Method of and apparatus for controlling a rolling mill
United States Patent Morooka 1 Apr. 8, 1975 I 1 METHOD OF AND APPARATUS FOR CONTROLLING A ROLLING MILL [57] ABSTRACT [75] Inventor: Yasuo Morooka, Hitachi. Japan [731 Assignee: Hitachi, Ltd.. Tokyo, Japan [22] Filed: Dec. 7. 1973 [2]] App]. No; 422.690  
 [30] Foreign Application Priority Data Dec. 1]. 1972 Japan 47-123342 [S2] U.S. Cl 72/11: 72/6 [51] Int. Cl BZlb 37/00 [58] Field of Search 72/12. 8. 9. 10. 11. I9.  
 I56] References Cited UNITED STATES PATENTS 3.474.668 10/1969 Mangan 73/159 3.509.751 5/1970 Shiraiwa et a1. 7l/l6 3.599.459 8/1971 Yeomans 72/8 3.793.859 2/1974 Sterrett et a1 72/20 X Primary Examiner-Mi1ton S. Mehr Attorney. Agenl. or Firm-Craig &amp; Antonelli A method of and an apparatus for controlling a rolling mill whereby the thickness distribution across the width of a strip steel produced by rolling can be made uniform and a desired flatness can be obtained in the strip steel. A thickness distribution and a length distribution across the width of the strip steel before being rolled are measured. and a thickness distribution desired to be obtained in the strip steel after being rolled is calculated based on the values obtained. At the same time. an expected roll opening distribution corresponding to the roll openings 51 and 52 on the device side and the operation side respectively and the roll bender forces 01 and 02 on the drive side and the operation side respectively is calculated based on an expected deflection and an expected flattening of the rolls. into a function of the factors 51 S2. 01 and 02. then the desired values of S1. S2. 01 and 02 are determined at a time when the dispersion of the difference between the thickness distribution desired to be obtained and the expected roll opening distribution is minimized. Control of the roll screw-down device and sired values of S1. S2, 01 and 02 respectively.  
 3 Claims. 5 Drawing Figures PATENIEDAPR 3.875.776  
 saamanrq FIG 3 T! DETECTOR MAIN CONTROL UNIT DETECTOR ANCILLARY ANCILLARY ANCILLARY CONTROL CONTROL CONTROL UNIT UNIT UNIT PRESSURE SOURCE PAIEIIIEIIIPII 875.778  
 IIIEI 3 I 4 FIG.4  
 MEASUREMENT OF H0O LIX) INPUT CALCULATION rm OF IIIx) FEEDBACK CONTROL I FEIED FORWARD CONTROL I MEASUREMENT ASSUMPTLON 0F h(x) h(x)= h(x) L I I CALCULATION OF PIX) [FORMULA II II] CALCULATION OF MIxI (FORMULA IIOI] ASSUMPTION OF SI,S2.QI.Q2  
 CALCULATION OF ywIx) [FORMULA (7)] CALCULATION OF SIxI [FORMULA (I211 CALCULATION OF hIxI [FORMULA (9)] CALCULATION OF MODIFICATION OF [FORMULAUSIORGTI] SZQLQZ NJENTEBAPR&#39; 197s 3.875.776  
 1 .1.7 i ui A F I G. 5  
 MEASUREMENT OF HIX). Ltx) CALCULATION m 0F hix) FEEDBACK CONTITOL I FEIIiD FORWARD CONTROL MEASUREMENT ASSUMPTION OF hm h(x)=h*(.x)  
 CALCULATION OF PIX) CALCULATION OF MIX) SOLVING EQUATIONS END METHOD OF AND APPARATUS FOR CONTROLLING A ROLLING MILL BACKGROUND OF THE INVENTION This invention relates to a method of and an apparatus for controlling by means of a control computer a rolling mill which permits to obtain as perfect a flatness as possible in a strip steel by using a roll bending device of the prior art.  
  The most important thing in rolling a strip steel is to obtain a desired flatness in the strip steel by working on the strip such that its thickness distribution in the longi tudinal direction is rendered uniform and irregularities extending across the width of the strip are eliminated. It is impossible to accomplish this object by using a working method relying on hunch. Thus, the practice of using a control apparatus of high precision. e.g., a control computer, for setting a rolling mill to a predetermined state as an attachment to the rolling mill has been popularized.  
  The use of the control apparatus of the type de scribed makes it possible to work on a strip steel such that the thickness distribution of the strip in the longitudinal direction is rendered uniform. However, it has hitherto been impossible to obtain a uniform thickness across the width of the strip steel and thus to obtain a desired flatness in the strip steel by using such apparatus. The reasons for this inability is presently to be described.  
  Methods of effecting control of a rolling mill with a view to obtaining a uniform thickness distribution in the longitudinal direction has long been studied and developed into a model which can be expressed by using numerical equations. On the other hand. it is only in recent years that methods of controlling a rolling mill in such manner that a desired flatness can be obtained across the width of a strip steel have been thought of. The latter methods have not yet been developed into a model which can be expressed by using numerical equations.  
  Of the latter methods, the method considered to be most effective in obtaining a uniform thickness distribution across the width of a strip steel is a roll bending method. This method consists in applying bending moments to opposite support ends of work rolls or backup rolls of a rolling mill to bend the rolls, so that the production of surface defects across the width of the strip steel can be prevented. As aforementioned, this method has not yet been elucidated so such an extent that it can be expressed in the form of a numerical model. Thus, it has been impossible to carry out calcu lation of set up procedures for the roll bending device, although it is possible to carry out calculation of set up procedures for the rolling mill by means of a control computer.  
  Accordingly, even in the case of a rolling mill provided with a control computer, it has hitherto been customary to use a shape control apparatus or flatness control apparatus ofthe feed-back type which operates independently of the control computer. Such shape control apparatus comprises a shape detector device adapted to detect the flatness or the shape of the surface of a strip steel, and a roll bending device adapted to be controlled according to the value of an output signal of the shape detector device. The value of the output signal may vary depending on the value at which the apparatus is first set It is thus impossible to effect adequate control ofa rolling mill with a view to obtaining a desired flatness in a strip steel unless the value at which the apparatus is set is based on good authority from the point of view of the theory of rolling.  
  More specifically. it has hitherto been customary to prepare beforehand a table of comparison of the bend ing moments applied to the rolls and the materials of strip steel to be rolled, and to manually set the control apparatus at a suitable roll bending by referring to this table. However, the values obtained in this way are not based on the theory of rolling in the strict sense of the term. so that it has been impossible to effect control adequately to obtain a desired flatness in a strip steel by this method of the prior art. Added disadvantages of the aforementioned conventional method are that the operation is troublesome and the precision with which control is effected is low, because the setting operation is performed manually.  
  In order to obviate the disadvantages of the afore mentioned method of the prior art and enable CtllCUltb tion of set up procedures for the roll bending device to be carried out by means of a computer. a proposal has been made to express the transverse cross-sectional shape of a strip steel taken along the width thereof by a numerical model consisting of the sum of a quadratic term and a biquadratic term of the transverse distance of the strip steel by assuming that the strip steel is symmetrical in its transverse cross-sectional shape with respect to the center line thereof across its width. The coefficient of each of the terms is obtained from a collinear graph which is prepared separately.  
  However, the proposed method is not reasonable in that the assumption is not warrantable that the trans verse cross-sectional shape of a strip steel which is essentially very complex can be expressed by a biquadratic curve which is symmetrical with respect to the center line across the width of the strip steel. Moreover, the numerical model itself is not obtained based on the theory of rolling. so that it does not agree with the actual cross-sectional shape of the strip steel. The fact that the coefficient of each term of the numerical model must be obtained manually from a collinear graph does not show that this method represents an advance over the prior art. The preparation of such collinear graph itself involves difficulty. In view of these disadvantages, the proposed method cannot be said to be a method suitable for practical use.  
  The present practice of effecting shape control in rolling a steel strip being as aforementioned, it would contribute greatly to the advance in the progress of the art ofshape control in rolling a steel strip if it were possible to elucidate shape control by means of the roll bending device by using numerical equations based on the theory of rolling and to carry out calculation of set up procedures for the roll bending device by means of a computer. It would also be possible to produce, with a high degree of efficiency, strip steels of a desired flatness and high precision. However, theoretically speaking, it is not possible to obtain a perfect flatness in a strip steel by using a roll bending device of the prior art. This is because the roll bending device of the prior art operates such that the load applied to a strip steel is concentrated on opposite marginal portions, in spite of the fact that the cross-sectional shape across the width of the strip steel constitutes very complex curves, so that it is impossible to produce deflection in the rolls which corresponds to the complex curves.  
 SUMMARY or THE INVENTION One object of this invention is to provide a method of controlling a rolling mill which permits to obtain as perfect a flatness as possible in a strip steel by using a roll bending device now in practical use.  
  Another object of the invention is to provide an up paratus adapted to carry the aforementioned method into practice.  
  According to the invention. the aforementioned objects are accomplished by first measuring a thickness distribution and a length distribution across the width of a strip steel before being rolled. then calculating. based on the values obtained. an ideal thickness distribution desired to be obtained in the strip steel after being rolled and. at the same time. calculating an expected roll opening distribution corresponding to the roll openings SI and S2 on the drive side and the operation side respectively and the roll bender forces Q1 and 02 on the drive side and the operation side respectively. based on an expected deflection and an expected flattening of the rolls. and finally calculating roll bender forces and roll screw-down amounts at the same time when the dispersion of the difference between the ideal thickness distribution desired to be obtained in the strip steel and the expected roll opening distribution is minimized. whereby the roll bending device and the roll screw-down device can be controlled by using the numerical values obtained.  
 BRIEF DESCRIPTION OF THE DRAWINGS FIG. I is a schematic view showing a stand of a fourhigh rolling mill at which a strip steel is being rolled by rolls to which bending moments aqe applied by a roll bending device in explanation of the principles of this invention;  
  FIG. 2 is a sectional view taken along the line ll ll of FIG. 2;  
  FIG. 3 is a block diagram of an apparatus adapted to carry the method according to the invention into practice. shown in association with an ordinary four-high rolling mill;  
  FIG. 4 is an explanatory view showing a procedure for calculating desired roll bender forces and roll screw-down amounts; and  
  FIG. 5 is an explanatory view showing an other procedure for calculating the desired roll bender forces and roll screw-down amounts.  
 DETAILED DESCRIPTION OF A PREFERRED EMBODIMENT The invention will now be described in detail with reference to the accompanying drawings. In FIG. I and FIG. 2. there is shown a stand of a rolling mill at which a strip steel is being rolled in explanation of the principles of this invention.  
  In FIG. 1, a strip steel 1 is disposed in roll pass between work rolls 2 and 2&#39; of a four-high rolling mill, with the work rolls 2 and 2 being supported at their backs by back-up rolls 3 and 3&#39; respectively. Rolling forces F1 and F2 are exerted on the roll necks of the back-up rolls 3 and 3&#39; respectively by a roll screw-down device (not shown). On the other hand. roll bender forces 01 and ()2 are exerted on opposite ends of the work rolls 2 and 2&#39; by a roll bending device [not shown]. Furthermore. the work rolls 2 and 2&#39; are subjected. at surfaces at which they are in engagement with the strip steel 1, to a reaction to a rolling load P(.i&#39;) per unit width. Thus. a deflection is produced in the work rolls 2 and 2&#39; and the back-up rolls as shown in the drawings.  
  Let us now analyze the relation between the crosssectional shape of the strip steel and the deflection curves of the rolls and the distribution of rolling pressures at the time rolling is carried out by establishing a coordinate axis v in a position which is perpendicular to the position in which the roll is divided into halves longitudinally thereof and by establishing a coordinate axis .\&#39;r in a direction at right angles to the coordinate axis y v or along the axes of the rolls in which no deflection is produced.  
  In FIG. 1, if a roll contact pressure distribution applied to the work rolls 2 and 2&#39; by the back-up rolls 3 and 3&#39;. a rolling pressure distribution applied to the steel strip 1 by the work rolls 2 and 2&#39;. a deflection curve of the back-up rolls 3 and 3&#39; produced at this time. and a deflection curve of the work rolls 2 and 2 produced at this time are called f(.\&#39;), P(.\&#39;), ybtx) and yn-(x) respectively. then the following relation holds with respect to the coordinate system .\&#39;y:  
 In a region [X] 5 w/Z In a region [X] w/2 EIwd4yw/dx4 0 Elbd4yb/dx4 0 where E: The modulus of elasticity of the rolls;  
 In: The coefficient of gyration of area of the work rolls;  
 lb: The coefficient of gyration of area of the back-up rolls;  
 W: The width of the strip steel; and  
 L: The overall length of the work rolls and back-up rolls.  
  If the condition of continuity is applied to the foundary between the aforementioned regions in order to solve the deflection equations, vw(.r) can be approximately expressed by the following formula:  
 where Qj is the roll bending pressure, j I is the roll bending pressure on the drive side andj 2 is the roll bending pressure on the driven side. and Pi is the rolling pressure per unit width detected at t points along the width of the strip steel and used instead of the rolling pressure distribution P(.r). The values P P P: can approximate the rolling pressure distribution P(.\&#39;). The value t may be five. a. h and c are constants determined by the width.  
  The aforementioned rolling pressure distribution P(.r) can be approximated by the following formula:  
 P(x) In&#34; 1x mx n.\&#39; p  
 where k, I. m, n and p are functions of rolling pressures per unit width and the width of the strip.  
  On the other hand. in FIG. 2 if a thickness distribution of the strip steel before being rolled. a thickness distribution after being rolled. a flattening distribution produced in the work rolls at the time of rolling. a roll opening distribution set when no load is applied to the rolls and an initial crown produced in the work rolls are called H(.\&#39;), 11(1). MU). S(.r and C\\(.\&#39;) respectively. then the expected width distribution along the width of the steel strip after being rolled htx can be obtained from the following formula:  
 It should be noted that the symbol h(.r) denotes not only an expected thickness of the strip but also an expected roll opening at the time rolling is actually performed. It should be further noted that the symbol vw(.r) denotes a deflection curve of the work roll and can be obtained from the equation (7). Since there are two working rolls acting on the strip, each of the rolls has a deflection curve \W(.\&#39;) and therefore the strip is affected with two t&#39;u&#39;(.r).  
  In the aforesaid formula. the flattening distribution, i.e., a distribution of compressive deformation of the work rolls in the axial direction M(.r) can be expressed as follows by the well-known Hitchcocks formula:  
 where C is the Hitchcocks constant and R is the radius of the work rolls.  
  Also, the rolling pressure P(.\&#39;) can be obtained from the following well-known rolling pressure calculation formula;  
 where kp is the resistance to deformation, Dp is the factor for relating plain strength to a roll gap and R is the radius of the rolls when they are flattened. These values can be calculated by using well-known formulas.  
  Also, the roll gap S(.\&#39;) and the initial crown in the rolls Cw(.r) can be expressed by the following formulas respectively:  
 SLY) 5 l 5 where 8,: The set value of screw-down position on the drive side;  
 S The set value of screw-down position on the oper&#39; ation side;  
 L: The overall length of the rolls;  
 (n-o: The initial crown in the central portion ofa roll;  
 and  
 .r: The length along the roll axis from the central portion of a roll.  
  By calculating the aforementioned formulas. there can be obtained the expected thickness distribution along the width of the steel lt(.r) as a function with re spect to the factors S1, S2, Q1 and 02. In other words, a transverse sectional shape ofa steel strip on the exit side ofa rolling mill during the process of a rolling operation in which roll bender forces are exerted on the rolls can be expressed in a function with respect to the factors S1, S2, 01 and Q2.  
  Let us now consider the requirement for producing a steel strip of a shape which embodies a theoretical ideal therein.  
  By positioning the origin of the coordinates on the center line across the width of strip steel before being rolled. let us call a length of the strip steel at a distance X along the width of the strip steel, a draft percentage of the rolling mill at that point, a mean length of the strip steel across the entire width of the strip steel. a mean draft percentage of the rolls across the entire width of the strip steel, a length of the strip steel at the distance X after rolling, and a mean value of the length across the entire width of the strip steel Lx, r(.\&#39;) Lm, rm. l.\&#39; and lm respectively. Since the strip steel after being rolled must have a uniform length across the entire width thereof in order to impart the best shape thereto. it is essential that l.t lm in the following formula:  
  Thus, the requirement for obtaining a most desirable transverse cross-sectional shape across the width of a strip steel would be that a roll draft percentage r(.r) at a point X along the width of the strip steel should satisfy the following formula:  
  With a roll draft percentage being called r(.r), a theoretical post-rolling thickness distribution h*(.\&#39;) can be expressed by the following formula:  
 Therefore, theoretically speaking, it would be possible to obtain a perfect flatness in a strip steel if the value ofQ could be determined such as to render the formula (l5) equal to the formula (9).  
  However. as aforementioned it is impossible to cause the rolls to produce a deflection such that the aforementioned requirement is met. Therefore, the optimum condition that can be satisfied by using the roll bending device now in actual use would be to determine the roll openings S1 and S2 on the drive side and the operation side respectively and the roll bender forces 01 and 02 on the drive side and the operation side respectively in or a function J given digitally by the following formula;  
  The Alitx) in the formulas l6) and (l7) can be expressed by the following formula; A/HX] S(.\&#39;) Zyu-U) ZCWU) M(.\&#39;) /r*(.\&#39;) lit The aforementioned requirement for obtaining as perfect a flatness as possible in a strip steel can be met in concrete form through a convergent calculation. The procedure of the convergent calculation is shown in FIG. 4. In calculating the formula (ll), h(.\) may be approximated by measured values or by h*(.\&#39;).  
  The aforementioned requirement may be obtained also as follows. In the formula lb) or formula (l7), J is a quadratic formula with respect to S1, 52, Q1 and 02. Accordingly, it is possible to obtain the best possible values for SI, S2, and Q2 by solving the four linear simultaneous equations obtained from J/6Q1 U. 6J/5Q2 0. 61/581 0, 81/552 0. That is. from 61/851 0; A SI A,. ,S2 A Ql A QZ B from 142151 A2252 .424Q2 B2 from 8J/5Ql 0; A SI A 52 A QI A 423 B from 8J/8Q2 U, A Sl A 52 A Ql +A .,Q4=B4 (1 where Ai and Bi are functions of rolling pressure distribution. the width of the strip and the thickness of the strip. The best possible values for S1. S2, 0] and 02 can be obtained by calculating the values of Aij and Bi and then solving the above equations. The procedure of the calculation is shown in FIG. 5. In actual practice, however. 51 S2 and Q! Q2 in many case. If this is the case, an answer which satisfies the aforementioned condition can be provided by solving the simultaneous equations.  
  ln H6. 3. there is shown, in association with an ordinary four-high rolling mill, an apparatus for carrying into practice the method based on the aforementioned principles of the invention. In the figure, the apparatus in schematic form is enclosed in a block 4 defined by dash-and-dot lines and comprises a main control unit 7 adapted to receive a supply of signals from a detector 5 for detecting the length L.\&#39; of a strip steel in several positions along the width of the strip steel before being rolled and another detector 6 for detecting the thickness distribution across the width of the strip steel before being rolled, and two pairs of ancillary control units 8, 8 and 9, 9&#39; receiving the supply of outputs of the main control unit 7. The main control unit 7 also receives from a rolling specifications setter 10 a supply of information on a desired thickness to be produced in a strip steel and the like.  
  The main control unit 7 receiving signals from the detectors 5, 6 and the rolling specifications setter 10 performs a calculation operation on the formulas (l) to (IR) either the formula (16) or (17) being used to obtain the roll opening on the drive side S1, roll opening on the operation side S2, roll bender force exerted on the drive side Q1 and roll bender force exerted on the operation side Q2 when the function J of the formula l7 or I6) is minimized. These values are supplied to the ancillary control units 7, 7&#39;, 9 and 9&#39; respectively.  
  Upon receipt of the signals, the ancillary control units 8, 8&#39; control a roll bending device 11, ll so that the forces exerted thereby may have values 01 and Q2. More specifically, the ancillary control units 8, 8&#39; transmit control signals to valve controllers 13, 13&#39; respectively which control the degree of opening of hydraulic pressure adjusting valves l4, 14&#39; for the roll bending device.  
  Likewise, the ancillary control units 9, 9&#39; control a roll screw-down device 12, 12&#39; respectively upon receipt of signals 81, $2 from the main control unit 7. More specifically, the ancillary control units 9, 9&#39; transmit control signals to valve controllers 130, I30 respectively which control the degree of opening of hydraulic pressure adjusting valve 140, respectively for effecting screw-down of the rolls. The roll bending device 11, ll and the roll screw-down device l2, 12&#39; are of the hydraulically operated type which receive a supply of fluid pressure from a fluid pressure source 15.  
  The detectors 5, 6 are not shown in concrete form. However, the detector 5 may be constructed such that it detects deterioration of the surface shape of the strip steel from a deviation in the length distribution Lx by optical means, by observing reflected microwaves or other known method and calculates backward the length distribution L1 based on these observations. The detector 6 may be of the known type, for example, which comprises a plurality of solenoids or the like arranged across the width of the strip steel for detecting the thickness thereof.  
  From the foregoing description, it will be appreciated that the present invention can contribute greatly to the advance in the progress of the art of controlling a rolling operation because it permits to fully automatically set, based on the theory of rolling and according to the surface shape and thickness distribution of a strip steel to be rolled, a roll bending device for operation which has hitherto been set manually and not based on the theory of rolling.  
  While the invention has been described with reference to a preferred embodiment thereof, it is to be understood that the invention is not limited thereto and the embodiment is shown and described for the purpose of illustration only, and that changes may be made therein without departing from the scope of the invention which is defined in the appended claims.  
 I claim:  
  1. A method of controlling a rolling mill comprising a roll screw-down device and a roll bending device for effecting control of the thickness and shape of a strip steel to be rolled by the rolling mill, comprising the steps of calculating an ideal thickness distribution lr*(.t) desired to be obtained in the strip steel after being rolled based on the values obtained by measuring the thickness distribution and the length distribution across the width of the strip steel before being rolled. calculating an expected roll opening distribution htx) based on an expected deflection of the rolls and an expected flattening of the rolls. and calculating roll bender forces and roll screw-down amounts when the dispersion of the difference between the expected ideal thickness distribution and the expected roll opening distribution are minimized whereby the roll bending device and the roll screw-down device can be controlled by the roll bender forces and the roll screwdown amounts obtained as aforementioned.  
  2. A method according to claim I wherein said step of calculating an expected roll opening distribution comprises the step of calculating a rolling pressure distribution based on the values obtained by measuring the thickness distribution and the length distribution across the width of the strip steel before being rolled and using said rolling pressure distribution for calculating the expected roll opening distribution.  
 3. A control arrangement for a rolling mill comprising: a roll screw down means. a roll bending means for effecting control of the thickness and shape of a strip steel to be rolled by the rolling mill. means for detect ing and supplying a signal of the length of the strip steel at at least one position along the width of the strip steel before being rolled, means for detecting and supplying a signal of the thickness distribution of the strips steel across the width of the strip before being rolled. a main control means for receiving the signals from said length and thickness detecting means and for calculating minimum values of dispersion of the differences of an expected roll opening distribution and an ideal thickness desired to be obtained in the strip steel. a pair of ancillary control means for supplying to the roll bending means output signals consistent with optimum roll bender forces based on an output of said main control means. and another pair of ancillary control means for supplying to the roll screw-down device output signals consistent with optimum roll screw-down amounts based on an output signal of said main control means. k