Solve Arc Length Problem: Find Circumference of Wing Section

[Gordon Arnaut]

Homework Statement

I'm trying to compute the circumference of a wing section. I have broken up the airfoil circumference into arc pieces and used cubic splines to come up with an equation for each piece.

For example, the arc nearest the leading edge of the wing is the function:

y = 0.0290X^{^3}-0.3334x^{^2}+1.8645x+1.4155

To find the length of this arc, I use the Arc Length Formula:


L = ∫_a^b √(1+[f'(x)])^{^2}dx


The integration limits are: a = 0.75, b = 5

Homework Equations

Alternate notation for the arc length formula gives the derivative as dy/dx:


L = ∫_a^b √(1+[dy/dx])^{^2}dx

The Attempt at a Solution

I started by getting the derivative of the function:

f'(x) = dy/dx = 3(0.0290X^{^2})-2(0.3334x)+1.8645

I then substituted u for the derivative term:

u = 3(0.0290X^{^2})-2(0.3334x)+1.8645

Here's where I get bogged down. How do I get du in terms of dx? I tried differentiating u but that gives negative numbers so something is wrong.

I'm wondering if I should try integrating in parts? Or maybe I should first try to factor that polynomial?

Sorry about the lack of math characters. I tried to use latex but it would not work in my browser.

 

[tiny-tim]

Sorry about the lack of math characters. I tried to use latex but it would not work in my browser.

Hi Gordon!

(have a square-root: √ and an integral: ∫ and try using the X² tag just above the Reply box )

It's not your browser … reality is at fault!

(There are server problems, and the LaTeX facility is off. )

/b
L = | sqrt(1+[dy/dx]^2)dx
/a
…
I started by getting the derivative of the function:

f'(x) = dy/dx = 3(0.0290X^2)-2(0.3334x)+1.8645

hmm … you seem to be trying to integrate

∫ √(1 + (ax² + bx + c)²) dx

I don't think there's any exact way of doing that.

 

[Gordon Arnaut]

Thanks Tiny-Tim.

I went back and put in those characters.

Why do you say there is not an exact way of integrating this expression? Can you explain?

Regards,

Gordon.

 

[tiny-tim]

Hi Gordon!

(you could have missed out the ^s also )

Why do you say there is not an exact way of integrating this expression? Can you explain?

Because it's the square-root of a quartic expression, and I just don't think there is (most square-roots don't have an "exact" integral).

 

[Gordon Arnaut]

Thanks again Tim.

I put in those missing superscripts. It's looking better now.

I hate to give up so quickly. What about integrating by parts? That way you could have a quadratic function in each part?

 

[tiny-tim]

I hate to give up so quickly. What about integrating by parts? That way you could have a quadratic function in each part?

If you're going to give up, then giving up quickly is best!

Integrating by parts would give you the square-root of a quadratic function in each part, which gives you a trig thing times another square-root

 

[Gordon Arnaut]

What about expressing the curve in terms of parameters?

Then we can use the formula for parametric equations:

L = ∫_b^a √(dx/dt)^{^2}+(dy/dt)^{^2} dt

 

[tiny-tim]

Try it if you like,

but my guess is you'll end up with the same problem, and have to use approximation methods and a computer.

 

[Gordon Arnaut]

Thanks, Tim.

I'm glad I asked. I actually have a solution to the problem and that is to use curves with second degree polynomials.

I can just cut the airfoil into smaller pieces and use curves defined by three points instead of four.

The result is a function that looks like this:

y = -0.2x^{^2}+1.69x+1.48

This should integrate nicely in the arc formula.

In fact these curves are much nicer to work with (parabolic, so they pass through the x-axis twice and therefore have two roots each).