Why crowned pulleys and bandsaw wheels?

[metric_taper]

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.

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.

[Paul Alciatore]

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

[old_toolmaker]

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!

[mklotz]

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

Rolling uphill

[Floradawg]

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.

[old_toolmaker]

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