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Research Articles

Governing Equations for Web Tension and Web Velocity in the Presence of Nonideal Rollers

[+] Author and Article Information
Carlo Branca

Ph.D. Candidate
e-mail: carlo.branca@okstate.edu

Prabhakar R. Pagilla

Professor
Fellow ASME
e-mail: pagilla@okstate.edu

Karl N. Reid

Professor
Fellow ASME
e-mail: karl.reid@okstate.edu

School of Mechanical and Aerospace Engineering,
Oklahoma State University,
Stillwater, OK 74078

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the Journal of Dynamic Systems, Measurement, and Control. Manuscript received July 25, 2011; final manuscript received August 23, 2012; published online December 21, 2012. Assoc. Editor: Jiong Tang.

J. Dyn. Sys., Meas., Control 135(1), 011018 (Dec 21, 2012) (10 pages) Paper No: DS-11-1228; doi: 10.1115/1.4007974 History: Received July 25, 2011; Revised August 23, 2012

Since rotating machinery is used to transport flexible materials (commonly known as webs) on rollers, it is common to observe periodic oscillations in measured signals such as web tension and web transport velocity. These periodic oscillations are more prevalent in the presence of nonideal elements such as eccentric rollers and out-of-round material rolls. One of the critical needs in efficient transport of webs is to maintain web tension at a prescribed value. Tension regulation affects almost all key processes involved during web transport including printing, registration, lamination, winding, etc. Governing equations for web tension and transport velocity that can accurately predict measured behavior in the presence of nonideal rollers are beneficial in understanding web transport behavior under various dynamic conditions and the design of suitable web tension and speed control systems. The focus of this paper is on modeling the effect of eccentric rollers and out-of-round material rolls on web tension and web transport velocity. The new governing equations for web velocity on an eccentric roller and web tension in spans adjacent to the eccentric roller are presented and discussed; a web span is the free web between two consecutive rollers. To solve these governing equations, the location of the entry and exit point of the web on the eccentric roller as it rotates and the length of the web spans adjacent to the eccentric roller are required; a procedure for obtaining this information is described. To corroborate the models and the developed approach, data from experiments on a large experimental web platform are compared with data from model simulations, and a representative sample of the results are presented and discussed.

FIGURES IN THIS ARTICLE
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© 2013 by ASME
Topics: Equations , Rollers , Tension
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References

Figures

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Fig. 2

Primitive element schematics

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Fig. 3

Two loop control scheme: the inner speed loop is used to regulate the angular velocity of the unwind roll and the outer tension loop provides correction to the reference velocity to maintain prescribed tension. This is often referred to in the literature as speed-based tension control system. The unwind block represents the dynamics of the unwind roll and the web dynamics block represents the governing equation for web tension.

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Fig. 4

Experimental and simulation results

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Fig. 5

Roller configurations

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Fig. 6

Eccentric idle roller: CG is the geometric center, CR is the center of rotation, e is the eccentricity, d0 is the distance between the centers of rotation of the two rollers, and d(t) is the distance between the geometric centers of the two rollers

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Fig. 7

Eccentric idle roller: CG is the geometric center, CR is the center of rotation, e is the amount of eccentricity, P is the web entry point, Q is the web exit point, den is the distance between the center of rotation and the web entry point, and dex is the distance between the center of rotation and the web exit point

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Fig. 8

The angle θen is determined by using the cosine law on the triangle with sides den, R, and e

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Fig. 9

Figure to determine the coordinates of CR in the frame having the x-axis aligned with the line joining the geometric centers of the rollers (for computation of den using Eq. (38))

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Fig. 10

Experimental data with the unwind roll at different radii (line speed = 200 fpm)

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Fig. 11

Experimental data; line speed = 200 fpm, unwind radius = 6.375 in

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Fig. 12

First peak with different web velocities

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Fig. 13

Experiments results with out-of-round material roll (line speed = 200 fpm)

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Fig. 14

Comparison between experimental and computer simulated data

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