Example
4.48” Pulley with 5/8” Shaft
2.00” Pulley with 5/8” Shaft
Distance between both pulley shaft centers =
Distance outer edge of both shafts – [ (½ X shaft 1st pulley diameter) + (½ X shaft 2nd pulley diameter) ]
In my case the Distance between both pulley shaft centers = 4”
4.48” X (constant 1.57) = 7.0336”
2.00” X (constant 1.57) = 3.1400”
7.0336” + 3.1400” = 10.1736”
2 X (Distance between both pulley shaft centers 4”) = 8”
10.1736” + 8” = 18.1736”
This is what I heard and seem to work out nice. Something else just noticed and sort of unique: the 2" pulley times the constant 1.57 = 3.14 as that is Pi (Greek Letter for that constant that it is)
MT
Correction: Although this is what I got from the help at Grainger, this is not completely correct unless the pulleys are of the same size.
I would add this to make up for the difference in size of pulleys. We are usually using different size pulleys to gear down the ratio so it is important to get it exact.
Additionally I realized what 1.57 constant I was told is. Since as above I saw that 2 X 1.57 = 3.14 which is Pi constant, I now know it was just the same as using [Circumference = 2 Pi X radius] and then dividing the circumference by 2.
The above I already knew and this using 1.57 was just another way of doing the same.
Here is what was missing for the two length portions of the belt between the pulleys at their greatest distance from shaft centers to be the exact. It’s not going to need trigonometry since we are dealing with right triangles an easier way is the Pythagorean Theorem.
Sorry for my drawing being not the best. Please imagine there being a second right triangle beneath the first one for the belt going along the bottom of the two pulleys. If I drew that it would get too cluttered.
Putting 1st pulley the smaller one first as I’m gearing down….
Starting with my pulleys a calipers measures outside edge of pulleys:
1st Pulley Diameter = 2”
1st Pulley Radius = ( 2” / 2 ) = 1”
2 X Pi 3.14 X Radius = Circumference
1st Pulley Circumference = ( 2 X Pi 3.14 X Radius 1”) = 6.28”
1st Pulley ½ Circumference = ( 6.26 / 2 ) = 3.14”
2nd Pulley Diameter = 4.48”
2nd Pulley Radius = ( 4.48” / 2 ) = 2.24”
2 X Pi 3.14 X Radius = Circumference
2nd Pulley Circumference = ( 2 X Pi 3.14 X Radius 2.24”) = 14.0672”
2nd Pulley ½ Circumference = ( 14.0672 / 2 ) = 7.0336”
In the right triangle in the picture attached:
a = absolute value of [ (1st pulley Diameter ) – (2nd pulley Diameter ) ]
a = absolute value of (2” – 4.48”)
a = absolute value of (-2.24”)
a = 2.24”
In the right triangle in the picture attached:
b = distance between centers of shafts on pulleys
In measuring with calipers the distance on farthest edges of shafts you just subtract ½ the shaft size of each of the two shafts and that is the distance between shaft centers.
For my motor bike I have b = 4”
b = 4”
The hypotenuse (longest side of the of the right triangle) = c
This is as in the picture c in the drawing.
This is where a more exact calculation is done.
Pythagorean Theorem [ (a X a ) + (b X b ) ] = (c X c )
Rearranging (c X c ) = [ (a X a ) + (b X b ) ]
Simplifying for c:
Square Root of ( c X c ) = Square Root of [ ( a X a ) + ( b X b ) ]
Square Root of ( c X c ) = Square Root of [ ( 2.24” X 2.24” ) + ( 4” X 4” ) ]
Square Root of ( c X c ) = Square Root of [ ( 5.0176” ) + ( 16” ) ]
Square Root of ( c X c ) = Square Root of [ 21.0176” ]
c = 4.5845“
c two times for the 2nd right triangle not pictured, but needed for the bottom of the pulley belt as well is:
c two times = ( 2 X 4.5845”)
c two times = 9.169”
Putting it all together:
Belt Length = ( ½ of 1st Pulley Circumference ) + ( ½ of 2nd Pulley Circumference) + ( c two times )
Belt Length = 3.14” + 7.0336” + 9.169”
Belt Length = 19.3426”
The error without using the additional steps and seeing that the length is longer is the difference between 19.3426” and previous length 18.1736” which amounts to 1.169” longer belt length.
I will write a program to do this so each time it will not be so tedious, and then measuring a few setups I would like to see if it is right on!
1.169" is significant so without the additional step the available adjustment may not be enough. In any case best to measure it and if the math agrees so be it
MT
