A primary factor in manufacturing high-quality cold-rolled sheet is the ability to accurately predict the required rolling force. Rolling force directly influences roll-stack deflections, which correlate to strip thickness profile and flatness. Accurate rolling force predictions enable assignment of efficient pass schedules and appropriate flatness actuator set-points, thereby reducing rolling time, improving quality, and reducing scrap. Traditionally, force predictions in cold rolling have employed deterministic, two-dimensional analytical models such as those proposed by Roberts and Bland and Ford. These simplified methods are prone to inaccuracy, however, because of several uncertain, yet influential, model parameters that cannot be established deterministically under diverse cold rolling conditions. Typical uncertain model parameters include the material's strength coefficient, strain-hardening exponent, strain-rate dependency, and the roll-bite friction characteristics at low and high mill speeds. Conventionally, such parameters are evaluated deterministically by comparing force predictions to force measurements and employing a best-fit regression approach. In this work, Bayesian inference is applied to identify posterior probability distributions of the uncertain parameters in rolling force models. The aim is to incorporate Bayesian inference into rolling force prediction for cold rolling mills to create a probabilistic modeling approach that learns as new data are added. The rolling data are based on stainless steel types 301 and 304, rolled on a 10-in. wide, 4-high production cold mill. Force data were collected by observing load-cell measurements at steady rolling speeds for four coils. Several studies are performed in this work to investigate the probabilistic learning capability of the Bayesian inference approach. These include studies to examine learning from repeated rolling passes, from passes of diverse coils, and by assuming uniform prior probabilities when changing materials. It is concluded that the Bayesian updating approach is useful for improving rolling force probability estimates as evidence is introduced in the form of additional rolling data. Evaluation of learning behavior implies that data from sequential groups of coils having similar gauge and material is important for practical implementation of Bayesian updating in cold rolling.

As the demand for thin and ultra-thin gage for cold-rolled sheet metals increases, cold rolling manufacturers and sheet metal producers must utilize technological advancements to stay competitive globally. To manufacture a competitive rolling mill, the mill designer must identify an appropriate mill configuration, roll sizes, and flatness control systems, etc. The design must be robust enough to accommodate broad product envelopes containing strips of different widths, gages, and materials. At the same time, the mill "sizing" must be competitive by avoiding over- or under-design. These two objectives, designing a robust mill and remaining competitive, tend to conflict, which establishes the need for optimal mill design based on rigorous multidisciplinary analysis.Conventionally, mill design is done using a 'sequential' design approach relying heavily on experience. The major limitation of sequential mill design is that it can lead to over-design or under-design since it segregates design tasks for the coupled components, making it difficult to consider many combinations of the design space. The result is a 'feasible' rather than 'optimum feasible' mill design. This work applies nonlinear programming to optimize the basic design of a 4-high reversing cold rolling mill, subject to static strength, fatigue, and strip geometric quality constraints. The work offers an improved multi-disciplinary design approach based on stress and fatigue analysis of rolls, and an efficient strip profile/flatness analysis to identify suitable design limits for roll bending devices.

Traditional methods of characterizing vertical-stack mill shape actuation behavior involves the range of actuated roll gap profile geometries within a 2-dimensional space that spans the quadratic curvature and a composite collection of even, higher order terms. Recent developments in describing shape actuation complexities and capabilities of highly coupled 20-high cluster mills offer an extension of the roll gap profile characterization in terms of the rolled transverse stress distribution (shape), spanning high orders of odd and even curvature. This paper examines the Shape Actuation Capabilities Envelopes (SACEs) of 4 / 6-high mills with a focus on their application in mill set-up and scheduling.

A primary factor in manufacturing high-quality cold-rolled sheet is the ability to accurately predict the required rolling force. The rolling force directly influences roll-stack deflections, which correlate to the resulting flatness quality of the rolled sheet. Increasingly high demand for thin and ultrathin gauge for cold-rolled sheet metals, along with the correspondingly larger sensitivity of flatness defects when rolling thin gauges, makes it more important to accurately and rapidly predict the rolling force before the rolling operation begins. Accurate rolling force predictions enable assignment of appropriate pass schedules and flatness mechanism set-points early in the rolling process, thereby reducing rolling time, improving quality, and reducing scrap. Traditionally, force predictions in cold rolling have employed two-dimensional analytical models such as those proposed by Roberts and by Bland & Ford. These simplified methods are prone to inaccuracy, however, because of several uncertain, yet influential, model parameters that are difficult to establish deterministically for wide-ranging products. These parameters include, for example, the average compressive yield strength of the rolled strip, frictional characteristics relating to low and high mill speeds, and the strain rate dependency of yield strength. Conventionally, these unknown parameters have been evaluated deterministically by comparing force predictions with actual rolling force data and using a best-fit regression approach. In this work, Bayesian updating using a probability mass function (PMF) is applied to identify joint posterior probability distributions of the uncertain parameters in rolling force models. It is shown that the non-deterministic Bayesian updating approach is particularly useful as new evidence becomes available in the form of additional rolling force data. The aim of the work is to incorporate Bayesian inference into rolling force prediction for cold rolling mills to create a probabilistic modeling approach which can also "learn" as new production data is added. The goal is a model that can better predict necessary mill parameters based on accurate probability estimates of the actual rolling force. The rolling force data used in this work for applying Bayesian updating is actual production data of grades 301 and 304L (low carbon) stainless steels, rolled on a 10-inch wide 4-high cold rolling mill. This force data was collected by observing and averaging load cell measurements at steady rolling speeds.

20-High rolling mills process high strength and/or very thin non-ferrous and ferrous metals using a complex, cluster arrangement of rolls. The 20-high roll cluster arrangement achieves specific flatness goals in the thin sheet by delivering maximum rolling pressure while minimizing the deflections of the small diameter rolls. 20-high mills also employ flatness control mechanisms with sophisticated actuators, such as those to shift intermediate rolls and deflect backup bearing shafts. The purpose of this is to compensate for variations in strip dimensional and mechanical properties which can cause poor flatness control quality from discrepancies in work-roll gap profile and distribution of rolling force. This suggests that the random property differences in the rolling parameters that substantially affect the flatness must be directly accounted for in flatness control algorithms in order to achieve strict flatness quality. The use of accurate mathematical models that account for the rolling pass target gage reduction can optimize the flatness control actuators and help gain an advantage in the thin gauge strip competitive global market. Based on the expected process parameter variations and nominal mill set-points (speed, tension, gage reduction, etc.), the mill's process control computer should determine the probability that target flatness control quality will be met for a required length of strip. The process computer should then either modify the number of rolling passes or adjust the thickness reduction schedule before rolling begins to secure an improved flatness probability estimate if the probability of achieving target strip flatness is too low for the required deliverable quality. Therefore, this research integrates 1) 20-high roll-stack mill mathematical modeling, 2) probability distribution data for random important rolling parameters, 3) reliability-based models to predict the probability of achieving desired strip flatness, and 4) optimization examples. The results can be used to reduce wasted rolled metal from poor flatness before rolling.

Optimal pass-scheduling on cluster-type cold rolling mills, use to process flat metals, presents added challenges over conventional (vertical-stack) mills due to the complexity of roll arrangements. Cluster-type rolling mills not only pose difficulties in modeling deflections occurring in the multi-roll stack, they also impose the burden of modeling more sophisticated mechanisms used to adjust rolling force distribution and achieve desired strip flatness. In a competitive global market for very thin gauge strip, an advantage is gained through use of efficient mathematical set-up models that can adequately optimize the flatness actuators according to the target gauge reductions for each rolling pass. The mill's process control computer should therefore determine a gauge reduction schedule leading to minimum number of passes, while simultaneously assigning nominal flatness control actuator set-points. Although recent developments in roll-stack deflection modeling using simplified, mixed finite element techniques have enabled more efficient roll-stack deflection modeling in 20-high and other cluster mills, the optimal pass-schedule problem is still complicated by the abundance of geometric and mechanical property variations in the strip or sheet to be processed. Furthermore, problems with strip flatness frequently arise because of uncertainties in roll diameter profiles resulting from variations in the roll grinding and roll wear patterns. In this paper, we extend recent work in pass schedule optimization (through improved roll-stack deflection) by applying First Order Reliability Methods to rigorously account for various rolling process uncertainties. The results allow predictive probability constraints for strip flatness to be included in the optimization problem, thus enabling mill operators some insight and control into the likelihood of achieving desired strip flatness for a given rolling pass schedule.

Besides the requirement to roll coiled sheet at high levels of productivity, the optimal pass scheduling of cluster-type reversing cold mills presents the added challenge of assigning mill parameters that facilitate the best possible strip flatness. The pressures of intense global competition, and the requirements for increasingly thinner, higher quality specialty sheet products that are more difficult to roll, continue to force metal producers to commission innovative flatness-control technologies. This means that during the on-line computerized set-up of rolling mills, the mathematical model should not only determine the minimum total number of passes and maximum rolling speed, it should simultaneously optimize the pass-schedule so that desired flatness is assured, either by manual or automated means. In many cases today, however, on-line prediction of strip crown and corresponding flatness for the complex cluster-type rolling mills is typically addressed either by trial and error, by approximate deflection models for equivalent vertical roll-stacks, or by non-physical pattern recognition style models. The abundance of the aforementioned methods is largely due to the complexity of cluster-type mill configurations and the lack of deflection models with sufficient accuracy and speed for on-line use. Without adequate assignment of the pass-schedule set-up parameters, it may be difficult or impossible to achieve the required strip flatness. In this paper, we demonstrate optimization of cluster mill pass-schedules using a new accurate and rapid strip crown model. This pass-schedule optimization includes computations of the predicted strip thickness profile to validate mathematical constraints. In contrast to many of the existing methods for on-line prediction of strip crown and flatness on cluster mills, the demonstrated method requires minimal prior tuning and no extensive training with collected mill data. To rapidly and accurately solve the multicontact problem and predict the strip crown, a new customized semi-analytical modeling technique that couples the Finite Element Method (FEM) with classical solid mechanics was developed to model the deflection of the rolls and strip while under load. The technique employed offers several important advantages over traditional methods to calculate strip crown, including continuity of elastic foundations, non-iterative solution when using predetermined foundation moduli, continuous third-order displacement fields, simple stress-field determination, and a comparatively faster solution time.