4

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? Longitudinal Equations from Etkin Reid

Thanks!

spacegirl1923
  • 435
  • 2
  • 8
  • 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 '21 at 12:44
  • 5
    @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 Hudec Jan 20 '21 at 12:48
  • @JanHudec Fair enough. Close vote retracted. – Ralph J Jan 20 '21 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 .... :) – CGCampbell Jan 20 '21 at 17:39

1 Answers1

5

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.

Peter Kämpf
  • 231,832
  • 17
  • 588
  • 929
  • 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 :) – spacegirl1923 Jan 20 '21 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. – CGCampbell Jan 20 '21 at 17:42
  • 1
    @CGCampbell can you be little more clear on what you mean? Are you complaining about the quality of the answer or the question? – spacegirl1923 Jan 20 '21 at 17:53
  • @spacegirl1923: Try this. Or this – Peter Kämpf Jan 20 '21 at 22:55
  • 2
    @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 '21 at 23:07
  • @spacegirl1923 Exactly what RalphJ said. Peter Kampf is that rare breed of a wicked smart and talented (obviously) engineer, who is also an avid pilot (and if I recall correctly, glider pilot) who has the ability to answer technical questions succinctly and in language understandable to his audience. You asked a question I could never understand and Peter answered it to your satisfaction. Not the first time he has done that, nor will it be the last, I am sure. It was my ham-handed way of complimenting him. – CGCampbell Jan 21 '21 at 12:08
  • @CGCampbell I hope my last edit made the answer more useful. Well argued criticism is always welcome. I have learned a lot here from researching answers to questions I would never have asked myself. – Peter Kämpf Jan 21 '21 at 13:39
  • @PeterKämpf you're first unedited answer was sufficient, the last paragraph above I assume is for other readers of this question :) the Princeton ref where it breaks down the effects of the dimensional derivatives on roots of the characteristic equation is really what I was hoping for! Twas feeling a bit lazy and didn't want to work out all the math for myself - thank you. It's an unconventional design, with several unique parameters, but I hadn't made the connection that + cma would move a set of eigenvalues out of the complex plane. Good to know! – spacegirl1923 Jan 21 '21 at 22:30
  • @RalphJ & CGCampbell - thanks lol :D being an infrequent visitor to this forum I hadn't picked up on that. – spacegirl1923 Jan 21 '21 at 22:32
  • @PeterKämpf are there cases where roots are real and both are negative. Such as, suppressing phugoid with flight control computer and it becomes non-oscillatory? – Kolom Jan 22 '21 at 03:38
  • 2
    @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. – Peter Kämpf Jan 22 '21 at 07:05