A Problem With Rolling Duct Tape

[NHPhysics]

Hey guys, I'm new here, but I have been using this site for a while and just decided to make an account.

Here is my problem, I hope someone can help me out:

I have a duct tape rolling machine which takes larger duct tape rolls and rolls them into smaller rolls. Because the machine is not regulated based on length of tape, I am trying to put together an equation that will allow me to input a length and get an output of the diameter of the roll.

I started out by taking the number of layers on the newly rolled duct tape roll by doing this: (D_o-D_i)/(2t)

Where Do is the total diameter of the new roll, Di is the core diameter of the roll, and t is the thickness of the tape.

I then took the average circumferance of the layers by doing this: pi(D_o+D_i)/2

This comes out to be the length of the material is L=(pi/4t)(D_o^2 - D_i^2)

I am pretty sure that is correct.

I then rearanged the equation so that the output is Do.

D_0=√((4L*t)/pi+D_i^2 )

Now, I am pretty sure all of this is correct, if you would like you could check the math, but I ran into problems when I tried to implement the equation.

I'll give you an example and then explain what I think the problem is:

Let's say I have used the rolling machine to roll a 1.625 inches (41.275 mm) roll with a core diameter of .375 inches (9.525 mm). The thickness of the duct tape rolled (based on the number on the store bought roll) is .012 inches (.3048 mm).

If I input this information into the equation, I get 163.625 inches (4389.2 mm) Although this seems reasonable, when I actually unrolled the tape, I was getting about 185 or so inches which is about 5 yards.

What I believe is the problem is that when the rolling machine pulls the tape off of the larger roll and rolls it onto a smaller roll, the tape is stretched or compressed due to the friction of the *sticky part* (I forget the word) of the tape.

Is there anyway I could implement this into the equation? Also, if the equation is wrong, could you suggest another way of finding the diameter based on length?

Thank You! If you need any more information, just ask.

 

[xts]

You got about 12% accuracy - not that bad.
Your machine probably just uses stronger tension, which compresses the soft (foam) part of the tape.
If you need better accuracy than you got - repeat your experiment several times, to check what is a relation of tape length to a diameter of the roll made by your machine. If it is stable - just use these coefficients, rather than those resulting from measuring original rolls.
Good luck!

 

[NHPhysics]

I appreciate the relply. I have taken a roll of tape that I rolled from the store bought roll (the same as the one I mentioned earlier) I took the equation and solved for t. When I solved, I got 0.01061348869456011 inches (0.269582613 mm) When I use this for larger rolls, it works nearly perfectly, but my problems comes when I begin to look at rolls of 26 to 50 inches in length.

I have another roll of .75 inches in diameter with the same core diameter and thickness and when I used this new thickness, I got a length of about 31.22 inches. When I unrolled it, it was about 36.5 inches.

Any suggestions why it wouldn't work for smaller rolls?

 

[xts]

My ad hoc theory: for low diameters, the tape is much more bended than for large diameters. Its length is defined by outer, hard layer. But as it bends, the soft part gets compressed, probably it crimps a bit, increasing its effective thickness.

Other point - your machine has probably coinstant torque rather than constant tension. So for low diameters the tension may be too high to keep the geometry stable.

Anyway - I would recommend to make a series of experiments, leading to a table: length->diameter, then use that table, eventually with some quadratic interpolation between its points

 

[NHPhysics]

Thank you again, that seems like the best approach to this. I like your theory by the way. At first it was sort of confusing, but it actually makes a lot of sense.

There are generally 3 different types of rolls that I am dealing with 26 inch, 50 inch, and 180 inch, so I guess I'll make a table and then average out the thickness of the 26 and 50 inch rolls to use for smaller diameters. I don't think I will be adding new diameters anytime soon, and if I am an inch or so off for the smaller rolls, that's fine.