One such real-world problem first came to my notice when I was a child growing up on an Iowa farm in the 1940s. At that time few farmers could afford grain-threshing equipment, and the custom was to share a threshing machine that would go from farm to farm.
The farmers brought the shocks of grain from the field to the machine, and then took the threshed grain to storage. The machine blew the straw into a huge pile. Too young to be asked to do any of this heavy work, I watched the whole process with fascination.
The threshing machine seemed huge, and was driven by a flat leather belt powered by a stationary farm tractor. The belt was perhaps a foot wide or more. Of course the thresher and tractor had to be properly aligned, and blocked so they wouldn’t move. What puzzled me was how that 20-foot-long belt could run all day and remain on the cylindrical pulleys without wandering off. The belt might not track properly until the tractor was properly positioned, but once that was accomplished, it usually needed no further attention. No one I asked knew the answer. “It just works,” they said.
At the university, we physics majors studied mechanics and even some material on machines. But this particular problem was never addressed by textbooks or in class. By then it had settled to the cobwebby recesses of my mind. After a career of teaching physics at the university level, I never saw this problem in any textbook. After retirement, I kept busy with a website, mostly about physics, and people would send me questions by email.
One of these asked about flat belt tracking.
Solving the Puzzle
Foolishly thinking I could brainstorm this myself, I first considered friction effects. Then I got smarter and sought out a real farm tractor that had the usual drum cylinder for flat leather belts. I noticed that this cylinder wasn’t of uniform diameter, but was of slightly larger diameter at its center. Other belt-driven machinery revealed the same thing. Aha!
Could this somehow be causing the belt to ride onto this elevated center section and stay there? But why would the belt prefer a larger diameter, and greater tension? It seemed counterintuitive, but one thing I’ve learned in my years in physics is that one’s naive intuition is usually wrong.
Back to the laboratory, and more experiments. At first I suspected friction had something to do with it. But the friction acts in the wrong direction, opposite the observed centering movement of the belt. Of course friction is important to the operation of the machine, because without it the belt wouldn’t turn. But perhaps something else was going on.
Along the way I remembered the old adage, “Sometimes an hour in the library can save you weeks in the laboratory.” I searched engineering mechanics books, a genre of literature I had seldom consulted before. And I came up blank. These books didn’t mention such mundane matters as pulley centering. The internet wasn’t much help either, except to describe this kind of pulley as a “crowned pulley,” and to give specifications for the elevation of the center as a function of the pulley radius.
It’s always easier to find information about something once you know its name. At a used-book sale I gravitated toward the engineering section, and there I found an older textbook, Elements of Mechanism by Peter Schwamb, Alynne L. Merrill, and Walter H. James, third edition (New York: John Wiley & Sons, 1921), and was astounded to see an excellent discussion that tackled the question of crowned pulleys head-on, giving a very good explanation that even a physicist could appreciate and understand. Today you can find it on Google Books.
(I love older textbooks, for they had real information and good explanations. This is the sort of book that libraries too frequently discard because it doesn’t get checked out, because it’s a textbook, and because it’s old. Newer textbooks, in spite of their obscene size, have so many glitzy color pictures that there isn’t enough space for the necessary words.)
Here’s what this textbook said:
If a belt is led upon a revolving conical pulley, it will tend to lie flat upon the conical surface, and, on account of its lateral stiffness, will assume the position shown in Fig. 52 [right]. If the belt travels in the direction of the arrow, the point a will, on account of the pull on the belt, be carried to b, a point nearer the base of the cone than that previously occupied by the edge of the belt; the belt would then occupy the position shown by the dotted lines. Now if a pulley is made up of two equal cones placed base to base, the belt will tend to climb both, and would thus run with its center line on the ridge formed by the union of the two cones. … The amount of crowning varies from about 5 inch on a pulley 6 inches wide to about ¼ inch on a pulley 30 inches wide.
The belt in contact with the truncated cone lies on the pulley relatively flat and undistorted in shape. But along the dotted line at point a, the belt moving upward makes first contact with the pulley. Just below a, the belt has a slight sidewise bend. But the important thing is that as the belt moves from a to b without slipping, it moves along the dotted line to a point farther up the incline of the cone, and this process continues until the belt rides onto the apex. It’s geometry!
Now let’s look at the case of a flat belt running over two cylindrical pulleys whose axles are misaligned. Will the belt crawl to the right (where the belt tension will be higher) or to the left (where the tension is lower)? After thinking it through, try it and you may realize that a different reason must be sought than the one we found above, since your intuition may have misled you again.
Quoting Elements of Mechanism:
When pulleys are located on shafts which are slightly out of parallel, the belt will generally work toward the edges of the pulleys which are nearer together. The reason for this may be seen from Fig. 53. The pitch line of the belt leaves pulley A at point a. In order to contain this point, the center plane of pulley B would have to coincide with XX1. Similarly, the belt is delivered from b on the under-side of pulley B, into the plane Y1Y. The result of this action is that the belt works toward the left and tends to leave the pulleys.
This passage is a little murky at first reading. The essential difference between the two cases can be seen in the diagrams. As seen on the previous page, in our Figure B (their Fig. 52), the belt makes contact with the pulley at line a. Note that the upper edge of the pulley (line b) makes an angle with line a.
Also, note that line a is perpendicular to the incoming portion of the belt.
In our Figure C (their Fig. 53), the belt makes contact with the pulley at line a. Note that the upper edge of the cylindrical pulley (line b) is parallel to line a. Also note that line a is not perpendicular to the incoming portion of the belt.
This is the important difference between the two cases. In Figure B, each new piece of belt coming onto the pulley is carried, without slipping, to a point higher (to the right) on the slope. In Figure C, each new piece of belt coming onto the pulley is “laid onto” the pulley at a point slightly lower (to the left) on the slope, and is carried around without slipping.
An example of a crowned pulley can be found in the drive wheel of a band saw. This demonstrates that the principle also works with steel belts, which are much more rigid laterally than leather or rubberized fabric belts.
What I like about this puzzle is that the behavior of the crowned pulley is counterintuitive; that most of the initial hypotheses you make will turn out to be wrong; and that some explanations of the crowned pulley seem so “right” — until you apply the same reasoning to the parallel shaft problem, and then it’s back to the drawing board.
I like puzzles that have several levels of apparent paradox and counterintuitive features. They teach us to use, but not to trust, our intuition, which is a good lesson. Intuition can sometimes be a part of the problem-solving process, but at some point, it must give way to “sweating the details” and being ruthlessly critical of “plausible-sounding” answers.
Test It Yourself
Even after reading the explanation, you may still be skeptical that it really works this way. Good! I urge basement tinkerers to try it with real belts and pulleys. I used Erector construction set parts, some wood turnings, metal cylinders, and rubber bands, only because they were handy.
Figure D, on the previous page, shows my two models. The model on the left has a wooden file handle as a crowned pulley. This serves to illustrate how the rubber band will migrate along the wooden pulley from the narrowest part at the left, to its largest diameter. The file handle has both convex and concave profiles. Placed in the concave part, the rubber band will quickly rise up the slope and will stabilize at a larger radius, even if that is the very narrow portion near the right end.
The second model (on the right) has two misaligned pulleys. In this model, it’s better to use wooden dowels, or to cover the metal cylinder surfaces with something like cloth tape, to prevent slipping. When turning, the belt migrates to the end where the pulleys are closest together.Related