Sunday, 7 August 2016

The True Story of P-Factor, for Pilots

P-Factor, or Propeller Factor (or asymmetric blade effect or asymmetric disc effect), is an important aerodynamic phenomena for pilots to understand, at least it is if your plane has a prop.

P-Factor is often explained to pilots in silly ways using false analogies (No! the blades do not have mouths, one does not take a bigger bite of air than the other!). At AirCrafty we absolutely detest false analogies and believe that anyone using them should be mailed third class to a dark place between two stars. What follows is the True Story of P-Factor:

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Below is an image of an aircraft whose 2-bladed prop has one blade (the descending blade) coming straight out of the screen and the other blade (obviously the ascending blade) going back through the screen. The dashed lines going through each blade represent the chord lines of each blade.
Angle of Attack (AoA), as all good pilots know, is the angular difference between the chord line of an airfoil and the relative wind. If the above prop began spinning while the aircraft was stationary on the ramp the Angle of Attack (AoA) would be the same as the angular difference between a vertical line and the chord line as shown below:
But if the aircraft were moving forward, finding out where the relative wind was coming from would be a little tougher, we'd have to add the velocity of the aircraft to the velocity of the prop as shown below:

Don't be intimidated by the busy graphic, stare at it for five minutes and it will make sense (...because the aircraft is moving forward the wind hits the prop from a more forward direction reducing the angle of attack on the prop blade by an amount equal to the BHA). 
Bonus! BHA stands for "Blade Helix Angle" a term you can use to impress some people at the public pool later today (the blade helix is traced by the prop tips of this C-130).

Now go watch this 3 minute YouTube video, the guy's accent is great and it'll help you solidify your grip on things before we move on.

OK so far we have an angle of attack on the blade but no P-Factor. Now we'll pitch the aircraft up and continue to fly straight and level (neither climbing nor descending) ...slow flight style; as we do this, P-Factor will emerge in mysterious fashion like consciousness emerging from a brain.

You've probably already gazed furtively down at the below graphic. Be intimidated by this busy graphic. Panic, hyperventilate, go through the five stages of grief, then recall the Nietzschen aphorism your dad taught you. Look! the busy graphic didn't kill you! Therefore you're stronger. With this new strength you will now understand P-Factor como un jefe.



There's a couple things going on here:

1. Because we pitched up, the velocity of each blade is no longer perpendicular to the velocity of the aircraft. This means a large component of the aircraft's velocity adds to the descending blade's velocity. For the ascending blade there is now a component of the aircraft's velocity that subtracts from the blade velocity. The addition of a component of the aircraft's motion to the descending blade and subtraction of a component of the aircraft's motion from the ascending blade is the first part of P-Factor. I left the old aircraft velocity vectors that are perpendicular to the blade velocity vector on the above graphic and made them a faint green color so you can still see how things were before we pitched up.

2. I also left the old relative wind vector for the descending blade on the above graphic (but left it off for the ascending blade) and made it a faint blue color, it is mostly hiding behind the new relative wind vector of the descending blade. Notice that the old relative wind vector isn't just shorter (and therefore the relative wind is slower) but it's also at a slightly different angle than the new relative wind vector (the not-faint blue arrow). This new relative wind vector is closer to vertical which means the BHA has been slightly reduced which means the angle of attack has been slightly increased. On the ascending blade the change in angle of attack is negligible in most cases (more on this in the Technical Section below). The increase in angle of attack on the descending blade combined with the negligible change in the angle of attack on the ascending blade is the second and final part of P-Factor.

I'll re-state it all together here...If the relative wind is not perpendicular to blade velocity, a component of the aircraft's velocity adds to the descending blade's velocity and subtracts from the ascending blade's velocity; at the same time, the angle of attack on the descending blade increases while the angle of attack on the ascending blade changes a negligible amount. The descending blade therefore has greater velocity and greater angle of attack than the ascending blade, therefore it creates more thrust than the descending blade, this asymmetric production of thrust causes a yawing moment (a yawing moment which is normally to the left and needs to be counteracted with right rudder), this yawing moment is P-Factor

And that, is the true story of P-Factor.

If you want to be able to quantify the affects of P-Factor rather than just understand it conceptually, and if you want to learn what "PAAoA" stands for, then continue on to the technical section below.

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PAAoA stands for Prop-Axis Angle of Attack. It is the angular difference between the axis about which the prop spins and the relative wind. The PAAoA, in combination with aircraft velocity and prop velocity, determine the magnitude of the moment we call P-Factor. The relationship between these three quantities is shown below:



The black dashed line is vertical, the relative wind (RW) velocity can be calculated using the Law of Cosines, C^2 = A^2 + B^2 - 2ABcos(c) modifying this for the present triangle and solving for relative wind gets us:
Once we have relative wind the BHA can be calculated using the Law of Sines. I won't bore you with the Law of Sines, here's the resultant equation solved for the BHA:

Now I'd like to curse Blogger four times for not building an equation tool into blogger and constantly monkeying with settings causing mathjax and various other LaTeX readers to be unreliable. Curse you Blogger, Curse you Blogger, Curse you Blogger, Curse you Blogger.

 $E=mc^2$.

UNDER CONSTRUCTION



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