roulis
Pas d'accord! Well not entirely d'accord anyway.
and the pilot gives a second oscillation to try to stop the dutch roll "problem", starting resonance.
My gripe is that the pilot giving a second oscillation is actually the problem since it is the pilot actions that in this case start off or aggravate the resonance.There are folks here that would think it presumptuous for a mere engineer, no matter how well qualified, to offer any advice on how to fly airplanes, but luckily Dave Davies has already done it, very concisely and very precisely:
The control of divergent dutch roll is not difficult so long as it is handled properly. Let us assume that your aeroplane develops a diverging dutch roll. The first thing to do is nothing - repeat nothing. Too many pilots have grabbed the aeroplane in a rush, done the wrong thing and made matters a lot worse. Don't worry about a few seconds delay because it won't get much worse in this time. Just watch the rolling motion and get the pattern fixed in your mind. Then, when you are good and ready, give one firm but gentle correction on the aileron control against the upcoming wing. Don't hold it on too long - just in and out - or you will spoil the effect. You have then, in one smooth controlled action, killed the biggest part of the roll. You will be left with a residual wiggle, which you can take out, still on ailerons alone, in your own time.
Don't attempt to correct the manoeuvre with rudder; as explained, the yaw is often suppressed and it is difficult to work out which way to apply the rudder at what particular moment and there is a good chance that the wrong rudder will be applied which will aggravate things very quickly. It is not difficult, however to apply the correct aileron control. Further, don't attempt to squash the dutch roll flat in one fell swoop but be content with taking out a big bite first time then sorting out the remainder next time
Clearly, Davies at least regards the correct method as a type of open loop process. Note also his remarks on the difficulty of suppressing dutch roll oscillations by the use of rudder.
I know very well that if a pilot decides to actively control a dutch roll using aileron then unless he properly adapts his internal transfer function to match the open loop dynamics of the airframe there is a risk that at some value of pilot gain the root locus of the closed loop system might cross over into the negative damping region. [That piece of jargon ridden technological gobbledygook translates as "If the pilot tries to chase the dutch roll and gets his timing wrong he could end up in a PIO or APC situation"]
From what you have written this might have been the case on that Learjet?
What we need is a new point of view that does not exist in our regulations of certification concerning RESONANCE. It could be builded on Machinbird's post and enhanced from his first text and his experience of PIO and high level flight experience. His post with Bode figure with numbers is a good beginning of what we have to think about the dynamic system.
In any case what is missing in our certification and knowledge is a quantified reference to dynamics of the transient parts of the plane's oscillation, Pio and other APC.
Here I do disagree with you. The object of certification is to avoid any resonance not devise rules to tame it. In fact the existing regulations do just that:
§ 25.181 Dynamic stability.
(b) Any combined lateral-directional oscillations (“Dutch roll”) occurring between 1.13 VSR and maximum allowable speed appropriate to the configuration of the airplane must be positively damped with controls free, and must be controllable with normal use of the primary controls without requiring exceptional pilot skill.
Part 23, against which I believe that Learjet to be certificated, is slightly different :
§ 23.181 Dynamic stability.
(b) Any combined lateral-directional oscillations (Dutch roll) occurring between the stalling speed and the maximum allowable speed (VFE , VLE , VN0 , VFC /MFC ) appropriate to the configuration of the airplane with the primary controls in both free and fixed position, must be damped to 1⁄10 amplitude in:
(1) Seven (7) cycles below 18,000 feet and
(2) Thirteen (13) cycles from 18,000 feet to the certified maximum altitude.
If that aeroplane was a susceptible to dutch roll problems as your posts suggest then it must be on the limit for certification surely? Arguing from that case to a need for another set of requirements strikes me as an extrapolation from the particular to the general - and rather like extrapolating from a single spot point, that is a dodgy process.