Two Pulley Belt Length

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Equation / Last modified by Administrator on 2017/01/31 07:06
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ID
MichaelBartmess.Two Pulley Belt Length
UUID
788be105-ed67-11e3-b7aa-bc764e2038f2

The Two Pulley Belt Length equation computes the length of a belt that goes around two pulleys, where the pulleys may be of unequal radii. 

INSTRUCTIONS: Choose your preferrred units and enter the following:

  • (R) The radius of the larger pulley
  • (r) The radius of the smaller pulley
  • (L) The distance between the axles

Belt Length: The calculator returns the belt length in inches.  However, this can be automatically converted to other length units via the pull-down menu.

See the video explanation of this equation's derivation.

TwoPulleyBeltLength-illustration.png

Notes

From the picture, see that the following describe the component parts of the solution:

`C_R` = half the larger pulley circumference = `pi / 2 * 2R`, where 2R = larger pulley diameter

`C_r` = half the smaller pulley circumference = `pi / 2 * 2r`, where 2r = larger pulley diameter

u = segment of larger circumference between half circle and tangent point of pulley = `theta/(2pi) * (pi * 2R)`

v = segment of smaller circumference between half circle and tangent point of pulley = `theta/(2pi) * (pi * 2r)`

b = length of belt between tangent point on two pulleys = `sqrt( L^2 - (R-r)^2 )`

`L * sin( theta ) = R - r`   ` =>`    `theta = asin( (R-r)/L )`

Belt Length = (`C_R + 2u`) + (`C_r - 2v`) + 2b

                     = (`piR + 2* thetaR`) + (`pir - 2* thetar`) +2b

                     = `pi ( R+r) + 2theta* (R-r) + 2b`

                     = `pi ( R+r) + 2* asin( (R-r) / L) * (R-r) + 2*sqrt( L^2 - (R-r)^2)`