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In the few rudder incidents with the B737, rudder-freeze caused the airplane to roll. From what I know, the rudder causes yaw while the aileron causes roll. Why in the case of these accidents/incidents would the rudder cause a roll?

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This has already been asked on physics.SE although not here, strangely. – Pondlife Jan 3 at 18:21
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Think about the Dutch roll that can be the result of a yaw perturbation. Yaw causes slip, slip causes roll, roll is reversed by the yaw stability. The yaw damper minimizes this sequence. Good article on stability and how effects are interlocked: Flightlab Ground School - 4. Lateral/Directional Stability. Related topic, page 4-13. – mins Jan 4 at 7:47
up vote 21 down vote accepted

There are many higher-order effects. For example, yaw causes a slight increase in airspeed over the outer wing of the turn and a decrease in airspeed over the inner wing. As a result, the outer wing will generate more lift than the other, introducing a rolling motion.

Yaw also causes side-slip which triggers several effects. Wings that are swept backwards, as on the B737, contribute to the roll towards the direction of the turn, because the outer wing gets effectively longer (visual explanation) and produces more lift. Further contributing to this roll is the outer wing having a higher angle of attack due to the dihedral effect (visual explanation).

Similar effects apply to other control surfaces. It's not true that rudder only causes yaw and ailerons only cause roll. Those are simplifications, even though aircraft are usually designed for these simplifications to hold fairly well.

There is also a direct effect. The image below didn't show the center of mass; I drew in a guess as a yellow circle. It's clearly below the rudder, which is true for most aircraft. This means applying rudder introduces a rolling torque, roughly equal to the sideways component of the force on the control surface, multiplied by the distance between rudder and the longitudinal axis. The magnitude of this product is visualised as the area of the red rectangle.

B737 front view

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The two effects - direct and yaw-induced appear to oppose each other, right? – Brian Drummond Jan 3 at 21:06
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@BrianDrummond: On many aircraft, in many flight conditions: yes, the direct effect is in the direction of the turn and the net torque of indirect effects is in the opposite direction, but that's not true in general. E.g., some less orthodox aircraft designs have a tail fin below the longitudinal axis. This tail would cause a plane to roll left when yawing right and vice versa. Also, many fighter aircraft have anhedral wings, reversing the dihedral effect to reduce stability. My guess is such exceptions are rare, but I don't have the numbers to back that up. – Marcks Thomas Jan 3 at 22:38
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" It's not true that rudder only causes yaw and ailerons only cause roll." that's why we call them coordinated turns – rbp Jan 4 at 13:49
    
Agreeing with @Marcks Thomas. If you look at the Predator drone, you'll note that it uses an "inverted vee" tail, where the control surfaces are part elevator, part rudder. If you deflect both surfaces to port, it will do two things: a) cause you to yaw to port, because the deflection is aft of the center of gravity, and b) cause you to roll to port because the deflection is BELOW the center of gravity. In that regard, the Predator typically doesn't need to use the ailerons to achieve a proper, coordinated turn. – Meower68 Jan 5 at 21:19

After stopping by at our physics friends, it's probably because of the swept wings (causing a lift differential) of the B737:

[...] 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. [...] 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.

The whole answer: Physics: What causes an aircraft to roll when rudder is applied

Note: Regarding the comments, this effect even occurs without swept wings. Swept wings only amplify the effect.

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Note, that A320 mechanical law (the last fallback when all electricity fails) only has elevator (stabilizer actually) and rudder control and relies on this effect for making turns. The aircraft was demonstrated fully controllable in this mode. – Jan Hudec Jan 4 at 8:37
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Although it does have an effect, even an aircraft without swept wings experiences a roll force when rudder is applied. – Ben Jan 4 at 8:49
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This is an incomplete answer, possibly misleadingly so. The lift differential is the key here, not the wing sweep. – egid Jan 4 at 20:38
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Agreed with the other comments. The wing sweep plays a part, but it's only a part and this happens in aircraft that don't have swept wings. For example, I've been in a Cherokee that was flown around the pattern with no aileron input. – reirab Jan 4 at 23:35

The reasons can be split in two categories: Direct and indirect.

Direct reasons are rolling moments which are created directly due to the rudder deflection and the side force on the vertical tail:

  1. Offset position of the vertical tail: Since the rudder is above the longitudinal axis of inertia, a side force will also cause a rolling moment. This is the main reason for the anhedral of the Lockheed F-104.
  2. Once the yawing motion starts, the forward-moving wing experiences a slight increase in dynamic pressure, so lift is increased on the forward moving wing and vice versa. This effect is inverse to flight speed and small in general.

Indirect reasons are caused by the resulting sideslip angle once the aircraft starts to yaw:

  1. Dihedral: Once the rudder-caused yaw produces a sideslip angle, the relative difference in angle of attack between both wings creates a rolling moment. This is proportional to the dihedral angle in flight, so you need to add wing bending to a static view of the aircraft to get an idea of the effective dihedral angle.
  2. Wing position: On a high wing, the changed flow around the fuselage in sideslip will create a difference in angle of attack at the root of the wing. This effect needs to be considered when selecting the proper dihedral angle; therefore high-wing aircraft have less dihedral than low-wing aircraft.
  3. Wing sweep: The reduced sweep angle of the windward wing will increase its lift curve slope, and the reverse happens on the opposite wing. This creates a strong rolling moment which is proportional to the sweep angle.

Once the sideslip angle increases, the rudder-induced side force will decrease until the aircraft reaches the trim point where the sideslip angle fully cancels the rudder deflection. But the aircraft is still yawing, so it will exceed the trim position and now the side force will act in the opposite direction. Proportionally to the side force, the tail-induced rolling moment will also change due to the buildup in sideslip.

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I kept looking for your name, I knew this one would attract you! Love that it's a simple comprehensive list of causes. Wish I could upvote more than once. – egid Jan 4 at 20:40
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@egid: I am currently at CES Unveiled and have little time for edits. Typed it this morning in the hotel room when I wasn't satisfied with the other answers. – Peter Kämpf Jan 4 at 21:15

When an aircraft yaws, one wing travels faster than the other creating more lift on that side. On a swept wing aircraft the lifting effect is even more pronounced. The extra lift on one side will cause it to roll.

enter image description here

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And the right wing is better exposed to the airflow than the left wing which is partially in the shadow of the fuselage. – mins Jan 4 at 7:40

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