JP7644343B2 - METHOD FOR CONTROLLING ROLLING OF METAL SHEET, ROLLING CONTROL DEVICE, AND METHOD FOR MANUFACTURING ROLLED METAL SHEET - Google Patents

Description

The present invention relates to a rolling control method for controlling the shape of a metal plate after rolling, a rolling control device for executing the rolling control method, and a method for manufacturing a rolled metal plate.

Various methods have been proposed to predict the shape of metal plates, such as thin or thick plates, after they have been rolled.

Patent Document 1 discloses a technology that improves prediction accuracy in the extrapolation region where no performance data exists, and further corrects errors in the rolling model. Specifically, a performance database that stores the manufacturing conditions of products manufactured in the past in association with information on the results of that manufacturing is used to calculate the similarity between each sample in the performance database and the required point (prediction target point), and creates a prediction formula for the vicinity of the required point by weighted regression using this similarity as a weight. This prediction formula improves prediction accuracy in the extrapolation region.

Patent Document 2 discloses a technology that separates the elongation strain (stress) distributed in the width direction of a metal plate during rolling into elongation strain that is geometrically transformed into a wave shape upon buckling and elongation strain that remains in the metal plate even after buckling, and predicts the shape of the metal plate.

Patent Document 3 discloses a technology for predicting the shape of a metal plate by determining the elongation strain inherent in the metal plate during measurement in addition to the shape feature quantities of the metal plate measured at the exit of the rolling mill, and measuring this as the true shape feature quantities imparted by the rolling mill by superimposing it on the shape feature quantities. Note that here, the plate threading direction, plate width direction position, and height direction displacement are measured at the exit of the rolling mill as geometric values, and the profile, steepness, and elongation strain difference are obtained as shape feature quantities.

Patent Document 4 discloses a method for controlling rolling of a metal plate, which includes the following first to third steps. In the first step, a provisional elongation-strain difference distribution of the metal plate during rolling is obtained under conditions that restrict the out-of-plane deformation of the metal plate, and a buckling critical strain difference distribution is obtained based on this provisional elongation-strain difference distribution, the thickness and width of the metal plate, and the tension acting on the metal plate at the exit side of the rolling mill. In the second step, when the provisional elongation-strain difference distribution exceeds the buckling critical strain difference distribution, the difference between the provisional elongation-strain difference distribution and the buckling critical strain difference distribution is obtained, and the difference is added to the provisional elongation-strain difference distribution to obtain the true elongation-strain difference distribution. In the third step, rolling conditions are set based on the true elongation-strain difference distribution, and the shape of the metal plate is controlled by rolling the metal plate.

JP 2008-112288 A JP 2005-153011 A JP 2012-218010 A Patent No. 6172401

However, the method disclosed in Patent Document 1 does not take into account nonlinear phenomena such as the buckling phenomenon of metal plates, and the nonlinear phenomena cannot be reflected in the prediction formula. If nonlinear phenomena are not taken into account, errors will occur in the model, making it impossible to accurately predict the shape of the metal plate after rolling.

The methods disclosed in Patent Documents 2 to 4 each predict the shape of a metal plate by taking into account the buckling phenomenon of the metal plate, and the prediction accuracy is improved compared to when the buckling phenomenon is not taken into account. In particular, the invention in Patent Document 4 is an extension of the inventions in Patent Documents 2 and 3, and by quantitatively grasping the correlation between the rolling load difference distribution and the elongation strain difference distribution in the plate width direction of the metal plate, which changes due to buckling, the true elongation strain difference distribution of the metal plate is obtained and the shape of the metal plate is predicted. However, as a result of careful investigation by the inventors, it was found that there is room for improvement in the prediction accuracy, as described below.

In the method disclosed in Patent Document 4, after a buckling judgment is performed on the metal plate, if the metal plate does not buckle (the provisional elongation strain difference distribution described above does not exceed the buckling critical strain difference distribution), the metal plate is rolled without changing the specified rolling conditions. On the other hand, if the metal plate buckles (the provisional elongation strain difference distribution exceeds the buckling critical strain difference distribution), the metal plate is rolled under rolling conditions set based on the true elongation strain difference distribution. Therefore, it is essential to improve the prediction accuracy of the shape of the metal plate.

The present invention was made in consideration of these points, and aims to accurately predict the shape of a metal sheet after rolling and freely control the shape of the metal sheet.

To achieve the above objective, the inventors predicted the shape of a metal sheet after rolling and investigated a method for controlling the shape of the metal sheet based on the predicted shape of the metal sheet, resulting in the following findings.

In the method disclosed in Patent Document 4, the true elongation and strain difference distribution of the metal plate is obtained by quantitatively grasping the correlation between the rolling load difference distribution and the elongation and strain difference distribution that change due to the above-mentioned buckling, and the shape of the metal plate is predicted. In other words, when the elongation and strain difference that is distributed in the width direction of the metal plate and that is converted into a wave shape and causes out-of-plane deformation is actually converted into a wave shape due to buckling of the metal plate, the load distribution corresponding to the elongation and strain difference is further converted into an elongation and strain difference and is contained in the metal plate. Then, the true elongation and strain difference distribution of the metal plate increases by the amount of this increased elongation and strain difference. In this way, the shape of the metal plate is controlled by predicting the true elongation and strain difference distribution of the metal plate.

The present invention is a further development of the invention of Patent Document 4. After intensive research, the inventors found that in the invention of Patent Document 4, an error occurs in the shape profile because it is assumed that the shape profile calculated by the buckling eigenvalue analysis remains the same even after buckling. Then, the inventors came up with the idea that the shape of the metal plate can be appropriately predicted by recalculating (eigenvalue analysis calculation using a theoretical buckling model) with the elastic strain difference reallocated due to the occurrence of the shape profile generated after buckling, rather than calculating with the plastic elongation strain difference of the evaluation target in the buckling eigenvalue analysis. That is, it was previously thought that the disclosed shape profile after buckling was determined by the plastic elongation strain difference used in the initial calculation, but as a result of intensive research by the inventors, it was found that the shape profile is determined by the elastic strain difference that is reallocated to the metal plate after the buckling shape occurs, that is, the elastic strain difference that exists in the plate even after buckling. Then, by predicting the true elongation strain difference from the determined shape profile, the shape of the metal plate can be controlled with higher accuracy. The present invention is based on the above findings, and the gist of the present invention is as follows.

According to the present invention, there is provided a method for predicting and controlling the shape of a metal plate, which includes the steps of: determining a true elongation and strain difference by buckling analysis using an elastic strain difference that exists in the metal plate even after buckling when predicting a wave shape profile after buckling based on a plastic elongation and strain difference imparted to the metal plate; and rolling the metal plate under rolling conditions set based on the true elongation and strain difference.

In the rolling control method for the metal plate, the step of determining the true elongation strain difference and the step of rolling the metal plate comprise determining a critical tension in the plate width direction at which the metal plate buckles based on a plastic elongation strain difference, which is the difference in the strain in the rolling direction of the metal plate during rolling under specified rolling conditions, the thickness of the metal plate, the width of the metal plate, and the tension acting on the metal plate at the exit side of the rolling mill, which is determined under conditions in which out-of-plane deformation of the metal plate is constrained by using an up-down symmetrical model that allows the displacement of the center of thickness of the metal plate to be in-plane displacement of a reference plane that passes through the midpoint of a line connecting the rotation centers of the upper and lower rolls and is parallel to the plate surface of the metal plate, and does not allow the displacement to be out-of-plane of the reference plane. The method may include a first step of calculating a buckling inherent strain difference, which is a strain difference; a second step of calculating a new buckling inherent strain difference using an elastic strain difference that is reallocated by the generation of buckling waves and remains in the metal plate even after buckling when the plastic elongation strain difference exceeds the buckling inherent strain difference, and calculating a true elongation strain difference by adding the difference between the plastic elongation strain difference and the new buckling inherent strain difference to the plastic elongation strain difference; and a third step of rolling the metal plate without changing the predetermined rolling conditions when the plastic elongation strain difference does not exceed the buckling inherent strain difference, and rolling the metal plate under rolling conditions set based on the true elongation strain difference when the plastic elongation strain difference exceeds the buckling inherent strain difference.

The metal plate rolling control method may further include a step of determining the plastic elongation strain difference.

In the second step of the rolling control method for the metal plate, a converted tension is calculated by converting the difference between the plastic elongation strain difference and the buckling inherent strain difference into a tension acting on the metal plate at the exit side of the rolling mill, and the true elongation strain difference is calculated by adding the elongation strain difference corresponding to the converted tension to the plastic elongation strain difference.

In the second step of the rolling control method for the metal plate, the rolling load difference in the width direction of the metal plate corresponding to the converted tension may be second-order differentiated in the width direction to obtain the elongation strain difference corresponding to the converted tension.

In the method for controlling rolling of a metal plate, the step of determining the true elongation strain difference and the step of rolling the metal plate comprise a first step of determining a rolling load difference, which is the difference in rolling load in the width direction of the metal plate during rolling under specified rolling conditions, and a plastic elongation strain difference, which is the difference in the strain in the width direction of the metal plate during rolling, under conditions in which the displacement of the center of thickness of the metal plate is allowed to be a displacement in the plane of a reference plane that passes through the midpoint of a line connecting the rotation centers of the upper and lower rolls and is parallel to the plate surface of the metal plate, and is not allowed to be a displacement out of the plane of the reference plane, under conditions in which the out-of-plane deformation of the metal plate is constrained; and a second step of determining a buckling inherent strain difference, which is the critical strain difference in the width direction at which the metal plate buckles, based on the plastic elongation strain difference, the thickness of the metal plate, the width of the metal plate, and the tension acting on the metal plate at the exit side of the rolling mill. When the plastic elongation strain difference exceeds the buckling inherent strain difference, a new buckling inherent strain difference is calculated using the elastic strain difference that is reallocated by the generation of buckling waves and remains in the metal plate even after buckling, and a buckling inherent load difference that is a rolling load difference corresponding to the new buckling inherent strain difference is calculated from the correlation between the rolling load difference and the plastic elongation strain difference, and the difference between the rolling load difference and the buckling inherent load difference is calculated, and there is no change in the crown ratio of the metal plate between the exit side and the entry side of the rolling mill. The method may include a third step of calculating a true elongation-strain difference by adding the buckling shape transformation elongation-strain difference and the plastic elongation-strain difference, assuming that the buckling shape transformation elongation-strain difference is greater than the buckling inherent strain difference, and a fourth step of rolling the metal plate without changing the specified rolling conditions if the plastic elongation-strain difference does not exceed the buckling inherent strain difference, and rolling the metal plate under rolling conditions set based on the true elongation-strain difference if the plastic elongation-strain difference exceeds the buckling inherent strain difference.

In the rolling control method for the metal plate, the step of calculating the true elongation strain difference and the step of rolling the metal plate are performed under conditions in which the out-of-plane deformation of the metal plate is constrained by using an upper-lower symmetrical model that allows the displacement of the center of thickness of the metal plate to be in-plane displacement of a reference plane that passes through the midpoint of a line connecting the rotation centers of the upper and lower rolls and is parallel to the plate surface of the metal plate, and does not allow the displacement to be out-of-plane of the reference plane, and is the difference in rolling load in the width direction of the metal plate during rolling under specified rolling conditions. A first step of calculating a plastic elongation strain difference, which is a difference in the width direction of the metal plate between the rolling load difference and the strain elongating in the rolling direction of the metal plate during rolling; a second step of calculating a buckling inherent strain difference, which is a critical strain difference in the width direction at which the metal plate buckles, based on the plastic elongation strain difference, the thickness of the metal plate, the width of the metal plate, and the tension acting on the metal plate at the exit side of the rolling mill; and a second step of calculating a buckling inherent strain difference, which is a critical strain difference in the width direction at which the metal plate buckles, based on the plastic elongation strain difference, the thickness of the metal plate, the width of the metal plate, and the tension acting on the metal plate at the exit side of the rolling mill. A third step of calculating a new buckling inherent strain difference using an elastic strain difference inherent in the metal plate, calculating a buckling shape transformation load difference corresponding to a buckling shape transformation elongation strain difference, which is the difference between the plastic elongation strain difference and the new buckling inherent strain difference, from the correlation between the rolling load difference and the plastic elongation strain difference, and deriving a new rolling load difference by superimposing the buckling shape transformation load difference on the rolling load difference, and assuming that the metal plate has a crown ratio change, determining the new rolling load difference; The method may include a fourth step of calculating a true elongation-strain difference by adding the plastic elongation-strain difference to the true elongation-strain difference and determining whether the true elongation-strain difference has converged, and a fifth step of rolling the metal plate without changing the predetermined rolling conditions if the plastic elongation-strain difference does not exceed the buckling inherent strain difference calculated in the second step, and rolling the metal plate under rolling conditions set based on the true elongation-strain difference if the plastic elongation-strain difference exceeds the buckling inherent strain difference calculated in the second step.

In the method for controlling rolling of a metal plate, it may be assumed that the new plastic elongation strain difference obtained in the third step is the plastic elongation strain difference obtained in the first step, and that the new buckling inherent strain difference obtained in the third step is the buckling inherent strain difference obtained in the second step.

In the method for controlling rolling of a metal plate, the metal plate may be deformed out of plane at the entry side of the rolling mill.

The method for controlling rolling of the metal plate may further include a step of measuring the shape of the metal plate after rolling using a shape meter installed on the exit side of the rolling mill, and a step of correcting the plastic elongation strain difference based on the difference between the actual elongation strain difference converted into out-of-plane deformation obtained from the measured shape of the metal plate and the predicted elongation strain difference converted into out-of-plane deformation.

According to another aspect of the present invention, a plastic elongation strain difference, which is the difference in the strain in the width direction of the metal plate that extends in the rolling direction of the metal plate during rolling under specified rolling conditions, is obtained under conditions in which the out-of-plane deformation of the metal plate is constrained by using an up-down symmetrical model that allows the displacement of the center of thickness of the metal plate to be in the plane of a reference plane that passes through the midpoint of a line connecting the rotation centers of the upper and lower rolls and is parallel to the plate surface of the metal plate, and does not allow the displacement to be out-of-plane of the reference plane; a buckling inherent strain, which is the critical strain difference in the width direction at which the metal plate buckles, is obtained based on the thickness of the metal plate, the width of the metal plate, and the tension acting on the metal plate at the exit side of the rolling mill. A rolling control device for a metal plate is provided, which includes: a calculation unit that calculates a true elongation-strain difference by adding the difference between the plastic elongation-strain difference and the new buckling inherent strain difference to the plastic elongation-strain difference when the plastic elongation-strain difference exceeds the buckling inherent strain difference; and a control unit that rolls the metal plate without changing the predetermined rolling conditions when the plastic elongation-strain difference does not exceed the buckling inherent strain difference, and rolls the metal plate under rolling conditions set based on the true elongation-strain difference when the plastic elongation-strain difference exceeds the buckling inherent strain difference.

According to another aspect of the present invention, a first step of calculating a buckling inherent strain difference, which is a critical strain difference in the width direction at which the metal plate buckles, based on a plastic elongation strain difference, which is the difference in the strain in the rolling direction of the metal plate during rolling under specified rolling conditions, in the width direction at which the metal plate buckles, and the thickness of the metal plate, the width of the metal plate, and the tension acting on the metal plate at the exit side of the rolling mill, and a first step of calculating a buckling inherent strain difference, which is the critical strain difference in the width direction at which the metal plate buckles, based on a top-bottom symmetrical model that allows the displacement of the center of thickness of the metal plate to be in the plane of a reference plane that passes through the midpoint of a line connecting the rotation centers of the top and bottom rolls and is parallel to the plate surface of the metal plate, and does not allow the displacement to be out of the plane of the reference plane, and a second step of calculating a buckling inherent strain difference, which is the critical strain difference in the width direction at which the metal plate buckles, based on a plastic elongation strain difference, which is the difference in the width direction of the strain in the rolling direction of the metal plate during rolling under specified rolling conditions, and a tension acting on the metal plate at the exit side of the rolling mill; A method for manufacturing a rolled metal plate is provided, which includes a second step of calculating a new buckling inherent strain difference using the elastic strain difference that is reallocated by the generation of buckling waves and remains in the metal plate even after buckling, and calculating a true elongation strain difference by adding the difference between the new buckling inherent strain difference and the plastic elongation strain difference to the plastic elongation strain difference, if the plastic elongation strain difference does not exceed the buckling inherent strain difference, rolling the metal plate without changing the specified rolling conditions, and rolling the metal plate under rolling conditions set based on the true elongation strain difference, if the plastic elongation strain difference exceeds the buckling inherent strain difference.

According to the present invention, the elastic strain difference that is redistributed due to the shape deformation after buckling is calculated, and recalculation is performed using this elastic strain difference (eigenvalue analysis calculation using a theoretical buckling model), so the shape of the metal plate (true elongation strain difference) can be predicted with high accuracy. As a result, by setting the rolling conditions based on this true elongation strain difference and rolling the metal plate under these rolling conditions, the shape of the metal plate after rolling can be freely controlled.

FIG. 1 is a diagram illustrating a target wave shape. FIG. 13 is a diagram showing a plastic elongation strain difference Δε _pl (y) and a rolling load difference ΔP(y) of a metal plate when the metal plate is rolled under conditions in which the out-of-plane deformation of the metal plate is restrained. This figure shows the buckling inherent strain difference Δε cr (y) and buckling shape transformation elongation strain difference Δε _{ts (y) that constitute the plastic elongation strain difference Δε pl} (y), and the buckling inherent load difference ΔP _cr (y) and buckling shape transformation load difference ΔP _ts (y) that constitute the rolling load difference _ΔP (y) when a metal plate is rolled under conditions in which the out-of-plane deformation of the metal _plate is constrained. FIG. 13 is a diagram showing a state after the buckling shape transformation elongation strain difference Δε _ts (y) and the buckling shape transformation load difference ΔP _ts (y) disappear when the metal plate is allowed to undergo out-of-plane deformation. FIG. 13 is a diagram showing how metal flows into a load-reduced region in the roll bite, increasing the difference in elongation strain in the metal sheet. 2 to 5, and are explanatory diagrams showing a schematic plan view of the relationship between the elongation strain difference of a metal plate and the rolling load in a conventional metal plate rolling control method, in which (a) shows the plastic elongation strain difference Δε _pl (y), (b) shows the buckling inherent strain difference Δε _cr (y) and the buckling shape transformation elongation strain difference Δε _ts (y), and (c) shows the true elongation strain difference Δε' _pl (y). 1 is a flowchart showing a conventional rolling control method. FIG. 1 is an explanatory diagram showing a shape prediction result of a metal sheet when a conventional rolling control method (shape prediction method) and FEM are used. FIG. 1 is an explanatory diagram showing the shape of a metal sheet when a conventional rolling control method (shape prediction method) and FEM are used. 2 to 5, and are explanatory diagrams showing a schematic plan view of the relationship between the elongation strain difference of a metal plate and the rolling load in the rolling control method of the metal plate of the present embodiment, in which (a) shows the plastic elongation strain difference Δε _pl (y), (b) shows the buckling inherent strain difference Δε _cr (y) and the buckling shape transformation elongation strain difference Δε _ts (y), and (c) shows the true elongation strain difference Δε' _pl (y). 2 is a flowchart showing a rolling control method according to the first embodiment. FIG. 2 is an explanatory diagram illustrating a rolling control method according to the first embodiment. FIG. 4 is an explanatory diagram showing a shape prediction result of a metal sheet in the case where the rolling control method according to the first embodiment is used. FIG. 4 is an explanatory diagram showing a shape prediction result of a metal sheet in the case where the rolling control method according to the first embodiment is used. 5 is a flowchart showing a rolling control method according to a second embodiment. 10 is a flowchart showing a rolling control method according to a third embodiment. FIG. 1 is a diagram showing a schematic diagram of a rolling line equipped with a rolling mill, a rolling control device, and a shape meter.

Embodiments of the present invention will be described below with reference to the drawings. In this specification and the drawings, elements having substantially the same functional configuration are designated by the same reference numerals to avoid redundant description.

<Symmetrical wave shape>
First, the target wave shape will be described with reference to FIG. 1. The X-axis is the longitudinal direction in which periodic waves occur, the Y-axis is the plate width direction, and the Z-axis is the plate thickness or wave height direction. The wave shown in FIG. 1 shows a typical ear wave. Note that periodic buckling waves (X direction in the figure) of waves represented by medium waves and quarter waves occurring at an intermediate position between ear waves and medium waves are also targets. The definition of steepness or flatness used as the size of the wave is expressed as a percentage by dividing the wave piece amplitude height H of the width direction edge part of the metal plate by the wave pitch P (period P) and multiplying it by 100. In addition, this wave shape is called a width direction position-wise shape profile (wave shape profile) cut out at the position indicated by the wave piece amplitude height H in the Y-Z cross section, and the shape profile obtained in the buckling analysis described later is a standardized profile of 0 to 1 without a height dimension.

<The principle behind the generation of elongation strain differences in metal sheets>
Next, the principle of the occurrence of a strain difference (hereinafter referred to as "elongation strain difference") in the longitudinal direction (rolling direction) of a metal plate when the rolled metal plate buckles (when out-of-plane deformation occurs in the metal plate) will be described with reference to Figures 2 to 6. Figure 6 corresponds to Figures 2 to 5 and is an explanatory diagram that shows a schematic plan view of the relationship between the elongation strain difference in a metal plate and the rolling load. As will be described later, Figure 6 is a relationship diagram between the elongation strain difference and the rolling load in a conventional method for controlling the rolling of a metal plate.

As shown in Fig. 2, a metal sheet H is rolled using a rolling mill 10 equipped with a pair of rolls. The X-axis in Fig. 2 indicates the longitudinal direction of the metal sheet H, and the metal sheet H is transported and rolled from the negative side of the X-axis to the positive side. The Y-axis in Fig. 2 indicates the width direction of the metal sheet H. Fig. 2 illustrates half of the metal sheet H in the width direction, i.e., from the center _Hc to the edge _He in the width direction of the metal sheet H.

FIG. 2 illustrates the plastic elongation strain difference Δε pl (y) in the width direction of the metal sheet H in the roll bite and the rolling load difference _ΔP (y) in the width direction acting in the vertical direction (Z-axis direction) of the metal sheet H when the metal sheet H is rolled under the condition that the out-of-plane deformation of the metal sheet H is restrained (i.e., the condition that the out-of-plane deformation of the metal sheet H is not allowed). The plastic elongation strain difference Δε _pl (y) is the difference in the elongation strain in the longitudinal direction of the metal sheet during rolling at the width direction position y based on the elongation strain of the edge H _e in the width direction when the rolled metal sheet H buckles (when the metal sheet has an out-of-plane deformation). In the following description, the definitions of the elongation strain and the elongation strain difference are the same as above. The rolling load difference ΔP(y) is the rolling load difference at the width direction position y based on the rolling load of the edge H _e in the width direction of the metal sheet H. In addition, the plastic elongation strain difference Δε _pl (y) and the rolling load difference ΔP(y) correspond to each other at a 1:1 ratio in the sheet width direction. In Fig. 2, the out-of-plane deformation of the metal sheet H is restrained, so that compressive stress is generated in the longitudinal direction immediately after the roll bite exit (bold arrow in Fig. 2). The relationship between the plastic elongation strain difference Δε _pl (y) and the rolling load difference ΔP(y) shown in Fig. 2 is shown in Fig. 6(a) as a schematic diagram.

As shown in Fig. 3, the plastic elongation strain difference Δε _pl (y) is separated into an elongation strain difference Δε _cr (y) that remains in the metal sheet H even after buckling (hereinafter referred to as "buckling inherent strain difference Δε _cr (y)") and an elongation strain difference Δε _ts (y) that is converted into a wavy out-of-plane deformation after buckling (hereinafter referred to as "buckling shape conversion elongation strain difference Δε _ts (y)"). Of these, the buckling inherent strain difference _{Δε cr} (y) is the limit strain difference beyond which the metal sheet H will buckle. In other words, the buckling inherent strain difference Δε _cr (y) is the critical strain difference in the sheet width direction at which the metal sheet H will buckle. Similarly, the rolling load difference ΔP(y) is separated into a rolling load difference ΔP _cr (y) (hereinafter referred to as "buckling inherent load difference ΔP _cr (y)") that corresponds 1:1 in the sheet width direction to the buckling inherent strain difference Δε _cr (y) and a rolling load difference ΔP _ts (y) (hereinafter referred to as "buckling shape transformation load difference ΔP _ts (y)") that corresponds 1:1 in the sheet width direction to the buckling shape transformation elongation strain difference Δε _ts (y). The buckling inherent strain difference Δε _cr (y), buckling shape transformation elongation strain difference Δε _ts (y), buckling inherent load difference ΔP _cr (y), and buckling shape transformation load difference ΔP _ts (y) shown in FIG. 3 are shown in FIG. 6(b) as a schematic.

Next, when the metal sheet H is allowed to deform out of plane, the buckling shape transformation elongation strain difference Δε _ts (y) is converted to out of plane deformation and disappears as shown in FIG. 4. Also, the compressive stress shown by the thick arrow in FIG. 2 decreases, and the apparent longitudinal tension acting on the metal sheet H increases (thick arrow in FIG. 4). Then, the rolling load corresponding to this tension, that is, the buckling shape transformation load difference ΔP _ts (y) corresponding to the buckling shape transformation elongation strain difference Δε _ts (y), disappears. When the buckling shape transformation load difference ΔP _ts (y) disappears, metal flows in the sheet width direction toward the load reduction region, that is, from the center H _c of the metal sheet H toward the edge H _e (thick arrow in FIG. 5) as shown in FIG. 5. As a result, according to the principle of constant volume, the elongation strain at the edge H _e of the metal sheet H increases according to the amount of metal flowing in the sheet width direction. That is, an increase in the elongation strain difference occurs corresponding to the disappearance of the buckling shape transformation load difference ΔP _ts (y) (thin arrow in FIG. 5). Therefore, as shown in FIG. 6(c), the true elongation strain difference Δε' _pl (y) in the metal plate H can be obtained by adding the elongation strain difference Δε _n (y) (hereinafter referred to as "buckling promoting strain difference Δε _n (y)") that increases corresponding to the disappearance of the buckling shape transformation load difference ΔP _ts (y) to the plastic elongation strain difference Δε _pl (y) when the out-of-plane deformation of the metal plate H shown in FIG. 2 is restrained. The buckling promoting strain difference Δε _n (y) is the elongation strain difference caused by the buckling of the metal plate H, and is a strain difference that is not observed when the out-of-plane deformation of the metal plate H is restrained, since buckling does not occur. In addition, the buckling shape transformation elongation strain difference Δε _ts (y) and the buckling promotion strain difference Δε _n (y) are both elongation strain differences corresponding to the buckling shape transformation load difference ΔP _ts (y), and these have the same distribution, but for convenience, different terms are used.

As described above, in terms of the rolling load difference and elongation strain difference in the width direction of the metal sheet H that change due to buckling, when the out-of-plane deformation of the metal sheet H is restrained, there is a correlation between the rolling load difference ΔP(y) and the plastic elongation strain difference Δε _pl (y) shown in Figure 6(a), and there is also a correlation between the rolling load differences ΔP _cr (y), ΔP _ts (y) and the elongation strain differences Δε _cr (y), Δε _ts (y) shown in Figure 6(b). On the other hand, when the out-of-plane deformation of the metal sheet H is permitted, there is a correlation between the rolling load difference ΔP _cr (y) and the elongation strain differences Δε _cr (y), Δε _ts (y), Δε _n (y) shown in Figure 6(c). Furthermore, the true elongation strain difference Δε' _pl (y) shown in Figure 6(c) is increased by the buckling promoting strain difference Δε n (y) compared to the plastic elongation strain difference Δε _pl (y) obtained under the conditions in which the out-of-plane deformation is restrained as shown in Figures 6(a) and ₆ (b), and the following formula (1) is derived.
Δε' _pl (y)=Δε _pl (y)+Δε _n (y) ...(1)

<Conventional rolling control method>
As described above, the present invention is a further development of the invention disclosed in Patent Document 4. Therefore, before describing the rolling control method according to the present embodiment, a conventional rolling control method disclosed in Patent Document 4 will be described. Fig. 7 is a flowchart showing the conventional shape prediction method.

(Step X1)
In step X1, a plastic elongation strain difference Δε _pl (y) of a metal plate to be evaluated at an arbitrary widthwise position y divided into N portions at a predetermined width in the widthwise direction (y direction) is set.

(Step X2)
In step X2, an eigenvalue analysis calculation is performed using a theoretical buckling model. The theoretical buckling model is a model that performs buckling analysis using a wave-shaped buckling equation created based on a model formulated with a triangular residual stress distribution, for example, as shown in the Journal of the Japan Society for Technology of Plasticity, Plasticity and Processing, Vol. 28, No. 312 (1987-1), pp. 58-66. In the theoretical buckling model, the buckling inherent strain difference Δε cr (y) is calculated based on the plastic elongation strain difference Δε _pl (y), the thickness of the metal plate, the width of the metal plate, and the tension acting on the metal _plate .

(Step X3)
In step X3, the occurrence of buckling is judged. The plastic elongation strain difference Δε _pl (y) is compared with the buckling inherent strain difference Δε _cr (y), and if Δε _pl (y) is equal to or less than Δε _cr (y), it is judged that the metal plate is flat and not buckled, and the process proceeds to step X4. On the other hand, if Δε _pl (y) is greater than Δε _cr (y), it is judged that the metal plate will buckle, and the process proceeds to step X5.

(Step X4)
In step X4, it is determined that the metal plate does not buckle and is flat. In this case, the shape of the metal plate is controlled by rolling the metal plate without changing the rolling conditions.

(Step X5)
In step X5, it is determined that the metal plate buckles. In this case, the difference between the plastic elongation strain difference Δε _pl (y) and the buckling inherent strain difference Δε _cr (y) is calculated, and this difference becomes the buckling promoting strain difference Δε _n (y) shown in Fig. 6 (c). Then, according to the above formula (1), the buckling promoting strain difference Δε _n (y) is added to the plastic elongation strain difference Δε _pl (y) to predict the true elongation strain difference Δε' _pl (y).

(Step X6)
In step X6, the rolling conditions are set based on the true elongation strain difference Δε′ _pl (y) predicted in step X5, and the shape of the metal sheet after rolling is controlled. Specifically, for example, the rolling conditions are set so that the true elongation strain difference Δε′ _pl (y) is equal to or smaller than the buckling inherent strain difference Δε _cr (y). In this way, the metal sheet after rolling does not buckle and becomes flat.

<Verification of conventional rolling control methods>
In the conventional method, the buckling eigenanalysis is performed as described above to calculate the buckling eigenstrain difference Δε _cr (y), and the true elongation strain difference Δε' _pl (y) is predicted to control the rolling of the metal plate. Here, the inventors have conducted extensive research and found that the shape profile predicted by the conventional buckling eigenanalysis has an error from the actual shape profile. Specifically, the conventionally predicted shape was verified by comparing it with the analysis result by the finite element method (FEM).

Figures 8 and 9 are explanatory diagrams showing the shape of a metal sheet when using the conventional method and FEM under the same metal sheet conditions and the same plastic elongation strain difference. Note that Figure 8 shows the plate thickness direction displacement during the longitudinal displacement of the plate width center part, i.e., the wave height direction displacement, and Figure 9 shows an enlarged view of the wave height profile of cross section A in Figure 8.

Referring to Figure 8, the wave pitch after buckling is almost the same when comparing the conventional method with FEM. On the other hand, referring to Figure 9, the wave height displacement in the width direction center part is almost the same when comparing the conventional method with FEM, but the wave height displacement in the area surrounded by the dotted line in Figure 9 is different, and it was found that an error occurs in the shape profile.

The inventors have conducted extensive research and found that the cause of the error in the shape profile is that the conventional method assumes that the shape profile calculated by the buckling eigenanalysis in step X2 remains the same even after buckling. The inventors have conceived that the shape of the metal plate can be appropriately predicted by recalculating the elastic strain difference (eigenvalue analysis calculation using a theoretical buckling model) that exists in the metal plate as a residual stress even after buckling, which is redistributed to the metal plate due to the generation of the buckling wave after buckling, rather than calculating by the plastic elongation strain difference of the evaluation target in the buckling eigenvalue analysis. Specifically, when the method of the present invention is used for the buckling eigenstrain difference Δε _cr (y) and the buckling eigenload difference ΔP _cr (y) calculated by the conventional method shown in FIG. 6, the shapes of the buckling eigenstrain difference Δε _cr (y) and the buckling eigenload difference ΔP _cr (y) of different shapes are obtained as shown in FIG. 10. In the following description, a method for controlling rolling of a metal sheet according to this embodiment will be described based on the above findings.

<Rolling control method according to the first embodiment>
A method for controlling rolling of a metal plate according to the first embodiment will be described. FIG. 11 is a flowchart showing the rolling control method according to the first embodiment. FIG. 12 is an explanatory diagram showing a schematic diagram of a method for predicting the shape of a metal plate in the rolling control method according to the first embodiment. In the first embodiment, steps A12 to A20 described later are repeated to perform convergence calculation. In the first embodiment, steps A11 to A13 correspond to the first step of the present invention, steps A14 to A21 correspond to the second step of the present invention, and steps A22 and A23 correspond to the third step of the present invention.

(Step A11)
In step A11, as shown in FIG. 12(a), the plastic elongation strain difference Δε pl (y) of the metal sheet to be evaluated at an arbitrary sheet width direction position y divided into N by a predetermined width in the sheet width direction (y-axis direction) is set. The plastic elongation strain difference _{Δε pl (} y) can be calculated using a known method, for example, the finite element method (FEM), the slab method, a physical model, or a regression equation of experiments or calculations. More specifically, the plastic elongation strain difference Δε pl (y) is obtained under the condition that the out-of-plane deformation of the metal sheet is restrained by using an up-down symmetrical model that allows the displacement of the sheet thickness center of the metal sheet to be a displacement in the plane of a reference plane that passes through the midpoint of a line connecting the rotation centers of the upper and lower rolls and is parallel to the sheet surface of the metal sheet, and does not allow the displacement out of the plane of the reference _plane . Note that N represents the number of nodes equally divided in the width direction. Hereinafter, N is the number of nodes.

Models predicting the rolled shape in step A11 have been worked on for some time. The sheet crown prediction formula required for actual operation is obtained for each rolling mill by a statistical method based on the calculation results by a numerical analysis method. For example, as shown in the Journal of the Japan Society for Technology of Plasticity, Plasticity and Processing, Vol. 25, No. 286 (1984-11), pp. 1034-1041, there is a method using a general-purpose sheet crown prediction formula on the delivery side of a rolling mill, which is derived by separating the sheet crown into factors that depend only on the elastic deformation conditions of the rolling mill and factors that depend on the plastic deformation conditions of the rolled material.
Using these, it is possible to obtain the sheet crown on the entry side and the sheet crown on the exit side of the rolling mill. Then, by multiplying the shape change coefficient ξ obtained by a separate experiment by the crown ratio change (Ch/h-CH/H), the plastic elongation strain difference Δε _pl (y) can be obtained. In other words, the plastic elongation strain difference Δε _pl (y) can be expressed by the following formula (2).
Δε _pl (y)=ξ×(Ch(y)/h(y)−CH(y)/H(y))...(2)
Here, CH is the crown on the entry side of the rolling mill, H is the plate thickness on the entry side of the rolling mill, Ch is the crown on the exit side of the rolling mill, and h is the plate thickness on the exit side of the rolling mill. Then, the plastic elongation strain difference Δε _pl (y) can be obtained based on the above formula (2).

(Step A12)
In step A12, in the first calculation of the iterative calculation, the plastic elongation-strain difference Δε _pl (y) is converted into a normalized elongation-strain difference Δε _{normal (y) normalized to a value between 0 (zero) and 1, with the maximum value being 1. In the second and subsequent calculations of the iterative calculation, the normalized elongation-strain difference Δε normal} (y) corrected in step A20 described later is used as _the normalized elongation-strain difference in the next and subsequent steps.

(Step A13)
In step A13, an eigenvalue analysis calculation is performed using a theoretical buckling model. The theoretical buckling model is a model that performs buckling analysis using a wave-shaped buckling equation created based on a model formulated with a triangular residual stress distribution, for example, as shown in the Journal of the Japan Society for Technology of Plasticity, Plasticity and Processing, Vol. 28, No. 312 (1987-1), p. 58-66. In the theoretical buckling model, if the normalized elongation strain difference Δε _normal (y), the thickness t of the metal plate, the width B of the metal plate, and the tension Ut acting on the metal plate are input, the buckling inherent strain difference Δε _cr (y) (criterion for buckling occurrence) similar to the plastic elongation strain difference Δε _pl (y), the wave pitch P generated by buckling, and the height profile (shape profile) W(y) normalized to 0 to 1 in the width direction cross section at the time of buckling are output. In the second and subsequent iterative calculations, the buckling inherent strain difference Δε _cr (y) is corrected by the elongation strain error Δε _er (y) in step A17 described below, and therefore does not have a similar form to the plastic elongation strain difference Δε _pl (y).

Here, the longitudinal stress acting on a metal plate is essentially a residual stress component that becomes 0 (zero) when integrated in the plate width direction and a tension (unit tension) component overlap. However, in the buckling model, the tension component is given separately, so here, it is assumed that the residual stress component becomes 0 when integrated in the plate width direction, that is, the residual stress component becomes 0 when the integrated value in the plate width direction or the average in the plate width direction is taken, and the distribution of the residual stress σ _res (y) and the distribution of the plastic elongation strain difference Δε _pl (y) are converted by the following formula (3).
Δε _pl (y)=-(σ _res (y)/E-Max(σ _res (y))/E) ...(3)

(Step A14)
In step A14, it is determined whether the calculation is the first calculation of the repeated calculations or the second or subsequent calculations. If it is the first calculation, the process proceeds to step A15, which will be described later, and if it is the second or subsequent calculations, the process proceeds to step A16, which will be described later. In step A15, it is determined whether buckling has occurred, but in the case of the second or subsequent calculations, since it has been determined that the metal plate will buckle, step A15 can be omitted.

(Step A15)
In step A15, the occurrence of buckling is judged. Specifically, the plastic elongation strain difference Δε _pl (y) is compared with the buckling inherent strain difference Δε _cr (y). If Δε _pl (y) is greater than Δε _cr (y), it is judged that the metal plate will buckle, and the process proceeds to step A16. Incidentally, as in the conventional method, the buckling inherent strain difference Δε _cr (y) has a similar relationship with the plastic elongation strain difference Δε _pl (y), so it may be judged at any position y in the plate width direction, but here, the comparison is made at their respective maximum values. On the other hand, if Δε _pl (y) is equal to or less than Δε _cr (y), it is judged that the metal plate will not buckle and is flat, and the process proceeds to step A22.

(Step A22)
In step A22, it is determined that the metal plate does not buckle and is flat. In this case, the shape of the metal plate is controlled by rolling the metal plate without changing the rolling conditions.

(Step A16)
In step A16, it is determined that the metal plate will buckle, and the shape of the metal plate is predicted using a shape prediction model. First, as shown in Fig. 12(b), the plastic elongation strain difference Δε _pl (y) is separated into a component that is converted into a shape deformation after buckling (buckling shape transformation elongation strain difference Δε _ts (y)) and a component that remains in the metal plate even after buckling (buckling inherent strain difference Δε _cr (y)). Then, the buckling inherent strain difference Δε _cr (y) is subtracted from the plastic elongation strain difference Δε _pl (y) using the following formula (4) to calculate the buckling shape transformation elongation strain difference Δε _ts (y).
Δε _ts (y)=Δε _pl (y)−Δε _cr (y) ...(4)

Next, assuming that the wave pitch P calculated in step A13 does not change with respect to the buckling shape transformation elongation strain difference Δε _ts (y) calculated by the above formula (4), the wave height width distribution h(x,y)=H×W(y)×Sin(2πx/P) obtained by multiplying the standardized height profile W(y) by the wave piece amplitude height H is predicted as the shape of the metal sheet after buckling. Here, the steepness λ(y) of the waveform after buckling is expressed by the wave piece amplitude height H, the standardized height profile W(y), and the wave pitch P for each sheet width direction position y, that is, it is expressed by the following formula (5) using the geometric shape elongation strain difference Δε _gs (y) that appears in the shape after out-of-plane deformation. However, in general, when expressing the size of the wave in rolling operation, it is often expressed as (5') as a double amplitude of the maximum amplitude H of the half amplitude wave. Moreover, this formula (5) is rearranged by the geometric shape elongation strain difference Δε _gs (y) to obtain formula (6).
λ(y)=2×H×W(y)/P=(2/π)√Δε _gs (y) ...(5)
λ=2×H/P ・・・・・・(5')
Δε _gs (P, H, W(y)) = (π/2×W(y)×2×H/P) ² ...(6)

Since the standardized height profile W(y) is a profile standardized with a value between 0 and 1, the geometric shape elongation strain difference Δε _gs (P, H, W(y)) can be calculated by assuming the maximum amplitude H of the single amplitude wave. If it is considered to be approximately the same as the buckling shape transformation elongation strain difference Δε _ts (y), the elongation strain difference Δε _er (y) that is not transformed into a wave shape as shown in FIG. 12(c) (hereinafter referred to as "elongation strain error Δε _er (y)") is expressed by the following formula (7).
Δε _er (y)=Δε _gs (P, H, W(y))−Δε _ts (y) ...(7)

(Step A17)
In step A17, the residual stress redistributed after the occurrence of the buckling wave, i.e., the elastic strain difference Δε _el ^* (y) inherent in the metal plate is calculated. Specifically, as shown in Fig. 12(d), the elastic buckling inherent strain difference (Δε _{el_cr} (y)) and the elastic elongation strain error (Δε _er (y)) are added together by the following formulas (8) and (8') to calculate the elastic strain difference Δε _el ^* (y) redistributed due to the occurrence of the buckling wave and inherent in the metal plate even after buckling. In this way, in step A17, the elastic strain difference Δε _el ^* (y) redistributed due to the shape deformation after buckling is calculated.
Δε _{el_cr} (y)=-(Δε _cr (y)-(ΣΔε _cr (y))/N) ...(8')
Δε _el ^* (y) = - {Δε _pl (y) - Δε _gs (P, H, W(y))} = Δε _{el_cr} (y) + Δε _er (y) ... (8)

(Step A18)
In step A18, the normalized elongation-strain difference Δε _{normal *(y) is determined again based on the residual stress distribution of the redistributed metal plate after buckling. Specifically, in the convergence calculation in which steps A12 to A20 are repeatedly performed as described below, the normalized elongation-strain difference Δε normal} ^* (y) having a corresponding value of 0 to 1 is determined using the elastic strain difference Δε _el ^* (y) inherent in the redistributed metal plate according to the following formula (9). Fig. ¹² (e) shows an example of this normalized elongation-strain difference _{Δε normal *} ⁽ y).
Δε _normal ^* (y)=-(Δε _el ^* (y)-Max(ε _el ^* (y)))/(Max(ε _el ^* (y))-Min(ε _el ^* (y)))...(9)

(Step A19)
In step A19, whether or not a solution has been found in the convergence calculation is judged by whether or not the value of the normalized elongation-strain difference Δε _normal ^* (y) after buckling and redistribution is equal to the value of the normalized elongation-strain difference Δε _normal (y) before redistribution. First, the error Δerror (y) of the normalized elongation-strain difference before and after redistribution is calculated for each sheet width direction position y by the following formula (10). That is, the error Δerror (y) is the amount of change in the normalized elongation-strain difference Δε _normal (y). Next, as shown in the following formula (11), the square error for each sheet width direction position y is integrated to calculate Δerror ^* , and this error Δerror ^* is compared with a predetermined threshold value. If the error Δerror ^* is greater than a predetermined threshold, it is determined that the reallocated normalized elongation-strain difference Δε _normal ^* (y) has not converged, and the process proceeds to step A20 to perform recalculation (eigenvalue analysis calculation using a theoretical buckling model). On the other hand, if the error Δerror ^* is equal to or smaller than a predetermined threshold, it is determined that the normalized elongation-strain difference Δε _normal ^* (y) has converged, and the process proceeds to step A21.
Δerror(y)=Δε _normal ^* (y)−Δε _normal (y) ...(10)
Δerror ^* = {Σ(Δerror(y)) ² }>Threshold value (11)

(Step A20)
In step A20, the normalized elongation-strain difference Δε _normal (y) used in the recalculation is corrected by using Δε _normal ^* (y) and Δerror (y) by returning to step A12. Specifically, the error Δerror (y) is multiplied by the relaxation coefficient α according to the following formula (12), and the result is added to the normalized elongation-strain difference Δε _normal (y) after reallocation, to reset the normalized elongation-strain difference Δε _normal (y) for recalculation. Here, when performing recalculation, if all the errors Δerror (y) are superimposed without multiplying by the relaxation coefficient α, the calculation result will vibrate, and the normalized elongation-strain difference Δε _normal (y) will not converge easily. Therefore, in this embodiment, the normalized elongation-strain difference Δε _normal (y) is made to converge easily by multiplying by the relaxation coefficient α. The relaxation coefficient α is arbitrary, but after extensive investigation, the inventors have confirmed that a value of 0.1 to 0.4 is appropriate.
Δε _normal (y)=Δε _normal (y)+α×Δerror(y)=Δε _normal (y)+α×{Δε _normal ^* (y)−Δε _normal (y)} ...(12)

Next, returning to step A12, the normalized elongation-strain difference Δε _normal (y) corrected in step A20 is used to carry out the subsequent steps A13 to A20. Then, steps A12 to A20 are repeated, and the convergence calculation is performed until it is determined in the convergence determination in step A19 that the normalized elongation-strain difference Δε _normal ^* (y) has converged.

(Step A21)
When it is determined in step A19 that the normalized elongation-strain difference Δε _normal ^* (y) has converged, the process proceeds to step A21. In step A21, the true elongation-strain difference Δε' _pl (y) shown in FIG. 6 is calculated. Specifically, the true elongation-strain difference Δε' pl (y) is calculated from the new buckling inherent strain difference Δε _cr (y) calculated from the elastic strain difference Δε _el ^* (y) redistributed to the metal plate even after buckling and the plastic elongation-strain difference Δε _pl (y) according to _the following formula (13). In addition, by developing this formula (13), the true elongation-strain difference Δε' _pl (y) can also be calculated from the plastic elongation-strain difference Δε _pl (y) and the buckling shape transformation elongation-strain difference Δε _ts (y).
Δε' _pl (y)=Δε _pl (y)+(Δε _pl (y)−Δε _cr (y))=Δε _pl (y)+Δε _ts (y) ...(13)

(Step A23)
In step A23, the rolling conditions are set based on the true elongation strain difference Δε′ _pl (y) calculated in step A21, and the shape of the metal sheet after rolling is controlled. Specifically, for example, the rolling conditions are set so that the true elongation strain difference Δε′ _pl (y) is equal to or less than the buckling inherent strain difference Δε _cr (y). In this way, the metal sheet after rolling does not buckle and becomes flat. Examples of the rolling conditions include the rolling load and the moment of the roll bender that controls the deflection of the roll. The rolling conditions can be set arbitrarily, and the true elongation strain difference Δε′ _pl (y) can be determined through this algorithm as necessary to control the shape of the metal sheet after rolling.

According to the first embodiment, the true elongation strain difference _Δε'pl (y) of the metal sheet is calculated from the plastic elongation strain difference _Δεpl (y) and the buckling inherent strain difference _Δεcr (y) or the buckling shape transformation elongation strain difference _Δεts (y). By calculating the elongation strain difference in this manner, it is possible to improve the prediction accuracy of the elongation strain difference compared to the conventional method. Therefore, by setting the rolling conditions based on the true elongation strain difference _Δε'pl (y), it is possible to freely control the shape of the metal sheet after rolling.

As mentioned above, the conventional method assumes that the shape profile is maintained even after buckling. However, in reality, the effect of the shape profile changing due to the residual stress distribution that is redistributed after buckling is not taken into account, resulting in an error in the shape profile as shown in Figure 9.

In this regard, the present inventors have found that the shape profile changes due to the shape deformation after buckling. According to the rolling control method of the first embodiment described above, the elastic strain difference Δε _e1 ^* (y) reallocated due to the shape deformation after buckling is calculated, and recalculation is performed using the elastic strain difference Δε _e1 ^* (y), so that the shape of the metal sheet can be appropriately predicted. As a result, it becomes possible to more appropriately control the shape of the metal sheet after rolling.

To verify the effect of the first embodiment (the effect of increasing the accuracy of shape prediction), the inventors compared the shape prediction results (converged solutions) obtained by performing the first embodiment with the shape prediction results obtained by performing a conventional method and the analysis results using FEM. The verification results are shown in Figs. 13 and 14. Fig. 13 shows the width-wise distribution of the elastic strain difference present in the metal plate after buckling, and Fig. 14 shows the width-wise distribution of the normalized profile.

Referring to Figures 13 and 14, the conventional shape prediction results are slightly different from the analysis results by FEM, whereas the shape prediction results of the first embodiment are almost identical to the analysis results by FEM. Therefore, it was found that the shape of the metal plate can be appropriately predicted according to the first embodiment.

In the first embodiment, the true elongation strain difference _Δε'pl (y) may be calculated based on the fluctuation of tension at the delivery side of the rolling mill due to buckling. Specifically, the buckling shape transformation elongation strain difference _Δεts (y) calculated in step A16 is converted to tension acting on the metal sheet. The change in the rolling load difference in the sheet width direction caused by the fluctuation of tension at the delivery side of the rolling mill is calculated, and the elongation strain difference _{Δεts '} (y) is calculated by second-order differentiation of _ΔPts (y) in the sheet width direction x as shown in the following formula (14). Then, as shown in the following formula (8), the true elongation strain difference _Δε'pl (y) is calculated by adding the elongation strain difference _Δεts' (y) calculated by formula (15) to the plastic elongation strain difference _Δεpl (y) set in step A11.
Δε _ts '(y)=d ² ΔP _ts (y)/dx ² ...(14)
Δε' _pl (y)=Δε _pl (y)+Δε _ts '(y)...(15)

In this way, the buckling shape transformation elongation strain difference Δε _ts (y) is converted into tension to obtain a transformed tension, and then the elongation strain difference Δε _ts '(y) corresponding to this transformed tension is obtained, so that the obtained elongation strain difference Δε _ts '(y) is closer to the actual phenomenon. Moreover, when obtaining the elongation strain difference Δε _ts '(y), the change in the rolling load difference ΔP _ts (y) is differentiated twice, so that the obtained elongation strain difference Δε ts '(y) is closer to the actual phenomenon. Therefore, the true elongation strain difference Δε ' _pl (y) of the metal plate can be predicted with even greater accuracy.

In the first embodiment, the plastic elongation strain difference Δε _pl (y) is set in step A11, but if the plastic elongation strain difference Δε _pl (y) is known or if an already obtained difference can be used, step A11 can be omitted. In this case, in step A13, the buckling inherent strain difference Δε _cr (y) is obtained using the known plastic elongation strain difference Δε _pl (y).

<Rolling control method according to the second embodiment>
A method for controlling rolling of a metal plate according to the second embodiment will be described. Fig. 15 is a flow chart showing the method for controlling rolling of the metal plate according to the second embodiment. In the second embodiment, step B11 corresponds to the first step of the present invention, steps B12 and B13 correspond to the second step of the present invention, steps B14 to B21 correspond to the third step of the present invention, and steps B22 and B23 correspond to the fourth step of the present invention.

(Step B11)
In step B11, a rolling load difference ΔP(y) acting on a metal plate to be evaluated at an arbitrary widthwise position y divided into N parts by a predetermined width in the widthwise direction and a plastic elongation strain difference Δε _pl (y) of the metal plate are set. The rolling load difference ΔP(y) and the plastic elongation strain difference Δε _pl (y) can be calculated using a known method, such as the finite element method (FEM), the slab method, a physical model, or a regression equation of experiments or calculations, as in the above step A11.

(Steps B12 to B20)
In steps B12 to B20, convergence calculations are performed to predict the shape of the metal plate. These steps B12 to B20 are similar to the above steps A12 to A20, and detailed description thereof will be omitted here.

(Step B22)
Step B22 is a case where it is determined in step B15 whether or not buckling has occurred that the metal sheet is not buckled and is flat. In such a case, the shape of the metal sheet is controlled by rolling the metal sheet without changing the rolling conditions.

(Step B21)
Step B21 is a step performed when it is determined in step B15 that the metal plate buckles and the normalized elongation-strain difference Δε _normal ^* (y) has converged. In step B21, the true elongation-strain difference Δε′ _pl (y) is calculated.

Here, in the above step B11, the rolling load difference ΔP(y) and the plastic elongation strain difference Δε _pl (y) are set, and the correlation between the rolling load difference ΔP(y) and the plastic elongation strain difference Δε _pl (y) is derived. In this step B21, based on this correlation, the buckling inherent load difference ΔP _cr (y) corresponding to the buckling inherent strain difference Δε _cr (y) calculated in step B13 is calculated. Then, the buckling shape transformation load difference ΔP ts (y) (ΔP _ts (y) = _ΔP (y) - ΔP _cr (y)) which is the difference between the rolling load difference ΔP(y) set in step B11 and the buckling inherent load difference ΔP _cr (y) calculated in this step B21 is calculated. Furthermore, assuming that there is no change in the crown ratio of the metal plate between the exit side and the entry side of the rolling mill, the buckling shape transformation elongation strain difference Δε _ts (y) is obtained from the buckling shape transformation load difference ΔP _ts (y) using a known method, for example, the finite element method (FEM), the slab method, a physical model, or a regression equation of experiments or calculations. Note that when the buckling shape transformation elongation strain difference Δε _ts (y) is obtained from the buckling shape transformation load difference ΔP _ts (y), the correlation between the rolling load difference ΔP (y) obtained in step B11 and the plastic elongation strain difference Δε _pl (y) may be used. Then, the true elongation strain difference Δε _{' pl} (y) is obtained by adding the buckling shape transformation elongation strain difference Δε _ts (y) to the plastic elongation strain difference Δε _pl (y) set in step B11 using the following formula (16).
Δε' _pl (y)=Δε _pl (y)+Δε _ts (y) ...(16)

(Step B23)
In step B23, the rolling conditions are set based on the true elongation strain difference Δε′ _pl (y) calculated in step B21, and the metal sheet is rolled to control the shape of the metal sheet. This step B23 is similar to the above-mentioned step A23.

The second embodiment is a modified example of the first embodiment. The first embodiment and the second embodiment differ in the method of calculating the increase in the elongation strain difference from the plastic elongation strain difference Δε _pl (y). In step A21 of the first embodiment, the increase in the elongation strain difference is calculated from the difference between the plastic elongation strain difference Δε _pl (y) and the buckling inherent strain difference Δε _cr (y), whereas in step B21 of the second embodiment, the increase in the elongation strain difference is calculated from the difference between the rolling load difference ΔP (y) and the buckling inherent load difference ΔP _cr (y). Therefore, in the second embodiment, the same effect as in the first embodiment can be enjoyed. That is, the true elongation strain difference Δε' _pl (y) of the metal plate can be predicted more accurately than in the past. Furthermore, by setting the rolling conditions based on the true elongation strain difference Δε' _pl (y), the shape of the metal plate after rolling can be freely controlled.

<Rolling control method according to the third embodiment>
A rolling control method for a metal plate according to the third embodiment will be described. FIG. 16 is a flowchart showing the rolling control method according to the third embodiment. As described later in this embodiment, i indicates the number of repeated calculations of steps C12 to C26 (hereinafter referred to as "first repeated calculations"), and j indicates the number of repeated calculations of steps C13 to C22 (hereinafter referred to as "second repeated calculations"). In the third embodiment, steps C11 and C12 correspond to the first step of the present invention, steps C13 and C14 correspond to the second step of the present invention, steps C15 to C24 correspond to the third step of the present invention, steps C25 and C26 correspond to the fourth step of the present invention, and steps C27 and C28 correspond to the fifth step of the present invention.

(Step C11)
The first iteration (i=1) of the first iteration begins.

(Step C12)
In step C12, a rolling load difference ΔP(i,y) acting on the metal plate and a plastic elongation strain difference Δε _pl (i,y) of the metal plate to be evaluated at an arbitrary width y divided into N parts in the width direction are set. This step C12 is similar to the above step B11.

(Steps C13 to C21)
Steps C13 to C21 are used to perform convergent calculations (second iterative calculations) to predict the shape of the metal sheet. These steps C13 to C21 are similar to steps B12 to B20, and detailed descriptions thereof will be omitted here. In the second embodiment, each parameter is a distribution of the sheet width direction position y and is marked with (y), but in the third embodiment, each parameter is a distribution of the sheet width direction position y and is the i-th iteration of the first iterative calculation and is marked with (i, y).

(Step C22)
If it is determined in step C20 that the normalized elongation-strain difference Δε _normal ^* (y) has not converged, the normalized elongation-strain difference Δε _normal (y) for recalculation is reset in step C21, and the process returns to step C13. At this time, in step C22, the number j of the second iterative calculation is incremented by one (j=j+1).

(Step C27)
Step C27 is a process performed when it is determined in step C16 that the metal sheet does not buckle and is flat, and the shape of the metal sheet is controlled by rolling the metal sheet without changing the rolling conditions.

(Step C23)
Step C23 is a case where, as a result of judging whether or not buckling has occurred in step C16, it is judged that the metal plate has buckled, and also, in step C20, it is judged that the normalized elongation-strain difference Δε _normal ^* (y) has converged. In step C23, the number i of the first iterative calculation is increased by one (i=i+1). Note that, although FIG. 16 illustrates the first iterative calculation (j=1), this number varies depending on the actual number.

(Step C24)
In step C24, a new rolling load difference ΔP'(i,y) is calculated. Here, in step C12, the rolling load difference ΔP(i,y) and the plastic elongation strain difference Δε _pl (i,y) are set, and the correlation between the rolling load difference ΔP(i,y) and the plastic elongation strain difference Δε _pl (i,y) is derived. In this step C24, based on this correlation, the buckling inherent load difference ΔP _cr (i-1,y) corresponding to the buckling inherent strain difference _{Δε cr} (i-1,y) calculated in step C14 is obtained. Then, a buckling shape transformation load difference ΔP ts (i-1, y) (ΔP ts (i-1, y) = ΔP (i-1, y) - ΔP _cr (i-1, y)) is calculated, which is the difference between the rolling load _{difference ΔP} (i-1, y) set in step C12 and the buckling inherent load difference ΔP _cr (i-1, y) calculated in step C24. Then, a new rolling load difference ΔP' (i, y) is calculated by adding the buckling shape transformation load difference ΔP _ts (i-1, y) to the rolling load difference ΔP (i-1, y) from the previous first iterative calculation using the following formula (17).
ΔP' (i, y) = ΔP (i-1, y) + ΔP _ts (i-1, y) ... (17)

In this third embodiment, it is assumed that the crown ratio of the metal sheet changes between the exit side and the entry side of the rolling mill. In other words, it is assumed that when the rolling load acting on the metal sheet fluctuates, the fluctuation in the rolling load causes the deflection of the rolls of the rolling mill to fluctuate, and the elongation strain of the metal sheet to fluctuate. Then, in step C24, a new rolling load difference ΔP'(i, y) is calculated.

(Step C25)
In step C25, the true elongation strain difference Δε′ _pl (i, y) is calculated.
Specifically, the buckling shape transformation elongation strain difference Δε ts (i-1, y) is obtained from the buckling shape transformation load difference ΔP _ts (i-1, y) obtained in step C24 using a known method, for example, the finite element method (FEM), the slab method, _{a physical model, or a regression equation of experiments or calculations. When the buckling shape transformation elongation strain difference Δε ts (} i-1, y) is obtained from the buckling shape transformation load difference ΔP _ts (i-1, y), the correlation between the rolling load difference ΔP (i, y) and the plastic elongation strain difference Δε _pl (i, y) obtained in step C12 may be used. Then, the true elongation strain difference Δε' _pl (i, y) is obtained by adding the buckling shape transformation elongation strain difference Δε _ts (i-1, y) to the new plastic elongation strain difference Δε _pl (i-1, y) using the following formula (18).
Δε' _pl (i, y)=Δε _pl (i-1, y)+Δε _ts (i-1, y) ...(18)

(Step C26)
In step C26, it is determined whether a solution has been found in the convergence calculation (first iterative calculation). First, the error Δer(y) between the true elongation strain difference _Δε'pl (i,y) and the previous plastic elongation strain difference _Δεpl (i-1,y) is calculated for each sheet width direction position y by the following formula (19). That is, the error Δer(y) is the amount of change in the plastic elongation strain difference _Δεpl (y). Next, as shown in the following formula (20), Δer ^* is calculated by integrating the squared error for each sheet width direction position y, and this error Δer ^* is compared with a predetermined threshold value. Then, if the error Δer ^* is larger than the predetermined threshold value, it is determined that the true elongation strain difference _Δε'pl (i,y) has not converged, and the process returns to step C12 to perform recalculation. At this time, as shown in the following formulas (21) and (22), the new rolling load difference ΔP'(i,y) is set as the rolling load difference ΔP(i,y), and the true elongation-strain difference _Δε'pl (i,y) is set as the elongation-strain difference _Δεpl (i,y). On the other hand, if the error Δer ^* is equal to or smaller than a predetermined threshold, it is determined that the true elongation-strain difference _Δε'pl (i,y) has converged, and the process proceeds to step C28.
Δer(y)=Δε' _pl (i, y)−Δε _pl (i−1, y) (19)
Δer ^* ={Σ(Δer(y)) ² }>threshold (20)
ΔP (i, y) = ΔP' (i, y) (21)
Δε _pl (i, y)=Δε' _pl (i, y) (22)

(Step C28)
In step C28, the rolling conditions are set based on the true elongation strain difference Δε′ _pl (i, y) calculated in step C25, and the metal sheet is rolled to control the shape of the metal sheet. This step C28 is similar to the above-mentioned step B23.
In addition, it may be assumed that the new plastic elongation strain difference obtained in the third step is the plastic elongation strain difference obtained in the first step, and that the new buckling inherent strain difference obtained in the third step is the buckling inherent strain difference obtained in the second step.

According to the third embodiment, the calculation is repeated assuming that the crown ratio of the metal sheet changes between the exit side and the entry side of the rolling mill. Therefore, the true elongation strain difference Δε′ _pl (i, y) can be predicted with higher accuracy.

<Other embodiments>
The above-described first, second and third embodiments are each implemented in a rolling line 1 shown in Fig. 17. The rolling line 1 has the above-described rolling mill 10 and a rolling control device 20 that controls the rolling mill 10. The rolling control device 20 has a calculation unit 21 and a control unit 22.

The calculation unit 21 first accepts input of provisional rolling conditions set in the rolling control device 20. Next, the calculation unit 21 performs calculations in steps A11 to A21 of the first embodiment, steps B11 to B21 of the second embodiment, and steps C11 to C26 of the third embodiment based on the rolling conditions that have been accepted as input. Then, in step A22 of the first embodiment, step B22 of the second embodiment, and step C27 of the third embodiment, if no change is made to the rolling conditions, the calculation unit 21 notifies the control unit 22 that no change is necessary to the provisional rolling conditions described above. On the other hand, in step A21 of the first embodiment, step B21 of the second embodiment, and step C25 of the third embodiment, when the true elongation strain difference Δε′ _pl (y) is calculated, the calculation unit 21 notifies the control unit 22 of the true elongation strain difference Δε′ _pl (y).

The control unit 22 sets the rolling conditions based on the true elongation strain difference Δε′ _pl (y) notified by the calculation unit 21. The control unit 22 derives rolling conditions such that the true elongation strain difference Δε′ _pl (y) is equal to or smaller than the buckling inherent strain difference Δε _cr (y), for example. Note that the calculation unit 21 may derive new rolling conditions.

When the control unit 22 receives a notification from the calculation unit 21 that no change in the rolling conditions is required, the control unit 22 outputs the original rolling conditions to the rolling mill 10 and controls the rolling mill 10 to control the shape of the metal sheet H after rolling. On the other hand, when new rolling conditions are derived as described above, the control unit 22 outputs the new rolling conditions to the rolling mill 10 and controls the rolling mill 10 to control the shape of the metal sheet H after rolling.

Furthermore, the control unit 22 determines whether or not to end rolling. If the control unit 22 determines not to end rolling, it returns the process to input of rolling conditions and performs step A11 in the first embodiment, step B11 in the second embodiment, and step C11 in the third embodiment onward. On the other hand, if the control unit 22 determines to end rolling, it ends this routine.

Further, in the rolling line 1, a shape meter 30 may be installed at the exit side of the rolling mill 10. The shape meter 30 measures the shape of the metal sheet H after rolling. As the shape of the metal sheet H, the longitudinal position and the sheet width direction position of the metal sheet H and the height displacement at those positions are measured. The measurement results of the shape meter 30 are output to the rolling control device 20. In the rolling control device 20, the calculation unit 21 corrects the buckling shape transformation elongation strain difference Δε _ts (y) based on the measurement result of the shape meter 30, and accordingly corrects the true elongation strain difference Δε' _pl (y). The correction of the true elongation strain difference Δε' _pl (y) follows the method described in Patent Document 3 (JP Patent Publication 2012-218010 A). That is, first, the actual buckling shape transformation elongation strain difference Δε _ts (y) is obtained based on the measurement result of the shape meter 30. The actual buckling shape transformation elongation strain difference Δε _ts (y) is compared with the buckling shape transformation elongation strain difference Δε _ts (y) predicted in the above embodiment, and the difference (error) E between them is taken as the model error, and learning and correction are performed on the plastic elongation strain difference Δε _pl (y) (rolling load difference ΔP(y)) obtained in steps A11, B11, and C12 based on this difference E. Specifically, after adding the difference E to the plastic elongation strain difference Δε _pl (y) (rolling load difference ΔP(y)) obtained in steps A11, B11, and C12, the following processes are performed to obtain the true elongation strain difference Δε' _pl (y). Then, in the control unit 22, the rolling conditions are corrected so that the shape of the metal sheet H becomes the target shape based on the correction result of the true elongation strain difference Δε' _pl (y) in the calculation unit 21. In this manner, the rolling conditions are feedback-controlled based on the measurement results of the shape meter 30.

The present invention can also be applied to the case where the metal sheet H is deformed out of plane at the entry side of the rolling mill 10. The inventors have investigated and found that when the metal sheet H is deformed out of plane at the entry side of the rolling mill, the difference in elongation strain in the metal sheet H after rolling is larger than when the metal sheet H is not deformed out of plane at the entry side of the rolling mill. In other words, the conventional method further deteriorates the shape prediction accuracy of the metal sheet. In contrast, in the present invention, the difference in elongation strain corresponding to the out-of-plane deformation at the entry side of the rolling mill can be included in the buckling shape transformation elongation strain difference Δε _ts (y), so there is no effect on predicting the true elongation strain difference Δε' _pl (y) in the metal sheet H. Therefore, even if the metal sheet H is deformed out of plane at the entry side of the rolling mill, the shape of the metal sheet H can be appropriately controlled.

Although the embodiments of the present invention have been described above, the present invention is not limited to these examples. It is clear that a person skilled in the art can come up with various modified or revised examples within the scope of the technical ideas described in the claims, and it is understood that these also naturally fall within the technical scope of the present invention.

The present invention is useful for predicting the shape of metal plates, such as thin or thick plates, after rolling and controlling the shape of the metal plates based on the prediction results.

Reference Signs List 1 Rolling line 10 Rolling mill 20 Rolling control device 21 Calculation unit 22 Control unit 30 Shape meter H Metal plate

Claims (11)

1. A method for predicting and controlling the shape of a metal sheet, comprising:
a first step of calculating a plastic elongation strain difference, which is the difference in the width direction of the strain that elongates in the rolling direction of the metal plate during rolling under specified rolling conditions, based on the thickness of the metal plate, the width of the metal plate, and the tension acting on the metal plate at the exit side of the rolling mill, and a buckling inherent strain difference, which is the critical strain difference in the width direction at which the metal plate buckles, obtained under conditions in which out-of-plane deformation of the metal plate is constrained by using a top-bottom symmetrical model that allows the displacement of the center of thickness of the metal plate to be in the plane of a reference plane that passes through the midpoint of a line connecting the rotation centers of the top and bottom rolls and is parallel to the plate surface of the metal plate, and does not allow the displacement to be out-of-plane of the reference plane;
If the plastic elongation strain difference exceeds the buckling inherent strain difference, a new buckling inherent strain difference is calculated using the elastic strain difference that is reallocated by the buckling wave generation and remains in the metal plate even after buckling, and a second step of calculating a true elongation strain difference by adding the difference between the plastic elongation strain difference and the new buckling inherent strain difference to the plastic elongation strain difference;
a third step of rolling the metal plate without changing the predetermined rolling conditions if the plastic elongation strain difference does not exceed the buckling inherent strain difference, and rolling the metal plate under rolling conditions set based on the true elongation strain difference if the plastic elongation strain difference exceeds the buckling inherent strain difference. The method for controlling rolling of a metal plate according to claim 1 , further comprising a step of determining the difference in plastic elongation strain. 3. The method for controlling rolling of a metal plate according to claim 1, wherein in the second step, a converted tension is obtained by converting a difference between the plastic elongation strain difference and the buckling inherent strain difference into a tension acting on the metal plate at the delivery side of the rolling mill , and the true elongation strain difference is obtained by adding the elongation strain difference corresponding to the converted tension to the plastic elongation strain difference. 4. The method for controlling rolling of a metal plate according to claim 3, characterized in that in the second step, a rolling load difference in the plate width direction of the metal plate corresponding to the converted tension is second -order differentiated in the plate width direction to obtain an elongation strain difference corresponding to the converted tension. 1. A method for predicting and controlling the shape of a metal sheet, comprising:
A first step of calculating a rolling load difference, which is the difference in rolling load in the width direction of the metal plate during rolling under predetermined rolling conditions, and a plastic elongation strain difference, which is the difference in strain elongating in the rolling direction of the metal plate during rolling, in the width direction of the metal plate during rolling, under a condition that out-of-plane deformation of the metal plate is constrained by using an up-down symmetrical model that allows the displacement of the thickness center of the metal plate to be in the plane of a reference plane that passes through the midpoint of a line connecting the rotation centers of the upper and lower rolls and is parallel to the plate surface of the metal plate, and does not allow the displacement to be out-of-plane of the reference plane;
A second step of determining a buckling inherent strain difference, which is a critical strain difference in the sheet width direction at which the metal sheet buckles, based on the plastic elongation strain difference, the sheet thickness of the metal sheet, the sheet width of the metal sheet, and a tension acting on the metal sheet at the delivery side of a rolling mill;
a third step of calculating a new buckling inherent strain difference using the elastic strain difference that is reallocated by the generation of buckling waves and remains in the metal plate even after buckling, calculating a buckling inherent load difference that is a rolling load difference corresponding to the new buckling inherent strain difference from the correlation between the rolling load difference and the plastic elongation strain difference, and calculating a difference between the rolling load difference and the buckling inherent load difference, and adding the buckling shape transformation elongation strain difference and the plastic elongation strain difference to calculate a true elongation strain difference, assuming that there is no change in the crown ratio of the metal plate between the exit side and the entry side of the rolling mill;
a fourth step of rolling the metal plate without changing the predetermined rolling conditions if the plastic elongation strain difference does not exceed the buckling inherent strain difference, and rolling the metal plate under rolling conditions set based on the true elongation strain difference if the plastic elongation strain difference exceeds the buckling inherent strain difference. 1. A method for predicting and controlling the shape of a metal sheet, comprising:
A first step of calculating a rolling load difference, which is the difference in rolling load in the width direction of the metal plate during rolling under predetermined rolling conditions, and a plastic elongation strain difference, which is the difference in strain elongating in the rolling direction of the metal plate during rolling, in the width direction of the metal plate during rolling, under a condition that out-of-plane deformation of the metal plate is constrained by using an up-down symmetrical model that allows the displacement of the thickness center of the metal plate to be in the plane of a reference plane that passes through the midpoint of a line connecting the rotation centers of the upper and lower rolls and is parallel to the plate surface of the metal plate, and does not allow the displacement to be out-of-plane of the reference plane;
A second step of determining a buckling inherent strain difference, which is a critical strain difference in the sheet width direction at which the metal sheet buckles, based on the plastic elongation strain difference, the sheet thickness of the metal sheet, the sheet width of the metal sheet, and a tension acting on the metal sheet at the delivery side of a rolling mill;
When the plastic elongation strain difference exceeds the buckling inherent strain difference, a new buckling inherent strain difference is calculated using the elastic strain difference that is reallocated by the buckling wave generation and is still present in the metal plate after buckling, and a buckling shape transformation load difference corresponding to the buckling shape transformation elongation strain difference, which is the difference between the plastic elongation strain difference and the new buckling inherent strain difference, is calculated from the correlation between the rolling load difference and the plastic elongation strain difference, and the buckling shape transformation load difference is superimposed on the rolling load difference to derive a new rolling load difference, and a third step of calculating the new rolling load difference on the assumption that the metal plate has a crown ratio change;
A fourth step of calculating a true elongation-strain difference by adding the buckling shape transformation elongation-strain difference and a new plastic elongation-strain difference, and determining whether the true elongation-strain difference has converged;
a fifth step of rolling the metal plate without changing the specified rolling conditions if the plastic elongation strain difference does not exceed the buckling inherent strain difference obtained in the second step, and rolling the metal plate under rolling conditions set based on the true elongation strain difference if the plastic elongation strain difference exceeds the buckling inherent strain difference obtained in the second step. The rolling control method for a metal plate according to claim 6, characterized in that the new plastic elongation strain difference obtained in the third step is assumed to be the plastic elongation strain difference obtained in the first step, and the new buckling inherent strain difference obtained in the third step is assumed to be the buckling inherent strain difference obtained in the second step. The method for controlling rolling of a metal sheet according to any one of claims 1 to 7 , characterized in that the metal sheet is subjected to out-of-plane deformation at the entry side of the rolling mill. Measuring the shape of the metal sheet after rolling using a shape meter installed on the delivery side of the rolling mill;
The rolling control method for a metal plate according to any one of claims 1 to 8, further comprising a step of correcting the plastic elongation strain difference based on a difference between an actual elongation strain difference converted into an out-of-plane deformation obtained from the measured shape of the metal plate and a predicted elongation strain difference converted into an out- of-plane deformation. A plastic elongation strain difference, which is the difference in the strain extending in the rolling direction of the metal plate during rolling under specified rolling conditions in the sheet width direction, is obtained under conditions in which the out-of-plane deformation of the metal plate is constrained by using a top-bottom symmetrical model that allows the displacement of the thickness center of the metal plate to be in-plane displacement of a reference plane that passes through the midpoint of a line connecting the rotation centers of the top and bottom rolls and is parallel to the sheet surface of the metal plate, and does not allow the displacement to be out-of-plane of the reference plane, the thickness of the metal plate ... a calculation unit that calculates a buckling inherent strain difference, which is a critical strain difference in the plate width direction at which the metal plate buckles, based on a plate width and a tension acting on the metal plate at the delivery side of a rolling mill, and when the plastic elongation strain difference exceeds the buckling inherent strain difference, calculates a new buckling inherent strain difference using an elastic strain difference that is reallocated after buckling, and calculates a true elongation strain difference by adding the difference between the plastic elongation strain difference and the new buckling inherent strain difference to the plastic elongation strain difference;
a control unit that rolls the metal plate without changing the specified rolling conditions when the plastic elongation strain difference does not exceed the buckling inherent strain difference, and that rolls the metal plate under rolling conditions set based on the true elongation strain difference when the plastic elongation strain difference exceeds the buckling inherent strain difference. a first step of calculating a plastic elongation strain difference, which is the difference in the width direction of the strain that elongates in the rolling direction of the metal plate during rolling under specified rolling conditions, based on the thickness of the metal plate, the width of the metal plate, and the tension acting on the metal plate at the exit side of the rolling mill, and a buckling inherent strain difference, which is the critical strain difference in the width direction at which the metal plate buckles, obtained under conditions in which out-of-plane deformation of the metal plate is constrained by using a top-bottom symmetrical model that allows the displacement of the center of thickness of the metal plate to be in the plane of a reference plane that passes through the midpoint of a line connecting the rotation centers of the top and bottom rolls and is parallel to the plate surface of the metal plate, and does not allow the displacement to be out-of-plane of the reference plane;
If the plastic elongation strain difference exceeds the buckling inherent strain difference, a new buckling inherent strain difference is calculated using the elastic strain difference that is reallocated by the buckling wave generation and remains in the metal plate even after buckling, and a second step of calculating a true elongation strain difference by adding the difference between the plastic elongation strain difference and the new buckling inherent strain difference to the plastic elongation strain difference;
a third step of rolling the metal plate without changing the specified rolling conditions if the plastic elongation strain difference does not exceed the buckling inherent strain difference, and rolling the metal plate under rolling conditions set based on the true elongation strain difference if the plastic elongation strain difference exceeds the buckling inherent strain difference.

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