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### What-If 0043: "Train Loop"

Posted: Tue Apr 30, 2013 1:30 pm UTC
Could a high-speed train run through a vertical loop, like a rollercoaster, with the passengers staying comfortable?

Good thing the makers of the first rollercoaster hadn't read this - would have been a shame if they got dissuaded!

### Re: What-If 0043: "Train Loop"

Posted: Tue Apr 30, 2013 1:34 pm UTC
I'm just not quite seeing the reasoning on this one.
Maybe I need coffee.
I'll go get some coffee.

### Re: What-If 0043: "Train Loop"

Posted: Tue Apr 30, 2013 1:50 pm UTC
So what if we used maglev, mined the loop inside a mountain (rather hard engineering, but might be lot cheaper than the support structures needed for other solutions) and as extra bonus if we pressurized the train and pumped the tunnel to vacuum we lose air resistance, on other hand we will also lose the jet engine and downforce...

Nah, that doesn't fix it either...

### Re: What-If 0043: "Train Loop"

Posted: Tue Apr 30, 2013 1:53 pm UTC
Sadly, they never took off. Fortunately, they never took off.

This, and "the little train that let everyone down" made me laugh out loud.

### Re: What-If 0043: "Train Loop"

Posted: Tue Apr 30, 2013 1:55 pm UTC
ctdonath wrote:I'm just not quite seeing the reasoning on this one.
Maybe I need coffee.
I'll go get some coffee.

Referring to the picture in Vroomfundel's post:

On the one hand, if the loop is too tall then the train won't make it to the top. It will slow down too much on the way up.

On the other hand, if the loop is too tight then the train and/or track will be damaged by g-forces (it doesn't take much g-force to damage a train track). And if there's one place you don't want to be when your vehicle is damaged, it's 900m above the ground with a horizontal speed of 400kph.

Randall's diagrams are fail-safe: when the plans don't work, the train just fails to make it to the top, rather than destroying the loop and killing everyone on board (and also everyone who the train lands on). This is uncharacteristic of Randall "Wipe out civilisation with a hairdryer" Monroe - I think he's losing his touch .

### Re: What-If 0043: "Train Loop"

Posted: Tue Apr 30, 2013 2:21 pm UTC
Yeah, I found this one of the less satisfying What-Ifs. I couldn't understand which limiting factors he was assuming or how the maths worked.

I get that a given maximum speed puts an upper bound on the loop size (otherwise the train will slide backwards/fall off at the top), and why a given speed and given maximum G-force tolerance put a lower bound on the loop size (otherwise the passengers/bogies/whatever will get crushed), but it's not at all clear why a fast train on a large loop might be okay while a slow train on a small loop wouldn't.

### Re: What-If 0043: "Train Loop"

Posted: Tue Apr 30, 2013 2:27 pm UTC
One challenge to the Joe Biden idea: there are no mountains between Delaware and D.C., unless Joe is willing to reroute the Amtrak eastern route to the Appalachians.

edit -- Thinking about it some more, an alternate idea: assuming you'd be drilling underground to place the loop anyway, as long as you've got some mechanism for pumping water out, you're not necessarily limited by the sea level. So, drill underground from ground level without a mountain, and use the energy you'd gain by going down to the bottom of the loop to help give you the energy to go over the loop. You can still add a jet engine to provide thrust to account for any energy lost to friction, but as long as the top of the loop is below ground level, you shouldn't need to add more energy other than to account for friction.

### Re: What-If 0043: "Train Loop"

Posted: Tue Apr 30, 2013 2:36 pm UTC
Maelin wrote:it's not at all clear why a fast train on a large loop might be okay while a slow train on a small loop wouldn't.

I think it's only because the trains don't bend vertically.

### Re: What-If 0043: "Train Loop"

Posted: Tue Apr 30, 2013 2:56 pm UTC
A roller coaster train weights a few tons spread out across maybe 40 feet and 16 axles, this train would have 8 cars and be fairly flexible. The El 17 weighs 64 tons, is 53 feet long and isn't flexible at all. The length of the train or the car is going to factor into how tight of a loop you can handle. Too tight and you're going to end up losing contact between the rail and all of your wheels which isn't going to be good.

The other thing to consider is that a heavy train traveling at high speeds is going to impact a tight loop as if that loop was more of a wall than anything else. Which is why the design cost of such a loop is going to be insane. Every bit of that loop is going to be subjected to force of a high speed passenger train trying to go through the track. Which is avoided by making the loop huge.

For a look at a roller coaster loop gone wrong: http://en.wikipedia.org/wiki/Son_of_beast

### Re: What-If 0043: "Train Loop"

Posted: Tue Apr 30, 2013 3:05 pm UTC
I'm disappointed that he gives numbers for various possible, impossible, limiting, 'great' loops, but then the final answer is just 'medium-sized' 'something like this' and 'fast'. Actually, we don't even get any numbers for speeds except for '35% faster'. I'd rather know a bit more about looping trains and less about where Joe Biden eats his sandwiches.

### Re: What-If 0043: "Train Loop"

Posted: Tue Apr 30, 2013 3:21 pm UTC
There's another alternative, albeit rather boring (and unlikely to be implemented): bolt everything down inside the train, put the passengers into rigid 5 or 7-point restraint systems, and replace the train wheels and tracks with an over/under dual wheel system. Oh, and make it a Cog Railway . Now the train can't leave the track even at 0.01 km/hr, and the cog drive guarantees forward motion regardless of orientation.

hah.

### Re: What-If 0043: "Train Loop"

Posted: Tue Apr 30, 2013 3:31 pm UTC
When he was toying with the restrictions why didn't he just modify the train as well? make it a lightweight high speed with short cabins so that it could go faster and turn tighter.

### Re: What-If 0043: "Train Loop"

Posted: Tue Apr 30, 2013 3:42 pm UTC

It can levitate ontop of the rails, thus it can also hang suspended upon rails. Vortex pinning 4 laif.

Its method of propulsion is also independent of contact with the rails and can reach staggeringly high speeds(, although that would kill the passengers, but its nice to know.)

Now the limitations of the system becomes:

1. Is the train strong enough to propel itself up a vertical surface or at least; does it have enough thrust vs original momentum to get to a point further up where it is strong enough
2. Is the loop itself strong enough to absorb the forces exerted on it?

Number 1 can be solved by slimming down the train some (put the propulsion system in the rails, supercooled magnets on the train, some refrigerant, all else is just a seat and a helmet for the passanger(s)), which, incidentally, also solves problem number 2.

No?

### Re: What-If 0043: "Train Loop"

Posted: Tue Apr 30, 2013 5:32 pm UTC
I think this comic is wrong.

Consider a circular loop with radius r, speed at bottom of loop v0, and speed at top of loop v1. Assume train of mass m is just coasting, i.e. no engines. By conservation of energy we have
(1/2)m v0^2 = (1/2)m v1^2 + mg(2r).
Multiply both sides by 2/(mr), yielding
v0^2 / r = v1^2 / r + 4 g.
Centripetal acceleration is velocity-squared over radius, so this means the centripetal acceleration at the top is 4 gees less than at the bottom. So if we enter at 5 gees centripetal acceleration the centripetal acceleration at the top is 1 gee, and hence the train stays on the track (barely). The gee-force felt at the bottom would be 6 gees (extra g from gravity), which is uncomfortable but perfectly survivable.

### Re: What-If 0043: "Train Loop"

Posted: Tue Apr 30, 2013 6:11 pm UTC
Whoever does manage to build one will need to include one of those restriction signs: "You Must Be At Least This Crazy/Brave/Just-Plain-Dumb" To Ride"

### Re: What-If 0043: "Train Loop"

Posted: Tue Apr 30, 2013 6:19 pm UTC
rhomboidal wrote:Whoever does manage to build one will need to include one of those restriction signs: "You Must Be At Least This Crazy/Brave/Just-Plain-Dumb" To Ride"

an unmatched quote sign creates tension. "

### Re: What-If 0043: "Train Loop"

Posted: Tue Apr 30, 2013 8:06 pm UTC
blueberry42 wrote:I think this comic is wrong.

Consider a circular loop with radius r, speed at bottom of loop v0, and speed at top of loop v1. Assume train of mass m is just coasting, i.e. no engines. By conservation of energy we have
(1/2)m v02 = (1/2)m v12 + mg(2r).
Multiply both sides by 2/(mr), yielding
v02 / r = v12 / r + 4 g.
Centripetal acceleration is velocity-squared over radius, so this means the centripetal acceleration at the top is 4 gees less than at the bottom. So if we enter at 5 gees centripetal acceleration the centripetal acceleration at the top is 1 gee, and hence the train stays on the track (barely). The gee-force felt at the bottom would be 6 gees (extra g from gravity), which is uncomfortable but perfectly survivable.

Indeed. Randall has not done his homework here.

You can do a loop at any speed.

### Re: What-If 0043: "Train Loop"

Posted: Tue Apr 30, 2013 8:50 pm UTC
Also, he didn't even bother to name the geometry of the classic roller coaster loop--the clothoid.

### Re: What-If 0043: "Train Loop"

Posted: Tue Apr 30, 2013 9:44 pm UTC

### Re: What-If 0043: "Train Loop"

Posted: Tue Apr 30, 2013 11:06 pm UTC
High-level backer you say?

Well \$#!+ apparently my first post can't contain a link. Google yourself some Hyperloop.

### Re: What-If 0043: "Train Loop"

Posted: Wed May 01, 2013 12:58 am UTC
This one is not very well done.
I would also like to see the work on why 9-15 gees are experienced going in a 70 meter track.

### Re: What-If 0043: "Train Loop"

Posted: Wed May 01, 2013 1:14 am UTC
blueberry42 wrote:I think this comic is wrong.

Consider a circular loop with radius r, speed at bottom of loop v0, and speed at top of loop v1. Assume train of mass m is just coasting, i.e. no engines. By conservation of energy we have
(1/2)m v0^2 = (1/2)m v1^2 + mg(2r).
Multiply both sides by 2/(mr), yielding
v0^2 / r = v1^2 / r + 4 g.
Centripetal acceleration is velocity-squared over radius, so this means the centripetal acceleration at the top is 4 gees less than at the bottom. So if we enter at 5 gees centripetal acceleration the centripetal acceleration at the top is 1 gee, and hence the train stays on the track (barely). The gee-force felt at the bottom would be 6 gees (extra g from gravity), which is uncomfortable but perfectly survivable.

How does that change what we see in the graphics - I mean, what's wrong with the interpretations, and what happens instead?

### Re: What-If 0043: "Train Loop"

Posted: Wed May 01, 2013 7:15 am UTC
Angelastic wrote:I'm disappointed that he gives numbers for various possible, impossible, limiting, 'great' loops, but then the final answer is just 'medium-sized' 'something like this' and 'fast'. Actually, we don't even get any numbers for speeds except for '35% faster'. I'd rather know a bit more about looping trains and less about where Joe Biden eats his sandwiches.

Another issue: Why limit ourselves to just one jet engine? If we are already ignoring the vertical clearance limits for rail tunnels, then we could mount three jet engines onto the locomotive--one dorsal, one port, and one starboard.

(I would link the BHG "What if we used more power" pic here but I forgot the URL).

AFAIK, we should be able to boost the thrust up until the train gets near its critical Mach number (probably near Mach 0.8 or so).

### Re: What-If 0043: "Train Loop"

Posted: Wed May 01, 2013 8:51 am UTC
Most rollercoasters use some potential energy -> kinetic energy conversion to make a loop: i.e. they go downhill more than they go up, so attaining the speed should not be difficult if you drop into an underground loop (as suggested earlier). Doing this would also improve the prospects of the loop surviving the experience, as the bedrock would make a reasonably good structure to build on. Add in a clothoid loop shape and lightweight, short train cars and it could be made to work: you only hit high gees at the start and end of the loop, and four g should be survivable for most seated passengers.
The real problem would be making a train that was light enough to do the loop, yet also include the motors to get it from A to B as part of a useful transit system. Oh, and the difficulty in funding the whole thing with ticket revenue from the small number of passengers you could carry on the train: strap-hanging is probably not an option...

### Re: What-If 0043: "Train Loop"

Posted: Wed May 01, 2013 11:14 am UTC
I'm quite the roller coaster enthusiast (although, shockingly, I haven't been to any park in over a year! :'( ) so this What-If was exciting for me. As soon as I saw the circular loop design, I had to scroll down to the bottom of the page to check if Randall had ended up going for the proper loop shape.

FYI, the 'proper' shape is commonly called a "teardrop loop". Pre-WWI roller coasters called 'Flip-flop' coasters (IIRC) were the first wooden coasters but unfortunately used a more circular loop design, which gave all the riders whiplash. Even today, though, the shape of the loop varies from coaster to coaster - some have more circular loops to give 'hang-time' at the top. (Without "upstop" wheels to keep the cars on the tracks, 'hang-time' wouldn't have been an option for pre-war coasters.) Someday, I'll get around to working out the formula for a 'perfect' loop - that is, one where the train enters at speed V and where the rider experiences a constant acceleration G + A throughout. No spoilers, please!

At the trainâ€™s top speed, that curve would create about two gees of acceleration. This might be survivable (for the passengers, at least, if not the train), but it would certainly not be comfortable.

I'm no expert so I might be mistaken, but 2G sounds perfectly comfortable to me. IIRC, normal, untrained people can usually survive up to about 4G sustained, maybe 10G 'spike', without blacking out, and at least 3G sus/5G spike without greying out. In roller coaster seats, 2G would be pretty comfortable even sustained for a few minutes - in an "enterprise" ride, which is a sort of inverting ferris wheel where the riders are not restrained but held inverted through G-force, IIRC in the spin-up period whilst it's horizontal, the riders are subjected to a constant 2G or so, and then in the upright period, the force oscillates between just under 1G and up to 3G, and it's mostly considered a pretty 'tame' ride.

I don't really get why the trains need such large vertical diameters if they're 'bendy' enough to go around them. If it's a question of the trains being able to survive the force, then surely building a larger loop would just worsen the problem? (Linear acceleration increases with radius so if anything, a larger loop would put more strain on the trains?) In any case, surely it's better to bolster the carriages with steel supports rather than jet engines?

I guess I don't really see why there should be a difference between normal trains and roller coaster trains. Assuming that they're 'bendy' enough to go around loops, and that the seats in trains aren't so drastically different to roller coaster seats (apart from the lack of restraints), then I don't see why roller coaster physics can't apply to normal trains.

EDIT: Also, just to say, the definition of a roller-coaster is disputed, but generally it's that it's a ride that is powered at the start by a lift hill or launch, and then uses that kinetic or potential energy to complete the rest of the circuit. So, having a train that uses a cog system to power itself around the loop isn't really a roller-coaster.
Come to think of it, if the train in the "What-If" is still using its engines/jets at the top of the loop, then 1) it isn't really a roller coaster and 2) it'll need some pretty good brakes to slow itself back down on the downwards section of the loop.

### Re: What-If 0043: "Train Loop"

Posted: Wed May 01, 2013 12:16 pm UTC
I'd just like to point out that the reason Joe Biden's commute was so dull has less to do with the length of the trip then it does with the fact that Joe is pretty much the only person in the country who rides AmTrak.

### Re: What-If 0043: "Train Loop"

Posted: Wed May 01, 2013 8:27 pm UTC
Sabbede wrote:I'd just like to point out that the reason Joe Biden's commute was so dull has less to do with the length of the trip then it does with the fact that Joe is pretty much the only person in the country who rides AmTrak.

The North-east corridor (DC to Boston) is actually well used and earns enough to pay for its operations (but not capital IIRC). The long-distance routes through unpopulated areas, on the other hand, are underutilized loss-making silliness presumably intended to gain Amtrak more political support in those areas.

### Re: What-If 0043: "Train Loop"

Posted: Thu May 02, 2013 5:05 am UTC
Yay, Lord of the Rings reference!

### Re: What-If 0043: "Train Loop"

Posted: Thu May 02, 2013 10:15 am UTC
blueberry42 wrote:I think this comic is wrong.

Consider a circular loop with radius r, speed at bottom of loop v0, and speed at top of loop v1. Assume train of mass m is just coasting, i.e. no engines. By conservation of energy we have
(1/2)m v0^2 = (1/2)m v1^2 + mg(2r).
Multiply both sides by 2/(mr), yielding
v0^2 / r = v1^2 / r + 4 g.
Centripetal acceleration is velocity-squared over radius, so this means the centripetal acceleration at the top is 4 gees less than at the bottom. So if we enter at 5 gees centripetal acceleration the centripetal acceleration at the top is 1 gee, and hence the train stays on the track (barely). The gee-force felt at the bottom would be 6 gees (extra g from gravity), which is uncomfortable but perfectly survivable.

So this is a lossless train (wind resistance, rolling resistance) which can bend at any angle?

First, assume a spherical cow...

### Re: What-If 0043: "Train Loop"

Posted: Thu May 02, 2013 1:26 pm UTC
blueberry42 wrote:I think this comic is wrong.

Consider a circular loop with radius r, speed at bottom of loop v0, and speed at top of loop v1. Assume train of mass m is just coasting, i.e. no engines. By conservation of energy we have
(1/2)m v0^2 = (1/2)m v1^2 + mg(2r).
Multiply both sides by 2/(mr), yielding
v0^2 / r = v1^2 / r + 4 g.
Centripetal acceleration is velocity-squared over radius, so this means the centripetal acceleration at the top is 4 gees less than at the bottom. So if we enter at 5 gees centripetal acceleration the centripetal acceleration at the top is 1 gee, and hence the train stays on the track (barely). The gee-force felt at the bottom would be 6 gees (extra g from gravity), which is uncomfortable but perfectly survivable.

As Blueberry42 shown, the maximum needed accelerations incurred by train/passengers is independent on loopsize, provided the minimum required initial velocity is used. This does depend on geometry though - While 6G is needed for a circular loop, a clothoid can allow the passengers and trains to only be subjected to 3.7G.

It's certainly possible for vehicles to go through vertical loops (i.e. google hot wheels vertical loop) which begs the question, what defines a vehicle as a train, and why does this defining property prevent it from going through tight loops [considering a 200m loop extremely tight].

The big problem with trains going through loops is placed on these lines in the what-if:
Most high-speed trains are limited to vertical curves with radii no shorter than 20 kilometers.

The reason for those limits isn't that trains aren't bendy enough. It's how fast they're going.

The rest is all about how since the speed the train needs goes like the square of the curve radius, and Randall, without telling us why, is fixated on really large radii (900m or more) it's 'impossible' to get the right speeds [which for a 900m radius is about 3 times the speed of sound, not exactly outside the scope of other what-ifs]

Following the link from "are limited" and reading through the specs, it seems like the 20km radius of curvature is required to prevent the passengers from feeling vertical accelerations higher than 0.050g, for their comfort, and while comfort was a part of the original question, it was removed at the beginning of the post so it seems like this condition is really not needed.

I find this particular what-if extremely sloppy, and would really have liked a more detailed explanation.

### Re: What-If 0043: "Train Loop"

Posted: Thu May 02, 2013 2:58 pm UTC
Patrik3 wrote: Someday, I'll get around to working out the formula for a 'perfect' loop - that is, one where the train enters at speed V and where the rider experiences a constant acceleration G + A throughout. No spoilers, please!

Fine, no spoilers, but if i may I'd like to suggest some additional parameters for you that will help your passengers be more comfortable: jerk and jounce. Jerk is the first derivative of acceleration, and jounce is the second (third and fourth derivative of position, respectively). I believe that the standard clothoid, aka "teardrop", loop shape already has this feature; it's important to rider comfort. Just like the fighter pilots on Battlestar Galactica get their heads knocked into the seat on launch, coaster riders get thrown around when the coaster tries to take them from zero acceleration to max acceleration instantaneously. If it helps you feel better about your design then declare the constant-G loop to start after the low-jerk transition curve

### Re: What-If 0043: "Train Loop"

Posted: Thu May 02, 2013 6:56 pm UTC
As a Norwegian, I have to say that this was not funny.

### Re: What-If 0043: "Train Loop"

Posted: Fri May 03, 2013 2:45 am UTC
tibfulv wrote:As a Norwegian, I have to say that this was not funny.

As a Norwegian-American with great pride in my Norwegian ancestry, I have to say it was funny. And, I can't see any way to take it in an insulting manner. (I also think it's not doable, but not for the reasons Randall has expressed.)

### Re: What-If 0043: "Train Loop"

Posted: Fri May 03, 2013 10:35 am UTC
vvn wrote:
tibfulv wrote:As a Norwegian, I have to say that this was not funny.

As a Norwegian-American with great pride in my Norwegian ancestry, I have to say it was funny. And, I can't see any way to take it in an insulting manner. (I also think it's not doable, but not for the reasons Randall has expressed.)

Sorry, bad joke. Just trying to parody the people who take offense at anything.

### Re: What-If 0043: "Train Loop"

Posted: Fri May 03, 2013 3:03 pm UTC
tibfulv wrote:Sorry, bad joke. Just trying to parody the people who take offense at anything.

As one of those people that take offense at anything, I take serious offense.

### Re: What-If 0043: "Train Loop"

Posted: Fri May 03, 2013 5:35 pm UTC
tibfulv wrote:Sorry, bad joke. Just trying to parody the people who take offense at anything.

As one of those people that take offense at anything, I take serious offense.

I take offense at people who take offense at trivial things.

Of course, that does mean that I frequently offend myself.

### Re: What-If 0043: "Train Loop"

Posted: Fri May 03, 2013 8:24 pm UTC
As someone who is not easily offended, I don't really mind anything in this conversation.

### Re: What-If 0043: "Train Loop"

Posted: Mon May 06, 2013 2:39 pm UTC
As someone who takes offence like a roller-coaster, my feelings have been going up and down.

### Re: What-If 0043: "Train Loop"

Posted: Mon May 06, 2013 7:44 pm UTC
There is one thing that wasn't mentioned. Trains don't have to be short. Admitted most high speed trains are only a few hundred meters long, freight trains can go on for kilometers. With extra engines and extra passenger cars, a train longer than the loop wouldn't slow down as much traversing it.