Method for controlling continuous rolling mill and control apparatus therefor

The invention provides a method for controlling a continuous rolling mill having at least ith and (i+1)th stands and a control apparatus therefor, wherein an exit strip thickness reference value of an automatic gauge control is changed in accordance with a predetermined strip length during size changing and roll gap is corrected so as to change a strip thickness at an exit side of the ith stand, when a size (e.g., thickness) is changed during rolling, and, at the same time, a roll peripheral speed of the ith stand is changed in correspondence with a change in forward slip (a change in entry strip thickness, a change in exit strip thickness, and a change in resistance to deformation) of the ith stand as well as changes in forward slip, exit strip thickness, entry strip thickness and roll peripheral speed of the (i+1)th stand, so as to make the strip speed at the exit side of the ith stand coincide with that at the entry side of the (i+1)th stand, thereby minimizing the interstand tension and performing smooth size changing.

BACKGROUND OF THE INVENTION 
I. Field of the Invention 
The present invention relates to a method for controlling a continuous 
rolling mill and to a control apparatus therefor, wherein exit strip 
dimensions, such as thickness, are changed during rolling so as to obtain 
rolled products having different dimensions. 
II. Description of the Prior Art 
In the iron and steel industry, manufacturing equipment tends to become 
large in an effort to improve productivity. Along with this tendency, the 
weight of the slab or the sheet bars supplied to continuous tandem mill 
plants is also increasing. Therefore, the unit weight of coils 
manufactured by such continuous tandem mills increases accordingly. 
However, since users require steel plates of various sizes, a control 
technique is required to change the exit strip size, that is, the size of 
a product, during rolling without degrading productivity. 
A method is conventionally proposed to control a continuous tandem cold 
mill as follows. The pass schedule before the size change is defined as 
schedule A. Under schedule A, the exit strip thickness, the roll gap, and 
the roll speed are not changed. The pass schedule after size change is 
defined as schedule B. The schedule operating during the transition period 
while changing the schedules from A to B is defined as schedule C. The 
schedules A to C are computed before rolling in accordance with setting 
operations (operations for setting the roll gap and the roll speed in 
accordance with the capacity of the equipment, the material used, the the 
strip to be rolled as the final product). When a size changing point X 
reaches each of the strands of the continouous tandem cold mill, the 
schedules A, C and B are performed in the order named, so that the roll 
gap and the roll speed are changed to obtain strips of different size. 
In the control method of the type described above, the roll gaps and roll 
speeds must be computed in advance for the schedules A, B and C. The 
control is independent of the rolling conditions and is performed in 
accordance with the preset values. Therefore, when changes such as 
resistance to deformation (caused by material hardness, material 
temperature or a friction coefficient between the roll and material), a 
difference between the entry and exit strip sizes, a forward slip, or a 
rolling force occur during the rolling process, an error will occur in the 
product. 
SUMMARY OF THE INVENTION 
The present invention has been made to eliminate the conventional 
drawbacks, and has for its object to provide a method for controlling a 
continuous rolling mill and a control apparatus therefor, wherein a 
deviation in an interstand tension is small when a size change is 
performed during rolling, even if rolling conditions spontaneously change 
during rolling at any stand of the continuous rolling mill, thus smoothly 
changing the strip size. 
In order to achieve the above object of the present invention, there are 
provided a method for controlling a continuous rolling mill having at 
least ith and (i+1)th stands and a control apparatus therefor, wherein an 
exit strip thickness reference value of an automatic gauge control is 
changed in accordance with a predetermined strip length during size 
changing and roll gap is corrected so as to change a strip thickness at an 
exit of the ith stand whenever a size (e.g., thickness) is changed during 
rolling, and, at the same time, a roll peripheral speed of the ith stand 
is changed corresponding to a change in forward slip (or to a change in 
entry strip thickness, a change in exit strip thickness, or a change in 
resistance to deformation) of the ith stand as well as changes in forward 
slip, exit strip thickness, entry strip thickness and roll peripheral 
speed of the (i+1)th stand, in accordance with the following equations, so 
as to make the strip speed at the exit side of the ith stand coincide with 
that at the entry side of the (i+1)th stand: 
EQU .DELTA.V.sub.Ri /V.sub.Ri =-.DELTA.f.sub.i /(1+f.sub.i)+.DELTA.f.sub.i+1 
/(1+f.sub.i+1)+ 
EQU .DELTA.h.sub.i+1 /h.sub.i+1 -.DELTA.H.sub.i+1 /H.sub.i+1 
+.DELTA.V.sub.Ri+1 /V.sub.Ri+1 
where 
V.sub.R : roll peripheral speed 
f: forward slip 
h: exit strip thickness 
H: entry strip thickness 
i, i+1: stand numbers 
.DELTA.: a small change 
and 
EQU .DELTA.f.sub.i =(.differential.f.sub.i 
/.differential.H.sub.i).DELTA.H.sub.i +(.differential.f.sub.i 
/.differential.h.sub.i).DELTA.h.sub.i +(.differential.f.sub.i 
/.differential.k.sub.i).DELTA.k.sub.i 
where 
k: strip resistance to deformation 
.differential.f/.differential.H: partial differential coefficient 
.differential.f/.differential.h: partial differential coefficient 
.differential.f/.differential.k: partial differential coefficient

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
Before an embodiment of the present invention is described in detail, the 
principle of the present invention will be described. The pass schedule 
before size change is defined as schedule A, under which the exit strip 
thickness, the roll gap and the roll speed of each stand remain fixed. The 
pass schedule after size change is defined as schedule B, under which the 
exit strip thickness, the roll gap and the roll speed of each stand have 
been changed. The duration of a size change during rolling, that is, the 
period during which schedule A is changed to schedule B, is defined as the 
transient duration. 
Assume that schedule A is changed to schedule B so as to change the exit 
strip thickness at the exit side of the final stand, as shown in FIG. 1. A 
hot strip finish rolling mill (seven stands) is used as a continuous 
rolling mill. The exit strip thickness of the seventh stand for schedule A 
is defined as h.sub.7A (mm), the exit strip thickness of the seventh stand 
for schedule B is defined as h.sub.7B (mm), and the strip length produced 
during size changing at the exit side of the seventh stand is defined as 
l.sub.7 (m). 
At the ith stand (i=1 to 7) of the finishing mill, the exit strip thickness 
of the ith stand for schedule A is defined as h.sub.iA (mm), the exit 
strip thickness of the ith stand for schedule B is defined as h.sub.iB 
(mm), and the strip length during size changing at the ith stand is 
defined as l.sub.i (m). If the mass flow of the strip passing through the 
ith stand during size changing is regarded as being equal to that of the 
strip passing through the seventh stand during size changing, the 
following equation is given: 
EQU l.sub.i (h.sub.iA +h.sub.iB)=l.sub.7 (h.sub.7A +h.sub.7B) (1) 
Therefore, 
EQU l.sub.i ={(h.sub.7A +h.sub.7B)/(h.sub.iA +h.sub.iB)}l.sub.7 (2) 
At the ith stand, at a point where the strip length at the exit is spaced 
apart by l.sub.Xi (m) from a size changing point X as shown in FIG. 2, the 
exit strip thickness h.sub.i of the ith stand is given by the following 
equation: 
EQU h.sub.i =h.sub.iA -{(h.sub.iA -h.sub.iB)/l.sub.i }l.sub.Xi (3) 
The method for controlling the continuous rolling mill according to the 
present invention includes first and second steps to be described 
hereinafter. In the first step, the exit strip thickness h.sub.i of the 
ith stand which is obtained by equation (3) is used as an exit strip 
thickness reference value of the automatic gauge control, so that the roll 
gap of the ith stand of the rolling mill can be spontaneously changed 
during rolling. In the example shown in FIG. 1, each time the size 
changing point X reaches the first stand to the seventh stand respectively 
of the finishing mill, the exit strip thicknesses at the exits of all the 
stands are respectively supplied as the exit strip thickness reference 
values to the automatic gauge controls of the stands. The roll gaps of the 
stands are respectively changed during rolling so as to change schedule A 
to schedule B. 
In the second step, when the strip thickness is changed during rolling, the 
interstand tension must not be changed even though the roll gaps of the 
stands are changed. For this purpose, the roll speeds of the stands must 
be changed simultaneously when the roll gaps are changed. At each stand of 
the continuous rolling mill, rolling conditions (resistance to deformation 
caused by material hardness, material temperature, or a change in the 
friction coefficient between the roll and the material; strip sizes at the 
entry and exit sides of the stand; forward slip; and rolling force) are 
spontaneously changed. In the second step of the present invention, data 
of resistance to deformation caused by material temperature and data of 
strip sizes at the entry and exit sides are supplied from the ith stand to 
the (i+1)th stand in synchronism with the exit strip speed. Thus, the roll 
speeds of the stands are controlled so as to obtain a constant mass flow 
between the ith and (i+1)th stands; this is called constant mass flow 
control hereinafter. 
The constant mass flow control will be described in detail. FIG. 3 shows a 
7-stand continuous rolling mill. Referring to FIG. 3, rolls 1 to 7 are 
respectively disposed at the first to seventh stands. Assume that, at the 
ith stand, the entry strip width is defined as B.sub.i, the exit strip 
width is defined as b.sub.i, the entry strip thickness is defined as 
H.sub.i, the exit strip thickness is defined as h.sub.i, the entry strip 
speed is defined as V.sub.i, and the exit strip speed is defined as 
v.sub.i. Also assume that, at the ith stand, the roll peripheral speed is 
defined as v.sub.Ri, the forward slip is defined as f.sub.i, the roll gap 
is defined as S.sub.i, and the rolling force is defined as P.sub.i. At the 
ith stand, the following equations are given: 
EQU v.sub.i =v.sub.Ri (1+f.sub.i) (4) 
EQU B.sub.i H.sub.i V.sub.i =b.sub.i h.sub.i v.sub.i (5) 
When a width deviation caused by rolling or the like is neglected, equation 
(5) can be represented as follows: 
EQU H.sub.i V.sub.i =h.sub.i v.sub.i (6) 
The exit strip thickness of the ith stand is given as follows: 
EQU h.sub.i =S.sub.i +(P.sub.i /M.sub.i) (7) 
where M.sub.i is the mill spring constant of the ith stand. 
In the constant mass flow control system, the roll peripheral speed at the 
ith stand (or at the (i+1)th stand) is corrected, so that the exit strip 
speed v.sub.i at the ith stand at an arbitrary time becomes equal to the 
entry strip speed V.sub.i+1 at the (i+1)th stand as follows: 
EQU v.sub.i =V.sub.i+1 (8) 
Even during a size change operation during rolling, the interstand tension 
between the ith and (i+1)th stands remains constant, thus performing 
stable size changing during rolling. 
When equation (6) is applied for the (i+1)th stand, it is represented as 
follows: 
EQU H.sub.i+1 .multidot.V.sub.i+1 =h.sub.i+1 .multidot.v.sub.i+1 (9) 
Equation (8) may be substituted into equation (9) to obtain the following 
equation: 
EQU H.sub.i+1 .multidot.v.sub.i =h.sub.i+1 .multidot.v.sub.i+1 (10) 
Equation (4) may be substituted into equation (10): 
EQU H.sub.i+1 .multidot.v.sub.Ri (1+f.sub.i)=h.sub.i+1 .multidot.v.sub.i+1 (11) 
The exit strip speed v.sub.i+1 of equation (11) is used for the (i+1)th 
stand with reference to equation (4), thus representing equation (11) in 
the following manner: 
EQU H.sub.i+1 .multidot.V.sub.Ri (1+f.sub.i)=h.sub.i+1 .multidot.v.sub.Ri+1 
(1+f.sub.i+1) (12) 
Equation (12) can be rewritten as: 
EQU v.sub.Ri ={(1+f.sub.i+1)/(1+f.sub.i)}(h.sub.i+1 /H.sub.i+1)v.sub.Ri+1 (13) 
Equation (13) is differentiated to obtain the small change .DELTA.v.sub.Ri 
as follows: 
EQU .DELTA.v.sub.Ri =(.differential.v.sub.Ri 
/.differential.f.sub.i).DELTA.f.sub.i +(.differential.v.sub.Ri 
/.differential.f.sub.i+1).DELTA.f.sub.i+1 + 
EQU (.differential.v.sub.Ri /.differential.h.sub.i+1).DELTA.h.sub.i+1 
+(.differential.v.sub.Ri /.differential.H.sub.i+1).DELTA.H.sub.i+1 + 
EQU (.differential.v.sub.Ri /.differential.v.sub.Ri+1).DELTA..sub.Ri+1 (14) 
where (.differential.v.sub.Ri /.differential.f.sub.i), 
(.differential.v.sub.Ri /.differential.f.sub.i+1), (.differential.v.sub.Ri 
/.differential.h.sub.i+1), (.differential.v.sub.Ri 
/.differential.H.sub.i+1), and (.differential.v.sub.Ri 
/.differential.v.sub.Ri+1) are partial differential coefficients, 
respectively. According to equations (13) and (14), 
EQU .DELTA.v.sub.Ri /v.sub.Ri =-.DELTA.f.sub.i /(1+f.sub.1)+.DELTA.f.sub.i+1 
/(1+f.sub.i+1)+ 
EQU .DELTA.h.sub.i+1 /h.sub.i+1 -.DELTA.H.sub.i+1 /H.sub.i+1 .DELTA.v.sub.Ri+1 
/v.sub.Ri+1 (15) 
Equation (15) is the fundamental relation for performing constant mass flow 
control. The roll peripheral speed of the ith stand is controlled 
corresponding to a change in the roll peripheral speed of the (i+1)th 
stand so as to satisfy equation (15). 
A case will be considered relating to the 7-stand hot strip finishing mill 
(FIG. 3). The seventh stand is generally the reference stand of the 
finishing mill. Since the speed of the seventh stand is the reference, 
equation (15) must be rewritten six times for i=1 to 6. 
The forward slip f.sub.i of equation (15) can be obtained by the following 
known equation (15)-1: 
EQU f.sub.i =tan.sup.2 {(.pi./8).sqroot.(h.sub.i /R.sub.i ') log.sub.e 
(1-.gamma..sub.i)+(1/2) 
EQU sin.sup.-1 .sqroot..gamma..sub.i -(1/2k.sub.i).sqroot.(h.sub.i /R.sub.i 
').multidot.(t.sub.b -t.sub.f)} (15)-1 
where 
R.sub.i ': flattened roll radius (mm) 
h.sub.i : exit strip thickness (mm) 
H.sub.i : entry strip thickness (mm) 
.gamma..sub.i : fractional reduction=(H.sub.i -h.sub.i)/H.sub.i 
t.sub.b : back tension stress (kg/mm.sup.2) 
t.sub.f : front tension stress (kg/mm.sup.2) 
k.sub.i : resistance to deformation (rolling force) (kg/mm.sup.2) 
The forward slip can also be obtained using another known equation other 
than equation (15)-1. 
Suppose neither the front tension stress t.sub.f nor the back tension 
stress t.sub.b changes. A change .DELTA.f.sub.i in the forward slip can 
then be approximated by the following equation: 
EQU .DELTA.f.sub.i =(.differential.f.sub.i 
/.differential.H.sub.i).DELTA.H.sub.i +(.differential.f.sub.i 
/.differential.h.sub.i).DELTA.h.sub.i + 
EQU (.differential.f.sub.i /.differential.k.sub.i).DELTA.k.sub.i (16) 
where k.sub.i is the strip resistance to deformation at the ith stand. A 
change .DELTA.f.sub.i+1 in the forward slip at the (i+1)th stand can be 
obtained in the same manner using equation (16). 
The change .DELTA.h.sub.i+1 can also be obtained in accordance with 
equation (7) as follows: 
EQU .DELTA.h.sub.i+1 =.DELTA.S.sub.i+1 +(.DELTA.P.sub.i+1 /M.sub.i+1) (17) 
A change .DELTA.h.sub.i in exit strip thickness at the ith stand can be 
obtained by equation (7): 
EQU .DELTA.h.sub.i =.DELTA.S.sub.i +(.DELTA.P.sub.i /M.sub.i) (18) 
Data of the change .DELTA.h.sub.i is supplied from the ith stand to the 
(i+1)th stand. That is, 
EQU .DELTA.H.sub.i+1 (t)=.DELTA.h.sub.i {t-(L.sub.i /v.sub.i)} (19) 
where t is time and L.sub.i is the distance between the ith and (i+1)th 
stands. As a result, the change .DELTA.H.sub.i+1 of equation (15) is thus 
known. 
Since the term .DELTA.v.sub.Ri+1 /v.sub.Ri+1 of equation (15) is a 
corrected portion of the roll peripheral speed of the (i+1)th stand, this 
portion is regarded as zero for the final master stand of the finishing 
mill. Therefore, the term .DELTA.v.sub.Ri+1 /v.sub.Ri+1 can be obtained 
from the roll speed correction value of the subsequent stand. 
The constant mass flow control method thus controls the variation 
.DELTA.v.sub.Ri /v.sub.Ri in the roll peripheral speed at the ith stand 
(i=1 to 6) of the finishing mill in accordance with equation (15). 
According to the present invention as described above, in any continuous 
rolling mill having at least two stands, the roll gap of the ith stand is 
controlled in accordance with the exit strip thickness at the ith stand 
when a size such as a strip thickness is changed during rolling. 
Simultaneously when the roll gap is changed, the roll peripheral speed of 
the ith stand is controlled, so that the exit strip speed at the ith stand 
is equal to the entry strip speed at the (i+1)th stand. As a result, any 
deviation in the interstand tension is kept from affecting the size 
changing operation during rolling, thus performing smooth rolling. 
Furthermore, even if a strip width is changed during rolling, a 
conventional automatic gauge control can be used. 
The control device according to the present invention will be described in 
detail with reference to FIGS. 4 and 5. FIG. 4 shows a case in which a 
7-stand hot strip finish rolling mill is used as a continuous rolling 
mill. Reference numerals 1, 2 and 3 denote first, second and third stands 
respectively; and 4, a roughing mill which is disposed in front of the 
first to third stands 1 to 3. It is noted that the fourth to seventh 
stands are not illustrated in FIG. 4, but they have the same structure as 
the first to third stands 1 to 3. 
Reference numeral 5 denotes a crop shear; 6, a thermometer of sheet bar; 7, 
8 and 9, main drive motors; 10, 11, 12 and 13, load cells which are 
respectively arranged at the roughing mill 4 and the stands 1, 2 and 3; 
14, 15, 16 and 17 are roll gap meters; 18, 19 and 20, speed regulators; 
21, 22 and 23, speed sensors (tachometers); 24, 25 and 26, adders; 27, 28 
and 29, roll speed reference voltages; 30, 31 and 32, constant mass flow 
computers; 36, 37, 38, 39, 40, 41 and 42, computing elements; 43, an 
adder; 44, 46, 48 and 50, delay apparatuses; 45, a cutoff length 
calculator; 47, 49 and 51, computing apparatuses; 52, 53 and 54, automatic 
gauge controls; and 55, a setting device for set-up calculation. 
In the control apparatus having the arrangement described above, when the 
leading end of the material (sheet bar) is clamped between the rolls of 
the roughing mill 4, a signal corresponding to the rolling force P.sub.R 
measured by the load cell 10 of the roughing mill 4 is supplied to the 
computing element 42. A signal corresponding to the roll gap S.sub.R 
measured by the roll gap meter 14 is also supplied to the computing 
element 42. The computing element 42 computes the exit material thickness 
h.sub.R of the roughing mill 4 in accordance with equation (20) given 
below: 
EQU h.sub.R =S.sub.R +(P.sub.R /M.sub.R) (20) 
where M.sub.R is a constant determined by the mill spring constant of the 
roughing mill 4. When the leading end of the material reaches the 
thermometer 6 disposed at the exit side of the roughing mill 4, a material 
temperature .theta..sub.R is measured. Data of the measured material 
temperature .theta..sub.R is supplied to the adder 43. Data of the 
material thickness h.sub.R is supplied from the computing element 42 to 
the adder 43. Data of the material thickness h.sub.R and data of the 
material temperature .theta..sub.R are supplied to the cutoff length 
calculator 45 through the delay apparatus 44. Data of the material 
thickness h.sub.R and data of the material temperature .theta..sub.R for 
each material portion having a predetermined length are produced from the 
adder 43 and are delayed in the delay apparatus 44 in synchronism with the 
leading end of the material. When the material is cut by the crop shear 5, 
the cuttoff length calculator 45 measures a cut portion of the material. 
Data of the material thickness h.sub.R and its temperature .theta..sub.R 
after cutting are supplied from the cutoff length calculator 45 to the 
computing apparatus 47 through the delay apparatus 46. Data of the 
material thickness h.sub.R1 is delayed by the delay apparatus 44 as 
follows: 
EQU h.sub.R1 (t)=h.sub.R {t-(L.sub.R1 /v.sub.R)} (21) 
where t is time, L.sub.R1 is the distance between the roughing mill 4 and 
the crop shear 5, and v.sub.R is the material speed. The temperature 
.theta..sub.R1 is given as follows: 
EQU .theta..sub.R1 (t)=.theta..sub.R {t-(L.sub.R1 /v.sub.R)} (22) 
Data of the entry strip thickness H.sub.1 for the first stand of the 
finishing mill is delayed by the delay apparatus 46 as follows: 
EQU H.sub.1 (t)=h.sub.R1 {t-(L.sub.R2 /v.sub.R)} (23) 
where L.sub.R2 is the distance between the crop shear 5 and the first stand 
1 of the finishing mill. The material temperature .theta..sub.R2 is also 
delayed by the delay apparatus 46 as follows: 
EQU .theta..sub.R2 (t)=.theta..sub.R1 }t-(L.sub.R2 /v.sub.R)} (24) 
However, since the material temperature drops between the roughing mill 4 
and the first stand 1 of the finishing mill, data of the temperature drop 
is processed by the delay apparatus 46, so that the material temperature 
.theta..sub.1 of the finishing mill is given as follows: 
EQU .theta..sub.1 ={(A.epsilon./h.sub.R)T.sub.1 +1/(.theta..sub.R2 +273).sup.3 
}.sup.-1/3 -273 (25) 
where A is a constant, .epsilon. is emissivity, and T.sub.1 is the delay 
time between the roughing mill 4 and the first stand 1 of the finishing 
mill. Data of the entry strip thickness H.sub.1 of the first stand 1 of 
the finishing mill and of the strip temperature .theta..sub.1 are 
temporarily stored in the computing apparatus 47. The stored pieces of 
data are represented by H.sub.1L and .theta..sub.1L. When the leading end 
of the strip is clamped between the rolls of the first stand 1 of the 
finishing mill, signals from the load cell 11 and the roll gap meter 15 
are supplied to the computing apparatus 47, so that the exit strip 
thickness h.sub.1 of the first stand 1 is given by the following equation: 
EQU h.sub.1 =S.sub.1 +(P.sub.1 /M.sub.1) (26) 
where S.sub.1 is the roll gap of the first stand 1, P.sub.1 is the rolling 
force of the first stand 1, and M.sub.1 is the mill spring constant of the 
first stand 1. The exit thickness h.sub.1 is also stored in the computing 
apparatus 47 with H.sub.1L and .theta..sub.1L. The stored data is 
represented by h.sub.1L. 
Pieces of data of the exit strip thickness h.sub.1 of the first stand 1 and 
the strip temperature .theta..sub.1 are delayed by the delay circuit 48 to 
the second stand 2 in synchronism with the material speed. Data of the 
entry strip thickness H.sub.2 of the second stand is delayed by the delay 
apparatus 48 as follows: 
EQU H.sub.2 (t)=h.sub.1 {t-(L.sub.1 /v.sub.1)} (27) 
where t is time, v.sub.1 is the strip speed at the exit of the first stand 
1, and L.sub.1 is the distance between the first stand 1 and the second 
stand 2. A delayed value .theta..sub.1 ' of the strip temperature 
.theta..sub.1 is given as follows: 
EQU .theta..sub.1 '(t)=.theta..sub.1 }t-(L.sub.1 /v.sub.1)} (28) 
Since the strip temperature decreases between the first and second stands 1 
and 2, a strip temperature .theta..sub.2 at the second stand 2 is 
corrected by the delay apparatus 48 as follows: 
EQU .theta..sub.2 =(.theta..sub.1 
'-.theta..sub.W)e.sup.-(2.alpha./c.gamma.).multidot.(L.sbsp.1.sup./h.sbsp. 
1.sup.v.sbsp.1.sup.) +.theta..sub.W (29) 
where .theta..sub.W is the cooling water temperature of the roll, c is the 
specific heat of the strip, .gamma. is the specific gravity of the strip, 
and .alpha. is the heat transfer coefficient of the finishing mill. 
The entry strip thickness H.sub.2 and the strip temperature .theta..sub.2 
at the entry of the second stand 2 can thus be obtained. In the same 
manner as described above, the entry strip thickness H.sub.i, the strip 
temperature .theta..sub.i and the exit strip thickness h.sub.i with 
respect to the ith stand can be obtained. These values for each stand are 
obtained as instantaneous values during rolling. 
The control will now be described for changing the roll gap when a size 
change is performed during rolling. The automatic gauge controls 52, 53 
and 54 of the stands of the finishing mill shown in FIG. 4 have the same 
structure. Only the automatic gauge control 52 of the first stand 1 will 
be described. The automatic gauge control 52 receives data of the exit 
strip thickness reference h.sub.1,REF, data of the gauge meter exit strip 
thickness h.sub.1 from the computing apparatus 47, data of the rolling 
force from the load cell 11, and data of the roll gap S.sub.1 from the 
roll gap meter 15. 
FIG. 5 shows the hydraulic type automatic gauge control 52. However, an 
electric type automatic gauge control may also be used. Referring to FIG. 
5, reference numeral 100 denotes an adder; 101, an integrator having a 
gain; 102, the roll gap preset value (S.sub.1,REF); 103, an adder; 104, 
the exit strip thickness reference (h.sub.1,REF) to be described later; 
105, the exit strip thickness (h.sub.1) of the first stand 1 which is 
obtained by the computing apparatus 47 shown in FIG. 4; 106, the rolling 
force (P.sub.1) from the load cell 11 shown in FIG. 4; 107, a relay which 
is closed when a predetermined time interval (e.g., 0.5 secs) has elapsed 
after the leading end of the sheet bar is clamped between the rolls of the 
first stand 1 and which is then immediately opened; 108, a memory for 
storing data of the rolling force P.sub.1 while the circuit is closed; 
109, an adder; 110, a multiplier; 111, a roll gap adder; 112, a PI 
controller; 113, a servo amp; 114, a hydraulic cylinder; and 115, a roll 
gap (S.sub.1) obtained by the roll gap meter 15 shown in FIG. 4. 
In the automatic gauge control having the structure described above, since 
the relay 107 is turned ON when the predetermined time interval (e.g., 0.5 
second) has elapsed after the leading end of the sheet bar is clamped 
between the rolls of the first stand 1, data of the rolling force P.sub.1 
designated by reference numeral 106 is stored in the memory 108. For the 
rest of the sheet bar, a difference between the rolling force P.sub.1 
designated by reference numeral 106 and a rolling force P.sub.1L whose 
data is stored in the memory 108 is calculated by the adder 109. An output 
from the adder 109 is multiplied by C/M.sub.1 by the multiplier 110, where 
C is any constant (e.g., 0.8) and M.sub.1 is the mill spring constant of 
the first stand 1. 
An output from the multiplier 110 is supplied to the roll gap adder 111, 
and an output from the roll gap adder 111 is supplied to the hydraulic 
cylinder 114 through the PI controller 112 and the hydraulic servo amp 113 
so as to correct the roll gap. A feedback signal from the roll gap meter 
15, and corresponding to the roll gap S.sub.1 designated by reference 
numeral 115, is supplied to the adder 111. 
The exit strip thicknesses reference h.sub.1,REF designated by reference 
numeral 104 is changed, so that the adder 100 is operated to obtain a 
difference between the exit strip thickness reference h.sub.1,REF or 104 
and the exit strip thickness h.sub.1 or 105. An output from the adder 100 
is integrated by the integrator 101. The integrated signal is then 
supplied to the roll gap adder 111 through the adder 103. Therefore, the 
roll gap is corrected so as to make the exit strip thickness h.sub.1 or 
105 of the first stand 1 coincide with the exit strip thickness reference 
h.sub.1,REF or 104. 
As described above, the automatic gauge control 52 corrects the roll gap 
S.sub.1 of the first stand 1 so as to make the exit strip thickness 
reference h.sub.1,REF or 104 of the first stand 1 coincide with the exist 
strip thickness h.sub.1 of the first stand 1. The above description is 
only made for the first stand 1, but also applies to the remaining stands. 
The exit strip thickness reference value will now be described in detail. 
An identical operation is performed at every stand, so that only the 
operation of the first stand will be discussed. The rolling schedule of 
the finishing mill is determined by the setting device 55 shown in FIG. 4. 
The rolling schedule includes values for the entry strip thickness, the 
exit strip thickness, the strip temperature, the strip width, the 
resistance to deformation, the entry tension, the exit tension, the roll 
radius, the forward slip, the rolling force, the roll gap, the roll speed, 
and the exit strip length during size changing of each stand. The rolling 
schedule is set for schedules A and B. 
The strip thickness and the strip length during size changing for schedules 
A and B are thus determined as described above. The strip thickness 
reference h.sub.REF in the size change during rolling is obtained as 
follows. The exit strip thickness reference value of the first stand of 
the finishing mill in the size change operation during rolling can be 
obtained by reference to equation (3) as follows: 
EQU h.sub.1 =h.sub.1A -{(h.sub.1A -h.sub.1B)/l.sub.1 }l.sub.x1 (30) 
where h.sub.1A is the exit strip thickness of the first stand 1 for 
schedule A, h.sub.1B is the exit strip thickness of the first stand 1 for 
schedule B, l.sub.1 is the exit strip length during size changing at the 
first stand 1, and l.sub.x1 is the exit strip length during size changing 
from the size changing point to the first stand 1. Therefore, data of 
h.sub.1A, h.sub.1B, and l.sub.1 of equation (30) is supplied from the 
setting device 55 to the computing element 39. The computing element 39 
receives data of the forward slip f.sub.1 from the computer 30 and 
performs the operation of (1+f.sub.1). The data of h.sub.1A, h.sub.1B, 
l.sub.1 and (1+f.sub.1) is supplied to the computing element 36. The 
computing element 36 also receives a signal N.sub.1 from the speed sensor 
21 of the main drive motor 7 of the first stand 1 to calculate l.sub.x1 : 
EQU l.sub.x1 =.intg.(1+f.sub.1)(2.pi.R.sub.1 /60)N.sub.1 dt (31) 
where R.sub.1 is the roll radius of the first stand 1. The exit strip 
thickness reference h.sub.1,REF of the first stand, which corresponds to 
the output from the computing element 16, is obtained by reference to 
equations (30) and (31): 
EQU h.sub.1,REF =h.sub.1A -(h.sub.1A -h.sub.1B)/l.sub.1 
.intg.(1+f.sub.1)(2.pi.R.sub.1 /60)N.sub.1 dt (32) 
Data of the exit strip thickness reference h.sub.1,REF of the first stand 1 
is supplied to the automatic gauge control 52. The exit strip thickness 
reference values of the remaining stands 2 to 7 are also obtained in the 
same manner as described above. 
The change in roll speed in the size change operation during rolling will 
be described below. The roll speed can be properly changed in accordance 
with the constant mass flow control. The values represented by equation 
(15) may be instantaneously calculated during rolling. 
The constant mass flow control will be described in detail. The roll speeds 
of all the stands can be changed in the same manner in accordance with the 
constant mass flow control, so that only a change in roll speed of the 
first stand will be discussed. According to equation (15), 
EQU .DELTA.v.sub.R1 /v.sub.R1 =-(.DELTA.f.sub.1 /1+f.sub.1)+(.DELTA.f.sub.2 
/1+f.sub.2)+ 
EQU (.DELTA.h.sub.2 /h.sub.2)-(.DELTA.H.sub.2 /H.sub.2)+(.DELTA.V.sub.R2 
/V.sub.R2) (32)-1 
Regarding the changes in forward slip which are numerators of the first and 
second terms on the right side, .DELTA.f.sub.1 of the first term can be 
directly obtained from equation (16) and .DELTA.f.sub.2 of the second term 
can also be obtained. The calculation of .DELTA.f.sub.1 is given in the 
equation shown below: 
EQU .DELTA.f.sub.1 =(.differential.f.sub.1A 
/.differential.H.sub.1A).DELTA.H1+(.differential.f.sub.1A 
/.differential.h.sub.1A).DELTA.h.sub.1 + 
EQU (.differential.f.sub.1A /.differential.k.sub.1A).DELTA.k.sub.1 (32)-2 
Referring to FIG. 4, values required to calculate equation (32)-2 are 
supplied as schedules A and B from the setting device 55 to the constant 
mass flow computer 30. The operation of the constant mass flow computer 30 
will be described below. 
Regarding the forward slip f.sub.1A of the first stand for schedule A, the 
following relation is given: 
EQU Z.sub.1A =H.sub.1A -h.sub.1A +(C.sub.1 M.sub.1 /B.sub.A)(h.sub.1A 
-S.sub.1A) (33) 
Let the f.sub.1C be: 
EQU f.sub.1C =1/2{.sqroot.h.sub.1A (H.sub.1A -h.sub.1A)/R.sub.1 Z.sub.1A 
(.pi./4.multidot.log.sub.e (h.sub.1A /H.sub.1A)- 
EQU 1/k.sub.1A (t.sub.bA -t.sub.fA))+sin.sup.-1 .sqroot..gamma..sub.1A }(34) 
for 
EQU .gamma..sub.1A =(H.sub.1A -h.sub.1A)/H.sub.1A (35) 
where the number 1 is the stand number, A refers to the schedule A, B.sub.A 
is the strip width, C.sub.1 is a constant, R.sub.1 is the roll radius, 
t.sub.bA is the back tension, t.sub.fA is the forward tension, 
.gamma..sub.1A is the fractional reduction, and k.sub.1A is the resistance 
to deformation. 
According to equation (15)-1, 
EQU .differential.f.sub.1A /.differential.H.sub.1A =2 tan f.sub.1C (1/cos.sup.2 
f.sub.1C)(.differential.f.sub.1C /.differential.H.sub.1A) (35)-1 
for 
##EQU1## 
Then, 
EQU .differential.f.sub.1A /.differential.h.sub.1A =2 tan f.sub.1C (1/cos.sup.2 
f.sub.1C)(.differential.h.sub.1C /.differential.h.sub.1A) (37) 
for 
##EQU2## 
Then, 
EQU .differential.f.sub.1A /.differential.k.sub.1A =2 tan f.sub.1C (1/cos.sup.2 
f.sub.1C)(.differential.f.sub.1C /.differential.k.sub.1A) (39) 
for 
EQU .differential.f.sub.1C /.differential.k.sub.1A =(1/2).sqroot.h.sub.1A 
(H.sub.1A -h.sub.1A)/(R.sub.1 Z.sub.1A).multidot. 
EQU (t.sub.bA -t.sub.fA)/k.sup.2.sub.1A (40) 
The partial differential coefficients of the forward slip f.sub.1A of the 
first stand with respect to the entry strip thickness H.sub.1A, the exit 
strip thickness h.sub.1A, and the resistance to deformation k.sub.1A can 
be obtained. 
In the same manner as described above, the pieces of data of the schedules 
A and B are supplied from the setting device 55 shown in FIG. 4 to the 
constant mass flow computer 31. The operation of the constant mass flow 
computer 31 is the same as that of the constant mass flow computer 30 of 
the first stand 1. The number 1 is replaced by the number 2 in equations 
(33) to (40). From the calculated results, the partial differential 
coefficients of the forward slip f.sub.2A of the second stand 2 with 
respect to the entry strip thickness H.sub.2A, the exit strip thickness 
h.sub.2A, and the resistance to deformation k.sub.2A are obtained. 
The pieces of data of the entry strip thickness H, the strip temperature 
.theta., and the exit strip thickness h which are obtained immediately 
before the size changing point X reaches each stand are stored in the 
computing apparatuses 47, 49 and 51. This data is defined as H.sub.1L, 
.theta..sub.1L and h.sub.1L for the first stand 1, and as H.sub.2L, 
.theta..sub.2L and h.sub.2L for the second stand 2. Similar definitions 
can be made for subsequent stands. 
The outputs corresponding to H.sub.1L, .theta..sub.1L and h.sub.1L and the 
instantaneous values H.sub.1, .theta..sub.1 and h.sub.1 are supplied from 
the computing apparatus 47 to the constant mass flow computer 30. At the 
second stand, the outputs from the computing apparatus 49 are supplied to 
the constant mass flow computers 30 and 31. Similar data transfer is 
performed at subsequent stands. 
The constant mass flow computer 30 computes the following changes with 
reference to equation (16): 
EQU .DELTA.H.sub.1 =H.sub.1 -H.sub.1L (41) 
EQU .DELTA.h.sub.1 =h.sub.1 -H.sub.1L (42) 
EQU .DELTA.k.sub.1 =k.sub.1 (.theta..sub.1)-k.sub.1L (.theta..sub.1L) (43) 
The resistance to deformation k.sub.1 of equation (43) is obtained from the 
following equation: 
EQU k.sub.1 
=0.00385(46.608-0.02987.theta.).times.(10.099+31.172.gamma.-29.842.gamma.. 
sup.2).times. 
EQU {11.153+2.7425 log 10.lambda.+0.68352(log 10.lambda.).sup.2 }(43)-1 
where .theta. is the strip temperature (material temperature) (.degree.C.), 
.lambda. is the strain rate (1/S), and .gamma. is the fractional reduction 
(-). In the same manner as described above, the constant mass flow 
computer 31 of the second stand 2 calculates the following values: 
EQU .DELTA.H.sub.2 =H.sub.2 -H.sub.2L (44) 
EQU .DELTA.h.sub.2 =h.sub.2 -h.sub.2L (45) 
EQU .DELTA.k.sub.2 =k.sub.2 (.theta..sub.2)-k.sub.2L (.theta..sub.2L) (46) 
The constant mass flow computer 30 of the first stand 1 instantaneously 
computes the change in forward slip .DELTA.f.sub.1 of the first stand in 
accordance with the following equation with reference to the value of 
equation (32)-2 from equations (33) to (43): 
EQU .DELTA.f.sub.1 =(.differential.f.sub.1A 
/.differential.H.sub.1A).DELTA.H.sub.1 +(.differential.f.sub.1A 
/.differential.h.sub.1A).DELTA.h.sub.1 + 
EQU (.differential.f.sub.1A /.differential.k.sub.1A).DELTA.k.sub.1 (47) 
In the same manner as described above, using the value of equation (16) 
which is obtained from equations (44) to (46) where the number 1 is 
replaced by the number 2, the constant mass flow computer 31 
instantaneously computes the change in forward slip .DELTA.f.sub.2 of the 
second stand 2 as follows: 
EQU .DELTA.f.sub.2 =(2f.sub.2A /2H.sub.2A).DELTA.H.sub.2 
+(.differential.f.sub.2A /.differential.h.sub.2A).DELTA.h.sub.2 + 
EQU (.differential.h.sub.2A /.differential.k.sub.2A).DELTA.k.sub.2 (48) 
The changes in forward slip of the subsequent stands can be obtained in the 
same manner as described above. 
Equation (15)-1 is used to obtain the forward slip f.sub.1 of the 
denominator of the first term of the right side of equation (32)-1. 
##EQU3## 
where R.sub.1 ' is the flattened roll radius. The constant mass flow 
computer 30 of the first stand 1 performs calculation in accordance with 
equation (49). The constant mass flow computer 31 of the second stand 2 
performs calculation in accordance with equation (49) where the number 1 
is replaced with the number 2. In other words, the constant mass flow 
computer 31 calculates the forward slip f.sub.2. As a result, the forward 
slips f of the stands are thus obtained. 
The values thus obtained are supplied to the constant mass flow computer 30 
of the first stand 1, and the constant mass flow computer 30 changes the 
roll peripheral speed in the size change during rolling as follows: 
EQU .DELTA.v.sub.R1 /v.sub.R1 =-{.DELTA.f.sub.1 /(1+f.sub.1)}+{.DELTA.f.sub.2 
/(1+f.sub.2)}+ 
EQU (.DELTA.h.sub.2 /h.sub.2)-(.DELTA.H.sub.2 /H.sub.2)+(.DELTA.v.sub.R2 
/v.sub.R2) (50) 
The values of the denominators of equation (50) correspond to those before 
size changing occurs. 
In the same manner, the constant mass flow computer 31 of the second stand 
2 computes a change in roll peripheral speed in the size change operation 
during rolling as follows: 
EQU .DELTA.v.sub.R2 /v.sub.R2 =-{.DELTA.f.sub.2 /(1+f.sub.2)}+{.DELTA.f.sub.3 
/(1+f.sub.3)}+ 
EQU (.DELTA.f.sub.3 /h.sub.3)-(.DELTA.H.sub.3 /H.sub.3)+(.DELTA.v.sub.R3 
/v.sub.R3) (51) 
The calculations for the subsequent stands are performed in the same manner 
as described above. The values are computed by the constant mass flow 
computer 32 in accordance with equations (50) and (51). The calculated 
results are supplied to the corresponding stand. When the 7-stand 
finishing mill is used, that is, when the seventh stand is regarded as the 
reference stand, the speed of the seventh stand is constant. In other 
words, .DELTA.v.sub.R7 /v.sub.R7 is zero. The output (corresponding to 
.DELTA.v.sub.R1 /v.sub.R1) from the constant mass flow computer 30 of the 
first stand 1 and the value (corresponding to v.sub.R1,REF) of the roll 
speed setter 27 are added by the adder 24. The sum data is then supplied 
to the speed regulator 18. The speed regulator 18 corrects the speed of 
the main drive motor 7, so that the roll speed can be changed in the size 
change operation during rolling. The operations for the remaining stands 
are performed in the same manner as described above. As a result, the size 
change operation during rolling can be stably performed, and a deviation 
in the interstand tension is very small. 
The present invention is not limited to the particular embodiment described 
above. The following modifications, for example, may be made within the 
spirit and scope of the present invention: 
(1) The constant mass flow control is only performed in the size change 
operation during rolling in the above embodiment. The constant mass flow 
control can be performed for ordinary rolling as well, thus providing 
further complete control. In this case, the pieces of data of the entry 
strip thickness, the strip temperature, the exit strip thickness and so on 
are stored when a predetermined time interval (e.g., 0.5 second) has 
elapsed after the leading end of the strip is clamped between the rolls of 
the stand, thereby performing the subsequent constant mass flow control. 
(2) In the above embodiment, data of the exit strip thickness reference 
value is supplied to the automatic gauge control of the corresponding 
stand when the size change is performed during rolling. However, the roll 
gap values in the size change operation during rolling and from schedule B 
are directly supplied to the hydraulic cylinder of the rolling mill 
(specifically, they are supplied as the roll gap S.sub.1 or 115) so as to 
change the roll gap. The roll speed may be changed in accordance with the 
constant mass flow control in the size change during rolling. 
(3) In the above embodiment, the speed of the reference stand in the size 
change operation during rolling is not changed. However, when the exit 
temperature control of the rolling mill is incorporated, the speed of the 
reference stand can be corrected by a difference between the reference 
temperature and the actual temperature. In this case, the stand speed is 
corrected corresponding to a percentage of the speed correction of the 
reference stand. 
(4) In the above embodiment, the roll speed is changed in accordance with 
the constant mass flow control of an upstream type. However, a downstream 
type control may also be utilized in which the speed on the downstream 
side is corrected. 
(5) The values of the rolling schedule in the size change operation during 
rolling or the size change after rolling can be supplied to an attachment 
of the continuous rolling mill, such as a guide, looper, and various types 
of measuring equipment. 
(6) The continuous rolling mill need only have a minimum of two stands. The 
resistance to deformation can also be obtained in accordance with the 
rolling force and the strip thickness at each stand of the continuous 
rolling mill. 
(7) The present invention may be applied to any continuous rolling mill for 
producing a steel wire, a steel rod, a steel product of any shape, or a 
steel plate. 
(8) The method for changing the roll gap in the size change operation 
during rolling can be applied to a single stand rolling mill, such as a 
plate mill and a reverse mill. 
The roll gap of the ith stand can be changed in correspondence with the 
exit strip thickness of the ith stand when a size change (e.g., thickness) 
during rolling is performed with a continuous rolling mill having at least 
two stands. Furthermore, simultaneous with the changing of the roll gap of 
the ith stand, the roll speed can also be changed in accordance with the 
constant mass flow control in which the strip speed at the exit of the ith 
stand is equal to that at the entrance of the (i+1)th stand. Even if 
rolling conditions such as resistance to deformation caused by strip 
hardness, strip temperature, or the friction coefficient between the roll 
and the strip, and such as exit and entry strip size, forward slip, and 
rolling force are instantaneously changed, smooth size changing is 
performed.