Why do crowned rollers work?

I’ve noticed the use of crown rollers in conveyor systems and the belt always goes towards the crown. The same occurs in a shooter. If there are two wheels with a gap in between, then the ball tends to deviate towards the wheels rather than the gap between the wheels. Why is this?

What physics or math allows crowned rollers to operate the way they do?

in modern times, we don’t explain stuff, we just post a link, because google works.

https://woodgears.ca/bandsaw/crowned_pulleys.html

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I take it the same principal is at work in a shooter as well. When the ball is in between two wheels, there is tension on one side of the ball which curves it towards the wheels. Right?

Now you’re trying to make me think! My brain isn’t up to that today, unfortunately.

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This video is the reason why I am curious as to why they work. I get the how, but I want to know the why.

I’ll take a stab at it…

The crowned roller has a larger radius, so while the angular velocity is the same, the linear velocity is larger. Now imagine 2 points of contact, one on the crowned part, and one on the non-crowned part of the roller. You now have 2 forces acting on the ball(or belt in your case), with one applying a larger force due to the larger velocity. Given that the crown is off-center, it will apply torque to the ball/belt and force it to rotate toward the crowned part (see insanely crude ASCII art below). As the ball/belt approaches the centered position, this lateral force reduces to zero.

|     ^|
|  ^  ^|
|  A  B|
|      |
|   -->|
Disclaimer: Terminology is likely wrong somewhere...

“Dammit Jim, I’m a programmer, not a mechanical engineer!” - Bones

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