Feb 11, 2014, 02:09 PM
Registered User
Discussion

# Thinking about dihedral, balance of yaw torques when circling,fin size,roll stability

The quote below from ShoeDLG deals with the yaw torque created by deflecting ailerons. See also the attachments that go with the post. (Thread was "Discussion adverse yaw" https://www.rcgroups.com/forums/show...1814572&page=7 )

The conclusion, as I understand it, was that on an aileron-controlled wing with zero dihedral, in circling flight at a constant bank angle, the wing would contribute a pro-skid yaw torque at speeds below best L/D, and a pro-slip yaw torque at speeds above best L/D.

But what happens if we have no ailerons, and we have dihedral, and we are controlling roll via rudder inputs? Is the situation very similar, or different?

In a constant-banked circle, we have to make the inside (slower) wing fly at a higher angle-of-attack than the outside (faster) wing. We do this via dihedral-- we fly with some sideslip (nose yawed to high side or outside of turn.)

Could it ever happen, while circling at a constant bank angle at a low airspeed (but assuming we are not flying so slowly as to stall a tip), that the inboard wing (low airspeed, high angle-of-attack) makes more drag than the outboard wing (high airspeed, low angle-of-attack)?

It would seem that this would happen whenever the airspeed (as measured at the aircraft centerline) is substantially lower than best L/D speed. As we slow down well below best L/D speed, and increase the angle-of-attack to hold lift constant, surely we also increase drag. This would seem to be what is going on with the inside wing. As we speed up closer to best L/D speed, and decrease the angle-of-attack to hold lift constant, surely we decrease drag (move closer to the best L/D angle-of-attack). This would seem to be what is going on with the outside wing.

This would mean that the wing was contributing a pro-skid influence during circling flight below the best L/D speed.

In this case, what happens if we reduce the size of the vertical fin? It seems that the wing's pro-skid influence will be less opposed, so we'll cause the plane to fly at a smaller sideslip angle, so there will be less difference in angle-of-attack between the wings, and the roll torques will no longer be in balance, and the bank angle will increase.

Do we EVER see this in practice? My understanding is that we virtually always see the opposite-- decreasing the size of the vertical fin increases roll stability-- increases the aircraft's tendency roll toward level, or decreases the aircraft's tendency to roll into a tighter turn, when flying at some bank angle X with the rudder centered.

What is the explanation for this apparent contradiction?

Of course a high-speed spiral dive may typically happen well above best L/D speed (for the bank angle), in which case we'd expect that reducing the size of the vertical fin would increase the slip angle and enhance roll stability (make the dihedral more effective)-- but do we ever see the opposite when circling slowly?

Expanding a bit more: take the simplified case of a negligible fuselage (very skinny) and a very large all-moving fin/rudder on a long moment-arm. To a first approximation the fin will be streamlined to the local flow. Since the fin is well aft of the wing, and the flight path and relative wind are curved, if the fin is streamlined to the local flow, the wing will experience a sideslip (relative wind has a component blowing from low wingtip toward high wingtip.) Imagine as a thought experiment that the dihedral angle is exactly right such that the roll torques are exactly balanced as the plane circles at airspeed X and bank angle Y, with the fin/rudder centered. Imagine that airspeed X is below best L/D speed, for the bank angle Y. The wing is (apparently) exerting a pro-skid influence, but its effect is negligible because the fin is so large.

Now if we reduce the size of the fin, won't the nose yaw inward? To keep flying at the original slip angle (as measured at the wing), won't we have to apply some outside rudder input? Because the aircraft is spirally unstable at that bank angle with the smaller fin?

If we don't see this in practice, why not?

I recently re-read Blaine Beron-Rawdon's articles on dihedral in "Model Aviation"-- a 2-part series of articles called "Spiral Stability and the Bowl Effect" (September and October 1990) and a series of 4 articles entitled "Dihedral, a 4-part series" (August through November of 1988). These articles deal mainly with radio-controlled sailplanes with no ailerons. The author felt that it was much easier to thermal at long distances if a sailplane was positively stable in roll-- i.e. required some amount of constant inside rudder, in proportion to the bank angle. The articles deal with how to achieve that-- what is the ideal dihedral angle, tail boom length, etc. Through most of the analysis, the author assumed that the vertical fin was large enough that the fin was completely streamlined to the flow, when the rudder was centered. He then went on to say that modifying these assumptions by reducing the fin size would increase the sideslip angle and increase the aircraft's spiral stability-- i.e. if the aircraft had been neutrally stable at some particular bank angle and airspeed with the larger fin, then making the fin smaller would cause the aircraft to tend to roll to wings-level, so that inside rudder would be required to hold the bank angle constant. While the articles are far from a rigorous aerodynamic treatment-- many assumptions are made-- they seem to support the idea that in circling flight, even at low airspeeds, the wing is contributing a pro-slip influence not a pro-skid influence. I still am having a hard time seeing why this should be so, when circling slowly.

I did notice that Shoe mentions that the difference in the distribution of lift force between the two wings would also typically contribute a yaw torque-- because the lift force is typically not aligned with the aircraft's z axis-- but surely there is more to it than that? That effect would vanish if, say, we simply mounted the wing to the plane at such an incidence that the aircraft's z-axis were completely perpendicular to the flight path at some particular angle-of-attack.

PS for simplicity it may be best to consider the constant-altitude case (power applied as needed)-- otherwise we have to consider the roll damping effect, i.e. the difference in angle-of-attack between the two wings induced by the fact that the flight path is descending-- a constant-bank descending turn involves a non-zero roll rate in the direction of the turn.

Steve

Quote:
 Originally Posted by ShoeDLG I finally got around to making some changes to the Vortex Lattice Code I've been using/developing. I analyzed a rectangular wing with no twist, no dihedral and no tail (a true plank) in a level 30 degree bank left turn at different speeds. The parameters I used were: Weight 1.0 lb. Span 5.0 feet Chord 0.5 feet Airfoil: NACA 63-009 Bank Angle: 30 degrees Aileron Span: full Aileron Hinge Line: unswept at 80% Chord I set conditions for a steady level turn for a couple of different speeds on either side of L/D_max (adjusted AOA for level flight equilibrium and aileron deflection for zero rolling moment). The first chart below shows the rolling moment coefficient, C_N, as a function of airspeed. A positive value of C_N (within the chosen coordinate system) means the airplane wants to yaw with its nose into the turn (skid). A negative value means it wants to yaw with its nose out of the turn (slip). You can see that somewhere around the speed for L_D_max, the yawing moment coefficient changes sign. This means that differential drag on the wings will cause a yaw into the turn at some speeds and a yaw out of the turn at others. To give a sense for how the lift and drag components are distributed along the span: -The second plot shows the lift distribution at 20 fps -The third plot shows the induced drag distribution at 20 fps -The fourth plot shows the total drag distribution at 20 fps -The left wing is on the inside of the turn The aileron deflection starts out at about 3 degrees at 20 fps, is less than a degree at 30 fps and goes down to 0.1 degree at 50 fps. C_N has a minimum value of about -0.00005 at about 40 fps (it's not surprising |C_N| gets smaller at higher speeds because the turn rate is going down as you go faster in a level, constant-bank turn). It's worth noting that this is not a purely drag-related effect. You would expect aileron deflection (inboard-downward/outboard-upward) to increase the drag on the inboard wing and decrease the drag on the outboard wing. It does, but the resulting lift differential makes the the wing want to yaw nose out of the turn. At higher speeds, aileron deflection creates a nose-out yawing moment, at lower speeds a nose-in moment. I didn't expect that. I'd be interested to see how close these results are to what AVL would predict.

### Images

Last edited by aeronaut999; Feb 11, 2014 at 02:40 PM.
Feb 11, 2014, 05:35 PM
B for Bruce
Quote:
 .....Now if we reduce the size of the fin, won't the nose yaw inward? To keep flying at the original slip angle (as measured at the wing), won't we have to apply some outside rudder input? Because the aircraft is spirally unstable at that bank angle with the smaller fin? If we don't see this in practice, why not?
I might not have the best answer as to the "why" but my experience with testing for fin area found that the exact opposite is the case.

There's still the spiral stability issue associated with the vertical tail volume coefficient to consider. As well the location of the CG has a very noticeable effect on the model's spiral stability.

We go back many years to when I was flying sailplanes far more often both for fun and contests. At the time I had a Top Flite Metric that became my main 2meter design due to an unforeseen "landing" with my previous model. In my desire to make it as efficient as possible I went for a rearward CG location to where it was just about 3 click critical on the trim lever.

A side effect of this was that the model needed a slight amount of inward rudder for shallow bank angles. Which indicates that it was side slipping slightly into the turn and trying to straighten itself out. But as the bank angle increased it got to where it needed no constant rudder, Just elevator to hold the turn. That occurred at around 25 to 30'ish degrees? (That memory thing again. We're talking 35'ish years ago). And steeper bank angles required holding a constant outward rudder to prevent tightening the bank angle. Which implies that at the steeper angles that it was side slipping outwards.

When I finally realized that the "3 click critical" CG location was a bit too much I added a little nose weight and that shifted the CG ahead to where I had 6 or 7 clicks of useable trim range between almost stalling and almost diving. As a result the bank angle where the inward rudder changed to neutral rudder to hold the bank changed to something up around 60 degrees.

Don't ask why. I don't know. But this same effect showed up later on a flat wing aileron model that I used as a vertical fin size test ship. In that case I started the model with a generous fin area and flew it at various CG and fin size combinations. To make this easy I started out too big. With a slightly forward CG it was more or less neutrally stable when level but anything other than the most mild turn required a little top aileron to avoid it tightening. I then shifted the CG back by 4'ish % and the model became totally unstable. It would try to fall off to either side when level and in a turn heavy amounts of top aileron were needed to hold the turn. It was a work load just to get it back on the ground.

At that point I cut away about 5% of the fin and rudder and tried again. Now the model handled pretty much the same as before I moved the CG back. But it was still no fun in the turns where anything more than about 3 to 5 degrees of bank needed a good amount of top aileron.

Cutting off another 5% of the area tamed this to some degree. Now the roughly 3 degrees of dihedral would level the model for 5'ish degree bank turns if I didn't hold a very small amount of inward aileron. The "neutral bank" angle where it needed equal corrections both inward and out occurred at around 15 to 20 degrees. More than that and I needed constant outward roll held.

Cutting another 5% off the fin found the sweet spot. Now the neutral bank angle was up around 45'ish degrees. And the model appeared to track well with relatively slight amounts of inward or outward control being needed to hold lower or higher bank angles.

Cutting off another 5% raised the neutral bank angle to around 60 to 70 degrees. But now the adverse yaw as I rolled into the turns was noticeable even despite the aileron differential. But once in the turns it wasn't bad.

Cutting a further 5% off caused the model to adversely yaw badly when rolling into the turns and the tail hung low in the turn all the time. It was noticeably sinking more in the turn as well with this size.

A new fin and rudder was then make up to bring the size back to the point where the neutral bank angle was 45'ish degrees and the model flown for a few years until I sold it to a buddy.

At one point I had recorded the area and calculated the vertical tail coefficients for each size step. But I lost the data many years ago and it hasn't surfaced again. So I can't offer up any hard numbers for this.

But it did show me that A) the spiral stability of a model is affected by the CG location and B) the fin size plays a big part in how the model tracks in a turn.

I seems intuitive to me that if I don't need to hold any rudder that the model is not side slipping inwards or outwards. Or at least all the oddities of the circular flight path on the inboard and outboard and wing to tail angles are cancelling out.
Feb 12, 2014, 07:24 AM
Registered User
Quote:
 Originally Posted by aeronaut999 In this case, what happens if we reduce the size of the vertical fin? It seems that the wing's pro-skid influence will be less opposed, so we'll cause the plane to fly at a smaller sideslip angle, so there will be less difference in angle-of-attack between the wings, and the roll torques will no longer be in balance, and the bank angle will increase. Do we EVER see this in practice? My understanding is that we virtually always see the opposite-- decreasing the size of the vertical fin increases roll stability-- increases the aircraft's tendency roll toward level, or decreases the aircraft's tendency to roll into a tighter turn, when flying at some bank angle X with the rudder centered.
If you're in a constant angle of bank turn with the rudder centered, the vertical fin / rudder will experience relative wind from the outside of the turn. Reducing the size of the vertical fin / rudder in this case would have almost exactly the same effect as applying rudder toward the inside of the turn. If the airplane has positive apparent dihedral effect, then applying rudder toward the inside of the turn will cause the bank angle to increase.

Quote:
 Originally Posted by aeronaut999 Do we EVER see this in practice?
With positive apparent dihedral effect, we ALWAYS see this in practice.

Quote:
 Originally Posted by aeronaut999 What is the explanation for this apparent contradiction?
I think the source of the apparent contradiction is tied up in the difference between trim and stability. Decreasing the size of the vertical fin will increase spiral stability. An increase in spiral stability does not necessarily imply the the airplane will stabilize at a smaller "controls fixed" or "controls free" bank angle.
Last edited by ShoeDLG; Feb 12, 2014 at 07:38 AM.
Feb 12, 2014, 10:36 AM
Registered User
Quote:
 Originally Posted by BMatthews I might not have the best answer as to the "why" but my experience with testing for fin area found that the exact opposite is the case.
Thanks for the notes-- that all seems consistent with the Beron-Rawdon articles. Moving the CG aft is complicated because by shortening the tail moment-arm you BOTH give the fin less power (this should tend to increase slip), and you also place it further forward (relative to the wing) so there is less curve in the flow between the wing and tail (this should tend to decrease slip). Your experience seems to suggest that the second effect was dominating. All assuming that the wing itself is tending to generate a pro-slip yaw torque. Unless something else was at work that I'm totally missing.

I'm a little surprised it was that sensitive to small CG changes.

One question I guess we have to ask is, do we really circle slower than best L/D speed (for the bank angle)? We certainly ought to-- but if so there is still something I'm not understanding as to how the spiral stability is working (questions in first post).

Steve
Feb 12, 2014, 10:57 AM
Registered User
Quote:
 Originally Posted by ShoeDLG If you're in a constant angle of bank turn with the rudder centered, the vertical fin / rudder will experience relative wind from the outside of the turn. Reducing the size of the vertical fin / rudder in this case would have almost exactly the same effect as applying rudder toward the inside of the turn. If the airplane has positive apparent dihedral effect, then applying rudder toward the inside of the turn will cause the bank angle to increase.
Shoe I'm not sure if you were speaking generally, or in the specific case of circling slower than best L/D speed (for the bank angle). The post above by BMathews, and the concepts presented in the Beron-Rawdon articles, seem to suggest the opposite of what you have stated. It seems to me it makes sense to start with the initial simplifying assumption that the fin is large enough to so dominate the yaw dynamics that it is completely streamlined to the flow. In that case you would surely agree that the fin has zero sideslip and the wing has some sideslip (flow toward the outside). Wouldn't you agree that whether the fin will start to feel a flow toward the inside, or a flow toward the outside, as we reduce the fin size, depends on whether the wing is making a pro-slip or a pro-skid yaw torque? Isn't that the only way the yaw torques can balance out? At least in the simplifying case where the fuselage is so skinny as to have a negligible effect on anything.

To me, saying that "the fin will feel a flow from the outside toward the inside" is the same as saying that the wing is making a pro-skid yaw torque. Do you agree? Is that what you were intending to say was going on? Generally, or only in the case where we are circling slower than best L/D speed?

It doesn't make sense to start our thought experiments by assuming that the wing is feeling no slip and asking what is the direction of flow at the tail. Because the wing has little to no inherent tendency to adopt any particular orientation to the flow. While the fin obviously has a powerful tendency to orient itself streamlined to the flow.

So, if the fin is feeling a flow from the outside toward the inside, and the wing is making a pro-skid yaw torque, yes I agree that reducing the size of the fin will be like applying inside rudder. The aircraft will tend to roll into a tighter turn. But BMathews seems to have reported the opposite.

Quote:
 Originally Posted by ShoeDLG I think the source of the apparent contradiction is tied up in the difference between trim and stability. Decreasing the size of the vertical fin will increase spiral stability. An increase in spiral stability does not necessarily imply the the airplane will stabilize at a smaller "controls fixed" or "controls free" bank angle.
Well, I was thinking about that. But the Beron-Rawdon articles treat the two as essentially the same. And doesn't that make sense? If (if) we want our model to require some inside rudder while thermalling, isn't it essential that it tends to seek a shallower bank angle if we center the rudder? When we have no ailerons and are controlling bank angle with rudder, I'm not seeing a distinction between rudder trim dynamics and rudder-fixed spiral stability dynamics.
Last edited by aeronaut999; Feb 12, 2014 at 11:13 AM.
 Feb 12, 2014, 11:14 AM Registered User You said it yourself: reducing the size of the vertical fin will reduce the sideslip angle at the wing. There's no doubt that if you reduce the sideslip angle by applying "inside rudder", the bank angle would increase (assuming positive apparent dihedral effect). Whether caused by "inside rudder' or a smaller vertical fin, a reduction in the sideslip will cause the bank angle to increase. This is completely independent of the yawing moment due to the wing.
 Feb 12, 2014, 12:37 PM B for Bruce The sensitivity to the small 1/4" shifts in the CG suggests to me that the effect isn't simply due to the shifts in moment arm distances. The effect is just too much to be simply based on that small a change in areas. Shoe, I don't know what changes as the bank angle increases but the effect where there's a point of neutral rudder was also VERY noticeable and very repeatable. With a given CG location and tail area size each configuration had a point where the rudder could be left alone and the model would track through the turn with only up elevator being held to avoid the model going into a spiral dive. When rudder was required at this point it was a case of small occasional nudges in or out as needed to respond to turbulence. Shallower banked open turns than this required some amount of inward rudder/aileron and tighter turns require a little outward rudder/aileron. Move the CG slightly or alter the size of the tail (or by inference I'm guessing that altering the tail moment arm) and this point of neutral roll/yaw requirement shifts to a new bank angle. This effect has been very noticeable over a variety of models I've flown with both RES and full house controls and flat to fairly flat wings. Interestingly it only shows up on models where I've set the CG rearwards to where it's riding close to the NP. Gentle Ladys and other models set up by beginners that I've helped tend to be balanced to a more stable setting for pitch. The forward CG on such models puts the neutral control bank angle up so high that I can't recall ever finding that same point on such models. It's likely still there but it's probably up around 70 to 80 degrees or some such silliness. Generally we don't see many novices doing thermal turns at such high bank angles though.... Latest blog entry: Sharpening carving knives for balsa or...
 Feb 12, 2014, 12:58 PM Registered User Bruce, Agree with all of your post. However, the behavior you describe provides no insight as to whether the yawing moment on the wing is nose-in or nose-out of turn.
 Feb 12, 2014, 01:38 PM Registered User Aero, I see what you're getting at... Start with a very large tail and the sideslip angle at the tail will be essentially zero regardless of what direction the yawing moment on the wing is. As you make the tail smaller, the direction the airplane will want to yaw will depend on the yawing moment on the wing. If the yawing moment on the wing is nose-into-turn, then the airplane will yaw nose-into-turn. If the yawing moment on the wing is nose-out of turn, then the airplane will yaw nose-out-of-turn. If the yawing moment on the wing is nose-into-turn then reducing the vertical tail size will cause a reduction in the side slip at the wing that will in turn reduce the rolling moment due to side slip. This will cause a roll into the turn. If the yawing moment on the wing is nose-out-of-turn then reducing the vertical tail size will cause an increase in the side slip at the wing that will in turn increase the rolling moment due to side slip. This will cause a roll out of the turn. I haven't seen anything to suggest that reducing the tail size (starting from a very large tail) will always cause an airplane to roll out of the turn.
 Feb 12, 2014, 02:33 PM Registered User Thread OP Yes now we are on the same page (re post immediately above). That is how I am thinking about the problem. Steve
Feb 12, 2014, 03:22 PM
B for Bruce
Quote:
 Originally Posted by ShoeDLG Bruce, Agree with all of your post. However, the behavior you describe provides no insight as to whether the yawing moment on the wing is nose-in or nose-out of turn.
Yaw or side slip?

Then there's the issue of the inner and outer wings moving at different velocities and the difference in lift that induces.

When the model is in a turn the fin will see an induced angle of attack due to the fin being behind the path of the CG as it works around the circle. I trying to visualize this and it seems like there are two factors that work together to generate this sideways angle of attack at the fin. First is the fact that the fin is following around in a larger diameter circle than that followed by the actual CG of the model. Second is the rotational rate of the CG which is constantly swinging the fin to the outside of the circle. Thus inducing an angle of attack.

But whichever effects there are at the fin during a turn they all produce an angle of attack at the fin which generates inward lift and an outward yaw angle. And inward lift will induce a side slip towards the inside and lift the inner wing. So our polyhedral or if the wing has any dihedral at all there will be a tendency to level the wings and open out the turn unless we hold some inward rudder.

Sound good?

So what is the inward rudder doing? I'd suggest that we need that inward rudder to alter the overall camber and angle of attack so that the vertical surface becomes "feathered" and effectively parallel to the turn circle. That way the wing won't be yawed.

But there's also the difference of inner to outer speed. The outboard wing SHOULD be trying to increase the bank angle. But on models with some or more dihedral the dominant effect is the dihedral wanting to recover to level. But in the end both things are happening. So perhaps the fin and rudder are moved to where it neutralizes most but not all of the yaw. Some very slight outward yaw angle to generate some outward roll is still left in the wing to counter the higher speed of the outer tip.

If this delicate balance sounds about right then it also suggests why flat wing models with little or no dihedral hit the neutral rudder point at lower bank angles than what I've seen in polyhedral models where the same neutral bank angle is much steeper. And possibly why really dead flat wing models NEED some outward aileron to generate the effective wing twist needed to counter the speed difference where even my shallow dihedral aileron models can get that "twist" through the rolling couple from the dihedral.
 Feb 12, 2014, 04:37 PM Senile Member Don't mean to throw a monkey wrench into this discussion, but when an airplane is in a constant altitude, constant bank turn there is no airspeed difference between the inside and outside wings. There is also no sideways airflow across the vertical stabilizer. The only rudder input needed is to offset adverse yaw caused by any application of aileron. Larry
 Feb 12, 2014, 04:42 PM Registered User When you get tired of that puzzle - explain how a flat model with no dihedral but lots of side area- does perfect turns with no banking involved . (Some don't think these qualify as aircraft ) The banking and rudder experiment becomes fun if you have a Night Vapor- and trim in a turn - then add -a little at a time - some elevator trim- or power These things should be required for any budding aircraft engineer -to see what is really happening. no guesswork
 Feb 12, 2014, 06:28 PM B for Bruce How do you figure that out Larry? The inner wing is covering a smaller diameter circle than the outer wing at any bank angle other than 90 degrees. So the outer wing has to cover more distance than the inner. To do so it has to have a higher airspeed at the outer wing tip than at the inner. Richard, your flat turning model doesn't describe a perfect turn though. It's a side slipping turn limited by the side lift off the fuselage. But the fuselage is such a low aspect ratio that there's still a lot of side slip. Latest blog entry: Sharpening carving knives for balsa or...
 Feb 12, 2014, 06:40 PM Registered User The fuselages are soooo large they really replace the wing as the load carrier when you turn em. the "perfect turn " doesn't really apply - I should think. most of these have effective areas equal to wing area and the aspect ratio is actually higher . maybe we are discussing different models .