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In discussion of aircraft roll stability in FAA paragraph on dihedral effect and in keel effect and in Why are high wing aircraft more stable the question of pendulum effect arises.

This is usually described as center of gravity displaced laterally from below center of lift during a roll in a high wing airplane causing a rolling moment which restores wings to a level orientation.

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In any explanation you give, please consider the case of a hang glider which depends on weight shifting to control roll and pitch. (Don't need to discuss pitch)

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  • $\begingroup$ All of the links in your question seem to point to answers that say "no, it's a myth". Are there any counterexamples? Right now I don't see any controversy. $\endgroup$ – BowlOfRed Jul 11 '18 at 23:58
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    $\begingroup$ @BowlOfRed I think the hang glider is a counterexample. $\endgroup$ – Pilothead Jul 12 '18 at 0:07
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    $\begingroup$ What is usually understood as a "pendulum effect" is the normal (centered) fuselage contributing to the stability, i.e. creating a rolling moment by the virtue of being lower. This is a fallacy. But once you have misaligned lift and weight (hang glider), i.e. the lift vector not going through C.G., you'll have a rolling moment. $\endgroup$ – Zeus Jul 12 '18 at 0:43
  • $\begingroup$ There is an aerodynamic effect (often called the Keel effect) that is sometimes called the "pendulum effect". There is also the idea that a plane that "hangs" below a high wing is stable due to gravity pulling it upright. I suspect that some things you find may be referring to one and some to the other. Are you asking about one of these concepts specifically? Wikipedia links it to the keel effect. $\endgroup$ – BowlOfRed Jul 12 '18 at 3:07
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    $\begingroup$ The fallacy is to think that the hangglider pilot shifts the center of gravity: He can't, because of the conservation of momentum. Instead, he shifts the wing and with it the center of lift to the opposite side. All rolling motion is around the center of gravity so the weight has no pull on the rolling moment (pun intended). $\endgroup$ – Peter Kämpf Jul 14 '18 at 22:42
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For a proper discussion, we should first define what a pendulum is. Only then can be established if such an effect can exist in airplanes.

Let's base the definition on Wikipedia. It says that

A pendulum is a weight suspended from a pivot so that it can swing freely.

Maybe it is also worth to look closer what a pivot is: A thing on which something turns.

So the pendulum is fixed to a fulcrum which keeps it suspended and allows it to swing freely. The ideal pendulum has all its mass in its massive bob and, therefore, pivot and center of gravity are not in the same place. If the center of gravity and the pivot would fall together, a pendulum could only rotate but not swing. And that swinging motion is what the pendulum is all about.

Now for airplanes: Here we have no pivot. All rotation can only happen around the center of gravity. This is equivalent to the pendulum with no length which is no pendulum any more. But what about hang gliders? Using Pilothead's sketch let's consider how a hangglider starts a rolling motion and compare that to how a glider does it. The top two sketches show each in steady straight flight while the bottom two sketches show both initiating a roll to the right:

Comparison of hang glider and sailplane

In both cases, a lateral imbalance is needed to create a rolling moment (red round arrow). While the pilot of the hang glider shifts the whole wing sideways, the glider pilot commands a lift difference between both wings using ailerons. Note the shift of the hangglider to the right and with it the center of lift in the lower left sketch: The center of gravity stays where it is due to the conservation of momentum while the pilot will shift slightly to the left. In case of the glider, no such lateral shift of the craft occurs; instead the distribution of lift is shifted in order to create the imbalance. The effect is the same: A lateral displacement between weight and lift which causes a rolling moment. Rolling happens around the center of gravity (because of conservation of momentum again) and in all cases there is no displacement between the pivot point and the working point of the weight, because both are the same: The center of gravity.

A hangglider is only superficially different from an airplane because the center of lift is actively shifted by displacing the wing laterally instead of a change in lift distribution (neglecting the influence of the pilot mass on the rigging of each wing), but in all cases the vehicle will rotate around its center of gravity. Weight has no lever arm to that center of gravity, so there can be no pendulum effect. Or keel effect, fo that matter.

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  • $\begingroup$ I've always understood that the keel effect is the roll-sideslip coupling due to pressure on surfaces above or below the cog (I.e. purely aerodynamic and not like a weighted keel). Is this not known as the keel effect, or is this not a real effect? $\endgroup$ – Sanchises Jul 15 '18 at 9:21
  • $\begingroup$ @Sanchises: Yes, that makes sense, but there is already the dihedral effect to occupy that slot. Granted, this name sounds as if only the wing is involved, but a fin or vertical tail (with a dihedral of 90°) shows the same effect, and for the same reason. I see no need to add yet another name, but admit that a real sailboat keel works in much the same way. In the Wikipedia article however, it is used as another name for the pendulum effect (but then it also confusingly talks of an aerodynamic side force). $\endgroup$ – Peter Kämpf Jul 15 '18 at 13:35
  • $\begingroup$ Ah I suppose indeed a vertical tail "keel" is indeed just 90° dihedral. The wiki article is useless I agree. Would you consider the fuselage or pontoons dihedral (or anhedral for pontoons)? $\endgroup$ – Sanchises Jul 15 '18 at 13:54
  • $\begingroup$ @Sanchises No, both the fuselage and pontoons would have too little aspect ratio and too little lever arm to create much of a rolling moment. On the other hand, a low fuselage is letting the wing react like it had a bit more dihedral - that is about it for fuselages, however. $\endgroup$ – Peter Kämpf Jul 15 '18 at 14:46
  • $\begingroup$ Notice both offset CG and center of lift to roll. Now, roll to 45 degrees and notice the loss of symmetry, CG and aerodynamic center are now not aligned WRT sky (VERTICAL LIFT) and ground (CG vector). End roll input. With the hang glider, the lift(drag) vector will pull its end up, the CG down until they are back in line. With the plane will realign the SAME WAY, using tail and dihedral effect roll torques. No mystery here. It gets better when the Sandringham Short starts turning. (found in ASE literature) $\endgroup$ – Robert DiGiovanni Oct 29 '18 at 18:57
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The pendulum effect is there in high wing a/c in theory but the effect is negligible because the moment arm of the center of mass is so short relative to lateral aerodynamic center. As with dihedral, for the effect to work you have to have sideslip, and the center of mass has to be below the lateral aerodynamic center of the sideslipping fuselage so that a rolling moment is created by gravity vs the lateral force.

Best way to imagine it is the take it to the extreme; imagine a rigidly installed weight on a solid pole extending 50 feet below the fuselage. If the airplane slideslips, there will be some lateral lift force generated opposite to the sideslip direction centered somewhere on the fuselage, and with the center of mass way below that due to the weight on the pole, the center of mass is going to want to get below the fuselage. On a normal high wing airplane, this effect is negligible if there at all.

Not for parasails however. Pendulum effect provides all of a parasail's lateral stability. If a parasail starts to sideslip, your ass wants to get below it because the center of mass is down where you are and the lateral aerodynamic center is up at the wing. The pendulum effect on a parasail is so strong that they are able to bank by skidding even though the skid is induced by increasing lift on the inside half of the wing (it's the accompanying drag that's doing the actual turning). In other words pendulum effect overpowers the opposite rolling moment of the lowered trailing edge.

On a high wing aircraft the main effect of the configuration is aerodynamic dihedral, which is the differential lift created by sideslip where spanwise flow is obstructed below the wing but not above by the T configuration, which tends to increase the lift of the low wing. A high wing aircraft can have sufficient self-leveling effect from aerodynamic dihedral that it can get away with no geometric dihedral, and the wing is straight, although most include some geometric dihedral as well.

A good example of this is the CL-215 waterbomber. The wings are straight and there is sufficient dihedral effect from the T wing placement to serve the roll stability needs of the aircraft's mission. But, when the conversion to the turboprop CL-415 was done, the flat nacelles of the PW123 engines were discovered to have a blocking effect on spanwise flow above the wing in sideslip that was equivalent to extending the fuselage above the wing, killing a lot of the aerodynamic dihedral effect. Any pendulum effect that was there wasn't significant (if anything, the center of mass of the 415 was lower because of the lighter engines vs the R2800s of the 215).

The fix for the CL415 was one of the biggest aerodynamic band-aids I've ever seen, those odd little wingtip extensions (they are NOT winglets) which create just enough dihedral effect in sideslip to restore what was lost from the addition of the fin-like nacelles.

https://upload.wikimedia.org/wikipedia/commons/9/9a/Canadair_CL-415_Filling.jpg

You see the opposite with high wing aircraft with sweep because there is a strong dihedral effect from the sweep itself. The combination of T configuration and sweep creates way too much dihedral effect. So, just about all high wing airplanes with swept wings have anhedral to cancel out the excess.

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    $\begingroup$ All that you describe except for the last paragraph is dihedral effect. This is a completely aerodynamic effect and has nothing to do with gravity or a pendulum. What the question is getting at is whether gravity can cause a roll moment due to CoG being lower than the center of lift. $\endgroup$ – TomMcW Jul 13 '18 at 17:13
  • $\begingroup$ The first sentence answered the question. There is an effect but it is too mild to be significant. The rest was an explanation of why, and what "aerodynamic dihedral" is about, which is the real benefit of the high wing configuration from a lateral stability standpoint. Seems pretty straightforward to me. $\endgroup$ – John K Jul 13 '18 at 20:50
  • $\begingroup$ But some people dispute that there even is such an effect, no matter how small. I understand all the effects due to sideslip, but does the fact that the cg is not directly below the center of lift create a roll moment? PK made the point that the lift vector is always radial to cg, therefore can create no roll moment, no matter how far away it is $\endgroup$ – TomMcW Jul 13 '18 at 21:05
  • $\begingroup$ Ok I see your point. I've edited my post. See if that helps. $\endgroup$ – John K Jul 14 '18 at 1:09
  • $\begingroup$ John, I totally agree with you on parasails, and it might help to line up 3 points, low wing (and tail), high wing (and tail), and parasail. The real key is the roll torque created by the lateral displacement (off setting) of CG and Clift for roll stability. The longer the torque arm, the stronger the "pendulum" becomes. It is as much lift(drag) pulling up as weight pulling down to restore to upright alignment. Not negligible in high wings, but lacking experimental data, the passion continues. $\endgroup$ – Robert DiGiovanni Oct 29 '18 at 20:43
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The "pendulum effect" as used in aerodynamic theory does NOT necessarily involve a weight that is free to swing. Maybe it's a poor phrase to describe what we are talking about, but it's the one that has become common. It's also a bit misleading because it implies that gravity itself is somehow exerting a direct roll torque on the aircraft. That is not actually the case-- gravity or weight acts at the CG and so does not exert a direct roll torque. Still, a low CG placement DOES tend to lead to a stabilizing roll torque that is much like the roll torque contributed by dihedral, sweep, etc. All of these effects contribute a stabilizing "downwind" roll torque in the presence of sideslip. All of these effects could be said to shift the aircraft's overall "effective dihedral"-- the relationship between sideslip and "downwind" roll torque-- in the more positive or less negative direction

Consider free-flight model airplanes with the wing on a pylon high above the fuselage. Consider paragliders which have a strong anhedral geometry to the arched wing and yet are generally quite stable roll in roll -- as illustrated by countless stories of both deliberate and accidental cloud-flying in such aircraft with minimal instrumentation and yet with acceptable outcomes.

Note that the multiple lines connecting the paraglider pilot to the wing essentially act like fixed struts due to the triangular geometry involved.

The key to understanding the "pendulum effect" lies in understanding that a turn generally involves some sideslip (for reasons that are not simple), and that during a sideslip, the aircraft's drag force has a sideways component relative to the aircraft's longitudinal axis, and we also generate aerodynamic sideforce ("sideways lift", acting perpendicular to the lift and drag vectors) as the airflow strikes the side of the fuselage, vertical tail, etc. Any sideways force that acts above or below the CG will contribute a roll torque.

In a paraglider, the same anhedral wing geometry that must contribute some amount of destabilizing "upwind" roll torque during a sideslip due to the difference in angle-of-attack between the left and right halves of the canopy or wing, also exposes a great deal of surface area to the sideways flow, high above the CG, contributing a stabilizing "downwind" roll torque-- the "pendulum effect". Obviously in a paraglider, the latter dominates over the former.

High-winged aircraft benefit from enhanced roll stability due to the "pendulum effect", though there is also an additional "downwind" roll torque created by interference between the fuselage and the wings. The latter may be absent if the wing is mounted on struts high over the fuselage-- the "parasol" configuration.

In a hang glider, the pilot hangs from a flexible strap typically connected near the aircraft CG. In such a case, a "pendulum effect" is only present when the pilot locks himself into place with his arms-- i.e. when he uses his muscles to make a roll input. When he is hands-free, his weight acts at the CG and there is no "pendulum effect", even though during a sideslip, his body does tend to displace slightly (a few inches) toward the "upwind" side of the control frame, just as a slip-skid ball would. Note that the pilot's tendency to swing slightly to the upwind side of the control frame during a slip, is simply a reflection of the sideways component of the wing's drag force plus the aerodynamic sideforce generated by the wing -- if these were zero the pilot would have no tendency to deflect toward the upwind side of the control-frame, and a slip-skid ball would remain centered. (In fact, in such a case the wind would actually blow the pilot toward the other side of the control frame-- the downwind side-- during a slip. The pilot's body would act more like a yaw string than a slip-skid ball! This is not what we observe in practice.)

In this answer, except where specifically stated otherwise, we'll consider the hang glider in the "hands-free" case-- i.e. when the pilot is exerting zero muscle force. The same dynamics do also affect the control inputs (muscle force) the pilot must exert to get a given result (e.g. roll rate), but we won't explore that very deeply in this answer.

On some older designs, the pilot's "hang strap" connected to the hang glider several feet below the "keel tube"-- in this case the pilot's weight acted well below the CG of the glider, and so the sideways aerodynamic forces generated by the wing during a slip did in fact contribute a stabilizing "pendulum effect". In such a case it's equally valid to look at the pilot and the glider as separate bodies, and note the roll torque generated by the sideways pull of the pilot's hang strap on the glider, or to view the glider and pilot as one system (with the mass of the pilot considered to be located at the point where the "hang strap" connects to the glider), and note the roll torque generated by the sideways aerodynamic forces acting above the CG of the whole system.

On many modern hang gliders, the pilot's "hang strap" actually connects to the glider partway up the kingpost, or on kingpost-less gliders, on a little stub sticking several inches above the keel tube. In this case the pilot's weight acts above the CG of the glider, so the interaction between the sideways aerodynamic forces in a slip, and the mass of the pilot, contributes a destabilizing "upwind" roll torque-- an "anti-pendulum" effect. This is done to increase maneuverability. Hang gliders experience substantial sideslip due to adverse yaw while rolling, so excessive "effective dihedral"-- contributing excessive "downwind" roll torque in the presence of sideslip-- is very undesirable and greatly limits the roll rate that can be achieved.

Note that the gull-shaped wing we see in many hang gliders-- especially in a view of the trailing edge-- contributes a dihedral geometry to the inboard part of the wing and an anhedral geometry to the outboard part of the wing. Even if the net result in terms of a pure dihedral effect is zero-- which may or not be the case-- this type of design does increase the total amount of surface area exposed to the sideways airflow during a sideslip. So the sideways aerodynamic force component during a slip will be larger with such a design, than with a completely flat wing. This is probably best described as an unintended consequence of a wing shape that evolved for other reasons. In some older hang gliders that had vastly more sail "billow"-- more "arch" to the trailing edge-- than today's designs, the pilot probably experienced a much larger tendency to "fall" toward the low or "upwind" side of the control frame during a sideslip, due to the increased side area exposed to the airflow.

As noted above, the relationship between bank angle, turning, and slipping is complex. It is driven in part by the increased drag experienced by the outboard wingtip in turning flight, due to its higher airspeed. It's a misconception to think that banking automatically generates a slip simply because weight now has a sideways component in the aircraft's frame of reference. It's also a misconception to think that banking automatically generates a slip simply because the lift vector is now tilted relative to the earth and the wing's lift vector now has a horizontal component-- that is true in any banked turn, slipping or not. Sometimes yaw rotational inertia may play a significant, albeit transient, role in causing sideslip. In actual flight, in many aircraft (including hang gliders) we can observe that slip is driven overwhelmingly by roll rate, and to a much smaller extent by yaw rate. One example of a case where we can often see substantial sideslip with no roll rate, is as we go over the top of a wingover, with a 90-degree bank angle, with no rudder input. A complete exploration of exactly what maneuvers will involve sideslip, and to what extent, and why, is far beyond the scope of this answer.

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  • $\begingroup$ second-to-last paragraph-- change "creates" to "contributes" $\endgroup$ – quiet flyer Oct 29 '18 at 17:41
  • $\begingroup$ ’exposes a great deal of surface area to the sideways flow, high above the CG, contributing a stabilizing "downwind" roll torque’ Can you explain this any better? I'm not understanding what you mean here. $\endgroup$ – TomMcW Oct 29 '18 at 17:58
  • $\begingroup$ Imagine an aircraft with totally flat wings, as viewed head-on. Now imagine an aircraft with gull-shaped wings, as viewed head on. Imagine that the wingtips are no higher or lower than the wing roots, in both cases. Won't the gull-shaped wing have more area exposed to the sideways flow during a sideslip, and thus generate more aerodynamic sideforce, than the flat wing? I guess by "sideways flow" I'm thinking specifically of the component that goes from one wingtip toward the other-- not necessarily parallel to the horizon, if the a/c is banked. $\endgroup$ – quiet flyer Oct 29 '18 at 18:03
  • $\begingroup$ sorry that was not a good comment, deleting $\endgroup$ – quiet flyer Oct 29 '18 at 18:07
  • $\begingroup$ It would produce more drag. Can't figure out where the side force would come from. Not saying it's not there, just saying I don't see what would provide it. $\endgroup$ – TomMcW Oct 29 '18 at 18:07

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