(I)=BR+
+ (21)
The average length of L, (i) across the strip width B, lj' is obtained n
'Z i, (i) l=··_·~--,-j--
The influence of the deflection of shape detection roller and the shape of the strip coil on the tensile strain of the strip Ss, (i) is obtained A
u€· J
(")_lj(i)-lj l
-
Simulation Calculation and Analysis
According to the measuring results of the shape measuring system of Hengshui Sheet Inc of Handan Iron and Steel Co, the simulation calculation and analysis are done. The width B and the thickness h of the strip are 890 and 2 rnrn , respectively, and the crown of the strip is 20 f.Lm. The calculation parameters are shown in Table 1. When the compensation for the' limbic measuring results is done, first of all, the positions of the strip edges are detected by the position sensor, and then, the actual width of the limbic measuring unit of the shape detection roller covered by the strip is calculated according to the strip width. Finally, the true tensile stresses of the strip edges are obtained according to Eqn. (3). Table 1
Calculation parameters
Outside radius of shape detection roller
mm
200
Inside radius of shape detection roller
135
Outside radius of coiler
300
Inside radius of coiler
158
(22)
n
J
2
i,
(23)
According to the planar deformation theory, the influence of the deflection of shape detection roller and the shape of the strip coil on the tensile stress of the strip Sa, (i) is obtained b.(jj(i)=l E 2b.€j(i)
-u
2, ... , n; j = l , 2, ... , m) (24) When coiling the j th layer strip, the compensation quantity caused by the shape detection roller deflection and the shape of the strip coil, i. e. , the tensile stress that should be subtracted from the shape measuring results, (jsj (y), can be expressed as (jsj(i) =b.(jj (i) (i=l, 2, ... , n; j=l, 2, ... , m) (i=l,
(25) When coiling the j th layer strip, a, and (jsj are subtracted from the shape measuring results 17m , and the true shape stress (j(i) can be expressed as
2. 1 Calculation of compensation quantity caused by transverse temperature difference of strip The transverse discrete values of t(y) are measured by the intelligent temperature apparatus. Then, they are fitted into the distribution curve according to Eqn. (4). The transverse temperature difference is about 10 'C, as shown in Fig. 4 (a). 17, is calculated and shown in Fig. 4 (b). It can be seen that the online shape stress curve appears to be a parabola, and at the center of the strip, the stress is the lowest. Thus, there is a trend that makes the strip form centre waves.
2. 2 Calculation of compensation quantity caused by deflection of shape detection roller Assuming that R Oj in Eqn, (21) is invariable, the compensation quantity caused by the shape detection roller deflection (jb is obtained. (jb is calculated under the shape conditions of flat, centre waves (transverse tensile stress difference of 21 MPa) and both edge waves (transverse tensile stress difference of 21 MPa) , and is shown in Fig. 5. It can be seen that whatever the shape state of the strip is, the ten-
Issue 6
Compensation Model for Shape Measuring of Cold Strip Rolling
sion stress measured by the shape meter all is low in the center and high in the both edges compared to the true tension stress, and the compensation quantitiesunder different conditions are nearly equal, that is, the size of the compensation quantity is not sensitive to the forms of the transverse distribution of the coiling tensile stress.
134 (a) 132 130
P
~
128 126 124
2. 3 Calculation of compensation quantity caused by shape of strip coil It is assumed that t. in Eqn. (21) is zero, and then the compensation quantity caused by the shape of the strip coil (jc is obtained. a., (j = 1, 2, "', m) is calculated under the shape conditions of flat, centre waves (transverse tensile stress difference of 21 MPa) and both edge waves (transverse tensile stress difference of 21 MPa), respectively. Fig. 6 shows the compensation quantity caused by the shape of the strip coil when coiling the l Oth , 20th, 30th, 40th, 50th, 60th, and 70th layers. It can be seen that when the shape of the strip is flat, the compensation quantity is very small and can be ignored. Under the shape conditions of centre waves and both edge waves, the compensation quantity is gradually increasing before coiling the 50th layer, and then, the compensation
20 15
10
~6'
5 0 -5
-10 -400
-200
0
200
400
B/mm
(a) Transverse distribution of strip temperature (t) (b) Compensation quantity caused by transverse temperature difference of strip (",,).
Fig. 4
• 25 •
;
Calculation of compensation quantity caused by transverse temperature difference of strip 0.25 0.15
~ ~
to
0.05 0 -0.05 -0.15 -400
o
-200
200
400 -400
-200
0
200
400 -400
-200
o
200
400
B/mm
(a) Flat;
Fig. 5
10
(a)
• 70th layer • 60th layer .. 50th layer ~ 40th layer • 30th layer • 20th layer • 10th layer
0.03 0.02
~0" -0.01 -0.02 -400
(b) Centre waves;
-200
0
(c) Both edge waves.
Compensation quantity caused by detection roller deflection Cab)
200
400
15
(b)
5
10
0
5
-5
0
-10
-5
-15 -400
-10 -200
0
200
400
-400
-200
B/mm
(a) Flat;
Fig.6
(b) Centre waves;
(c) Both edge waves.
Compensation quantity caused by shape of strip coil Co',)
0
200
400
26 •
quantity is gradually stabilized with the Increase In the coiling layers.
2. 4
General compensation quantity of shape measuring and CIsj are added, and the general compensation quantity of shape measuring CId is obtained, as shown in Fig. 7. It can be seen that when the shape of strip is flat, the general compensation quantity of shape measuring is mostly caused by the transverse temperature difference of the strip, and the compensation quantity caused by the shape detection roller CIt
20 (a)
~e
• • .. • • • •
10
0
-10 -400
70th layer 60th layer 50th layer 40th layer 30th layer 20th layer 10th layer
deflection and the compensation quantity caused by the shape of the strip coil are very small correspondingly and can be ignored. Under the shape conditions of centre waves and both edge waves, the general compensation quantity of shape measuring varies largely, which is gradually decreasing under the shape conditions of centre waves and which is gradually increasing under the shape conditions of both edge waves, before coiling the 50th layer. Sith , the general compensation quantity of shape measuring is gradually stabilized with increase of the coiling layers.
12
30
8
20
4
10
0
0
-3
-10
-6 -200
0
200
400
(a) Flat;
Fig.7
3
Vol. 17 .
Journal of Iron and Steel Research. International
-400
-200
0 B/mm
(b) Centre waves;
200
400
-20 -400
1) The size of the compensation quantity caused by the deflection of shape detection roller is not sensitive to the forms of the transverse distribution of the coiling tensile stress. which can be calculated according to one or several kinds of easy waves. 2) When the shape is flat, the compensation quantity caused by the shape of the strip coil is very small, which can be ignored; under the conditions of centre waves and both edge waves, the compensation quantity is gradually increased before coiling the 50th layer, and then. the compensation quantity is gradually stabilized with the increase in the coiling layers, there is no need to recalculate. 3) The true shape stress of the strip is obtained after subtracting general compensation quantity of shape measuring from the shape measuring results and ignoring the influence of the transverse temper-
0
200
400
(c) Both edge waves.
Synthetic compensation quantity of shape measuring
Conclusions
-200
(ad)
ature difference of the strip, the deflection of the shape detection roller, and the shape of the strip coil. References , [lJ
xu
[2J
LI Ru-jia. The Temperature Impacts and Its Compensation Measures During the Strip Flatness Measuring and Controlling [n. Heavy Machinery, 1995(5), 26 (in Chinese). JIA Chun-yu , SHANG Zhi-dong, Compensation of Plank Shape Target Curves for Cold-Rolling Strip Journal of Iron and Steel Research, 2000, 12(4), 64 (in Chinese). LIU Hong-min. Three-Dimensional Rolling Theory and Its Applications [MJ. Beijing, Science Press, 1999 (in Chinese). LIAN Jia-chuang, LIU Hong-min. Gauge and Shape Control [MJ. Beijing, Weapon Industry Press, 1996 (in Chinese). LIAN Iia-chuang , LI Jun-hong , SUN Ii-quan. Calculating Drum's Unit Pressure by the Iteration of Layer by Layer [n. Heavy Machinery, 2001(6), 30 (in Chinese).
[3J
Le-jiang. Flatness Control in Cold Strip Rolling and Mill Type Selection [MJ. Beijing, Metallurgical Industry Press, 2007 (in Chinese).
[n.
[4J [5J [6J
[7J
LIAN Iia-chuang, The Calculation for Unit Pressure at the Coilers Drum [n. Chinese Journal of Mechanical Engineering, 1980, 16(3): 78 (in Chinese).
ScienceDirect JOURNAL OF IRON AND STEEL RESEARCH, INTERNATIONAL. 2010, 17(6): 21-26
Compensation Model for Shape Measuring of Cold Strip Rolling YU Bing-qiang,
SUN Ya-bo,
LIU Hong-min,
YOU Lei,
PENG Yan
(Engineering Research Center of Rolling Equipment and Complete Technology of Ministry of Education, Yanshan University, Qinhuangdao 066004, Hebei, China) Abstract: Some unavoidable factors in the process of cold strip shape measurement interfere with the shape meter, so the shape measuring results cannot reflect the true shape of the strip and the measuring precision is low. The influences of the measuring error of the strip edges, the transverse temperature difference of the strip, the deflection of shape detection roller, and the shape of the strip coil on the shape measuring results were analyzed in detail, and the corresponding compensation models were established. The simulation calculation and analysis were carried out on a cold strip mill, and a number of disciplinarian cognitions were obtained. Key words: cold strip rolling; shape measuring; compensation model
During cold strip rolling, the additional stresses are generated in the rolled strip owing to the existence of some interferers, so the online shape is changed, and the shape measuring results cannot accurately reflect the true shape of the strip. After the strip with good online shape is offline, the shape is becoming bad, so the precision of shape measuring should be ensured through compensation for these interferers. The experiential models for compensating various additional stresses are given in Ref. [1 J, and their precision depends on the compensation coefficients and is difficult to be ensured. In this study, the influences of the measuring error of the strip edges, the transverse temperature difference of the strip, the deflection of shape detection roller, and the shape of the strip coil on the shape measuring results were analyzed in detail, and corresponding compensation models were established. Finally, the simulation calculation and analysis were carried out on a 1050 HC cold mill of Hengshui sheet Inc of Handan Iron and Steel Co as an example, and a number of disciplinarian cogrutions were obtained.
1
Compensation Models for Shape Measuring
1. 1 Compensation model for measuring error of strip edges When the width of the strip is not equal to the effective measuring width of the shape detection roller, a portion of the limbic measuring unit of the shape detection roller is covered by the strip edges. The tensile stress of the strip calculated according to the conventional arithmetic is less than the true tensile stress of the strip, and the compensation should be done on the limbic shape measuring results. As shown in Fig. 1, the stress of the strip by which the measuring unit of the shape detection roller is covered is assumed to be T;, and it can be obtained according to the pressure acting on the measuring unit of the shape detection roller by the strip, P;, as follows:
T;=~ 2sin
(1)
2
e
where is the rounding angle, that is, the size of the shape detection roller circle covered by the strip.
Foundation Item: Item Sponsored by National Science and Technology Support Plan of China (2007BAF02BI0): Provincial Natural Science Foundation of Hebei of China (E2006001038) Biography: YU Bing-qiang0963-), Male, Doctor; E-mail: ybingq@ysu. edu, en; Received Date: August 21, 2009
Vol. 17
Journal of Iron and Steel Research. International
• 22 •
strip; E is the elastic modulus of the strip; and t is the average value of t(y) across the strip width B.
Fig. 1
Mechanical model of measuring unit of shape detection roller
When the measuring units of the shape detection roller are completely covered by the strip, the tensile stress of the strip (1m (i) is (1m
Ti ( l") = bh
l•
= 2 , 3 ,"', n 1 r:
(2)
where, b is the width of the measuring unit of the shape detection roller; and h is the thickness of the strip. For. the strip edges, the limbic measuring units of the shape detection roller are not always covered by the strip completely. Thus, the tensile stresses of the strip edges should be calculated according to the actual width of measuring unit of the shape detection roller covered by the strip edges, there is
(1m(i)=b~h
i=l, n
(3)
where b, is the actual width of the limbic measuring unit of the shape detection roller covered by the strip edges. 1. 2 Compensation model for transverse temperature difference of strip During cold strip rolling, the transverse distribution of the thermal elongation of strip is uneven because the temperature difference exists along the strip width direction, so the thermal stress is generated. The compensation of the transverse temperature difference of the strip should be done on the shape measuring results[Z-3]. The transverse distribution of the measured or calculated strip temperature t (y) can be expressed as the quartic polynomial t(y) = to+tzyZ + t4y4 (4) where to' tz' t 4 are fitting coefficients. The compensation quantity caused by the transverse temperature difference of the strip, i. e. , the thermal stress that should be subtracted from the shape measuring results, (1,(y) , can be obtained (1,(y)=a, • E· [t-t(y)] (5) where, a, is the linear expansion coefficient of the
1.3 Compensation model for deflection of shape detection roller and shape of strip coil During shape measuring, the shape detection roller is pressed by the strip. The deflection of shape detection roller occurs, and the upper surface of the shape detection roller shows the concave state. Then, the transverse distribution of the elongation of the strip is changed, and the additional stress is generated. At the same time, the transverse distribution of the tensile stress of the strip is changed. Thus, the transverse distribution of pressure measured by shape meter is not transformed from the forward tension which reflects the true shape, and the measuring error is formed. Thus, the compensation of the shape detection roller deflection should be done on the shape measuring results. After the rolled strip with crown is coiled, the crown of the strip coil is formed, and the outside radiuses of the strip coil are not equal along the strip width direction. Thus, the transverse distribution of the elongation of the strip is changed, and the additional stress is generated. In the meanwhile, the transverse distribution of the tensile stress of the strip is changed, and the measuring error is formed. The compensation of the shape of the strip coil should also be done on the shape measuring results. After the tensile stresses are measured by the shape meter, its transverse distribution (1m (y) IS expressed (y) =bo+bzYz +b4y4 (6) where bo' bz' and b, are fitting coefficients. 1. 3. 1 Calculation of shape detection roller deflection The shape detection roller deflection is calculated with the influence coefficient method[4-5]. The mechanical model is shown in Fig. 2. The transverse distribution of the unit width con(1m
Ls
c
y
Fig. 2
Mechanical model of shape detection roller
Issue 6
• 23 •
Compensation Model for Shape Measuring of Cold Strip Rolling
tact pressure acting on the shape detection roller q(y) is obtained as follows:
q(y)=20'm(y)h(y)sin[
~]
(7)
where h(y) is the transverse distribution of the strip thickness. It is assumed that n units (measuring units) of the shape detection roller are covered by the strip. And q(y) and the strip are also divided into the same units. The deflection of the ith unit of the shape detection roller, i. e. , the deflection in the point Yi caused by all the unit width contact pressures qj (j = 1, 2, "', n) can be obtained
. . L>
fi= ~aijbqj
(i=2,
(8)
, n-1)
j=l
~aijbsqj
(i=1, n)
(9)
j=l
where aij is the bending influence coefficient of the shape detection roller. 1.3.2 Calculation of outside radius of strip coil Because the axial stress of the strip coil is very small, which can be divided into n small strip coils along the axial direction according to the measuring units of the shape detection roller, and the iteration of layer by layer[6-7] is taken to calculate the outside radius of the strip coil. During coiling, the radial displacement of the ith small strip coil u, can be expressed as
of the ith small strip coil Ui,k.j can be obtained _ri,k,j[ + ( l - ) p Ui,k,j - -E O'Oi u i,k,j
+ r..«, Pi,k+l,j-Pi,k,j] h ,.«, tlk,)
2, "', j-1) (2) where ri,k,j' hi,k,j are the coiling radius and the strip thickness of the kth layer strip of the ith small strip coil, respectively, when coiling the jth layer strip; Pi.s.,» Pi,k+l,j are the radial pressure of the inside and outside surfaces of the kth layer strip of the ith small strip coil, respectively, when coiling the jth layer strip. Therefore, the radial displacement increment of the kth layer strip of the ith small strip coil for a period of coiling time from the (j -1) th layer strip to the jth layer strip can be expressed as (k=l,
tiUi,k,j
(3)
Eqn. (11) and Eqn. (12) are combined with Eqn, (3), and Pi,k,j can be obtained from Eqn. (4). 1 Pi,k,j
=
2
ri,k,j - _ -n.«: -( 1 -u ) E Ehi,k,j
+
ri,k,j [
[tiUi,k,j +Ui,k,j-l -
Pi,k+l,j
1]
(4)
E O'Oi rr,k,j h.z,ktj. It can be seen that r..s.; and Pi,k,j can be tained, synchronously. Then, when coiling the layer strip, the outside radius of the ith strip R Dj can be expressed as RDj(i)=ri,j,j (i=1, 2, "', n; j=l, 2, "',
r, [ O'Oi +(1-) + r, dPi,j] u, -_ E U Pi.s dri
m)
(5)
(i =
1, 2, "', n; j = 1, 2, "', m) (0) where, O'Oi is the coiling tensile stress of the ith small strip coil; Pi.} is the inner radial pressure of the j th layer strip of the ith small strip coil; u is the Poisson coefficient of strip; r, is the coiling radius of the ith small strip coil; and m is the number of layers of the strip coil. For calculation, dr, is set to be equal to hi' where hi is the strip thickness of the ith small strip coil, and dpi,j is the radial pressure difference between the inside and outside surfaces of the jth layer strip of the ith small strip coil. When coiling the (j -1) th layer strip, the radial displacement of the kth layer strip of the ith small strip coil Ui,k,j-l can be expressed as
1.3. 3 Calculation of compensation quantity caused by deflection of shape detection roller and shape of strip coil The additional stress caused by the deflection of shape detection roller and the shape of the strip coil is acting on a section of the strip from the mill exit to the coiler and the outermost layer strip of the strip coil. It is assumed that the strip is always contacting with the upper surface of the shape detection roller under O'm during coiling, the longitudinal profile figure of the ith unit is shown in Fig. 3. After the shape detection roller is bended , the Coller
Mill exit
Shape detection roller 02
01
-r
Ui,k,j-l -_ri,k,j-l[ O'Oi + ( l - u) p i,k,j-l + (k=l,
objth coil
2, "',
j-1)
(1)
In the same way, when coiling the jth layer strip, the radial displacement of the kth layer strip
Fig, 3
Longitudinal profile of the ith unit
• 24 •
Vol. 17
Journal of Iron and Steel Research, International (j(i)=(jm(i)-(jt(i)-(jsj(i)
geometric relationships in Fig. 3 are as follows:
I; COSAo a=arctan -=arctan +1 ._, a2 a20 S1I1I\o
(16)
_ [R+ R Oj (i) ]sina j3- arccos
(17)
y= ~
(18)
e
eo -
(j=1, 2,
«
m)
(26)
i
I ' iCOSAO
eo -
-(j3-a)
S= arctan
liCOSAo
a 10 -
I' _, i S1I1I\o
(19)
B=y-S
(20)
where, eo, ajO' and a20 are the center distances before the shape detection roller is bended; e , aj' and a2 are the center distances after the shape detection roller is bended; and R is the outside radius of the shape detection roller. When coiling the jth layer strip, for the ith unit, the length of the strip from the mill exit to the coiler and the outermost layer strip of the strip coil lj (i) is expressed as [ R+Ro/I) . ] tanj3+alO - l;sinAo S cos RtanS+2n:R oj (i)
.
lj(I)=BR+
+ (21)
The average length of L, (i) across the strip width B, lj' is obtained n
'Z i, (i) l=··_·~--,-j--
The influence of the deflection of shape detection roller and the shape of the strip coil on the tensile strain of the strip Ss, (i) is obtained A
u€· J
(")_lj(i)-lj l
-
Simulation Calculation and Analysis
According to the measuring results of the shape measuring system of Hengshui Sheet Inc of Handan Iron and Steel Co, the simulation calculation and analysis are done. The width B and the thickness h of the strip are 890 and 2 rnrn , respectively, and the crown of the strip is 20 f.Lm. The calculation parameters are shown in Table 1. When the compensation for the' limbic measuring results is done, first of all, the positions of the strip edges are detected by the position sensor, and then, the actual width of the limbic measuring unit of the shape detection roller covered by the strip is calculated according to the strip width. Finally, the true tensile stresses of the strip edges are obtained according to Eqn. (3). Table 1
Calculation parameters
Outside radius of shape detection roller
mm
200
Inside radius of shape detection roller
135
Outside radius of coiler
300
Inside radius of coiler
158
(22)
n
J
2
i,
(23)
According to the planar deformation theory, the influence of the deflection of shape detection roller and the shape of the strip coil on the tensile stress of the strip Sa, (i) is obtained b.(jj(i)=l E 2b.€j(i)
-u
2, ... , n; j = l , 2, ... , m) (24) When coiling the j th layer strip, the compensation quantity caused by the shape detection roller deflection and the shape of the strip coil, i. e. , the tensile stress that should be subtracted from the shape measuring results, (jsj (y), can be expressed as (jsj(i) =b.(jj (i) (i=l, 2, ... , n; j=l, 2, ... , m) (i=l,
(25) When coiling the j th layer strip, a, and (jsj are subtracted from the shape measuring results 17m , and the true shape stress (j(i) can be expressed as
2. 1 Calculation of compensation quantity caused by transverse temperature difference of strip The transverse discrete values of t(y) are measured by the intelligent temperature apparatus. Then, they are fitted into the distribution curve according to Eqn. (4). The transverse temperature difference is about 10 'C, as shown in Fig. 4 (a). 17, is calculated and shown in Fig. 4 (b). It can be seen that the online shape stress curve appears to be a parabola, and at the center of the strip, the stress is the lowest. Thus, there is a trend that makes the strip form centre waves.
2. 2 Calculation of compensation quantity caused by deflection of shape detection roller Assuming that R Oj in Eqn, (21) is invariable, the compensation quantity caused by the shape detection roller deflection (jb is obtained. (jb is calculated under the shape conditions of flat, centre waves (transverse tensile stress difference of 21 MPa) and both edge waves (transverse tensile stress difference of 21 MPa) , and is shown in Fig. 5. It can be seen that whatever the shape state of the strip is, the ten-
Issue 6
Compensation Model for Shape Measuring of Cold Strip Rolling
sion stress measured by the shape meter all is low in the center and high in the both edges compared to the true tension stress, and the compensation quantitiesunder different conditions are nearly equal, that is, the size of the compensation quantity is not sensitive to the forms of the transverse distribution of the coiling tensile stress.
134 (a) 132 130
P
~
128 126 124
2. 3 Calculation of compensation quantity caused by shape of strip coil It is assumed that t. in Eqn. (21) is zero, and then the compensation quantity caused by the shape of the strip coil (jc is obtained. a., (j = 1, 2, "', m) is calculated under the shape conditions of flat, centre waves (transverse tensile stress difference of 21 MPa) and both edge waves (transverse tensile stress difference of 21 MPa), respectively. Fig. 6 shows the compensation quantity caused by the shape of the strip coil when coiling the l Oth , 20th, 30th, 40th, 50th, 60th, and 70th layers. It can be seen that when the shape of the strip is flat, the compensation quantity is very small and can be ignored. Under the shape conditions of centre waves and both edge waves, the compensation quantity is gradually increasing before coiling the 50th layer, and then, the compensation
20 15
10
~6'
5 0 -5
-10 -400
-200
0
200
400
B/mm
(a) Transverse distribution of strip temperature (t) (b) Compensation quantity caused by transverse temperature difference of strip (",,).
Fig. 4
• 25 •
;
Calculation of compensation quantity caused by transverse temperature difference of strip 0.25 0.15
~ ~
to
0.05 0 -0.05 -0.15 -400
o
-200
200
400 -400
-200
0
200
400 -400
-200
o
200
400
B/mm
(a) Flat;
Fig. 5
10
(a)
• 70th layer • 60th layer .. 50th layer ~ 40th layer • 30th layer • 20th layer • 10th layer
0.03 0.02
~0" -0.01 -0.02 -400
(b) Centre waves;
-200
0
(c) Both edge waves.
Compensation quantity caused by detection roller deflection Cab)
200
400
15
(b)
5
10
0
5
-5
0
-10
-5
-15 -400
-10 -200
0
200
400
-400
-200
B/mm
(a) Flat;
Fig.6
(b) Centre waves;
(c) Both edge waves.
Compensation quantity caused by shape of strip coil Co',)
0
200
400
26 •
quantity is gradually stabilized with the Increase In the coiling layers.
2. 4
General compensation quantity of shape measuring and CIsj are added, and the general compensation quantity of shape measuring CId is obtained, as shown in Fig. 7. It can be seen that when the shape of strip is flat, the general compensation quantity of shape measuring is mostly caused by the transverse temperature difference of the strip, and the compensation quantity caused by the shape detection roller CIt
20 (a)
~e
• • .. • • • •
10
0
-10 -400
70th layer 60th layer 50th layer 40th layer 30th layer 20th layer 10th layer
deflection and the compensation quantity caused by the shape of the strip coil are very small correspondingly and can be ignored. Under the shape conditions of centre waves and both edge waves, the general compensation quantity of shape measuring varies largely, which is gradually decreasing under the shape conditions of centre waves and which is gradually increasing under the shape conditions of both edge waves, before coiling the 50th layer. Sith , the general compensation quantity of shape measuring is gradually stabilized with increase of the coiling layers.
12
30
8
20
4
10
0
0
-3
-10
-6 -200
0
200
400
(a) Flat;
Fig.7
3
Vol. 17 .
Journal of Iron and Steel Research. International
-400
-200
0 B/mm
(b) Centre waves;
200
400
-20 -400
1) The size of the compensation quantity caused by the deflection of shape detection roller is not sensitive to the forms of the transverse distribution of the coiling tensile stress. which can be calculated according to one or several kinds of easy waves. 2) When the shape is flat, the compensation quantity caused by the shape of the strip coil is very small, which can be ignored; under the conditions of centre waves and both edge waves, the compensation quantity is gradually increased before coiling the 50th layer, and then. the compensation quantity is gradually stabilized with the increase in the coiling layers, there is no need to recalculate. 3) The true shape stress of the strip is obtained after subtracting general compensation quantity of shape measuring from the shape measuring results and ignoring the influence of the transverse temper-
0
200
400
(c) Both edge waves.
Synthetic compensation quantity of shape measuring
Conclusions
-200
(ad)
ature difference of the strip, the deflection of the shape detection roller, and the shape of the strip coil. References , [lJ
xu
[2J
LI Ru-jia. The Temperature Impacts and Its Compensation Measures During the Strip Flatness Measuring and Controlling [n. Heavy Machinery, 1995(5), 26 (in Chinese). JIA Chun-yu , SHANG Zhi-dong, Compensation of Plank Shape Target Curves for Cold-Rolling Strip Journal of Iron and Steel Research, 2000, 12(4), 64 (in Chinese). LIU Hong-min. Three-Dimensional Rolling Theory and Its Applications [MJ. Beijing, Science Press, 1999 (in Chinese). LIAN Jia-chuang, LIU Hong-min. Gauge and Shape Control [MJ. Beijing, Weapon Industry Press, 1996 (in Chinese). LIAN Iia-chuang , LI Jun-hong , SUN Ii-quan. Calculating Drum's Unit Pressure by the Iteration of Layer by Layer [n. Heavy Machinery, 2001(6), 30 (in Chinese).
[3J
Le-jiang. Flatness Control in Cold Strip Rolling and Mill Type Selection [MJ. Beijing, Metallurgical Industry Press, 2007 (in Chinese).
[n.
[4J [5J [6J
[7J
LIAN Iia-chuang, The Calculation for Unit Pressure at the Coilers Drum [n. Chinese Journal of Mechanical Engineering, 1980, 16(3): 78 (in Chinese).