How do you Calculate Belt Length?

[Gabriel]

Thank you all for helping me with this. Could you help me understand how the formula above works? I think that its the same process that I described in my first post. It seems to me that if you're taking pi/2(D1 + D2), what you're doing is taking half of the circumference of the two pulleys, which seems to me like you're assuming that the belt connects to exactly half of each pulley. Please correct me if I'm wrong, but a belt will only connect to half of each pulley if the pulleys have the same diameter.

I guess for engineering purposes this estimation is more than sufficient, but I'm curious about the mathematics of it.

I was thinking that if you marked the point where the belt intersects each pulley and drew an angle, than the angle describing how much of the smaller pulley connects to the belt would be the same as the angle that describes how much of the larger pulley does NOT connect to the belt. If you could find this angle, than you could take its arc length and solve part of the problem. Than you would just have to find the length of the lines between the pulleys which you might be able to do with the pythagorean theorem. I'm just not sure how to get there. Anybody have any ideas?

[Joe Matt]

Use geometry...

Measure the distance between the two pulleys, write down and multiply it by 2. (56 * 2= 112 in).

Now, take one pulley (6 inches) then find the circumference of it (2*pi*r).

2*pi*r= 37.699 -> 38 in around.

Then divide that by 2 (the belt only contacts half the pulley) and you get 19 in. Repeat your steps for 2.

112in + 19in + 63in = 194 in will be needed. Of course, it's totally theoretical.

That's for when the belt touches half the pulley, so now you have to do some more findings. First you need to find the angle of the opening where the pulley doesn't touch. Divide that by 360, then you'll get a decimal, convert that to a fraction, then multiply that by the total circumference of the pulley. Subtract that number from the total circumference and that's the length the belt touches the pulley.

[tkwetzel]

Originally Posted by JosephM

Use geometry...

Measure the distance between the two pulleys, write down and multiply it by 2. (56 * 2= 112 in).

Now, take one pulley (6 inches) then find the circumference of it (2*pi*r).

2*pi*r= 37.699 -> 38 in around.

Then divide that by 2 (the belt only contacts half the pulley) and you get 19 in. Repeat your steps for 2.

112in + 19in + 63in = 194 in will be needed. Of course, it's totally theoretical.

I thought of it that way, but with a small change that will be more accurate because it compensates for the extra length between the two pulleys along the line that the belt travels.

I looked at the line that the belt travels between the two pulleys as the hypotonuse of a right triangle. The longest side is 56" (the centerline bewtween the pulleys) and the shorter side is 14" (large radius minus the smaller radius). Then you use the pythagrean therom a^2+b^2=c^2
I included my math so anyone can check it.

56^2+14^2=c^2
3136+196=c^2
3332=c^2
c=57.72...

So you would multiply that number by two and add it to half the circumference of each of the pulleys. Using the numbers in JosephM's post...the distance would come out to roughly 197.5" of total belt length.