Centroid & MOI of conic section

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mcrae0104

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Anyone have a convenient equation to find the centroid of a section of a non-elliptical conic thin shell and its moment of inertia (Ix)?
01.jpg
If the curve is elliptical, like the one above, I can find Ix of the fuselage skin by subtracting the Ix of the inside of the skin from the Ix of the outside of the skin. I can also work out its centroid, which I will need to find the neutral axis of the whole fuselage section buildup at any station (upper skin + lower skin + longerons).

02.jpg
However, I've used non-elliptical sections, so I've made life more difficult because I don't believe the y-bar and Ix equations apply any longer. I suppose I could integrate these curves graphically but it would be tedious. Any suggestions?

Edit: It occurs to me that I will need the torsion constant, J, in addition to I.
 
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BJC

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Anyone have a convenient equation to find the centroid of a section of a non-elliptical conic thin shell and its moment of inertia (Ix)?
View attachment 86484
If the curve is elliptical, like the one above, I can find Ix of the fuselage skin by subtracting the Ix of the inside of the skin from the Ix of the outside of the skin. I can also work out its centroid, which I will need to find the neutral axis of the whole fuselage section buildup at any station (upper skin + lower skin + longerons).

View attachment 86485
However, I've used non-elliptical sections, so I've made life more difficult because I don't believe the y-bar and Ix equations apply any longer. I suppose I could integrate these curves graphically but it would be tedious. Any suggestions?
To find the centroid, make a test piece the shape of your (fuselage cross section?) that is large enough to easily handle, and find the point at which it balances.


BJC
 

flyboy2160

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It looks like you built the sections up from curves and lines.( :p If you had equations, you could use the C word.) There are CAD systems that will do the section calcs. EAA allows a free download of the student version of SOLIDWORKS. The paid version will do these sections calcs, but I don't know if the free one does. There might even be other free CAD or math programs that will do this.

If the sections include ellipses, you could use the formulas in the book Roark's Formulas for Stress and Strain. But that will also be a pain, since you'll have to use the off axis theorem to transpose the inertias.
 
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mcrae0104

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It looks like you built the sections up from curves and lines.( :p If you had equations, you could use the C word.)
The section is not constructed from curves (arcs) and lines; it is a spline--ref. Raymer on conic lofting.

Alas, I only have an equation for the rho value of 0.414 and could integrate to find the area in that case...

$$ \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 $$

Once we change rho to another value, the equation no longer follows the standard form of an ellipse (because it is no longer an ellipse). Bummer.
 
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pictsidhe

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Depends on the equation for the curve. I would either derive an exact expression mathematically, or if that is beyond me, feed it into a spreadsheet. Those are really, really good at eliminating tedium. They can be verified against something that you can derive or look up the exact expression for.
 

mcrae0104

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Proppastie for the win!!! Thank you.

It didn't even occur to me that ACAD might do this without having to jump over to SW. I got to learn some handy new commands: REGION and MASSPROP.
04.jpg 03.JPG

Now to solve that torsional constant...
 
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proppastie

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I move the item from the centroid as shown to 0,0,0 the numbers are correct then.
 

wsimpso1

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Excel and SolidWorks are your friends.

You can even run four different sizes and use Linest() function to come up with a curve fitted formula for any given family.

BIllski
 

mcrae0104

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I move the item from the centroid as shown to 0,0,0 the numbers are correct then.
Yes, good point. Shown above is I with respect to the axis drawn, not the neutral axis of the section.
 
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