Mathematical model for cold rolling and temper rolling process of thin steel strip

A mathematical model for cold rolling and temper rolling process of thin steel strip has been developed using the influence function method. By solving the equations describing 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 is 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 show good agreement with the measured data and the validity and precision of the model are proven.     

轧机工作辊有限元分析

四辊轧机在工作过程中,工作辊既承受工件的变形抗力,又承受轧机轧制力和弯辊力的共同作用,造成辊身应力状态复杂。为保证轧辊的合理与优化设计,采用有限元方法针对轧辊工作过程进行分析,提出了一种有限元模型建模方法,准确计算出工作辊在工作过程中的变形和应力分布。计算结果为轧辊的结构改进与优化设计提供依据,以提高轧机的设计水平。     

三辊非对称滚弯成形的数值仿真及试验验证

基于有限元法对三辊非对称滚弯成形工艺进行了研究,对比分析了三辊非对称滚弯成形和三辊对称滚弯成形过程中,变形区应力场、板材上表面的塑形应变场及卷制力的变化规律。仿真结果表明：侧辊位移进给量相同的工况下,三辊非对称滚弯成形的卷制力大于三辊对称滚弯成形的卷制力;三辊非对称滚弯变形区的纵向应力和径向应力均大于三辊对称滚弯成形的纵向应力和径向应力;三辊非对称滚弯成形板材压弯段的成形质量高于三辊对称滚弯成形的成形质量。最后,经三辊非对称滚弯试验验证,有限元模型的成形误差为6.8%,有较高的精度。     

Roll shape design for foil rolling of a four-high mill

In this paper, a mathematical model for a four-high mill is proposed to analyze the elastic deformation of the backup and work rolls, including their respective axial deflection and surface flatness. The contact pressure between the backup and work rolls and that between the work roll and foil are functions of the roll’s position in the axial direction. In this analytical model, the rolls and foil are divided into many small regions and finite difference or matrix methods are used to derive the deflections of the work and backup rolls as a function of the force density in the foil width direction. A camber shape at the surfaces of the backup and work rolls is designed to counteract the rolls’ elastic deformation and to make the rolled foil as flat as possible. The effects of the selected roll materials upon the rolled thickness distribution are also discussed. A thickness variation within 1 μm can be obtained for a foil of 50-μm thick and 40-mm wide under reduction of 32 % using a WC work roll with a camber radius of 150 m.     

Roll System and Stock’s Multi-parameter Coupling Dynamic Modeling Based on the Shape Control of Steel Strip

The existence of rolling deformation area in the rolling mill system is the main characteristic which distinguishes the other machinery. In order to analyze the dynamic property of roll system’s flexural deformation, it is necessary to consider the transverse periodic movement of stock in the rolling deformation area which is caused by the flexural deformation movement of roll system simultaneously. Therefore, the displacement field of roll system and flow of metal in the deformation area is described by kinematic analysis in the dynamic system. Through introducing the lateral displacement function of metal in the deformation area, the dynamic variation of per unit width rolling force can be determined at the same time. Then the coupling law caused by the co-effect of rigid movement and flexural deformation of the system structural elements is determined. Furthermore, a multi-parameter coupling dynamic model of the roll system and stock is established by the principle of virtual work. More explicitly, the coupled motion modal analysis was made for the roll system. Meanwhile, the analytical solutions for the flexural deformation movement’s mode shape functions of rolls are discussed. In addition, the dynamic characteristic of the lateral flow of metal in the rolling deformation area has been analyzed at the same time. The establishment of dynamic lateral displacement function of metal in the deformation area makes the foundation for analyzing the coupling law between roll system and rolling deformation area, and provides a theoretical basis for the realization of the dynamic shape control of steel strip.     

Modi cation of Roll Flattening Analytical Model Based on the Plane Assumption

Roll flattening is an important component in the roll stack elastic deformation,which has important influence on controlling of the strip crown and flatness. Foppl formula and semi?infinite body model are the most popular analyti?cal models in the roll flattening calculation. However,the roll flattening calculated by traditional flattening models has a great deviation from actual situation,especially near the barrel edges. Therefore,in order to improve the accuracy of roll flattening,a new model is proposed based on the elastic half plane theory. The calculation formulas of roll flatten?ing are deduced respectively under the assumptions of plane strain and plane stress. Then,the two assumptions are combined through the method of introducing an transition coe cient,and the distribution rules of roll flattening for di erent rolling force,flattening width,roll length and roll diameter are analyzed by using the FEM analysis software Marc. Regarding the ratio of the length to roll end and the roll diameter as variable to fit the transition coe cient,the new model of roll flattening is established based on the elastic half plane theory. Finally,the transition coe cient is fitted to establish the model. Compared with the traditional models,the new model can e ectively improve the cal?culation deviation in the roll end,which has important significance for accurate simulation of plate shape,especially for the distribution of rolling force between rolls.     

Coupled dynamic modeling of rolls model and metal model for four high mill based on strip crown control

The crown is a key quality index of strip and plate,the rolling mill system is a complex nonlinear system,the strip qualities are directly affected by the dynamic characteristics of the rolling mil.At present,the studies about the dynamic modeling of the rolling mill system mainly focus on the dynamic simulation for the strip thickness control system,the dynamic characteristics of the strip along the width direction and that of the rolls along axial direction are not considered.In order to study the dynamic changes of strip crown in the rolling process,the dynamic simulation model based on strip crown control is established.The work roll and backup roll are considered as elastic continuous bodies and the work roll and backup roll are joined by a Winkler elastic layer.The rolls are considered as double freely supported beams.The change rate of roll gap is taken into consideration in the metal deformation,based on the principle of dynamic conservation of material flow,the two dimensional dynamic model of metal is established.The model of metal deformation provides exciting force for the rolls dynamic model,and the rolls dynamic model and metal deformation model couple together.Then,based on the two models,the dynamic model of rolling mill system based on strip crown control is established.The Newmark-β method is used to solve the problem,and the dynamic changes of these parameters are obtained as follows:(1) The bending of work roll and backup roll changes with time;(2) The strip crown changes with time;(3) The distribution of rolling force changes with time.Take some cold tandem rolling mill as subject investigated,simulation results and the comparisons with experimental results show that the dynamic model built is rational and correct.The proposed research provides effective theory for optimization of device and technological parameters and development of new technology,plays an important role to improve the strip control precision and strip shape quality.     

An investigation of contact problem between strip and work roll with a smooth straight surface during cold rolling

Under the assumption of elastic roller, and the neglect of heat transfer problem, this paper explores the distribution of elastic deformation of the work roll with initially smooth straight surface in the rolling area during the rolling process, and analyzes the effect of this deformation distribution on the strip shape after rolling.

Based on the large deformation—large strain theory, the update Lagrangian formulation (ULF) and the incremental principle were used to develop 3D elastic-plastic analytical model of aluminum strip rolling. The flow stress was considered the function of strain and strain rate to infer the governing equation containing the effect of strain rate.

Besides, in this article, we propose an iteration procedure to calculate the contact force between strip and work roll, and the contact deformation of the work roll. In the treatment of the contact problem, the work roll was regarded as a rigid body at the beginning of each step. The incremental displacement and contact force of strip nodes were first calculated, and then entered into the first iteration loop. The contact forces of strip nodes were allocated to the work roll to calculate work-roll elastic deformation. The elastic deformation of work-roll nodes would be compensated to the incremental displacement of the strip nodes (the work-roll deformation increment was added on the incremental displacement of strip node). Then the new contact nodal forces were derived from the finite element program of strip. The above procedures were repeated to incorporate the new contact nodal forces into the first iteration loop to obtain the new strip node incremental displacement. An error range was also designated as the comparison standard to compare the displacement increments of the strip from two iterations to determine whether the iteration was completed.

This paper considers the initial shape of work roll is straight and smooth. The elastic deformation distribution on the work roll surface in the rolling area is analyzed to serve as the basis for work-roll radius compensation. Finally, deformation of strip shape is also discussed.     

Analysis of thrust force in a work roll shifting mill

In a work roll shifting mill, thrust force can cause troubles in the operation of the mill. Based on theoretical analysis and practical application, the mechanics and control approach of the thrust force have been analyzed. Research shows that the relative displacement vector of friction pair is a criterion for controlling the thrust force during work roll shifting. This friction pair is caused by contact between the work roll and the backup roll in the deformation zone. The thrust force is determined by relative displacement vector, and it can be controlled effectively through decreasing the relative displacement vector. Using vector analysis, the effects of the work roll shifting and crossing on mechanics and practical application are also discussed. The experiment shown in this paper verifies that the work roll crossing can decrease the thrust force which is caused by its shifting during rolling.     

棒材高精度二辊矫直质量控制策略研究

高精度矫直辊辊型设计及工艺参数控制是二辊矫直机的核心技术。为了满足高精度棒材矫直要求,在辊型设计过程中依据矫直辊磨损状态、弹塑性理论、塑性变形硬化系数求解模型及中性层偏移理论,提出“凹三凸二”分段辊型的矫直辊组合形式及辊型设计方法。在此基础上,根据棒材与矫直辊的局部线接触状态,建立棒材二辊矫直倾角及辊缝估算模型,进而探讨二辊矫直工艺参数（倾角、辊缝、矫直速度和导板间距）的变化规律和控制策略,并给出具体设计实例和有限元模拟分析。从矫直力、塑性变形深度、直线度及残余应力等角度,证明该辊型设计方法和工艺控制策略的有效性,且现场试验矫后的棒材直线度小于1 mm/m,完全满足棒材高精度矫直标准。     

Study on the improved accuracy of strip profile using numerical formula model in continuous cold rolling with 6-high mill

The quality requirements for thickness accuracy in cold rolling continue to become more stringent. In cold rolling mill, it is very important that the rolling force calculation considers rolling conditions. The rolled strip thickness was predicted using calculated rolling force. However, the prediction of strip thickness in cold rolling is very difficult; in particular, for 6-high mill with shifted intermediate roll (IMR), the accuracy of thickness is not good. In this study, to improve the accuracy of rolled strip thickness, the roll gap flattening can be given based on Hertz contact theory, with contact between rolls and the smooth cylindrical rolls for the rolling elastic deformation. Also, the distribution of the roll gap flattening may be calculated using the contact force of unit transverse length. The strip profile at the continuous cold rolling is calculated by using the numerical analysis model considering the initial strip profile before cold rolling. Hence, we propose that the numerical model can predict the rolled strip profile more quickly and accurately and be applicable to the field. The results of the proposed numerical model were verified by FE-simulation and cold rolling experiments of 6-high mill with five stands.     

Analysis of the roll hunting force due to hardness in a hot rolling process

A hot rolling operation is performed to alter the thickness of a metal by passing the material through a pair of rollers, forming a gap that is somewhat narrower than the thickness of the material. Therefore, the quality of the product is a function of the pressure applied by the rollers. However, in this process, a roll hunting force occurs in which the rolling force is irregularly changed during the rotation of the rollers due to various complex mechanisms, which include roll surface hardness, difference in rotational speed between rolls, heat treatment conditions, and roll wear. In this study, roll wear tests were conducted to analyze the roll hunting force caused by variation in the hardness of the work roll. The friction coefficient of the work roll was then examined based on hardness. Then, a two-dimensional finite element model was constructed to investigate the roll hunting force as a function of the change in friction coefficient of the work roll. This finite element model was verified in relation to the theoretical rolling expression. Finite element model analysis was performed for three friction coefficients, and the effect of the roll hunting force was determined based on the reduction ratio and temperature. In addition, the wear depth of the work roll by the hardness was predicted. The influence of the abrasion of the work roll on the hunting force was analyzed.

     

冷连轧机轧制力的影响因素

通过将轧制变形区划分为微单元,对变形区内轧制力影响因素的变化规律进行研究。根据微单元塑性变形方程式,建立单位轧制压力的计算模型。针对变形抗力、摩擦因数等非线性因素,建立基于神经网络校正系数和工作参考点方法的数学模型,分析其在变形区内的变化规律。针对轧辊压扁和轧制力计算的耦合现象,采用计算机迭代计算的模型完成轧制力的解耦计算。工业轧机实际数据验证了采用微单元方式计算轧制力预报精度高,可用于生产实践。     

Modified slab analysis of asymmetrical plate rolling

In this paper the general case of asymmetrical plane strain rolling due to unequal roll diameter, unequal surface speed of the rolls and different contact friction is considered. An analytical model based on the slab method of analysis was used and further developed to obtain the characteristics of asymmetrical sheet rolling and to predict strip curvature. This method describes an enhancement to the rolling theory where friction becomes an integral part of deformation mechanics in the roll gap. To verify the validity of the proposed analytical model, the analytical rolling force, torque and curvature were compared with experimental and analytical results of other investigators. Very good agreements are found. Capability as well as the simplicity of the proposed model in predicting more accurate theoretical results for the rolling force, torque and curvature makes it suitable for the online control application.     

HC轧机辊系变形及中间辊移动量与工作辊弯辊力最佳配合的研究

本文从理论和实验上研究了HC轧机的辊系变形,并采用优化方法计算了工作辊最佳弯辊力以及中间辊移动量与工作辊弯辊力的最佳配合,为HC轧机的研制,推广及使用提供了基础。     

An upper bound method for analysis of three-dimensional deformation in the flat rolling of bars

A new upper bound method for the analysis of three-dimensional deformation in the flat rolling of bars is proposed. In the conventional upper bound method, stress distribution is not calculated. Hence, a new method of calculating stress distribution is proposed in which the hydrostatic stress in each element and the contact stress on the contact surface between material and roll in each element are calculated. In the conventional upper bound method, the structure of the computer program also depends highly on the kinematically admissible velocity fields assumed. Hence, a new method of analysis in which the structure of the computer program depends minimally on the kinematically admissible velocity fields assumed is proposed. In the analysis of strip rolling, the calculated roll force and roll torque agree with the roll force and roll torque calculated from Sims’ slab method. In the analysis of the flat rolling of bars, the calculated width spread agrees with the width spread obtained by experiments reported in the literature. The effects of a reduction in thickness, roll radius, material width-to-height ratio, front tension, back tension, and front and back tensions on the width spread, forward slip, roll force and roll torque are demonstrated. Hence, the validity of the new upper bound method is confirmed.     

快速辊系变形在线计算方法研究

工作辊和支撑辊弹性变形计算是板带轧机板形设定模型的关键,其计算速度与结果精度是当前制约辊系变形在线计算的主要因素。物理建模方面,在传统影响函数法的基础上,通过引入工作辊中心倾斜角度这一变量,改进辊系弹性变形简化模型,使辊系在非对称工况下也能满足弯矩平衡条件。在辊系变形数值计算方面,根据Timoshenko深梁理论建立挠曲平衡微分方程,采用有限差分法对四辊轧机辊系变形进行数学建模,计算辊系的角位移和挠曲变形。该有限差分模型采用递推方式进行挠曲计算,避开了影响函数法中的大量矩阵乘法运算,计算效率有较大提高。分别通过与影响函数法、有限元法对非对称工况的计算结果进行对比,验证基于有限差分法的快速辊系变形计算方法具有较快的计算速度和较高的精度。     

A thermal analysis of strip-rolling in mixed-film lubrication with O/W emulsions

Increase of both roll and strip surface temperatures can significantly affect a rolling process, roll conditions and strip mechanical properties. A comprehensive thermal analysis in cold rolling, especially in a mixed film regime, is needed to understand how thermal fields develop in roll and strip during rolling. It requires a simultaneous solution of the mixed film model for friction in the roll bite and the thermal model for roll and strip thermal fields. This paper presents a numerical procedure to analyse strip rolling process using lubrication with oil-in-water (O/W) emulsions. The thermal model includes the effect of heat generation due to the strip deformation and frictional shear stress at the asperity contacts. The numerical analysis employs a coupled thermal model and a mixed film lubrication model for calculating the friction and the asperity deformation in the bite. The thermal model considers the initial temperatures of the roll and strip, temperature rise due to the strip plastic deformation and friction. While the O/W mixed-film lubrication model takes into account the effect of surface roughness and oil concentration (%vol) of the emulsion. The thermal effect is analysed in terms of strip surface temperature and roll temperature, which are influenced by rolling parameters such as reduction, rolling speed, oil concentration in the emulsion. The results of the parametric study indicate that the effect of oil concentration on the thermal field is relatively small compared to that of reduction ratio and rolling speed. The reduction ratio increases the maximum interface temperature in the roll bite. In the mixed film regime, rolling speed also increases the maximum interface temperature and alters the temperature field of the strip. The numerical procedure was validated against known experimental data and can readily be extended to hot rolling or used to analyse roll strip temperature subjected to different cooling system.     

Coupled approach for flatness prediction in cold rolling of thin strip

The paper presents a predictive model of the flatness defects, which appear during rolling of thin plates, the origin of which is the roll stack thermo-elastic deformation. The combination of the elastic deflection, the thermal crown and the roll grinding crown results in a non-parallel bite, and if the deformed roll transverse profile is not an affinity of the incoming strip profile, differential elongation results and induces high stresses in the outgoing strip. The latter, combined with the imposed strip tension force, result in a net post-bite stress field which may be sufficiently compressive locally to promote buckling. A variety of non-developable shapes may result, generally occurring as waviness (centre waves, wavy edges, quarter-buckles, etc.). This problem is most of the time addressed in a decoupled way, i.e. as a post-processing of the residual stresses computed by a strip rolling model; the present paper on the contrary describes a fully coupled approach of in-bite plastic deformation and post-bite buckling. For this purpose, a simple buckling criterion has been introduced in a FEM model of strip and roll deformation, Lam3/Tec3; its implementation is documented in details. The capabilities and limits of the present approach are described and discussed. Characterised by its coupled approach, it is primarily devoted to cases where on-line (under tension) manifested defects occur. It is shown that the impact of the post-bite, post-buckled stress field on the in-bite stress and strain fields is quite small in the cases investigated; however, subtle changes appear in the velocity field at bite exit, and this is sufficient to transform completely the post-bite stress field, which is found in much better agreement with measurements if such a coupled treatment is used.     

冷轧机板形控制系统弯辊力预设定模型开发

针对某公司六辊冷轧机,采用影响函数法建立了辊系变形计算模型并开发了相应的计算程序。利用所建辊系变形模型结合轧机的板形控制策略,建立了针对轧机的理论和在线两种弯辊力预设定模型。实例计算表明,所开发模型精度符合实际要求,可用于生产实际,为提高轧机板形控制预设定精度,改善轧机产品的板形质量提供了理论依据。