Question about interpreting VSPAero steady stability output data

[Cody]

Hello, 

I have a quick question about the VSPAero .stab output files for a steady stability analysis. My understanding is that stability derivatives, as well as information such as neutral point location and static margin, should be steady with respect to angle of attack. Yet, when I conduct a multiple aoa steady analysis for an aircraft, I get varying stability derivatives (with respect to alpha) and XNP / SM values for each aoa point analyzed. I am not sure which numbers to use, then, for further analysis. Does anyone have advice on this front? 

[Rob McDonald]

Stability derivatives are not actually constant in general.

CL_alpha is reasonably constant -- up until stall, but clearly it can not be constant thereafter.  There are other situations that cause a non-linear cl vs. alpha curve -- any of these will cause CL_alpha to be non-constant.

CD_alpha is never constant.  We generally expect CD vs. alpha to be a parabola.

When linearizing the equations of motion (typically about an equilibrium point), we treat them as constant because it allows us to simplify the equations to ease the analysis.  If you are doing traditional linearized stability analysis, then you should calculate the derivatives about your equilibrium point and go from there.

However, if you are performing an analysis where larger perturbations are common -- or where you will not have a single equilibrium point (say a flight simulator) -- then you will likely need to build a more complex aerodynamic model to feed to your simulation.  It might work with tables of (non-constant) aerodynamic derivatives -- or it might skip the derivatives entirely and just work with tabulated aero data.

Rob

[Cody]

I understand, thanks Rob. This makes sense for the non-linear curves of the stability derivatives. However, the one thing I don't understand is what about specifically static margin / XNP? I thought these values are defined explicitly by not varying with angle of attack, since XNP the point where CMy doesn't change with alpha. I'm not sure how to estimate a useful value for XNP from OpenVSP if it changes with aoa.

[Rob McDonald]

At some point, you figure out that everything they told you was a lie...

Look in 'Theory of Wing Sections', at some of the airfoil data in the back.  Stick to symmetrical airfoils to keep things simple.

Look at the curve of CM_c/4 vs. alpha.  Theory says it should be zero and constant.  What happens at stall?

For a cambered airfoil, CM_c/4 should not equal zero -- so the curve should be offset.  If the AC is not at c/4, then we would expect to add a term that is proportional to cl - which should be a straight line with alpha, so we would expect CM_c/4 to perhaps be a straight line.

Look through the data -- I suspect you'll find some that are curved - and not just near stall.

I prefer to think about static margin as dCM/dCL -- and this is how VSPAERO calculates it (look at the bottom of the stab file).  Run a target alpha - perturb one degree, run again, use those two points to compute dCM/dCL.

There is more fun to be had when you're working with a configuration with a large deltaZ between the lifting surfaces and the CG.  Think about a high-wing aircraft with a heavy cargo pod rigidly strapped to the belly of the aircraft.  What about a low-wing aircraft with a heavy pod high above the CG?

Rob

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[Cody]

Hahaha, I think you may be right, the simplifications I used in college seem to be failing everywhere... I took a look at the Theory of Wing Sections, and you're right, at stall the CM_c/4 curve wildly plummets and before then it isn't even a straight line for cambered aircraft.

I'm not sure what you are getting at concerning deltaZ between lifting surfaces and CG, though. My instincts would be that the moment arm around CG is the same regardless of the height of the lifting surface, since lift acts along that line anyway - how does this effect these stability factors majorly? 

[Brandon Litherland]

Don't just think about the lift.  Think about the drag too.  Or even the thrust from the propellers/engines. Or interference with the tail.  Free Body Diagrams are your best friend.

[Cody]

Ah, good point, in retrospect it should have been very obvious. Drag from the wing (or engine thrust, any force parallel to freestream) causes a pitch-up moment around the CG if the CG is well below the wing, like Rob's first scenario, and vice versa if the wing is below the CG, like Rob's second scenario. Not sure how this can effectively be played into XNP calculations, which must be tremendously complicated by this additional consideration since now the neutral point location is also a function of CD... 

[Rob Niewoehner]

Warren Phillips treats the problem of high or low CG in his second edition of “Mechanics of Flight.” The induced drag term has a nonlinear contribution to pitching moment that we ignore when vertical distance between wing and CG are small.

Amphibs and hanggliders are particularly susceptible.