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November 01, 2005
Favorite Analogies #3: The Belt Equation and Dog Leashes
I always tell people that "My dog knows the belt equation". When he's out for a walk on his leash and wants to sniff the local news posted by a previous dog at a telephone pole, he doesn't sniff on the near side of the pole. He always walks around to the other side, wrapping his leash around the pole with about a 180 degree wrap. Why? He knows that if I want him to stop, the friction that develops between the leash and the pole will make it harder for me to pull him.
The belt equation is used every day on every converting line. This simple exponential equation says the friction force required to slide a tensioned web relative to a cylinder will be related to tension, wrap angle, and friction coefficient. (continued...)
It if often written as:
T(high)/T(low) = exp^(mu*theta)
T(high) = tension on side overcoming friction
T(low) = tension side with friction helping it
exp = natural log e (about 2.71)
mu = coefficient of friction or traction
theta = wrap angle in radians
(180 degrees = 3.14 radians)
How is this used? For a tensioned web to slip over a cylinder, the high tension side must overcome both the tension on the other side and the frictional force that is created by the tensioned web on the cylinder. As the slip point is neared, the tension changed during the entire wrap moving exponentially from the low tension to the high tension. In some regards, the friction creates more friction. The friction increases the tension of the wrapped web on the high tension side, creating more friction.
When is the belt equation used in web handling?
This is one of the most used equation in web handling. When does a idler roller slip? When does an unnipped drive roller slip? When does a flat belt slip? Also, lane by lane across a web, can a tension differential be supported?
This last one is often forgotten. Too often the web is viewed as a string with a single tension and crossweb variations are forgotten. To understand some peculiar problems with steering guides and shear wrinkles, it good to understand how the crossweb tension of a bending web may crossover a roller. This can happen in a case where looking at average tensions, you would feel safely inside the belt equation's no slip criteria, but if you look at high tension lanes and low tension lanes, you can see how crossweb tension variations viewed lane by lane or inch by inch crossweb will exceed the belt equation.
I don't think my dog has figured this last part out.
tjw
Posted by Tim Walker at November 1, 2005 07:03 AM
