For a sense of scale, 60 mph travels one mile in 60 seconds, or 0.016 miles per second. Hyperloop goes 760 mph, which is 0.211 miles per second. (Jeez, that's fast.) So it only takes about 10 seconds for the car to traverse 2 miles, or in other words to traverse whatever effect such a motorized roller would have on the path of the vehicle. Like, if the roller causes the track to move "outwards," for example toward the ocean, by a little bit, then the pod is going to move "a little bit toward the ocean and then back" in 10 seconds.
That seems like it might be reasonable. But I suppose the question is whether a 7,700 lb pod travelling at 760 mph can tolerate, say, a 2-meter deviation in its trajectory in 10 seconds. Meaning, one meter toward the ocean, and then one meter back towards its original course. Can the pod bank against the walls by a displacement of 0.2m per second? If you hold out your arms to the length of a meter, 0.2m is pretty small, but it's not too small. And the pod would be going sideways by that amount per second.
1-meter deviation due to thermal expansion is extreme. 0.25m per mile might be a more realistic deviation. So that would be a 1-meter deviation total: 0.5m toward the ocean, then 0.5m back, which would still mean the pod needs to scrape against the inner wall by 0.1m per second.
I wonder what the limits on the banking are...
I agree, that would not work. I was suggesting that motorized rollers move the track longitudinally instead of sideways. Essentially they would take up the slack and actively push it to the end stations, counteracting the friction from the passive rollers. You could use thermometers and position sensors to model the stress on the tube.
That's an interesting idea, but are you sure that thermal expansion imparts enough force against 175 miles of steel in order to push it all the way to the station? 175 miles of steel would be incredibly heavy. It seems like the heating/cooling might even cause the steel to weaken enough such that, rather than rolling the tube towards the station, it would start to scrunch like an accordion. But I wish a structural physicist would step in with some calculations to prove whether or not 175 miles of steel on rollers could be pushed by thermal expansion without compromising hyperloop's structural stability.
Maybe we could ask /r/askscience?
EDIT: After re-reading your comment, I think you're saying the rollers would actually provide the force necessary to move 175 miles of steel, rather than letting the expansion passively push it. But those rollers would have to grip onto something in order to do that. They wouldn't be rollers so much as tank treads.
Hmm... Do you think that might work? Could 175 electric motors working in unison provide enough force to tank treads that bite into the concrete pillar below, and into the steel tube above, in order to successfully shift 175 miles of steel? And the system has to have temperature sensors every mile, and those sensors must not ever report incorrect readings or else the whole tube breaks. And those motors must never break down or run out of power...
I'm sounding pessimistic, but honestly I have no idea whether that would work.
EDIT 2: Ah, I think I've figured out why it can't work. It's not a straight tube. It bends all along the route. http://i.imgur.com/3TavjCY.png
So motorized rollers would actually need to bend the tube in order to successfully transmit the thermal expansion. In other words, when a piece of steel tubing is initially created, it's bent with a certain curvature. But if rollers are to transmit the expansion all the way to the station, the exact location of where the steel is bent needs to be dynamically defined at runtime, rather than statically assigned at compile time. Did I phrase that very clearly? It would help to be in-person, or maybe I could draw a picture if needed.
It doesn't seem like rollers could dynamically bend the steel all along the 175 mile track in order to carry the expansion.
Nope. That's why you may need to push it! :)
Assume the rollers have the Crr of a passenger train wheel (0.0020). If each section weighs 193 tonnes (23 mm thick by 2.23m diameter by 100' by 8 g/cm³), then each roller will provide ~400 N of friction. Over 175 miles (worst case) that works out to 3.56 meganewtons of resistance to motion.
For a thin-walled cylinder, buckling stress is modeled using the large deflection approximation[1] as:
σ_x = 0.238 Et/r
Where σ_x is the bucling pressure, E is the material's Young's modulus, t is the wall thickness, and r is the radius. Plugging in 190 GPa, 20 mm, and 1.125 m yields…
58 meganewtons, or about 16 times as strong as the predicted frictional force
I guess they may not need active control after all!
But what about curves? Won't it result in a lot of sideways force? Needs more calculations…
[1] The weaker of the two. http://www.dtic.mil/dtic/tr/fulltext/u2/a801283.pdf
Basically, what do you think about the fact that it's not a long, straight tube? The rollers would need to dynamically bend the steel tube in order to carry the expansion all the way to the station, wouldn't they? Is that possible?
Even if there aren't any rollers, the tube still needs to re-bend itself somehow. But how? That would mean steel curves near the endpoints would move a whopping 150 meters in total. So it seems like part of the tube manufactured as a curve in summer would need to become a straight length of pipe by winter.
First yours, since it's much cooler. The minimum curve radius is 3.67 km, or a length difference of 18 mm between the left and right side over the length of one pylon.
Assuming that all sections start as straight tubes, how much sideways force would that apply on the pylon? For that we need to calculate the "flexural stiffness"[1], which for a hollow cylinder is given as:
EI = E pi/4 (D⁴ - d⁴)
Where E is Young's modulus, and D and d are the outer and inner diameter.
Therefore the bending moment[2] M is:
M = EIκ = E pi/4 (D⁴ - d⁴)κ
Where κ is the curvature, equal to 1/r.
Finally, muddling through the equations on the bending moment article (and treating the tube as a 200' beam with a load in the center) yields:
f = M / (100')²/(200') [not sure about this one]
A whopping 300 tonnes•f of lateral force on the pylons! Probably doable, but it means if your tubes are constructed straight your pylons are going to be buttressed at a ~60° angle. Also, note that this only applies to the slip distance (~150m, or three pylons), since the tube sections well within the curve will never move enough to be in a non-curved section of track, so they can be built curved from the beginning.
Also you can vary the curvature smoothly[3], which can substantially reduce the forces involved. For example, if you eased into the curve over 30 pylons and it only moved by 3, the sideways force would be reduced by 10x (since the curvature is always "about right"). Further calculations here are left as an exercise to the interested reader. :D
As for my own objection… Even on the sharpest turns successive pylons will only vary from straight line by 0.15°, so the lateral forces caused by expansion and contraction shouldn't exceed 9 tonnes force. Piddly by comparison.
[1] http://www.explorelifeonearth.org/cursos/CB-18%20simple%20st...
[2] https://en.wikipedia.org/wiki/Bending_moment#Example
[3] i.e. the curve doesn't start suddenly, but gradually; this is how actual tracks/roads are laid out. https://en.wikipedia.org/wiki/Euler_spiral
TL;DR as long as you ease into the curves (which is good for comfort anyway), it looks totally doable even without motorized rollers.
Rail lines have more curves which can take up that slack, and the positional tolerances are much more forgiving. Because of its high speed operation the hyperloop is both incredibly straight and incredibly sensitive to kinking.
You make a great point! I didn't quite grok it before.
>Can that expansion really be pushed 175 miles north and 175 miles south simultaneously?
>Surely at some point the weight of the tube overcomes the steel's desire to expand, so the result would be the tube either scrunching up like an accordion or going in the lateral direction.
Agreed. I have a feeling some sort of active control (basically just motorized rollers) will be needed every couple of miles to prevent "sun kinks". I haven't done the math on that though. It might be sufficient that a large hollow tube shape is much more resistant to buckling than a rail shape.
"Sun kinks": http://www.climatecentral.org/news/climate-change-warp-railr...