Yes, I'm aware of this post. Unfortunately, its author failed to discuss the solution from the Hyperloop paper. The author incorrectly assumed the tube is attached rigidly to the pillars and then proceeded to criticise this wrong assumption.
"But the real problem isn’t with the slip joints at the ends; it’s with the pillar supports along the way. Musk thinks the dampers, “tucked away inside each pylon,” need to accommodate only “the small length changes between pylons.” Wrong."
In other words, the original hyperloop white paper assumes that the movement at each pylon is small, and that the only large-scale movement is at the ends. But this is wrong. There is large-scale movement throughout the system. The thermal expansion motion reaches a maximum at the ends, but it doesn't magically go away as soon as you leave the station. If the tube at the station is moving 100m then the tube near the station is moving 100 meters minus epsilon.
The later parts of Hyperloop paper are more precise. I've already quoted "longitudinal slip for thermal expansion" from section 4.2.3. And, unfortunately, leancrew.com article author missed the proposed solution for the problem they spent many words talking about.
> If the tube at the station is moving 100m then the tube near the station is moving 100 meters minus epsilon.
Yes, of course. I agree that the problem of 100m thermal expansion is real and needs to be solved. And I'd like some arguments about why won't Hyperloop "longitudinal slip" solution work (where tube could freely move past the pillars).
You could argue that writing "pillars which constrain the tube in the vertical direction but allow longitudinal slip for thermal expansion" and "slip joints at stations…" is not detailed enough. I've concluded this "longitudinal slip" means some kind of linear bearing on top of each pillar, where straight tube could slowly move without constraints in the direction of the cumulative thermal expansion, while being constrained vertically.
v v
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tube
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v v
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tube
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Leancrew author missed the "longitudinal slip" part, because most of the article proves that thermal expansion accumulates. Of course it does! Thus the "longitudinal slip" solution.
Yes, Hyperloop paper could be more detailed and more clear. Indeed, it confused leancrew author.
Do you anyone knowledgable (in mechanical engineering?) who could comment on "longitudinal slip" solution? How feasible is it?
Indeed.
> How feasible is it?
Not even remotely feasible. You would need to have bearings at each column. Friction slip joints would introduce way too much stress.
Remember: the tube is not only moving hundreds of meters at the ends, it's moving >>1m along (very nearly) its entire length.
That's the kind of discussion I'd love to have! Of course, you need linear bearings [1] at each column. But what kind of bearings? Ball or roller bearings? [2] Plain/solid bearings? [3] Something else? What are the forces the tube would produce while expanding? What would be the linear expansion speed? What bearings would be able to sustain those speeds and forces? Would the tube and the pillars be able to sustain them? Do bearings with necessary parameters even exist? How expensive would they be? So on and so on.
Unfortunately, I haven't seen such discussion yet. I could try doing the calculations myself (starting at [4], for example). But I'm afraid I would make some silly newbie mistake in the process.
> Remember: the tube is not only moving hundreds of meters at the ends, it's moving >>1m along (very nearly) its entire length.
Yes, of course the tube is moving by hundreds of meters along most of its length. The questions are: how fast? with what load against the pillar?
[1]: https://en.wikipedia.org/wiki/Linear-motion_bearing
[2]: https://en.wikipedia.org/wiki/Ball_bearing
[3]: https://en.wikipedia.org/wiki/Plain_bearing
[4]: http://www.pbclinear.com/Blog/Calculating-PV-for-a-Plain-Bea...
But what the hell, I need to get my mind off politics, let's do the math.
The fatigue limit stress for steel is about 200MPa. You'd want a 50% safety margin, so let's limit it to 100MPa (which also make the math a little easier). The maximum stress is in the middle of the run, where you are pushing or pulling against N/2 pylons on both sides. Pylons are spaced every 30m. LA to SF is ~450 km. Let's call it 300 to make the math easier. So that's 30k pylons.
There are two different tube designs: passenger only, and passenger plus vehicle. Let's just do passenger only. The tube diameter is 2.23m with a wall thickness of 20-23mm, so the cross-sectional area of the steel is 0.14-0.16 m^2. Let's call it 0.15. So the maximum tensile load we can tolerate is 15 mega-newtons, or about a kilonewton of differential stress per pylon.
The mass of the tube segment borne by each pylon is 0.15 x 30m x 8000kg/m^3 (density of steel) = 36 tons. The weight of this segment is about 350 kilonewtons. So you'd need a coefficient of friction no greater than 1/350. Interestingly, that seems to be just about what you can get out of an industrial roller bearing, so there is no way that you can get anywhere close to that from a sliding bearing.
I wasn't able to find cost figures for roller bearing capable of handling a 36 ton load. I'm sure they exist. I'm equally sure that they're pretty frickin' expensive and require a lot of maintenance.
It looks like our estimation of "Hyperloop thermal expansion problem solution" feasibility now comes down to your last sentence:
> I wasn't able to find cost figures for roller bearing capable of handling a 36 ton load.
I don't know, railroad car steel wheel? Max load per axle for train cars: 26 - 40 ton. Searching on alibaba.com for "train car wheelset" and "train car bogie" never comes up with more than $2500 per piece. Maintenance seems doable, if we look at millions of train car running around the planet.
> These are the questions that would need to be answered by a hyperloop proponent, not me. The burden of proof is on the person making the claim. The only claim I'm making is that no one has proposed a solution. The two sentences that the Hyperloop white paper devotes to this issue is not a solution.
Yeah, I agree with that sentiment. Luckily, we are not the people making an actual practical decision on Hyperloop. Like you said, we are just trying to get our minds off politics.