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Why is the sideslip angle, $\beta$, defined as

$$\beta=sin^{-1}\left(\frac{v}{|{\bf v}|} \right)$$

and not the same as freestream azimuth angle $\psi$

$$\psi=tan^{-1}\left(\frac{v}{u} \right)$$

where ${\bf v}=u \hat {\bf b}_x+v \hat {\bf b}_y+w \hat {\bf b}_z$ is the airspeed of the aircraft in body coordinates, $\hat {\bf b}$, and $|\cdot|$ is euclidean norm of $\bf v$. $\psi$ can be interpreted as the angle between $\hat {\bf b}_x$ and the projection of $\bf v$ onto the bodies $x$-$y$ plane. What is the physical interpretation of $\beta$?

Thanks in advance!

Notation

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    $\begingroup$ Is it helpful to consider the case of knife-edge flight, i.e. flight at a 90-degree bank angle, with the side of the fuselage supporting most of the aircraft weight? $\endgroup$ Jun 19, 2019 at 18:39
  • $\begingroup$ @quietflyer, thank you, it is a thought provoking suggestion but I am struggling to gain any insight from it haha $\endgroup$
    – eball
    Jun 19, 2019 at 19:09
  • $\begingroup$ Likewise I am not familiar enough with this highly mathematical approach to provide you a quick answer-- $\endgroup$ Jun 19, 2019 at 19:21
  • $\begingroup$ Where did you find these definitions, please? $\endgroup$ Jun 19, 2019 at 21:29
  • $\begingroup$ For $\beta$ see PDF page 341: google.com/url?sa=t&source=web&rct=j&url=http://…. I don't have a reference for the freestream azimuth $\psi$...but it seems to be how most people think of sideslip. $\endgroup$
    – eball
    Jun 20, 2019 at 20:41

2 Answers 2

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Dispensing with the math, because it makes my head hurt...

Azimuth refers to geographic direction relative to a compass reference point, so azimuth angle refers to the deviation between the body's longitudinal axis and its horizontal path parallel to the earth's surface (heading vs track along the ground).

Sideslip angle refers to the deviation, referenced along the lateral axis of the body (parallel to the wings in other words), between the body's longitudinal axis and the airflow, independently of direction relative to the earth.

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I think this answers my question. If anyone wants more details please let me know:

The sideslip angle, $\beta$, is the angle which the stability coordinate system needs to be rotated about the $z$ stability axis after being rotated by the angle of attack, $\alpha$, about the body $y$-axis. The sideslip angle is equal to the freestream azimuth angle $\psi$ when the angle of attack is zero.

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