# Two Pulley Belt Length

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Equation / Last modified by Administrator on 2017/01/31 07:06
"belt length" =
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MichaelBartmess.Two Pulley Belt Length
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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. # 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)