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Longitudinal Dihedral

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Longitudinal Dihedral

Old 12th Mar 2005, 19:41
  #21 (permalink)  
 
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No, no, no.

I'm sorry, but changing the tailplane incidence on an all-moving tail aircraft WITHOUT changing the CG, which is possible by countering with elevator - is NOT going to change the stability of the aircraft.

You cannot assume that the same change in AoA occurs at tail and at the wing; the downwash gradiet has a very significant input into this calculation.

Plus the centre of pressure is not the correct thing to be talking about when dealing with this - it's the aerodynamic centre and the neutral point which matter for this kind of work.

I'll try to put a more point-by-point response together....

ok, Beagle's post with italic comments

Draw a diagram with the wing centre of pressure at distance a ahead of the centre of mass and the tailplane centre of pressure at distance b behind the centre of mass. Fairly obviously, for things to be in balance, then the lift generated by the wing, Lw and the lift generated by the tail, Lt, must be such that:

Lw x a = Lt x b.

Not true - this is assuming that there is no pitching moment about the CP, which is by no means guaranteed. Or, rather, it's assuming the CP doesn't move when the disturbance is introduced later in the analysis

Or, in other words:

CLw x rho V Sw x a = CLt x rho V St x b

Which can be re-arranged to CLw/CLt = St x b/Sw x a

Now, imagine our original diagram is given a small disturbance. Fairly obviously, to stop the whole thing tipping on its arse, then the increase in lift from the tail, dLt at the tail centre of pressure must be greater than the increase in lift from the wing, dLw, at the wing centre of pressure. That is:

actually, the pitching moment restoring the static situation (which generally comes from the tail) must be greater than the destabilishing moment from the wing...

dLt x b > dLw x a Or, in other words:

dCLt x rho V St x b > dCLw x rho V Sw x a

Which simplifies to:

dCLt x St x b > dCLw x Sw x a

Now, for aerofoils in the cruise range concerned, the lift curve slope may be assumed to be linear and the same for both wing and tail aerofoils, hence dCLt = dCLw.

You cannot RPT cannot assume that the same alpha disturbance occurs on both wing and tail - due to downwash. So you cannot make the leap that the delta CL on both surfaces is the same, regardless of what you assume about the lift-curve slopes

So that:

St x b > Sw x a

Going back a few steps, we already showed that CLw/CLt = St x b/Sw x a. So it is clear that, for stability, CLw > CLt and this is achieved by ensuring that the wing is rigged at a greater incidence than the tailplane.

OK a bit simplistic but far more algebraically correct than the so-called CFS explanation which is utter hogwash!

And a tip for you boffins - K.I.S.S

Unless your tail and/or wing are operating in the non-linear range then changing the setting angle of either will NOT affect the stability, because stability is determined by:

the lift curve slope and pitch moment slopes of the wing WITH alpha
the tailplane lift curve slope
the downwash gradient (again, alpha)
and the relative position of these surfaces in the fore/aft sense, plus their relative sizes.

as long as the various slopes are not impacted, I can change the tail setting angle and stability will not be in ANY way affected


===============================

OK, let\'s keep it simple, as suggested.

An aircraft with wing and tail located as previously is trimmed in steady level flight. Ignoring the pedantics of CP and pitching moment reference points, let\'s say that

Lw * a = Lt * b

Let us now disturb the aircraft from the trimmed position by introducing an instantaneous change in alpha, da.

The effect of an increase in wing aoa of da is to change the wing downwash by a corresponding amount, de (where de=da * downwash_gradient)

Therefore, the change in angle of attack seen by the tailplane is : d(at)=da-de

Assuming now that the lift-curve slopes of the two airfoils are identical, and equal \'k\', we get:

the change in wing lift: dCL=k * da

the change in tail lift: dCLt=k * d(at) = k * (da-de)

Since de and da are related by the downwash gradient, if we denote that as d_g we get:

dCLt = k * ( da - da*d_g) = k * da * (1-d_g)

Therefore the total change in pitching moment about the cg would be:

+dCL * a - dCLt * b

= k * da * a - k * da * (1-d_g) * b

= k * da * [a - b *(1-d_g) ]


Notice that nowhere in that calculation of the resulting pitching moment does the actual absolute AoA on either surface play a role - what matters is the gradient with alpha of the aerodynamic characteristics.

Additionally, this is hideously simplified; to do it properly would mean starting from using something other than CPs AND would need some decent greek symbols!

Last edited by Mad (Flt) Scientist; 12th Mar 2005 at 20:16.
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Old 12th Mar 2005, 20:19
  #22 (permalink)  
 
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Take it back to basics and imagine we're talking about a basic toy glider.

It works for that; it has to as they have no pilots! As does a large wing dihedral angle.

But both are fairly crude methods of achieving stability; no-one is saying that positive longitudinal dihedral ('decalage') is essential, just that it is one way of ensuring a measure of basic longitudinal stability in some types of aeroplane.

The movements of Centre of Pressure with the extremely small perturbations of which we speak would be small enough to be ignored.

'Downwash gradient'? Did they call it something else at CFS?

"Hey, Wilbur, why don't we just stick to bikes?"
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Old 12th Mar 2005, 20:32
  #23 (permalink)  
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This is all happening too fast for me. Two glasses of the Coonawara shiraz and the debate has gone out of control behind my back.

I am now happy in my own world that longitudinal dihedral does not directly affect stability. However, I think I have convinced myself that an aircraft in level flight cannot hold a trimmed attitude without a degree of longitudinal dihedral.

That is why models, with fixed control surfaces, have it.

In BEagles book, the rise in restoring tail moment will be larger than the unstable wing moment so long as b x St is greater than a x Sw ( for equal increases in alpha).

Dick W
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Old 12th Mar 2005, 20:33
  #24 (permalink)  
 
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Downwash gradient is the relative change in the downwash field (behind the wing, specifically as experienced by the tailplane) with change to wing AoA.

It's a VERY significant factor in assessing the longitudinal stability of a configuration - as a rule of thumb, for every one degree change in angle of attack at the wing, there's about half a degree of downwash change.

What that means is that the wing 'straightens' out the disturbance before it gets to the tail, and so SIGNIFICANTLY reduces the effectiveness of the tail in countering any disturbances.

In fact, it depends on the relative position of wing and tail - if one were to have a tail located directly next to the wing, the downwash gradient would be very powerful - taken to extreme, one can conceive of the tail being so close to the wing, and the flow being so 'straightened' by the wing, that the tail would not actually experience any AoA change at all. Consider T/E flaps - whatever my AoA is at the leading edge, the flow angles at the flaps are very much invariant, because the wing defines the flow, not the AoA.

A similar phenomon occurs on canard-deltas, especially 'close coupled' designs like Eurofighter, where the downwash field from the canard has a marked effect on the AoA and efficiency of the wing unless great care is taken.

And I'm sorry, but I see no way for this to contribute to the longitudinal stability of an aircraft, except in very special circumstances that aircraft do not encounter routinely. (near-stalled surfaces)

I'm trying to conceive if it has an effect on speed stability, but I can't visualise how that might work, either.

=============

an aircraft in level flight cannot hold a trimmed attitude without a degree of longitudinal dihedral.
Sure it can. Hawk has an all-moving tail. Hawk has a wing incidence of +1 deg. I guarantee I can find a flight condition where the stab angle is also +1 degree - in fact, I'll be able to find a whole family of speeds/weights/cgs where the tail angle to trim is +1. The tail range goes either side of +1 on Hawk, if it couldn't ever trim at +1 we wouldn't have bothered with the tail having the travel it does.
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Old 12th Mar 2005, 22:26
  #25 (permalink)  
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Sorry, I was relying on little pencil sketches. I'll drop that one.

Thank you all once again for a most informative set of posts. I raised this here because of a query on the Wannabees forum that I could not have answered with any confidence. I have no idea at this stage what, if any, the JAR answer is, but sorting that out is for another day.

Dick W
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Old 12th Mar 2005, 23:01
  #26 (permalink)  
 
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Ok, let me see if I got this right.

This type of angle diference is (I dare say) rougly calculated for a speed well bellow cruise, where lift issues are more critical, and so turning the a/c more stable due to a constant pressure on the back of the frame against a slightly forward CG downforce. That would explain why at high speed (cruise) the a/c body is about 2 deg. nose up. Seems logical enough to me but is this right?

GD&L
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Old 12th Mar 2005, 23:08
  #27 (permalink)  
 
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Well, there's a bit of a difference opinion here

But my answer is "no". The wing and tail setting angles are primarily cruise performance issues; there are other criteria, but the goal will be to get an efficient cruise condition. Of course, whether it's high or low speed cruise, the weight and so forth will affect the actual practical implications. If the designers are working with an optimistic weight forecast, using quite high cruise speeds and perhaps relatively short missions (hence low fuel weights) they'll end up with a relatively low incidence for the wing. If the actual aircraft comes out heavier and thristier, AND if operators tend to fly at lower cruise speeds for economic reasons, then you'll end up with more lift needed in cruise than was expected, and so higher deck angles.
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Old 13th Mar 2005, 09:00
  #28 (permalink)  
 
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The CFS explanation is certainly flawed in that it ignores the effect of downwash over the tailplane and the fact that zero lift angle of attack is rarely zero degrees. Like all classroom explanations it is a simplifcation that is used to make the subject more digestible, by limiting the number of variables being considered. But I believe that this makes the explanation incomplete rather than totally invalid.

The main function of the taiplane is to provide longitudinal stability. To do this its angle of attack must change as the aircraft pitches away from its trimmed condition.

As pointed out by MFS in his initial post, the downwash from the wing reduces the amount by which the talplane angle of attack changes as the aircraft pitches up or down. This reduces the stabilising effects of the tailplane.

This problem could be reduced by moving the tailplane to a different position such as on the top of the fin, where it will be less affected by the downwash from the wings. But longitudinal dihedral might provide an alternative solution to this problem.

To see how this might work we need to start by looking at a single aerofoil and simplify the matter by ignoring upwash and downwash. If we assume that the CL/Alpha graph is a straight line then each degree change in angle of attack will produce the same numerical change CL.

Let's suppose that this is plus 0.1 for each degree increase in alpha. If the aerofoil is initially at 4 dgrees above its zero lift angle, then its CL will be 4 x 0.1 = 0.4. If we increase the alpha by one degree we get a CL of 0.4 + (1 x 0.1) = 0.5. This represents a 25% increase in CL. If the aerofoil is being used as a tailplane (and we ignore changes in it C of P position) this will produce a 25% change in its contribution to the aircraft pitching moment.

Now imagine that the aerofoil is initially 2 degrees above its zero lift alpha. Its initial CL will be 0.2 and increasing alpha by 1 degree will produce a CL of 0.3. This is a 50% increase.

This means that the percentage by which CL changes for each degree change in angle of attack is greatest when the aerofoil is close to its zero lift angle.

It is possible that Longitudinal dihedral takes advantage of this effect by ensuring that the tailplane is closer than the wing, to its zero lift angle. I believe that this is the basis of the CFS explanation.

We can see the overall result if we now bring together the effects of downwash and longitudinal dihedral. The downwash from the wing reduces the amount by which the angle of attack of the tailplane varies as the aircraft pitches in flight. This reduces the stablising effect of the tailplane. Longitudinal dihedral increases the percentage by which tailplane lift changes for each degree change in its angle of attack. This compensates for the destabilising effects of the downwash from the wings.

It should be possible to vary the relative magnitudes of these two competing effects by adjusting the amount of longitudinal dihedral. For a logitudinally stable aircraft the stablising effect of the tailplane should increases as the deviation from the trimmed condition increases.

The use of trimmable tailplanes or stabilators, does not render the above argument invalid. As airspeed increases the downwash from the wings decreases. This reduces the amount of longitudinal dihedral that is required. This also enables us to minimise tailplane drag in cruise flight.
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Old 13th Mar 2005, 17:38
  #29 (permalink)  
 
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Two things.

The role of the tailplane is to provide both stability and control. Even if the aircraft were stable tail-off, one would still need a pitch control surface....

And the percentage of lift argument is, I'm sorry, nonsense.

Think of this case:

Aircraft with a wing, tailplane, and reaction jet at the end of a long pole - not unlike the pitch 'puffer' on Harrier.

Case 1: pitch puffer off, tailplane generating 0.4 CLt to trim the aircraft. Change of alpha at the wing creates a change in alpha at the tail of 1 degree, which gives a delta CLt of 0.1 and a delta pitching moment of 0.1*k

Case 2: pitch puffer ON, tailplane now only generates 0.2CLt to trim because the 'puffer' is doing the rest of the work. Same change in alpha and alpha tail, same delta CLt of 0.1, same delta pitching moment of 0.1*k

What matters for pitch stability is the change in pitching moment with change to aircraft alpha (or aircraft lift if you prefer - the classical textbook way of showing stability is the dCm/dCL plot). In both of those cases the tailplane contribution to the change in Cm with the same change in alpha (and CL on the wing) is IDENTICAL.

The percentage change of the tailplane lift is utterly irrelevant to the pitch stability consideration.

Unless the tailplane is operating in the non-linear lift-curve slope range - which means it's FAR too close to stall for design comfort - there is no benefit to any particular tailplane angle.

Those people with access to a trimming taiplane aircraft could prove it if they wished. Find a flight condition where the stab to trim with no column input is in the middle of the trim range. trim the stab nose-up and apply nose-down elevator to hold the trim alpha, then do the same in the other direction - the aircraft short period response - it's pitch stability - will be the same whatever the combination of tail and elevator angles used to trim.

If tail angle affected stability, I would see different dCm/dCL gradients when I tested an aircraft in ther wind tunnel at the range of tail settings - something that is routinely done to establish tail effectiveness and downwash characteristics; in the normal operating range, where the curves are linear, there is NO effect and the curves are parallel, even with huge changes in tail angle - 15 to 20 degrees or more.
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Old 14th Mar 2005, 11:23
  #30 (permalink)  
 
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Hallelujah!

The longitudinal dihedral stability (CFS) nonsense is finally exposed.
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Old 15th Mar 2005, 14:53
  #31 (permalink)  
 
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Just one other thought to throw in. If you have a given range of elevator travel with respect to the tailplane, an increase in longitudinal dihedral by increasing the leading edge down incidence on the tailplane will allow a further forward c.g. limit (as the tailplane will generate a larger down force for a given AoA) whilst maintaining essentially the same degree of control. Therefore, the aircraft may be flown with a greater static margin or a greater degree of longitudinal static stability if longitudinal dihedral is increased. So, to go back to Dick's original question, an increase in longitudinal dihedral allows an aircraft to be flown with a further forward c.g position thus resulting in an increase in static stability.

Perhaps we are into the semantics of the English language?
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Old 15th Mar 2005, 17:58
  #32 (permalink)  
 
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That's reaching. To use the same example, should we say that an increase in elevator travel or elevator area also increases longitudinal stability, because that also would allow a further forward cg position.

Also, increased tail angle of attack won't help if you're near to using the tailplane lift capability, as you'll just be closer to the stall in that case; so even the increase in trim power comes with a caveat; it's certainly not a simple "increased long dihedral = increased stability" situation.

Aside

Just to be clear, we are simplifying things in that the detailed aerodynamics of the tailplane WILL vary with respect to the tailplane incidence angle; factors such as the degree to which is is sealed against the body/fin/bullet will influence the efficiency of the tail, and that saelant may be affected by the incidence (typically, it'll be best sealed at the zero position, given 'normal' geometry).

Also, the strength of the downwash field experienced by the tail is itself a function of the position of the tail; one could argue that the critical position is where the tail LE is relative to the wing, not the tail pivot point; so a change in incidence MIGHT cause a small change in the downwash field; this could easily go either way, depending on the actual configuration geometry, so again no rough-and-ready rule is going to work.
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Old 16th Mar 2005, 20:54
  #33 (permalink)  
 
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Just found this document that could throw some light to this topic.

Edited to add another
PDF document with good readings about this subject.

GD&L

Last edited by GearDown&Locked; 16th Mar 2005 at 21:06.
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Old 17th Mar 2005, 00:51
  #34 (permalink)  
 
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Second link is not working.

But that first one is the classical dihedral effect i.e. in the roll axis. Otherwise known as Lv or Cl-beta or Cr-beta depending on where you learned your flight dynamics. Nothing whatsoever to do with the concept of 'longitudinal dihedral' which is being discussed (or rather, I hope, debunked) here.
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Old 17th Mar 2005, 14:28
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'longitudinal dihedral' link 1

'longitudinal dihedral' link 2

These links above DO speak about the subject. Enjoy.

GD&L
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Old 17th Mar 2005, 18:32
  #36 (permalink)  
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Both those links invoke the discredited percentage or proportionality argument.

Personally, I believe MadFlt Scientist when he says that fundamentally longitudinal dihedral does not enhance longitudinal stability.

Dick W
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Old 17th Mar 2005, 23:50
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If the percentage argument had any validity - and it doesn't! - then an aircraft flying at zero wing CL - i.e. approximately a zero 'g' bunt case - would be fantastically unstable longitudinally, because any change to the wing lift would be an INFINITELY LARGE PERCENTAGE CHANGE.

While a zero 'g' trim cannot be sustained indefinitely, due to the large and immovable object we all live on getting in the way eventually, it is perfectly possible to hold a zero 'g' point for quite some time - as anyone who's seen the footage taken from inside the various airliners that have been used for weightlessness experiments can attest. ('Vomit comet', I believe was one such vehicle)

Alternatively, shall we consider the impact of the percentage of lift argument on the directional axis. At zero sideslip, the sideforce on the forward fguselage - which is destabilising - is zero. Any change in beta will cause a sideforce here; once again, an INFINITE percentage increase. I think it will be widely acknowledged that aircraft are not, in general, directionally unstable at zero sideslip.
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Old 20th Mar 2005, 00:44
  #38 (permalink)  
 
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I've been following this for a while, and wish to pose a question: What would be the effect of the loss of part or all of the horizontal tail?

For the answer, see: http://aviation-safety.net/database/...?id=19770514-0

What does this tell us about the lift vector on the stab in stabilized flight?
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Old 20th Mar 2005, 03:18
  #39 (permalink)  
 
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The effect is a loss of trim capability and of stability. Which, if either, is critical, depends on the flight conditions. Aircraft have survived loss of control surfaces before, and they have crashed too.
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Old 20th Mar 2005, 16:04
  #40 (permalink)  
 
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Loss of 1/2 stab

It tells me that the horizontal stab has a negative lift vector, and when half the stab is missing, the remaining side could not keep the nose from tucking under very rapidly. At least, not at approach speed.

Had it failed at descent speed, the crew might have survived a while longer, until they tried to slow down.

Incidentally, this accident set off a worldwide Aging Aircraft campaign to develop new inspection techniques for continued airwortiness.
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