CN114632826B - Method for setting rolling force and rolling moment of asynchronous rolling of hot rolled steel strip - Google Patents
Method for setting rolling force and rolling moment of asynchronous rolling of hot rolled steel strip Download PDFInfo
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- B—PERFORMING OPERATIONS; TRANSPORTING
- B21—MECHANICAL METAL-WORKING WITHOUT ESSENTIALLY REMOVING MATERIAL; PUNCHING METAL
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- B21B37/00—Control devices or methods specially adapted for metal-rolling mills or the work produced thereby
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Abstract
The invention belongs to the technical field of rolling production, and particularly relates to a method for setting rolling force and rolling moment of asynchronous rolling of a hot-rolled steel strip. The method comprises the following steps: s1: setting parameters; s2: determining the neutral angle gamma of the fast roll f And slow roll neutral angle gamma s (ii) a S3: obtaining a calculated value of the total rolling force F of the asynchronous rolling; s4: obtaining a calculated value of the total rolling moment T of the asynchronous rolling; s5: and setting the final calculated values of the total asynchronous rolling force F and the total rolling moment T for the rolling force and the rolling moment of the asynchronous rolling of the hot rolled steel strip. The invention provides a method for setting and calculating rolling force and rolling moment in an asynchronous rolling process of a hot-rolled steel strip. The method can be used for setting the rolling parameters in the asynchronous hot rolling process under various conditions of different linear speeds of the rollers, different diameters of the rollers and the like, and has high calculation precision.
Description
Technical Field
The invention belongs to the technical field of rolling production, and particularly relates to a method for setting rolling force and rolling moment of asynchronous rolling of a hot-rolled steel strip.
Background
The asynchronous rolling is a new technology which effectively reduces rolling pressure, reduces rolling passes and improves efficiency. The method mainly comprises three asymmetric rolling forms: the diameters of the upper and lower rollers are unequal, the speeds of the upper and lower rollers are unequal, and the friction between the upper and lower rollers and the surface of a rolled piece is unequal. In the asynchronous rolling, neutral points of an arc contacted by an upper roller and a lower roller are deviated towards two sides, a rubbing rolling effect is formed in a deformation area between the two neutral points, and strong additional shearing deformation is generated, so that the plastic flow of metal is accelerated.
The Chinese invention patent with the application number of 201911124336.8 discloses an asynchronous hot continuous rolling method for preparing a hot-rolled ultrathin steel strip, and different asynchronous rolling modes are adopted on a hot continuous rolling unit with a 5-7 stand to realize the rolling production of the hot-rolled steel strip with the ultrathin specification of 0.6 mm.
The Chinese invention patent with the application number of 201911123929.2 discloses an asynchronous cold continuous rolling method for preparing cold-rolled ultrathin steel strips, and different asynchronous rolling modes are adopted on a 5-rack cold continuous rolling unit to realize the rolling production of the cold-rolled steel strips with the ultrathin specification of 0.09 mm.
The Chinese patent with the application number of 202110393259.7 discloses a rolling force setting method for asynchronous rolling of steel strips. The patent assumes that the contact friction between the upper and lower working rolls and the strip steel conforms to the Coulomb friction law, and assumes that the roll diameters of the upper and lower working rolls are different and the speeds of the upper and lower working rolls are different. And giving out a neutral angle formula of the fast roller and the slow roller, and respectively calculating the rolling force of the front sliding area, the rolling area and the rear sliding area.
In the above-mentioned patents for the asynchronous hot continuous rolling and the asynchronous cold continuous rolling of extremely thin steel strip, the assumption of coulomb friction conditions and plastic conditions of uniform compression is adopted. For the cold rolling process, this assumption does not cause a large rolling force calculation error. However, in the hot rolling process, since the rolling stock and the roll are in a stuck state by contact friction, if the above assumption is adopted, a large error occurs in the calculation of the rolling force.
The invention provides a method for calculating rolling force and rolling moment in an asynchronous hot continuous rolling process by adopting a plastic condition hypothesis of a full adhesion friction condition and uneven compression.
Disclosure of Invention
Aiming at the defects of the prior art, the invention provides a method for setting the rolling force and the rolling moment in the asynchronous rolling process of a hot rolled steel strip. The asynchronous rolling can be the asynchronous rolling of the two working rolls with different roll radiuses and different roll linear speeds or the combination of the two working rolls. Under the condition that the linear speeds of the rollers are different, the roller with the higher linear speed is called a fast roller, and the roller with the higher linear speed is called a slow roller.
The rolling force model of the asynchronous rolling process established by the invention needs the following basic assumptions:
(1) The contact arc of the elastically deformed roll is circular, and the roll radius (R) of the fast roll (the parameters related to the fast roll in the invention are all represented by f subscript) and the roll radius (R) of the slow roll (the parameters related to the slow roll in the invention are all represented by s subscript) are respectively f 、R s ) And roller flattening radius (R' f 、R′ s ) The ratio of (A) to (B) is constant, i.e.
(2) The width-thickness ratio of the steel strip is large, the width can be ignored during rolling, and the steel strip is deformed according to a plane;
(3) The horizontal force is considered to be unevenly distributed along the height direction of the section, so the resultant force f of the horizontal force is used for replacing the horizontal force;
(4) Adopting the assumption of full adhesion friction, the friction stress of the fast roller and the slow roller is tau f =τ s =k;
The following approximate conditions were also used:
(5) Radian of contact arc of whole rolling deformation zoneAre all small, so for any point in the whole rolling deformation zone, there is
As shown in figure 1, the upper roll is a fast roll, the lower roll is a slow roll, the rolling deformation zone of the steel strip is divided into three parts on the contact arc of the upper roll, and the three parts are a rear sliding zone, a rolling zone and a front sliding zone from an inlet to an outlet. On the contact arc of the deformation zone of the steel strip and the fast roll, the central line of the working roll is taken as a zero point, the inlet direction is directed, and the radian is 0-gamma f The partial contact arc is the contact arc of the front sliding area and the radian gamma f ~γ s Part of contact arc is contact arc of rubbing area, radian gamma s The partial contact arc at α is the contact arc of the back sliding zone, where γ f Is the neutral angle of the fast roller, gamma s Is the neutral angle of the slow roll, alpha is the bite angle of the fast roll, and are all expressed by radian.
Referring to FIG. 1, a coordinate system is established with the center line of the roll as the y-axis and the rolling direction as the x-axis, a microcell dx is taken at a position which is away from the center line x of the roll in the horizontal direction,the camber value of the included angle between the micro unit position and the central line of the roller on the contact arc of the deformation zone and the upper roller (simpleCalled the arc value of the point contact arc), the force analysis is carried out along the horizontal direction to obtain:
where df is the change in the resultant force f in the horizontal direction at x in the unit width, back and forth of the microcell. k is yield shear stress, namely the friction stress of the fast roller and the slow roller at the microcell under the full-adhesion friction condition, and 2k is the deformation resistance at the microcell.
The direction of the friction force of the different deformation zones differs, as expressed by the sign of the expression (1) before the last term 2kdx of the expression (1): the front sliding area is positive, the rear sliding area is negative, and the last item in the rubbing rolling area formula (1) is directly 0.
From the geometric relationships and approximation (5) of FIG. 1, one can derive
Wherein R' is the flattening radius of the roller,is the thickness of the steel strip at dx, h i 、h o Respectively the inlet thickness and the outlet thickness of the steel strip, and alpha is the biting angle of the steel strip on the corresponding roller.
For convenience of description, subscripts of parameters such as contact arc radian, biting angle, friction coefficient and flattening radius of a certain point of the fast roll are omitted, namely, the subscripts are directly usedα=α f And R '= R' f For slow roll, these parameters are related to fast roll, specifically
A compound of the formula (2) and(3) In the formula (1), the unit rolling pressures of the upper and lower rolls (i.e., the radial compressive stresses p of the rolls) are considered to be equalThen:
according to the Orowan plasticity condition at full adhesion
By substituting formula (5) for formula (4)
The frictional stress k at the microcell is constant, the above formula can be rewritten as
Wherein, the last term of the equation is that the front sliding area is taken, the back sliding area is taken, and the rolling area is zero.
The two sides of the formula (7) are simultaneously pairedIntegral, the solution of the differential equation for unit rolling pressure is:
the integral constant C in the equation (8) may be determined by boundary conditions of the forward slip region, the rolling region, and the backward slip region.
In the formula, 2k o Is the exit-side deformation resistance, p o Is the unit rolling pressure (vertical stress) on the outlet side, σ o Is the outlet unit tension (horizontal stress). Substituting equation (9) into equation (8) can obtain integral constant C corresponding to the forward sliding region, and thus obtain unit rolling pressure p of the forward sliding region I Is composed of
In the formula, 2k i Is the resistance to deformation on the inlet side, p i Is the unit rolling pressure, σ, of the inlet side i Is the unit tension of the inlet. When the formula (11) is substituted into the formula (8), the unit rolling pressure of the backward sliding region can be obtained
Due to the continuity of the boundary conditions, the unit rolling pressure isThe number of points is equal, that is,
in the formula (I), the compound is shown in the specification,is a neutral angle gamma f The thickness of the steel strip at the point.
The unit rolling pressure of the rolling area obtained by finishing is
in the same way, the unit rolling pressure isThe positions are still equal, so that the unit rolling of the rolling area is realizedThe control pressure can also be expressed as
it can be seen that in order to achieve the above calculation of the unit rolling pressure, the neutral angle parameter γ also needs to be determined f And gamma s . At any point x in the rolling area (the radian of the included angle between the contact arc of the fast roll and the central line of the roll is) The unit rolling pressures calculated by the equations (14) and (15) are equal, that is:
simplified to
The metal satisfies the principle of equal second flow at any point in the deformation zone, and the relationship between the neutral angles of the fast and slow rolls can be expressed as:
wherein m is the differential velocity ratio, v f And v s The roll linear speeds of the fast roll and the slow roll are respectively.The thicknesses of the steel strips at the neutral angles of the fast roller and the slow roller are respectively.
The joint formula (17) and the formula (18) can obtain a calculation expression of the neutral angles of the fast roller and the slow roller:
in the asynchronous rolling process, the total rolling force F of the asynchronous rolling can be expressed as the sum of the total rolling forces of the front sliding area, the rolling area and the rear sliding area, and the integral calculation of the unit rolling pressure is performed in sections:
rolling moment T of the fast and slow rolls f And T s Respectively as follows:
the total rolling moment T is:
in the formula, it is obviousThe integral operation of (A) is also carried out by three sections of a front sliding area, a rolling area and a rear sliding area, namely
In the actual process of asynchronous rolling, most parameters of the calculation process (for example)Inlet and outlet thickness h of the steel strip i 、h o Width of steel strip B, different speed ratio m, coefficient of friction of work roll μ s 、μ f Roll diameter R of work roll s 、R f Unit tension sigma of steel strip inlet and outlet i 、σ 0 Etc.) can be determined by the actual rolling condition, and the deformation resistance 2k at a certain point x of a contact arc in the asynchronous rolling process of the steel strip can be calculated by the following formula:
wherein, sigma is initial deformation resistance, MPa; ε is the true strain at point x, i.e., ln (h/h) int ) H is the thickness of the steel strip at the point and can be calculated according to the contact arc radian of the point by the formula (3), h int The thickness of the incoming material of the steel strip;is the rolling speed, m/s; t is the rolling temperature, DEG C; b 0 ~b 4 Is the undetermined coefficient. The initial deformation resistance and undetermined coefficient are constants according to specific rolling process and steel strip types and can be obtained through an experimental mode. It can be seen that the arc of contact of formula (24) isA function of the correlation. Resistance to deformation on the inlet side 2k i Resistance to deformation at exit side 2k o The inlet and outlet thicknesses h of the steel strip can be respectively adjusted i 、h o And substituting to calculate.
Another key parameter in the above calculation is the roll flattening radius R' of the fast rolls. The mutual coupling relationship exists between the rolling force and the flattening radius of the roller, so that an accurate mode is to numerically solve the rolling force in an iteration mode and recalculate the flattening radius until the relative error of the two flattening radii before and after meets a certain precision requirement, and then the iteration can be stopped. The iterative calculation flow of the rolling force model is shown in fig. 2.
The iteration convergence condition is as follows:
in the formula, R 0 ' calculating value of the flattening radius at this time, mm; r' is the calculated value of the last flattening radius, namely the flattening radius value used in the calculation, namely mm; epsilon R For iterative calculation accuracy, 10 is generally taken -3 Can meet the requirement, and can be not more than 10 -3 The numerical value of (c). Meanwhile, in order to prevent the calculation from falling into an infinite iteration loop, a maximum number of iteration loops, such as 5, may be given to guarantee the model calculation time.
For the asynchronous hot continuous rolling process, the method can be used for calculating the setting calculation of the hot rolling force of a certain rolling pass.
The overall idea and derivation process of the calculation of the present invention are described above, and the following description is made for the specific steps of the present invention, and the method of the present invention includes the following basic steps:
s1: setting parameters:
setting parameters of the asynchronous rolling process, wherein the parameters comprise:
width of steel strip B, incoming thickness of steel strip h init Thickness h of steel strip entrance i Outlet thickness h of steel strip o Modulus of elasticity E of steel strip s Elastic modulus E of work roll wr Differential speed ratio m of work rolls, inlet unit tension σ i Outlet unit tension σ o Radius of the fast roll R f = R, slow roll radius R s Ratio of radius of fast and slow rollsRolling rateRolling temperature t;
s2: determining the fast roll neutral angle gamma f And slow roll neutral angle gamma s (ii) a Specifically, the calculation can be performed by the method in the above formula (19);
s3: obtaining a calculated value of the total asynchronous rolling force F:
wherein R' is the flattening radius of the fast roll, p I 、p II 、p III The unit rolling pressures of the front sliding area, the rolling area and the rear sliding area are respectively;
p II the calculation is performed using one of the following two equations:
Wherein:the arc of contact of the fast roller isThe thickness of the steel strip at the position (b) is calculated using the following formula:
2k is the arc radian of the contact arc of the fast rollerIs resistant to deformation of the position ofForce, 2k o For resistance to deformation at the exit side of asynchronous rolling, 2k i Is the deformation resistance of the asynchronous rolling inlet side; can be calculated using equation (24) above;
alpha is the biting angle of the steel strip deformation zone on the fast roll, and is calculated according to the following formula:
s4: obtaining a calculated value of the total rolling moment T of the asynchronous rolling:
s5: and setting the final calculated values of the total asynchronous rolling force F and the total rolling moment T for the rolling force and the rolling moment of the asynchronous rolling of the hot rolled steel strip.
In order to make the calculation result more accurate, the flattening radius needs to be coupled with the rolling force, and specifically, an iterative calculation method can be adopted, and the flow is shown in fig. 2:
recalculating the fast rolling flat radius R 'from the current F value after calculating the current calculated values of F and T from the steps S2 to S4 using the current fast rolling flat radius R' o And judging whether the current iterative calculation process meets the iterative convergence condition: if the iteration convergence condition is met, the calculation is finished, and the calculated values F and T obtained in the steps S3 and S4 of the current iteration calculation process are final values and are used in the step S5; if the iterative convergence condition is not met, recalculating the current obtained flattening radius R' o Iterating back to the step S2, and performing next iteration calculation as a new fast rolling flat radius R。
The initial value of the fast rolling flat radius R', namely the roll radius R can be directly adopted as the fast rolling flat radius in the first calculation. The iterative convergence condition isε R For iterative calculation accuracy, take not more than 10 -3 The numerical value of (c).
Recalculating crush radius R 'from current F value' o One method of (1) is as follows:
in the formula (I), the compound is shown in the specification,E wr is the elastic modulus of the roll, upsilon s For Poisson's ratio, 0.3 may be taken.
The invention has the beneficial effects that: the invention provides a method for setting and calculating rolling force and rolling moment in an asynchronous rolling process of a hot-rolled steel strip. The method can be used for setting the rolling parameters in the asynchronous hot rolling process under various conditions of different linear speeds of the rollers, different diameters of the rollers and the like. The calculation precision is high, and the error value of the calculated rolling force and the actually measured rolling force is within 10 percent.
Drawings
FIG. 1: the division of the steel strip rolling deformation zone and the force analysis of the micro-units are shown.
FIG. 2: the flow schematic diagram of the iterative calculation method of the rolling force and the rolling moment in the asynchronous rolling process of the steel strip.
Detailed Description
Example 1
Taking a five-stand four-roller hot continuous rolling unit as an example, the diameter of a working roller of the unit is 600-750 mm, and the maximum rolling force of the unit is 32MN. The initial thickness of the hot-rolled steel strip is 35mm, the width is 1001mm, and the chemical components in percentage by weight (less than or equal to) are as follows: c:0.18, si:0.18, mn:0.68, cr:0.02, nb:0.006, mo:0.002, ti:0.029, P:0.014, S:0.008 and the finishing temperature of 910 ℃. The differential speed ratio of each rolling pass, the pre-rolling thickness, the post-rolling thickness, the rolling force calculated by using the method of the invention and the actually measured rolling force are shown in table 1.
TABLE 1 EXAMPLE 1 differential speed ratio, pre-Rolling thickness, post-Rolling thickness, calculated Rolling force and measured Rolling force
As can be seen from Table 1, the error between the asynchronous hot rolling force of the hot rolled steel strip calculated by the method and the actually measured asynchronous hot rolling force is within 10.8 percent, the error between the calculated value of the rolling moment and the actually measured value is within 19.9 percent, and the precision is high.
Example 2
Taking a CSP six-rack hot continuous rolling finishing mill unit of a certain factory as an example, the diameters of working rolls of racks F1-F3 of the unit are 720-800 mm, the diameters of working rolls of racks F4-F6 are 540-600 mm, and the maximum rolling force is 32MN. The initial thickness of the hot-rolled steel strip is 40.57mm, the width is 1510mm, and the chemical components by weight percentage (less than or equal to percent) are as follows: c:0.087, si:0.13, mn:1.35, cr:0.02, V:0.01, nb:0.033, P:0.01, S:0.004 and the finishing temperature is 880 ℃. The differential speed ratio, the pre-rolling thickness, the post-rolling thickness, the rolling force calculated by using the method of the invention and the actually measured rolling force of each rolling pass are shown in table 2.
TABLE 2 example 2 differential speed ratio, pre-rolling thickness, post-rolling thickness, calculated rolling force and measured rolling force
As can be seen from Table 2, the error between the asynchronous hot rolling force of the hot rolled steel strip calculated by the method and the actually measured asynchronous hot rolling force is within 9.0 percent, the error between the calculated value of the rolling moment and the actually measured value is within 15.6 percent, and the precision is high.
Example 3
Taking a 2150mm seven-frame four-roller hot continuous rolling unit as an example, the diameter of a working roller of the unit is 570-750 mm, the diameter of a supporting roller of the unit is 1300-1450 mm, and the maximum rolling force of the unit is 32MN. The initial thickness of the hot-rolled steel strip is 40.67mm, the width is 1304.9mm, and the chemical components by weight percentage (less than or equal to percent) are as follows: c:0.074, si:0.17, mn:1.61, cr:0.02, V:0.01, nb:0.036, P:0.013, S:0.0019 and the finishing temperature of 863 ℃. The differential speed ratio, the pre-rolling thickness, the post-rolling thickness, the rolling force calculated by using the method of the present invention and the actually measured rolling force of each rolling pass are shown in table 3.
TABLE 3 EXAMPLE 3 differential speed ratio, pre-Rolling thickness, post-Rolling thickness, calculated Rolling force and measured Rolling force
As can be seen from Table 3, the error between the asynchronous hot rolling force and the actually measured asynchronous hot rolling force of the hot rolled steel strip calculated by the method is within 8.6 percent, the error between the calculated value and the actually measured value of the rolling moment is within 15.1 percent, and the precision is high.
Example 4
Taking a 2150mm seven-frame four-roller hot continuous rolling unit as an example, the working roller of the unit is 570-750 mm, the diameter of the supporting roller is 1300-1450 mm, and the maximum rolling force of the unit is 32MN. The hot-rolled steel strip has the initial thickness of 32mm and the width of 1250.5mm, and comprises the following chemical components in percentage by weight (less than or equal to percent): c:0.032, si:0.009, mn:0.24, cr:0.012, V:0.0011, nb:0.0006, P:0.009, S:0.006 and the finishing temperature of 900 ℃. The differential speed ratio, the pre-rolling thickness, the post-rolling thickness, the rolling force calculated by using the method of the present invention and the actually measured rolling force of each rolling pass are shown in table 4.
TABLE 4 example 4 differential speed ratio, pre-rolling thickness, post-rolling thickness, calculated rolling force and measured rolling force
As can be seen from Table 4, the error between the asynchronous hot rolling force of the hot rolled steel strip calculated by the method and the actually measured asynchronous hot rolling force is within 8.9%, the error between the calculated value of the rolling moment and the actually measured value is within 20.7%, and the precision is high.
Claims (8)
1. A method for setting rolling force and rolling moment of asynchronous rolling of a hot rolled steel strip is characterized by comprising the following steps:
s1: the parameters are set up in such a way that,
setting parameters of the asynchronous rolling process, wherein the parameters comprise:
width B of steel strip and incoming thickness h of steel strip init Thickness of entry of steel strip h i Outlet thickness h of steel strip o Modulus of elasticity E of steel strip s Modulus of elasticity E of work roll wr Differential speed ratio m of work rolls, inlet unit tension σ i Outlet unit tension σ o Radius of the fast roll R f = R, slow roll radius R s Ratio of radius of fast and slow rollsRolling speed of rollingThe rolling temperature t;
s2: determining the neutral angle gamma of the fast roll f And slow roll neutral angle gamma s ;
S3: obtaining the calculated value of the total rolling force F of the asynchronous rolling,
wherein R' is the flattening radius of the fast roll, p I 、p II 、p III Are respectively frontThe unit rolling pressure of the positions of the sliding area, the rolling area and the post-sliding area,
p II the calculation is performed using one of the following two equations:
Wherein:the arc of contact of the fast roller isThe thickness of the steel strip at the position (d) is calculated using the following formula:
2k is the arc of contact of the fast roll is2k, resistance to deformation o For resistance to deformation on the outlet side of asynchronous rolling, 2k i Is the deformation resistance of the asynchronous rolling inlet side;
alpha is the biting angle of the steel strip deformation zone on the fast roll, and is calculated according to the following formula:
s4: obtaining a calculated value of the total rolling moment T of the asynchronous rolling,
s5: the final calculated values of the total asynchronous rolling force F and the total rolling moment T are used for setting the rolling force and the rolling moment of asynchronous rolling of the hot rolled steel strip;
wherein, the calculation of the total rolling force F and the total rolling moment T of the asynchronous rolling needs the following basic assumptions that (1) the contact arc of the roller after elastic deformation is circular, and the radius R of the fast roller and the slow roller f 、R s And roll flattening radius R' f 、R′ s The ratio of (A) to (B) is constant, i.e.(2) The width-thickness ratio of the steel strip is large, the widening can be neglected during rolling, and the steel strip is deformed according to a plane; (3) The horizontal force is considered to be unevenly distributed along the height direction of the section, so the resultant force f of the horizontal force is used for replacing the horizontal force; (4) Adopting the assumption of full adhesion friction, the friction stress of the fast roller and the slow roller is tau f =τ s = k; (5) Radian of contact arc of whole rolling deformation zoneAre all very small, so for any point in the entire rolling deformation zone, there are
3. the method for setting the rolling force and the rolling moment for the asynchronous rolling of the hot rolled steel strip as claimed in claim 1, wherein the deformation resistance is calculated by:
4. The method for setting the rolling force and the rolling moment of the asynchronous rolling of the hot rolled steel strip according to claim 1, wherein the values of the fast rolling flat radius R', the asynchronous rolling total rolling force F and the asynchronous rolling total rolling moment T are obtained in an iterative manner, and the method comprises the following specific steps:
flat radius R' in the current use of fast rolls, according toAfter calculating the current calculated values of F and T in the steps S2 to S4, recalculating the fast rolling flat radius R 'according to the current value of F' o And judging whether the current iterative computation process meets the iterative convergence condition: if the iteration convergence condition is met, the calculation is finished, and the calculated values F and T obtained in the steps S3 and S4 of the current iteration calculation process are final values and are used in the step S5; if the iteration convergence condition is not met, recalculating the current obtained flattening radius R' o And (5) iterating back to the step S2, and performing the next iteration calculation as a new fast rolling flat radius R'.
6. The method for setting the rolling force and the rolling moment for the asynchronous rolling of the hot rolled steel strip as claimed in claim 5, wherein the iterative calculation precision ε R Not more than 10 -3 。
7. The method for setting rolling force and rolling moment in asynchronous rolling of hot rolled steel strip as claimed in claim 4, wherein the recalculation of the fast rolling flat radius R 'according to the current F value' o The method comprises the following steps:
8. The method for setting the rolling force and the rolling torque for the asynchronous rolling of the hot rolled steel strip as claimed in claim 4, wherein the initial value of the fast roll flattening radius R' is the fast roll radius R.
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