Shape control apparatus for flat material
BACKGROUND OF THE INVENTION 
     This invention relates to a shape control apparatus for flat material, and more particularly to a shape control apparatus which can form hot rolled steel into an acceptable shape. 
     Generally, as control apparatus of this type, there have been known apparatus wherein the temperature distribution of a hot rolled steel plate in the widthwise direction is measured to determine a widthwise load distribution which, in turn, is used to operate shape controllers such as a roll bending device and a roll coolant device to produce to produce rolled products with a desired shape from flat material. 
     In conventional shape control apparatus of this type, however, no consideration is given to the thermal crown, which varies with time, and the roll wear These are important factors in shape forming and, as a result, the failure to consider these factors leads to the disadvantage that defective shapes arise as time passes or as the number of rolled products increases. 
     SUMMARY OF THE INVENTION 
     This invention has the object of eliminating such disadvantages, and relates to a shape control apparatus for producing rolled products from flat material in which a thermal crown magnitude and a roll wear magnitude in the widthwise direction of the rolls are respectively predicted from the number of rolled products, rolling time interval between a number of rolled products and rolling weight after a changer in the vertical spaceing of the rolls. A rolling load distribution is determined from a widthwise temperature distribution of the flat material determined from temperatures of the material at a plurality of points including a widthwise center of the material lying on the incoming side of the rolls. An optimum roll bending force is determined on the basis of the above results and is used for controlling a roll bending device, whereby rolled products having an acceptable shape are produced from flat material even when time passes and the number of rolled products increases. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a block diagram of a shape control apparatus embodying this invention; 
     FIG. 2 is a graph showing the thermal crown magnitude of rolls in the widthwise direction of a flat material; 
     FIG. 3 is a graph showing the relationship between the thermal crown magnitude at the center of the rolls in the lengthwise direction and the number of rolled products; 
     FIG. 4 is a graph showing the wear magnitude of the rolls in the widthwise direction of the flat material; 
     FIG. 5 is a graph showing the relationship between the roll wear magnitude at the roll center and the rolling weight; 
     FIG. 6 is an explanatory diagram showing the load distribution of a rolling mill in the state in which rolls curve; 
     FIG. 7 is a flow chart for computing the curvature magnitude of the rolls; and 
     FIG. 8 is a flow chart for computing an optimum bending force. 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     The principle of this invention is as described below. 
     An arbitrary time after the rearrangement of rolls in a hot rolling line, the thermal crown magnitude Y T  (x) of the rolls is symmetric with respect to the center of the rolls in the lengthwise direction thereof and can be substantially expressed by a quadratic equation as illustrated in FIG. 2. Illustrated as a function of rolling time or the number of rolled products and, considering a rolling time interval typically occurs between the rolling of different numbers of rolled products, the thermal crown magnitude Y T  (0) at the roll center is as illustrated in FIG. 3. It is seen from FIG. 3 that: 
     (1) The thermal crown magnitude changes rapidly after a change in the vertical spacing of the rolls. 
     (2) As the rolling proceeds, the change in the thermal crown magnitude becomes slower. 
     (3) When a rolling interval such as rolling cessation is introduced of a long duration, the thermal crown magnitude decreases because of a decrease in the temperature of the rolls, and is changed rapidly again after rolling resumes following the interval. 
     In view of the above, the thermal crown magnitude Y T  (x) is expressed by the following equation on the basis of the number N of rolled products and the rolling time interval between the rolling of products after a change in the vertical spacing of the rolls: 
     
         Y.sub.T (x)=(A.sub.T x.sup.2 +B.sub.T x+C.sub.T)·{1-exp (-D.sub.T ·N.sub.E)}                                       (1) 
    
     
         N.sub.E =(N.sub.E.sup.N-1 +1)·exp (-E.sub.T ·T) (2) 
    
     where Y T  (x) . . . the thermal crown magnitude of the rolls, 
     x . . . the coordinate value of the rolls in the longitudinal direction thereof, 
     A T , B T , C T , D T , E T  . . . constants, 
     N E  . . . the equivalent number of rolled products, 
     N E   N-1  . . . the equivalent number of rolled products preceding by one product, 
     τ . . . the period of time of a rolling interval since the rolling of the preceding product. 
     Next, a roll wear magnitude Y W  (x) will be described. An arbitrary time after a change in the vertical spacing of the rolls, roll wear magnitude is also symmetric with respect to the roll center as illustrated in FIG. 4, and it can be expressed by a biquadratic equation. 
     In addition, when the wear magnitude Y W  (0) at the roll center is illustrated versus rolling weight W after a change in the vertical spacing of the rolls, a substantially proportional relation exists and is illustratively shown in FIG. 5. 
     In view of the above, the roll wear magnitude Y W  (x) can be expressed by the following equation on the basis of the rolling weight W after a change in the vertical spacing of the rolls: 
     
         Y.sub.W (x)=(A.sub.W x.sup.4 +B.sub.W x.sup.3 +C.sub.W x.sup.2 +D.sub.W X+E.sub.W)*W                                              (3) 
    
     where 
     Y W  (x) . . . roll wear magnitude, 
     A W , B W , C W , D W , E W  . . . constants, 
     W . . . rolling weight after the change in vertical spacing of the rolls. 
     Next, the curvature magnitude of rolling mill rolls will be described. Usually, a dynamic equation concerning the roll curvature is expressed by the following: ##EQU1## where Y B  . . . the curvature magnitude of a roll axis, 
     E . . . the modulus of longitudinal elasticity of the rolls, 
     I . . . the second moment of area of the rolls, 
     α . . . constant, 
     G . . . the modulus of transverse elasticity of the rolls, 
     A . . . the cross-sectional area of the rolls, 
     P(x) . . . distributed rolling load in the axial direction of the rolls. 
     In order to solve Eq. (4), the load distribution P(x) and boundary conditions may be given. 
     FIG. 6 shows a rolling load distribution in a quadruple rolling mill in the state in which rolls curve. In FIG. 6, the x-axis represents coordinates in the direction of a roll axis (in the widthwise direction of a flat material), while the y-axis represents coordinates indicative of the curvature of the roll axis. 
     A flat material 1 is rolled by upper and lower work rolls 2a and 2b. Under this condition, a load distribution P 1  (x) arises between the flat material 1 and the upper work roll 2a. Simultaneously, a load distribution P 2  (x) arises between the upper work roll 2a and an upper backup roll 3a. Letter P in the figure indicates a rolling load which is detected by a load detector which produces rolling force representing signals processed in the apparatus, and letter F indicates a bending force which acts between the upper and lower work rolls 2a and 2b. Thus, the difference between the rolling load P and bending force F is the rolling weight W. 
     When the balance of the forces is considered in FIG. 6, ##EQU2## where b . . . the width of the flat material. P 1  (x) can be evaluated by knowing the widthwise temperature distribution of the flat material 1: ##EQU3## where R&#39; . . . deviating roll radius, 
     Δh . . . rolling reduction, 
     Q P  . . . reduction force function, 
     K . . . deformation resistance, 
     K o , n, m, α . . . constants, 
     ε . . . strain, 
     ε o  . . . strain velocity, 
     T . . . temperature. 
     In addition, when the load distribution between the upper work roll 2a and the upper backup roll 3a is indicated and the balance of the forces is considered: ##EQU4## holds where L . . . the length of the rolls. 
     In general, Eq. (4) can be solved by computing apparatus utilizing processing steps of a flow chart shown in FIG. 7. 
     As stated before, when the rolling load distribution P 1  (x) is obtained, the roll curvature Y B  can be computed. It is therefore necessary to know the temperature distribution of the flat material in the widthwise direction thereof. 
     The widthwise temperature distribution of the flat material or steel plate in the hot rolling line can be expressed by the following quadratic equation in the light of the fundamental equation of thermal conduction: 
     
         T(X)=T.sub.o -a·x.sup.2                           (9) 
    
     where T o  . . . plate temperature at the center in the widthwise direction of the plate, 
     x . . . distance (coordinate) from the center of the width of the plate, 
     a . . . constant. 
     This can be computed by measuring the temperatures of at least two points including the center of the width of the plate and producing temperature representing signals which are processed in the apparatus. 
     In principle according to the present invention, an optimum roll bending magnitude for producing an acceptable shape of the steel plate favorable is determined for control of a roll bending device on the basis of a computed thermal crown value Y T  (x) in the widthwise direction of the rolls and a computed roll wear value Y W  (x) in the widthwise direction, which are based on the rolling history consisting of the number of products rolled and time intervals between rolling of products after a change in the vertical spacing weight of the rolls, and the roll curvature magnitude Y B  (x) which is based on the rolling load determined from the temperature distribution computed from detected temperatures. 
     In judging the acceptable shape of the plate, the total value Y(x) is determined from the computed values of thermal crown Y T  (x), the computed roll wear value Y W  (x) and the roll curvature magnitude value Y B  (x); 
     
         Y(x)=Y.sub.T (x)-Y.sub.w (x)+Y.sub.B (x)                   (10) 
    
     A criterion that the square deviation of the total value from Y=0 is minimized determines the optimum bending force F OPT  : ##EQU5## The optimum bending force F OPT  can be computed in accordance with a flow chart shown in FIG. 8. 
     Now, one embodiment of this invention will be described with reference to FIG. 1. 
     Referring to the figure, numeral 1 designates a flat material or steel plate, symbols 2a and 2b upper and lower work rolls, and symbols 3a and 3b upper and lower backup rolls. A thermal crown magnitude calculating means, herein shown as a calculator 4, receives hysteresis data signals on the period of time of the rolling interval between rolled products and the number of rolled products counted after the change in spacing of rolls to determine the thermal crown magnitude Y T  (x) in accordance with Eq. (1). A roll wear calculating means, herein shown as a roll wear calculator (5), receives data on the rolling weight after rearrangement of the rolls to determine the wear magnitude Y W  (x) in accordance with Eq. (3). Both quantities Y T  (x) and Y W  (x) are determined only once before the steel plate 1 is bitten into the rolling mill between the rolls. Shown at numeral 6 is a thermometer for measuring the temperatures of the steel plate at at least two points including the widthwise center of the part of the steel plate 1 located on the incoming side of the rolling mill and produces temperature-representing signals. A temperature distribution determining means, herein shown as a calculator 7, determines the widthwise temperature distribution from the temperature outputs provided by the thermometer 6 in accordance with Eq. (9). A rolling load distribution determining means, herein shown as a rolling load distribution calculator 8, is connected to the temperature distribution determining means to determine the widthwise rolling load distribution on the rolls from the widthwise temperature distribution in accordance with Eq. (6). 
     An optimum bending force determining means, herein shown in part as a calculator 9, receives the output values Y T  (x), Y W  (x) and P 1  (x) of the respective calculators 4, 5 and 8 as input values to determine the optimum bending force F OPT  to be applied to the rolls in accordance with the flow chart of FIG. 8. 
     The optimum bending force determining means further includes a controlling means, herein shown as a bending controller 10, receiving the output value of the optimum bending force F OPT  to control bending in consideration of the time at which the point of the steel plate 1 measured by the thermometer 6 reaches the rolling mill. 
     Thus, this embodiment takes into account, not only the roll curvature based on the temperatures distributed in the widthwise direction of the plate, but also the thermal crown magnitude of the rolls and the roll wear magnitude based on the rolling history information after a change in spacing of the rolls, so that an acceptable shape control is possible even when the number of rolled products increases or when rolling is stopped. 
     As set forth above, according to this invention, a thermal crown magnitude and a roll wear magnitude in the widthwise direction of rolls as based on rolling history information i.e., number of products rolled and time intervals of rolling, after a change in vertical spacing of the work rolls are respectively determined, while a rolling load distribution is determined from the widthwise temperature distribution of a flat material determined from temperatures of the material at a plurality of points lying on the incoming side of a rolling mill. An optimum roll bending force is determined on the basis of the results as calculated above and is used for bending control, so that a flat material having an acceptable shape can always be produced even when the number of rolled products increases or when rolling is stopped.