Synergistic optimization of shape-force and dynamic coupling effect in bending roll systems during accelerated rolling

Abstract

During the acceleration phase of continuous cold rolling mill production, transient transitions in rolling speed can lead to dynamic imbalance in the rolling force field. This imbalance induces fluctuations in strip thickness and shape distortions (such as edge waves, center waves, and other defects), which have become critical bottlenecks restricting the quality of high-end cold-rolled products and production line efficiency. Based on the ABAQUS platform, this study constructs a three-dimensional elastoplastic finite element model of a six-roll continuous cold rolling mill, systematically revealing how Work Roll Bending (WRB) and Intermediate Roll Bending (IRB) couple to affect the dynamic response of strip thickness distribution, flatness evolution, cross-sectional crown, and rolling force field. The results demonstrate that increasing WRB shifts the strip thickness distribution from “thick center, thin edges” to “thin center, thick edges,” significantly improving secondary wave defects. Increasing IRB promotes more uniform thickness distribution, but its control capability over quaternary wave defects is relatively weaker. Bending force effectively suppresses shape fluctuations by adjusting rolling force distribution and inter-roll pressure. For the acceleration process, a composite control strategy integrating dynamic bending force compensation and tension cooperative control is proposed, and a segmented speed interval optimization model is established. Simulations show that through segmented speed interval optimization and dynamic compensation, edge wave defects during the acceleration phase are reduced, and thickness uniformity is improved. Optimizing the strip production process during acceleration can effectively enhance production efficiency and reduce production losses. The research outcomes, by establishing a multi-objective coordinated dynamic shape control system, provide theoretical support and engineering implementation pathways for the intelligent process optimization of continuous cold rolling mills during acceleration.

Introduction

During the acceleration of the rolling process, the difficulty of shape control increases because the high rolling speed can cause fluctuations in the thickness and shape of the strip. The rolling force also fluctuates with changes in rolling speed, leading to unstable contact pressure between the rollers and the strip, which in turn affects the uniformity of the strip shape¹. Fluctuations in rolling force can lead to shape defects in the strip during rolling, such as edge waves and center waves. When the rolling force increases, it can cause uneven deformation in the center or edges of the strip, resulting in edge or center waves. Fluctuations in rolling force directly affect thickness control during the rolling process; sudden changes in rolling force can alter the stiffness of the rolling mill, thereby changing the roll gap. This can lead to fluctuations in the exit thickness of the strip, affecting the dimensional accuracy of the product. During accelerated rolling, fluctuations in rolling force can cause uneven thickness at the head and tail of the strip, affecting the stability of subsequent rolling².

The influence of rolling speed on the roll gap is very significant and is the main variable affecting the roll gap model during acceleration and deceleration. The influence rate of rolling speed on the roll gap exceeds 31.5%, indicating that it is the most influential factor among all controllable rolling parameters on the roll gap. As the rolling speed increases, the rolling force and roll gap continuously decrease, leading to uneven thickness and shape distribution of the strip during the acceleration phase, which cannot meet production requirements and reduce production efficiency³. Thickness deviation is an important indicator for evaluating cold rolling control systems and the quality of the final product. Controlling the thickness deviation of the strip is crucial for ensuring product quality. During the cold rolling process, changes in rolling speed can cause significant fluctuations in factors such as the friction coefficient, thereby affecting the rolling force and the final strip thickness. During the acceleration and deceleration phases of cold rolling, although the entry thickness deviation is small and the Automatic Gauge Control (AGC) system is operational, significant changes in rolling force due to variations in rolling speed lead to severe fluctuations in exit thickness deviation. These changes in thickness deviation directly affect the shape of the strip. Under non-steady-state conditions, controlling thickness deviation becomes more complex, especially during acceleration and deceleration, when traditional mechanical models struggle to accurately describe the process⁴.

Generally, techniques such as roll bending^5,6,7,8, roll shifting⁹, segmented cooling¹⁰, roll crossing¹¹, and tension control^12,13 can be used to regulate the shape of cold-rolled strips. Lianjie Li¹⁴ et al. established a three-dimensional finite element model to analyze the impact of roll bending force on the shape of strips under continuous cold rolling conditions. The adjustment capability of work roll bending on the secondary flatness of the strip is at least 2.5 times that of intermediate roll bending. A multi-stand model for continuous rolling can effectively analyze the trends of secondary and quartic flatness and improve strip shape by optimizing roll bending in the TCR. Wang, ZH¹⁵ et al. established a predictive model using genetic algorithms and artificial neural networks, and developed a GA-ANN model using MATLAB software. The GA optimizes the initial weights and biases of the ANN, with the optimization process including population initialization, selection, crossover, and mutation operations. The GA-ANN model showed high prediction accuracy with correlation coefficients of 0.9825 and 0.9776 on the training and test sets, respectively, and mean absolute errors (MAE) of 7.7672 and 8.8906. Sun, W¹⁶ et al. used ABAQUS to establish a multi-pass simulation model to study the effect of bending force on the elongation deviation of DP980 strip on stands S1 to S4 of a continuous cold rolling mill. The simulation results showed that the effects of work roll bending and intermediate roll bending on strip elongation deviation differ, leading to the development of a new linear combination control strategy for bending force to control quarter waves. Wu Changsheng¹⁷ et al. proposed a set of tension system optimization setting techniques to address issues of rolling capacity exceeding limits and production quality defects in a 5-stand continuous cold rolling process. By establishing models for rolling pressure, rolling power, slippage, and thermal scratches, comprehensive parameter analysis was conducted, providing theoretical support for quality control in the continuous cold rolling process. Field application results showed that the optimized tension system effectively reduced production quality defects and improved production efficiency.

Kozhevnikov¹⁸ et al. used simulation technology to analyze and study dynamic conditions. They developed an effective technique based on a dynamic model to derive characteristics from the rolling process and the rolled strip. Jin¹⁹ et al. constructed a deformation resistance model for DP600 using the rolling-tension method and studied the trends of secondary and quartic shape components in four types of incoming strip shapes. Sajan Kapil²⁰ et al. used the finite element method to model the work rolls of a four-high rolling mill and studied changes caused by different process parameters. They investigated transient and steady-state changes in roll gap, exit thickness profile, exit stress, and strip force along the length of the strip. Finite element analysis can predict the strip shape and stress–strain at the rolling exit. Hiroshi Utsunomiya²¹ et al. used the finite element software ABAQUS to establish a rolling model for composite plates and conducted non-steady-state analysis. First, they studied the effect of rolling conditions on necking through finite element analysis under simple conditions. They also conducted multiple cold rolling analyses on three types of composite plates to study the necking behavior of composite plates.

This paper selects intermediate roll bending and work roll bending for shape control and uses tension control as an auxiliary method. Finite element analysis²² is utilized, with a continuous rolling mill model established in ABAQUS, and explicit analysis²³ in ABAQUS is used for cold rolling simulation.

Establishment of finite element

Material data i.e. rolling parameter selection

To determine the deformation of the strip under cold rolling acceleration conditions and the influence of intermediate roll bending force on strip thickness and shape, and to optimize the distribution and compensation of bending forces across different stands through reasonable methods, a high-precision finite element model is indispensable (Fig. 1).

Fig. 1

Schematic diagram (or 3D model) of the six-roll cold rolling mill.

The ABAQUS software was used to establish and study a three-dimensional model, creating a single-stand model. The overall model consists of seven components: six rolls and one strip. Compared to the strip, the plastic deformation of the rolls is relatively small. To better study the changes in the strip, the rolls were set as elastic bodies, and the strip was set as a plastic body. Table 1 lists the parameters of the components of the six-roll mill. To better simulate the rolling conditions of the strip, rigid plates were added to both ends of each roll. Constraints were applied to the rigid plates, and rotational speed and bending force were applied through the rigid plates. The overall model is shown in Fig. 2.

Table 1 Cold rolling model parameters.

Fig. 2

Model of a six-roll cold mill.

Since the rolls are set as elastic bodies, plastic deformation is not considered. The strip uses the actual material properties of the factory’s raw materials for simulation. The density and elastic properties of the rolls and the strip are the same, with a density of 7.85 g/cm³, a Young’s modulus of 210,000 MPa, and a Poisson’s ratio of 0.3. The plastic strain of the strip was obtained through tensile tests.

The relationship between stress and strain is given by the formula: σ = Eε, where σ is stress, E is the material’s elastic modulus, and ε is strain. The stress–strain data of the strip material from the tensile test are shown in Table 2 The yield strength of the material is 380 MPa. When the stress exceeds 380 MPa, the specimen undergoes significant and uniform plastic deformation. To increase the strain, the stress must be increased. When the stress reaches 530 MPa, the uniform deformation stage of the specimen ends, and the ultimate tensile strength of the material is determined to be 530 MPa. Beyond this point, the specimen begins to undergo non-uniform plastic deformation, forming a neck, and the stress decreases until the specimen fractures, the specific changes are shown in Fig. 3, and The plastic strain corresponding to its yield stress is shown in Table 3.

Table 2 Reduction schedule.

Fig. 3

Stress–strain curve.

Table 3 Material stress–strain.

Using the data transfer method in ABAQUS, the simulation data from the first stand is used as the input for the second stand, and the strip model data from the first simulation is transferred to the second simulation model. This process is repeated to simulate a five-stand continuous rolling process. This approach effectively studies the changes in strip thickness and shape during the acceleration phase of the five-stand continuous rolling process. Through data transfer, the deformation state of the strip after rolling in each mill stand (such as thickness and residual stress) can be directly used as the initial conditions for the next mill stand, avoiding the re-computation of complex contact and large deformation processes with poor convergence, thereby significantly improving computational efficiency. Establish independent finite element models for each rolling stand. Execute the simulation for the first stand to generate the result file (Job-1.odb), then map the comprehensive strip state at the exit of Stand-1—including geometry, stress, strain, plastic strain, and temperature fields—onto the entry strip of the Stand-2 model as its initial condition. This sequential workflow proceeds as follows: After building the Stand-1 model and obtaining Job-1.odb, construct the Stand-2 model and import the deformed strip geometry via File → Import → Part in the Part Module, selecting Job-1.odb as the source. Subsequently, define a predefined field in the Stand-2 model using the exported strip state from Job-1.odb to initialize Job-2. Repeat this process iteratively until all five stands (Job-1 through Job-5) are simulated, thereby establishing a chained multi-stand rolling analysis.

Grid division

Good mesh generation can significantly improve the accuracy and efficiency of finite element simulations. The types of mesh elements include first-order elements, which have nodes only at the corners of the elements and use linear interpolation during calculation, with two integration points in one degree of freedom direction, such as first-order tetrahedrons and first-order hexahedrons. These are simple and quick to generate but can be coarse and imprecise, leading to issues like shear locking. Second-order elements, which have nodes at both ends and the middle of each edge and use quadratic interpolation during calculation, with three integration points in one degree of freedom direction, such as second-order tetrahedrons and second-order hexahedrons, provide better simulation results.

This paper uses second-order hexahedral elements for mesh generation. The strip is divided into uniform, equally sized second-order hexahedral elements. For the rolls, the mesh generation for the work rolls, intermediate rolls, and backup rolls uses larger elements in the middle and smaller elements around the circumference, as shown in Fig. 4. The work rolls are further subdivided, with the contact area between the work rolls and the strip refined using a three-to-one mesh, as shown in Fig. 5. This further improves the accuracy of the strip rolling simulation, allowing for better analysis of the strip rolling acceleration phase.

Fig. 4

Overall mesh of the rolling mill model.

Fig. 5

Refined mesh (three-to-one transition) at the work roll-strip contact zone.

Preliminary rolling results

Using production data from a 1420 mm cold tandem rolling line at a steel plant, the strip width was 1100 mm, entry thickness 2.1 mm, exit thickness 2.0 mm, and the strip material was the same as that used for the aforementioned tensile tests. The data corresponds to a stable rolling stage with settings of WRB = 200 KN and IRB = 200 KN. The reduction amount in the control simulation was precisely adjusted to ensure the simulated rolling thickness fell within a reasonable error margin. The resulting simulated exit flatness distribution of the strip was obtained and compared with actual measurements, as summarized in Table 4.

Table 4 Comparison of simulated and actual values.

By precisely controlling the simulation reduction amount, the simulated strip exit average thickness was successfully replicated at 2.102 mm. The error relative to the target value of 2.0 mm was 2.53%, which falls within the acceptable error range for the simulation, thereby validating the model’s effectiveness in macroscopic thickness control. The simulated transverse thickness distribution reflects the actual distribution trend. The minimum thickness error was 1.55%, but the maximum thickness point error was 2.24%, indicating room for improvement in the model’s simulation of local effects like edge thickening. For exit flatness, the simulated result was 8.5 I-unit, resulting in a relative error of 25.00% compared to the actual value of 6.8 I-unit. This error level is significantly higher than the thickness errors; nevertheless, the simulated flatness meets production requirements.

A preliminary simulation of the model was conducted to observe the stress distribution under load. Without controlling other parameters such as the reduction amount, a preliminary simulation was performed on the fifth stand. Experiments were conducted with intermediate roll bending forces of 100 KN, 200 KN, 300 KN, 400 KN, and 500 KN, while keeping the work roll bending force constant at 200 KN. The study focused on the variation of rolling force over time under an acceleration of 360 mm/s² in line speed. This acceleration value of 360 mm/s² was used for shape correction on one production line, representing half of the normal rolling acceleration employed by a specific factory aiming to improve strip quality under higher acceleration conditions. After applying tension compensation correction, the rolling acceleration was essentially around the typical minimum normal rolling speed of 500 mm/s².

Figure 6 shows the variation of rolling force over time when the intermediate roll bending force is 300 KN. As can be seen from Fig. 6, as time increases, the rolling speed continuously increases, and the rolling force gradually decreases, with the rolling force–time curve slowly flattening. At 0 s, the rolling force is approximately 1,732,000 N, and by 0.2 s, it decreases to 1,517,000 N. The rolling speed begins to increase at 0.02 s, whereas the change in rolling force slightly lags behind the change in rolling speed. The rolling force starts to decline with the increasing speed only around 0.04.

Fig. 6

Rolling force over time at for 300KN intermediate roll bends.

Figure 7 shows the variation of rolling force over time under different intermediate roll bending forces. It can be observed that the higher the load, the greater the initial rolling force. The initial rolling force for 100 KN is about 1,612,000 N, while for 500 KN, it is about 1,908,000 N. Over time, the rolling force for each curve gradually decreases. The rate of decrease varies under different loads, with curves under higher loads decreasing faster.

Fig. 7

Plot of rolling force versus time for different intermediate roll bending forces.

Figure 8 shows the force distribution diagram of the entire rolling mill during the rolling process. The color bar ranges from 1.600 × 10⁻¹ MPa to 5.064 × 10² MPa, with red areas indicating stress concentration at the neck of the work rolls. Figures 9 and 10: The horizontal direction represents the strip width, and the vertical direction corresponds to the strip length. The strip exit side is positioned at the bottom of the figures. Figure 9 shows the residual stress map of the strip after rolling, where colors from blue to red represent stress from low to high. Most areas are blue and green, indicating low stress, but some areas show red and yellow, indicating higher stress. This figure illustrates the contour plot of plastic strain in the strip after rolling. The numerical values range from 0 to 0.14, with the color gradient from blue to red representing the increase in strain from low to high. The predominant blue and green colors across most areas indicate relatively low strain levels. Changes in rolling force, resulting from the increased rolling speed, have introduced variations and fluctuations in the final thickness and flatness of the rolled strip. Therefore, the latter section of the strip, which underwent more stable rolling conditions, was selected for further investigation.

Fig. 8

Force diagram of the whole mill.

Fig. 9

Stress cloud diagram.

Fig. 10

Strain cloud diagram.

Adding intermediate roll bending force under acceleration

Influence of work roll bending roll on thickness and flatness under accelerated conditions

Cold rolling bending refers to the process of applying a certain bending force to the rolls during cold rolling, causing them to deform and change their convexity and deflection. This is done to control the flatness and thickness distribution of the strip. The bending force is applied through hydraulic cylinders at the necks of both ends of the rolls, allowing adjustments to the magnitude and direction of the bending force.

To study the influence of work roll bending force (WRB) on the thickness and flatness of the rolled strip, experiments were conducted with work roll bending forces of − 100 KN, 0 KN, 100 KN, 200 KN, and 300 KN. During these experiments, the intermediate roll bending force was kept constant at 200 KN. The study focused on the effects of bending force under an acceleration of 360 mm/s² in line speed over a 0.15-s acceleration period.

The influence of Work Roll Bending (WRB) force on the strip thickness distribution is shown in Fig. 11. Overall, the strip thickness increases with increasing bending force, while the thickness changes at the edges are more pronounced than those in the center section. When the WRB force increases from − 100 KN to 200 KN, the rolled strip exhibits a profile characterized by a thicker center and thinner edges. However, when the WRB force increases from 200 to 300 KN, the thickness at both edges becomes greater than that in the center. Adjusting the WRB force can effectively modify the uniformity of strip thickness, but it also slightly increases the overall strip thickness. The influence of work roll bending force (WRB) on strip flatness distribution is shown in Fig. 12. Figure 12 illustrates the flatness of the rolled strip under different work roll bending forces. It can be observed that the flatness mostly exhibits an upward-opening parabolic shape. As the work roll bending force increases, the flatness values at the edges of the strip decrease, while the flatness values in the middle increase. The flatness transitions from an upward-opening parabola to a downward-opening parabola. The bending force compensates for the roll gap convexity caused by the elastic deformation of the roll system, improving edge wave defects in the strip. Adjusting the work roll bending force can enhance the flatness quality of the finished strip.

Fig. 11

Work roll Influence of distributionbending force (WRB) on strip thickness.

Fig. 12

Work roll Influence of distribution of bending force (WRB) on the flatness of metal strips.

Effect of intermediate roll bending roll on thickness and flatness under acceleration

To study the influence of intermediate roll bending force (IRB) on the thickness and flatness of the rolled strip, experiments were conducted with intermediate roll bending forces of 100 KN, 200 KN, 300 KN, 400 KN, and 500 KN. During these experiments, the work roll bending force was kept constant at 200 KN. The study focused on the effects of bending force under an acceleration of 360 mm/s² in line speed over a 0.15-s acceleration period.

The influence of Intermediate Roll Bending (IRB) force on the strip thickness distribution is shown in Fig. 13.Overall, the strip thickness increases with increasing bending force, while the thickness changes at the edges are more pronounced than those in the center section. When the IRB force increases from 100 to 200 KN, the rolled strip exhibits a profile characterized by a thicker center and thinner edges. However, when the IRB force increases from 200 to 500 KN, the thickness distribution shifts to thicker edges and a thinner center. Adjusting the IRB force can effectively modify the uniformity of strip thickness, but it also slightly increases the overall strip thickness. The influence of intermediate roll bending force (IRB) on strip flatness distribution is shown in Fig. 14. Figure 14 illustrates the flatness of the rolled strip under different intermediate roll bending forces. It can be observed that the flatness exhibits a downward-opening parabolic shape. As the intermediate roll bending force increases, the flatness values at the edges of the strip decrease, while the flatness values in the middle increase. The bending force compensates for the roll gap convexity caused by the elastic deformation of the roll system, improving edge wave defects in the strip. Adjusting the intermediate roll bending force can enhance the flatness quality of the finished strip.

Fig. 13

Intermediate roll Effect of distributionbending force (IRB) on strip thickness.

Fig. 14

Intermediate roll Effect of distributionbending force (IRB) on the flatness of metal strips.

Effect of roll bending rolls on crown work under acceleration

The crown of cold-rolled strip steel is closely related to its shape, making it an important factor in determining whether the strip meets production requirements. Generally, the cross-section of cold-rolled strip steel can be considered symmetric, and Chebyshev polynomials are used to describe its cross-sectional curve shape.

$$f\left( x \right) = a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{4} ,$$

(1)

x : Normalized transverse coordinate along the strip width;$a_{0}$, $a_{1}$, $a_{2}$, $a_{3}$: a₂, a₄, …: Polynomial coefficients.

Quadratic Crown: Refers to the quadratic polynomial component of the strip’s cross-sectional shape, usually corresponding to the strip’s quadratic wave shape (such as edge waves or center waves). It reflects the difference in thickness between the center and edges of the strip and is an important indicator of the overall crown. The quadratic crown CW₂ can be expressed using the quadratic component of the polynomial:

$$CW_{2} = - \left( {a_{2} + a_{4} } \right),$$

(2)

Quartic Crown: Refers to the quartic polynomial component of the strip’s cross-sectional shape, corresponding to the strip’s quartic wave shape (such as 1/4 waves or combined edge-center waves). It mainly describes the thickness variation at the 1/4 position of the strip, reflecting the complexity of the cross-sectional shape. The quartic crown CW₄ can be expressed using the quartic component of the polynomial:

$$CW_{4} = - \frac{{a_{4} }}{4},$$

(3)

When the quadratic crown is too large, the center thickness of the strip becomes significantly greater than the edge thickness, leading to center waves. Conversely, when the quadratic crown is too small, the edge thickness becomes greater than the center thickness, leading to edge waves. The ability to control the quadratic crown decreases with the reduction in strip width, making it challenging to control the crown for narrow strips. The quartic crown mainly affects the thickness variation at the 1/4 position of the strip. When the quartic crown is too large, significant thickness variations occur at the 1/4 position, resulting in quartic waves. The ability to control the quartic crown is relatively limited and is primarily determined by factors such as the roll contour angle.

Changes in work roll bending force can cause variations in the strip crown. To study the influence of work roll bending force (WRB) on the crown of the rolled strip, experiments were conducted with work roll bending forces of − 100KN, 0 KN, 100 KN, 200 KN, and 300 KN. During these experiments, the intermediate roll bending force was kept constant at 200 KN. The study focused on the effects of bending force under an acceleration of 360 mm/s² in line speed over a 0.15-s acceleration period.

Figure 15 shows the influence of work roll bending force on the quadratic and quartic crown. As can be seen, the quadratic crown of the strip decreases as the work roll bending force increases. The quartic crown decreases with the work roll bending force from − 100 to 200 KN but increases when the bending force increases from 200 to 300 KN. This indicates that work roll bending has a certain improvement effect on both edge waves and 1/4 waves.

Fig. 15

Effect of work roll bending roll force on crown.

Influence of intermediate roll bending rolls on crown under acceleration

Intermediate roll bending also affects the crown to some extent. To study the influence of intermediate roll bending force (IRB) on the crown of the rolled strip, experiments were conducted with intermediate roll bending forces of 100 KN, 200 KN, 300 KN, 400 KN, and 500 KN. During these experiments, the work roll bending force was kept constant at 200 KN. The study focused on the effects of bending force under an acceleration of 360 mm/s² in line speed over a 0.15-s acceleration period.

Figure 16 shows the variation of the strip’s quadratic and quartic crown as the intermediate roll bending force increases from 0 to 500 KN. It can be observed that both the quadratic and quartic crown decrease as the intermediate roll bending force increases. The change in quadratic crown is more significant than that in quartic crown, indicating that intermediate roll bending has a greater influence on quadratic waves than on quartic waves.

Fig. 16

Effect of intermediate roll bending force on crown.

Influence of roll bending roll on rolling force work under accelerated conditions

During the strip rolling process, the distribution of rolling force directly affects the shape of the loaded roll gap, thereby altering the shape of the strip at the exit. The distribution of rolling force also influences the strip’s flatness. By analyzing the rolling force, potential strip shape defects during the rolling process can be identified and adjusted. To comprehensively evaluate the effectiveness of the rolling mill’s shape control methods, it is essential to analyze the distribution of rolling force under different shape control conditions, such as center waves and edge waves.

Under constant parameter conditions, the intermediate roll bending force was set to 200 KN, and simulations were conducted by varying the work roll bending force. Bending forces of − 100 KN, 0 KN, 100 KN, 200 KN, and 300 KN were applied at the necks of both ends of the work rolls.

Figure 17 shows the rolling force per unit width under different work roll bending forces. As observed in Fig. 17, as the work roll bending force increases from − 100 to 300 KN, the rolling force in the middle section of the strip gradually decreases, while the rolling force at the edges increases. The difference in rolling force between the edges and the center also changes. When the work roll bending force reaches 300 KN, the rolling force decreases from the center toward the edges of the strip. This is because excessive work roll bending force causes significant deflection of the work rolls, reducing the rolling force at the edges. The results indicate that work roll bending can improve the distribution of rolling force and modify the strip shape.

Fig. 17

Rolling force per unit width for roll bending forcesdifferent work.

Influence of intermediate roll bending on rolling force under accelerated conditions

Under constant parameter conditions, the work roll bending force was set to 200 KN, and simulations were conducted by varying the intermediate roll bending force. Bending forces of 100 KN, 200 KN, 300 KN, 400 KN, and 500 KN were applied at the necks of both ends of the intermediate rolls.

Figure 18 shows the rolling force per unit width under different intermediate roll bending forces. As seen in Fig. 18, as the intermediate roll bending force increases, the distribution of rolling force per unit width becomes more uniform. As the intermediate roll bending force increases from 100 to 500 KN, the overall rolling force decreases, but the rate of change in the rolling force at the center is slower than that at the edges. The results indicate that intermediate roll bending can improve the distribution of rolling force and modify the strip shape.

Fig. 18

Rolling force per unit width for bending forcesdifferent intermediate roll.

Effect of bending on inter-roll pressure work roll under accelerated conditions

Analyzing inter-roll forces in cold rolling helps improve product quality, extend equipment life, enhance production efficiency, save energy, optimize processes, enhance safety, and support research and development.

Under constant parameter conditions, the intermediate roll bending force was set to 200 KN, and simulations were conducted by varying the work roll bending force. Bending forces of − 100 KN, 0 KN, 100 KN, 200 KN, and 300 KN were applied at the necks of both ends of the work rolls.

Figures 19 and 20 illustrate the distribution of inter-roll contact pressure between the work roll and intermediate roll, as well as between the intermediate roll and backup roll, under varying work roll bending forces. The work roll and intermediate roll are in direct contact, while the work roll and backup roll interact indirectly. Applying bending forces at both ends of the work roll alters the inter-roll pressure between the work roll and intermediate/backup rolls. As the work roll bending force increases from − 100 KN to 300 KN, Fig. 19 reveals that the inter-roll pressure between the work roll and intermediate roll follows a downward-opening parabolic curve. With increasing bending force: Pressure gradually increases at the roll edges; Pressure gradually decreases at the roll center; The magnitude of pressure change is smaller at the center than at the edges. Under negative bending force: A significant pressure gap exists between the center and both edge zones; Peak contact pressure values are higher. Under positive bending force: Contact pressure distribution becomes more uniform; Peak pressure values are lower. Figure 20 shows that the inter-roll pressure between the intermediate rolls and backup rolls forms an upward-opening parabola. Compared with the roller pressure in contact with the working roller and the intermediate roller, there is no significant change overall. The roller force between the intermediate roller and the support roller shows an upward-opening parabolic curve, and its roller relay increases overall as the bending roller force increases. The change in the roller force at the edges is slightly greater than the change in the roller force in the middle section.

Fig. 19

The roll separating force between the work roll and intermediate roll.

Fig. 20

The roll separating force between intermediate roller and support roller.

Effect of bending on inter-roll pressure intermediate roll under acceleration

Under constant parameter conditions, the work roll bending force was set to 200 KN, and simulations were conducted by varying the intermediate roll bending force. Bending forces of 100 KN, 200 KN, 300 KN, 400 KN, and 500 KN were applied at the necks of both ends of the intermediate rolls.

Figures 21 and 22 illustrate the distribution of inter-roll contact pressure between the intermediate roll and the work roll, as well as between the intermediate roll and the backup roll, under varying intermediate roll bending forces. The intermediate roll maintains direct contact with both the work roll and the backup roll. Applying bending forces to both ends of the intermediate roll significantly alters the inter-roll pressure between these rolls. As the intermediate roll bending force increases from 100 to 500 KN. Figure 21 demonstrates that the inter-roll pressure between the intermediate roll and work roll follows a downward-opening parabolic curve, exhibiting these changes: Pressure gradually decreases at the roll edges; Pressure gradually increases at the roll center; The magnitude of pressure change is smaller at the center than at the edges. As bending force increases, the pressure gap between the center and edge zones widens, with peak pressure values continuously rising. Figure 22 shows that the inter-roll pressure between the intermediate rolls and backup rolls forms an upward-opening parabola. As the intermediate roll bending force increases, the overall inter-roll pressure increases, with the change in pressure being greater at the edges than at the center. However, compared with the roll pressure between the working roll and the intermediate roll, the roll force between the edge rolls varies greatly, and as the bending force increases, the degree of variation remains essentially the same.

Fig. 21

The roll separating force between work roll and intermediate roll.

Fig. 22

The roll separating force between intermediate roller and support roller.

Optimisation of bending roll forces under acceleration

Due to the acceleration phase, fluctuations in rolling speed disrupt rolling stability. By compensating the bending force, the bending state of the rolls can be dynamically adjusted to counteract these adverse effects, thereby maintaining the thickness uniformity and shape quality of the strip. This is a crucial measure in the cold rolling process to ensure product quality and stable production. Traditional bending force compensation methods, which primarily provide a fixed bending force compensation value for constant-speed operation, are insufficient for acceleration conditions. Therefore, a dynamic bending force compensation value must be established.

To study bending force compensation for each stand separately, the acceleration phase is divided into n + 1 equal segments. Dynamic bending force compensation for each segment can effectively improve the shape value. Table 5 presents the Work Roll Bending (WRB) and Intermediate Roll Bending (IRB) force settings on various stands used in this simulation.

Table 5 Preset values for bending force of intermediate rollers for each frame.

The work roll bending compensation for each stand is calculated as follows. ΔS_n(i): Work roll bending force compensation value for the ii-th speed segment of the stand. Y: Influence of stand speed on intermediate roll bending force. v_max: Maximum speed of the stand. v(i): Speed of the stand at the ii-th segment. S_n′(i): Set work roll bending force of the stand. S_n: Set work roll bending force of the stand. G(x)_i: Target function for dynamic shape changes. ϕ_z: Work roll bending force increase rate. ζ₁,ζ_2,ζ₁,ζ₂: Work roll bending compensation coefficients.

$$\left\{ \begin{gathered} \Delta S_{n} \left( i \right) = Y\zeta_{1} e^{{\zeta_{2} v\left( i \right)}} \left[ {v_{\max } - v\left( i \right)} \right] \hfill \\ S_{n}{\prime} \left( i \right) = \Delta S_{n} \left( i \right) + S_{n} \hfill \\ S_{n}{\prime}_{\min } < S_{n}{\prime} < S_{n}{\prime}_{\max } \hfill \\ F\left( x \right) = \min \sum\limits_{i = 1}^{5} {G\left( x \right)_{i} } \hfill \\ \varphi_{z} = \frac{{\Delta S_{n} (i) - \Delta S_{n} (i + 1)}}{{S_{n} (1) - S_{n} (n + 1)}}, \hfill \\ \end{gathered} \right.$$

(4)

The intermediate roll bending compensation for each stand is calculated as follows. ΔS_n(i): Intermediate roll bending force compensation value for the ii-th speed segment of the stand. Y: Influence of stand speed on intermediate roll bending force. v_max: Maximum speed of the stand. v(i): Speed of the stand at the ii-th segment. S_n′(i): Actual intermediate roll bending force of the stand. S_n: Set intermediate roll bending force of the stand. G(x)_i: Target function for dynamic shape changes. ψ_z: Intermediate roll bending force increase rate. ζ₁,ζ_2,ζ₁,ζ₂: Intermediate roll bending compensation coefficients.

$$\left\{ \begin{gathered} \Delta S_{n} \left( i \right) = Y\xi_{1} e^{{\xi_{2} v\left( i \right)}} \left[ {v_{\max } - v\left( i \right)} \right] \hfill \\ S_{n}{\prime} \left( i \right) = \Delta S_{n} \left( i \right) + S_{n} \hfill \\ S_{n}{\prime}_{\min } < S_{n}{\prime} < S_{n}{\prime}_{\max } \hfill \\ F\left( x \right) = \min \sum\limits_{i = 1}^{5} {G\left( x \right)_{i} } \hfill \\ \psi_{z} = \frac{{\Delta S_{n} (i) - \Delta S_{n} (i + 1)}}{{S_{n} (1) - S_{n} (n + 1)}} \hfill \\ \end{gathered} \right.$$

(5)

A reasonable target shape function G(x) is established for each stand. Under constraints, an optimal set of values X = (× 1, × 2, × 3, × 4)X = (× 1, × 2, × 3, × 4) is sought to minimize shape fluctuations. The Powell optimization method is used to calculate the bending force compensation values. For example, by dividing the speed into eight segments, the work roll bending force increase rate and intermediate roll bending force increase rate are calculated, as shown in the figure below.

Figures 23 and 24 display the bending force compensation curves, with Fig. 23 showing the work roll bending (WRB) force increase rate and Fig. 24 presenting the intermediate roll bending (IRB) force increase rate. Analysis of WRB Compensation Curve (Fig. 23): The WRB compensation value for each stand increases monotonically with rising speed segments. Compensation rates are lower at low-speed segments but rise significantly at high-speed segments. Stands 1–4 exhibit higher compensation rates (0.1–0.3%), undertaking primary deformation and demonstrating strong sensitivity to speed variations. Stand 5 (final stand) has the lowest compensation rate, primarily serving shape control functions. Analysis of IRB Compensation Curve (Fig. 24): The overall IRB compensation rate is lower than WRB, as intermediate rolls mainly provide support functions, requiring minimal speed compensation. Compensation rate trends across stands are similar to WRB, with Stands 1–4 maintaining higher rates than Stand 5, consistent with WRB patterns (Fig. 25).

Fig. 23

Work roll bending force compensation curve.

Fig. 24

Intermediate roll bending force compensation curve.

Fig. 25

Plate shape values before and after bending roll force compensation.

In wide strip rolling, bending force compensation can improve edge shape by adjusting the bending degree of the rolls, reducing edge thinning and edge wave defects, thereby enhancing product quality. By optimizing the bending force compensation strategy, effective shape control can be achieved under different rolling schedules, improving the production efficiency and flexibility of the rolling mill. Under steady-state normal rolling conditions, the strip exhibits optimal flatness performance, with overall flatness higher than both pre-compensation and post-compensation states. However, after roll bending compensation, the flatness has already approached that of normal rolling conditions, indicating that the roll bending force provides a certain degree of compensation for strip shape.

Simulation shows that increasing the bending roll force of the working roll can effectively adjust the thickness distribution from “thick in the middle and thin at the edges” to “thin in the middle and thick at the edges,” significantly suppressing secondary waviness. Therefore, the WRB compensation rate is significantly improved in the high-speed section to cope with the dynamic changes in rolling force distribution during acceleration. The intermediate roll bending force has limited control over fourth-order waves but positively influences rolling force uniformity. Therefore, the IRB compensation rate is relatively low and tends to stabilize, primarily serving as auxiliary support and overall force field balancing.

Tension under acceleration optimisation

During the cold rolling process, the strip passes through multiple rolls. Due to factors such as uneven roll surfaces and equipment precision errors, the strip is prone to lateral deviation. The inter-stand tension keeps the strip under a certain level of tension, similar to straightening it with an external force, effectively preventing the strip from deviating during rolling. This ensures that the strip passes smoothly through each roll along the correct path, maintaining the stability of the rolling process. Tension also stabilizes fluctuations in rolling force. During cold rolling, the rolling force is influenced by various factors, such as material inhomogeneity and roll wear. Inter-stand tension stabilizes the deformation conditions of the strip as it enters the rolls, thereby reducing fluctuations in rolling force. When the rolling force is stable, the reduction amount of the rolls on the strip remains relatively stable, which helps ensure uniform strip thickness.

Additionally, tension can control the strip shape. Proper tension can achieve shape correction through stretching. For example, when edge waves occur, increasing the tension appropriately can improve the strip shape. Table 6 presents the front and back tension values applied on different stands in this simulation.

Table 6 Preset values for front and rear tension for each rack.

To address the interference caused by rolling speed fluctuations on rolling stability, the cold rolling automatic control system often compensates by adjusting the front and back tension of the stands when rolling speed changes. This aims to counteract the changes in friction conditions caused by speed variations, ensuring that the rolling force remains stable during the process. Traditional tension compensation methods, which provide a fixed tension compensation value for constant-speed operation, are insufficient for acceleration conditions.

By dividing the acceleration phase into n + 1 equal segments and applying dynamic tension compensation for each segment, the strip shape can be significantly improved.

The compensation is calculated using the following formula. K_q(i): Influence coefficient of front tension on rolling force for the ii-th segment. K_h(i): Influence coefficient of back tension on rolling force for the ii-th segment. F_zzc(i): Rolling force difference for the ii-th speed segment. F_qzw(i): 10% of the front tension set value. F_hzw: 10% of the back tension set value. F_zb(i): Tension compensation value. F_zzw: Maximum stable rolling force. F_zzfp(i): Average rolling force for the ii-th segment. φ_z: Rate of change of tension compensation value.

$$\left\{ \begin{gathered} K_{q} \left( i \right) = \frac{{F_{zzc} \left( i \right)}}{{F_{qzw} }} \hfill \\ K_{h} \left( i \right) = \frac{{F_{zzc} \left( i \right)}}{{F_{hzw} }} \hfill \\ F_{zb} (i) = \frac{{F_{zzw} - F_{zzfp} (i)}}{{K_{q} \left( i \right) + K_{h} \left( i \right)}} \hfill \\ \varphi_{z} = \frac{{F_{zb} (i) - F_{zb} (i + 1)}}{{F_{zb} (1) - F_{zb} (n + 1)}}, \hfill \\ \end{gathered} \right.$$

(6)

The rate of change of tension compensation for different racks can be obtained in Fig. 26.

Fig. 26

Tension compensation curve.

As shown in Fig. 26, the tension compensation amount continuously increases, with the largest increase occurring at the beginning of acceleration. The increase in tension compensation gradually decreases over time. Since the rolling speed increases from the first stand to the fifth stand, the increase in tension compensation also gradually increases with the rolling speed.

The tension compensation for the five stands is studied to verify its effectiveness in shape compensation. To study the influence of tension on the shape of the rolled strip, an intermediate roll bending force of 300 KN and a work roll bending force of 200 KN are used. The study focuses on the influence of shape under an acceleration of 360 mm/s² over a 0.15-s acceleration period.

Figure 27 shows the shape values before and after tension compensation. As can be seen, dynamic tension compensation effectively improves the strip shape, reducing edge wave defects. The improvement in the middle part of the strip is more significant than at the edges.

Fig. 27

Plate shape values before and after tension compensation.

Dynamic compensation measures are applied to the front and back tension of the stands across the entire rolling speed range. For different rolling speeds, corresponding tension compensation values are set to ensure precise tension compensation under various speed conditions, significantly enhancing the stability of cold-rolled strip production.

The simulation shows a decreasing trend in rolling force over time, indicating significant changes in friction conditions during acceleration, leading to dynamic variations in tension requirements. Therefore, tension compensation increases most significantly during the initial acceleration phase to quickly stabilize the rolling force, preventing strip deviation and thickness fluctuations.

Industrial application verification and discussion

To verify the engineering effectiveness of the dynamic bending roll force and tension compensation strategy proposed in this paper, this study further applied it to the acceleration process control of a 1420 mm five-stand cold rolling mill at a domestic steel mill and conducted a comparative analysis with the traditional fixed parameter control strategy. The rolling mill is equipped with advanced profile gauges and thickness gauges capable of real-time monitoring of strip profile and thickness deviations. For the industrial trial, we selected cold-rolled material with a width of 1100 mm, an inlet thickness of 2.5 mm, and an outlet thickness of 0.8 mm, which are similar to the material and specifications used in the simulation. Control group: Acceleration process control using the original fixed bending roll force and tension setpoints of the production line. Experimental group: The segmented speed interval optimization model proposed in this paper was applied to dynamically compensate the bending roll force (WRB/IRB) and frame-to-frame tension based on real-time rolling speed. Table 7 presents a comparative analysis of the data before and after the improvement.

Table 7 Data comparison table.

After implementing the dynamic compensation strategy proposed in this paper, the strip shape deviation and thickness variation in the acceleration section were reduced by approximately 36% and 37%, respectively. These data strongly demonstrate that the strategy effectively suppresses quality fluctuations caused by the acceleration process, significantly improving the cross-sectional uniformity and dimensional accuracy of the strip. The edge reduction control rate improved from 78.5 to 88.2%, an increase of nearly 10 percentage points. This indicates that the dynamic compensation strategy effectively alleviates excessive thinning in the edge regions of the strip by optimizing roll gap shape and force distribution, which is of great significance for producing high-precision, high-quality cold-rolled products. The most direct economic benefit is reflected in a 60% reduction in scrap rate. This means a sharp decrease in the number of downgraded products and scrap caused by deviations in plate shape or thickness, not only reducing production costs but also minimizing production interruptions caused by quality incidents, thereby enhancing the overall operational efficiency and stability of the production line.

Conclusion

• 1. Influence of bending force on strip thickness and flatness. Increasing the work roll bending force (WRB) shifts the strip thickness distribution from “thick in the middle and thin at the edges” to “thin in the middle and thick at the edges.” Increasing WRB significantly improves strip flatness, reducing edge and center wave defects. Increasing the intermediate roll bending force (IRB) makes the strip thickness distribution more uniform, but the thickness changes at the edges are greater than in the middle. IRB has a stronger control effect on secondary waves but limited control over quaternary waves.

• 2. Influence of bending force on strip crown. secondary crown: As WRB and IRB increase, the secondary crown gradually decreases, indicating that bending force can effectively suppress center and edge waves. Quaternary Crown: Increasing WRB reduces the quaternary crown, while increasing IRB slightly increases it.

• 3. Influence of bending force on rolling force and inter-roll pressure. Increasing WRB and IRB makes the rolling force distribution more uniform, reducing the difference in rolling force between the edges and the center of the strip. Increasing bending force changes the inter-roll pressure distribution between the work rolls and intermediate rolls, as well as between the intermediate rolls and backup rolls, thereby affecting the roll gap shape.

• 4. Dynamic compensation. By optimizing the distribution of bending force across segmented speed intervals, shape fluctuations during the acceleration phase are significantly improved, reducing edge thinning and wave defects. By optimizing the inter-stand tension settings across segmented speed intervals, the rolling force distribution is stabilized, strip deviation is reduced, and thickness uniformity is enhanced.