Hyperloop

[Ben W]

It would be good if you could you provide some numbers for this assertion.

Thermal expansion would be almost entirely along the length of the tube which is its greatest strength. The tube would pass through the pylon bushings practically in a straight line. How did you arrive at the result that there would be a buckling danger?

The very best ball bearings have a coefficient of friction of about 0.001, which means that if you have a perfectly rigid object resting on one and you push horizontally from one side, 0.1% of the force will be absorbed by the bearing through friction, and 99.9% will propagate to the other side. In the case of the Hyperloop, assume that each pylon has a coefficient of friction of 0.001 (incredibly optimistic), and consider 100km of level straight hypertube track, with pylons every 30 meters. That's 3300 pylons. Now push with 100 pounds of force on one end of the 100km tube. Only 3 pounds of force will be transmitted all the way to the other side; 97 pounds of force will end up compressing the tube in between. If the residual 3 pounds of force is not enough to overcome the initial static friction at the far end of the tube, then all 100 pounds of force will compress the tube, and the far end will not move. In the case of an 800km Hyperloop, you could detonate a nuclear bomb at one end, and the other end would not move. (I don't think even shockwaves would get all the way through.)

So with some plausible configurations of temperatures across the hyperloop, you will have two widely spaced sections that are still, and a central region that wants to expand/contract. The force of the expansion/contraction has to be sufficient to propagate hundreds of kilometers through the pylons all the way to the still endpoints, and overcome the static friction there. That is a ridiculously huge amount of force. One of the major expenses of high-speed rail is securing it down tightly enough, every few feet, to overcome the thermal forces that want to buckle the rail. (In practice, high-speed rail is under incredible tension throughout.)

But we can't just put the Hyperloop under incredible tension. Its path is not perfectly straight; it has significant curves. When you have a curved section of tubing and apply compression or tension from the endpoints, it creates a gigantic lateral force in the middle. (Try it with a piece of spaghetti.) The Hyperloop alpha design is utterly unequipped to deal with such lateral forces. (That will pull strongly one way in the morning, and push equally strongly the other way in the evening.) This is quite unlike the transient lateral forces experienced in an earthquake, which the pylons are designed for. So the only solution is to reduce the tension/compression forces at a local level through expansion joints, or else to actively adjust the entire tube position to keep the tension/compression minimized.

Do you happen to have some numbers that contradict this? I would love to be wrong about it!

- - - Updated - - -

As before: the oil industry long ago learned this long-axis thermal expansion problem not to exist. I've got an 815-mile long steel tube 700 yards from where I live called the Trans-Alaska Pipeline.

40 years on and it hasn't buckled yet.

Have you seen the expansion joints in the Trans-Alaska Pipeline? Good luck going through them at 760 miles an hour ;-)

(caption for the photo: "The pipeline zigzags so it can shrink and expand as temperature changes, and also flex during an earthquake, without breaking.")

 

[RDoc]

The very best ball bearings have a coefficient of friction of about 0.001, which means that if you have a perfectly rigid object resting on one and you push horizontally from one side, 0.1% of the force will be absorbed by the bearing through friction, and 99.9% will propagate to the other side.

Well as a start, that's not what the coefficient of friction is. It's the ratio of the normal force to the frictional force, it has nothing to do with the force parallel to the surface.

 

[RDoc]

Airlocks

I was thinking about how the airlocks might work. It seems to me that the car would be in a much closer fitting tube to minimize the volume of air. Assuming the cargo/passenger car is 50m long, 2.5m wide, 1.6m high plus the suspension which I'm assuming is full length and .5m high, all surfaces with a clearance on the sides of 1 cm, that gives a total volume of just over 5 m3. With the pumps specified, that would allow the airlock to be evacuated in about 13 sec. It could be done faster if there were a vacuum reservoir, but it doesn't seem necessary, although if they could be synchronized, an outgoing airlock could partially pump down an incoming airlock. For the incoming car, the re-pressurization could be almost instantaneous of course, but it would still have to be pumped down for the next car.

Incidentally, the smaller tube idea wouldn't have to be just for the airlock. Once in urban areas where the velocity would be pretty low, the tube could be smaller (not as small as the airlock) since the aerodynamic issues would be far less than for the high speed lengths which might help with things like right of ways or harbor crossings.

 

[Ben W]

Well as a start, that's not what the coefficient of friction is. It's the ratio of the normal force to the frictional force, it has nothing to do with the force parallel to the surface.

Technically true; I typed too fast. But the frictional force does push back parallel to the surface, so it has essentially the same effect. Let's look at it again.

The Hyperloop (per tube) weighs about 1000 metric tonnes per kilometer, so the entire length of the tube weighs about 800,000 metric tons. (8x10^8 kg). While the tube is moving, the force required to overcome dynamic friction (and keep the pipeline moving steadily), assuming a coefficient of friction of 0.001, is about 8x10^9N * 0.001 = 8x10^6 N. The cross-sectional area of the steel tube is roughly 0.2 square meters. The compressive force on the hollow tube is thus 4x10^7N / m^2, or about 6000psi.

With stringers, the tube might be able to withstand this amount of compression without deformation. (Construction steel I-beams, a much more efficient shape, can handle about 50,000psi.) However, the loop will have curves in it with ~10km radius. Each 30m-spaced pylon subtends an arc of about 0.2 degrees on such a curve. With 6000psi compression on the tube, the lateral force on each pylon will be on the order of 10^5N (equivalent to 10 metric tons at 1g). A 30m section of hyperloop weighs about 30 metric tons, so the net effect is that the loop will have to be braced at a 20-degree angle one direction in the morning, and a 20-degree angle the other direction in the evening, to prevent it jumping off the track. The alpha pylon design doesn't appear capable of handling sustained lateral forces of this magnitude.

Obviously the Hyperloop engineers can fill in these numbers with actual ones, particularly the friction coefficient, and I hope they will. But also note that if the bearings fail on a few pylons, the net friction skyrockets. A few stuck pylons could take down the entire system.

Another consequence of the alpha design is that the tube would essentially have to be constructed from the midpoint outwards, because welding together large sections with independently moving endpoints would be nearly impossible. With expansion joints or active motion control, the loop could be assembled in many places at once.

Anyway, please do check my numbers. I'd love to be wrong about this. It's been a while since my systems engineering days

Also, let's hope the Hyperloop tube never breaks at the top of the Grapevine. 50 miles of hyperloop sliding unchecked downhill into Sylmar would not be a pretty sight!!

 

[rcml]

Would it be practical/cheaper to partially insulate the tube and then actively control its temperature (by heating it up at night) in order to minimize temperature differences?

 

[Ben W]

Would it be practical/cheaper to partially insulate the tube and then actively control its temperature (by heating it up at night) in order to minimize temperature differences?

I thought about that. It would have to be maintained at the maximum expected temperature (with a safety margin), which would take a lot of power. Probably much easier to actively move it along the track as it expands/contracts. Insulation may be a good idea anyway to minimize temperature swings, though it would have to be weatherproofed. Keeping the tube in the shade (under the solar panels) will be a helpful start.

 

[RDoc]

Technically true; I typed too fast. But the frictional force does push back parallel to the surface, so it has essentially the same effect. Let's look at it again.
...

Well, actually for bucking, a large diameter tube is far better than an I beam, orders of magnitude better. I beams are designed for lateral bending loads, not longitudinal compressive loads. Anyway, that's not a very good model since the tube is laterally supported all along it's length which greatly increases resistance to buckling.

I don't see how you got the calculation for pylon lateral force and bracing. The tubes go through the pylon essentially at right angles even in the curves, so the compression of the tube has no effect on the pylons. I think a better model would be to consider the increase in radius of the bend due to lengthening of tube, then calculate the force distributed along the length of the curve required to bend the tube back. For a 10km radius curve of 60 deg and 40C temp increase, that would increase the arc length by about 2.7m which would try to push the center of the curve out by about 1 m. The pylons would have to exert enough force distributed along the 2.5 km half length to bend the tube by less than a meter. That doesn't sound like a lot.

 

[Ben W]

Well, actually for bucking, a large diameter tube is far better than an I beam, orders of magnitude better. I beams are designed for lateral bending loads, not longitudinal compressive loads. Anyway, that's not a very good model since the tube is laterally supported all along it's length which greatly increases resistance to buckling.

I don't see how you got the calculation for pylon lateral force and bracing. The tubes go through the pylon essentially at right angles even in the curves, so the compression of the tube has no effect on the pylons. I think a better model would be to consider the increase in radius of the bend due to lengthening of tube, then calculate the force distributed along the length of the curve required to bend the tube back. For a 10km radius curve of 60 deg and 40C temp increase, that would increase the arc length by about 2.7m which would try to push the center of the curve out by about 1 m. The pylons would have to exert enough force distributed along the 2.5 km half length to bend the tube by less than a meter. That doesn't sound like a lot.

There's a big difference between "essentially" and "perfectly" ;-) Consider the macaroni-shaped section of tube between three pylons (A, B, C) in a 10km curve. The compressive force at each pylon is 8x10^6N, longitudinally along the direction of the tube. At pylons A and C, the compressive force points toward the center pylon B at a 0.2-degree angle. (10km = 1 radian = 58 degrees; 30m = 0.2 degrees). To balance these forces , the center pylon B must exert a compensating force that balances the lateral (0.2-degree) component of the compressive forces directed at it. The center pylon thus experiences a lateral force of 2 * 8x10^6N * sin (0.2°) = 56,000N, equivalent to about 6 metric tons at 1G, and must resist this force continuously.

And the curves would have to shift much more than a meter to accommodate the overall expansion, because they would have to absorb the expansion of the long adjacent straight sections as well. (The straight sections have nowhere to flex.)

 

[AudubonB]

Have you seen the expansion joints in the Trans-Alaska Pipeline? Good luck going through them at 760 miles an hour ;-)

(caption for the photo: "The pipeline zigzags so it can shrink and expand as temperature changes, and also flex during an earthquake, without breaking.")

Seen them many, many times. I take your point. However, those zigzags also don't exist in many places as well: both TAPS and many other pipelines go vast stretches as straight-line segments.

Let's try another approach: consider the need for the Hyperloop to possess a "perfect" tube: as far as I have understood the proposal, it is not necessary. If correct, then slip joints at whatever desired interval will easily provide whatever countermeasures may be needed for expansion and contraction. The ability of these joints to keep sufficient integrity to maintain the Low-P required is well within current technology.

 

[palmer_md]

it is not the forces of a normally travelling train that you need to calculate against. It is the forces required when all the moving trains (in 30 second intervals worst case) are required to deploy the emergency stop measures in order to bring the system to a quick orderly stop for some reason.

 

[RDoc]

There's a big difference between "essentially" and "perfectly" ;-) Consider the macaroni-shaped section of tube between three pylons (A, B, C) in a 10km curve. The compressive force at each pylon is 8x10^6N, longitudinally along the direction of the tube. At pylons A and C, the compressive force points toward the center pylon B at a 0.2-degree angle. (10km = 1 radian = 58 degrees; 30m = 0.2 degrees). To balance these forces , the center pylon B must exert a compensating force that balances the lateral (0.2-degree) component of the compressive forces directed at it. The center pylon thus experiences a lateral force of 2 * 8x10^6N * sin (0.2°) = 56,000N, equivalent to about 6 metric tons at 1G, and must resist this force continuously.

And the curves would have to shift much more than a meter to accommodate the overall expansion, because they would have to absorb the expansion of the long adjacent straight sections as well. (The straight sections have nowhere to flex.)

Good point, the pylons would have to essentially turn the force along the tube. Here is my calculation assuming a 40C temp rise:

Assuming a pylon bearing with a rolling resistance of .005 and a total tube weight of 1.3MT, the compression midway down the tube would be about 3.2 MN at the center, falling off to near zero as you go toward both the ends, so 6.4 MN at the very center for the force from both sides. Suppose there were a 60 degree turn at the center with a radius of 10000m. That would be a length of 5.2km and 175 pylons. Each pylon would have to turn the force by about .003 radians (.524/175), so the turning force would be about 38000N. Now compare that to the force needed to resist a car going by at .5g lateral acceleration, about 127000N. The force required to turn the expansion force is certainly not minor as you point out, but in the worst case, it's about 30% of that when a car went by. With a traversing car, the force vector would be about 900000N at an angle of 2.5 degrees. With no car it would be about 611000 at an angle of 3.6 degrees.

The long expansion of the straight section doesn't have to be zeroed out, all that's needed is to bend the tube around the curve so the increased length is passed through the curve to the other side, but the above case would cover a tangential start to the curve as it applies to anywhere on the curve. In practice it would be much less at the start and end of the curve since the they wouldn't start tangentially as that would impose an instantaneous acceleration, There would be a lead in of gradually decreasing radius to get to the final minimum radius to minimize the jerk (second derivative).

WRT buckling: The greatest danger would be at the midpoint of the line as the force falls off linearly towards the ends. I'm not going to attempt to calculate the buckling limit, but would point out that there are two very large diameter tubes welded together and supported in 2 dimensions pretty continuously. In the worst case, the tube will compress about 4mm over the length of a section which seems pretty minor to me, so while it might be a problem, it's certainly not an obvious one IMHO. If you want to try calculating the buckling limit I'd be quite interested.

 

[DonPedro]

Buckling would have to be done by FE analyses to get interesting results, I think. Otherwise you are going to assume a perfect cylinder, which is not going to be the case. Gravity, ambient pressure and potentially operating loads will lower the buckling limit significantly.

I think that also the dynamic behavior of large sections of the tube could be interesting. You could get some sort of ringing/standing waves and other dynamic phenomena. Probably not a big deal to handle, but certainly has to be checked during design.

 

[IceWendigo]

"You Must Construct Additional Pylons!"

 

[ElSupreme]

"You Must Construct Additional Pylons!"

If they used Overlords they wouldn't have to worry about land costs, or earthquakes!

I hope they call the ends Depots.

 

[vfx]

For stopping and keeping the tube airlocked, the car slows by diverting the air from the air bearings to an inflatable bladder at the back of the car to seal the tube.

 

[rolosrevenge]

If they used Overlords they wouldn't have to worry about land costs, or earthquakes!

I hope they call the ends Depots.

Why all this talk about pylons? A nydus canal is basically hyperloop, with the advantage of being biodegradable.

 

[Ben W]

Good point, the pylons would have to essentially turn the force along the tube. Here is my calculation assuming a 40C temp rise:

Assuming a pylon bearing with a rolling resistance of .005 and a total tube weight of 1.3MT, the compression midway down the tube would be about 3.2 MN at the center, falling off to near zero as you go toward both the ends, so 6.4 MN at the very center for the force from both sides. Suppose there were a 60 degree turn at the center with a radius of 10000m. That would be a length of 5.2km and 175 pylons. Each pylon would have to turn the force by about .003 radians (.524/175), so the turning force would be about 38000N. Now compare that to the force needed to resist a car going by at .5g lateral acceleration, about 127000N. The force required to turn the expansion force is certainly not minor as you point out, but in the worst case, it's about 30% of that when a car went by. With a traversing car, the force vector would be about 900000N at an angle of 2.5 degrees. With no car it would be about 611000 at an angle of 3.6 degrees.

Apples and oranges, I think. The car at full speed passes each pylon in less than a tenth of a second, so the force it imparts is essentially a near-instantaneous impulse.It would be interesting to calculate at what point in advance of the capsule arriving, the pylon begins to "feel" the lateral force of the oncoming capsule. The system is fairly rigid after all, so it could be that the lateral force of the capsule is effectively spread over multiple pylons at once. The maximum instantaneous force on a single pylon (from the capsule) would then be considerably less than 900000N. Also, each pylon only needs to "course-correct" the capsule by about 5cm laterally relative to its straight-line trajectory from the previous pylon.

The long expansion of the straight section doesn't have to be zeroed out, all that's needed is to bend the tube around the curve so the increased length is passed through the curve to the other side, but the above case would cover a tangential start to the curve as it applies to anywhere on the curve. In practice it would be much less at the start and end of the curve since the they wouldn't start tangentially as that would impose an instantaneous acceleration, There would be a lead in of gradually decreasing radius to get to the final minimum radius to minimize the jerk (second derivative).

"all that's needed is to bend the tube around the curve"... so then it just passes the extra length along to the next curve? That begs the question. If there are 20 miles of curves and 80 miles of straight sections, then each curve has to accommodate 5x its own length worth of expansion/contraction if the straight sections can't. I do agree that actively assisting the expansion/contraction of the tube is one way to work around the thermal expansion issues.

WRT buckling: The greatest danger would be at the midpoint of the line as the force falls off linearly towards the ends. I'm not going to attempt to calculate the buckling limit, but would point out that there are two very large diameter tubes welded together and supported in 2 dimensions pretty continuously. In the worst case, the tube will compress about 4mm over the length of a section which seems pretty minor to me, so while it might be a problem, it's certainly not an obvious one IMHO. If you want to try calculating the buckling limit I'd be quite interested.

The tube would probably remain reasonably tubular, though it's not clear what tolerances the system has for the tube being deformed. The lateral forces on the pylons (and the tendency for the tube to want to jump the tracks) are a much more significant problem I think.

 

[RDoc]

"all that's needed is to bend the tube around the curve"... so then it just passes the extra length along to the next curve? That begs the question. If there are 20 miles of curves and 80 miles of straight sections, then each curve has to accommodate 5x its own length worth of expansion/contraction if the straight sections can't. I do agree that actively assisting the expansion/contraction of the tube is one way to work around the thermal expansion issues.

That's not the design. All the expansion is taken up at the two ends, they move by a maximum of about 320 m at each end assuming 80C temp difference. None of the thermal expansion is taken up anywhere in the system except at the ends.

 

[Ben W]

it is not the forces of a normally travelling train that you need to calculate against. It is the forces required when all the moving trains (in 30 second intervals worst case) are required to deploy the emergency stop measures in order to bring the system to a quick orderly stop for some reason.

These forces are purely longitudinal, and small relative to the steady-state compression due to thermal expansion. The thermally-induced compression is about 8000000N, and the additional compression caused by a 26000kg capsule decelerating at 0.5g is only about 125000N, or 1.5% of that. In a worst-case scenario, if the 50 pods (at 30-sec intervals) in the first half of the tube immediately decelerate, and the 50 pods in the second half of their journey keep going, then the compression on the midpoint of the tube could rise about 75% for the 60 seconds it takes all 50 capsules to brake to a stop. If this is problematic, then the 50 pods in the second half could brake as well to offset the central compression. Let's also look at conservation of momentum. The 100 pods weigh a combined 2600 metric tons; the tube overall weighs 800000 metric tons. With an average capsule speed of 800kph, all braking to a halt, the entire hypertube will then want to start sliding toward the destination at 2.6kph. Without active control of the hypertube movement along the pylons, this could be a serious problem.

- - - Updated - - -

That's not the design. All the expansion is taken up at the two ends, they move by a maximum of about 320 m at each end assuming 80C temp difference. None of the thermal expansion is taken up anywhere in the system except at the ends.

Right. I was referring to your comment: "For a 10km radius curve of 60 deg and 40C temp increase, that would increase the arc length by about 2.7m which would try to push the center of the curve out by about 1 m. The pylons would have to exert enough force distributed along the 2.5 km half length to bend the tube by less than a meter." What I'm getting at is that the deviation you have to compensate for (through lateral force on the pylons) is effectively 5 meters, not 1 meter, since the pylons along the curves have to compensate for expansion on the much longer straight sections as well. If you had an 800km straight hyperloop with a single 10-km-radius curve in the middle, the tube would "want" to jump 100m off the tracks there every day due to thermal expansion.

 

[IceWendigo]

In curves would the pod rotate so that the lateral-G is felt as coming from the bottom of the seat?

If you are sideways and feel 1G comming from what appears to be down, is it relatively smooth or is there discomfort like you are being tossed around?


If so, how much of a curve could be built (in places where a line is not possible) so that you would get tossed around less than a roller coaster?