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Camber Control System of Plate Rolling
Copyright © IFAC 10th T riennia l Wo rld Congress . Mun ICh. FRG. 198i
CAMBER CONTROL SYSTEM OF PLATE ROLLING Y. Tanaka*, K. Ohmori*, T. Miyake*, M. Inoue*, K. Nishizaki** and S. Tezuka*** .. *,\1 iwshiIlW ~rllfk\ . !\al/'a;aki Sll'ti COl j }(Jm lill ll . !\lIfll.lhiki. j a/m ll """ H I-Tech RfSNlrrh IlIsl il ll lr . !\ml'tl,\ aki SII'l'l Corp()w l iu lI . Ch iba, j a/HIII ***S ili('U1I D!'paflllll'lll . !\ml'tlsaki SI!'!'! COfporal ill ll . Uuwr!a-k ll. T IIIIYII. j apall
Abstract. Un~que measuring .and con~rol ~echniques of plate camber have been developed ana ut1llzed 1n the plate mlll of Mlzushlma Works. Features of the techniques are as follows: (1). Plate ~amber can be measured regardless of sideslipping or turning of a plate dUrlng rolhng. (2) Any nonuniform pl~te camber profile can be expressed accurately in the form of the n-th order polynomlal. (3) ~e measured camber is used to calculate the aimed wedge, to which the actual wedge lS fed back to control system, thereby reducing the plate camber in the next pass. A new control system, which . ado~ted the a~ve-ment~o~ techniques, has been succ~ssfully used and con~r1butlng to an lncrease ln Yleld and a reduction in poor gradlng and extra processlng of plates in the mill. Keywords.
Rolling mills; Computer control; PID control; Sensors; Process control.
INRTODOCTIOO
incorporated into the plate rolling process, and has been contributing to yield i~rovement, reduction in poor grading and eff1ciency increase in the shearing process. In this paper, we mens ioned the camber measuring and cotrolling methods in the plate mill as well as their effectiveness to a commercial plate rolling.
Control of plate camber (edgewise curvature) is one of the IIPst i~rotant technical ~roblems to be solved in the fleld of steel rolllng. Plate camber results in lowering of yield, and causes poor grading and lower efficiency in the shearing process. In the past, a method of preventing happenings of camber during rolling was such that a rolling cperator adjusted the difference of roll gaps bet~en the right side and the left! when he notlCed the presence of camber on hlS visual in~pectioI,l. As it depended on his experience and sklll, th1S method fell short of correcting plate camber completely. Many studies (Hayashi, 1977; CTYKa'l, 1978; Nakaj ima, 198Oa) have been made in recent years to find out the camber generation mechanism in the rolling and further to dvelop the camber control techniques. The methods of preventing the camber are classified into two kinds. One is meandering control (Nakajima, 1980b, 1981). This is a method of suppressin~ camber by pre~en~ing the occurrence of sideslipplng (sllpplng of the plate-roll contacting position in rolling) as the cause of camber. The defect in this method lies in the impossibility of correcting the camber which has alrady been generated. The other is a method in which the camber is measured during rolling and it is corrected in the subsequent rolling pass.
CAMBER MEASUREMENI'
In. camber measurement, the following problems eXlsted. (1) Measuring must be made as closely as possible to the mill in order not to lower rolling efficiency. (2) Measuring must be free from the influence of sideslippin~ of the plate during rolling. (3) Measurlng must express faithfully the camber curvature distribution in the longitudinal direction, because this camber curvature is not always uniform in !he whole plate length. (4) Measured values should be transformed to a simple algebraic equation so that it can be immediately used for the subsequent control. The three devices to measure the central ~sition in the width direction of the plate (herelnafter called the "off-center meters") are installed in the le~th direction of the rolling line, and by expresslng the camber profile of the ~late by the n-th order polynomial, the above-mentlOned problems have been solved. Details of this procedure are described below.
At the plate Mill of Mizushima Works, we have been studying the camber control us ing the latter method. As a result, we have developed unique methods for camber measuring and controlling. This camber control system has already been AC
II- F
Principle of Measurement The.princi~le
of measurement is explained on the baS1S of Flg. 1. It shows that the x-axis and yaxis correspond to the lenght and width
14 7
Y. Tanaka directions of the plate, respectively, and that off-center meters are installed each at locations Xa , Xb and Xc' Distance between Xa and Xc and between Xa and Xb are denoted by L1 and L2, respectively. . Now it is supposed that the camber prof1le of the plate can be expressed by the following n-th order polynomioal,
y
= F(x) =
aa
+ a1x + .... + anXn
Durin~
rolling, ,this plate pro~resse~ in,the positive direction of the x-ax1s, wh1le 1t may sideslip and rotate. To simplify the , explanation/ it is assumed here that there 1S no such sideshpping and rotation. Assume that the position of a plate measured by three off-center meters is expressed by three points A, B and C, and that a straight line (l) passing A and B crosses another straight line X = Xc at D. Then, the distance M between C and D is expressed as follows,
where y = H(x) is an equation of straight line (I), which is expressed as follows.
1'1 al.
Then two measured off-center values F(x1) and F(x2) and a3, ••• , an, which have been already derlved, are substituted into eq. (6), and the simultaneous equations are solved to obtain aa, and al.
az,
Now, in the explanation made heretofore, it was
assumed that there was no sideslipping and rotation of the plate, but in the actual rol~ing operation, such movements may occur. Accord 109 to the present method, however, their adverse effect can be disregarded. The reasons for this are explained below. First, no adverse effect is taken by the sideslipping of the plate, because the value of M will not change, even if y = F(x) makes parallel movement in the y-axis direction. Next/ the effect of rotation of the steel plat~ is m1nimal and can be neglected. Actual rotat1on angle 8f the plate on the rolling table amounts to 0.3 at the most, and such a magnitude of rotation will only bring about changes in the M value by mere 1 mm maximum. Conse9uently, there is no possibility that the rotat1on of the plate will exercise an adverse effect on the estimation of the coefficient of y = F(x). We confirmed separately by simulation that the effect of rotation is negligible. Configuration of the Equipment
As the plate moves in the positive direction of the x-axis, the value of M changes. If the value of M is denoted by G(s), curved line m = G(s~ will express changes of Mbrought by the mot1on of the plate, where's' is the motion distance from the ~resent position. When eqs. (1) and (3) are Subst1tuted into eq. (2), and further, variables xa, Xb and Xc are altered into xa - ~, Xb - s and x - s, G(S) will be obtained, and 1t becomes the 10110wing (n-2)th order polynomial, G(s) =
ho
+ b1S + ••••• + bn-2Sn-2
(4)
where coefficients ho, b1, ••• , bn-2 are expressed as the linear combination of coefficients a2, a3, ••• ,an' Namely, n
bi =f=Cij(xa,L1,L2) · aj J=2
(i=O,l, .. ,n-2) (5) Measurement Result
In the above equation, Cij is a function of xa , L1 and L2' From the above, it is possible to obtain the polynomial y = F(x), which denotes the camber profile, by the following procedure: (1) M is observed, which changes as the plate moves. Then, changes in M are approximated by the (n-2)th order polynomial as shown in eq. (4), and 1:>0, b1, ... , bn-2.are obtained. . (2) Slmultaneous equal10ns (5) are solved 1n order to obtain a2, a3, ••• , an' (3) As the shape of the plate is expressed by eq. (1), the following two equations become valid at two arbitrary measuring points Xl and x2'
aa
+
a1 x1
+
The arrangement of instruments for the camber meter (camber measuring equipment) is shown in Fig. 2. The instruments consist of three offcenter meters. In Fig. 3, the system configuration of the camber meter is shown. The central position in the width direction of the plate is measured by three off-center meters, and on the basis of the measurements, distance M is calculated by an upper-rank microcCXll>uer. These distance values are transmitted to the process computer~ where first they are approximated to the (n-2)th order polynominal, and then coefficients aa, aI, •• • , an of the n-th order polynomial y = F(x), which expresses pla~e camber are calculated. The camber prohle thus determined is fully used in camber control and displayed in patterns on the CRT screen as guidance to the operator.
n . Eaj X1J j=2
An example of measurement result by the camber meter is shown in Fig. 4. In Fig. 4 (a), the measured value of distance M and its approximated curve are shown; in Fig. (b), the measured and calculated values of the camber profile are shown. In these figures/ M and the camber profile have been approximated by the 4-th and 6th order ~lynomials, res~ctively. ThrOl.!gh the use of thiS camber measurmg method/ bendmg shapes having various curvatures whiCh are generated during rolling can be faithfully expressed. A histogram of measurement errors with the cClIlber meter is shown in Fi~. 5, where errors are evaluated by converting them into an error of camber values (Ca in the figure) per 15 m of plate length. This camber meter has attained the accuracy as indicated by an average error value of 0.79 mm and a standard deviation of 2.1 mm.
(6) CAMBER CCNfROL
In plate rolling, givin~ the wedge (difference in thickness between the rlght and left edges of the
Camber Control System of Plate Rolling
plate) produce the difference in elongation between the right and left edges of tne plate, and so plate camber ha{lpenes. This {ilenanenon can conversely be applled to correctlng the camber. First, from the measurement result of camber, the next-psss aimed wedge is calculated which lS necessary for correcting the camber. Later, rolling control is effectd in the next pass, so that the aimed wedge can be satisfied. In the following, models necessary for camber control and the detailed control method will be described. Models for Camber Control To controll camber, the following two models have been prepared: (1) Model showing the relation between the camber, which occurs as a result of rolling, and the wedge of the plate: This model is used for calculating, from the camber {lrofile of the plate, the aimed wedge which lS necessary for correcting the camber. (2) Model for observing the wed~e during rolling: This model is used for controlllng the wedge according to the target. The above models are described in detail below. Relation forrrula between camber and wedge. When no conslderatlon lS ~lven to the threedimensional deformatlon such as the widthdirectional metal flow during rolling, the relation between the camber and the wedge is inevitably determined by the difference in elongation between the right and left edges of the rolled material, and can be expressed by the following equation,
PI = 2. (PO
E
+
!1
h
hdf
l'!)
Pw
B
= (- - - ) Kn
Kw
+
B
+
10
Sdf ~
la
4Cwsin (~ Tt) sin(~ Tt) W
hdf
=
Cl.1 Pdf + Cl.2Sdf + 0.30 + Ci.4
(9)
where Pdf is rolling load difference (work-side load - drive-side l,?ad), aoo Ci.l to a4 . a~e coefficients determlned by rolllng Coooltlon. Camber Control Method This section concretely describes the method of controlling the camber, after it has been measured. First, the measured camber y = F(x), whi~h h~s been described by the n-th order polynomlal lS conve~ted into curvature to by the following equatlOn, F" (x) (10)
p(x) =
(8)
?Yl
where Pw is work-side rolling load, l'!) is driveside rolling load, Kw is work-side mill constant, Kn is drive-side mill constant, Sdf is d~fference between right and left roll gaps (work-sld~ gap drive-side gap), ~ is work roll crown, W lS roll barrel length, 10 lS distance between roll chocks, and <5 is side slipping value (deviation of the central position in the width direction of the ?late from the central position of the rol11ng mill. It is also called the "off-center value". Deviation towards the work side is taken as "Positive".). Actually, the effects of roll deflection, roll flattening, etc., are exercised. Therefore, the authors performed simulation by a sp~it model. (Shoet, 1968) and prepared ~ correctlon equatl0t:l. On the basis of the simulation results, the basic model equation (8) is corrected and the following equation (9) has been obtained,
H
A~
149
hdf h
H
where PI is delivery cl:'rvatur~, Po ~s entry . curvature, lis elongatlOn ratlO, 11Cf; lS changes m wedge ratio, B is plate ~idth, h is d~livery' thickness, H is entry thlckness, hdf lS dell very wedge (work-side plate thickness - drive-side plate thickness), and ~f is entry wedge (workside plate thickness - drive-side plate thickness). Wedge observation model. This model is used for observlng the wedge dur10g rolling. Since this model is used in the dynamic control system, calculation speed is required, and so it is important to simplify the model fo~la. It:l forming this model, careful attentlon was glven to maintaining accuracy and also simplifying the model equation. The deformation condition of rolling mill is shown in Fig. 6. A basic model as shown below is obtained from the kinetic relation between the rolled material and the mill,
where F'(x) = dF(x)/dx, F' '(x) = dZy(x)/dx 2 . Eq. (10) expresses a curvature at an arbitrary point x in the length direction of the plate: Next, the aimed wedge hdf*(x) of the succeedlng pass is calculated to correct the camber of curvature to ' hdf*(x)
~f(x)
= ( -- -
Bp(x) ) h
(11)
H
where hdf*(x) is· aimed wedege at positi,?n x in the length direction of the plate, ~f l~ current wedge at position x in the le~th direction of the plate, and h is delivery-side aimed plate thickness. . Eg. (11) can be obtained by equating the de11very slde camber to1 to 0 in eq. (7). In the actual control {lrocess, the plate is sub-divided in ~he length dlrection and aimed wedges of all sectlons are calculated. When the succeeding pass begins! the feedback control system shown in Fig. 7 lS constructed and wedge control is ~ffected to ge~rate t~e aimed wedge. At this tlme, the equatlOn (9) lS used for observ ing the wedge. The scre~own position is adjusted depeooing upon the deviation from the aimed wedge. If the wedge is controlled accord~ng to t~e aimed wedge, the plat~ after the pass wlll acqulre a camber less proflle.
Y. Tallaka
APflication of Camber Control to Ccmrercial Ra hng Actual ~rocess of camber control. Camber is measure ln the dlrectlon at arrow (A) in Fig. 2, and controlled in the direction of arrow (B). After camber measurement, the aimed wedge is calculated by the process computer on the basis of eqs. (10) and (11), and the result is transmitted to the lower-rank microcomputer. When the control pass begins, the wedge control shown in Fig. 7 is executed. . Instruction value Sdf* of the roll gap dlfference between right and left hand side and aimed wedge hdf* are provided with limit values to prevent abnormal control. Also for the controller, PI (proportion and integral) control function used. At present, the plate mill has no sideslip detecting sensor and the sideslipping value cannot be measured. Therefore, an additional model is formulated which expresses the rolling load difference caused by sideslippin~, and a wedge observation model is used in whlch the term of the sideslipping value is eliminated from eq.
1'10/.
Nakajima, K., T. Kajiwara, T. Kikuma, T. Kimura, H. Matsurroto! and M. Tagawa (1980a). Study of sideslipplng control method in hot strip rolling. The Proc. 1980 Japanese Stri~ Con£. TeChnology ot Plastlc~, 6 6~ Naka]lma, K., T. Ka]lWara, T. Kl , T. Kimura, H. Matsurroto, M. Tagawa, I. Masuda, and K. Yoshirroto (1980b). Study of sideslipping control method in hot strip rolling. The Proc. 31th Japanese ~oint Con£. TechnOTOgy ot Plastlclty., 471 4 4. Naka]lma, k., T. Kajiwara, T. Kikuma, T. Kimura, H. Matsurroto, and M. Tagawa, Y. Shirai, K. Yoshirroto (1981). Study of sideslipping control method in hot strip rolling. The Proc. 1981 Japanese S/)ring Con£. TechnoIogy ot plastlc lty., 147 15 . Akuta, T., arid H. Takahashi (1981). The Proc. 20th Japanese SICE Science Conf., 413=414. Shoet, K. N., ana N. A. Townsena (968). JISI., Q, 1088.
(9) •
Control results. An example of control effected on a IS-mm-thlck plate is shown in Fig. 8: The camber which amounted to 42.5 mm at the tlme of measurement has been corrected down to 12 mm by camber control. It clearly indicates the effectiveness of the camber control. Effectiveness of the camber control execution in the commercial rolling process is shown in Fig. 9. This is the rolling result at the same roll chance. When the camber control is utilized! dispersion has been reduced to half, indicatlng the effe~tiveness of the camber control. Camber has been suppressed within 5 mm per 15 mm of the plate length.
y
-'- -_~.
L -_ _
_
____'''
.r. : P ositioll o f :\n. 1 (l/f .n .:
Hayashi, C., and T. Kohno (1977). Analysis of camber growth mechanism in hot strip mills. Tetsu to Hagane., l, 21-24. A. r.
( 1978)
3,
58~60.
0
JlPyr
(Jlf
L't' n ll'!
Fig.
tlwlL'r
( l 'llkr 1l1l' IL' r ll,dl' l
Principle of Measurement
Off -ce ll te r mete rs L\)
Pl a te-
(rO\\" 11
me ter
Fig. 2 Arrangement of Instrument for
REffiREtU:S
HaKJlOHeHllblX
P o~;tlOll Di ~\, _ :!
( ,- : l' u~i l jull ot r\(J:~ off ·\.\'JllL'!
We have developed unique camber measuring and controlling methods and realized their applications to commercial operations. The outstand ing features of the camber meter and the camber control are summarized below. (1) Camber can be measured at a location very near to the mill site without being adversely affected by sideslipping and turning of the plate during rolling. (2) Through expressing the plate camber by means of the Il""th order polynomial, the actual camber profile can be faithfully expressed and the measured camber profile can be immediately reflected on the control. (3) From the measured result of the camber, the aimed wedge is calculated, and the wedge control is performed at the succeeding pass, thereby correcting the camber. The above-rrentioned system is now srroothly operating, and is eff~cti~e in redu:ing the . camber, thereby contrlbutlng to an lmprovement ln yield, reduction in poor grades and enhancement of efficiency in the shearing process.
'E'·~,'~' r.
CONCWSIOO
CTYKa'l
_ __
npoKaTKe .'HCTOB B BaJlKax,
I(
JlPyry,
Tsvetllye Metally .,
Camber Heasurement and Control
Camher C()llll"Ol S\'Sll'm ()f Plall' R()lIillg
~IE!.I'J.i\C
I SI
:;:;0
Fig. 3 System configuration for camber measurement and control
-20
E E ..,..
-28
<:
C -36 ~
~
>
-44
..
12 16 Sampling point of M (a ) Locus of J[
.-
60 ~ 0.
s,"
• :'leasured - Approximated curve q·th order pol ynomial)
20
•...\.CLl::i ; GU710er p:-o{i ie - \leasL:re-: ?f0!: lc -.\":~:: C:::i:;.:;e:- :7:e:~: \ 6·~h '':'f Ce:- ;: c·l:. ~0;:::a;1
r.
._ _ _~J/I\\
- j"
III ~1I1id
lint' Ilo ·1cl:tdcd
Il, uhll li ll£ loaded
Fig. 4 An example of camber profile measurement
Fig. 6 Wedge model on the assumption that rolls are rigid
.
~~~l --1 . ·/~ ~ I ' - -1~ '1]---'~ ' :'{' it
"'_Cl ~J -",~~~-;;If "'" -'."1 2
l
__ . __ Error or C,l pe:- 1.):n pla te !c:lg th
Fig. 5 Histogram of measurement error wi th camber meter
lF~~"'"
~=:fiJ [5, I ~,.t l g; IJ '
""",,,.1
("PIlI'llll'l!!
~------' ~_ _ _ _ _...J
- - -- - - -
Fig. 7 Block diagram of camber control
Y. T ana ka et al.
152
50r-~~-:-:-:----:-----------
(a) Be(ore control
u
Co",'" - "',,""
J ]
E E
~-5U
:~
50 r - 7 7")-:A-:(7"te-,.-c-o-n-:-tr-o-:-I-----------, 1b
.,.o
CitlHlwr - 12 111111
O~~==~~~~~=-~-- I -50~O-----L------;2~U-----L-----:-4LU· ~----- ---
Plate length
(,11)
Fig. 8 An example of cani>er control
~
III
:::
1
\IS
~o o
Ca
I
I
o Nllt cc Illtrollefl • CUlltru lied
1t{.r--. 0 0 J~~' ---· ....
11111 - - - - - - - - -
± •• ••
0
o~-t-;_· o
- 11111 - - - - - - - - -
III
,;'
••
r. "e
·0
oo-to-.. . . . . . 0,>,
JU
40
l 'l;. l e lell)(lh (Ill)
Fig. 9 Effect of camber control
c .,11
CAMBER CONTROL SYSTEM OF PLATE ROLLING Y. Tanaka*, K. Ohmori*, T. Miyake*, M. Inoue*, K. Nishizaki** and S. Tezuka*** .. *,\1 iwshiIlW ~rllfk\ . !\al/'a;aki Sll'ti COl j }(Jm lill ll . !\lIfll.lhiki. j a/m ll """ H I-Tech RfSNlrrh IlIsl il ll lr . !\ml'tl,\ aki SII'l'l Corp()w l iu lI . Ch iba, j a/HIII ***S ili('U1I D!'paflllll'lll . !\ml'tlsaki SI!'!'! COfporal ill ll . Uuwr!a-k ll. T IIIIYII. j apall
Abstract. Un~que measuring .and con~rol ~echniques of plate camber have been developed ana ut1llzed 1n the plate mlll of Mlzushlma Works. Features of the techniques are as follows: (1). Plate ~amber can be measured regardless of sideslipping or turning of a plate dUrlng rolhng. (2) Any nonuniform pl~te camber profile can be expressed accurately in the form of the n-th order polynomlal. (3) ~e measured camber is used to calculate the aimed wedge, to which the actual wedge lS fed back to control system, thereby reducing the plate camber in the next pass. A new control system, which . ado~ted the a~ve-ment~o~ techniques, has been succ~ssfully used and con~r1butlng to an lncrease ln Yleld and a reduction in poor gradlng and extra processlng of plates in the mill. Keywords.
Rolling mills; Computer control; PID control; Sensors; Process control.
INRTODOCTIOO
incorporated into the plate rolling process, and has been contributing to yield i~rovement, reduction in poor grading and eff1ciency increase in the shearing process. In this paper, we mens ioned the camber measuring and cotrolling methods in the plate mill as well as their effectiveness to a commercial plate rolling.
Control of plate camber (edgewise curvature) is one of the IIPst i~rotant technical ~roblems to be solved in the fleld of steel rolllng. Plate camber results in lowering of yield, and causes poor grading and lower efficiency in the shearing process. In the past, a method of preventing happenings of camber during rolling was such that a rolling cperator adjusted the difference of roll gaps bet~en the right side and the left! when he notlCed the presence of camber on hlS visual in~pectioI,l. As it depended on his experience and sklll, th1S method fell short of correcting plate camber completely. Many studies (Hayashi, 1977; CTYKa'l, 1978; Nakaj ima, 198Oa) have been made in recent years to find out the camber generation mechanism in the rolling and further to dvelop the camber control techniques. The methods of preventing the camber are classified into two kinds. One is meandering control (Nakajima, 1980b, 1981). This is a method of suppressin~ camber by pre~en~ing the occurrence of sideslipplng (sllpplng of the plate-roll contacting position in rolling) as the cause of camber. The defect in this method lies in the impossibility of correcting the camber which has alrady been generated. The other is a method in which the camber is measured during rolling and it is corrected in the subsequent rolling pass.
CAMBER MEASUREMENI'
In. camber measurement, the following problems eXlsted. (1) Measuring must be made as closely as possible to the mill in order not to lower rolling efficiency. (2) Measuring must be free from the influence of sideslippin~ of the plate during rolling. (3) Measurlng must express faithfully the camber curvature distribution in the longitudinal direction, because this camber curvature is not always uniform in !he whole plate length. (4) Measured values should be transformed to a simple algebraic equation so that it can be immediately used for the subsequent control. The three devices to measure the central ~sition in the width direction of the plate (herelnafter called the "off-center meters") are installed in the le~th direction of the rolling line, and by expresslng the camber profile of the ~late by the n-th order polynomial, the above-mentlOned problems have been solved. Details of this procedure are described below.
At the plate Mill of Mizushima Works, we have been studying the camber control us ing the latter method. As a result, we have developed unique methods for camber measuring and controlling. This camber control system has already been AC
II- F
Principle of Measurement The.princi~le
of measurement is explained on the baS1S of Flg. 1. It shows that the x-axis and yaxis correspond to the lenght and width
14 7
Y. Tanaka directions of the plate, respectively, and that off-center meters are installed each at locations Xa , Xb and Xc' Distance between Xa and Xc and between Xa and Xb are denoted by L1 and L2, respectively. . Now it is supposed that the camber prof1le of the plate can be expressed by the following n-th order polynomioal,
y
= F(x) =
aa
+ a1x + .... + anXn
Durin~
rolling, ,this plate pro~resse~ in,the positive direction of the x-ax1s, wh1le 1t may sideslip and rotate. To simplify the , explanation/ it is assumed here that there 1S no such sideshpping and rotation. Assume that the position of a plate measured by three off-center meters is expressed by three points A, B and C, and that a straight line (l) passing A and B crosses another straight line X = Xc at D. Then, the distance M between C and D is expressed as follows,
where y = H(x) is an equation of straight line (I), which is expressed as follows.
1'1 al.
Then two measured off-center values F(x1) and F(x2) and a3, ••• , an, which have been already derlved, are substituted into eq. (6), and the simultaneous equations are solved to obtain aa, and al.
az,
Now, in the explanation made heretofore, it was
assumed that there was no sideslipping and rotation of the plate, but in the actual rol~ing operation, such movements may occur. Accord 109 to the present method, however, their adverse effect can be disregarded. The reasons for this are explained below. First, no adverse effect is taken by the sideslipping of the plate, because the value of M will not change, even if y = F(x) makes parallel movement in the y-axis direction. Next/ the effect of rotation of the steel plat~ is m1nimal and can be neglected. Actual rotat1on angle 8f the plate on the rolling table amounts to 0.3 at the most, and such a magnitude of rotation will only bring about changes in the M value by mere 1 mm maximum. Conse9uently, there is no possibility that the rotat1on of the plate will exercise an adverse effect on the estimation of the coefficient of y = F(x). We confirmed separately by simulation that the effect of rotation is negligible. Configuration of the Equipment
As the plate moves in the positive direction of the x-axis, the value of M changes. If the value of M is denoted by G(s), curved line m = G(s~ will express changes of Mbrought by the mot1on of the plate, where's' is the motion distance from the ~resent position. When eqs. (1) and (3) are Subst1tuted into eq. (2), and further, variables xa, Xb and Xc are altered into xa - ~, Xb - s and x - s, G(S) will be obtained, and 1t becomes the 10110wing (n-2)th order polynomial, G(s) =
ho
+ b1S + ••••• + bn-2Sn-2
(4)
where coefficients ho, b1, ••• , bn-2 are expressed as the linear combination of coefficients a2, a3, ••• ,an' Namely, n
bi =f=Cij(xa,L1,L2) · aj J=2
(i=O,l, .. ,n-2) (5) Measurement Result
In the above equation, Cij is a function of xa , L1 and L2' From the above, it is possible to obtain the polynomial y = F(x), which denotes the camber profile, by the following procedure: (1) M is observed, which changes as the plate moves. Then, changes in M are approximated by the (n-2)th order polynomial as shown in eq. (4), and 1:>0, b1, ... , bn-2.are obtained. . (2) Slmultaneous equal10ns (5) are solved 1n order to obtain a2, a3, ••• , an' (3) As the shape of the plate is expressed by eq. (1), the following two equations become valid at two arbitrary measuring points Xl and x2'
aa
+
a1 x1
+
The arrangement of instruments for the camber meter (camber measuring equipment) is shown in Fig. 2. The instruments consist of three offcenter meters. In Fig. 3, the system configuration of the camber meter is shown. The central position in the width direction of the plate is measured by three off-center meters, and on the basis of the measurements, distance M is calculated by an upper-rank microcCXll>uer. These distance values are transmitted to the process computer~ where first they are approximated to the (n-2)th order polynominal, and then coefficients aa, aI, •• • , an of the n-th order polynomial y = F(x), which expresses pla~e camber are calculated. The camber prohle thus determined is fully used in camber control and displayed in patterns on the CRT screen as guidance to the operator.
n . Eaj X1J j=2
An example of measurement result by the camber meter is shown in Fig. 4. In Fig. 4 (a), the measured value of distance M and its approximated curve are shown; in Fig. (b), the measured and calculated values of the camber profile are shown. In these figures/ M and the camber profile have been approximated by the 4-th and 6th order ~lynomials, res~ctively. ThrOl.!gh the use of thiS camber measurmg method/ bendmg shapes having various curvatures whiCh are generated during rolling can be faithfully expressed. A histogram of measurement errors with the cClIlber meter is shown in Fi~. 5, where errors are evaluated by converting them into an error of camber values (Ca in the figure) per 15 m of plate length. This camber meter has attained the accuracy as indicated by an average error value of 0.79 mm and a standard deviation of 2.1 mm.
(6) CAMBER CCNfROL
In plate rolling, givin~ the wedge (difference in thickness between the rlght and left edges of the
Camber Control System of Plate Rolling
plate) produce the difference in elongation between the right and left edges of tne plate, and so plate camber ha{lpenes. This {ilenanenon can conversely be applled to correctlng the camber. First, from the measurement result of camber, the next-psss aimed wedge is calculated which lS necessary for correcting the camber. Later, rolling control is effectd in the next pass, so that the aimed wedge can be satisfied. In the following, models necessary for camber control and the detailed control method will be described. Models for Camber Control To controll camber, the following two models have been prepared: (1) Model showing the relation between the camber, which occurs as a result of rolling, and the wedge of the plate: This model is used for calculating, from the camber {lrofile of the plate, the aimed wedge which lS necessary for correcting the camber. (2) Model for observing the wed~e during rolling: This model is used for controlllng the wedge according to the target. The above models are described in detail below. Relation forrrula between camber and wedge. When no conslderatlon lS ~lven to the threedimensional deformatlon such as the widthdirectional metal flow during rolling, the relation between the camber and the wedge is inevitably determined by the difference in elongation between the right and left edges of the rolled material, and can be expressed by the following equation,
PI = 2. (PO
E
+
!1
h
hdf
l'!)
Pw
B
= (- - - ) Kn
Kw
+
B
+
10
Sdf ~
la
4Cwsin (~ Tt) sin(~ Tt) W
hdf
=
Cl.1 Pdf + Cl.2Sdf + 0.30 + Ci.4
(9)
where Pdf is rolling load difference (work-side load - drive-side l,?ad), aoo Ci.l to a4 . a~e coefficients determlned by rolllng Coooltlon. Camber Control Method This section concretely describes the method of controlling the camber, after it has been measured. First, the measured camber y = F(x), whi~h h~s been described by the n-th order polynomlal lS conve~ted into curvature to by the following equatlOn, F" (x) (10)
p(x) =
(8)
?Yl
where Pw is work-side rolling load, l'!) is driveside rolling load, Kw is work-side mill constant, Kn is drive-side mill constant, Sdf is d~fference between right and left roll gaps (work-sld~ gap drive-side gap), ~ is work roll crown, W lS roll barrel length, 10 lS distance between roll chocks, and <5 is side slipping value (deviation of the central position in the width direction of the ?late from the central position of the rol11ng mill. It is also called the "off-center value". Deviation towards the work side is taken as "Positive".). Actually, the effects of roll deflection, roll flattening, etc., are exercised. Therefore, the authors performed simulation by a sp~it model. (Shoet, 1968) and prepared ~ correctlon equatl0t:l. On the basis of the simulation results, the basic model equation (8) is corrected and the following equation (9) has been obtained,
H
A~
149
hdf h
H
where PI is delivery cl:'rvatur~, Po ~s entry . curvature, lis elongatlOn ratlO, 11Cf; lS changes m wedge ratio, B is plate ~idth, h is d~livery' thickness, H is entry thlckness, hdf lS dell very wedge (work-side plate thickness - drive-side plate thickness), and ~f is entry wedge (workside plate thickness - drive-side plate thickness). Wedge observation model. This model is used for observlng the wedge dur10g rolling. Since this model is used in the dynamic control system, calculation speed is required, and so it is important to simplify the model fo~la. It:l forming this model, careful attentlon was glven to maintaining accuracy and also simplifying the model equation. The deformation condition of rolling mill is shown in Fig. 6. A basic model as shown below is obtained from the kinetic relation between the rolled material and the mill,
where F'(x) = dF(x)/dx, F' '(x) = dZy(x)/dx 2 . Eq. (10) expresses a curvature at an arbitrary point x in the length direction of the plate: Next, the aimed wedge hdf*(x) of the succeedlng pass is calculated to correct the camber of curvature to ' hdf*(x)
~f(x)
= ( -- -
Bp(x) ) h
(11)
H
where hdf*(x) is· aimed wedege at positi,?n x in the length direction of the plate, ~f l~ current wedge at position x in the le~th direction of the plate, and h is delivery-side aimed plate thickness. . Eg. (11) can be obtained by equating the de11very slde camber to1 to 0 in eq. (7). In the actual control {lrocess, the plate is sub-divided in ~he length dlrection and aimed wedges of all sectlons are calculated. When the succeeding pass begins! the feedback control system shown in Fig. 7 lS constructed and wedge control is ~ffected to ge~rate t~e aimed wedge. At this tlme, the equatlOn (9) lS used for observ ing the wedge. The scre~own position is adjusted depeooing upon the deviation from the aimed wedge. If the wedge is controlled accord~ng to t~e aimed wedge, the plat~ after the pass wlll acqulre a camber less proflle.
Y. Tallaka
APflication of Camber Control to Ccmrercial Ra hng Actual ~rocess of camber control. Camber is measure ln the dlrectlon at arrow (A) in Fig. 2, and controlled in the direction of arrow (B). After camber measurement, the aimed wedge is calculated by the process computer on the basis of eqs. (10) and (11), and the result is transmitted to the lower-rank microcomputer. When the control pass begins, the wedge control shown in Fig. 7 is executed. . Instruction value Sdf* of the roll gap dlfference between right and left hand side and aimed wedge hdf* are provided with limit values to prevent abnormal control. Also for the controller, PI (proportion and integral) control function used. At present, the plate mill has no sideslip detecting sensor and the sideslipping value cannot be measured. Therefore, an additional model is formulated which expresses the rolling load difference caused by sideslippin~, and a wedge observation model is used in whlch the term of the sideslipping value is eliminated from eq.
1'10/.
Nakajima, K., T. Kajiwara, T. Kikuma, T. Kimura, H. Matsurroto! and M. Tagawa (1980a). Study of sideslipplng control method in hot strip rolling. The Proc. 1980 Japanese Stri~ Con£. TeChnology ot Plastlc~, 6 6~ Naka]lma, K., T. Ka]lWara, T. Kl , T. Kimura, H. Matsurroto, M. Tagawa, I. Masuda, and K. Yoshirroto (1980b). Study of sideslipping control method in hot strip rolling. The Proc. 31th Japanese ~oint Con£. TechnOTOgy ot Plastlclty., 471 4 4. Naka]lma, k., T. Kajiwara, T. Kikuma, T. Kimura, H. Matsurroto, and M. Tagawa, Y. Shirai, K. Yoshirroto (1981). Study of sideslipping control method in hot strip rolling. The Proc. 1981 Japanese S/)ring Con£. TechnoIogy ot plastlc lty., 147 15 . Akuta, T., arid H. Takahashi (1981). The Proc. 20th Japanese SICE Science Conf., 413=414. Shoet, K. N., ana N. A. Townsena (968). JISI., Q, 1088.
(9) •
Control results. An example of control effected on a IS-mm-thlck plate is shown in Fig. 8: The camber which amounted to 42.5 mm at the tlme of measurement has been corrected down to 12 mm by camber control. It clearly indicates the effectiveness of the camber control. Effectiveness of the camber control execution in the commercial rolling process is shown in Fig. 9. This is the rolling result at the same roll chance. When the camber control is utilized! dispersion has been reduced to half, indicatlng the effe~tiveness of the camber control. Camber has been suppressed within 5 mm per 15 mm of the plate length.
y
-'- -_~.
L -_ _
_
____'''
.r. : P ositioll o f :\n. 1 (l/f .n .:
Hayashi, C., and T. Kohno (1977). Analysis of camber growth mechanism in hot strip mills. Tetsu to Hagane., l, 21-24. A. r.
( 1978)
3,
58~60.
0
JlPyr
(Jlf
L't' n ll'!
Fig.
tlwlL'r
( l 'llkr 1l1l' IL' r ll,dl' l
Principle of Measurement
Off -ce ll te r mete rs L\)
Pl a te-
(rO\\" 11
me ter
Fig. 2 Arrangement of Instrument for
REffiREtU:S
HaKJlOHeHllblX
P o~;tlOll Di ~\, _ :!
( ,- : l' u~i l jull ot r\(J:~ off ·\.\'JllL'!
We have developed unique camber measuring and controlling methods and realized their applications to commercial operations. The outstand ing features of the camber meter and the camber control are summarized below. (1) Camber can be measured at a location very near to the mill site without being adversely affected by sideslipping and turning of the plate during rolling. (2) Through expressing the plate camber by means of the Il""th order polynomial, the actual camber profile can be faithfully expressed and the measured camber profile can be immediately reflected on the control. (3) From the measured result of the camber, the aimed wedge is calculated, and the wedge control is performed at the succeeding pass, thereby correcting the camber. The above-rrentioned system is now srroothly operating, and is eff~cti~e in redu:ing the . camber, thereby contrlbutlng to an lmprovement ln yield, reduction in poor grades and enhancement of efficiency in the shearing process.
'E'·~,'~' r.
CONCWSIOO
CTYKa'l
_ __
npoKaTKe .'HCTOB B BaJlKax,
I(
JlPyry,
Tsvetllye Metally .,
Camber Heasurement and Control
Camher C()llll"Ol S\'Sll'm ()f Plall' R()lIillg
~IE!.I'J.i\C
I SI
:;:;0
Fig. 3 System configuration for camber measurement and control
-20
E E ..,..
-28
<:
C -36 ~
~
>
-44
..
12 16 Sampling point of M (a ) Locus of J[
.-
60 ~ 0.
s,"
• :'leasured - Approximated curve q·th order pol ynomial)
20
•...\.CLl::i ; GU710er p:-o{i ie - \leasL:re-: ?f0!: lc -.\":~:: C:::i:;.:;e:- :7:e:~: \ 6·~h '':'f Ce:- ;: c·l:. ~0;:::a;1
r.
._ _ _~J/I\\
- j"
III ~1I1id
lint' Ilo ·1cl:tdcd
Il, uhll li ll£ loaded
Fig. 4 An example of camber profile measurement
Fig. 6 Wedge model on the assumption that rolls are rigid
.
~~~l --1 . ·/~ ~ I ' - -1~ '1]---'~ ' :'{' it
"'_Cl ~J -",~~~-;;If "'" -'."1 2
l
__ . __ Error or C,l pe:- 1.):n pla te !c:lg th
Fig. 5 Histogram of measurement error wi th camber meter
lF~~"'"
~=:fiJ [5, I ~,.t l g; IJ '
""",,,.1
("PIlI'llll'l!!
~------' ~_ _ _ _ _...J
- - -- - - -
Fig. 7 Block diagram of camber control
Y. T ana ka et al.
152
50r-~~-:-:-:----:-----------
(a) Be(ore control
u
Co",'" - "',,""
J ]
E E
~-5U
:~
50 r - 7 7")-:A-:(7"te-,.-c-o-n-:-tr-o-:-I-----------, 1b
.,.o
CitlHlwr - 12 111111
O~~==~~~~~=-~-- I -50~O-----L------;2~U-----L-----:-4LU· ~----- ---
Plate length
(,11)
Fig. 8 An example of cani>er control
~
III
:::
1
\IS
~o o
Ca
I
I
o Nllt cc Illtrollefl • CUlltru lied
1t{.r--. 0 0 J~~' ---· ....
11111 - - - - - - - - -
± •• ••
0
o~-t-;_· o
- 11111 - - - - - - - - -
III
,;'
••
r. "e
·0
oo-to-.. . . . . . 0,>,
JU
40
l 'l;. l e lell)(lh (Ill)
Fig. 9 Effect of camber control
c .,11