How dihedral creates roll stability, controlling roll with yaw inputs in aircraft with dihedral, and some thoughts on "pendulum stability" in paragliders and hang gliders

How dihedral creates roll stability, controlling roll with yaw inputs in aircraft with dihedral, and some thoughts on "pendulum stability" in paragliders and hang gliders

August 8, 2007 edition
Steve Seibel
www.aeroexperiments.org

Fundamentals: turns in rudderless aircraft involve sideslip

All turns in rudderless aircraft involve some sideslip. If the aircraft banks left and turns left, there will be a slight left-to-right component in the relative wind or airflow over the aircraft. In other words the nose of the aircraft will be pointing slightly to the right of the actual direction of the flight path and airflow at any given moment. In other words the nose of the aircraft is pointing slightly toward the high side or outside of the turn. This is called a "slip". The root cause of this slip is the increased airspeed, and increased drag, experienced by the outboard wingtip, which creates a yaw torque opposite the direction of the turn, making the aircraft heading "lag" behind the actual direction of the flight path at any given moment. In aircraft without rudders, this slip exists both in intentional turns, and in accidental turns, where turbulence has rolled the aircraft into an unintended bank. In aircraft with rudders, this slip should only exist in unintentional turns: in an intentional turn the pilot will apply a touch of "inside" rudder to swing the nose into line with the actual direction of the flight path and relative wind, eliminating the slip.

Fundamentals: dihedral creates a "downwind" roll torque

When there is a sideways component in the airflow over an aircraft, dihedral will create a roll torque in the "downwind" direction. For example, if the aircraft has dihedral and there is a left-to-right component in the relative wind or airflow, the relative wind will tend to strike the undersurface of the left wing harder than it strikes the undersurface of the right wing. In other words the left wing will experience a higher angle-of-attack than the right wing. This will create a right roll torque.

Fundamentals: sweep creates a "downwind" roll torque

Sweep creates a dihedral-like effect. When there is a sideways component in the airflow over an aircraft, sweep will create a roll torque in the "downwind" direction. For example, if there is a left-to-right component in the relative wind or airflow, the relative wind will meet the leading edge of the left wing more "squarely" then it meets the leading edge of the right wing. In relation to the airflow, the right wing is "more swept" and the less wing is "less swept". This means that the left wing will generate more lift than the right wing. This will create a right roll torque. As with dihedral, this is a "downwind" roll torque.

Fundamentals: effective dihedral and effective anhedral

A wing may have both anhedral and sweep.

A wing has "effective dihedral" if it has both sweep and dihedral, or if dihedral alone, or if it has some anhedral but also has enough sweep that it behaves as if it has dihedral. A wing has "effective dihedral" if generates a roll torque in the "downwind" direction whenever there is a sideways component in the airflow over the aircraft.

A wing has "effective anhedral" if it has anhedral and no sweep, or if it has enough anhedral, in relation to whatever sweep is present, that it behaves like an simple unswept wing with anhedral. A wing has "effective anhedral" if generates a roll torque in the "upwind" direction whenever there is a sideways component in the airflow over the aircraft.

A wing with both sweep and anhedral may exhibit "effective dihedral" at high angles-of-attack and "effective anhedral" at low angles-of-attack. If a wing with both sweep and anhedral exhibits "effective dihedral" throughout the flight envelope, it will exhibit more "effective dihedral" at high angles-of-attack than at low angles-of-attack. If a wing with both sweep and anhedral exhibits "effective anhedral" throughout the flight envelope, it will exhibit less "effective anhedral" at high angles-of-attack than at low angles-of-attack. The reason for this is that the dihedral-like effect of sweep -- i.e. the "downwind" roll torque generated by sweep in the presence of a sideways airflow -- is greater at high angles-of-attack than at low angles-of-attack. The roll torque created by dihedral or anhedral in the presence of a sideways airflow is relatively independent of angle-of-attack.

In the remainder of this article, we won't talk explicitly about sweep. When we talk about "anhedral" or "dihedral", we'll really mean "effective anhedral" or "effective dihedral". The reader can keep things simple by picturing a simple unswept wing with anhedral or dihedral, but the same concepts apply to the more complex wing shapes described above, with the caveat that the flight characteristics might become quite different if we make a change in angle-of-attack.

Fundamentals: dihedral creates a stabilizing roll torque

When an aircraft is turning to the left, and there is no rudder, or the pilot is not using the rudder, there will be a slight left-to-right component in the relative wind or airflow over the aircraft. If the aircraft has dihedral, the dihedral will create a roll torque to the right. This will tend to bring the aircraft back to wings-level. Anhedral would create a roll torque in the opposite direction. If the aircraft is banked but a rudder or some other means is used to ensure that the aircraft meets the relative wind exactly head-on with no sideslip, i.e. with no sideways component in the airflow over the aircraft, then the aircraft's dihedral or anhedral geometry will not create any roll torque.

Fundamentals: effect of a vertical fin on roll stability

We can increase an aircraft's yaw stability or "weathervane effect" by enlarging the vertical fin, or by adding a vertical fin to a tail-less swept-wing aircraft like a hang glider. This will decrease the amount of sideslip that we see during turning flight. (This is especially true if the tail acts at a fairly short moment-arm compared to the wingspan; if the tail boom is long compared to the wingspan then the tail may respond to the curving airflow of the turn in a way that increases sideslip?)

If the aircraft has dihedral, the decrease in sideslip will decrease the amount of rolling-out torque created by the dihedral. In other words, with less slip, the dihedral has less effect on the aircraft's roll stability. The aircraft will become less spirally stable or more spirally unstable. If we see an aircraft become less spirally stable or more spirally unstable as we enlarge the vertical fin, this suggests that the aircraft has effective dihedral.

If the aircraft has anhedral, the decrease in sideslip will decrease the amount of rolling-in torque created by the anhedral. In other words, with less slip, the anhedral has less effect on the aircraft's roll stability. The aircraft will become less spirally unstable. If we see an aircraft become less spirally unstable as we enlarge the vertical fin, this suggests that the aircraft has effective anhedral.

Fundamentals: controlling roll through yaw in an aircraft with dihedral

In an aircraft with dihedral, if we use a rudder or some other means to yaw the aircraft to the right with respect to the actual direction of the flight path and relative wind, so that the nose of the aircraft points to the right of the actual direction of the flight path and relative wind, this will create a left-to-right component in the airflow over the aircraft. If the aircraft has dihedral, the sideways component in the airflow will interact with the dihedral to create a roll torque in the "downwind" direction: toward the right. In an aircraft with a large amount of dihedral, the rudder serves as an effective roll control. In an aircraft with anhedral, the rudder can serve as a "backwards" roll control-- left yaw can create a right roll torque.

Fundamentals: the "pendulum effect" mimics dihedral:

If an aircraft has the CG located far below the main wing, i.e. if the CG is located below most of the aerodynamic surface of the aircraft, this creates a "pendulum effect" that behaves like dihedral. If the aircraft experiences a sideways component in the airflow, the "pendulum" geometry will create a "downwind" roll torque. For example if there is a left-to-right component in the airflow, the drag forces and aerodynamic sideforces, acting high above the CG, will tend to roll the aircraft to the right. If the aircraft enters a left turn with some sideslip-- i.e. with some left-to-right component in the airflow-- the aircraft will tend to roll toward the right. This is a stabilizing effect, just as we see in an aircraft with dihedral. And the "pendulum" geometry allows us to control roll with yaw, just as dihedral does: if the aircraft yaws to the right with respect to the actual direction of the flight path and airflow, so that there is left-to-right component in the airflow over the aircraft, the aircraft will experience a roll torque toward the right, just as would an aircraft with dihedral. If the aircraft is banked but a rudder or some other means is used to ensure that the aircraft meets the relative wind exactly head-on with no sideslip, i.e. with no sideways component in the airflow over the aircraft, then the aircraft's "pendulum" geometry will not create any roll torque.

Paragliders have a strong "pendulum" geometry:

In a paraglider the pilot is "tied in" to the wing: the pilot can't swing from side to side in relation to the wing, at least not without making lines go slack. The pilot might as well be connected to the wing with solid struts. This creates pendulum stability in pitch and roll. The pendulum stability in roll looks much like dihedral. Yaw the paraglider sideways and the drag load from the wing, acting high above the CG, now contains a roll component that makes the glider want to roll in the same direction as it was yawed. This creates a strong dihedral effect even though the wing has an anhedral shape. In a turn with some sideslip, the drag force acting high above the CG has a sideways component that tends to roll the aircraft back toward wings-level, even though the wing itself has an anhedral shape.

Hang gliders do not have a "pendulum" geometry:

In a hang glider the pilot is not "tied" to the wing: he is free to swing from side to side and front to back. When the pilot is not exerting a muscle force to hold himself in place on the control bar then there is no pendulum effect at all. If we ask if the glider is stable or unstable in roll, we want to know what the glider does when the pilot is not exerting a muscle force on the bar. We are not really interested in any pendulum effect that may exist when the pilot is gripping the bar tightly. If it ever happens that the when the glider yaws or slips, the drag load on the wing contains a strong sideways component, or strong aerodynamic sideforces are created, then the pilot will tend to fall toward the "upwind" side of the control bar, i.e. toward the direction of slip or away from the direction of yaw. (The pilot is tending to behave like a slip-skid ball: in a left turn with some slip, the pilot would fall to the left side of the control frame). In this situation the pilot will have to exert a muscle force if he wants to keep himself at the center of the control bar. The direction of the muscle force will create a dihedral-like effect, rolling the glider in the "downwind" direction, opposite the direction of any sideslip. But this really doesn't "count" as a pendulum stability effect. All the roll torque has to come through the pilot's muscles -- that's not stability, that's the pilot making a roll input. The fact that the pilot happens to hang to the low side of the bar doesn't really put any constraint on the choices that are available to the pilot-- he can still make a roll input in either direction or he can make no roll input at all. So we don't have pendulum stability in hang gliders, at least in roll. If we tied the pilot to the center of the control bar it would be different--then we would mimic a paraglider, where the pilot and wing really are one fixed system, with a CG well below the wing.

Sideways forces are small in hang gliders:

My own experience is that I never tend to hang more than a few inches to either side of the centerline of the control bar, and in most cases the tendency to hang off-center is undetectable. This indicates that sideways drag components and aerodynamic sideforces are low, even when the hang glider is yawed a bit sideways. This may have been less true in older gliders with lots of billow: if you drag an old billow-cruiser sideways through the air it would seem this would have to generate some amount of significant sideways load on the glider, which would translate into a significant tendency for the pilot to "fall" to the low side of the bar.

When sideways forces are small, only a small amount of pendulum roll stability would exist even in the case where the pilot was tied to the center of the control bar.

A hang glider with "pendulum" stability:

Things are a bit different with this RC trike. The servo gears hold the trike car in a fixed position; the trike car is not free to flop from side to side. And, the aircraft did show strong dihedral-like properties-- it would fly hands-off just fine, and when I bent the little fin to the side to make a rudder-like effect it always rolled in the same direction of the yaw, not the opposite. The same was true when I dragged a ribbon from one wingtip. It's interesting to wonder if that would have been the case with the same wing (or a scaled-up full-sized version) and a freely hanging pilot. Quite possibly so-- all that sweep should create a strong dihedral effect.

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