# How to calculate the stability derivative of Pitch Moment with respect to AOA rate?

How to separate the stability derivatives: $C_{M\dot \alpha}$ and $C_{Mq}$.

I can imagine a $q=const, \dot \alpha=0$ movement like this But what a $\dot\alpha=const, q=0$ movement would be look like?

• What is the actual question here? Is what the $\dot\alpha = const, q=0$ movement looks like, or is it how do you derive $C_{M\dot\alpha}$ and $C_{Mq}$ from flight test data, i.e. what manoeuvres do you need to perform to acquire the necessary data? – DeltaLima Oct 10 '15 at 13:10
• @DeltaLima I mean what the movement of $\dot \alpha=const,q=0$ would look like – CatDog Oct 25 '16 at 23:38
• Wow, one year and 16 days to respond to a clarification question... Welcome back! – DeltaLima Oct 26 '16 at 7:29

$q = 0$ means no change of pitch angle. You also have a constant rate change in AoA ($\dot{\alpha} = const$).

The difference between the pitch angle ($\theta$) and AoA ($\alpha$) is the flight path angle ($\gamma$).

$\theta = \alpha + \gamma$

From this it follows that the flight path angle changes opposite to the change of change of AoA.

$\dot{\gamma}=-\dot{\alpha}$

This is not a manoeuvre that will happen in real life. An approximation of such a situation can happen when after a brief but strong pull-up the pitch angle is kept constant while the path angle of the aircraft is still increasing. The rate of change of the AoA will not be constant in such a manoeuvre.

• Why not in real life? If the aircraft slows down so lift decreases, it will accelerate downwards without changing pitch. Voilá, the angle of attack increases and the pitch stays constant. – Peter Kämpf Oct 9 '15 at 21:52
• @PeterKämpf I agree with what you say and thought of that situation as well, but it does not change the angle of attack at a constant rate as per the original statement. – DeltaLima Oct 9 '15 at 21:59
• @Federico Thanks for the correction, I'll buy you a beer soon :-) – DeltaLima Oct 9 '15 at 22:00
• Depends on the rate of speed change. Yes, this is artificial, but so is the condition of a constant angle of attack change. By adjusting the speed, you can tailor the downward acceleration and thus the AoA rate. – Peter Kämpf Oct 10 '15 at 4:23

If you add wind (a gust), the sought-after situation would be flying into a vertical gust. If the vertical wind speed changes linearly with the distance flown and the aircraft has sufficient inertia and/or low static stability, you get no pitch movement and a constant angle-of-attack rate, albeit only for a short time.

• Thanks, I can imagine the situation. Adding wind is really helpful to make an intuitive image. – CatDog Oct 25 '16 at 23:46