What is the derivative of the angle of attack?

Suppose a flying missile's velocity vector has components $$u, v, w$$, as illustrated below.

From my knowledge of flight dynamics, the angle of attack is

$$\alpha = \arctan \left(\frac{w}{u}\right)$$

where $$u, w$$ are time-varying. Then,

$$\frac{d\alpha}{dt} = \frac{1}{1+\frac{w^2}{u^2}} \left(\frac{w}{u}\right)'= \frac{w'u - wu'}{u^2+w^2}$$

Some books indicate

$$\frac{d\alpha}{dt} = q + \frac{Z}{mu}$$

where $$q$$ is the second component of angular velocity, $$Z$$ is the third component of the total force, and $$m$$ is the mass of missile. How can these two formulas be the same?

They are two equations describing two different states.

The first equation is the momentary state of $$w$$ and $$u$$ without regards as to what causes variation of $$w$$ and $$u$$.

The second equation is the change in angle of attack as an inertial response to a force or moment acting on the missile.

• I get the idea but how could we derive the second equation explicitly? – Dat May 5 at 15:42

Assuming that there isn't an error in the books you've read:

Both formulas are true. It's true that

$$\frac{d\alpha}{dt} = \frac{w'u - wu'}{u^2 + w^2},$$

and it's also true that

$$\frac{d\alpha}{dt}= q + \frac{Z}{mu}.$$

From this, we can conclude that

$$\frac{w'u - wu'}{u^2 + w^2} = q + \frac{Z}{mu}.$$

Both of these two formulas describe the way that the angle of attack changes, but they describe it in different ways.

I'm not sure exactly how the $$q + Z/mu$$ formula was derived, but it seems to be related to the fact that angle of attack is related to pitch and the flight path. Specifically, for flight with zero bank angle, the angle of attack is the pitch angle minus the climb angle. It looks like the $$q$$ term comes from the rate of change of pitch, and the $$Z/mu$$ term comes from the rate of change of climb angle.

• Yes, I believe they are both true but how can we formulate from one to another? I don't understand how to derive formula 2 – Dat May 4 at 0:35
• @Dat I added another paragraph describing my best guess about the $q + Z/mu$ formula. – Terran Swett May 4 at 17:43
• I see, I understood the q term but still can't get the Z/mu term. The second formula is from en.wikipedia.org/wiki/Flight_dynamics_(fixed-wing_aircraft) – Dat May 5 at 12:46

I think the equations are not referred to the same axys.

The first equation refers to the angle of attack, the angle between both components of the velocity refered to the body axys (x along the body), using a non-inertial frame (as moves with the missile) . Well, actually following some comments, is not actually the angle of attack, it will be the angle of attack in the specific case of v=0.

The second equations looks to be referring to the rotation of the missile referred to an inertial frame of an external observer (for example, Earth), so actually, it looks like the actual movement of the body frame refered to that observer in a specific rotation component .

Imagine an example, that I am able to provide the right thurst to the missile in such a way that I keep u, w and q constant rotating in a circle.

In this case, the angle of attack will be constant and its increase 0 with time but the second alpha will be increasing/decrasing with a constant increase/decrease, also this change will be linked to the circle rotation.

I think you can also make the math and relate both equations (you will need more components!) taking care of inertial forces.

Final comment, as you are introducing as well a second term in teh second alpha equation, it really looks that it is an specific case in the movement of missile.