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When continuous rudder is applied in a typical light aircraft during straight and level flight at "normal" flying speeds and altitudes, the primary effect is that the aircraft will yaw to the left - so it's flying "sideways" to some extent.

It is commonly understood amongst pilots that the secondary effect in most aircraft, is that the aircraft will also start rolling in the direction of yaw (i.e. applying left rudder causes left yaw, and left roll).

What causes this roll? And does the Dihedral angle have anything to do with it?

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I remember now... I experience it each time when I fly in San Andreas (Err... I meant the game) –  Waffle's Crazy Peanut Jan 14 '13 at 13:19
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3 Answers 3

Rody and Mike almost got it right. :)

Most aircraft are designed with swept wings. That is the primary mechanism that gives the roll effect to an airplane that may only receive a yaw input. if you look at this picture:

A typical swept wing aircraft from Wikipedia commons

You can see that both wings have a backwards sweep to them. Now, if you introduce a yaw to the aircraft, one wing will extend out more directly into the wind-stream, while the other wing will be even more swept. This effectively makes one wing longer, and the other wing shorter. Like in this image (this image is actually displaying a more extreme case that also involves boundary layer separations, but that is beyond this answer):

Wing AoA in a Yaw from a public training site

Actually, this picture displays it exactly as I was taught in USAF flight school::

Yaw effects on lift of swept wing aircraft from Langley Flight School, standard textbook image

The longer wing will generate more lift, and the shorter one will generate less lift. And since there is unequal lift around the roll axis, the airplane will roll, and continue to roll.

Of course, with more lift comes more drag, so that will counter the lift and pull the wing back (Causing an effect known as "Dutch Roll"). Many aircraft have a device called a "yaw damper" to counter this (or else you will feel quite queasy flying). Dutch roll is demonstrated by this GIF:

Wikipedia Commons Dutch Roll GIF

The reason that Mike's answer is not totally correct is from Figure 2 in his wiki link. Note that the words "non-zero" are included in this diagram:

Wikipedia Commons Dihedral Roll causation diagram

That means that theoretically, if one were able to yaw an aircraft with dihedral perfectly, the aerodynamic forces would not be the causal factor. Also, most aircraft that have a dihedral or anahedral configuration also have a wing sweep, so that is the overall factor that is at play.

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The small Cessnas I know have dihedral, but not any sweep, that I'm aware of. Diagram C is really confusing. I'm sure the sweep of the forward wing increases roll moment just because it sticks out further, but it says the trailing wing has more AOA and the leading wing has less. How can that be (assuming there is upward dihedral angle)? My sense of geometry says the opposite. If the dihedral angle were negative (anhedral) then I could see it. Maybe that's how your fighters are made. –  Mike Dunlavey Jan 12 '13 at 1:39
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@MikeDunlavey The last diagram also gets to the problem of "Dutch Roll" because there is a lift/drag relationship that plays into the whole situation. An aircraft is constantly in a state of dynamic change. –  Larian LeQuella Jan 12 '13 at 2:29
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@ColinK true, but the dihedral (or anehedral) effects are a much smaller part of the equation (unless extreme) than the sweep. And if you look at figure two in the wiki link, the key words are "non-zero sideslips" meaning that theoretically if you induce yaw with zero bank at the same time, you don't get that rolling effect induced because of the dihedral (albeit nearly impossible to do in the real world). –  Larian LeQuella Jan 12 '13 at 3:54
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Hey @Larian, I guess we're going to disagree here :) My understanding is any rudder displacement, regardless of bank, is going to produce a roll moment in an aircraft with dihedral. I was taught, and it was definitely handy for hands-off flying, that I could hold course with just my feet, in a 172 (that seems to have no sweep). –  Mike Dunlavey Jan 12 '13 at 17:11
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@MikeDunlavey You are partially correct, don't get me wrong on that regard, however it's more than just the effects from dihedral. The fact that the rudder is above the plane of the center of gravity in the airplane also plays into the effect of the roll you get. In attempting to isolate just one factor in a dynamic and complex situation like an airplane flying, you get A LOT of other considerations to deal with. –  Larian LeQuella Jan 12 '13 at 19:18
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There's quite a few factors at play here, and granted, it's been a while since my last flight dynamics class :)

After a few iterations with the other answerers below, here is a (still not yet complete) list of reasons the effect will occur:

  1. While yawing, the left and right wings will have slightly different speeds. This difference in wing speed causes slightly more lift on one wing than on the other, thus inducing a rolling moment (more lift usually also means more drag, so this effect dampens the yawing motion). In a spirally unstable aircraft, any non-zero initial roll will grow due to said effect, in principle without bounds (the aptly named "graveyard spiral").

  2. Rudder deflection causes an aerodynamic moment in the yaw direction, but as rudders on many aircraft are built into the vertical vertical tail section, this moment also has a small arm w.r.t. the roll axis, thus causing a rolling moment.

  3. Large yaw angles could create an asymmetrical interference between the fuselage and both wings; one wing will be directly in the airflow, while the other is in the "wind shadow" of the fuselage. This causes a difference in lift (and larger induced drag on the shadowed wing), causing a rolling moment.

  4. An aircraft's design may employ a dihedral angle, which is mostly there to stabilize the spiral mode. However, after applying a(n) (impulsive) yaw angle, the aircraft will side slip, causing a different angle of attack on the two wings. This will again cause a difference in lift between the two wings, therefore causing a rolling moment (which will eventually stabilize).

  5. An aircraft's design may employ wing sweep to mitigate adverse effects in the transsonic and supersonic regimes. When an aircraft with swept wings is side-slipping after applying a yaw, the relative wind on the outward wing will be more aligned with the chord line of the wing's cross section than the inward wing. The net effect is the same as if the wind speed would be faster on the outward wing then on the inward wing, therefore, a rolling moment is caused. Note that this is only true for backward-sweep angles; the adverse is true for forward-swept angles (which is part of the reason not many aircraft have a forward sweep angle).

  6. More to come.

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I'm not sure I'm convinced: While the aircraft is yawing, i.e. while the nose's compass heading is changing, the wings will have different speeds, and therefore one will generate more lift. However, this condition (different wing speeds) only lasts a few seconds, yet the roll continues? To me it seems like this will account for only a portion of the roll. –  Johan Jan 11 '13 at 10:48
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@Johan: It will initiate a small roll. For roll-unstable aircraft, any non-zero roll angle will amplify and grow due to the same effect, irrespective of whether there is still yaw or not. –  Rody Oldenhuis Jan 11 '13 at 10:53
    
Right... perhaps I should have mentioned that I have your typical light aircraft in mind. I believe that these normally do have a dihedral angle. I just scanned the wiki page on aircraft modes you refer to, and what I understand from there, is that a dihedral should damp out that effect? –  Johan Jan 11 '13 at 10:56
    
I edited the question to clarify that I'm talking about light aircraft, and to add the secondary question "does the dihedral have anything to do with this effect" –  Johan Jan 11 '13 at 11:06
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@Johan: Slightly, yes. Of course (if I remember correctly), the value of the dihedral angle that perfectly corrects parasitic roll for some applied yaw rate depends on the specific value of that yaw rate. The dihedral angle is there mostly to make the aircraft stable, i.e., the spiral instability is much less likely to occur. But this does not mean no roll at all will occur -- it only means that the induced roll will slowly go back to zero, rather than grow beyond control. –  Rody Oldenhuis Jan 11 '13 at 11:08
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I want to replace my former answer with this one.

The on-line book by John S. Denker has a very good chapter on this subject.

He outlines a number of reasons why there is slip-roll coupling:

  • dihedral effect

  • wing-sweep effect

  • updraft/downdraft at wing roots caused by off-center flow

  • "wind shadow" caused by off-center flow

  • torque about roll axis from rudder displacement and other surfaces exposed to asymmetric wind

  • propwash (which actually causes some adverse roll moment)

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Congratulations, Mike! This is the only answer which contains all factors, all other answers are not complete. Makes me wonder why you did not get more points ... maybe this topic belongs to Aviation StackExchange. –  Peter Kämpf Jun 6 at 20:45
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