grumpy wrote:
The roller diameter doesn't enter into it,
Sure it does. Larger roller, larger surface contact area between roller and belt.
I disagree.
A larger roller does mean more area - but friction force does not depend on area, only on the normal force and the coefficient of friction. With a larger roller ( and, the same initial belt tension) the force per unit area between the belt and the roller is lower.
This might help to understand what's happening:
The belt will slip on the drive roller when the ratio of the tension in the top of the belt to the tension in the bottom of the belt exceeds this value:
e^(f*theta); [ this should be read as "e raised to the ( f x theta) power ]
where e is the naperian number ( about 2.72), f is the coefficient of friction between the belt and the roller, and theta is the wrap angle in radians. (Note that the drum diameter does not enter into this calculation .) This formula is from my old copy of Shigley's "Mechanical Engineering Design" but you could probably find it somewhere on the web. It's also pretty easy to derive.
As an example, for a friction coefficient f=0.5 ( a conservative value) , and a wrap angle of pi radians ( 180 degrees) this number (maximum tension ratio) is about 4.8. To continue the example, if you tensioned the roller separation spring at 10 pounds, you will, initially, have 5 pounds of tension in both the top and bottom sections of the belt.
As you sand something and load the belt, the tension in the lower section will stay constant at 5 pounds ( it's controlled only by the tensioning spring at the idler roller) , and the tension in the upper section of the belt ( between the sanded part and the drive roller) will rise. If you load the belt enough so that this upper belt tension reaches 4.8 times the lower belt tension of 5 pounds ( about 25 pounds) the belt will start to slip on the drive roller.
If you're still convinced roller diameter makes a difference, try this: Take a length of rope and a 5 pound weight and find a tree with horizontal branches of different diameters ( and similar bark surface). Hang the rope over a small branch ( with the weight hanging free) and see how hard you have to pull down on the other end to get the rope to slip on the branch( and lift the weight) . Now, do the same thing with a larger diameter branch. It will take the same force to drag the rope over the larger branch - as long as each branch is large enough to allow the rope to flex completely around the branch to get the same 180 degree wrap. You could try this in your shop with dowels and string, instead, if that were easier.
Phil