Length of belt around two pulleys

[GPerk]

I need to calculate the length of a belt (or chain or wire) around 2 circles (wheels, pulleys). The 2 circles will not be the same size. To do this I think I need to find the points on the circles where two lines are tangent to the circles.

One approach is as follows:
Let X1, Y1, and R1 be the center and radius of circle 1.
And X2, Y2, R2 for circle 2. Assume circle 1 is the smaller of the two and they do not intersect.
1. Draw a line(L1) from X2,Y2 thru X1,Y1 and extend it beyond the smaller circle. (This line is NOT the tangent line unless the circles are the same size.)
2. Draw a line(L2) thru X1,Y1 perpendicular to L1.
3. Draw a line(L3) thru X2,Y2 perpendicular to L1.
4. Draw a line(L4) thru the 2 points where the L2 crosses the edge of circle 1 and and L3 crosses edge of circle 2. Extend this line so it crosses L1 at Point P.
5. The distance from P along L1 to X1,Y1 is the same as the distance from P to the tangent on circle 1. Likewise for circle 2 giving the tangent segment.
6. Same as step 5 for the other tangent segment
7. Calculate total length of the belt as 2 * tangentsegment + contact arc on circle 1 + contact arc on circle 2.

My question is: Is this the best approach to this problem?
And, is this algorithm available in some programming language on the net?

[mendhak]

Correct me if I'm wrong, but I tried this myself.

Since the point of contact will be on a line perpendicular to the center of the circle, that point will be (x1,y1+r1). And the other circle's point will be (x2,y2+r2).

Using that,

2 * ( SQRT ( (x2-x1)^2 + ((y2+r2)-(y1+r1))^2))
+ (PI * r1)
+ (PI * r2)

[opus]

I haven't got the solution(yet), but I think your approach has a mistake.
Only if the circles are of the same radius, the tangent to both circles runs thru the point on the perpendicular line to your L1.If the circles are of differnet radii, the point will be offset by a little!
Look at the picture

[zaza]

Opus is correct - the tangent line is offset on both circles.

The easiest way to tackle this is to transform to a set of coordinates in which one of the tangents lies along the x axis - this is just a rotation. Then, from the attached bmp, you can see that the length of the tangent is simply X1-X2. To work out the lengths of the arcs around the pulleys you need the angle of the intercept. This can be easily worked out because all the angles in blue and green are the same - hence it is simply 2 * atan( (Y2-Y1) / (X1-X2) ). The length of the arc thus has the same ratio to the circumference of the circle as the angle does to 360 degrees. Do this for both circles, and add on twice the length of the tangent. Remember to transform the lengths back again afterwards if you change the scale.

zaza

[Rassis]

If R1 and R2 are the radius of the two circles such that R1>R2 and D is the distance between the centres of the two circles, then:



For instance:

- If R1 = 10, R2 = 5 and D = 30, L is equal to 107,959
- If R1 = R2 = R = 10 and D = 30, L is equal to 122,832 which is the same as 2(Pi)R+2D = 2 x (Pi) x 10 + 2 x 30 = 122,832

Rui

[damasterjo]

Thats what I was thinking too.

[kwtan]

C = INTERSHAFT DISTANCE (CENTER POINT DISTANCE OF TWO PULLEY)
DP= DIAMETER LARGE PULLEY
dp= DIAMETER SMALL PULLEY
LP= BELT LENGTH

[opus]

Welcome to the forum
did you look at the dates of the previous posts, this "homework"-question was due last year!