# What happens when the eigenvalues of the longitudinal A matrix in dynamic stability analysis, are NOT two sets of complex conjugate pairs?

The complex conjugate pairs of eigenvalues of the longitudinal A matrix characterize phugoid and short period dynamic responses, this information is well established.

My question is - what happens when the eigenvalues consist of 2 real roots and 1 complex conjugate pair? Can this happen (assuming no error on input, or does it only happen with an error?) and if it does what does this say about the dynamics of the system? Examining the eigenvectors I'm wondering how to interpret two eigenvectors that consist of all real values?

Thanks!

• Does this have anything to do with aviation? I have no idea. Looks like it might be better suited to Math.SE, perhaps.
– Ralph J
Jan 20, 2021 at 12:44
• @RalphJ, it definitely has something to do with aviation, specifically aircraft design. Definitely not appropriate for math.se, because it concerns physical interpretation. physics.se, might work, but it's not off topic here either. Jan 20, 2021 at 12:48
• @JanHudec Fair enough. Close vote retracted.
– Ralph J
Jan 20, 2021 at 15:49
• So, I saw this in the Close queue, which didn't show any answers and thought, most people will go to XKCD for an obligatory answer. I remember I thought, how long before we get a Peter Kampf obligatory answer. Got through the queue, came here and .... :) Jan 20, 2021 at 17:39

what happens when the eigenvalues consist of 2 real roots and 1 complex conjugate pair?

This means that the one complex conjugate root is still an oscillation but the other has split into two aperiodic movements. One of them should be a positive and the other a negative real value. While the negative root signifies a damped, aperiodic movement, the positive root shows that the airplane has one instable, divergent mode.

Can this happen?

Yes, for example when the center of gravity is moved aft of the neutral point, the phygoid will split into two real roots. The positive one describes the divergent behavior when static stability has been lost.

Backgound:

The equations of motion describe how acceleration, speed and pitch rate of the airplane react to small changes of the control parameters (elevator angle and throttle setting). They are second order differential equations and can be solved similar to systems of quadratic equations where the coefficients form the elements of a square matrix. The eigenvalues of the matrix describe frequency and damping of the movement which is caused by changes to the control parameters while the eigenvectors describe how acceleration, speed and pitch rate are affected in detail.

Note that this only concerns longitudinal movement (forward-back, up-down and pitch). The equations can be expanded to cover roll, yaw and left-right movement as well, but the question only concerns the simpler longitudinal part.

• That would make sense in the case I'm evaluating Cm_alpha is positive. Do you have any references for this, a text, paper, link etc I'd like to learn more about this! Is that the only scenario in which this occurs? Thanks :) Jan 20, 2021 at 16:49
• +1, but probably really shouldn't be ... after all, the upvote description is "this answer is useful" ... seeing that I can read English words all in sentences, it's an answer; it's probably a valid answer.... but to me? Not very useful.... well done Peter, as always. Jan 20, 2021 at 17:42
• @CGCampbell can you be little more clear on what you mean? Are you complaining about the quality of the answer or the question? Jan 20, 2021 at 17:53
• @spacegirl1923. The last 5 words are the key to CGCampbell's comment -- most folks who fly airplanes have no idea what eigenvalues are & what they're used for. Peter, being both a pilot and an engineer, is an exception to that generalization. Added to which, he's wicked-smart on this stuff & exceptionally willing to help out with questions that most of us don't begin to comprehend. Nothing wrong with the question -- if Peter believes that it's worth the time to answer, then your question was fine. And I'd trust an answer you get from Peter any day of the week, and twice on Sundays!
– Ralph J
Jan 20, 2021 at 23:07
• @Kolom: Of course, with computer control everything is possible. Until the unstable frequency exceeds the rates of the servos needed to move the control surfaces quickly enough, that is. But I wonder how a complex software can be modelled by a second order inhomogeneous differential equation so it can be shoehorned into the system of equations ;-) Sometimes it is just enough to increase the damping term (first derivative) and have a negative second order derivative (negative feedback) but that doesn't do the limits of software any justice but assumes it works perfectly in all cases. Jan 22, 2021 at 7:05