Notes on theory--"effective dihedral" in flex-wing hang gliders
This page was last updated June 20, 2014.
In this article, we'll see how sweep and anhedral compete with each other to determine the "effective dihedral" of a flex-wing hang glider, or any other aircraft with both sweep and anhedral. We'll see that the results of this "competition" depend strongly on angle-of-attack-- the dihedral-like influence of sweep is much stronger at high angles-of-attack than at low angles-of-attack, while the influence of anhedral is independent of angle-of-attack.
Let's start by looking at how dihedral works to generate a roll torque whenever an aircraft experiences a sideways component in the airflow or relative wind. When an aircraft with dihedral is not pointing directly into the relative wind or airflow-- which in flight, has little or nothing to do with the external, meteorological wind-- the "upwind" wing experiences an increase in angle-of-attack and lift and tends to rise, while the "downwind" wing experiences a decrease in angle-of-attack and lift and tends to descend. Some aircraft use a rudder as the sole or primary means of roll control-- such as this MX Quicksilver ultralight airplane or this "Gentle Lady" radio-controlled sailplane (upper aircraft in photo). The rudder yaws the aircraft sideways to the relative wind or airflow, and then the aircraft rolls toward the "downwind" wing-- in the direction of the deflected rudder. Aircraft that use the rudder as the sole or primary means of roll control almost always have a great deal of dihedral. In a few unusual cases the "downwind" roll torque in response to yaw or slip may be created mainly by sweep, or by a high-wing wing configuration.
For a given yaw or slip angle and a given airspeed, the roll torque created by dihedral is independent of the overall average-angle-of-attack. This means that at any given airspeed, for any given yaw or slip angle, the roll torque created by dihedral is independent of the average lift coefficient and the G-loading. The rudder will roll the aircraft toward the "downwind" wingtip whether the G-loading is positive, zero, or negative-- the "Gentle Lady" sailplane shown above may be flown in sustained inverted flight with no reversal of roll control. Or if we are looking only at the plus 1G condition, we can observe that the direction of the yaw torque created by dihedral always acts in the same direction-- toward the "downwind" wing-- whether the lift coefficient is high and the airspeed is low, or vice versa. (We're assuming here that the angle-of-attack isn't so high that we might actually stall the upwind wing-- this might indeed create an "upwind" roll torque.)
All the same is true for an aircraft with anhedral rather than dihedral, except now the roll torque acts in the opposite direction-- toward the "upwind" wingtip. If we put the Gentle Lady's wing on upside-down, we'd have to give left rudder to turn right, and vice versa. (Also, the upside-down airfoil would be terribly inefficient in upright flight, but that's another subject-- we'll assume we have abundant ridge lift for this thought experiment.)
A swept wing generates a roll torque in somewhat the same manner as does a wing with dihedral. When the aircraft is yawed to point sideways to the relative wind or airflow, the "upwind" wing becomes "less swept" relative to the airflow and experiences an increase in lift coefficient, while the "downwind wing" becomes "more swept" relative to the airflow and experiences a decrease in lift coefficient. The increase or decrease in lift coefficient can be modelled by multiplying or dividing the lift coefficient by some "efficiency" factor related to the sweep angle and the sideslip angle. The trouble is, if both wings are meeting the air at the zero-lift angle-of-attack--if the aircraft is executing a zero-G maneuver-- then the sideslip will not generate any roll torque at all. After all, zero multiplied or divided by some "efficiency" factor is still zero. If both wings are meeting the air at a negative-lift angle-of-attack-- if the aircraft is in sustained inverted flight-- then the "upwind" wing will again be "more efficient", but it will be more efficient at creating negative or downward lift. Now the aircraft will roll in the "upwind" direction, not the "downwind" direction. If an aircraft has sweep and no dihedral, and the pilot is trying to use the rudder for roll control, he'll find that the rudder's effect has reversed in negative-G flight. He'll find that a sustained rudder input has no roll effect at all in zero-G flight.
In summary, at high angles-of-attack, a swept-wing aircraft behaves as if it has lots of dihedral-- sideslip generates a strong "downwind" roll torque. At low angles-of-attack, a swept-wing aircraft behaves as if it has only a little dihedral-- sideslip generates a weak "downwind" roll torque. At the zero-lift angle-of-attack, a swept-wing aircraft behaves as if it has no dihedral-- sideslip generates no roll torque. At a mildly negative-lift angle-of-attack, a swept-wing aircraft behaves as if it has a small amount of anhedral-- sideslip generates a weak "upwind" roll torque. At a strongly negative-lift angle-of-attack, a swept-wing aircraft behaves as if it has a lot of anhedral-- sideslip generates a strong "upwind" roll torque.
What if an aircraft has both sweep and anhedral? Let's confine ourselves to the plus-1-G scenario-- we are flying right-side-up in unaccelerated flight. The sweep will contribute the strongest dihedral-like effect-- the most "downwind" roll torque-- when the angle-of-attack is high, i.e. when the airspeed is low. The sweep will contribute very little dihedral-like effect when the angle-of-attack is very low, i.e. the airspeed is very high. If the outer wingtips have enough washout that they meet the air at a negative-lift angle-of-attack during high-speed flight, then sweep will actually contribute an anhedral-like, upwind roll torque from those parts of the wing. Even if we remove the complication of washout entirely, sweep will create much more dihedral-like "downwind" roll torque at high airspeed than at low airspeed.
Meanwhile, for any given yaw or slip angle, anhedral will create the same difference in angle-of-attack between the "upwind" and "downwind" wings whether the airspeed is low and the angle-of-attack is high, or vice versa. Anhedral will create an "upwind" roll torque regardless of the airspeed and angle-of-attack.
What will be the net result of this "competition" between sweep and anhedral? At a very low angle-of-attack-- a very high airspeed-- the effect of sweep will be negligible and anhedral will always "win". A sideways airflow-- a sideslip-- will always create a roll torque toward the "upwind" wingtip in this part of the flight envelope. We can say that the "effective dihedral" is "negative" in this part of the flight envelope. At progressively higher angles-of-attack and progressively lower airspeeds, this roll torque will become weaker and weaker, until a sideways airflow creates zero roll torque. We can say that the "effective dihedral" is zero in this part of the flight envelope. At still higher angles-of-attack and lower airspeeds, the sweep will start to dominate, contributing a progressively stronger "upwind" roll torque. We can say that the "effective dihedral" is "positive" in this part of the flight envelope.
When we use the phrase "effective dihedral" in this context, we mean something very specific-- the direction and amount of roll torque generated by a sideways airflow over the aircraft. If the "effective dihedral" is strongly positive, a sideways airflow generates a strong "downwind" roll torque, tending to raise the "upwind" wingtip. If the "effective dihedral" is neutral, a sideways airflow over the aircraft generates no roll torque, and neither wingtip tends to rise. If the "effective dihedral" is strongly negative, a sideways airflow generates a strong "upwind" roll torque, tending to raise the "downwind" wingtip.
Speaking strictly in terms of the roll torque generated by sideslip, an aircraft with both sweep and anhedral will behave as if it has only anhedral at high airspeeds and low angles-of-attack. The "effective dihedral" will be negative. It will behave as if it is a straight, unswept wing at some lower airspeed and higher angle-of-attack. The "effective dihedral" will be zero. And it will behave as if it has only dihedral at some still lower airspeed and still higher angle-of-attack. The "effective dihedral" will be positive.
More precisely, the upper end of this spectrum is limited by the stall. If the aircraft has lots of anhedral and only a little sweep, it may reach the stall angle-of-attack before ever reaching the "neutral" point. In this case we'll never see a dihedral-like "downwind" roll torque. The "effective dihedral" will never actually be positive. We'll just see a weaker "upwind" roll torque at high angles-of-attack and low airspeeds than at low angles-of-attack and high airspeeds. On the other hand, if we have only a modest amount of anhedral and lots of sweep, we may see a dihedral-like "downwind" roll torque over the entire normal 1-G flight envelope. To reach the very low angle-of-attack where anhedral would dominate over sweep and "effective dihedral" would become negative in 1-G flight, we may need to exceed the red-line airspeed. Alternatively, we might reach this part of the flight envelope if we engage in a semi-ballistic (reduced-G) maneuver or ballistic (0-G) flight or sustained inverted (negative-G) flight.
Flex-wing hang gliders have ample anhedral, especially in the outer portions of their wings, which are far from the CG and well positioned to generate lots of "upwind" roll torque in a sideslip. See the related article on this website entitled "Notes on sail billow, dihedral, and anhedral in flex-wing hang gliders" for more on how sail billow creates an anhedral geometry in the outboard portions of the wings, above and beyond the "airframe anhedral" or droop in the leading-edge tubes relative to the keel tube. Even a glider with zero "airframe anhedral" or slight "airframe dihedral"-- meaning that the leading-edge tubes actually rise relative to the keel tube-- may have ample anhedral in the outboard portions of the sail, to due the billowed shape of the sail. Flex wing gliders also have sweep, which generally contributes a dihedral-like "downwind" roll torque in a sideslip, especially at high angles-of-attack. At very low angles-of-attack the sweep may contribute a net "upwind" or anhedral-like roll torque, because the outboard portions of the wing-- again, far from the CG and well positioned to generate lots of roll torque in a sideslip-- are meeting the air at a negative angle-of-attack, due to washout. What is this net result of all this? Is the "effective dihedral" of a flex-wing hang glider positive, negative, or zero near the min-sink angle-of-attack? What about near the stall? What about at lower angles-of-attack, when the bar is well pulled-in? What about with the bar absolutely "stuffed" or pulled in as far as it can go, to the full reach of the pilot's arms-- and with the pilot "balled" up with knees bent and pulled over the bar?
Some answers to those questions are given in the related article in this website entitled "A hands-on exploration of 'effective dihedral' in flex-wing hang gliders". To summarize this article, in-flight experiments using a controllable rudder to induce yaw and slip, and also using a wing-mounted drogue chute to induce yaw and slip, suggest that flex-wing hang gliders typically have nearly zero "effective dihedral" near the min. sink angle-of-attack. In some cases the "effective dihedral" at this angle-of-attack can be observed to be very mildly positive, and in other cases, the "effective dihedral" at this angle-of-attack can be observed to be very mildly negative. As the bar is pulled further in, the "effective dihedral" becomes distinctly negative. It is always distinctly negative by the time the glider has accelerated to 10 mph above the min. sink angle-of-attack-- in fact often this is true at only 5 mph, or less, above the min. sink angle-of-attack. The "effective dihedral" becomes even more strongly negative at even higher airspeeds and lower angles-of-attack-- the glider really wants to roll toward the upwind wingtip in this part of the flight envelope.
The theory and results given above, and the more detailed results given in "A hands-on exploration of 'effective dihedral' in flex-wing hang gliders", beg the following question-- "How can a flex-wing hang glider's 'effective dihedral' possibly become less negative or more positive as we pull in the bar? After all, dihedral tends to make an aircraft roll towards wings-level, and anhedral tends to make an aircraft roll toward a steeper bank angle. And we know from everyday experience that when circling with a low airspeed / forward bar position/ high angle-of-attack, we may need to high-side the bar. Little if any low-siding will be required even in a "beginner"-friendly glider in this part of the flight envelope. On the other hand, we always need to low-side the bar quite strongly to hold the bank angle constant with the bar well pulled-in and the angle-of-attack low and the airspeed high. Doesn't this seem to indicate that the glider has strongly positive "effective dihedral" when the bar is well pulled-in and the airspeed is high and the angle-of-attack is low?"
A brief answer-- it's easy to carry out simple ground experiments that show that a glider's "effective dihedral" is strongly negative at low angles-of-attack, where the wing is still supporting at least some of it's own weight, but is not lifting the pilot off his feet. No special rudder or other yaw device is required-- we can just stand holding the glider somewhat cross to the wind. At low angles-of-attack-- but not so low that the glider is pressing down on our shoulders with its full weight-- the upwind wing will want to drop and the downwind wing will want to rise. The glider's anhedral is dominating over sweep-- the "effective dihedral" is negative. In circling flight-- as opposed to standing still on the ground-- we introduce other variables that have nothing to do with "effective dihedral". As a result of these other variables, we can't we judge the direction and magnitude of a glider's "effective dihedral" simply by looking at the roll trim in circling flight.
Remember, "effective dihedral" describes the direction and magnitude of the roll torque generated when the aircraft is exposed to a sideways (sideslipping) airflow. This is not the same thing as the aircraft's tendency to roll into or out of a turn.
What are some other variables that influence a glider's roll trim in circling flight?
1) In circling flight, we have a small amount of sideslip, so the wing's positive or negative "effective dihedral" will contribute some small amount of downwind or upwind roll torque. But we also have a relative wind or airflow that is curved--just like the flight path through the airmass is curved. The relative wind or airflow also varies in speed from one part of the aircraft to the other. The high wingtip is moving faster than the low wingtip, and so the high wingtip generates more lift than the low wingtip, and this contributes a rolling-in torque. This effect is much stronger at low airspeeds where the turn radius is small than at high airspeeds where the turn radius is large. For a given flight speed and turn radius, this effect is much more pronounced in aircraft with a large wingspan than in aircraft with a small wingspan. More comprehensively, we have to think in terms of the "effective span"-- if we give the outer portions of the wings a lot of washout so that they are meeting the air at the zero-lift angle-of-attack (or in case of sideslip, so that the average angle-of-attack of the outer portions of the left and right wings is zero), then these outer portions don't "count" as part of the aircraft's "effective span" in the context of the roll torque generated by the difference in airspeed between the left and right wings. In turning flight, the difference in airspeed between these non-lifting portions doesn't contribute any rolling-in torque at all. In fact, in an aircraft with washout, at very low angles-of-attack (high airspeeds) the outer portions of each wing are meeting the air at a negative-lift angle-of-attack, and making negative (downward) lift. In this case, the difference in airspeed between these negatively-lifting portions of the left and right wing actually contributes a rolling-out torque, not a rolling-in torque. At very low angles-of-attack, we can say that the wing's "effective span" has actually become negative, in the context of the roll torque generated by the difference in airspeed between the left and right wings. By this we mean that the angle-of-attack is so low that the negatively-lifting portions of each wing are so large that the rolling-out torque generated by the difference in airspeed between the negative-lifting outer portions of each wing is actually larger than the rolling-in torque generated by the difference in airspeed between the positively-lifting inner portions of each wing. The "effective span" of flex-wing hang gliders is undoubtedly negative in this sense when the bar is well pulled-in, especially with the VG loose. After all, with the VG loose (and sometimes even with the VG tight), when we pull the bar all the way in and "ball up" to shift our weight further forward and pull our knees over the bar, the outer portions of each wing make so much negative lift that the lower sidewires go slack and the upper side wires (in the case of a kingposted glider) become tight, even in constant-speed 1-G flight.
2) Also, when the glider is descending relative to the airmass, it must constantly roll toward the low wingtip just to hold the bank angle constant. To visualize this more clearly, imagine a very steep downward flight path-- almost vertically downward-- the maneuver is essentially a rolling helix or corkscrew, with the direction of roll the same as the direction of turn. To hold the bank angle constant, the aircraft must continually roll toward the low wingtip. This sets up a "roll damping" effect-- a resistance to rolling-- that operates on the entire wingspan, not just the lifting parts. At low angles-of-attack, the outer tips may be making zero lift or negative lift, due to washout, but they'll still contribute a "roll damping" effect. (As a thought experiment, imagining the tremendous "roll damping" effect that would be generated by a very long horizontal extension projecting outward from each wingtip, set at the angle-of-attack that would deliver zero lift in non-turning, non-rolling flight.) This "roll damping" or resistance to rolling tends to make the bank angle decrease. When the bar is pulled well in and the sink rate is very high, this effect is very important. Adding power via a motor--as with a powered harness or a trike-- changes this dynamic. A constant-banked powered climb is a rolling corkscrew or helix with the direction of roll opposite the direction of turn. Now the aircraft must continually roll toward the high wingtip just to hold the bank angle constant. And now the aircraft's "roll damping" tends to make the bank angle increase. For any given airspeed and turn radius, the aircraft will have a much stronger tendency to roll toward a tighter bank angle in a steep powered climb than in an unpowered descent. In some cases a hang glider flown with a powered harness may be almost uncontrollable at full power in a steep turn at a typical thermalling airspeed-- the tendency to roll toward the inside wing in a turn may be just too strong. This can be solved by simply reducing the power until the situation is sorted out. Similarly, at some higher airspeed and lower angle-of-attack where the wing distinctly tends to roll out of the turn with power off, the aircraft may tend to hold a constant bank angle or to roll slowly toward a steeper bank angle with power on.
Neither of these effects have anything to do with the roll torque generated by a sideways airflow or sideslip. They don't fall within the realm of "effective dihedral". But they do greatly influence the aircraft's handling in turning, circling flight. Many aircraft are designed with a mild amount of positive "effective dihedral"-- they tend to roll in the same direction as a sustained rudder input-- yet also tend to roll toward a steeper bank angle during a perfectly "coordinated" turn with zero sideslip, due to one or both of effects noted above. This is particularly true of long-spanned, lightly-loaded, slow-flying airplanes like sailplanes.
In short, we can't rely on an aircraft's behavior in circling flight to tell us whether the "effective dihedral" is positive or negative. In circling flight, sideslip is typically modest and so "effective dihedral" only exerts a modest effect, and other confounding variables come into play.
Another question is begged by the notes above: "How can a flex-wing hang glider's 'effective dihedral' be near zero at the minimum sink angle-of-attack in flight, yet be distinctly positive at roughly the same angle-of-attack during ground-handling?" I don't have a definitive answer to that question, but I'll offer some ideas in the near future in another article on this website. In short, it seems likely that when the glider is not fully loaded on the ground-- when a substantial portion of the pilot's weight is still carried by his feet-- the glider has significantly less sail billow, and therefore significantly less "airframe anhedral". After all, when the outboard portions of the leading-edge tubes bow aft under load, this allows an increase in sail billow. Again, for more on the relationship between sail billow and anhedral, see the related article on this website entitled "Notes on sail billow, dihedral, and anhedral in flex-wing hang gliders". Of course, the outboard portions of the leading-edge tubes also bow upwards under load, which would decrease anhedral. To really explore these relationships further, we probably need to carry out careful experiments in a wind tunnel or on a moving platform, where we can hold the angle-of-attack constant while we vary the airspeed, and thus the aerodynamic load generated by the glider. One thing is clear-- unless a pilot has help from a "wire crew", it is impossible for him to keep the glider yawed substantially cross to the relative wind and airflow, unless a substantial portion of his own weight is still carried on his feet rather than by the wing. He needs to have good traction on the ground to exert enough yaw torque to overpower the glider's natural "weathervane" tendency and point the nose significantly cross to the wind.
And here is yet another related question-- "If a flex-wing hang glider's 'effective dihedral' is near zero at the minimum-sink angle-of-attack in flight, won't it require substantial high-siding to hold the bank angle constant while circling at this angle-of-attack? At the minimum-sink angle-of-attack, the sink rate is low, and so effect #2 described above-- the roll damping associated with the roll rate required just to hold the bank angle constant-- will be small. Meanwhile, effect #1 described above-- the fact that the high wingtip is moving faster, and tending to create more lift, than the low wingtip-- will be significant. If this isn't counteracted by a significant amount of positive "effective dihedral", won't the glider tend to roll to steeper bank angle?"
Again, I don't have a definitive answer to this question. Any asymmetry in lift between the left and right wings-- whether due to "roll damping", or due to the difference in airspeed between the high and low wingtips, or some other factor, will be somewhat reduced by the keel's tendency to shift to even out the trailing-edge tension. But that's not a satisfying answer-- after all, we know that we often do have an asymmetry in lift between the two wings, and thus a requirement for high-siding or low-siding. It's possible that something about the aerodynamics of curving, turning flight tends to shift the keel tube toward the high wingtip even when the difference in loading between the two wings is minimal. If we plotted out a graph of keel position versus load asymmetry, and we found that the resulting line or curve did not pass through the origin of the graph but rather was biased toward the high wingtip, this could explain the experimental results described here. The idea is that near the minimum-sink angle-of-attack, the high wing would create more lift than the low wing if the keel were fixed, but when the keel is free to move, something shifts it toward the high wingtip far enough to make the glider roughly neutral in roll. This hypothesis is completely conjectural at this point-- I intend to do some filming of the keel / crossbar position during circling flight in smooth air to explore this idea further. If no such bias in keel position toward the high wingtip is found, this will cast some doubt on some of the other experimental results above-- namely the idea that a glider's "effective dihedral" may be near zero at the minimum-sink angle-of-attack, even on a glider such as a Wills Wing Falcon that doesn't require high-siding while circling at this angle-of-attack.
Thanks for visiting Steve Seibel's "Aeroexperiments" website.