A primary factor in manufacturing high-quality cold-rolled sheet is the ability to accu rately 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, twodimensional 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 materi al' s strength coefficient, strain-hardening exponent, strain-rate dependency, and the rollbite 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 mate rials. 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.

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.

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 ultra-thin 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.