Length of a portion of a circle

[jamie516]

This is PhD level work, and mathematics isn't really my thing, so thank you so much in advance if you can help me! I am trying to find the length of a curved fibre for a composites process I am working on. The fibre geometry is as follows: The length between 2 points is 200mm, but the fibre is curved by 4mm eccentricity. So from point a to b follows:

the fibre curves by 4mm up here
a __________________ b (200mm)

with 4mm offset in the centre of the line, making the length greater than 200mm, but I don't know what that length is. I would also like to know what the radius of a circle with 4mm of circle curvature at the perimeter separated by 200mm is?

 

[jamie516]

Here is an image which will hopefully make the problem clearer

 

[nomather1471]

http://www.loadtr.com/465889-aasasasasasas.htm

 

[jamie516]

OK, what if I want to change the values of 4 and 200, how did you work that out?

 

[jamie516]

and that doesn't give me the values between the two points of the line (or fibre in this case)

 

[zgozvrm]

Your end-to-end fiber distance (200mm) represents a chord of the circle. The radius, r, of the circle through the center of the chord will be perpendicular to the chord and divide the chord into 2 equal lengths. Since your fiber is offset by 4mm at the center, you have a radius that is divided by the chord so that the left side (in nomather1471's diagram) is r-4 mm long and the right side is 4 mm long (r-4 + 4 = r).

Using the Pythagorean theorem, you have a right triangle with sides 100 mm and (r-4) mm and a hypotenuse of r mm. Therefore, you have:
$$r^2 = (r-4)^2 + 100^2$$

So, for any distance D (measured from A to B) and an offset C at the middle of the fiber you have:
$$r^2 = (r-C)^2 + (D/2)^2$$

which can be simplified to:
$$r = \frac{C}{2} + \frac{D^2}{8C}$$

Of course, all of this assumes a constant curvature of the fiber (the radius is constant).

 

[jamie516]

Thanks zgozvrm, I have that now. I just need to work out what the radius would be for different fibre lengths, nomather1471 how did you get that diagram?

 

[zgozvrm]

Are you saying that you want to be able to determine the radius for different fibre lengths at a fixed distance (a to b)? Or are you looking to find the offset, given the distance and fiber length?

 

[jamie516]

I need to calculate the the arc length for different fibre lengths and offsets, so I always have the fibre length and offset.

 

[jamie516]

Sorry, I realize having sat down and worked through the problem that I was being very stupid , and to work out what r is is obvious what nomather1471 posted applies to all fibre lengths

 

[zgozvrm]

I need to calculate the the arc length for different fibre lengths and offsets, so I always have the fibre length and offset.

The arc length IS the fibre length!

I think you meant to say was that you need to calculate the CHORD length for different fibre lengths and offsets (i.e. the length of the line segment between the ends of the fibre when curved at a constant radius such that the midpoint of that line segment is at a distance from the midpoint of the fibre equal to the offset)

 

[jamie516]

Actually what I meant to say is that I want to calculate the fibre length or arc length, and I know the chord length.

 

[zgozvrm]

Okay, then once you find the radius r, using:

$$r = \frac{h}{2}+\frac{d^2}{8h}$$

Where d is the length of the chord and h is the height of the offset, you can find the circumference C of the circle by using:
$$C = 2 \pi r$$

Now, all that's left is to find out what ratio of your arc (the fibre length) is to the circumference. That ratio will be the same as the ratio of the twice the angle between the blue radius line (in nomather1471's diagram) and the red radius line divided by 360 degrees, or more simply, the angle between the blue and red radii divided by 180 degrees. Multiply that ratio by the circumference of the circle to get the arc length you're looking for.

Let's call the angle between the red radius line and the blue radius line alpha, represented by the symbol:
$$\alpha$$

We know that

$$\sin{\alpha} = \frac{d/2}{r}=\frac{d}{2r}$$ , therefore

$$\alpha = \arcsin{\frac{d}{2r}}$$

So the ratio is calculated by

$$\frac{\alpha}{180}=\frac{\arcsin{\frac{d}{2r}}}{180}$$ and your arc length is then

$$C \times \frac{\alpha}{180}=C \times \frac{\arcsin{\frac{d}{2r}}}{180}=\frac{C}{180} \times \arcsin{\frac{d}{2r}}=\frac{2 \pi r}{180} \times \arcsin{\frac{d}{2r}}=\frac{\pi r}{90} \times \arcsin{\frac{d}{2r}}$$






To summarize: Given the chord length, d (the distance from one end of the curved fibre to the other) and the offset, h (the distance from the center of the chord to the center of the fibre), first find the radius, r with the formula:

$$r = \frac{h}{2}+\frac{d^2}{8h}$$

Then find the arc length (the length of the fibre) with:

$$\frac{\pi r}{90} \times \arcsin{\frac{d}{2r}}$$