ISIJ International, Vol. 45 (2005), No. 8, pp. 1173–1178 Mathematical Model for the Thin Strip Cold Rolling and Temper Rolling Process with the Influence Function Method Yuli LIU and Won-Ho LEE1) Dept. of Process Eng. & Dev., QUAD Eng. Inc., 75 Scarsdale Road, Toronto, Ontario, M3B 2R2, Canada. 1) POSCO Tech. Res. Lab., Pohang P.O. Box 36. 790-785, Korea. (Received on February 8, 2005; accepted on April 26, 2005 ) A mathematical model for the thin strip cold and temper rolling process has been developed using the influence function method. By solving the equations describing the roll gap phenomena in a unique procedure and considering more influence factors, the model offers significant improvements in accuracy, robustness and generality of the solution for the thin strip cold and temper rolling conditions. The relationship between the shape of the roll profile and the roll force was also discussed. Calculation results show that any change increasing the roll force may result in or enlarge the central flat region in the deformation zone. Applied to the temper rolling process, the model can well predict not only the rolling load but also the large forward slip. Therefore, the measured forward slip, together with the measured roll force, was used to calibrate the model. The model was installed in the setup computer of a temper rolling mill to make parallel setup calculations. The calculation results showed good agreement with the measured data and the validity and precision of the model were proven. KEY WORDS: steel cold rolling; mathematical model; roll force. rolling process in which there are relatively large entry and exit elastic zones. (2) The pressure profile iterative loop was taken as the basic iterative loop. The boundaries of different zones are determined with an extra iterative loop. Under some circumstances, it is very difficult to determine the boundaries of different zones5) because the iterative processes to determine the boundaries would diverge or not converge to unique values. (3) The solution procedure was divided into three regimes, namely, Regime I (with no central flat zone), Regime II (with a small central flat zone and one pressure peak) and Regime III (with a large central flat zone and two pressure peaks). For the different regimes there are different algorithms. Before using the model to calculate a certain rolling case, we have to decide which regime the certain rolling case belongs to. Although the boundaries between the different regimes were given through multi-variable regression analysis in the foil rolling case,3) it can not be applied to other cases. Several applications of Fleck’s work have since followed by Dixon et al.,6) Domanti et al.,7) Gratacos et al.,8) Montmitonnet et al.,9) Haeseling et al.,10) and Liu et al.11) Some improvements were made, but the above mentioned problems remained unsolved. In this paper, the aforementioned shortcomings were eliminated. The strip elastic deformation at the entry and exit was taken into consideration. The roll profile iterative loop was taken as the basic loop, so that the boundaries of different zones can be directly determined according to the 1. Introduction Severe cases of strip rolling, such as thin strip cold rolling and temper rolling etc., cause well known problems with simplified classic circular arc models. They may have convergence difficulties or bad sensitivity to process parameters. In order to solve these problems, some researchers have attempted to develop more realistic models by the use of influence functions describing the roll deformation (see, for example, Jortner et al.,1) Grimble et al.2) These models gave some prediction improvement, but they failed to converge for thinner or harder materials as ever. A major development in modeling of the thin strip and foil rolling process with the influence function method was achieved by Fleck et al.3) By assuming that there is a region of roll flattening where the roll surface is flat and parallel, they solved the non-convergence problems. In the flat region, no further reduction takes place and the shear or frictional stresses at the roll/strip interface remain at values below that predicted by the Coulomb friction law. Therefore, the roll pressure in this region can be obtained by inverting the roll profile to roll pressure relationship defined by the roll deformation influence function. This model has provided useful reference results, however it has several shortcomings: (1) The strip elastic deformation zones at the entry and exit were neglected, which results in an under estimation of the load. Because of the ‘elastic plug’ effect,4) the predicted load by this model may be much lower than the actual one. Besides, the model could not be used in light reduction 1173 © 2005 ISIJ ISIJ International, Vol. 45 (2005), No. 8 roll profile. The extra boundary determination iterative loop was eliminated, and it was relatively easy to determine the boundaries of the different zones. Besides, which regime the rolling case belongs to could also be directly determined by observing the shape of the roll profile. Therefore, it would not be necessary to determine the regimes and choose the calculation procedure before hand. After calibration using the production data, the newly developed model was applied to the actual temper rolling process and the validity and precision of the model were also proven. 2. Main Equations A geometric sketch of the deformation zones is shown in Fig. 1. The geometric parameters are defined with a polar coordinates system. The roll-gap model is based on two relationships which enable a gauge profile to be calculated from a known pressure profile and conversely a pressure profile to be calculated from a gauge profile. The elastic deformation of the work roll is related to the roll pressure via the linear integral equation ∫ Fig. 1. differential equations. In the entry and exit elastic regions it is2): dp 2a  µpν ⫹ p(1⫺2ν ) tan β 2(h⫺ h1 ) tan βE  ⫽⫺ ⫹   dθ h  1⫺ν h1 (1⫺ν 2 )  ...........................................(4) φ a(θ ) ⫽ U (θ ⫺ t ) p(t )dt ⫹ R ...................(1) 0 where t : a(q ) : R: p(t) : U(q ⫺t) : and, applying the yield criteria of the difference of vertical and horizontal stresses equal to yield stress in the plastic regions2): integration variable, which varies from 0 to f ; work roll radius at point q ; undeformed radius of work roll; Normal pressure profile; Jortner’s influence function9,10); dp dy 2a ................(5) ⫽⫺ ( µp⫺ y tan β ) ⫹ dθ h dθ where n : Poisson’s ratio; E : Young’s modulus of elasticity; h1 : unstrained gauge in a elastic region; (h⫺h1)/h1 is the elastic strain in vertical direction; R∆φ U (θ ⫺t ) ⫽ πE     1⫺ cos(θ ⫺t )  ⫻ (1⫺ν 2 )cos(θ ⫺t ) ln  ⫹ 2    1⫹ cos(θ ⫺t )     β⬇ ⫺ (1⫺ν ⫺2ν 2 ) sin(θ ⫺t ) dh ; 2adθ y : yield stress; m : friction coefficient between rolls and the strip. In the central flat region, the discrete pressure profile can be obtained by solving the following linear equation system (all parameters are expressed in discrete form):   1⫺ cos(θ ⫺t )     1⫹ cos(θ ⫺t )  ⫻  tan⫺1  ⫹ tan⫺1      sin(θ ⫺t )     sin(θ ⫺t )   ...........................................(2) j2 where D f : segmental angle. The gauge profile is determined by the deformed roll radius according to the equation ∑ (U h(q )⫽Rs⫺2a(q ) cos(q ⫺f 0) ...................(3) ⫺ i , j ⫺U e , j ) p j ⫽ j1 Rs ⫺ hi Rs ⫺ he ⫺ 2 cos(θi ⫺φ0 ) 2 cos(θ e ⫺φ0 ) j1⫺1 ∑ d where h(q ) : gauge profile; Rs : distance between centers of the work rolls; f 0 : angle measured to the line of work roll centers. In above equations, the f 0 is a pre-determined value, which must be larger than possible contact angle after roll centers considering roll flattening and elastic recovery. Otherwise, the calculation will fail. However, larger f 0 will increase the calculation time. Normally take f 0⫽ (0.5⬃1.5)√苶 (h1⫺h2)/R will be sufficient. The pressure profile is related to the gauge profile via © 2005 ISIJ Sketch of the deformation zones in the roll gap. e (U i , j ⫺U e , j ) p j ⫺ ∑ (U i , j ⫺U e , j ) p j i ⫽ j1,L, j2 j 2⫹1 ...........................................(6) where j1 : the first node in the flattened region; j2 : the final node in the flattened region; e : the node at entry plane; d : the node at exit plane. The intersections of roll profile, determined by a(q ), and strip profile h(q ), will determine the node at entry plane and exit plane during iteration. Once the pressure and gauge profiles are determined, the roll force, P, and torque, T, can be calculated through inte1174 ISIJ International, Vol. 45 (2005), No. 8 Fig. 2. The calculation procedure of new model. gration of the pressure and friction force distribution: ∫ φ P⫽ a(θ ) 0 ∫ Before applying the present model to actual temper rolling process, the reliability and convergence of calculation result were checked with various rolling conditions. Here, some typical calculated results are introduced. Figure 3 is an example of foil, which is ultra thin material, cold rolling process calculation. Three reduction rates were used in the calculation. Though very thin entry thickness was used, there was no convergence problem, in this model. It can be seen that as the reduction increases from 50 to 70%, the pressure profile changes from one peak (regime II) to two peaks (regime III) , and the length of the flat region and the roll force are also increase more than two times. It reveals that the same calculation procedure could be used regardless of deformation regimes in the present model. Figure 4 shows the effect of the friction coefficient upon the gauge and pressure profile under temper rolling conditions. It can be seen that not only the pressure profile but also the gauge profile is greatly influenced by the friction coefficient. At low friction conditions, the contact arc between the roll and the strip is nearly circular, but at high friction conditions, a flat region appears in the central part of the contact arc. Similar situation can also be seen when we change the yield stress of the strip. As shown in Fig. 5, the contact arc changes its shape from nearly circular to non-circular and a central flat region finally appears, as the yield stress increases. Besides, the elastic recover zone is also greatly in- cos β [ p(θ ) ⫹ τ (θ ) tan β ]dθ ..........(7) cos ξ φ T⫽ a 2 (θ )[τ (θ ) ⫹ p(θ ) tan ξ ]dθ ...............(8) 0 where x : angle between un-deformed roll and horizontal axis. At the same time, the forward slip, f, can also be approximately, by neglecting the minor influence of elastic recovery to forward slip, calculated with the following formula: f⫽hn /h2⫺1 ................................(9) where hn : strip gauge at a flat region or a neutral point; h2 : strip gauge at the exit side. 3. Calculation Procedure and Analysis of New Model The main flow chart of the calculation procedure is shown in Fig. 2. The iterative calculation procedure was introduced to solve the equations describing the roll-deformation and the strip-stress distribution. No matter which regimes the rolling conditions belong to, the same calculation procedure can be used. That is one of the most important advantages in the present model. 1175 © 2005 ISIJ ISIJ International, Vol. 45 (2005), No. 8 Fig. 3. Thickness and pressure profile of foil cold rolling R⫽89 mm, friction coefficient⫽0.03, s y⫽230 MPa, entry thickness⫽0.03 mm, E⫽230 GPa, (a) reduction⫽30 %, (b) reduction⫽50 %, (c) reduction⫽70 %. Fig. 4. Thickness and pressure profile under different friction coefficients R⫽205 mm, s y⫽196 MPa, entry thickness⫽ 0.505 mm, reduction⫽0.46 %, rolling speed⫽37 m/min, entry tension⫽2 496 kg, exit tension⫽3 616 kg, width⫽ 1 073 mm, (a) m ⫽0.12, (b) m ⫽0.15, (c) m ⫽0.18, (d) m ⫽0.21. Fig. 5. Thickness and pressure profile under different yield stress, R⫽205 mm, entry thickness⫽0.505 mm, reduction⫽0.46 %, width⫽1 073 mm, rolling speed⫽37 m/min, entry tension⫽2 496 kg, exit tension⫽3 616 kg, m ⫽0.12, s y⫽210 MPa, (b) s y⫽240 MPa, (c) s y⫽270 MPa, (d) s y⫽300 MPa. fluenced by the yield stress. Another factor, which can influence the thickness and pressure profile, is the entry and exit tension. Figure 6 shows the thickness and pressure profile under different tensions. As the tensions decrease, the pressure increases and the length of the central flat region also increases. The above mentioned calculation results show that not only the decreasing of entry thickness, but also other factors, such as increasing the reduction, friction and yield stress as well as decreasing tension, can result in a central flat region. That means any factor that can cause an increase of the roll force may result in a central flat region in the deformation zone. Through the analysis of rolling phenomena, it was confirmed that the new model has a calculating stability and can give physically reasonable result for wide range of temper rolling conditions. © 2005 ISIJ 4. Model Calibration and Application Result In order to use the theoretical model as a setup model in the elongation control system of a temper rolling mill in POSCO, the model was calibrated against the production data collected from the same mill. The friction coefficient, m , and the constrained yield stress, K, were taken as the calibration factors. Because the roll force, P, and the forward slip, f, can be measured in the temper rolling mill, and they can also be calculated by the theoretical model, we used these two measured data to determine the calibration factors K and m . Equating the measured roll force Pm and forward slip fm with their calculation formula, which were derived from the theoretical model, we got two coupled equations. Solving the coupled equations with the iterative method, we obtained the constrained yield stress, K, and the friction coefficient m for each coil. 1176 ISIJ International, Vol. 45 (2005), No. 8 Fig. 6. Thickness and pressure profile under different tensions, R⫽205 mm, entry thickness⫽0.505 mm, reduction⫽ 0.46 %, width⫽1 073 mm, rolling speed⫽37 m/min, s y⫽196 MPa, m ⫽0.12. (a) Entry and exit tension⫽8 100 kg, (b) entry and exit tension⫽5 400 kg, (c) entry and exit tension⫽2 700 kg, (d) entry and exit tension⫽0 kg. Table 1. Fig. 7. The values of the calibration factors for each group. Calculation of constrained yield stress and friction coefficients. Because present model can well predict the large forward slip in the temper rolling process, we got reasonable results using the above method. A flow chart of the calibration calculation is shown in Fig. 7. After the constrained yield stress and friction coefficients were calculated for all coils, we divided the coils into groups according to the steel grades and the constrained yield stress. The average constrained yield stress and average friction coefficient of each group were used as the final values of the calibration factors for this group. Table 1 shows the values of the calibration factors for each group. After calibration, the model was installed in the plant together with the data acquisition system to make parallel calculations and comparisons. An evaluation sample that consists of the valid data of 3 724 coils is adopted to make the evaluation and comparison. Figure 8 shows a comparison of the measured roll forces and the roll forces calculated by a present model. It can be seen that the predicted roll forces match the measured values well and the model has relative high prediction accuracy. A statistical analysis shows that the correlation coeffi- Fig. 8. Roll force measured and predicted by the present model. cient between the measured roll forces and the roll forces predicted by the present model equals 0.8899. In order to show the prediction accuracy improvement made by the present model, the regression model, which had been used as the setup model in the same temper rolling mill before the present model was developed, was also evaluated with the same data. Figure 9 shows a comparison of the measured roll force and the roll force calculated by the regression setup model. As we can see, the samples in Fig. 9 are more scattered than those in Fig. 8. The correlation coefficient between the measured roll force and the roll force calculated by the regression setup model is 0.7175. Besides, the average calculated roll force is lower than that of the measured roll force. This indicates that the regression setup model underestimated the roll force in general. By comparison, we can see that the present theoretical model is much more accurate than the regression setup model. From a comparison with 1177 © 2005 ISIJ ISIJ International, Vol. 45 (2005), No. 8 Fig. 9. fore, it would not be necessary to determine the regimes and choose the calculation procedure before hand. By solving the equations describing the roll gap phenomena in a unique procedure, the new model offered significant improvement in the accuracy, robustness and generality of the solution for thin strip cold and temper rolling conditions. The model was installed in the process computer of an actual temper rolling mill to make setup calculations. The calculation results showed good agreement with the measured data and the validity and precision of the model were proven. Roll force measured and predicted by the regression model. REFERENCES the measured data and comparison with the old setup model, the validity and precision of the present model are proven. 1) 2) 3) 5. Conclusions 4) 5) A new mathematical model for the thin strip cold and temper rolling process has been developed by modification of the influence function method. 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