Belt Length Calculation

belt length trigonometry

I’m sure that if I look around I can find a belt length calculation tool, but that would spoil the fun of me making the calculation and then checking it is correct.

spindle for cnc machine with o-ring belt

The reason for the calculation is that I have made a replacement spindle for the Genmitsu 3018-pro and my intention is to use belt drive.

As this is quite small I’m going to use an o-ring for the belt, therefore I need to know the equivalent ID of a circular belt.

Two circles of r1 and r2 a distance d apart, what is the length of a belt that wraps around them?

The belt runs tangential to the two pulleys, hence the radial lines on each circle are at right angles to line of length b.

belt length trigonometry

In order to really understand this I needed to turn the diagram around. Now with the right angles at the bottom I could see that I could add another line, so creating a rectangle and a right angle triangle.

belt length diagram

Now I can see that b2 = d2 – (r1 – r2)2

So I have the length of the straight part of the belt calculated. Now I need to work out the length of belt that wraps around each pulley.

The angles a1 and a2 sum to 180°

arc1 = 2 pi r1 (360 – 2a1) / 360

arc2 = 2 pi r2 (360 – 2a2) / 360

The total belt length is then = arc1 + arc2 + 2b

This is all easier in a simple excel worksheet and I’ve made that available. This also calculates the equivalent inner diameter of the belt, assuming it is circular. Hence you can find an o-ring that could work as a belt.

excel belt length calaculator

This breakdown of the calculation is useful as I now have some other parameters such as the contact length with each pulley and the free length of the belt. Therefore I could calculate the maximum torque that the belt could take along with the first resonance. I will have a think and add these in.

About Nigel 358 Articles
I've been making models since I was around 7 years old and using a lathe from the age of 11, a self taught engineer with a passion for making model engines.

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