The real purpose of dihedral and anhedral: creating an aerodynamic coupling between yaw and roll

August 6 2005 edition
Steve Seibel
www.aeroexperiments.org
steve at aeroexperiments.org

 

NOTE August 2006: all the content in this section has now received a fresher treatment in the "Semi- Unconventional Aerophysics Tutorial" pages. Many items that are not yet covered in the main text pages of the SUAT section are covered briefly on the page entitled "Pool of images for Semi-Unconventional Aerophysics Tutorial Pages." This older material is still accurate to the best of my knowledge except for one point: I now feel that the suggestion that increasing wingtip washout (as opposed to increasing sail billow) will tend to create an anhedral geometry was unwarranted.

 

This article consists primarily of some excerpts from the longer article entitled "How washout and billow increase the net geometric anhedral of a swept wing, and other related topics". The passages marked "*" are excerpts from this longer article and the passages marked "**" are additional notes.

 

*It's a misconception that anhedral or dihedral create a roll torque because the left wing becomes "more horizontal" than the right wing, or vice versa, or one wing ends up with a "more vertical lift vector" or a "greater projected area", when the aircraft banks to one side.

*In reality, in the aircraft's reference frame, banking can be thought of as a change in the direction of the weight vector, acting at the CG, while all the aerodynamic force vectors (drag, lift from left wing, lift from right wing, etc) initially remain unchanged in the aircraft's own reference frame, at least if we imagine that the aircraft has not yet began to move sideways through the airmass. This situation cannot produce a roll torque. If the aircraft does not move sideways through the air, anhedral or dihedral cannot produce a roll torque when the aircraft banks.

*However, if the aircraft begins moving sideways through the airmass, then anhedral or dihedral will create a roll torque. We'll explain the full reason for this in just a bit.

*The roll torque created by anhedral or dihedral when an aircraft is tipped into a bank is entirely dependent upon the fact that immediately after leaving the wings-level condition, the aircraft will initially begin to move sideways through the air to at least a small degree. This sideways motion is created by the fact that the flight path has started to curve, because the banked wing is generating a sideways (centripetal) force, but the aircraft's yaw stability or "weathervane effect" has not yet exerted sufficient yaw torque to overcome the aircraft's yaw rotational inertia and create the yaw rotation that is required to keep the nose of the aircraft aligned with the direction of travel at any given instant, i.e. to keep the nose of the aircraft pointing straight into the changing direction of the relative wind. So the flight path starts to curve, the heading of the aircraft initially tends to remain constant, and this creates a sideways airflow over the aircraft.

*If the aircraft is banking because the pilot is using weight-shift or ailerons to make an intentional roll control input, then the nose will tend to initially yaw in the "wrong" direction due to adverse yaw, and again this will create a sideways airflow over the aircraft.

*The main purpose of a rudder is to prevent this sideslip due to adverse yaw and yaw rotational inertia when the pilot is intentionally banking the aircraft. Most rudderless aircraft--unless they are using spoilerons for roll control--will experience some sideslip whenever the pilot is intentionally entering a turn, as well as when the aircraft is tipped into a bank by turbulence.

*The dynamics that we've been discussing up to this point take place mainly while the aircraft's bank angle is changing, or immediately after the aircraft's bank angle has changed. But even after quite a few seconds have passed with no further change in bank angle, and the aircraft's yaw stability mechanisms have had plenty of time to act, there will still typically be a very slight sideslip when a rudderless aircraft is banked, because a bank always creates a curvature in the flight path (i.e. the aircraft is turning), and when the flight path is curving, the outboard, faster-moving wingtip tends to experience more drag than the inboard, slower-moving wingtip. This yaws the nose toward the outside or high side of the turn, at which point the drag torques from the left and right wings, plus any "weathervane" yaw torque from the vertical tail (if present), again come into balance. In future articles we'll use the concept of "airflow curvature" to explore this balance of yaw torques in more detail, and we'll examine how the position and size of the vertical fin (if present) affects the slight amount of sideslip that we typically see in a stabilized, constant-bank turn in a rudderless aircraft. This is a rather complex subject.

*Therefore if a pilot wishes to keep the nose of the aircraft pointing directly into the airflow during a stabilized, constant-bank turn, he'll typically need to apply just a touch of inside rudder, especially at low airspeeds where the turn radius will be small and the difference in airspeed between the two wingtips will be the most pronounced.

*If there is no rudder, or if the pilot doesn't apply inside rudder, this continued, slight sideslip means that dihedral will continue to create a rolling-out (stabilizing) roll torque even when the bank angle is no longer increasing. Also, this continued, slight sideslip means that anhedral will continue to create a rolling-in (destabilizing) roll torque even when the bank angle is no longer increasing. But we're jumping the gun here: the reason a roll torque arises when a sideways airflow is present is given below.

*Anhedral or dihedral create a roll torque whenever an aircraft moves sideways through the airmass, regardless of whether the sideways motion is due to the fact that the aircraft has gained a velocity component toward one wingtip and the aircraft's yaw stability mechanisms have not yet yawed the nose into alignment of the new direction of travel, or the direction of the flight path has not changed but the nose of the aircraft has yawed to point to the left or the right of the direction that the aircraft is actually moving through the airmass. Both cases have the same end result--the aircraft moves sideways through the airmass, so there will be a sideways component in the airflow over the aircraft (relative wind). In both cases, anhedral or dihedral will generate a roll torque. And in both cases the aircraft's yaw stability mechanisms will be attempting to yaw the nose to point directly into the relative wind, which would end the sideslip or skid.

*The roll torque created by dihedral or anhedral during a sideslip or skid is caused by the fact that the left wing is experiencing a higher or lower angle-of-attack than the right wing, due to the sideways component in the airflow (relative wind). This is the central point of this entire article.

*To understand this, look at these photographs of wings and models of wings, taken from the side. (Photo 1: Superfloater ultralight sailplane with dihedral.) (Photo 2: model of wing with dihedral) (Photo 3: model of wing with anhedral). In many of them, you can see the underside of the left wing and the top side of the right wing, or vs. vs.. If the aircraft were moving directly toward the camera, which would involve an extreme angle of sideways motion through the airmass, i.e. an extreme sideslip angle, then clearly there would be a very large difference in angle-of-attack between the left and right wings: one wing would be generating a positive lift force and the other side of the wing would be generating a negative lift force. In real-life situations, involving much smaller angles of sideways motion (sideslip), neither wing will be flying at a negative angle-of-attack, but the difference in angle-of-attack between the left and right wings will still exist to a smaller degree. This will create a roll torque. Use your imagination and visualize what kind of roll torque will be created as a small sideways airflow component flows over these wings. You will see that the left wing will be developing more lift than the right wing, or vs. vs., due to the difference in angle-of-attack between the left and right wings.

*Of course the outboard parts of the wing, near the tips, will tend to generate larger roll torques than the inboard parts of the wings, near the roots. This is because the outboard areas are further from the CG, and so they act at a greater moment-arm.

*Use your imagination again (diagrams will be included in future editions of this article): think about which direction the wing's lift force will be tilted when wing gets banked to one side by a bit of turbulence. Let's assume that the wing hasn't started slipping sideways just yet, so anhedral or dihedral haven't yet come into play--there's no need to think about the left and right wings individually yet, so just focus on the sideways tilt of the aircraft's entire, net lift vector as the aircraft banks. Now consider that the aircraft will be "pushed" sideways through the air by this tilted lift vector, at least for a few seconds, as we've already discussed above, until the nose has a chance to start clocking smoothly around the horizon and the aircraft ends up in a full-fledged turn with minimal sideslip. Now, as the aircraft is slipping sideways through the airmass, any anhedral or dihedral that is present will create a roll torque, as we've already seen by looking at the photos of the wings with anhedral and dihedral. Now think about the direction of this roll torque that is created by the sideways airflow over an anhedral or dihedral wing, due to the aircraft's initial, sideways motion through the airmass, immediately after a one wing drops due to a bit of turbulence. With a bit of thought we can see that when a wing drops, the aircraft will move sideways through the air in such a way that dihedral will end up creating a stabilizing roll torque that tends to return the aircraft to wings-level, and anhedral will end up creating a destabilizing roll torque that tends to roll the aircraft to a steeper bank angle. Again, this roll torque is dependent upon the fact that when one wing drops, the aircraft will sideslip a bit (due to yaw rotational inertia) before settling into a full-fledged turn, and this will create a sideways component in the airflow (relative wind).

*We can also see that if the aircraft has very high degree of yaw stability--i.e. a very strong "weathervane effect"--due perhaps to a very large vertical fin--then when it gets tipped into a bank by a bit of turbulence, instead of slipping sideways to any appreciable degree, it will tend to quickly "weathervane" into a full-fledged turn, with the nose clocking smoothly around the horizon and little sideways component in the airflow over the wing. So increasing the size of the vertical fin, or increasing the aircraft's "weathervane" stability torque by increasing the amount of sweep in the wing, will tend to minimize the roll torque created by any anhedral or dihedral that is present, and this will have a definite effect on the aircraft's roll stability or spiral stability. If the aircraft has dihedral, then a large vertical fin will tend to decrease the aircraft's roll stability or increase the aircraft's roll or spiral instability. If the aircraft has anhedral, then a large vertical fin will tend to increase the aircraft's roll stability or decrease the aircraft's roll or spiral instability.

*For some diagrams and additional notes on the way that the stabilizing effect of dihedral is entirely dependent upon the fact that an inadvertent bank always leads to a sideslip, see section 9.3 of John S. Denker's superb "See how it flies" website. Note however that Denker focusses on a slightly different driving mechanism for the sideslip than the two factors that we've emphasized here--he invokes the "long-tailed slip effect" while we've focussed on yaw rotational inertia and the difference in airspeed between the inboard and outboard wingtips. Denker actually views the vertical tail as driving the sideslip that arises during an unintentional bank, rather than mitigating the sideslip! Following Denker's logic, in an aircraft with a relatively small wingspan, and a relatively small amount of rotational inertia in the yaw axis, and a relatively long tail moment arm, increasing the size of the vertical tail could actually end up having the opposite effect as we've described above, in the context of unintentional banks due to turbulence. (This idea wouldn't apply to long-spanned flying-wing aircraft like hang gliders and trikes). However, with any aircraft, a large vertical tail will also minimize the adverse yaw that arises from a pilot's control inputs, which will increase the responsiveness of an aircraft with dihedral and decrease the responsiveness of an aircraft with anhedral, as we'll see below.

*Just as dihedral tends to return an aircraft to wings-level whenever a wing drops in turbulence and the aircraft starts sliding sideways, so too does dihedral tend to make an aircraft respond sluggishly to a pilot's roll inputs (unless a rudder or spoilerons are being used to overcome adverse yaw and/or yaw rotational inertia and avoid a sideslip).

*Just as anhedral tends to make an aircraft roll into a tighter bank whenever a wing drops in turbulence and the aircraft starts sliding sideways, so too does anhedral tend to make an aircraft respond more quickly to a pilot's roll inputs (at least in cases where a rudder or spoilerons are not being used to overcome adverse yaw and/or yaw rotational inertia and avoid a sideslip.)

*The roll torque created when a wing with anhedral or dihedral experiences a sideways airflow leads to a "coupling between yaw and roll". What does this mean? If a pilot applies heavy right rudder to yaw the nose of the aircraft toward the right, or if adverse yaw or any other phenomenon yaws the nose of the aircraft toward the right, then the aircraft will be moving sideways through the air. The nose will not be pointing the same direction that the aircraft is actually moving through the airmass. There will be a sideways component in the airflow (relative wind) over the aircraft, blowing from the left wingtip toward the right wingtip. Then dihedral or anhedral will create a roll torque:

*If the wing has dihedral, then when the pilot's rudder input (or adverse yaw or any other factor) yaws the nose of the aircraft toward the right (in relation to the actual direction of the flight path and airflow) as described above, the resulting left-to-right sideways component in the airflow will cause the left wing to experience a higher angle-of-attack than the right wing, and this will create a roll torque toward the right. So an initial yaw toward the right ends up creating a roll torque toward the right. This is a "positive coupling" between yaw and roll.

*In the same situation, if the wing has anhedral rather than dihedral, the left-to right sideways component in the airflow will cause the right wing to experience a higher angle-of-attack than the left wing, and this will create a roll torque toward the left. So an initial yaw toward the right ends up creating a roll torque toward the left. This is a "negative coupling between yaw and roll.

*Therefore, over-enthusiastic use of a rudder or wingtip drag device to "skid" the nose in the direction of the intended turn (as opposed to using the "right" amount of rudder to overcome adverse yaw and keep the nose aligned with the flight path and airflow (relative wind), or using no rudder and allowing the nose to swing the "wrong" way due to adverse yaw) will create a helpful roll torque in the case of a wing with dihedral--this is why many aircraft such as Gentle Lady RC sailplanes, Dragonfly tugs and even Cessna 152's can be controlled with the rudder alone, without use of the ailerons. (Granted this technique is not always very efficient, and in some cases can be an invitation to a spin).

*By the same logic, over-enthusiastic use of a rudder to "skid" the nose in the direction of the intended turn (as opposed to using the "right" amount of rudder overcome adverse yaw and keep the nose aligned with the flight path and airflow (relative wind), or using no rudder and allowing the nose to swing the "wrong" way due to adverse yaw) will create an unfavorable roll torque in the case of a wing with anhedral. In an aircraft with anhedral, it is very possible for a left rudder input to cause a right bank followed by a right turn--this vividly demonstrates that "yawing" (i.e. swinging the nose to one side in relation to the direction of the flight path and relative wind) and "turning" (i.e. creating a curvature of the flight path) are not at all the same thing! And by the same logic, on an aircraft with anhedral, adverse yaw will actually create a helpful roll torque! All this will be discussed in much more detail in the areas of this website that deal with experiments with rudders on flex-wing hang gliders with anhedral.