METHOD FOR CALCULATING THE ROLLING LOAD DISTRIBUTION, METHOD FOR CALCULATING THE ROLLING LOAD, METHOD FOR CALCULATING THE CONTACT ARC LENGTH, AND ROLLING METHOD

Abstract

A method for calculating the rolling load distribution of a rolling mill 10, comprising: calculating the contact arc length of at least one of the elastic recovery and elastic reduction regions based on Hooke's law under the plane strain condition of a rolled material 1, the force balance within the belt-roll contact of the rolling mill 10, and the stress distribution in the rolling direction approximated by a function that depends on the position coordinate in the rolling direction of the rolling mill 10; and calculating the rolling load distribution of at least one of the elastic recovery and elastic reduction regions based on the contact arc length.

Description

Rolling load distribution calculation method, rolling load calculation method, contact arc length calculation method, and rolling method

[0001] The present disclosure relates to a rolling load distribution calculation method, a rolling load calculation method, a contact arc length calculation method, and a rolling method.

[0002] The rolling load of a rolling mill determines the thickness and shape of the strip at the exit of the rolling stand. Therefore, accurate prediction of the rolling load is important from the viewpoint of stable operation and quality assurance.

[0003] Conventionally, various methods have been proposed for predicting and calculating rolling loads with high accuracy (for example, Patent Document 1 and Non-Patent Document 1).

[0004] For example, the method shown in Non-Patent Document 1 is commonly used for predicting the rolling load in the elastic region. Here, the term "elastic region" is used as a general term for both the elastic recovery region and the elastic reduction region.

[0005] Figure 4 shows a schematic diagram of the elastic recovery region and the elastic reduction region. The elastic recovery region is the elastic region on the exit side of the rolling mill. The elastic reduction region is the elastic region on the entry side of the rolling mill. The elastic region is the region where the rolled material is in contact with the rolling rolls but is elastically deformed rather than plastically deformed.

[0006] Japanese Patent Application Laid-Open No. 2006-55881

[0007] "Theory and Practice of Plate Rolling" Iron and Steel Institute of Japan, pp. 41-43

[0008] The method disclosed in Non-Patent Document 1 calculates the load distribution in the elastic region and the contribution to the rolling load due to the elastic region by ignoring the frictional stress in the elastic region and by approximating that the rolling direction stress is constant in the elastic region. Here, the contribution to the rolling load due to the elastic region corresponds to the integral of the load distribution in the elastic region.

[0009] Non-Patent Document 1 discloses a method for calculating the rolling load distribution in the elastic recovery region using the following formula:

[0010] Furthermore, Non-Patent Document 1 discloses a method for calculating the rolling load distribution in the elastic reduction region using the following formula:

[0011] Furthermore, Non-Patent Document 1 discloses a method of integrating the rolling load distribution in the elastic reduction region to calculate the contribution of the elastic reduction region to the rolling load. Furthermore, Non-Patent Document 1 discloses a method of integrating the rolling load distribution in the elastic reduction region to calculate the contribution of the elastic reduction region to the rolling load.

[0012] Furthermore, Non-Patent Document 1 discloses a method for calculating the contact arc length of the elastic recovery region using the following formula:

[0013] Furthermore, Non-Patent Document 1 discloses a method for calculating the contact arc length of the plastic region using the following formula:

[0014] Furthermore, Non-Patent Document 1 discloses a method for calculating the contact arc length in the elastic pressure region using the following formula:

[0015] Non-Patent Document 1 obtains the load distribution and contact arc length in the elastic region through these calculations. Therefore, for the plastic region, if a model such as that by Bland & Ford is used, Non-Patent Document 1 can obtain the load distribution over the entire contact region.

[0016] However, as mentioned above, the method disclosed in Non-Patent Document 1 ignores the effect of frictional stress in the elastic region, and therefore has the problem of large approximation errors under rolling conditions that increase frictional stress. Examples of rolling conditions that increase frictional stress include rolling with dull rolls with large roll roughness and rolling of high-strength high-tensile steel.

[0017] To address this issue, Non-Patent Document 1 discloses a method that uses a strict theoretical formula based on Airy's stress function. However, this method has the problem that it is not suitable for online calculations using a process computer because it is necessary to calculate a numerical solution of a differential equation to obtain the load distribution in the elastic region.

[0018] Furthermore, Patent Document 1 discloses a method for improving the prediction accuracy of the rolling load in rolling with dull rolls by introducing a correction parameter into a theoretical formula for determining the roll flattening radius. However, this method has the problem that the correction parameter must be determined in advance for various rolling conditions, and furthermore, the physical meaning of this correction parameter is unclear.

[0019] An object of the present disclosure is to quickly and accurately calculate the rolling load distribution in the elastic region or the contribution of the elastic region to the rolling load even under rolling conditions in which frictional stress increases.

[0020] [1] A method for calculating a rolling load distribution in a rolling mill, comprising: a step of calculating a contact arc length of at least one of an elastic recovery region and an elastic reduction region based on Hooke's law in a plane strain state of a material to be rolled, a balance of forces in a roll bite of the rolling mill, and a rolling direction stress distribution approximated by a function that depends on a position coordinate in the rolling direction of the rolling mill; and a step of calculating a rolling load distribution of at least one of the elastic recovery region and the elastic reduction region based on the contact arc length.

[0021] [2] In the step of calculating the rolling load distribution, the rolling load distribution in the elastic recovery region is calculated based on the following formula (1), and the rolling load distribution in the elastic reduction region is calculated based on the following formula (2). However, in formula (1), the following formulas (3) and (4) hold: However, in formula (2), the following formulas (5) to (7) hold: However, in formulas (1) to (7), p ０ (x) is the rolling load distribution in the elastic recovery region, and p １ (x) is the rolling load distribution in the elastic reduction region, x is the position coordinate in the rolling direction, E' is the Young's modulus of the rolled material in the plane strain state; μ is the coefficient of friction; ν' is the Poisson's ratio of the rolled material in the plane strain state; and R' is the roll radius after flattening.

[0022] [3] The rolling load distribution calculation method according to the above [1] or [2], wherein in the step of calculating the contact arc length, the contact arc length in the elastic recovery region is calculated based on the following formula (8), and the contact arc length in the elastic reduction region is calculated based on the following formula (9). However, in formula (8), the following formulas (10) to (13) hold: However, in formula (9), the following formulas (14) to (20) hold: However, in the formulas (8) to (20), E' is the Young's modulus of the rolled material in the plane strain state, k ０ is the yield stress at the roll bite exit, k １ is the yield stress at the roll bite entrance, R' is the roll radius after flattening, ν' is the Poisson's ratio of the rolled material in the plane strain state, μ is the coefficient of friction, and E' is the Young's modulus of the rolled material in the plane strain state.

[0023] [4] A method for calculating the rolling load of a rolling mill, comprising: a step of calculating a contact arc length of at least one of an elastic recovery zone and an elastic reduction zone based on Hooke's law in a plane strain state of the material to be rolled, a balance of forces in the roll bite of the rolling mill, and a rolling direction stress distribution approximated by a function that depends on a position coordinate in the rolling direction of the rolling mill; and a step of calculating a contribution to the rolling load by at least one of the elastic recovery zone and the elastic reduction zone based on the contact arc length.

[0024] [5] In the step of calculating the contribution to the rolling load, the contribution to the rolling load due to the elastic recovery region is calculated based on the following formula (1), and the contribution to the rolling load due to the elastic reduction region is calculated based on the following formula (2). However, in formula (1), the following formulas (3) and (4) hold: However, in formula (2), the following formulas (5) to (7) hold: However, in the formulas (1) to (7), E' is the Young's modulus of the rolled material in the plane strain state; μ is the coefficient of friction; ν' is the Poisson's ratio of the rolled material in the plane strain state; and R' is the roll radius after flattening.

[0025] [6] The rolling load calculation method according to the above [4] or [5], wherein in the step of calculating the contact arc length, the contact arc length in the elastic recovery region is calculated based on the following formula (8), and the contact arc length in the elastic reduction region is calculated based on the following formula (9). However, in formula (8), the following formulas (10) to (13) hold: However, in formula (9), the following formulas (14) to (20) hold: However, in the formulas (8) to (20), E' is the Young's modulus of the rolled material in the plane strain state, k ０ is the yield stress at the roll bite exit, k １ is the yield stress at the roll bite entrance, R' is the roll radius after flattening, ν' is the Poisson's ratio of the rolled material in the plane strain state, μ is the coefficient of friction, and E' is the Young's modulus of the rolled material in the plane strain state.

[0026] [7] A method for calculating a contact arc length of a rolling mill, comprising the steps of calculating the contact arc length of at least one of an elastic recovery region and an elastic reduction region based on Hooke's law in a plane strain state of the rolled material, the balance of forces in the roll bite of the rolling mill, and a rolling direction stress distribution approximated by a function that depends on the position coordinate in the rolling direction of the rolling mill.

[0027] [8] In the step of calculating the contact arc length, the contact arc length in the elastic recovery region is calculated based on the following formula (1), and the contact arc length in the elastic pressure reduction region is calculated based on the following formula (2). However, in formula (1), the following formulas (3) to (6) hold: However, in formula (2), the following formulas (7) to (13) hold: However, in the formulas (1) to (13), E' is the Young's modulus of the rolled material in the plane strain state, k ０ is the yield stress at the roll bite exit, k １ is the yield stress at the roll bite entrance, R' is the roll radius after flattening, ν' is the Poisson's ratio of the rolled material in the plane strain state, μ is the coefficient of friction, and E' is the Young's modulus of the rolled material in the plane strain state.

[0028] [9] A rolling method, comprising: changing a roll gap based on the rolling load distribution calculated by using the rolling load distribution calculation method according to any one of [1] to [3] above, while reflecting a current lubrication condition.

[0029]

[10] A rolling method, comprising: changing a roll gap based on the rolling load calculated by using the rolling load calculation method according to any one of [4] to [6] above, while reflecting a current lubrication condition.

[0030] According to the method of the present disclosure, even under rolling conditions in which frictional stress increases, the rolling load distribution in the elastic region or the contribution of the elastic region to the rolling load can be calculated quickly and accurately.

[0031] 1 is a diagram showing an example of a rolling mill facility to which a rolling load distribution calculation method, a rolling load calculation method, and a contact arc length calculation method according to an embodiment of the present disclosure are applied; FIG. 2 is a diagram showing the results of calculating the rolling load distribution using the method of the present disclosure and the results of calculating the rolling load distribution using a conventional method; FIG. 3 is a table comparing the results using the method of the present disclosure and the results using the conventional method; and FIG. 4 is a schematic diagram of an elastic recovery region and an elastic reduction region.

[0032] Hereinafter, embodiments of the present disclosure will be described with reference to the drawings.

[0033] FIG. 1 is a diagram illustrating an example of a rolling mill facility to which a rolling load distribution calculation method, a rolling load calculation method, and a contact arc length calculation method according to an embodiment of the present disclosure are applied.

[0034] The rolling mill facility includes rolling mills 10-1 to 10-5, a control device 20, a process computer 30, and an online computer 40.

[0035] When there is no particular need to distinguish between the rolling mills 10-1 to 10-5, they may be simply referred to as the rolling mills 10. While five rolling mills 10, 10-1 to 10-5, are shown in Figure 1, this is just an example. The number of rolling mills 10 may be any number equal to or greater than one.

[0036] The rolling mill 10 includes rolls. The rolling mill 10 rolls the material 1 to be rolled by the rolls. The material 1 to be rolled is, for example, a steel plate. The material 1 to be rolled may also be a non-ferrous metal such as aluminum or titanium.

[0037] The control device 20 is a device that controls the rolling mill 10. The control device 20 controls the roll gap of the rolls of the rolling mill 10. The control device 20 also controls the roll speed of the rolls of the rolling mill 10.

[0038] The process computer 30 may be a general-purpose computer such as a workstation, a personal computer, etc. Alternatively, the process computer 30 may be a dedicated computer configured to function as the process computer 30 of the rolling mill facility shown in FIG.

[0039] The process computer 30 executes the rolling load distribution calculation method, the rolling load calculation method, and the contact arc length calculation method according to the present disclosure. As a result, the process computer 30 can calculate the rolling load distribution in at least one of the elastic recovery region and the elastic reduction region. Furthermore, the process computer 30 can calculate the contribution to the rolling load by at least one of the elastic recovery region and the elastic reduction region. Furthermore, the process computer 30 can calculate the contact arc length in at least one of the elastic recovery region and the elastic reduction region.

[0040] Here, the elastic recovery region is an elastic region on the delivery side of the rolling mill 10. The elastic reduction region is an elastic region on the entry side of the rolling mill 10. In this embodiment, the term "elastic region" is used as a general term for both the elastic recovery region and the elastic reduction region.

[0041] The details of the calculation of the rolling load distribution, the calculation of the contribution to the rolling load, and the calculation of the contact arc length by the process computer 30 will be described later.

[0042] The process computer 30 controls the control device 20 based on the calculated rolling load distribution, contribution to the rolling load, and contact arc length, and controls the roll gap of the rolling rolls of the rolling mill 10 and the roll speed of the rolling rolls of the rolling mill 10.

[0043] The online computer 40 may be a general-purpose computer such as a workstation, a personal computer, etc. Alternatively, the online computer 40 may be a dedicated computer configured to function as the online computer 40 of the rolling mill facility shown in FIG.

[0044] The online calculator 40 reverse-calculates the coefficient of friction and other data from actual rolling data such as the rolling load of the rolling mill 10, the thickness of the material 1 to be rolled, and the tension of the material 1 to be rolled, as well as preset data such as the roll diameter of the rolling rolls of the rolling mill 10.

[0045] The online computer 40 calculates the friction coefficient and the like using preset data such as the roll diameter of the rolling rolls of the rolling mill 10, which is set based on the calculation results of the process computer 30, thereby enabling stable and high-speed calculation of the friction coefficient.

[0046] Furthermore, the online computer 40 can feed back the calculation results of the friction coefficient and the like to the process computer 30. The process computer 30 can use the calculation results of the friction coefficient and the like fed back from the online computer 40 for calculation of the rolling load distribution, etc.

[0047] (Calculation of Contact Arc Length) A process in which the process computer 30 calculates the contact arc length of the rolling mill 10 will be described.

[0048] The process computer 30 calculates the contact arc length of at least one of the elastic recovery region and the elastic reduction region based on Hooke's law in the plane strain state of the rolled material 1, the balance of forces within the roll bite of the rolling mill 10, and the rolling direction stress distribution approximated by a function that depends on the position coordinates in the rolling direction of the rolling mill 10.

[0049] The process computer 30 calculates the contact arc length in the elastic recovery region based on the following formula (1), and calculates the contact arc length in the elastic pressure release region based on the following formula (2).

[0050] However, in formula (1), the following formulas (3) to (6) hold: However, in formula (2), the following formulas (7) to (13) hold: However, in the formulas (1) to (13), E' is the Young's modulus of the rolled material in the plane strain state, k ０ is the yield stress at the roll bite exit, k １ is the yield stress at the roll bite entrance, R' is the roll radius after flattening, ν' is the Poisson's ratio of the rolled material in the plane strain state, μ is the coefficient of friction, and E' is the Young's modulus of the rolled material in the plane strain state.

[0051] <Approximation by First-Order Polynomial> Next, a process will be described in which the process computer 30 calculates the contact arc length in the elastic recovery zone by approximating the stress in the rolling direction in the elastic recovery zone with a first-order polynomial relating to the position coordinate in the rolling direction. Note that the position coordinate in the rolling direction is set to 0 at the roll bite exit.

[0052] The process computer 30 approximates the stress q in the rolling direction by a first-order polynomial relating to the position coordinates in the rolling direction using the following equation (14). However, in formula (14), x ０ is the outlet yield point.

[0053] From the balance of forces in the roll bite of the rolling mill 10, the following equation (15) holds.

[0054] In the above equation (15), it is assumed that the plate thickness is minimum at the delivery yield point as shown in the following equation (16).

[0055] Furthermore, the above equation (15) is approximated as in the following equation (17).

[0056] Substituting the assumption of the above formula (16) and the approximation of formula (17) into formula (15) yields the following formula (18).

[0057] Therefore, at the delivery yield point, the following equation (19) holds.

[0058] On the other hand, the yield condition at the delivery yield point can be expressed as the following equation (20).

[0059] The plate thickness h(x) is expressed by the following equation (21).

[0060] Then, the following equations (22) and (23) approximately hold at the delivery yield point.

[0061] When the above equation (23) is solved for x0, the following equation (24) is obtained.

[0062] The process computer 30 can calculate the contact arc length in the elastic recovery region using the above equation (24).

[0063] In addition, if the friction coefficient μ is set to 0 in the above equation (24), a result that matches the conventional result in which friction is ignored can be obtained.

[0064] Next, a process will be described in which the process computer 30 calculates the contact arc length in the elastic reduction region by approximating the stress in the rolling direction in the elastic reduction region with a first-order polynomial relating to the position coordinates in the rolling direction.

[0065] The process computer 30 approximates the stress q in the rolling direction with a first-order polynomial relating to the position coordinates in the rolling direction using the following equation (25). However, in formula (25), x １ is the entry yield point, and l is the contact arc length.

[0066] From the balance of forces in the roll bite of the rolling mill 10, the following approximate equation (26) can be obtained.

[0067] On the other hand, when x is approximately equal to l, the following equations (27) and (28) hold.

[0068] Then, the following equation (29) is obtained.

[0069] As in the case of the elastic recovery region, the following equation (30) is obtained from the relationship at the boundary between the elastic region and the plastic region.

[0070] From the above formula (30), the contact arc length in the elastic pressure region can be obtained as shown in the following formula (31).

[0071] The process computer 30 can calculate the contact arc length in the elastic pressure region using the above equation (31).

[0072] <Approximation Using a Cubic Polynomial> Next, a process will be described in which the process computer 30 calculates the contact arc length in the elastic recovery region by approximating the stress in the rolling direction in the elastic recovery region with a cubic polynomial related to the position coordinates in the rolling direction.

[0073] The process computer 30 approximates the stress q in the rolling direction in the elastic recovery region by the following equation (32).

[0074] Here, the rolling load distribution p in the elastic recovery region is approximated by a second-order polynomial of the following equation (33).

[0075] In the above equation (33), a quadratic function is selected, but this is just an example, and other functions may be used. For example, approximation may be performed using a linear function or an elliptic function. Regardless of which function is selected, the following discussion will be almost the same.

[0076] At this time, the stress q in the rolling direction can be expressed by a third-order polynomial using the following equation (34).

[0077] Also, the value at the delivery yield point is obtained by the following equation (35).

[0078] On the other hand, p(x ０ ) = q(x ０ ) + k ０ Therefore, by substituting this into the right-hand side of the above equation (35), we get q(x ０ ) is solved to obtain the variable x as shown in the following equation (36): ０ We obtain a rational function expression for

[0079] Therefore, the following equations (37) to (39) are obtained.

[0080] Here, as shown in the following equations (40) and (41), β ０ Define η and

[0081] Then, the cubic equation in the above equation (39) can be expressed as follows: ０ This is expressed as the following equation (42) as a cubic equation for

[0082] When the above equation (42) has three real solutions, p and q are defined as in the following equations (43) and (44).

[0083] Then, the solution to be found for the above equation (42) is expressed as the following equation (45).

[0084] Therefore, the process computer 30 can calculate the contact arc length in the elastic recovery region by the following equation (46).

[0085] Next, a process will be described in which the process computer 30 calculates the contact arc length in the elastic reduction region by approximating the stress in the rolling direction in the elastic reduction region with a third-order polynomial relating to the position coordinates in the rolling direction.

[0086] The process computer 30 performs calculations similar to those in the elastic recovery region in the elastic compression region, and obtains a value at the inlet yield point as shown in the following equation (47).

[0087] On the other hand, the rolling load distribution p(x １ ) = q(x １ ) + k １ Therefore, by substituting this into the above equation (47), we obtain p(x １ ) is solved to obtain the following equation (48).

[0088] In addition, the rolling load distribution p(x １ ) can also be expressed as the following equation (49).

[0089] Therefore, the following equation (50) is obtained.

[0090] Here, as shown in the following equations (51) to (54), β １ , d, c, b are defined.

[0091] Furthermore, p and q are expressed as in the following equations (55) and (56).

[0092] Therefore, β1 is expressed as in the following equation (57).

[0093] Therefore, β １ From the definition of (a), the process computer 30 can calculate the contact arc length in the elastic pressure region by the following equation (58).

[0094] (Calculation of Rolling Load Distribution) A process in which the process computer 30 calculates the rolling load distribution of the rolling mill 10 will be described.

[0095] The process computer 30 calculates the rolling load distribution in at least one of the elastic recovery region and the elastic reduction region based on the calculated contact arc length.

[0096] The process computer 30 calculates the rolling load distribution in the elastic recovery region based on the following formula (59), and calculates the rolling load distribution in the elastic reduction region based on the following formula (60). However, in equation (59), the relationships of the following equations (61) and (62) exist: However, in equation (60), the following equations (63) to (65) hold: However, in the formulas (59) to (65), p ０ (x) is the rolling load distribution in the elastic recovery region, and p １ (x) is the rolling load distribution in the elastic reduction region, x is the position coordinate in the rolling direction, E' is the Young's modulus of the rolled material in the plane strain state; μ is the coefficient of friction; ν' is the Poisson's ratio of the rolled material in the plane strain state; and R' is the roll radius after flattening.

[0097] <Explanation of Rolling Load Distribution Using Differential Equations> Next, calculation of the rolling load distribution will be explained using differential equations. First, calculation of the rolling load distribution in the elastic recovery region will be explained.

[0098] In the elastic recovery region, the relationship of the following equation (66) holds as a differential equation regarding the rolling load.

[0099] Here, E ０ and a ０ is defined as the following equations (67) and (68).

[0100] Then, the above equation (66) can be expressed as the following equation (69).

[0101] The initial value problem (p(0)=0) for the above equation (69) can be solved analytically, and the solution is expressed as the following equation (70).

[0102] Next, the calculation of the rolling load distribution in the elastic reduction region will be explained.

[0103] In the elastic reduction region, the relationship of the following equation (71) holds as a differential equation regarding the rolling load.

[0104] Here, E １ , a １ , z are defined as in the following equations (72) to (74).

[0105] Then, the above equation (71) can be expressed as the following equation (75).

[0106] The initial value problem (p(0)=0) for the above equation (75) can be solved analytically, and the solution is expressed as the following equation (76).

[0107] The approximate solution p for the rolling load distribution in the elastic recovery region is ０ Based on (x), q(x) in the elastic recovery region can be approximated as in the following equation (77).

[0108] Also, at the delivery yield point, p ０ (x ０ ) = q ０ (x ０ ) + k ０ If we note that, the transcendental equation shown in the following equation (78) holds.

[0109]

[0110]

[0111] The above equation (79) is truncated to an appropriate degree to obtain x ０ For example, if we cut off the equation to the first order, we get the following equation (80).

[0112] A similar calculation can be performed for the elastic pressure region.

[0113] (Calculation of Contribution to Rolling Load) A process in which the process computer 30 calculates the rolling load of the rolling mill 10 will be described.

[0114] The process computer 30 calculates the contribution to the rolling load due to at least one of the elastic recovery region and the elastic reduction region based on the calculated contact arc length.

[0115] The process computer 30 calculates the contribution to the rolling load due to the elastic recovery region based on the following formula (81), and calculates the contribution to the rolling load due to the elastic reduction region based on the following formula (82). However, in equation (81), the relationships of the following equations (83) and (84) exist: However, in equation (82), the following equations (85) to (87) hold: However, in the formulas (81) to (87), E' is the Young's modulus of the rolled material in the plane strain state; μ is the coefficient of friction; ν' is the Poisson's ratio of the rolled material in the plane strain state; and R' is the roll radius after flattening.

[0116] <Explanation using definite integral about contribution to rolling load> The contribution to the rolling load by the elastic recovery region is expressed as p ０By a definite integral with the elastic recovery region of (x) as the integration interval, it is expressed as an elementary function as shown in the following equation (88).

[0117] In addition, the contribution to the rolling load due to the elastic reduction region is １ By a definite integral with the elastic pressure region of (x) as the integration interval, it is expressed as an elementary function as shown in the following equation (89).

[0118] As described above, the rolling load distribution calculation method, rolling load calculation method, and contact arc length calculation method according to this embodiment calculate the contact arc length of at least one of the elastic recovery zone and the elastic reduction zone based on Hooke's law in the plane strain state of the rolled material 1, the balance of forces in the roll bite of the rolling mill 10, and the rolling direction stress distribution approximated by a function dependent on the position coordinate in the rolling direction of the rolling mill 10. Then, based on the calculated contact arc length, the rolling load distribution of at least one of the elastic recovery zone and the elastic reduction zone is calculated. Furthermore, based on the calculated contact arc length, the contribution to the rolling load due to at least one of the elastic recovery zone and the elastic reduction zone is calculated. In this way, in order to take into account the influence of frictional stress in the elastic zone, the rolling direction stress distribution is approximated by a function dependent on the position coordinate in the rolling direction, thereby reducing calculation errors under rolling conditions in which frictional stress is large. Examples of rolling conditions in which frictional stress is large include rolling with dull rolls with large roll roughness and rolling of high-strength high-tensile steel.

[0119] Furthermore, the method according to this embodiment provides an analytical approximate solution similar to the method shown in Non-Patent Document 1, which ignores the influence of frictional stress in the elastic region. Therefore, the method according to this embodiment can calculate the rolling load distribution and the contribution to the rolling load due to the elastic region without going through a calculation process for obtaining a numerical solution of an equation, as in the method based on Airy's stress function shown in Non-Patent Document 1. Therefore, the method according to this embodiment is suitable for online calculation in the process computer 30. Furthermore, the method according to this embodiment can perform calculations using a known theoretical formula for obtaining the roll flattening radius without introducing the correction parameters shown in Patent Document 1. Therefore, the method according to this embodiment has the advantage that it is not necessary to calculate the correction parameters in advance.

[0120] Furthermore, according to the method of this embodiment, in order to take into account the influence of frictional stress in the elastic region, the rolling direction stress distribution is approximated by a function that depends on the position coordinate in the rolling direction, so that the influence of frictional stress in the elastic region can be taken into account in the calculation of the rolling load distribution.

[0121] Furthermore, the method according to this embodiment provides an analytical approximation solution for the rolling load distribution in the elastic region, and also provides analytical approximation solutions for the contact arc length in the elastic region and the plastic region. Therefore, by combining the method according to this embodiment with a model that provides an analytical approximation solution in the plastic region (such as a model by Bland & Ford), it is possible to calculate the rolling load distribution at high speed. Therefore, for example, the method can be used to visualize the rolling state by back-calculating the friction coefficient, deformation resistance, etc. based on the rolling load and forward slip in real time based on actual rolling data.

[0122] The method according to this embodiment can also be combined with a model that provides a numerical solution (such as the numerical solution of Karman's differential equation) rather than an analytical solution in the plastic region. In this case, instead of performing convergent calculations of the contact arc length in the elastic region using the Newton-Raphson method or the like, the contact arc length in the elastic region calculated by the method according to this embodiment can be used to speed up load calculations.

[0123] In this way, the method according to this embodiment can quickly and accurately calculate the rolling load distribution in the elastic region or the contribution of the elastic region to the rolling load even under rolling conditions in which frictional stress increases.

[0124] Furthermore, when calculating a rolling load distribution using the rolling load distribution calculation method according to this embodiment, the rolling load distribution may be calculated by reflecting the current lubrication conditions. In this case, the actual load is acquired, and the friction coefficient is changed using the rolling load distribution calculation method according to this embodiment to estimate a friction coefficient at which the estimated load and the actual load match, and the estimated friction coefficient may be used as a friction coefficient reflecting the current lubrication conditions, etc., and may be applied when estimating subsequent rolling loads. In this case, the friction coefficient may be set based on a table and a model formula based on operation factors. Furthermore, the roll gap may be changed based on the rolling load distribution calculated in this manner. Furthermore, when calculating a rolling load using the rolling load calculation method according to this embodiment, the calculation may be performed by reflecting the current lubrication conditions in the same way, and the roll gap may be changed based on the calculation result.

[0125] (Example) Figure 2 shows the results of calculating the rolling load distribution using the method of the present disclosure and the results of calculating the rolling load distribution using a conventional method. In Figure 2, the models of the friction coefficient and deformation resistance are fixed. Figure 2 also shows the results of applying the method of this embodiment to the calculation of the rolling load distribution assuming rolling with dull rolls.

[0126] In Fig. 2, "Bland & Ford" shows the results of calculating the rolling load distribution by a conventional method. "Bland & Ford" calculates the rolling load distribution in the elastic region by a known method, and calculates the rolling load distribution in the plastic region by the known Bland & Ford.

[0127] 2, "Bland and Ford 1" is the result of calculating the contact arc length in the elastic region by approximating the stress q in the rolling direction with a first-order polynomial, and then calculating the rolling load distribution in the elastic region based on the calculated contact arc length, according to the method of this embodiment. Note that "Bland and Ford 1" uses the well-known Bland and Ford method for calculating the rolling load distribution in the plastic region.

[0128] 2, "Bland and Ford 3" is the result of calculating the contact arc length in the elastic region by approximating the stress q in the rolling direction with a cubic polynomial, and then calculating the rolling load distribution in the elastic region based on the calculated contact arc length, according to the method of this embodiment. Note that "Bland and Ford 3" uses the well-known Bland and Ford method for calculating the rolling load distribution in the plastic region.

[0129] In addition, in FIG. 2, "Karman" indicates the result of calculating the rolling load distribution as a strict numerical solution using the Karman equation.

[0130] Referring to FIG. 2, it was confirmed that the calculation results of the rolling load distribution by "BlandAndFord1" and "BlandAndFord3" to which the method of this embodiment was applied were almost equivalent to the results of "Karman" which calculated a strict numerical solution.

[0131] FIG. 3 shows a table comparing the results of the method of this embodiment with the results of the conventional method.

[0132] In FIG. 3, "Bland & Ford" and "Bland & Sims" are the results obtained by the conventional method.

[0133] In Fig. 3, "Bland & Ford 1" and "Bland & Sims 1" are the results of calculations in which the stress q in the rolling direction is approximated by a first-order polynomial in the method of this embodiment. In Fig. 3, "Bland & Ford 3" and "Bland & Sims 3" are the results of calculations in which the stress q in the rolling direction is approximated by a third-order polynomial in the method of this embodiment.

[0134] In Fig. 3, "Karman" represents the result of calculating a strict numerical solution. In Fig. 3, "Karman3" represents the result of calculating the stress q in the rolling direction in the elastic region by approximating it with a third-order polynomial.

[0135] 3, the calculation time when the rolling load is calculated by the method of this embodiment is 1 / 100 or less of the time required by "Karman" which calculates a strict numerical solution. Furthermore, the result of calculating the rolling load by the method of this embodiment is almost the same as the result of calculating the rolling load by "Karman" which calculates a strict numerical solution.

[0136] The present disclosure is not limited to the above-described embodiments. For example, multiple blocks shown in the block diagrams may be integrated, or one block may be divided. Instead of executing multiple steps shown in the flowcharts in chronological order as described, each step may be executed in parallel or in a different order depending on the processing capacity of the device executing each step, or as needed. Other modifications are possible within the scope of the present disclosure.

[0137] 1 shows a configuration in which the process computer 30 and the online computer 40 are separate devices, these devices may be configured as a single device. In this case, the process computer 30 may have the functions of the online computer 40.

[0138] For example, the method according to the present disclosure can also be applied to the case where Orowan's theory, which takes into account the influence of shear stress, is used to calculate the plastic region.

[0139] REFERENCE SIGNS LIST 1 Rolled material 10 Rolling mill 20 Control device 30 Process computer 40 Online computer

Claims

1. A method for calculating the rolling load distribution of a rolling mill, comprising: a step of calculating a contact arc length of at least one of an elastic recovery zone and an elastic reduction zone based on Hooke's law in a plane strain state of a rolled material, a balance of forces in the roll bite of the rolling mill, and a rolling direction stress distribution approximated by a function that depends on a position coordinate in the rolling direction of the rolling mill; and a step of calculating a rolling load distribution of at least one of the elastic recovery zone and the elastic reduction zone based on the contact arc length.

2. The rolling load distribution calculation method according to claim 1, wherein in the step of calculating the rolling load distribution, the rolling load distribution in the elastic recovery region is calculated based on the following formula (1), and the rolling load distribution in the elastic reduction region is calculated based on the following formula (2). However, in formula (1), the following formulas (3) and (4) are expressed: However, in formula (2), the following formulas (5) to (7) are expressed: In the formulas (1) to (7), p ０ (x) is the rolling load distribution in the elastic recovery region, p １ (x) is the rolling load distribution in the elastic reduction region, x is the position coordinate in the rolling direction, E' is the Young's modulus of the rolled material in the plane strain state, μ is the coefficient of friction, ν' is the Poisson's ratio of the rolled material in the plane strain state, and R' is the roll radius after flattening.

3. The rolling load distribution calculation method according to claim 1 or 2, wherein in the step of calculating the contact arc length, the contact arc length in the elastic recovery region is calculated based on the following formula (8), and the contact arc length in the elastic reduction region is calculated based on the following formula (9): However, in formula (8), the following formulas (10) to (13) hold: However, in formula (9), the following formulas (14) to (20) hold: However, in the formulas (8) to (20), E' is the Young's modulus of the rolled material in the plane strain state; k ０ is the yield stress at the roll bite exit, k １ is the yield stress at the roll bite entrance, R' is the roll radius after flattening, v' is the Poisson's ratio of the rolled material in the plane strain state, μ is the coefficient of friction, and E' is the Young's modulus of the rolled material in the plane strain state.

4. A method for calculating the rolling load of a rolling mill, comprising the steps of: calculating a contact arc length of at least one of an elastic recovery zone and an elastic reduction zone based on Hooke's law in a plane strain state of a material being rolled, a balance of forces in the roll bite of the rolling mill, and a rolling direction stress distribution approximated by a function that depends on a position coordinate in the rolling direction of the rolling mill; and calculating a contribution to the rolling load by at least one of the elastic recovery zone and the elastic reduction zone based on the contact arc length.

5. A rolling load calculation method according to claim 4, wherein in the step of calculating the contribution to the rolling load, the contribution to the rolling load due to the elastic recovery region is calculated based on the following formula (1), and the contribution to the rolling load due to the elastic reduction region is calculated based on the following formula (2). However, in formula (1), the following formulas (3) and (4) are expressed: However, in formula (2), the following formulas (5) to (7) are expressed: However, in the formulas (1) to (7), E' is the Young's modulus of the rolled material in the plane strain state, μ is the coefficient of friction, ν' is the Poisson's ratio of the rolled material in the plane strain state, and R' is the roll radius after flattening.

6. The rolling load calculation method according to claim 4 or 5, wherein in the step of calculating the contact arc length, the contact arc length in the elastic recovery region is calculated based on the following formula (8), and the contact arc length in the elastic reduction region is calculated based on the following formula (9): However, in formula (8), the following formulas (10) to (13) hold: However, in formula (9), the following formulas (14) to (20) hold: However, in the formulas (8) to (20), E' is the Young's modulus of the rolled material in the plane strain state; k ０ is the yield stress at the roll bite exit, k １ is the yield stress at the roll bite entrance, R' is the roll radius after flattening, v' is the Poisson's ratio of the rolled material in the plane strain state, μ is the coefficient of friction, and E' is the Young's modulus of the rolled material in the plane strain state.

7. A method for calculating the contact arc length of a rolling mill, comprising the steps of calculating the contact arc length of at least one of an elastic recovery region and an elastic reduction region based on Hooke's law in a plane strain state of a rolled material, a balance of forces in the roll bite of the rolling mill, and a rolling direction stress distribution approximated by a function that depends on a position coordinate in the rolling direction of the rolling mill.

8. The method for calculating a contact arc length according to claim 7, wherein in the step of calculating the contact arc length, the contact arc length of the elastic recovery region is calculated according to the following formula (1), and the contact arc length of the elastic compression region is calculated according to the following formula (2): However, in formula (1), the following formulas (3) to (6) are expressed: However, in formula (2), the following formulas (7) to (13) hold: However, in the formulas (1) to (13), E' is the Young's modulus of the rolled material in the plane strain state; k ０ is the yield stress at the roll bite exit, k １ is the yield stress at the roll bite entrance, R' is the roll radius after flattening, v' is the Poisson's ratio of the rolled material in the plane strain state, μ is the coefficient of friction, and E' is the Young's modulus of the rolled material in the plane strain state.

9. A rolling method, comprising the steps of: changing a roll gap based on the rolling load distribution calculated by using the rolling load distribution calculation method according to any one of claims 1 to 3, while reflecting a current lubrication condition.

10. A rolling method, comprising the steps of: changing a roll gap based on the rolling load calculated by using the rolling load calculation method according to any one of claims 4 to 6, while reflecting a current lubrication condition.