Train Wheels Are Cones
awesci.com
awesci.com
Walking into rail design was hilarious. I worked on motorcycles and did some car stuff. I figured it was obvious, and sorta dismissed this assignment as a joke. Nope. My dismissive intuitions were just flat out wrong. It kinda leaves an impression on you to sorta avoid saying you know for sure before putting in some amount of work.
(For the curious motorcyclist, I recommend "Honda Common Service Manual" as a starting point.)
Turns out designing a new system to fit requirements is orders of magnitudes harder than fixing a system somebody else designed.
You see this in software all the time. Anyone can follow a tutorial. But can you start from scratch and build something novel? Can you build it such that others can maintain long after you’re gone? That’s hard.
Same with cooking. Anyone can follow a recipe. But can you design a recipe?
Making it scale economically is where all the math gets involved.
Black body radiation never quite sat right with me. One of the few subject areas where I just resigned myself to memorizing the formulas and moving on with life. Same with non-integer dimensional spaces for the most advanced partial differential course I took. I can visualize 2 million dimensional spaces just fine professor. But one and a half? What does this even mean?
Think about it like this. If I have a 2d field (x,y) and I enforce every point's x value to be 0, I pretty much just made the x degree of freedom redundant, and can now call the field a 1d field. If instead, I enforce every 3rd point's x value to be 0, I've now got a 1+2/3 dimensional space. Because there is some redundancy, I no longer get the full entropy that 2 dimensions provide.
Here's a paper which discusses fractional degrees of freedom: Effective degrees of freedom of a random walk on a fractal by AS Balankin · 2015 · Cited by 42 — This allows us to define the fractional dimensional space allied ... number of effective dynamical degrees of freedom on the fractal
https://pubmed.ncbi.nlm.nih.gov/26764671/
To hammer the relationship home, consider the holographic principle, which is based on observation of black holes, and states that our reality only needs 2 spatial dimensions instead of 3. Both Hawking and Susskind have eluded to this being the case solely because of the symmetries in the laws of physics. The symmetries cause a massive redundancy/pattern in the field values over the 3d space, such that we should theoretically be able to predict the state of the entire field given only the values at two-thirds of the volume.
Therefore, you can imagine a 2d surface which contains the state of our universe, and some kind of computational (possibly geometric/algebraic) projector, which understands the redundancies, reads the 2d surface, and renders a sparse 3d volume. In the case of our universe, the projection operation might be extraordinary complex, requiring a deep understanding of the laws of physics and the redundancies they induce into the underlying state that they operate on.
If you happen to come back, what do you mean by this partial quote,
"Hardy couldn't have been more wrong about the innocence of pure mathematics."
I'm just getting more interested in math the older I get.
Thanks in advance.
I graded very well in math through college, but later in life, I went back and explored more how all the concepts relate. In college, you are sort of fed calculus through a fire hose, and you just have to 'accept it' and move on. And you are left wondering, how were these ideas, these conclusions, reached? Until you go back through and see the long history of infinite series and see the various attempts to codify solutions. The problem is, as a student, you cannot possibly spend that much time deriving the whole solution from scratch and still hope to finish a degree in four years. As Carl Sagan said “If you wish to make an apple pie from scratch, you must first invent the universe.”
Which is why you can never stop learning. I am in my mid 60s, and I still learn something new regularly. Something big at least once a year, something smaller at least once a month. Never stop learning.
It was originally thought that Einstein's theory of special relativity was just a cool mathematical idea. But it was much much more. Mathematics has consequences that are so deep that even now it is rearing its head in AI..differentiable loss functions.
So I don't think anyone can be blamed for not getting it intuitively.
Max Planck wrote about quantized energy emissions from black bodies first in 1900. With this assumption, the spectrum of black body radiation could be derived successfully. That won him the Nobel Price in 1919. Albert Einstein postulated that light itself was quantized in one of his famous series of papers in 1905. This paper won him the Nobel Price in 1922.
[Side note: confusingly, Max Planck was awarded the 1918 Nobel Price and Albert Einstein was awarded the 1921 Nobel Price. This happened because the committee decided in 1918 and again in 1921 that none of the candidates met their standards and withheld the price for later.]
Thanks for sharing.
Edit: The last paper only describes fractional dimensions in a limited sense, because the way the author constructs the coordinate system still involves a positive integer number of coordinates.
One of my favorite lines from an early CB750 manual was "Pleased to be applying the 26mm spanner to the castellated nut. Thank you."
How can you not follow instructions when they are worded like that?
I learned about train wheels decades ago, when I briefly ran through train history as an interest. What I found interesting at the time was how thoroughly this was understood right at the advent of the age of steam. Rail engineers understood this from the beginning.
Another thing is that since the wheels ride an a narrow portion, going straight most of the time, wheel wear distorts the conical section, which leads to the wheels rubbing (instead of rolling) in corners, which accelerates the wear. So rail wheel wear is non-linear. Once the wear reaches a point wear it causes slipping/rubbing, it leads to positive feedback on the rate of wear. A failure mode that isn't obvious until you thing about it.
A simple corner on a motorway/autobahn/highway/etc has at least three curves in plan - in, out and shake it all about. OK, in and out and the main bend itself. The idea is that you need to transition the various forces into and out of the turn as safely as possible. A right angle turn is not a simple: straight -> quarter circle -> straight. As well as that, we have to consider line of sight and overtaking and water runoff and the effect of wind on high sided vehicles and ... and ... . Oh and of course these turns happen in 3D. I'm quite a fan of Holden Hill in Devon on the A38 - get your speed right in either direction and it feel effortless but you need enough power too. Purists in the UK will probably point at Snake Pass and most of Welsh roads and the like but I know Holden Hill.
Anyway, roads are sodding complicated. Why not curl up with this little number: https://www.standardsforhighways.co.uk/prod/attachments/c27c...
That's the design standard for major UK roads. Note things like crest and sag curves, worrying about kerbs and so on.
How would you rate humans for doing this right, in general , everywhere they have?
I do know that with a few exceptions, English roads do what the posted signs say they will do.
Holden and Telegraph hills and that area in general have roads that are millenia old or their course was generally decided quite a while back and repurposed every now and then. Starting off life as livestock drovers trails and the like. The old Britons may have cut back the trees and the like a bit. The romans did proper engineering. After them things went a bit vague for a while. etc.
Have a look at this: https://www.romanobritain.org/7-maps/map_counties_roads_town... - the Fosse Way runs "under" the A38 for quite a way and may be the A380 too, south of Isca Dumnoniorum (Exeter.) I'm now in Yeovil and the Fosse is the A37 here. The stretch from here to Ilchester (Lendiniae) is called Roman Road and is straight and flat and prone to flooding!
I think the Holden Hill roads have been done quite well. The hill is a nightmare shape! Each square metre will belong to someone, who would have to be bought out or worked around. There's a race course at the top and a fuel station, houses, and who knows what else on it. The traffic volume can be huge, especially in summer.
My mum was from Devon and her description of driving from Newton Abbot (Ipplepen) to London in the sixties is pretty ... different to now. It often took two days by the time you decided to stop after sitting in traffic for hours (now: about three hours.) She also drove to Edinburgh and beyond quite often and that really took some time. I used to commute from Plymouth to Chertsey (near enough London) and that took around four hours in the mid nineties.
Next time you drive out that way or anywhere for that matter, cast an eye on the place. See the boundaries, think of the history. In Devon, look at the depth of the hedges - its called "Devonshiring": the dense hedging "walls" with lanes running through them. Many of those hedges are bronze age or older. Modern Devon and Cornwall are roughly what was Dumnovaria (according to the Romans) which is suspiciously similar to modern Brittany and rather closer to Wales (Brythonic) than what becomes modern England eventually. The Corns wave a flag and a language that sadly was only properly native up until about 50 years ago when the last native speakers died. It is being revived and is a Brythonic language, like Welsh (Cymraeg), Scottish (Gaelic), Irish (Gaeilge) and the rest. There is also Cumbric (Cumbria) and others. Devon was largely subsumed by Wessex (West Saxons) earlier than Cornwall and hence is a bit more English (whatever that means.)
The reason that I'm wittering on about history is those roads are seriously old and have a context. Imagine who else has trod those roads back in the day. For example the patron saint of Germany is St Boniface. He's from Crediton.
Here's a lovely example of Arrow: https://youtu.be/mEcQyO77p7E?t=63
Note how jarring the transitions are between elements. Meanwhile, here's a nice smooth B&M ride: https://youtu.be/6Ee3pfpo1eQ?t=86
For optimum comfort of the rider, when the track banks, the center of rotation should be at the rider's chest. Arrow fails to do this, and places the center of rotation between the rails. This causes any banking to create a significant lateral force on the rider, which whips the rider's head into the over-the-shoulder retraint. In a proper heartlined roll, the track appears to "slip" from under the train, like this: [0]
In the video I linked, the turnaround after the two vertical loops was particularly brutal and painful.
What makes it even worse is a poor design of the wheel assemblies. The wheels on the inside of the rail that keep the car centered are designed to allow a small gap between them and the rail, and they're not spring-loaded or dampened in any way, which causes extreme levels of hunting oscillations [0].
Their roller coaster Drachen Fire [1][2] was so rough and painful that they stopping running the ride out after only 6 years.
[0] https://rcdb.com/3475.htm#p=42548
[1] https://en.wikipedia.org/wiki/Hunting_oscillation
https://woodgears.ca/bandsaw/crowned_pulleys.html
In the train's case (if we ignore the gap between the wheels), the pair of wheels work like a crowned pulley, and the track finds center like a belt.
I usually demonstrate it with two solo cups put mouth-to-mouth, to make a pair of facing cones that represents the motorcycle tire. The starting condition is that you're above parking lot speeds, and the bike is stable and is dynamically inclined to stay perfectly upright. To go left, you turn the bars right to upset the stable bike onto the left cone, and it goes left. To go right you turn the bars left and it upsets the bike onto the right cone, and goes right.
Just be careful when doing this and don't fall.
And then will you notice yourself doing it. It's quite remarkable. All those years you thought you turned the handlebars into the turn. You've actually been turning them the other way subconsciously in order to lean into the turn.
You’re talking about a motorcycle?
I didn’t know any of them had cruise control. I think the parent post is talking about a push bike.
But on the off chance there is a push bike with cruise control…
There's always dirt riding, track riding, and mountain biking to scratch the itch without traffic!
The feeling of riding a motorcycle is truly amazing and I absolutely miss it, but the risk factor is simply too high for me to be worth it.
I started riding in the dirt at age 15, racing 3/8 mile dirt track at 16, and motocross at 17. Got my first street bike at 19. Rode about 100K miles until I moved from a rural part of the country to a heavily urban area, and gave it up because it was no fun in heavy traffic.
Avoiding alcohol and drugs, definitely. Wearing good safety gear, also definitely. I trashed one helmet in my life, walked away with zero injury. Equipment includes proper shoes, pants, jacket. I had one high speed slide, head never touched the ground, but I lost a couple patches of skin, and that was with leather. Modern materials are better. Denim is worthless. Also, carefully choose who you ride with. I had a number of peers that I would not ride with, they were accidents waiting to happen, and I didn't want to get caught up in their dumb ass mistakes.
However, it’s so subtle that in practice you don’t feel them turn back to the direction of the turn and it fully feels like you’re steering right to turn left.
*Edit*: Maybe I’m wrong about this. There’s another comment below that says that the counter steer does happen through leaning alone.
The way I experience countersteer is that it pulls the bottom of the bike towards the outside of the curve, in order to lean or maintain a lean. If you lean with balance it’s not needed.
Is it possible to turn a bicycle at higher speeds without leaning? Ie turn like a car by just turning the front wheel. I've tried it a few times and it's been very difficult.
If you consider the whole system (bike and rider) as a point mass on a stick, it's still leaned over (centre of mass of not above the tyres) in that case.
The equilibrium position (lean angle) of this feedback system can be shifted by the rider either by applying force to the handlebars, or shifting the centre of mass away from the middle (leaning your body to the side).
It's possible to steer by only shifting weight, but a bike with locked handlebars would not stay upright (no feedback mechanism). Applying force to the handlebars is the way to get maximum control authority over the system.
Edit: more detail... the tilt angle is defined by the lateral offset between the centre of mass and the wheel contact patch. The rider shifting his weight achieves this by moving the centre of mass, while using the handlebars moves the wheels to the side while leaving the centre of mass (mostly) unmoved. Of course, the wheels' contact patch can be moved further and faster laterally (by steering) than the rider's weight can be shifted (which is limited by the rider's flexibility).
Sudden maneuvers are an essential survival skill on two wheels, and it's important to understand how to steer rapidly. A rider who just steers intuitively is vulnerable to a phenomenon known as "target fixation": becoming mentally focused on an obstacle and steering into it. Having a rational understanding of how to steer allows the rider to exert conscious control over the bike's direction in these situations.
Whatever happens must be too subtle to notice, and/or completely subconscious.
It comes intuitively if you've ever ridden a bike at more than a sedate speed.
At some point, someone noticed racing bikes counter steer really severely so you can easily see the wheel is actually pointing the opposite way. But the reality is that even at "won't fall over" speed on a bicycle, you're already doing the exact same thing. When you lean, you counter steer. Otherwise you'll high-side like a missile.
> I have asked dozens of bicycle riders how they turn to the left. I have never found a single person who stated all the facts correctly when first asked. They almost invariably said that to turn to the left, they turned the handlebar to the left and as a result made a turn to the left. [...] I have never found a non-scientific rider who had particularly noticed it and spoke of it from his own conscious observation and initiative.
The existence of counter-steering is still controversial to some riders, to the point where machines like the "No B.S. Bike" were created to demonstrate it as a necessary effect: https://soundrider.com/archive/safety-skills/nobsbike.aspx
To turn, you lean in the direction you want to go. But what way does the handle bar turn when you do that? It counter steers! You will fall if it doesn't (which is essentially what the No B.S. Bike demonstrates).
The first time I read about counter steering, I thought "man, this'll take forever to practice" until I realized it's really the only way one ever does it.
I think a lot of people assume those are different things. But they aren't. You simply can't turn at speed without counter steering, regardless of how you visualize the mechanics.
You are spot on for people who think pushing with their foot on the inner foot peg has an effect. I'm sure it does, but it's a lot of wasted effort given that the end result you are looking for is the bar turning the opposite direction, and you'll be doing that whether you are conscious of it or not. It takes far less effort to simply not think about it, and do what comes natural since your arms and hands will inevitably do the right thing without any "different" theories interfering.
It's also why 3 wheel bikes are notorious for throwing the rider high-side if they corner too quickly. It stops you from leaning, so there is no ability to counter steer. You end up behaving more like a London double-decker bus in the turn.
In particular people say you can't turn left without first turning right and vice versa, which is obviously nonsense - if that were true as stated then it would be impossible to ever turn at all.
Intuitively it feels weird, but logically it makes sense - if I'm too close to something, I can't turn briefly towards it in order to get away from it.
The speed matters because as the speed is faster, a smaller angle change in the handlebars corresponds to a bigger sideways tilt motion of the bike.
Particularly good parts are the explanation of fire and trees ("trees come out of the air"): https://youtu.be/nYg6jzotiAc?t=440 and the explanation of the mirror problem, i.e. how does a mirror know to reverse left and right but not up and down: https://youtu.be/nYg6jzotiAc?t=1976
In this case - just reposting the Feynman video - it's fine, but in other cases it leads to a lot of uninformed or unnecessary discussion, sometimes speculating on some hypothetical that the article already answers.
Wing Dihedral - https://en.wikipedia.org/wiki/Dihedral_(aeronautics)
Crowned Pulleys - https://woodgears.ca/bandsaw/crowned_pulleys.html
[1]: https://news.ycombinator.com/item?id=28350423
[2]: https://news.ycombinator.com/item?id=28350667
The problem with this is that newer technologies were not competing with rail purely on the technological merits, but with rail's massive amount of competition in suppliers and economies of scale. It turns out that with not a lot of effort, you can apply much the same savings and technological improvements to traditional rail, except without the huge added expense of converting to a standard you are currently incompatible with. Newer technologies have the reverse problem, in that they have very few, sometimes even a single supplier, and are basically custom projects with all the expense that entails. So the monorails, PRT, maglev, and other weird system are mostly unique specimens, or very few in number.
I've yet to see a photo of a BART wheel up close, nor an explanation as to why they supposedly chose a cylindrical wheel. I've looked. I can't imagine the design decision happened without some knowledge of why conical wheels were used in the past. Call me skeptical.
Or, looked at the other way, when the track curves, then the axle becomes uncentered.
(PhD was 'Residual stress in rails', for what that's worth. Judging from the profiles of the rails I saw, direct contact with the wheel flange plays a substantial role in keeping the train in place on curved track. But on roughly straight track, I'm satisfied that the argument about conicity applies).
The London Underground has some lines that are horrifically loud. The squealing must surely be at dangerous sound levels. I’d always assumed it was the flange against the rail, and you appear to be confirming that?
But it mostly (totally?) happens on very tight curves. It shouldn't happen much or at all on gentler curves.
(Of course, this is circular, because I'm kind of defining "gentler" and "tight" based on whether they cause flange squeal. Still, there's a point - there is something like a threshold of curve tightness where flange squeal becomes much more probable.)
I surmised this running 2 motors up a 3%(?) grade with 20k ton gross at 10mph. It's about the only explanation I could come up with is that the running gear was twisting under the gravity and the energy being put down to work against it. It might also just be a stringline sort of effect dragging the motors to one side of the track and pressing the flange. Maybe one of the rail engineers will come holler at me for my poor trainhandling skills.
The conical section of the wheels is mostly intended to prevent hunting on straight track, and the shape can't be made too aggressive without increasing the wear on wheels on rails. So on curves the superelevation is added to provide the extra force required.
Because conical wheels do increase wear and can contribute to oscillation in their own way, there have been experiments with cylindrical wheels especially on higher-speed trains---BART is a well known example. It ultimately didn't work very well and so they have been re-trueing the wheels to a non-cylindrical profile, although still not quite a traditional conical one. Basically in higher-speed operation the re-centering effect is too significant and causes one wheel to "chatter," which over time creates a significant vibration in the rail. Trouble is cylindrical wheels tend to cause the same thing to happen on the other side. It was a very hard problem before computer modeling became available.
I've never heard about that theory as for why superelevation/cant is supposedly being used until now.
Given that most of the time you'll end up with a remaining net force to the outside of the curve even after application of cant, it doesn't seem to make that much sense, either.
I think for low-speed freight the balance needs to be pretty close on to ideal to meet regulations, e.g. FRA regulations give calculations for acceptable ranges. But since it's dependent on running speed it's hard to get correct for freight and passenger mixed operation which is the subject of this FRA report that has a lot of detail on the calculations: https://railroads.dot.gov/sites/fra.dot.gov/files/fra_net/19...
I see what you mean with regards to how it's also described on Wikipedia – only I've got some currentish (European) literature in front of me which claims that cant and the resulting cant deficiency/excess are only of secondary importance with regards to wheel and rail wear (the main factors are simply the curve radius itself and the construction of the running gear of the trains operating over the curve), and as such the main importance of cant is simply ride comfort. Likewise it also claims that according to some practical experiments done by some infrastructure operators, no link could be found between occurrences of cant excess for slower moving heavy freight trains and increased maintenance requirements (Which interestingly somewhat contradicts the corresponding supposition given in your FRA document...).
This also matches the evolution of the design rules on the German national railways – in the 80s there still used to be a relatively elaborate system of determining the allowable cant excess for slower moving trains depending on the annual tonnage of that kinds of trains, but since then at some point that system got dropped and has been radically simplified: The regular cant is simply 55 % of the equilibrium cant and it's up to the design engineer to deviate from that value if necessary (when the speed distribution varies from that of a normal mixed-traffic route).
Interestingly all of that somewhat contradicts the statements given in your linked FRA document. To some extent this can probably be explained by European freight trains being shorter, somewhat lighter (lower axle loads) and also nowadays slightly faster than their American counterparts, and also due to traditionally using somewhat higher allowable cant deficiency values, especially with regards to passenger rolling stock. It likely doesn't explain everything, though, but I don't know enough, either, to reconcile those two differing points of view.
If the wheel pairs were independent then it wouldn't matter how fast each wheel in a pair rotates.
Yes, the cars' weight rests on the end of each axle via a "bogie" that holds the suspension and brakes and such, and then the multi-axle bogie itself rotates on a center pin:
https://en.wikipedia.org/wiki/Bogie#Components
https://en.wikipedia.org/wiki/List_of_railroad_truck_parts#A...
The axles are tough. Each axle weighs about 1 ton if I remember correctly. Each wheel can be reworked on a lathe several times (either with the wheel set removed or in situ on a drive-through floor-mounted lathe). After a few years, the diameter of the wheel is out of spec, and new ones are pressed on the axle. Axles can last about 75 years.
Technical study
https://assets.new.siemens.com/siemens/assets/api/uuid:2dbbe...
http://cs.trains.com/ctt/f/95/t/79912.aspx
Fast angle wheels first came out when MPC took over Lionel. The wheels are not squared off where they ride on the rail. They are angled to the flange. "Fast angle" is a toolmaker's term for adding an angle to a surface so the part can be quickly removed from the tool without marring the surface during manufacture. Hence the term "fast angle wheel" was coined by Lionel employees.
The fast angle did more than benefit manufacture. Because the wheels are fixed to the axel, it benefits them on curved track. The wheelsets can drift to a point where one wheel diameter point touching the rail is slightly larger than the opposite wheel diameter point touching the rail. This reduces friction because the outside rail is longer in circumference than the inside rail. Especially sharp 031 or 027 curves. If you look closely, you can see the cars lean into the curves as the outside wheels drift to a larger diameter.
Of course, pneumatic tires have cones that adjust their shape on the fly...
But yes, historically the awful screeching around corners was because BART used cylindrical wheels. It's also, apparently, why they can't run all night -- the tracks need nightly maintenance due to the grinding.
https://www.apta.com/wp-content/uploads/Resources/mc/rail/pr...
https://www.bart.gov/sites/default/files/docs/New%20wheel%20...
May it be for the same reason? If so, then the fix seems straightforward.
https://books.google.ch/books/about/BART.html?id=ubbwDwAAQBA....
The cylindrical wheel decision is closely related to the decision to use Indian/broad gauge, which was expected to provide a smoother ride as well as allowing more support equipment to be mounted under the car where it would produce less vibration.
Both are decisions that have not stood the test of time, although the choice of Indian gauge cannot practically be reversed. But I think the discussion around this often pays the original designers far too little credit: BART was intentionally a highly innovative design with numerous aspects that were somewhat experimental. BART's automated control system, for example, was such a debacle that BART initially operated with signal towers and the control system required nearly complete replacement. But it was a completely trailblazing design, and the same missteps would have to be made somewhere. BART was used once again as a test platform for an innovative radio control scheme in the 2000s, evidence of which can still be seen mounted trackside on the SFO wye.
Many lessons learned from BART's performance have contributed to later designs around the world, including notably the DC Metro which was built just shortly after by some of the same contractors.
It would probably be very expensive, and might not have ROI, but couldn't you lay narrower gauge rail in between the current rails, then modify or replace rolling stock to use the smaller gauge... and once done, remove the old broad gauge rail?
Looking at some random BART rail images, laying the new rail would be difficult; some places have equipment between the rails, other places have concrete between the rails. It would probably need to be a very long project; early stages could just be verifying which sections would be feasible to add narrower gauge to and making sure new construction would allow for it and when rework is already happening, consider working room for narrower gauge into the maintenance. You could really only progress sensibly once at least one line was nearly ready.
I guess the question would have to be, would all the expense and time it would take to switch to a narrower gauge, be justified the improvement in user comfort (I assume) and benefits of using more standard equipment.
It would be a lot of expense, for not a whole lot of benefit in the short term, and BART probably doesn't have enough money with reduced COVID commuting trends to even think about paying for something like this when they could instead build more rail and get more butts in more seats.
This is wider than most metro trains (especially 19th century systems), but BART isn't a metro system anyway.
[1] https://en.wikipedia.org/wiki/Loading_gauge#Standard_loading...
My recollection is that this was also related to running (relatively) high-speed trains on elevated tracks in windy areas, and so wanting additional lateral stability.
I was on a commuter train which derailed because it LOST a wheel. I don't understand why they aren't at least locked in mechanically. That wheel went rolling at 80MPH and blasted straight through trees near the tracks.
This feels like a really bumpy road at high speed, and stop if the train driver reduce the speed just a little bit.
I can’t find anything from either - though did read about the institutional racism dated by Feynman. Imagine being the person who questions his suitability for a Phd.
He didn't mention shotgunning beers.
How many posts were there at the time of writing? Did your comment influence the subsequent voting? Would garbagetime have been naturally downvoted if given enough time? Is hacker news actually declining in quality, or is it just tendency to favor good things when remembering the past?
Does any of this matter at all?