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omega_minus
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- TL;DR Summary
- A question on roll speed with compressible material deflecting over its surface
Hello,
I am trying to solve a problem at work that at first I thought was easy but has proven trickier than I originally thought:
If I have a strip of material traveling in a straight line and it is tangent to a roll turning below it, the roll would need to turn at the surface speed of the material to avoid it skidding. The speed is simply v=rw, where v is the strip linear speed, r is the radius of the roll and w is the rotational velocity (in radians/sec). However, if the material bends at the roll (deflection) I am not so sure about the relationship. (See attached image)
Consider that the material is rather thick. As is deflects there is an inner and outer radius of the material. I'd think the material (steel in this case) must compress on the inner radius and stretch on the outer surface. This means now the underside and top side are moving at different rates. If the material is not to skid (not to move relative to the surface of the roll), what speed should the roll turn if the incoming strip speed remains fixed?
My initial thought is that it turns at the speed given by the previous relation. But different "layers" of the compressing/stretching strip are moving at different velocities (strip velocity plus or minus the stretching and compressing rates). So if the roll turns at the speed of the surface layer there'd be an instantaneous jump in roll speed for any deflection whatsoever that wouldn't seem to be a function of wrap angle. This is counterintuitive (but that doesn't mean it can't be right). I have also considered that the strip could only be stationary to the roll's surface at one point since the underside compresses coming in and must then stretch out again on leaving the deflection roll. But this doesn't help me to see the answer either.
Can anyone give me an idea of how to solve this problem? I've searched all over google and the forums but I may not be looking with the best search terms.
Thanks
I am trying to solve a problem at work that at first I thought was easy but has proven trickier than I originally thought:
If I have a strip of material traveling in a straight line and it is tangent to a roll turning below it, the roll would need to turn at the surface speed of the material to avoid it skidding. The speed is simply v=rw, where v is the strip linear speed, r is the radius of the roll and w is the rotational velocity (in radians/sec). However, if the material bends at the roll (deflection) I am not so sure about the relationship. (See attached image)
Consider that the material is rather thick. As is deflects there is an inner and outer radius of the material. I'd think the material (steel in this case) must compress on the inner radius and stretch on the outer surface. This means now the underside and top side are moving at different rates. If the material is not to skid (not to move relative to the surface of the roll), what speed should the roll turn if the incoming strip speed remains fixed?
My initial thought is that it turns at the speed given by the previous relation. But different "layers" of the compressing/stretching strip are moving at different velocities (strip velocity plus or minus the stretching and compressing rates). So if the roll turns at the speed of the surface layer there'd be an instantaneous jump in roll speed for any deflection whatsoever that wouldn't seem to be a function of wrap angle. This is counterintuitive (but that doesn't mean it can't be right). I have also considered that the strip could only be stationary to the roll's surface at one point since the underside compresses coming in and must then stretch out again on leaving the deflection roll. But this doesn't help me to see the answer either.
Can anyone give me an idea of how to solve this problem? I've searched all over google and the forums but I may not be looking with the best search terms.
Thanks