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Research Papers

Spatially Dependent Transfer Functions for Web Lateral Dynamics in Roll-to-Roll Manufacturing

[+] Author and Article Information
Edison O. Cobos Torres

Department of Mechanical Engineering,
Texas A & M University,
College Station, TX 77843
e-mail: orlando.cobos@tamu.edu

Prabhakar R. Pagilla

Professor
Department of Mechanical Engineering,
Texas A & M University,
College Station, TX 77843
e-mail: ppagilla@tamu.edu

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT,AND CONTROL. Manuscript received September 11, 2017; final manuscript received May 2, 2018; published online June 18, 2018. Assoc. Editor: Mazen Farhood.

J. Dyn. Sys., Meas., Control 140(11), 111011 (Jun 18, 2018) (12 pages) Paper No: DS-17-1457; doi: 10.1115/1.4040216 History: Received September 11, 2017; Revised May 02, 2018

Spatially dependent transfer functions for web span lateral dynamics which provide web lateral position and slope as outputs at any location in the web span are derived in this paper. The proposed approach overcomes one of the key limitations of the existing methods which provide web lateral position only on the rollers. The approach relies on taking the Laplace transform with respect to the temporal variable of both the web span lateral governing equation and the boundary conditions on the rollers, and solving the resulting equations. A general web span lateral transfer function, which is an explicit function of the spatial position along the span, is obtained first followed by its application to common guide configurations. The approach also significantly simplifies the consideration of shear (relevant to short spans), in addition to bending, which has been found to be difficult to handle in past studies. We first develop spatially dependent lateral transfer functions by considering only bending which is relevant to most web handling situations, and then add shear to the formulation and develop spatially dependent lateral transfer functions that include both bending and shear. Results from model simulations and pertinent discussions are provided. The spatially dependent transfer functions derived in this paper are a significant improvement over existing lateral transfer functions and provide mechanisms to analyze web lateral behavior within spans, study propagation of lateral disturbances, and aid in the development of closed-loop lateral control systems in emerging applications that require precise lateral positioning of the web.

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References

Seshadri, A. , and Pagilla, P. R. , 2013, “ Modeling Print Registration in Roll-to-Roll Printing Presses,” ASME J. Dyn. Syst., Meas., Control, 135(3), p. 031016.
Shelton, J. J. , 1968, “ Lateral Dynamics of a Moving Web,” Ph.D. dissertation, Oklahoma State University, Stillwater, Ok. https://shareok.org/bitstream/handle/11244/30409/Thesis-1968D-S545l.pdf?sequence=1&isAllowed=y
Shelton, J. J. , and Reid, K. N. , 1971, “ Lateral Dynamics of a Real Moving Web,” ASME J. Dyn. Syst., Meas., Control, 93(3), pp. 180–186. [CrossRef]
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Seshadri, A. , and Pagilla, P. R. , 2010, “ Optimal Web Guiding,” ASME J. Dyn. Syst., Meas., Control, 132(1), p. 011006.
Seshadri, A. , and Pagilla, P. R. , 2012, “ Adaptive Control of Web Guides,” Control Eng. Pract., 20(12), pp. 1353–1365. [CrossRef]
Brown, J. , 2005, “ A New Method for Analyzing the Deformation and Lateral Translation of a Moving Web,” Eighth International Conference on Web Handling, Stillwater, OK, June 5–8, pp. 39–58. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.508.2705&rep=rep1&type=pdf
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Figures

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Fig. 1

Web span consider for modeling

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Fig. 2

Web behavior at roller entry (normal entry condition)

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Fig. 5

Effect of shear: (a) free body forces and (b) slope

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Fig. 6

Two-span, three-roller system

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Fig. 7

Evolution of web lateral position and slope for 4 points along the span for a sinusoidal roller rotation of θ0=0.01 sin(3t) of roller R1

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Fig. 8

Evolution of web lateral position and slope for 4 points along the span for a sinusoidal roller rotation of θL=0.01 sin(3t), zL=X1θL of roller R2

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Fig. 9

Evolution of web lateral position and slope for 4 points along the entry span for a sinusoidal disturbance of y0=0.002 sin(3t) on R1, with proportional–integral (PI) control of the RPG

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Fig. 10

Evolution of guide roller rotation (θL(t)) with PI control in the plane of the entry span

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Fig. 11

Projection of the guide roller rotation into the plane of the exit span; θ0R2(t)=(cos β)θL(t) and β is the web wrap angle on the guide roller

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Fig. 12

Evolution of web lateral position and slope for 4 points along the exit span for 89 deg wrap angle

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