Need a double check - Real world problem - algebra and summation algebra

[MoofyPup]

Hey there, everyone.

I'm an engineer/designer who is working on an interesting little problem. I need to design a drum/reel/spool that can hold a length of filament/rope, L, in as small a space as possible.

So, I'm trying to minimize the interior and exterior diameters of the drum and the filament when completely wrapped together.

I did some calculations yesterday and I think I am taking the correct approach. I am not 100% confident in my math though.

I'm attaching my work in the links below. Hopefully, it's clear enough that this all makes sense. Let me know if it is not.

I'd love some feedback on either the method or the calculations.

Thanks,
Matt

https://dl.dropboxusercontent.com/u/96463620/scan/Drum_Wrap_page_1.pdf


https://dl.dropboxusercontent.com/u/96463620/scan/Drum_Wrap_page_2.pdf


https://dl.dropboxusercontent.com/u/96463620/scan/Drum_Wrap_page_3.pdf

 

[wjm11]

Hey there, everyone.

I'm an engineer/designer who is working on an interesting little problem. I need to design a drum/reel/spool that can hold a length of filament/rope, L, in as small a space as possible.

So, I'm trying to minimize the interior and exterior diameters of the drum and the filament when completely wrapped together.

I did some calculations yesterday and I think I am taking the correct approach. I am not 100% confident in my math though.

I'm attaching my work in the links below. Hopefully, it's clear enough that this all makes sense. Let me know if it is not.

I'd love some feedback on either the method or the calculations.

Thanks,
Matt

https://dl.dropboxusercontent.com/u/96463620/scan/Drum_Wrap_page_1.pdf


https://dl.dropboxusercontent.com/u/96463620/scan/Drum_Wrap_page_2.pdf


https://dl.dropboxusercontent.com/u/96463620/scan/Drum_Wrap_page_3.pdf

Matt,

Others here can likely assist you better than myself; however, I would offer some comments:

In your first pdf, part 1), you state W = H/d, where W is the number of wraps. This is true for the first layer. The second layer might have only W-1 wraps. View the "Circle stacking in a square" to see what I mean: http://en.wikipedia.org/wiki/Circle_packing_in_a_square . For example, you have shown 10 wraps equaling 1 layer in your sketch. The second layer might only have 9 wraps die to the filament dropping into the crevices of layer one.

Secondly, it appears you've changed your definition of "Wrap" between parts 1) and 2). The part 2 diameter formulas are for layers, not wraps as defined in part 1. I believe that error propagates through your calculations, but check to be sure.

Hope that helps.

 

[Deleted member 4993]

Following up on Wjm's note:

For the second layer - the wrap diameter increases by [√(3)/2]*d(where the cord diameter = d)