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Review Article

Dynamical Models of the Camber and the Lateral Position in Flat Rolling

[+] Author and Article Information
Andreas Steinboeck

Automation and Control Institute,
Vienna University of Technology,
Gußhausstraße 27-29/376,
Vienna 1040, Austria
e-mail: andreas.steinboeck@tuwien.ac.at

Andreas Ettl

Christian Doppler Laboratory for Model-Based
Process Control in the Steel Industry,
Automation and Control Institute,
Vienna University of Technology,
Gußhausstraße 27-29/376,
Vienna 1040, Austria
e-mail: ettl@acin.tuwien.ac.at

Andreas Kugi

Professor
Christian Doppler Laboratory for Model-Based
Process Control in the Steel Industry,
Automation and Control Institute,
Vienna University of Technology,
Gußhausstraße 27-29/376,
Vienna 1040, Austria
e-mail: kugi@acin.tuwien.ac.at

1Corresponding author.

Manuscript received March 29, 2017; final manuscript received June 29, 2017; published online August 2, 2017. Editor: Harry Dankowicz.

Appl. Mech. Rev 69(4), 040801 (Aug 02, 2017) (14 pages) Paper No: AMR-17-1024; doi: 10.1115/1.4037177 History: Received March 29, 2017; Revised June 29, 2017

In flat rolling, the lateral position of the product in the rolling mill and the camber (curvature of the product centerline seen in top view) are key process variables. We explore how their evolution can be analytically modeled based on nonlinear geometric relations, material derivatives, balance equations, constitutive equations for the material flow in the roll gap, and a change of coordinates to obtain a time-free formulation. Based on example problems, we verify the developed novel model and further illustrate the mechanisms behind it. Finally, a literature review on models in this field reveals that there is not yet a consensus on the correct analytical model of the evolution of the camber in flat rolling. The literature review shows that most published models are special cases of the model developed in this paper.

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Figures

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Fig. 1

Top view of mill stand and flat product

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Fig. 2

Stress resultants in the product at the entry and exit port of the roll gap

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Fig. 3

Stand-alone simulation model

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Fig. 4

Tangential rolling (p′(0−)=p′(0+)=0) with κ(0) = const., Δh+/h¯+=Δh−/h¯−, and λ = 1.4

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Fig. 5

Straight rolling (ω− = 0) of a cambered product with κ(0) = const., Δh− = 0, Δh+ = 0, and λ = 1.4

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Fig. 6

Straight rolling (ω− = 0) of an initially straight product (κ(0) = 0) with φ(0−)=−0.1  rad, Δh+/h¯+−Δh−/h¯−=0.15, and λ = 1.4

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Fig. 7

Straight rolling (ω = 0) of an initially straight product (κ(0 − ) = 0) with φ(0−)=−0.1  rad, Δh− = 0, Δh+ = 0, and λ = λ0 + 1 where λ0 = 1 and λ1 = 0.2/m

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Fig. 8

Rolling of an initially straight product (κ(0) = 0) with upstream angular velocity ω−=(−0.1  rad/m)v¯−, φ(0−)|X=0=0, Δh− = 0, Δh+ = 0, and λ = 1.4

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Fig. 10

Top view of the deformation (elongation) of an infinitesimal trapezoidal section of the product in the roll gap

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