Numbers on Staggering, and new Buttressing Idea

 Qualitative drawings of the friction-buffers sometimes miss an important details, and I believe this is true of the final illustration of my last post. Here, I hope to simply drive this train to thought to its final conclusion in broad terms.

Specific Numerical Methods for the "Ruler" Shape

Doing this took a spreadsheet (intent is that it is publicly visible).

Let's start out writing down some criteria. I was inconsistently using an incorrect criteria that "radius of sheet termination is where its velocity is within +/- 3 m/s from its neighbor". This is not correct, so I'll write the correct criteria in a bullet point:

• constraint: All sheets at all point must move +/- 3 m/s relative to all its neighbors, at all points.
• consequence: As you widen the opening of one sheet, the relative velocity of its revealed neighbors must be +/- 3 m/s
• for some sheets: this is the same as being within 3 m/s of its neighbor
• for other sheets: they must divide up the flow between sheet numbers far away from its position

These are stranger than they look, and give the reason for the "ruler" shape I outlined before. However, I only did that qualitatively so I'll just do the same in a quantitative since here. You can refer to the spreadsheet for numbers, but for completeness I need to give the algorithm because it's not as simple as an equation. Start with the scenarios of interest:

• 125 meter radius, 8 sheets
• 250 meter radius, 16 sheets
• 500 meter radius, 32 sheets

These are chosen because they satisfy the 2^N number, and also because they're close enough to previously published numbers such as to be presumed acceptable in terms of power consumption. Now, for each scenario, for each sheet:

• Calculate the statically-obvious values for each sheet (1-8, 1-16, 1-32)
• The radius of the sheet in the middle cross section
• The angular velocity of the sheet
• For the 0th and Nth entry you can enter in the tube's angular velocity and 0 angular velocity
• Go to the N/2th sheet
• This sheet will have the 0th and Nth sheet as it's revealed neighbors, so those are entered as the 2 relevant neighbor angular velocities
• Divide each remaining numerical segment into 2, and fill in revealed neighbors for its apparent neighbor
• Example: the N/4th sheet has 0th and N/2th neighbor as neighbors
• Repeat this process to fill in the rest of all the neighbor angular velocities
• From those revealed neighbor angular velocities, find the point that the neighbors are moving 3 m/s relative to each other, and that's how wide the sheet's hole can be 

This is very weird, but it still produces the same "ruler" shape. Here, I report the height from the floor which corresponds well to the illustrations from my last post.

R=125 meter scenario

R=250 meter scenario

R=500 meter scenario

In all these cases you still get the "ruler" shape... however, even for the smallest "ticks" on the ruler, there is still significant height from the floor, about half-way. This just seems to be how the math falls out, you can reason out why if you try hard.

Material Reduction from Staggering

Comparing to the taper-nested reference, we use less material, because the only real observation here is that some openings for some sheets can be made larger (thus, using less material). So let's slap an improvement number on those.

		Sheet area
R	sheets	no staggering (km3)	staggering (km3)	improvement
125	8	2.850	2.916	2.25%
250	16	21.213	21.707	2.27%
500	32	162.888	166.682	2.28%

It's as well-established prior observation on this blog (almost from the first post) that you have to use more dividers (and thus divider area) as you go to bigger radii. I didn't adjust for this, meaning that 500 meters gives bigger livable area in addition to needing more supporting material, so keep that in mind.

The "no staggering" case presumes that all sheets have a hole which matches the size of the access hole exactly. Then there are a number of route geometry work to sum up all the areas of all the sheets, and that's what I did for both the "no staggering" case and with the data shown in the plots.

It is kind of interesting that there is more improvement for larger sizes, but in absolute terms it is marginal in all cases. This will bring me into the economic argument later.

Decent Illustrations of Staggering with Curvature

With this, I can try again with the prior paradigm to draw the friction buffers. I still don't aim for exact numbers, but wanted to get something representative. As you might have guessed, doing anything other than the N=8 scenario would be tedious, so all drawings would map to the R=125 meter scenario.

This gives us a couple of things we were looking for, with some drawbacks.

• Good:
• Friction buffer sheets legitimately have outward curvature everywhere, without any dramatic conceit like a rigid floor integrated (last post)
• The access window is acceptably small-looking, it's an single circle as opposed to a long tunnel, should flow air & people & goods well
• Bad:
• Space inefficient, as you can see big open spaces inside some sheets, mostly an artifact of striving for an exactly circular shape for the inward curve
• The connection angles are terrible at several connection points... I can barely tell what's happening in some places

How can we clean this up? Look back at the plots, you see that sheets 0 (the hull), 4, and 8 all have to come in almost exactly up to the access window. We have not yet taken advantage of the taper-nested idea, so we will do that. However, we want to do it in a way that maintains the direction of outward curvature and gives nice nearly-right connection angles which is important for the air ingress management system.

But visually, this is the thing that didn't work because of the circles. So we will only partially relax that. We will make the mentioned sheet number, 4 (the hull is already straight), and like the hull, straighten it out. This will have a partially-straight segment (might have to be rigid) but the rest of those layers outside of this area will still work like the other dividers.

Doing this, with an eye to connection angle, we'll also return to the slanted hull shape. Here we go:

This is fanciful. To allow the new straight segment we had to re-introduce a long access tunnel. Tradeoffs:

• Good:
• Compared to last, we have good solid connection angles
• Bad (mostly everything else):
• Re-introduced a long access tunnel
• Even less space-efficient

Practical Considerations Leading to Buttressing Design

At this point, we will lean into rigidity, but with a more clear-eyed vision of where we'll go. But first let's better formalize a requirement instead of the hand-waving "connection angle" mention.

• Requirement: we are able to hold the friction-dividers in place
• Radially:
• wobble can be partially dampened by contact near the end cap
• non-periodic bulk forces, like air currents, micro-gravity forces, can be resisted by the edge of the friction-buffer pushing against something
• Axially:
• here too, bulk forces must be resisted by giving it something to push against

If you look at past sketches, it seems clear that taper-zero could have a problem, since it's balloons pushing against balloons. The case of taper-nested seems like this requirement isn't a problem, but you have to pull up the numbers to see that most sheets have to terminate very near to the access window... which isn't necessarily a problem for radial stabilization, but axially there's nothing to push against. It's not obvious that this is a problem but it could be a problem.

To accomplish the ideal for all the stated goals, I'll double-down on the rigid segments for certain flag-leader sheets, and further sacrifice the circular shape. I'll mark the rigid parts with a bold line. This still uses smooth circular curvature in places, but also demands a custom shape.

Instead of a long access tunnel, we will simply angle the sheet N=4 inward. By keeping it at an angle, we still satisfy both the radial & axial push requirement.

Trade-offs here:

• Good:
• Minimal clutter around access window
• Vastly more space efficient, with the theoretical "triangle" wedge shape starting to emerge
• Able to buttress the friction buffers against movement in all directions
• Bad:
• The rigid segment (only for N=4) must now move at a different speed, this requires a stronger direct mechanical coupling over this moving joint for the mechanical forces involved

On the bad, I want to be clear this isn't just a different speed for the outside air (which is stationary), but is between the inner tube and the outside air. So it is offering a "checkpoint" kind of speed so that the slower sheets can push against it and the faster sheets can push against the hull.

If you refer back to the graphs, you can logically predict that very large sizes will have more checkpoint friction-buffers, all requiring mechanical connection and rigid parts. For extreme sizes, you might require not only several checkpoint sheets, but a tree-like network of these extending out at +45 degree and -45 degree angles, every one at a unique speed.

I'll end it here, but want to add one more claim that the benefits of de-cluttering the access window are likely to be greater at greater sizes. Because at those sizes you are managing more sheets with roughly the same spacing between them. So balancing mechanical thrust on those sheets precisely becomes a much bigger deal. In short, when you get big enough you will want buttressing for the friction-buffer sheets. That seems reasonable and intuitive.

My sketch here is awful, they would all be smooth surfaces in practice. 3-D printed, I'm sure. That's likely overkill for small tubes, but this is at least a path to a scalable design that I can endorse, and that's important.