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Two Pulley Belt Length

belt length=π⋅(R+r)+2⋅asin(R-rL)⋅(R-r)+2⋅√L2-(R-r)2
(R)Radius of Larger Pulley
(r)Radius of Smaller Pulley
(L)Axel Separation

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 units and enter the following:

  • (R) Radius of Larger Pulley
  • (r) Radius of Smaller Pulley
  • (L) Distance between Axles

Belt Length: The calculator returns the belt length in inches rounded to the nearest mil (1/1000th inch).  However, this can be automatically converted to other length units via the pull-down menu.

See the video explanation of this equation's derivation.

/attachments/788be105-ed67-11e3-b7aa-bc764e2038f2/TwoPulleyBeltLength-illustration.png

Notes

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

CR = half the larger pulley circumference = π2⋅2R, where 2R = larger pulley diameter

Cr = half the smaller pulley circumference = π2⋅2r, where 2r = larger pulley diameter

u = segment of larger circumference between half circle and tangent point of pulley = θ2π⋅(π⋅2R)

v = segment of smaller circumference between half circle and tangent point of pulley = θ2π⋅(π⋅2r)

b = length of belt between tangent point on two pulleys = √L2-(R-r)2

L⋅sin(θ)=R-r   ⇒    θ=asin(R-rL)

Belt Length = (CR+2u) + (Cr-2v) + 2b

                     = (πR+2⋅θR) + (πr-2⋅θr) +2b

                     = π(R+r)+2θ⋅(R-r)+2b

                     = π(R+r)+2⋅asin(R-rL)⋅(R-r)+2⋅√L2-(R-r)2