Maxcess International -Fife, MAGPOWR, and Tidland present Converting Blog

How I got to the answers:

1. I sketched out a bowed roller geometry thinking 'I can do this...'

2. Gave that up and went to my old OK-State U 'Web Handling Applications' seminar notes and found an impressive set of triple nomagraphs and how to figure spreading using dimensionless numbers Kc, Ke, and Ks. This was an approach from the pre-Excel days (a.k.a. the slide rule days).

3. Next to Ron Swanson's landmark IWEB '97 paper on spreader and anti-wrinkle roller (where Swanson cops out and refers poor readers to Delahoussaye '93).

4. On to Delahoussaye '93 where there are NO EQUATIONS SHOWN, just finite element model outputs (I should have known better).

5. Wait, I should have tried on of Dave Roisum's books, such as "Mechanics of Web Handling"..., but the equations aren't there either.

6. So back to my sketches. Worked up some similar triangles from measuring bow, face width, and entry spans. But I wasn't sure about these.

7. Sent my first draft equations to the old WH contact to look over my work. Good, but not exact agreement with using the OSU nomagraphs.

8. Sent my notes also to an old comrade (names are withheld to protect the helpful). He pointed out that the CRC handbook had the equations for radius from arc length and 'bow', not quite what I'd derived, but surprisingly close (though he noted my triangles may not have been similar).

9. OK, back to the sketch pad, figured out how the CRC equation is derived, went on to use the corrected bow radius calcs to estimate spreading in the entry span and spreading on the roller. Whew. I'm done.

The result:

Bowed Roller Radius of Curvature:
given chord length F, bow depth of the chord B, bowed roller radius of curvature Rc (from CRC) = F^2/8B

Bowed Roller Pre-Roller Spreading:
given also entry span length L, pre-roller spreading is nearly is a maximum of 8BL/F^2 (multiple this by 100 for percent spreading). This will reduce slightly if you account for reduced effective radius from the entry wrap orientation. This assumes straight line tracking from the upstream roller, but since web's a beam-like, a better spreading estimate would be 2/3rds of this or 5.33(BL/F^2)

Bowed Roller On-Roller Spreading:
given also bowed roller radius Rr, and the wrap angle of the bowed roller, alpha (in radians, and assuming this angle splits the bow plane), the lateral strain while on the roller will max out at Rroller*tan(alpha)/Rbow, which will often never be fully for stiffer webs since there may not be enough friction available to create this amount of lateral strain. (Substituing in the Rbow equation, the strain on the roller is 8*B*Rroller*tan(alpha)/F^2.

Both of these last equations are simplified, but mostly due to standard small angle sin(beta) = tan(beta) = beta, causing little damage for angles less than 30, even 60 degrees.

Combined, the total spreading pre-roller and on-roller will be (5.33L + 8*Rr*tan(alpha))(B/F^2) where the on-roller spreading will usually be 5-10% of the total spreading.

Fun with algebra, huh?

tjw