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I understand that rudder deflection causes several things to happen. Primarily the aircraft yaws, and for a positive (trailing edge left) rudder deflection there will be a positive side force generated and negative yawing moment to yaw the plane nose left. I also understand that there is a direct secondary effect of rudder in that the side force acting at a distance above the roll axis of the aircraft creates a rolling moment (positive rolling moment to starboard for a positive rudder deflection) and there is a yaw-induced effect where as the aircraft yaws (in the case of positive rudder deflection the yaw is to port) the outer (starboard) wing experiences a greater flow velocity and therefore generates more lift which generates a negative rolling moment (to port).

My question is what is the overall rolling moment for some rudder deflection, is the direct effect stronger or is the yaw-induced effect stronger.

For context I am a third year aeronautical engineering student trying to answer this question: "About which axis does the secondary effect of rudder deflection act? About this axis which direction (sign) would a moment generated from a negative rudder defection act, for a conventional tailplane and rudder (relevant control derivative is positive)? Explain any reasoning behind your answer."

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    $\begingroup$ If you have progressed to the third year of your studies without developing an instinctive feeling for the effects of the rudder being above the roll axis then I suggest you should consider taking up the design of boats. $\endgroup$ Jan 8 at 0:01
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    $\begingroup$ The total resultant effect on roll from any rudder input, is a complex issue, dependant on several other factors, (AOA, airspeed, wing anhedral/dihedral and sweep, etc.), but generally, it will be in the same direction as rudder input, (right rudder will generate right roll and left rudder, left roll) $\endgroup$ Jan 8 at 12:19
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    $\begingroup$ It would help a lot to build your own aircraft and try it. Imagine a very tall thin balsa rudder on your glider, then a square one with equal area. Then try varying degrees of wing dihedral. A small hill and some glue is all you need (paper works too). $\endgroup$ Jan 8 at 13:56
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    $\begingroup$ @CharlesBretana The z axis positive direction is defined down in the kinematic coordinate system. The signs of deflections, forces and moments are correct. $\endgroup$ Jan 8 at 16:00
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    $\begingroup$ Normally, rolling will be dominated by dihedral. The rudder will have only a minor influence, but it controls the yaw angle which produces the sideslip which in turn is necessary for dihedral to do its job. $\endgroup$ Jan 8 at 16:10

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If you consider the roll torque generated by the overall shape of the aircraft (including the difference in forward airspeed between the left and right wings) due to the yaw rate induced by the rudder to fall within the scope of your question (a "secondary effect" of the rudder),

then should you not also consider the roll torque generated by the overall shape of the aircraft (especially including dihedral wing geometry, wing sweep, and high vs low wing design) due to the sideslip angle induced by the rudder to also fall within the scope of your question (i.e. a "secondary effect" of the rudder)?

It's not obvious to me how expansive the homework question is trying to be when it mentions the "secondary effect" of rudder. Perhaps it is only focusing on the direct roll torque created by the rudder, independent of yaw rate and independent of sideslip angle. In which case the answer is fairly obvious for most aircraft.

But your own specific question here to ASE seems to be whether the "direct" roll torque created by the rudder is typically stronger or weaker than the yaw-rate-related roll torque created by the rudder. Without directly answering your question, I'd suggest that it may help you understand the overall problem if you realize that the sideslip-related roll torque created by the rudder is typically stronger than either of those other two things.

(It may help you better understand the distinction between yaw-rate-related effects and sideslip-related effects, if you consider the case of a sustained constant-heading sideslip. To hold the bank angle constant in this maneuver, most aircraft need a fairly significant aileron input "opposite" the rudder input, which tells you that the sideslip-induced roll torque created by the rudder is dominating over the "direct" roll torque created by the rudder.)

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