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Research Papers

Probabilistic Force Prediction in Cold Sheet Rolling by Bayesian Inference

[+] Author and Article Information
Andrew W. Nelson, John C. Wendel

Parks College of Engineering,
Aviation, and Technology
Saint Louis University,
3450 Lindell Boulevard,
St. Louis, MO 63103

Arif S. Malik

Parks College of Engineering,
Aviation, and Technology
Saint Louis University,
3450 Lindell Boulevard,
St. Louis, MO 63103
e-mail: amalik8@slu.edu

Mark E. Zipf

I2S, LLC, 475 Main Street,
Yalesville, CT 06492
e-mail: mzipf@istmg.com

1Corresponding author.

Contributed by the Manufacturing Engineering Division of ASME for publication in the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript received May 23, 2013; final manuscript received April 11, 2014; published online May 21, 2014. Assoc. Editor: Brad L. Kinsey.

J. Manuf. Sci. Eng 136(4), 041006 (May 21, 2014) (11 pages) Paper No: MANU-13-1232; doi: 10.1115/1.4027434 History: Received May 23, 2013; Revised April 11, 2014

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.

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References

Figures

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Fig. 1

Depiction of rolling force model parameters (R′ is equivalent, deformed roll radius)

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Fig. 2

Roll-bite, indicating a neutral point and the influence of friction on rolling pressure distribution along the contact arc

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Fig. 4

Posterior PMF for data set 1, first update (coil 1, pass 1)—Roberts model (Fm = 68.86 ton, bin width = 1.12 ton)

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Fig. 5

Posterior PMF for data set 1 (coil 1, pass 1) forth repeated update—Roberts model (Fm = 68.86 ton, bin width = 1.12 ton)

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Fig. 16

Posterior mode (PM) probabilities with Roberts and B&F force models for first and final passes of coils 1–4 in Table 1

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Fig. 3

Prior PMF for data set 1 (coil 1, pass 1)—Roberts model (Fm = 68.86 ton, bin width = 1.12 ton). Note very large range in force value due to initial, uniform Prior PDF.

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Fig. 8

Posterior PMF for data set 6 (coil 1, pass 6)—Roberts model (Fm = 77.65 ton, bin width = 1.12 ton)

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Fig. 9

Posterior PMF for data set 7 (coil 2, pass 1)—Roberts model (Fm = 64.46 ton, bin width = 1.12 ton)

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Fig. 10

Posterior PMF for data set 9 (coil 2, pass 3)—Roberts model (Fm = 64.46 ton, bin width = 1.12 ton)

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Fig. 11

Posterior PMF for data set 10 (coil 3, pass 1)—Roberts model (Fm = 64.44 ton, bin width = 1.12 ton)

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Fig. 12

Posterior PMF for data set 13 (coil 3, pass 4)—Roberts model (Fm = 54.94 ton, bin width = 1.12 ton)

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Fig. 13

Posterior PMF for data set 14 (coil 4, pass 1)—Roberts model (Fm = 56.40 ton, bin width = 1.12 ton)

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Fig. 14

Posterior PMF for data set 21 (coil 4, pass 8)—Roberts model (Fm = 38.09 ton, bin width = 1.12 ton)

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Fig. 6

Relative error in posterior mode force prediction for repeated learning with all six passes of Coil 1

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Fig. 7

Posterior mode (PM) probability for repeated learning with all six passes of coil 1

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Fig. 15

Relative error of posterior mode (PM) force value with Roberts and B&F force models for first and final passes of coils 1–4 in Table 1

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Fig. 17

Relative error of posterior mode (PM) force value for material-related Bayesian updating strategies on type 304 stainless coils, Nos. 3 and 4 in Table 1 (Roberts force model)

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Fig. 18

PM probability for material-related Bayesian updating strategies on type 304 stainless coils, Nos. 3 and 4 in Table 1 (Roberts force model)

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