# On which axis do we measure angle of attack and airspeed?

The lift value depends on the relative airspeed of the plane. That's what every website says without anymore explainations.

But as we are talking about lift, are we talking about the relative speed ONLY on the axis of the chord line of the airfoil ? Or the relative speed, not related to any particular axis ?

Now about angle of attack (AOA ):

Is the AOA measured on a specific axis ? Or is it the absolue angle between the airspeed Vector and the chord Line ? Or with other words : does AOA = 20° when the plane is on a 20° side slip ?? That's seems wrong to me but if we relate to the words of wikipedia for example thats what is being said.

Airspeed is always the total magnitude of the free-stream incident on the wing/airplane. In zero sideslip, angle of attack is exactly the angle between free-stream and body x-axis (could be chord for an isolated wing). In non-zero sideslip, it's nuanced; see below.

By industry standard (Ref. Etkins, Dynamics of Flight; Stevens, Aircraft Control and Simulation), the angle of attack ($$\alpha$$) and the angle of sideslip ($$\beta$$) are defined as Euler rotation from the coordinate axis attached to the free-stream such that the speed vector is $$\begin{bmatrix}V_a & 0 & 0\end{bmatrix}^T$$, where $$V_a$$ is the airspeed magnitude, to the body frame (where x-axis aligns with the chord):

$$\begin{bmatrix}u_a \\ v_a \\ w_a\end{bmatrix} = \begin{bmatrix}\cos\alpha\cos\beta & -\cos\alpha\sin\beta & -\sin\alpha \\ \sin\beta & \cos\beta & 0 \\ \sin\alpha\cos\beta & -\sin\alpha\sin\beta & \cos\alpha \end{bmatrix}\begin{bmatrix}V_a \\ 0 \\ 0\end{bmatrix}$$

where $$u_a$$, $$v_a$$, $$w_a$$ are the incident speeds in the body frame.

Simplify, and we have:

$$\alpha = \tan^{-1}\frac{w_a}{u_a}$$

$$\beta = \sin^{-1}\frac{v_a}{V_a}$$

(Image Ref. Etkins, Dynamics of Flight)

Note that when sideslip is small (whereby $$\cos{\beta}\approx1$$), we can reduce the above to:

$$\alpha \approx \sin^{-1}\frac{w_a}{V_a}$$

Therefore, taking angle of attack as the projected angle between chord and free-stream is pretty darn good.

• That's awesome ! Thank you very much. Curious that all the websites are went to did not talk about that. I understood, thank you very much. Apr 23 '20 at 17:18
• @Anselme See the updated last paragraph. This clears up why most people don't go to the nuance.
– JZYL
Apr 23 '20 at 17:20
• Yep, I just saw it. I am running a flight simulation I made and I was computing the lift while always considering the sideslip as 0. But that's completely wrong at sometimes, performing dynamic stalls for example. Thanks Apr 23 '20 at 17:22
• Sorry, I was a bit to confident about your answer. But something still doesnt make sense to me. Let's take an extreme case where the plane is 90° from the relative airspeed. Then w is 0 and the angle of attack (alpha) will still be 0, multiplied by the relative velocity squared, this gives us a prettygood amount of lift,as if it was flying forward. But as no airflow is going in the chord direction NO LIFT is generated. So our model is broken. Do we need to take the velocity parallel to the wind chord ? Apr 23 '20 at 18:12
• @Anselme You're saying there's a 90deg sideslip? Put it to a wind tunnel and you'll see how much lift there is (in CL). Very small!
– JZYL
Apr 23 '20 at 18:45

Angle of attack is always measured as the angle between the chord line and the relative wind. Different aircraft orientations will cause differences in angle of attack between the two or more wings on an airplane or the rotor on a helicopter. For example, stalling an airplane while in uncoordinated flight can lead to a spin as the inner wing achieves a greater angle of attack than the outer wing.

• Thank you for your answer. "Angle of attack is always measured as the angle between the chord line and the relative wind." So that means a plane yawing 20° from the flight path while being straight level and maintaining sufficient speed will be stalling ?? That does not make any sense to me. Apr 23 '20 at 15:55
• It might be stalling, but it might not. It depends on the AoA. The yaw angle is not the same as AoA. (which has already been defined) Apr 23 '20 at 16:00
• It has been defined by : "Angle of attack is always measured as the angle between the chord line and the relative wind". So if the relative wind comes from a 30° yaw angle, then AOA is the SAME as yaw angle. Or maybe the definiton of the AOA works only with 2D vectors and not 3D one ? Apr 23 '20 at 16:02
• @Anselme, Ok, sorry... I see where your confusion is now, and perhaps I wasn't clear. AoA is not a measurement of spanwise flow, it is a measurement of the angle of flow that is parallel to the chord. In other words, you would see this angle from a side view, but not a planform view. (Reference figures in other answer) Does this make sense? Apr 23 '20 at 23:58
• Nice, now all is clear ! Thanks Apr 24 '20 at 6:54