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I found an equation to calculate angle of attack based on the velocity vector in the body frame of the aircraft.

$\arctan(v_x/v_z) = \alpha$

1) Is this correct?

When I input in a $v_z$ approaching zero, the value approaches $\infty$ which means that angle of attack moving only in the x direction is 90 degrees since $\arctan(x)_{x\rightarrow \infty}= \pi/2$

2) Is there supposed to be an assumption that you never calculate Aoa of zero with this?

3) Am I misinterpreting what the velocities and angles represent?

Overall I am just a bit confused, because I kind of expected something that wouldn't mess up at level flight.

Boto
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1 Answers1

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It is exactly the other way round:

$$ \alpha = \mathrm{arctan} \left( \frac{v_z}{v_x} \right) $$

You can simply derive it from the trigonometry in a triangle: sketch In this image the aircraft is moving to the right (along the arrow labelled v) and the $x$-axis is pointing along the aircraft body.

The resulting function looks like this: angle of attack

  • For $ v_z = 0 $, you get $ \alpha = \arctan(0) = 0 $.

  • For $ v_x \to 0 $, you get $ \alpha \to \frac{\pi}{2} = 90^\circ $.

Bianfable
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  • It looks like you are using a Vx and Vy defined relative to the horizon, not the aircraft body frame. It looks like what you are describing is the climb or glide angle, not the angle-of-attack. You can fix the first diagram by relabelling your V as Vx, and draw your Vz perpendicular to Vx and pointing generally downward (and a little forward) , and relabel your Vx as V. This portrays the AoA of an aircraft in the illustrated pitch attitude in level flight ( i.e. horizontal trajectory). Of course the illustrated AoA would be rather extreme. – quiet flyer Aug 01 '19 at 18:02
  • Anyway for a diagram redrawn this way, your comments about the correct formula are correct- it needs to be arctan (Vz/ Vx). The comment about an angle-of-attack of 90 degrees seems a little odd, though technically true. – quiet flyer Aug 01 '19 at 18:03
  • Reference for direction of x, y, and z in aircraft body frame: https://en.m.wikipedia.org/wiki/Flight_dynamics_(fixed-wing_aircraft)#Reference_frames – quiet flyer Aug 01 '19 at 18:04
  • @quietflyer You are absolutely right! I fixed the sketch. Thanks :) – Bianfable Aug 01 '19 at 18:12
  • Also if Vz is zero, using Vz as defined in question (aircraft body frame), this means angle-of-attack is zero. Says nothing about whether or not aircraft is in level flight. So, this answer could use some improvement, even though it did give the correct formula for angle-of-attack. – quiet flyer Aug 01 '19 at 18:13
  • You are welcome; looks like you were fixing as I was continuing to type more. Anyway you are welcome. – quiet flyer Aug 01 '19 at 18:14
  • I removed the false comments about level flight. That would only have been true in the incorrect reference frame... – Bianfable Aug 01 '19 at 18:18
  • This is only true without wind, of course...? – Cpt Reynolds Aug 01 '19 at 19:08
  • This is a good start building a backup AOA system that does not require a physical AOA sensor (that can break). Just as human logic and laws allowed fewer lifeboats on the Titanic, the current disaster should lead to more redundancy. This will work in wind because the triangle can be drawn based on IAS and rate of ascent/descent (if any). From there angle of nose to horizon is compared with direction of flight relative to horizon (with incidence of wing to nose, for you B-52 fans, added). Note this can be done with existing instruments many times a second by computer. – Robert DiGiovanni Aug 01 '19 at 22:39
  • @RobertDiGiovanni but think about what happens, say, if you deploy fuselaged- mounted spoilers or even wing - mounted spoilers, or for that matter just change power setting, all of which could in theory have no influence on angle-of-attack. So , clearly for this vector diagram to relate to AoA, Vz must be relative to airmass not ground (relates to updrafts/downdrafts), and also must be measured in the exact direction drawn ( perpendicular to Vx), not vertically. Also good point about diagram as drawn assuming zero incidence. Vx also must be relative to airmass (i.e.airspeed). – quiet flyer Aug 01 '19 at 23:38
  • @quiet flyer there would be some error in what I proposed from updraft or downdraft, but it would be in the ball park and a good 3rd reference to compare. It seems they need to make their AOA sensors a bit sturdier. I'd think about putting an extra just outside my window where a rear view mirror would go in my car. – Robert DiGiovanni Aug 02 '19 at 05:29