You called this a PIO and gave the G- rego tailscrape at LHR as an exemplar of the same. Neither of them are caused by a PIO/APIO. Now, if you provide evidence that Gz and pitch rate gets out of sync with the SSC and elevator deflection then it might be of interest.
I'm not saying the tailscrape at LHR was caused by PIO.
I'm saying that there is something that caused the PIO.
Something that should not have happened, that caused the PIO.
And the same thing that caused the PIO, also caused the tailstrike.
First, do you agree that the triangles on the elevator deflection are clear signs of PIO ?
However, there is not a PIO problem, prior to the tail strike. There is a "flight control dynamics problem"
I zoomed in on the curves available here.
https://assets.publishing.service.go...XWBC_09-22.pdf
If you put the two curves together, you get this picture :
https://i.gyazo.com/7805953bd1bc0dfc...89e5c86214.png
You can see that the elevator follow closely the sidestick input, up until a point...
There is one nose down input, approximately 5 seconds (two and a half squares) before the red event (tail strike). Third black vertical line before the red event.
After that, there is one pitch up order. It is followed by the elevators, but late. And not to the extent that you would expect.
Then, the stick is released a bit. There is still some delay in that.
The stick is pulled again. This time there is a half second delay at the beginning of the pulling.
Then in the middle of the nose-up elevator movement, the blue line slows down. The delay is now one second between the elevators and sidestick pitch order.
As soon as the elevators reach full aft, the pilot releases his input up until the tailstrike. While the pilot releases his input, the elevators remain generally almost at full aft. You can even see, at the event, the pilot order is towards nose down. Has been in a nose down direction for 2 seconds. And the current position of the pilot order is neutral.
However, the elevators are near full aft, and are going full aft. Doing exactly the opposite of what the pilot had been ordering for two seconds.
In the flare, and on the ground, as well as after takeoff, the SSC gives a simple proportional deflection for the elevator deflection
Hence, this is clearly not the case.
Maybe in theory, not in practise.
The deed is done. The PIO occurs just after this moment, but it's already too late.
The triangles were a good indication that something weird was happening before them. And looking into it, it is the case.
There is a lot more to say about this. Notably, what would have happened if the elevators had followed the nose down tendency in the two seconds before tailstrike ? Since it's a very short duration, it's likely the pilot input would have been very similar.
It is possible to plot pitch, pitch rate, pitch rate rate (which is proportional to pitching moment), but I'm fairly sure that everything will be very consistent. It is aerodynamics, except if one aileron was broken in half, for sure everything is going to be consistent.
The discrepancy is between sidestick order and elevator response. The discrepancy during a PIO itself is due to elevator maximum speed being reached. The discrepancy before that is less obvious.
In any case, I think it wiser to pull up seriously when you hear the thrust reaching maximum. Not before that...
As for the rest, your post was really interesting. Some general comments :
I really like the APU missing incident.
We do not have the same notations so it's sometimes not easy to follow. What would V19 be, for example ?
You talking about the C* law shows in-depth knowledge of airbus FBW. It's fun to see how reality is different than the simple stuff told in the FCOM. It would be interesting to teach this in engineering school. Do you mean there is a PI filter on top of this equation, or that just having this equation plugged into the system is enough to make it work PI-like ?
For normal cases, going into all the matrix thingys and even the non linear effects is not really necessary. The simple stuff, pitch rate rate times pitch inertia equals pitch moment, without taking into account the other terms, is more than enough. In this case there is no "cross axis effect", all the angles are small..