ter-intuitive; (2) most of the initial hypotheses you make will turn out to be wrong; and (3) some explanations of the crowned pulley seem so "right"until you apply the same reasoning to the parallel shaft problem, then it's "back to the drawing-board". I like puzzles that have several levels of apparent paradox and counter-intuitive features. They teach us not to trust our intuition, which is a good thing. 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.
When a flat belt runs over two pulleys, only one of them needs to be crowned to achieve lateral stability. The amount of curvature required in actual machinery is small. [The usual figure for the amount of crowning is 1/8 inch per foot of pulley width. Thus, the difference in maximum and minimum radii of a crowned 6-inch wide pulley would be 1/16 inch.] The method works for belts of leather or rubberized fabric that have some elasticity. You can demonstrate the assertions above by making a small model with axles and rubber band belts. A wooden turning, or a tapered tool handle works well for one pulley.The driving pulley is the one usually crowned. Crowning both pulleys can cause instabilities in tracking.
The large diameter idler wheel of a shop band saw has a rubber crowned tire. The metal blade is made to run true, on top of the crown, by adjusting the axle tilt of that wheel. This metal belt is not flexible, and the principle at work here is a bit different than for a leather or rubber belt, though the result seems superficially to be the same. Consider this a separate puzzle.
Though you will find much descriptive material on crowned pulleys on the internet, I have not yet found a good explanation of the physics of how it works. Several sites identify the factors that are important, and come very close to an explanation. Old textbooks on mechanics from a century ago sometimes have good treatments of this. Only one book in my personal library even mentions pulley crowning, and it was written in 1885. I will post my explanation here, after readers have had a chance to puzzle it out for themselves. Correspondence is invited, and if a reader supplies a really good explanation, I won't have to write up mine.
One wonders when the advantages of pulley crowning were discovered, and how it came about. Hans-Peter Gramatke informs me that:
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