how to calculate aspect ratio on elliptical wing
12-30-2003, 05:07 PM
#51
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RE: how to calculate aspect ratio on elliptical wing
In the case of a linearly tapered wing panel (a trapezoid), it turns out that the MAC is equal in length to the chord at the panel's spanwise centroid, and that the quarter chord of the section at the panel's spanwise centroid is also at the panel's AC. This allows you to find the wing MAC and AC geometrically by breaking the wing up into a series of linearly tapered panels. I suspect that most CAD programs are not set up to perform: integral(chord^2*dA) or integral(x*chord*dy), so I don't think there's an easy way to get them to spit out the MAC or the location of the AC.
Maybe Im not clear. Im trying to relate this to CG placement. If our job was to balance any arbitrary wing planform exactly on the center of the its area (50% MAC), I would think we would a) determine the area centroid of each wing panel b) connect the centroids by a horizontal line passing through the panel centroids c) the total wing would be balanced anywhere on that line - be it a point on the root chord, the tip chord or at any position along the wing panel. End of job, no equations required. See pic as illustration.
So now (the complication) if I want to balance the wing at some user defined forward % of this centroid, 25% for example, isnt the centroid of the overall area of some value in this determination? If not, recognizing I can easily slice up the wing into N-elements in any orientation & determine the individual centroids of those elements, isnt it some sort of moment type equation applied to those elements that will determine the desired 25% point? I recognize the equations work for simpler geometry, but they dont help much for non-uniform curves. But the cad programs must have very accurate algorithms as proven by their ability to compute the area & centroids of simple geometric shapes to high precision. They dont 'know' whether the bounding curve is a circle, square or seagull wing - its just a curve.
Maybe Im not clear. Im trying to relate this to CG placement. If our job was to balance any arbitrary wing planform exactly on the center of the its area (50% MAC), I would think we would a) determine the area centroid of each wing panel b) connect the centroids by a horizontal line passing through the panel centroids c) the total wing would be balanced anywhere on that line - be it a point on the root chord, the tip chord or at any position along the wing panel. End of job, no equations required. See pic as illustration.
So now (the complication) if I want to balance the wing at some user defined forward % of this centroid, 25% for example, isnt the centroid of the overall area of some value in this determination? If not, recognizing I can easily slice up the wing into N-elements in any orientation & determine the individual centroids of those elements, isnt it some sort of moment type equation applied to those elements that will determine the desired 25% point? I recognize the equations work for simpler geometry, but they dont help much for non-uniform curves. But the cad programs must have very accurate algorithms as proven by their ability to compute the area & centroids of simple geometric shapes to high precision. They dont 'know' whether the bounding curve is a circle, square or seagull wing - its just a curve.
12-30-2003, 05:19 PM
#52
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RE: how to calculate aspect ratio on elliptical wing
Tallpaul, neat picture. This brings up another point. In the case up upturned tips (dihedral, ployhedral or swoopy upturn winglets), does one still use the planform (projected outline) area? Or how do you deal with the 'area' that is going into the z-axis?
12-30-2003, 09:03 PM
#53
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RE: how to calculate aspect ratio on elliptical wing
Full-scales seldom have the dihedral angles many models do, so the effect of dihedral is usually minimal on area.
Really odd shapes with lots of dihedral, the projected plan view is the norm for figuring area, as a rolled attitude generally isn't considered for wingloading computations.
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Finding the centroid of an area can be used for c.g. purposes, IF you can also determine the m.a.c. This should be centered on the exact point of the centroid.
Once the location of the m.a.c. is established, the 25% or whatever is easily found.
It's finding the length of the m.a.c. that creates the problem, as it isn't always nicely positioned between the leading and trailing edge at the spanwise location of the centroid.
Really odd shapes with lots of dihedral, the projected plan view is the norm for figuring area, as a rolled attitude generally isn't considered for wingloading computations.
.
Finding the centroid of an area can be used for c.g. purposes, IF you can also determine the m.a.c. This should be centered on the exact point of the centroid.
Once the location of the m.a.c. is established, the 25% or whatever is easily found.
It's finding the length of the m.a.c. that creates the problem, as it isn't always nicely positioned between the leading and trailing edge at the spanwise location of the centroid.
