V-belt

Calculate a length of the V-belt when the diameter of the pulleys is:

D1 = 600 mm
D2 = 120 mm
d = 480 mm

Result

l =  2185 mm

Solution:

$d_{1}=600 \ \text{mm} \ \\ d_{2}=120 \ \text{mm} \ \\ d=480 \ \text{mm} \ \\ \ \\ r_{1}=d_{1}/2=600/2=300 \ \text{mm} \ \\ r_{2}=d_{2}/2=120/2=60 \ \text{mm} \ \\ \ \\ a=r_{1} - r_{2}=300 - 60=240 \ \text{mm} \ \\ \ \\ A=\arctan(a/d)=\arctan(240/480) \doteq 0.4636 \ \text{rad} \ \\ \ \\ A_{1}=\pi + 2 \cdot \ A=3.1416 + 2 \cdot \ 0.4636 \doteq 4.0689 \ \text{rad} \ \\ A_{2}=\pi - 2 \cdot \ A=3.1416 - 2 \cdot \ 0.4636 \doteq 2.2143 \ \text{rad} \ \\ \ \\ l_{1}=r_{1} \cdot \ A_{1}=300 \cdot \ 4.0689 \doteq 1220.6664 \ \text{mm} \ \\ l_{2}=r_{2} \cdot \ A_{2}=60 \cdot \ 2.2143 \doteq 132.8578 \ \text{mm} \ \\ \ \\ b=\sqrt{ d^2-a^2 }=\sqrt{ 480^2-240^2 } \doteq 240 \ \sqrt{ 3 } \ \text{mm} \doteq 415.6922 \ \text{mm} \ \\ \ \\ l=l_{1} + l_{2} + 2 \cdot \ b=1220.6664 + 132.8578 + 2 \cdot \ 415.6922 \doteq 2184.9086 \doteq 2185 \ \text{mm}$

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