Bayesian-Based Probabilistic Force Modeling in Cold 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 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.