Physics of Nip Rollers: Understanding Force & Torque

[tomizzo]

So I posted a question regarding nip and torque and I'm starting to realize that I need to know more about the mechanics of a nip roller to fully understand them. Here a pic I already posted just to give you a visual

http://i.imgur.com/ReGYoVm.png

Are nips meant to completely isolate tension zones? How does the force applied to the nip rollers affect torque? What is the purpose of apply more force between nip rolls? Does increased force mean less slippage of web between nip rollers?

I've tried google, but I can't seem to find a good document talking about the physics of a driven nip roller as well as the applications of them.

Any help?

 

[eigenperson]

So, I am not even remotely an expert on this, but I'll do my best at analyzing the case where the belt stretches only negligibly. Maybe someone will come by to tell me that I'm completely wrong, but hopefully not.

***

Let's start with a simplification. Instead of rollers and a moving belt, imagine that you have some people, Alice and Bob, holding a rope. Bob is standing to the right of Alice, about 2 m away. Let's have Alice and Bob tug on the section of the rope between them (call it AB) until it's under a tension of 100 N. To do this, Bob pulls to the right with a force of 100 N, and Alice pulls to the left with a force of 100 N.

Now Charlene comes up and stands 2 m to the right of Bob. She picks up the rope there and pulls on it. As long as Bob is cooperative, she can achieve any kind of tension that she wants on section BC of the rope -- either less than 100 N, equal to 100 N, or greater than 100 N -- while maintaining a tension of 100 N on section AB.

If Charlene wants a tension of exactly 100 N on section BC of the rope, then Bob can just wander away and let Alice and Charlene pull against each other.

If C wants a tension of, say, 60 N on section BC, then Bob will have to adjust the tension by pulling to the right with a force of 40 N. So Alice is pulling 100 N to the left, Bob is pulling 40 N to the right, and Charlene is pulling 60 N to the right.

If C wants a tension of 130 N on section BC, then Alice has to pull to the left with a force of 100 N, Bob has to pull to the left with a force of 30 N, and Charlene has to pull to the right with a force of 130 N.

And so on.

So in principle these two sections of the rope can have totally different tensions. Obviously if Charlene specifies an extremely high tension for section BC, the rope could break, or she or Bob could be unable to supply the required force (either because the rope slips through their hands or because they are pulled off their feet). But other than that, there is no limitation on the tension of section BC just because section AB has a tension of 100 N.

***

Now, that is a static rope. But it does not really matter if the rope is static or not. Of course, in the scenario I've just described, all the three people are going to have a bad case of rope burn if the rope is moving, so let's replace them with bicycle gears (sprockets), and the rope with a bicycle chain. The situation is still exactly the same.

To a first approximation, the chain cannot stretch and is moving at a constant velocity through the whole system -- a chain cannot slip against sprockets! To create the desired tension, we have to adjust the torques of the gears, not the speeds. Now, from the perspective of the gear, there are two sources of torque: the chain, and the drive system. The torque on the gear from the chain is equal to the radius of the gear times the difference in tension. Because the gear must rotate at a constant angular velocity (no slipping), the drive system must provide equal but opposite torque. This is analogous to how Alice, Bob, and Charlene had to tug on the rope in the previous example.

For example, assume we want a tension of 100 N on sections AB and BC. Also assume the chain is totally slack going into gear A and coming out of gear C. Finally, assume that the radius of the gears are 10 cm. Then the drive system at gear A must provide a torque of 10 Nm in the opposite direction from the chain's motion, while the drive system at gear C must provide a torque of 10 Nm in the direction of the chain's motion. Gear B doesn't need a drive system at all in this case -- it can just be allowed to spin. Notice that this provides a tension of 100 N on both sections of chain, no matter what the speed of the chain is. This is analogous to the example where sections AB and BC of the rope were both at 100 N tension, and if the velocity of the chain just so happens to be 0, it is identical.

If we want a tension of 100 N on AB and 60 N on BC, then we must arrange the drive systems so that gear A is torqued at -10 Nm, gear B is at +40 Nm, and gear C is at +60 Nm. Again, this is exactly the same as the rope situation. You can check for yourself that there's no net torque on any of the sprockets, as must be the case, and that all the tensions are right, and that it doesn't matter what the speed of the chain is.

***

Finally, we move to your example with nip rollers. Here there is an additional complication because the nip rollers transmit forces to the belt by friction (instead of directly, like the sprockets do). So to get the nip rollers to transmit the appropriate force to the belt, it is not enough to set the drive system correctly, since the roller could just slip. You also have to apply a great enough normal (squeezing) force so that the entire frictional force desired can actually be transmitted.

Example: If the coefficient of friction between the roller and the belt is 0.1, and you want to impart a force of 100 N to the belt, you must put a normal force of at least 1000 N on the belt. Note: This calculation uses the Coulomb model of friction, which may or may not be accurate enough for applications. I don't know.

If you provide a large enough squeezing force, there is no slippage, and the system behaves just like the sprockets. If you don't put enough force on the rollers, you won't get the desired amount of tension no matter how fast you spin the rollers. (That's not to say that changing the speed of the rollers does nothing in this case -- it will liberate large amounts of heat energy, for example -- but one thing it will not do is tension the belt.)

So this is why you apply a large force between the nip rollers -- to prevent the belt from slipping and enable you to impart a large force to the belt.

********

In summary, my attempt to answer your questions:

Are nips meant to completely isolate tension zones?

I don't know what they're meant to do, but in theory, they could.

How does the force applied to the nip rollers affect torque?

Assuming by "the force applied" you mean the "squeezing" force, it does not affect torque unless the squeezing force is insufficient to allow the full required friction force to be transmitted between the roller and the belt.

What is the purpose of apply more force between nip rolls?

If the force between the nip rolls isn't big enough, it isn't possible to apply enough force to the belt to create the desired tension.

Does increased force mean less slippage of web between nip rollers?

If you are experiencing slippage, increasing the "squeezing" force will decrease the slippage. With enough squeezing there will be no slippage at all. If you are not experiencing slippage, then you already have enough force between the nip rollers, and increasing this force will do nothing.