How Crowned Pulleys Make Belts Track

Most people interested in machines know that adding a slight crown to a pulley tends to keep flat belts centered. But it is not obvious how it works.
belt_tracking_dia

As shown in the sketch above, introduction of a slight crown to the pulley causes the belt to bend when it is displaced laterally. Consider a point on the belt which comes into contact with the pulley at point A and sticks. Once it sticks to the pulley, it travels perpendicular to the axis of rotation of the pulley, coming to the point A' after one quarter of a revolution of the pulley. The belt has now moved to the position indicated by the dashed lines. Based on this argument, we see that a belt tends to climb the surface of a pulley whose surface is inclined relative to its axis of rotation. I came across this explanation in a fantastic old book called \emph{Elements of Mechanism, 3rd Edition} by Schwamb, Merrill, and James, published in 1921 by Wiley.

Crowned pulleys work well in practice provided that the belt material is not very stiff and that the belt width w is not too large. Based on this understanding, one can compute how much crown to put into the pulley for a given tension and belt material.