Research Papers

Displacement of Multiple, Coupled Timoshenko Beams in Discontinuous Nonlinear Elastic Contact, With Application to Rolling Mills

[+] Author and Article Information
Arif S. Malik1

Department of Aerospace and Mechanical Engineering,  Saint Louis University, 3450 Lindell Boulevard, St. Louis, MO 63103amalik8@slu.edu

Jantzen L. Hinton

Department of Mechanical and Materials Engineering,  Wright State University, 3640 Colonel Glenn Highway, Dayton, OH 45435jantzenhinton@hotmail.com


Corresponding author.

J. Manuf. Sci. Eng 134(5), 051009 (Sep 10, 2012) (10 pages) doi:10.1115/1.4007185 History: Received September 25, 2011; Revised July 13, 2012; Published September 10, 2012; Online September 10, 2012

Introduced is an efficient numerical method to calculate the displacements of structures involving multiple, shear-deformable beams in lengthwise nonlinear elastic contact. Applicability of the method is general, and includes structures such as stacked beams, pipes, or machine components. A particularly relevant application, which is discussed and demonstrated in this paper, is the prediction of deflections and contact conditions between the rolls and strip in rolling mills used to process flat metals. The presented nonlinear displacement calculation method involves iteration using an efficient, simplified mixed finite element formulation. Nonlinearities in the general coupled beam problem can arise either from changing contact conditions between the lengthwise-coupled shear-deformable beams (opening/closing of gaps) or from nonlinear elastic coupling-stiffness between the beams. Unlike the conventional problem of end-connected beams resting on elastic foundations, this work presents an element stiffness matrix formulation and sample nonlinear solution techniques for the problem involving multiple beams that are mutually coupled along their lengths by intervening, nonlinear elastic foundations. Important practical examples of the method are given, based on production cold rolling mill data for type 301 stainless steel on a four-high mill. The examples demonstrate efficiency and value of the nonlinear coupled beam formulation in identifying appropriate machined roll profiles to produce desired strip flatness quality. The cold rolling applications presented are characterized by nonlinear Hertz foundations with stiffness-hardening, and by discontinuous contact conditions arising from nonuniform, machined roll diameter profiles. Solution efficiencies for the new formulation are compared using modified Newton–Raphson, average stiffness, and direct substitution techniques. While the average stiffness method is relatively slower, its characteristics may make it beneficial for transfer function development in flatness control systems. Application of the method to cold rolling mills illustrates the importance of including discontinuous contact nonlinearities, and enables reduced trial and error to determine suitable roll profiles. Presented is a new efficient variational formulation for computing displacements in structures involving multiple beams laterally coupled with nonlinear elastic foundations and changing contact conditions. Practical value of the method is demonstrated for identification of machined roll profiles in cold rolling mills in order to achieve desirable flatness of rolled products.

Copyright © 2012 by American Society of Mechanical Engineers
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Figure 1

(a) Well-established beam on single elastic foundation problem; (b) multiple beams coupled length-wise by nonlinear elastic foundations in between them (no prior formulation)

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Figure 2

Four-high single-stand reversing cold rolling mill

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Figure 3

Roll configurations of some cold rolling mills (viewed from roll ends)

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Figure 4

(a) Cylindrical Timoshenko beams coupled with nonlinear Herztian elastic foundations and (b) the same configuration as Fig. 4(a), but with initial foundation gaps present

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Figure 5

Nonlinear finite element simulation of displacement for two rolls in elastic contact (3D 1/4 symmetric model, ∼750,000 elements, 6.3 GB memory, and >10 h PC CPU time)

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Figure 6

Example model for problem involving Timoshenko beams with multiple nonlinear coupled elastic foundations as applied to rolling mills (upper symmetric model)

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Figure 7

Example of poor strip flatness (waviness), due to excessive relative difference between entry and exit strip thickness profiles

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Figure 8

(a) Parameters of laterally coupled beam-foundation structure used for Mid America Stainless rolling mill case studies (note only upper symmetric section of mill is shown) and (b) ten-node model discretization; (c) 71-node discretization (meshes not to scale)

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Figure 9

Normalized Hertz foundation stiffness moduli k2 , k2−1 , and k2−2

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Figure 10

(a) Converged contact force distributions (upper plot) and entry/exit strip thickness profiles (lower plot) for case 1 work-roll taper gradient of 0.1818 mm/m using a 71-node model discretization. (b) Converged contact force distributions (upper plot) and entry/exit strip thickness profiles (lower plot) for case 2 work-roll taper gradient of 0.5455 mm/m using a 71-node model discretization. (c) Noniterative contact force distributions (upper plot) and entry/exit strip thickness profiles (lower plot) for case 3 work-roll taper gradient of 0.9091 mm/m using a 71-node model discretization. (d) Converged contact force distributions (upper plot) and entry/exit strip thickness profiles (lower plot) for case 3 work-roll taper gradient of 0.9091 mm/m using a 71-node model discretization.

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Figure 11

Test strip to illustrate flatness for work-roll taper gradient of 0.1818 mm/m (case 1 in Table 5)

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Figure 12

Mesh convergence for case 3: Work-roll taper gradient of 0.9091 mm/m (a) % change in strip exit thickness at center and edge locations, (b) % difference relative to results of finest mesh (1091 nodes) (Note: Converged values are same for all iterative methods)



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