Why crowned pulleys and bandsaw wheels?

This post is not about building a physical tool but I think that is a valid post here because it is about a mental tool, a tool of understanding that is an important part of many real hardware tools.

As a teenager in the 1960s I had a Burgess 3 wheeled bandsaw, which like the milling machine of an earlier post, has followed me around the world and I still use it today. For some reason, which I don't recall, I built a 2 wheeled machine. Maybe I wanted full width at full height. Back then like Vyacheslav Nevolya I used to cast stuff out of scrap aluminium and that's how I made the new bandsaw wheels. Even though I was aware that pulleys for flat belt driven machinery and commercial band saws etc. had at least one crowned pulley, but the arrogance of youth lead me to believe that I knew better. It was obvious that a hollow pulley or bandsaw wheel would give better alignment. Arrogant I might have been back then but I was also quick to learn when I was wrong and I soon remachined the hollowed out wheel to a crown shape and had no further problems of flying bandsaw blades.

Attachment 15698
Here is a pic. of the little Burgess as it is today. I don't seem to have photo of the homemade bandsaw.

As an inquizitive lad I searched books (yes, the internet has not always existed) in the library for an explanation of this seemingly counter intuitive behaviour, but found no answer. So I adopted my usual approach with such mysteries, that is I would lay awake in bed turning it over in my mind. One night it all clicked into place and I was a happy boy. However, back in October of this year during discussions with friends the subject resurfaced but I had forgotten the explanation. So back to a couple of sleepness nights to rediscover the mechanism. I hate to leave such matters hanging.

So here is the flat belt on crowned pulleys explanation. I had to get it written down before I forget again and have to go through the mental gymnastics when the topic next resurfaces.
The "trick" to it all, is to realise that the belt velocity is constant across its width but the pulley velocity is not. Then the relative velocities between pulley and belt work out in accordance with observation.
It is easy to look at it in reverse and think that the part of the belt on the larger diameter is moving faster than the part of the belt on the smaller diam. That leads to the conclusion that the belts would fly off a crowned pulley, which I think is the intuitive reaction of most people. We know different though.

To get an idea of the mechanisms at work we only need look at the crowned pulley not any parallel pulleys in the loop.
Firstly, let us consider only a slice of half the width of the belt passing over just one side of the pulley as in Fig.1.

Attachment 15699
Fig.1.

If we consider the linear velocities across the width of the belt, say V1, V2 and V3, shown near the bottom of Fig.1, we can see that they must be equal or the belt would be tearing itself apart. We known that is not happening. In other words the linear forward velocity of all points on the belt must be equal.
Now consider the circumferencial velocity of the pulley surface at points P1 and P2. Due to different diameters at these points it is obvious that P! has a higher velocity than P2.
The only way that we can reconcile this with a belt which has a constant velocity across its surface is to realize that over most of the contact area between belt and pulley there is slip of varying magnitude.
Fig.2 shows us a view looking from "A" in Fig.1.

Attachment 15700
Fig.2.

It is reasonable to assume that somewhere around half way between points 1 and 2 the belt and pulley surface will be travelling at matched velocities. So at point 2 the belt will be travelling faster relative to the pulley surface and at point 1 the belt will be slower than the pulley. This means that the belt will be trying to climb up the crown of the pulley, as in Fig.3. Think of a car in a curve, the outer wheels will be travelling faster that the inner wheels. The belt is like the car and the pulley surface is like the road surface.

Attachment 15701
Fig.3.

Let us now put a new full width belt on the pulleys in place of our sliced belt as in Fig.4.

Attachment 15702
Fig.4.

Imagine that we did not put the belt on properly and it is a little offset to the right. We have seen that the crowned pulley produces a force tending to force the belt towards the crown. The wider the section of the belt on one side of the pulley the greater the difference in velocities between the outer edge and the inner edge and hence the greater the force. So in Fig.4 we can see that the force "F" on the right hand side is greater than "f" on the left side. This force imbalance will make the belt move toward the left until f = F or in other words the belt becomes centralized over the pulley. That is it automatically corrects any errors of alignment.
This is in accord with our experience.

Intuition often leads many (including me in my naive and inexperienced youth) towards the idea that a hollow pulley would centralise a belt. However, anyone who has tried this will have learnt very quickly that it doesn't work. Looking at Fig.5 we can easily see why.

Attachment 15703
Fig.5.

As in the preceding explanation we can see that if the belt is not perfectly centred there will be unbalanced lateral forces acting on each side of the belt. This time the forces will act outwards rather than inwards as before. So when F > f as shown the belt will be flung off, which is exactly what anyone knows who has tried such pulleys. It reinforces any intial alignment error rather than correcting it.

QED.

Thanks Tony
Brilliant explanation.
As a lad I worked in an old brickworks with flat belts driving everything, I guess they sort of knew but without knowing exactly why.
You mentioned sleepless nights, now if you could just explain how it can happen that one can fall asleep chewing on a problem only to wake up with the answer, it's happened to me often.
I had one of those bandsaws once and swapped it for an old motorbike my cousin had in boxes, here it is now (best deal I ever made).

[ATTACH=CONFIG]15708[/ATTACH

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Originally Posted by olderdan

Thanks Tony
Brilliant explanation.
As a lad I worked in an old brickworks with flat belts driving everything, I guess they sort of knew but without knowing exactly why.

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Dan,
Glad you like it. My problem has always been that I have never been satisfied knowing that stuff happens, I have to know why. It has cost me a lot of sleep over the decades.

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Originally Posted by olderdan

I had one of those bandsaws once and swapped it for an old motorbike my cousin had in boxes, here it is now (best deal I ever made).

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Great deal obviously.

Thanks Tony for the brillant talk !
I think it's the same for me, I fully understand you.
BR
Christophe

I agree Tony just knowing that something works or does not work in not enough. Being born severely dyslexic, my early years in school were a nightmare, so much so that my teachers classed me as educationally stupid but then again in 1959 and 60/61 country school teachers were not well versed on dyslexia. I accidentally discovered a way to counter it in my 3rd grade with tinted cellophane then became voraciously hungry to learn things, so much so that I became addicted to knowledge even though I attended school I had become autodidactic. This served me well enough to eventually be able to retire as an industrial design engineer with a PHD in engineering philosophy.I remain so through this day to an extent even though I can no longer remember as much of my math as I once had and never achieved the level that mklotz and others have. So I really appreciate these explanations found here in these forums

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Originally Posted by Frank S

.......... even though I can no longer remember as much of my math as I once had and never achieved the level that mklotz and others have. So I really appreciate these explanations found here in these forums

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Except for some stuff in quantum physics and cosmology, I believe that if you can't explain something in simple non-maths terms then you don't fully understand it. Many things that I have developed everyday explanations for would be much easier for me and much more concise if explained in terms of maths but that sometimes blurs a physical understanding. Gyroscopic effects are my favourite example of that. Pick up a physics book and it is bound to have a mathematical derivation of the standard relationship between torque and spin velocities. However, even most engineers and scientists that I have discussed this with don't have a good grasp of the relationship of the simple maths to the seemingly magic reactions you get when playing with bicycle wheels. I have developed a straightforward explanation of this and those who have seen it light up like a light bulb. They finally make the connection. It is equally enlightening to those with and without the maths understanding.
I am far away from a maths-phobe. I have spend a lot of my working life developing mathematical models to simulate a wide variety of dynamic physical systems, mainly but not exclusively for land and air vehicles. Sometimes the results of such work help develop a physical understanding where none previously existed. Unless that comes I never feel that I understand anything, and I don't like that.

Thank you Tony for your intuitive explanation for why pulleys with a center crown work so well in keeping the belts centered.

By the way, after reading your crowned pulley posting, I went on to see one of my favorite YouTube channels Keith Appleton, an expert in restoring model steam engines. Near the end of Keith's recent video "REBUILDING A VINTAGE MODEL OPEN STEAM LAUNCH - PART #3" he shows a rare tour of his shop. On his workbench is the same style Burgess bandsaw. I think both of you have had this type of bandsaw forever and great to see tools owned and maintained for over 50 years by the same owner.

Regards, Paul

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Originally Posted by Paul Jones

Thank you Tony for your intuitive explanation for why pulleys with a center crown work so well in keeping the belts centered........... he shows a rare tour of his shop. On his workbench is the same style Burgess bandsaw. I think both of you have had this type of bandsaw forever and great to see tools owned and maintained for over 50 years by the same owner.

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Paul,
It is amazing how good those little bandsaws were/are. The only thing that I have modified are the blade guides. I often think that a motor upgrade would help but it has never been a big enough problem to motivate me to do it.
I believe that Black and Decker bought the company or maybe just the rights to make the saw. I tried one once and it was nothing like as good as the original, it had been cheapened down.
Talking of keeping tools for a long time, I still have a number of hand tools that were my father's. He wasn't involved in hobbies which required tools like I am but he had a few tools for jobs around the house. he never had a single power tool. I use almost daily a small ball pane hammer and several pliers etc. that were his. Probably 80/90 years old.
When I started working at 15, I worked with an 85 year old, he had two sons who had no interest in dirty hands stuff and one day he invited me to his home, told me that he would not be using several tools again and insisted that I take them. I still have and use several that must be well over 100 years old. The throw-away idea didn't exist when these tools were made and I expect that in due course my son or grandson will continue with them, but they'll have to wait until I'm 6' under before I'll be giving them up.

Tony,

Wonderful to know you have such an appreciation for keeping, using and preserving the older hand tools from family and friends, plus the ones you acquired along the way. I too have kept all of my fathers hand tools he used for repairing and improving things around our homes. My dad always included me as his helper for this work around the house until I was in college and he acknowledged my craftsman skills far exceeded his craftsman skills.

My dad was a captain in the merchant marine and at sea for long periods so he left me in charge of making repairs and improvements to our houses by providing a long list of things to build or repair. He also paid me well rather than hiring general contractors and I used the money to buy more tools. By sixteen, I built my first metal lathe but I think the most important tool that allowed me to be very creative was a Rockwell-Delta table saw and jointer that my dad gave me on my 18th birthday in the 1960's. My parents had just purchased a new waterfront home on Bainbridge Island and asked me to do the finish construction for built-in cabinets in hallways, dining room, bedrooms and closets, fireplace surrounds, and to completely finish the basement, and add fencing and custom gates and doors. The table saw and jointer was in use everyday well past sun down and I learned at lot about wood joinery. For me this was all fun and not work, and I still have the table saw and jointer, and now use it for our own custom home. Give your kids good tools and computers, and let their imaginations take over.

Thank you for the discussions and ideas here on HMT.

Paul

Hi Tony
I to love my old tools, I have all my fathers spanners and gadgets that he made, my uncles bench vice and things that retiring toolmakers would hand to workmates, micrometers etc. I feel a connection when I use them and I think its all about respect.

I have been wondering lately about drum brakes after your post of your grinding machine, I have made a Tls conversion for my Triumph (spare backplate), the shoes are floating and the action seems disappointingly spongy, maybe not such a good idea. I do always skim the shoes on a fixture to -.020” drum dia.
The subject has got me wondering about heat expansion of drums that are laced into the wheel and how it effects spoke tension and wheel stability. Is this why Vincents chose to have bolt on drums instead?.
I have a feeling you will know the answer.

Attachment 15720

Attachment 15721

Regards
Olderdan

Dan,

I have only just seen your message. I haven't yet figured out how to keep track of comments on my posts yet. This is the only forum that I am active on, so there is still much to learn.
I know that many people do as you have, that is machine the shoes a little undersize. The problem with that approach is that the shoes don't have the same curvature as the drum until they have worn a bit. That contributes to spunginess.
What I do is, shim the shoes out from the cams and then machine the shoes to drum size. Then with the shims removed the backplate/shoe assembly will fit and when the brake is applied the shoes and drum will match. I use an expanding mandrel to hold the back plate when machining the shoes to ensure concentricity.
Another thing that I do differently from the norm is to do with the cam lever actuation. The norm is to have a rigid adjustable rod between the two levers as you have done. Getting that adjustment spot on for equal pressure on each shoe is not easy and any error will cause spunginess also. I can't take credit for an alternative method because I first saw it on the Earles forked BMWs that I had in the 1960s. As soon as I saw it I saw how clever it was and it made perfect sense. You have the outer cable connected to one of the cam levers (instead of fixed to the backplate) and the inner cable to the other cam lever. Now the force on the two levers is balanced. This method demands that the cams are symmetrical, or else designed for the job, because it will be the outer edge of one cam that operates the shoe and the inner edge of the other cam. That lack of symmetry slightly changes the leverage on the shoe for a given torque on the external cam lever. That imbalance is not too important but can easily be fixed by making the two cam levers with slightly different lengths. IIRC BMW used equal length levers but I make the correction with different lengths.
I hope that convoluted explanation makes sense. I'll try and dig up some sketches or photos. Otherwise Mr Google may know about it.

PS. The full cable force is applied to each cam lever, whereas the rigid rod system has to share the application force between the two. So for equal lever lengths on the two systems, the rigid rod system will brake less for a given cable force. However, there are two sides to leverage - force and displacement. Although the cable only system produces more force on the cam levers it does this at the expense of double overall cable movement, so it will feel more powerful but also more spungy, the answer to this is to shorten the cam actuation levers to return to the original overall leverage.

I accidentally posted this in the clutch rectification post in the first place. Time to correct my silly error.

I found some pix of the BMW drum brake activation.

Attachment 15784
You can see the outer cable connected to the left side lever and the inner cable connected as usual to the inner cable

Attachment 15785
Here you can see the BMW solution, which I had forgotten, to the problem that I mentioned of slightly different leverage of the cam on each shoe. Basically they offset one of the cams from the usual position. to ensure that the cam contact point on each shoe was the same.
There is some other cleverness in there also. With this cable on each lever system all the springs do is pull the shoes closer relative to each other so one shoe may be pulled away from the drum while the other is left rubbing. The solution adopted was to have one spring stronger than the other and have an adjustable shoe stop. So the stronger spring would pull one shoe onto the stop so it couldn't withdraw any more and then the other shoe would be pulled back until stopped by the cable adjustment in normal fashion. In my opinion all DLS brakes should be done that way.

There is some free software on my web site about drum brakes which you might find of interest, it is freeware.

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Originally Posted by tonyfoale

Gyroscopic effects are my favourite example of that. Pick up a physics book and it is bound to have a mathematical derivation of the standard relationship between torque and spin velocities. However, even most engineers and scientists that I have discussed this with don't have a good grasp of the relationship of the simple maths to the seemingly magic reactions you get when playing with bicycle wheels. I have developed a straightforward explanation of this and those who have seen it light up like a light bulb. They finally make the connection. It is equally enlightening to those with and without the maths understanding.

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I have spent many years working with and repairing Gyros, and I understand the theory of how they work but I am often asked "how do they work" and it is very difficult to explain this to those without the basics. I would be very interested to read your straightforward explanation so that I can explain it better. I hate the blank looks I often get when I try to explain it all. Of course the new Piezo, LED, Laser "Gyros" so common today, have stolen the real Gyro's name and work differently.

Great explanation on why pulleys need to be crowned, I understand at last.
Now can you tell me which wheels on a 2" x 72" Belt Grinder with 4 wheels, (2 on the platen, 1 for tracking, and 1 being the driven wheel),need to be crowned. There is much debate on this on the web. Some say all 4 wheels, some say just the tracking wheel, some say just the driven wheel and some say both the driven and tracking wheels need to be crowned.

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Originally Posted by Moby Duck

Great explanation on why pulleys need to be crowned, I understand at last.
Now can you tell me which wheels on a 2" x 72" Belt Grinder with 4 wheels, (2 on the platen, 1 for tracking, and 1 being the driven wheel),need to be crowned. There is much debate on this on the web. Some say all 4 wheels, some say just the tracking wheel, some say just the driven wheel and some say both the driven and tracking wheels need to be crowned.

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There is always some misalignment between pulleys due to manufacturing tolerances etc, so if you crown them all then each will be trying to align the belt on itself. So you'll end up with the effects from each pulley fighting the effects from the others, thereby putting some unwanted and unnecessary stress in the belt.
If you only crown one then that one will align the belt which will then find its own position on the other pulleys provided that they are wide enough. Of course with a tiltable tracking wheel you can adjust the belt centrally over the other pulleys.
So which wheel track with? I think that the answer now becomes clear, it is unlikely to be a good idea to use a crown wheel to align the belt and other to then realign it elsewhere.
So it's one crowned pulley which has the tracking control.

Strange that this topic came up yesterday, because then I pulled that little Burgess saw apart (which features at the beginning of this topic) to give it a 50 year service. The drive belt which the blade runs on is badly chewed up and the top pulley has seen better days. 2 new belts should arrive today, they should see me OK good for the next 100 years. The wheels are made of a hard plastic with no rubber tyres so I shall machine the rims smooth again and glue on some tyres cut from an inner tube. I'll only be crowning the top wheel.

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Originally Posted by Moby Duck

I have spent many years working with and repairing Gyros, and I understand the theory of how they work but I am often asked "how do they work" and it is very difficult to explain this to those without the basics. I would be very interested to read your straightforward explanation so that I can explain it better. I hate the blank looks I often get when I try to explain it all. Of course the new Piezo, LED, Laser "Gyros" so common today, have stolen the real Gyro's name and work differently.

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OK, I'll made that a separate post though, so watch out for it.

I agree about the name gyro being hijacked by totally different devices. When the Segway came out the press reports stated that it was gyro stabilised. As you will know it would have had to be bigger and heavier to do that. In reality it used an IMU, inertia sensors, to detect its balance state and then apply torque to the wheels to correct balance errors. No gyros in sight.

Thanks, you have confirmed what I already believed about the best wheel to crown but I needed your opinion to be sure.

I shall watch for the Gyro post.

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Originally Posted by Moby Duck

Thanks, you have confirmed what I already believed about the best wheel to crown but I needed your opinion to be sure.

I shall watch for the Gyro post.

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The gyro explanation has been posted here http://www.homemadetools.net/forum/g...ut-maths-65745

Tony Foale: Found this video on youtube;
https://www.youtube.com/watch?v=TNuzi-jMXoY&t=416s

I don't completely agree with him on a couple of things. First, I believe even without the bowing of the rubber band it would still climb to the center because there is more tension on the upper side (because it is stretched further) and that alone should pull it back up to the center. Second I don't see any real comparison between the molecules of a rubber band and the chains being shaken around with his hand. A good visual analogy maybe, but not what is really going on.

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Originally Posted by Floradawg

I don't completely agree with him on a couple of things. First, I believe even without the bowing of the rubber band it would still climb to the center because there is more tension on the upper side (because it is stretched further) and that alone should pull it back up to the center. Second I don't see any real comparison between the molecules of a rubber band and the chains being shaken around with his hand. A good visual analogy maybe, but not what is really going on.

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His 'spring' analogy is not optimal. I posted this as it is a good demonstration of Tony's post description of tracking.
This self centering belt requires that it have some compliance, so there is differential tension in the band.
I have a 13" wide stationary belt sander (one of those cheap import machines sold 25 years ago by many import tool sellers, mine happens to be Woodtek). This never tracked right, as the fools that manufactured it, crowned the rubber driven shaft, so it caused a concave surface in the wood that was being fed through it. I fixed that, and then used duct tape to crown the idler pulley that was a spring loaded tension pulley as well. It would track for a while, then suddenly when the belt got hot, would take off one way or the other. So I removed the convex crown, and made a concave crown, and this has worked very well for years now. But these belts are very stiff, and made of some fiber cloth that does not stretch much.
I had to fix many other design and manufacturing defects in this machine as well, mainly to the power feed belt system that had the idler shaft at an angle to the main platen. So many issues with this that were clearly human errors of machine creation. But with a machine shop, I could fix all of them and happy with the surface finish it produces. The only improvement would be some mechanism that would dither the belt back and forth so it did not track the same groove making abrasive on the belt to the work. Really only a problem with medium to course grits.

This touches on a favorite mechanical principle of mine which perhaps can be called "self steering". The most common application of this in today's world may be railroad wheels, but it goes back much further than that, perhaps even before recorded history. But let me start with railroad wheels. Railroad locomotives and cars do not and never had any manually operated steering mechanisms. They just follow the tracks. But, HOW?

(OK, some modern train locomotives do have steering mechanisms, but they are very rare and are used to solve unique problems.)

Many people who have observed the wheels on railroad cars and even on the locomotives and have assumed that the flanges on the inside edges of those wheels are what keeps them on the track and also what guides those wheels around curves. But this is not so. In fact, if those flanges were used for steering, they would wear down so fast that the railroad repair shops would have an awful time replacing them and railroad wrecks from worn wheel rins would be so common that most cities would completely ban rail lines from anywhere near occupied places, like houses or places of business. In my youth I observed a streetcar line in New Orleans which had a very sharp curve due to the limitations of the streets it was on. At that corner, the normal steering mechanism for the railroad style wheels on the streetcars could not cope with that sharp curve and the flanges did take up the slack. The streetcars had to go very slow and you could hear the loud screeching of steel against steel whenever a streetcar went around one of those curves. You could also see a visible amount of wear on the track at that point. I am sure they did regular inspections to be sure to replace that track before it wore to a dangerous extent. Here is that corner and those streetcar tracks are still there today. You can observe the different color of the blacktop pavement, probably due to frequent maintenance on the tracks.

https://www.google.com/maps/place/St...!4d-90.1339998

No, those flanges are only there as a last ditch, emergency measure when the real steering mechanism fails. They are in fact, a fail-safe and ONLY a fail-safe.

No, there is a much better and more clever mechanism that is used to steer railroad wheels around curves and which also keeps them CENTERED on the rails even on straight track so that the wheel flanges almost never touch the rails. The secret, if you can call something in plain sight a secret, is the shape of the "flat" part of the wheel, which is not really flat. The rolling surface of railroad wheels are shaped with a taper instead of being flat. This taper is arranged so that each wheel has a smaller diameter on it's outside edge and a larger diameter at it's inside edge which is next to that emergency flange. And railroad wheels are also rigidly attached to each end of an axle so both the left and right wheels turn at exactly the same speed: they are in locked in angular alignment. The result of these two details of RR wheel construction result in the following behavior. When the two wheels are perfectly centered on the track, both of them contact the rails at points where they have the same diameter (on those tapers) and they roll straight down the track with no problems. But, when the pair of wheels moves, for whatever reason, to one side of the track, then the wheel on that side of the track rides up on the tapered section and rides on the rail on that side at a point where it has a larger diameter. And at the same time, the other wheel rides across the track to a point of contact with the other rail where the taper produces a smaller diameter. The larger diameter will have a larger circumference so that wheel, which is rotating at the exact same speed as the other wheel and is strongly constrained not to slip on the rail, will try to travel a greater distance. At the same time the wheel that is riding on a smaller diameter, which produces a smaller circumference, and which is also constrained as before, will try to travel a shorter distance. This difference in the distance traveled by the two wheels will cause a steering of the combination toward the opposite side of the track which means back toward the center. When the wheels deviate to the other side, the opposite effect occurs and the wheel steers in the other direction, again back toward the center.

Thus, long before the flange on one of the wheels can reach contact with the rail, the two wheels on their axle will self steer back to the center.

When a curve in the track is encountered, the same thing happens. But it is the curve in the track that starts the process as one wheel starts to ride higher on it's taper and the other wheel rides lower. The pair of wheels self steers into the curve in the track and, once again, the wheel flanges do not normally come into contact with the track. The streetcar example I mentioned above is a very unusual exception to this rule.

And the designers of railroad wheels took or borrowed this idea from something that was much more ancient, the common, barrel shaped barrel. For many hundreds of years barrel makers produced their barrels with the shape that is commonly called barrel shaped. They have curved sides that have a larger diameter at the center of the barrel and smaller diameters at the top and bottom. This shape certainly makes it easier to produce a water proof barrel as the metal rings that hold it together can be hammered on while hot and tightening the joints as they cool. And round barrels can be rolled across a floor which is easier then carrying them. But I am sure it was not long before someone tried to roll them along some kind of makeshift track. And then they noticed that a barrel shaped object could be placed on a pair of level, wood rails and given a push. It would them continue to roll for a great distance without falling off those wood rails. Hence, the barrel is not only a great shape for making a water tight container, but that container can also be easily moved for a distance.

When the industrial age came upon us, this self steering idea was not forgotten. Early machines used leather belts to transmit power from a power source like a water wheel or later a steam engine to the devices and machines that needed that power. These leather belts were run on what looked like flat pulleys, but those pulleys were, like the railroad wheels, NOT ACTUALLY FLAT. They were made with what was called a crown. That means that they had a curved surface from one side to the other with a larger diameter at the middle and smaller diameters at the edges. This is much like the egg or sphere shaped pulleys in your demonstration, but just the very middle of them.

So, now the situation is somewhat reversed from that of the railroad wheels or the barrels. The pulley is fixed in one left-right position and it's rotational axis is fixed so it can not steer itself. But the leather belt is somewhat like the tracks that the RR wheels or barrels rode on. Again slippage is not pronounced but if the leather belt moves off center then one edge of the belt will be traveling a greater distance then the other edge. As that edge that is on the greater diameter of the pulley approaches, it will be pulled into the pulley faster than the edge on a smaller diameter. This will pull the belt at an angle to the pulley such that the part of the belt that is approaching the pulley will be angled toward the center of the pulley and it will then ride higher on the crown. This moves the belt toward the center just as the RR wheels were moved toward the center of the track. This does not depend on any stretch or bulges in the leather belt and in fact is more due to it's stiffness than how much it flexes. The overall, approaching section of the belt is being rotated (or steered) so that it rides up on the crown of the pulley.

This self steering of a belt that is transmitting power is so pronounced that I have two examples of it on my metal lathe. The spindle is driven by a flat belt that rides on crowned pulleys both on the spindle and on the counter (intermediate) shaft and by a Vee belt that rides in a standard Vee pulley on the motor but on a crowned pulley on the counter shaft. It is no surprise that the flat belt, which is about 1" wide works well on two crowned pulleys, but I was surprised that a Vee belt that has an inside edge of only about 3/8" width would also ride well on a crowned pulley. Both of these pulley crowns look to the naked eye like flat surfaces so it does not take much of a crown to produce this effect.

https://www.scientificamerican.com/a...wheel-science/

https://gizmodo.com/the-reason-train...lev-1793213013

Paul,
That was very well explained! I ran across a similar explanation some time ago in a lengthy model railroading article. I don’t recall where I saw it, but it was an eye opener for me. It is just mind boggling to get a grasp on some of the engineering marvels that have appeared throughout history!

And, with a bit of manipulation, you can get the barrel to appear to roll uphill...

https://www.homemadetools.net/forum/...g-uphill-51682

Wow! Thank you for the detailed explanation of this subject. I had read about how the wheels travel a greater distance by riding out on the greater diameter of the wheel on an outside curve with a solid axle to wheel bond without need of a differential. Very interesting stuff to me. Very dull to some.

I was in awe the first time I learned I learned this also. It makes perfect sense to me now.