Longitudinal Dihedral

[Dick Whittingham]

Longitudinal Dihedral

Old definition: Angle between the angles of incidence of the wing and tailplane. Positive if the wing incidence is greater.

All-moving tails or tails with trim systems that move the stabiliser have variable angles of incidence. Therefore the longitudinal dihedral is variable in flight. I suppose you could define a basic longitudinal dihedral in these cases by measuring the tailplane incidence at the designed zero deflection position.

Why would you design longitudinal dihedral into an aircraft? I have found two internet entries. One says it is to increase longitudinal stability, but the explanation of the way it works is fundamentally flawed. The other is on a Cranfield site that says it does improve longitudinal stability , but does not say why or how.

I have my own ideas, but would like to hear from any or all experts out there. What is the purpose or function of longitudinal dihedral

Dick W

[DFC]

I think you are asking about the following which is a very simple case describing what I think you are talking about;

Bolt on the wings at an incidence of 4 degrees.

Bolt on the horizontal stabilizer at an angle of incidence of 2 degrees.

Aircraft flying along level (wing 4deg angle of attack and tail 2 deg angle of attack) and gets disturbed, nose up slightly.

Momentum carries the aircraft along original path thus increasing angle of attack on both the wing and the tail.

Say direction of the relative airflow changes by 2 degrees.
Thus angle of attack of the wing increases from 4 to 6 degrees - a 50% increase.
Angle of attack on tail increases from 2 to 4 degrees..........a doubling of the angle of attack.

That increase in angle of attack of the horizontal stabilizer increases lift and thus the tail rises to restore the original situation.

Is that what you are after?

Regards,

DFC

[Mad (Flt) Scientist]

DFC

Since the zero-lift angle of attack of a cambered airfoil isn't zero, you can't equate a % change in local AoA into a % change in lift/force.

Plus one would typically have to account for the delta downwash at the tail, due to the presence of the wing, which is going to roughly halve the delta AoA seen by the tail anyway.

As to the original question - why would one design for a different angle of incidence for the mainplane and tailplane...

The main driver, I would suggest, for the wing incidence is to obtain the best compromise between the fuselage deck angle for minimum drag, and the airfoil AoA for best L/D. Constraints on this optimisation would include ensuring that any geometric limits for takeoff/rotation/landing/flare were not penalising. (Barring a Crusader-style wing!)

Having thus arrived at a ideal wing incidence (almost as a \'tail-off\' consideration) the tail incidence is constrained by the desire for a minimum drag configuration at high speed - where we may desire that our mid c.g. cruise weight configuration has faired elevators and the best efficiency for our tail in creating trim lift - and the desire for the most efficient tail for purposes of low speed trim to ensure our field performance is not unduly affected by Vmu or trim limits.

Chances are, we\'re going to end up with a small positive root incidence on the wing, and a near-zero tail incidence.

In considering the effects the \'longitudinal dihedral\' might have on longitudinal S&C, let\'s consider a simplified case : symmetric wing and tail airfoils, and a fixed H-stab. Let\'s assume the wing is set at 2 degrees, and the tail at zero.

In cruise flight, assuming the body AoA is zero, we\'ll have a wing AoA of 2 degrees, a downwash of about 1 degree, and hence a tail AoA of -1 degree, giving us a bit of download for trim. (Which is good, assuming we\'re a nice simple +ve static stability type).

If we introduce a disturbance x to the body AoA, the wing AoA will be disturbed by X also, while the tail AoA will change by X/2, approximately (downwash ratio of 0.5 assumed previously). If our aircraft is stable the pitch axis should restore to the undisturbed condition.

Now, let\'s magically redesign the aircraft, and increase the wing incidence to 4 degs. Since a wing AoA of 2 deg is what we need for trimmed flight, that means we\'ll be at a body AoA of -2 degs for cruise. The tail AoA will be -3 degs now (-2 body -1 downwash) which means that to be in trim we\'ll need some elevator. Lets assume the elevator does indeed get adjusted.

Now, disturb this aircraft by X degrees. Again, delta of X to the wing AoA, delta of X/2 to the tail AoA, which means, all other things being equal, the aerodynamic response will be identical.

The only thing we\'ve changed is that the tail is going to be at -3+X/2, not at -1+X/2. As long as the tail lift curve is linear over a sufficiently large range, and the disturbances are relatively small, there will be no mathematical differences. If the tail is significantly nonlinear, it may matter - but in this case I\'d expect the case where the tail incidence is minimised to be the best case - and that\'s not directly a function of increasing or decreasing long-l dihedral, but comes from a consideration of the trim requirements.

In short, I can see no stability arguments in favour of increased or decreased long-l dihedral; other factors matter more.

[FE Hoppy]

Doesn't this have more to do with contolability rather than stability?
I'm thinking about high alpha situations where the stab alpha is + and thus a large elevator input is required to provide the required down force. max deflection is aerodynamicaly limited. so the designer has a choice of a bigger elevator and hence stab or some "decalage"sp. and smaller stab/elevator.

[Dan Winterland]

I concur with DFC. It's simple and the standard CFS A2 answer.

It's quite noticeable on the aircraft I did my A2 on, the Chipmunk. Wings at 4 degrees, tailplane at zero. One thing we noticed was that the students often bounced on their first solo. This was due to the c of g being further forward (without the weight of their often quite portly instructors) a greater elevator input being required, therefore greater longitudianl dihedral, therefore more pitch stability being encountered and touching down mainwheels first - leading to a bounce.

[Mad (Flt) Scientist]

I'm sorry, but an increase of body AoA of 2 degrees will produce an increase of 2 deg AoA on the wing, but will NOT produce a 2 degree AoA increase at the tail unless the downwash gradient is ZERO - which is only possible if the wing has fallen off.

[barit1]

A fresh viewpoint:

Q. Why is it that when the CG is moved aft, the airplane is more efficient (albeit less stable)?

A. Because the aft CG means the horizontal stab doesn't have to work so hard - lower CL & Cd.

How can this be true if the stab is at a positive effective AOA? I was always taught the stab is at a negative AOA (effectively, allowing for the wing downwash). This implies the CG is forward of the wing CP to balance the bird.

This creates speed stability. A positive speed disturbance means both the wing lift (up) and tail lift (down) are increased, causing a nose-up moment to restore the trimmed speed.

Or do I need a revised aero text?

[Mad (Flt) Scientist]

The latter.

The wing lift is not directly relevant.

What matters is the aircraft pitching moment, which is comprised of the wing (or wing-body if you prefer) pitching moment, and the tail lift * tail arm. Those make up the aircraft tail-on pitching moment, and it's THAT which matters.

(The reason wing lift isn't the key is that a wing with a nose down pitching moment - i.e. any cambered airfoil) will have an increased nose-down PM with increased speed at fixed alpha - even if the lift reference is at the cg.

[BOAC]

Q. Why is it that when the CG is moved aft, the airplane is more efficient (albeit less stable)?

A. Because the aft CG means the horizontal stab doesn't have to work so hard - lower CL & Cd.
This maybe a vast over-simplification of your question, barit1 - and I apologise if it is, but the MAIN reason why this is more efficient is that the tailplane lift is contributing to the overall lift component whereas if the tail has to produce a nose up force, it is not. Therefore the wing works less 'hard' therefore less induced drag. As 'mad' says, the wing pitching moment is almost irrelevant in the balance equation as it is acting MORE OR LESS around the cg. Aft movement of cg REDUCES the pitching moment of the tail and therefore REDUCES pitch stability.

[BEagle]

Having just typed out a complete explanation from my much-copied A2 notes, that BASTARD PPRuNe 'you are not logged in' registration bollocks sent it off to cyberspace.

Why, oh why, must we have this bloody nonsense? It is an utter pain in the arse....


I might type it out again. On the other hand.....it is sufficient to say that the 'CFS' explanation is utter rot.

[Dick Whittingham]

Thank you all for your inputs.

I have to admit that it was the "CFS" explanation that I described as fundamentally flawed. As far as I can see the main reason for designing in a longitudinal dihedral would be to achieve a cruise set-up with the wing at alpha for best TAS/Drag and the tail at minimum drag alpha, to minimise trim drag. This would rely on the wing/body moment at cruise alpha being near zero, achieved by careful CG positioning. By its nature this implies a longitudinal dihedral of about 4deg.

How does this seem to you all?

Dick W

[Mad (Flt) Scientist]

Sounds too large at 4 degs, if we're talking about high-subsonic cruise.

I'd expect a cruise alpha in the range of 0-2 deg body, with a wing setting angle (incidence) of no more than 2 deg more. That means the wing alpha is going to be between 2 and 4 deg. Assuming, as ever, a downwash ratio of 0.5, that implies that a tailplane AoA of zero will be achieved with a tail incidence of about 0-1 deg:

For a body AoA of A and a wing incidence of W, the tail AoA as a function of Tail incidence will be:

tail AoA=A+T-(A+W)*0.5 (1)

So for a minimum tail AoA=0, we get:

T=(A+W)*0.5 - A = (W-A)*0.5

Therefore for A=0, W=2: T=1
for A=2, W=2 : T=0)

If the expression (1) is rewritten in terms of longitudinal dihedral D, where D=W-T, then we get:

tail AoA=A+ (W-D)-(A+W)*0.5

or

tail AoA = (A+W)*0.5 - D

so for a min drag tail AoA of 0:

D=(A+W)*0.5

Which for typical values, again, of A and W in the 0-2 deg range each, gives a D also in the 0-2 deg range

[barit1]

As far as I can see the main reason for designing in a longitudinal dihedral would be to achieve a cruise set-up with the wing at alpha for best TAS/Drag...

While that may be the objective, it's often missed by a mile. Ever notice the deck angle at cruise?

The fuselage is a pretty inefficient lifting body, and yet at typical cruise Mach, it's at maybe 1 or 2 degrees noseup. Shouldn't it be level? I think it's because the aero designer optimizes the wing incidence for unrealistically high Mach, and then misses that calculation on the low side. So, in order to cruise at real world conditions (crude oil approaching $60/bbl), the galley carts roll aft.

The DC-10 could definitely have benefitted from another 2 degrees incidence.

[Mad (Flt) Scientist]

Cruise body AoA isn't the only constraint on wing incidence, as I mentioned above. You also have to consider field performance issues, both drag/lift optimisation and the need to get a decent Vmu limit. Then there's the issue of avoiding an overly nose-low approach, with attendant nose gear landing concerns.

[BEagle]

This time I’ll draft in Word first!

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.

Or, in other words:

CLw x ½ rho V² Sw x a = CLt x ½ rho V² 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:

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

dCLt x ½ rho V² St x b > dCLw x ½ rho V² 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.

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

[Dick Whittingham]

BEagle

Well, that's a blow! I am going to copy that to Word and work on it.

Dick W

[Paul Wilson]

BEagle - top tip for PPruning,

No need to type into Word, when you have finsished typing your post, just highlight the whole lot then 'right click' and select copy. If your post fails because of the dreaded not logged in, log in then right click where you would type and select paste. I just do it automatically for every post.

Of course you could always find your Pprune cookie and use an editing tool to change the expiry date to next century, but thats too hard for me.

[John Farley]

Of course you could say that the job of CFS is to provide the finest pilots for the RAF not to design aircraft.

If you give the studes not quite correct info then only the best survive out in the real world and so the standards of the RAF become second to none. Clever eh?

Mind you dunno how it all is today.

[Dick Whittingham]

C'mon fellas, I'm a retired CFS*, and here I am asking for help.

BEagle's equations are internally consistent, but I still have an emotional problem with the concusions. If you consider an aircraft with an all-moving trimming tail the the tail angle for trim will be widely diferent at low and high speeds in level flight. This is the same thing as having widely different longitudinal dihedral at each extreme. Is the aircraft then more stable at low speed than at high speed?

At low speed the tail angle for trim would surely be more nose down. If this increased longitudinal dihedral makes it more stable, could you get away with an aft CG by flying slower?

Dick W

[John Farley]

Dick

It may help to remember that the fixed tailplane has an elevator which of course will need to move when speed is changed. Same like the whole caboodle moving these days.

JF

[Mad (Flt) Scientist]

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 ½ rho V² Sw x a = CLt x ½ rho V² 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 ½ rho V² St x b > dCLw x ½ rho V² 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!

[BEagle]

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?"

[Dick Whittingham]

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

[Mad (Flt) Scientist]

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.

[Dick Whittingham]

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

[GearDown&Locked]

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

[Mad (Flt) Scientist]

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.

[Keith.Williams.]

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.

[Mad (Flt) Scientist]

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.

[Blue Heeler]

Hallelujah!

The longitudinal dihedral stability (CFS) nonsense is finally exposed.

[LOMCEVAK]

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?

[Mad (Flt) Scientist]

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.

[GearDown&Locked]

Just found this document (http://www.aoe.vt.edu/~cwoolsey/Courses/AOE3134/Lectures/AOE3134.Lecture10.pdf) that could throw some light to this topic.

Edited to add another
PDF document (http://www.b2streamlines.com/EffectiveDihedral.pdf) with good readings about this subject.

GD&L

[Mad (Flt) Scientist]

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.

[GearDown&Locked]

'longitudinal dihedral' link 1 (http://www.worldwideschool.org/library/books/tech/engineering/TheAeroplaneSpeaks/chap6.html)

'longitudinal dihedral' link 2 (http://www.av8n.com/how/htm/aoastab.html)

These links above DO speak about the subject. Enjoy.

GD&L

[Dick Whittingham]

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

[Mad (Flt) Scientist]

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.

[barit1]

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/record.php?id=19770514-0

What does this tell us about the lift vector on the stab in stabilized flight?

[Mad (Flt) Scientist]

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.

[barit1]

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.

[bookworm]

= 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.

Nah, you're just hiding the dependence. The "gradient with alpha of the aerodynamic characteristics" (by which you mean k, I guess) is always positive in the unstalled regime. If that's all that matters, we wouldn't care about all this aircraft stability stuff, because they'd always be stable.

But the problem is the rest of it, the [a - b *(1-d_g) ]. It's that bit that swaps sign and gets nasty when you move the CG. So how do we know what the sign is? The easiest way is to use the criterion that the total pitching moment is zero. i.e. that

a*CL - b*CLt = 0

And once you substitute that back in for a and b, you get:

k * da * a * [1 - CL/CLt *(1-d_g) ]

so the ratio of lift coefficients rears its ugly head, and suddenly it matters again whether CL is greater than or less than CLt, which, with the same k for wing and tail, is the proportionality argument again. Or more precisely it matters whether CL*(1-d_g) is greater than or less than CLt, which just means that we have to modify the proportionality argument to take account of the downwash gradient.

[Mad (Flt) Scientist]

No.

Because a and b are the geometric position of the lift and tail lift forces relative to the cg. What that is saying is that

(1) the further aft I place the tail, the more stabilising it is.

(2) the further aft I place the tail, the less trim lift I need from the tail.

BUT you cannot say that therefore the less trim lift there is, the more stabilising it is. They are two effects, both caused by the geomtric position of the tail, but NOT linked to each other.

Again, if I were to offload the tail by introducing a reaction force as a trim system (the Harrier pitch puffer I postulated earlier) then I could vary CLt as I desired, and destroy the CL*a=CLt*b relationship, but the contribution of the tail to the stability of the system would not change.

Alternatively, if I were to deploy trailing edge wing flaps that (magically) did not move the wing CP, such that a and b did not vary, I would need a very different tail lift to trim the increased CL from the wing, but the stability calculation WOULD NOT CHANGE ONE IOTA.

If we were to stop talking about CPs and use pitching moment in the equations, and AC/neutral point type discussions, it all becomes rather more clear.

[bookworm]

BUT you cannot say that therefore the less trim lift there is, the more stabilising it is. They are two effects, both caused by the geomtric position of the tail, but NOT linked to each other.

I think you demonstrate the link most eloquently in the previous two lines! I can say that if I load the aeroplane so as to require a greater (with the correction factors for d_g) lift coefficient from the tail than from the wing, it will be unstable.

Alternatively, if I were to deploy trailing edge wing flaps that (magically) did not move the wing CP, such that a and b did not vary, I would need a very different tail lift to trim the increased CL from the wing, but the stability calculation WOULD NOT CHANGE ONE IOTA.

That case satisfies both your criterion, and the proportionality criterion, which is that CL*(1-d_g) is greater than CLt. Since CLt is proportional to CL to get the tail lift to the correct trimmed value, if it is stable for the CL where d_g is greatest then it will be stable for other CL values.

[Mad (Flt) Scientist]

If I change Cm0, I change the trim requirement. i.e. CLt.

I do not change CL.

I therefore change the CL:CLt ratio.

But a Cm0 shift does not change the stability (or lack thereof) in any way.

Therefore there is no causal relationship between CL:CLt and stability.

-----------

A practical aircraft has a pitching moment when the wing/body generates no lift. Therefore a tail lift term is required for trim at CL(wing)=0

If the proportionality argument held water, it would be absolutely impossible to hold a stable condition here, because any change in wing lift is infinite as a proportion, while the corresponding tail lift proportional change is finite.

------

Fundamentally, the Lw*a=Lt*b relation is a gross over-simplification - as I mentioned before, that's not the way it's done to actually determine stability, because of issues like Cm0.

[Mad (Flt) Scientist]

Additionally, I should address the logical fallacy you have tripped over:

If the tail arm is increased, then
(a) aircraft more stable
(b) smaller tail lift to trim
does not in any way imply that
smaller tail lift to trim makes the aircraft more stable.

To illustrate, let us suppose a different logical relationship:

The more money I earn, then
(a) the happier I am and
(b) the more tax I pay.

If I were to follow the logic you wish to use, that would mean that (b) implies (a) and therefore:

the more tax I pay, the happier I am

and that I could become happier if my tax rate were increased.

I think it safe to say that paying more tax does not, of itself, make me happier.

Simply, if X => (a) and X => (b) that tells us NOTHING about the causal relationship between (a) and (b)

[GearDown&Locked]

MFC

Following your tail positioning problem, I suspect that if we could change the tail position of a balanced (read stable) airframe closer to its main wing without changing its proportions we would find it hard to trim and hard to stablize; something like a coil spring with a weak shock absorber: the tail would be bouncing the airframe all the time at the smallest aerodynamic variation. Conversely, if the tail is pushed aft the airframe would have a greater resistance to pitch changing.

GD&L

[Mad (Flt) Scientist]

OK, how about this attempt to nail the proportionality argument:

It is not uncommon that you will have a DOWNLOAD on the tail in order to trim. As the cg is moved forward, the download required to trim becomes greater, therefore the tail lift (as a magnitude) also becomes greater.

According to the proportionality argument, increasing the tail lift in this way will reduce the stability of the aircraft, because the effectiveness of my tail is somehow related to the absolute amount of lift it is creating.

While in reality a forward movement of the cg produces an INCREASE in longitudinal stability.

What the proportionality argument is claiming (I think!) is that there is a 'sweet spot' where tail lift to trim is ZERO and any movement away from that point reduces the stabilising effect of the tailplane, whether that movement be forward or aft. This is, I'm sorry, nonsense.

[Dick Whittingham]

I hesitate to intervene again, but let us use a bit of logic on the “percentage” argument, which seems to be:

Proposition 1: If wing alpha is greater than tail alpha then an increase in the same number of degrees on both wing and tail – a positive gust – will give a higher percentage increase of tail moment than of wing moment.

Proposition 2: If the aircraft was flying before the gust an with increased longitudinal dihedral, with tail alpha less than in Proposition 1, with wing alpha the same, then in the gust the percentage increase in tail moment over the percentage increase in wing moment will be even higher.

Proposition 3: The effect given in Proposition 2 must lead to a proportionally greater nose down tail moment over the nose up wing moment and therefore an increase in longitudinal stability.

Conclusion: Increased longitudinal dihedral increases longitudinal stability

The argument falls because relative percentage changes in wing or tail alpha in gusts do not affect the relationship of the wing and tail moments. An incremental increase in wing or tail alpha gives exactly the same incremental increase in the wing and tail moments whether the alpha increment is from 2deg to 4deg, 4deg to 6deg or 6deg to 8 deg. The conventional Cl/alpha graph is linear in this regime.

Now, the killer. Consider the case where tail alpha is at zero before a positive gust. any increase in tail alpha will then be an infinite percentage increase. Under the “percentage” argument this should lead to an infinite increase in longitudinal stability, and the aircraft should hold an unchanging and unchangeable attitude. Clearly this is nonsense.

For an argument to hold it must be true in all cases, including the extremes. The percentage argument gives a clearly false conclusion in the zero tail alpha case, and this destroys the whole theory

Dick W

[bookworm]

If the tail arm is increased, then
(a) aircraft more stable
(b) smaller tail lift to trim
does not in any way imply that
smaller tail lift to trim makes the aircraft more stable.

Let's address this issue of "logic" first then.

You correctly point out that an equality is not equivalent to a causality. That, however, does not mean that the equality is invalid.

An example that we regularly use in aerodynamics: the relationship of drag with speed whose form was recently debated at length in another thread. We use a relationship between drag and speed, typically D(v) = A*v^2 + B/v^2, on a regular basis for example to determine optimum glide speeds etc.

Is there a causality there? On the downward sloping part of the curve, does increasing speed "cause" a decrease in drag ? It certainly doesn't do so directly. But with the assumption of 1G flight, and a dependence that is linked through the induced drag coefficient, we are usually more than happy to accept the equality as a useful working model.

To return to your whimsical but well chosen example:

The more money I earn, then
(a) the happier I am and
(b) the more tax I pay.

If I were to follow the logic you wish to use, that would mean that (b) implies (a) and therefore:

the more tax I pay, the happier I am

But you would surely agree that, assuming the premises you present to be true, if I wanted, without further information, to find someone who is happy, it would be sensible to look for someone paying a lot of tax?

So back to stability...

If I have an aircraft with an all-moving tailplane, does pulling on the stick and decreasing the AOA of the tail to give it more longitudinal dihedral improve the stability? No, of course not.

If I'm designing an aircraft and I want to know how far away to put the tail, a reasonable guide (subject to modification by effects such as downwash gradient, differences in aerofoil shape, span etc.) is to put it far enough away that it can achieve the tail lift required at a lower lift coefficient than the wing, i.e. while maintaining longitudinal dihedral. Similarly, if I look at aircraft in flight, they will tend to have longitudinal dihedral because the tail is placed with enough moment to provide stability.

Are we converging?

Consider the case where tail alpha is at zero before a positive gust. [and similarly for MFS\'s point on downward lift from the tail]

I think you\'re losing sight of the wood for the trees here -- if the tail is producing zero or negative lift, there clearly is longitudinal dihedral (in the broad sense that we\'ve been using it in this thread), and there is stability (in our simple model) because the centre of lift is at or behind the centre of gravity.

The naive view, probably still held by a majority of pilots ;), is that this negative tail lift is a requirement for stability. We all agree that it\'s not.

So the only cases worth discussing are the ones where the tail is producing positive lift, which is where the "proportionality argument" fits in.

[Mad (Flt) Scientist]

First off, the logic issue.

If the tail arm is increased, then
(a) aircraft more stable
(b) smaller tail lift to trim
does not in any way imply that
smaller tail lift to trim makes the aircraft more stable.


In other words, these are statements of causality, and are unidirectional. An increased tail arm causes the aircraft to be more stable. Making an aircraft more stable does not increase the tail arm (I could, for example, make the tail bigger instead). An increased tail arm reduces the tail lift to trim, but reducing the tail lift to trim does not make the tail arm longer (I could, for example, change the Cm0 by changing the camber, moving flaps, etc.).

In fact, if I were to double the tail area I would halve the tail CL to trim, but the actual dimensional magnitude of the lift would be the same. But the aircraft would be MUCH more stable, due to the increased tail area. According to the proportionality argument, the aircraft should be EXACTLY as stable, because my tail lift is unchanged.


An example that we regularly use in aerodynamics: the relationship of drag with speed whose form was recently debated at length in another thread. We use a relationship between drag and speed, typically D(v) = A*v^2 + B/v^2, on a regular basis for example to determine optimum glide speeds etc.

Is there a causality there? On the downward sloping part of the curve, does increasing speed "cause" a decrease in drag ? It certainly doesn't do so directly. But with the assumption of 1G flight, and a dependence that is linked through the induced drag coefficient, we are usually more than happy to accept the equality as a useful working model.

You example here is actually:

Increased speed causes increased dynamic pressure.
Increased dynamic pressure causes lower AoA for 1'g' flight., and in turn.
Lower AoA causes lower CD

And

Increased dynamic pressure causes increased drag for a given value of CD.

Since increased speed is the root of both logical branches, for the case where the increased dynamic pressure CD>drag effect is actually less powerful than the reduction in CD, I can say that reducing speed DOES reduce the drag force.

What I cannot say, and this is analogous to "smaller tail lift to trim makes the aircraft more stable" is:

"increased drag for a given value of CD" causes "lower CD" or vice versa.

The two ends of the branches cannot be related to each other. Because neither branch can be reversed. Lower CD does not imply lower AoA (there are other ways to get a lower CD) and a higher drag for a given CD can be obtained through e.g. area changes, not just Q.

To return to your whimsical but well chosen example:

quote:The more money I earn, then
(a) the happier I am and
(b) the more tax I pay.

If I were to follow the logic you wish to use, that would mean that (b) implies (a) and therefore:

the more tax I pay, the happier I am



But you would surely agree that, assuming the premises you present to be true, if I wanted, without further information, to find someone who is happy, it would be sensible to look for someone paying a lot of tax?

No. Because in that case I might end up with someone in a high tax country, rather than in a zero tax country. Because these are not the only variables.

If only the variable discussed existed, the logical relations would be reversible, and you could link everything together. Because there are multiple other things affecting all the examples, you cannot simply reverse the logic.

=============================================
So back to stability...Indeed

If I have an aircraft with an all-moving tailplane, does pulling on the stick and decreasing the AOA of the tail to give it more longitudinal dihedral improve the stability? No, of course not.

So, longitudinal dihedral has no effect on stability. Good. But....

If I\'m designing an aircraft and I want to know how far away to put the tail, a reasonable guide (subject to modification by effects such as downwash gradient, differences in aerofoil shape, span etc.) is to put it far enough away that it can achieve the tail lift required at a lower lift coefficient than the wing, i.e. while maintaining longitudinal dihedral. Similarly, if I look at aircraft in flight, they will tend to have longitudinal dihedral because the tail is placed with enough moment to provide stability.

Are we converging?

We were, until that last paragraph ;)

Longitudinal dihedral - relative angle of wing and tail - as a very rough and ready design guide, may have some value in indicating typical design choices which provide a good balance between the wing/body integration desire, the desire for a low-ish AoA for the tail in typical conditions and decent trim/control power. But it says absolutely NOTHING about stability.

If Im deciding where to position a tail during the design stage, I will look at the trim cases (can I trim to the stall, can I trim at the extremes of cg, can I trim at high speed), at the control power cases (can I rotate, can I demonstrate Vmu) and at the stability criteria (am I stable enough at aft cg, am I too stable at forward cg). Assuming the only design variable I have is tail position fore/aft, I will push it as far back as I need to, simplifying, be able to trim the stall at forward cg and be stable enough at aft cg. The former is concerned with the balance between the download that the tail can generate and the pitching moments (from weight and Cm-wb) that it must counter. Knowing the CL I believe I can safely get from the tail, I will work out the minimum distance I move back to get that CLtail. The only consideration that tail setting angle will get is to ensure that the angle-of-attack of tha tail remains low enough that I will not prematurely stall the surface.

For the stability (aft cg) case what concerns me is the tailplane lift curve slope. Where I am on the curve is of little interest to me, provided I remain broadly linear. Obviously, it\'s nice to be near the middle of the curve, because I\'m furthest from non-linearity. But if the tail is linear over a decent alpha range, exactly where I end up in the linear range does not matter. Because what affects the tail contribution to stability is the magnitude of the pitching moment generated in response to a given disturbance, and that simply depends on lift curve slope and tail arm.

Consider the case where tail alpha is at zero before a positive gust. [and similarly for MFS\'s point on downward lift from the tail]

I think you\'re losing sight of the wood for the trees here -- if the tail is producing zero or negative lift, there clearly is longitudinal dihedral (in the broad sense that we\'ve been using it in this thread), and there is stability (in our simple model) because the centre of lift is at or behind the centre of gravity.

I don\'t know how you equate a tail download with "longitudinal dihedral" - it\'s being used to refer to the setting angle of the wing and tail surfaces relative to each other. I can have a negative or positive long-l dihedral and positive or negative tail lift in any combination; just move the elevator up and down as required.

And Im not using a simple model where there is no pitching moment from the wing. The presence of a zero-lift pitching moment is common to almost all aircraft and fundamental to exposing the flawed nature of the proportionality argument and longitudinal dihedral as a cause of stability.

So the only cases worth discussing are the ones where the tail is producing positive lift, which is where the \"proportionality argument\" fits in.

Again, I can change the tail angle without changing the tail lift. Just move the elevator. Proportionality still makes no sense.

And, by the way, are we using proportionality of angle-of-attack, or of lift coefficient, or of lift force itself? Because I can mess with various of those by introducing downwash, or changing tail size, or changing the Cm0.

[bookworm]

On the issue of logic, I think we'll have to leave the readers to draw their own conclusion as to what is and is not reasonable when it comes to statements involving relationships between variables in systems of many variables.

I don't know how you equate a tail download with "longitudinal dihedral" - it's being used to refer to the setting angle of the wing and tail surfaces relative to each other. I can have a negative or positive long-l dihedral and positive or negative tail lift in any combination; just move the elevator up and down as required.

Well of course you can, but how successful are your designs where the tail surface is set for positive incidence in cruise but the tail is nevertheless required to produce negative lift? That doesn't strike me as very efficient! DW raised in the very first post the issue of all-moving surfaces, where there's no such option, and there are trim systems that achieve their variability by variation of the incidence angle of the entire surface. It seems reasonable to me to consider the zero-control-deflection case.

And, by the way, are we using proportionality of angle-of-attack, or of lift coefficient, or of lift force itself?

Perhaps this is where we are at crossed purposes? The "proportionality argument" means to me the criterion of

dCL/da /CL < dCLt/da /CLt

For a case where dCL/da = dCLt/da this reduces to

CLt < CL

and for symmetric surfaces with lift coefficient proportional to AOA it reduces to

a_tail < a

i.e. longitudinal dihedral. I believe this to be the algebraic version of the argument Keith Williams put with numbers.

This can be simply modified to recognise the effect of the downwash gradient. It does not assume zero pitching moment from the wing, but is does assume that variation of pitching moment with a is negligible. This is undoubtedly a simplification, but again it can be modified by considering the moment about the aerodynamic centre. In most cases, that's small compared to the moment from the tail.

Because what affects the tail contribution to stability is the magnitude of the pitching moment generated in response to a given disturbance, and that simply depends on lift curve slope and tail arm.

Indeed it does, and if the way you want to do the sums is to make the product of those greater than some other number that you've calculated using other parameters, that's fine. Just don't be surprised if it ends up with a criterion that is approximately equivalent to the proportionality criterion.

[Mad (Flt) Scientist]

OK, yet a fourth criteria. I confess I'm getting utterly confused as to what, exactly, is now being claimed for longitudinal dihedral, and even what it's being defined as.

The thread started with a question:

Longitudinal Dihedral

Old definition: Angle between the angles of incidence of the wing and tailplane. Positive if the wing incidence is greater.

Why would you design longitudinal dihedral into an aircraft?

I fail to see how the latest criteria:

dCL/da /CL < dCLt/da /CLt

relates to that (there's nothing there about tail or wing setting angles, just (indirectly)angles of attack, which aren't the same thing)

More importantly, I'm not sure what the new criteria is supposed to represent. Is it a criteria for stability?

And how does it work for, e.g. CL approaching zero, for a case where pitching moment is non-zero and therefore a Clt is required for trim? How does it apply to a case where the trim tail load is zero? How does it apply in inverted flight, where the lift coefficients on both wing and tail reverse sign?

[bookworm]

I fail to see how the latest criteria ... relates to that (there's nothing there about tail or wing setting angles, just (indirectly)angles of attack, which aren't the same thing)

MFS

It is undoubtedly true to suggest that one could achieve the same tail lift coefficient with different tail setting angles for fixed horizontal tail surfaces, simply by requiring the use of elevator deflection. One could, in principle, select an angle for the tail with negative longitudinal dihedral as measured by that incidence, where the lift coefficient criterion is satisfied by having the elevator sticking up like an air brake.

But in selecting a tail setting angle, a design that requires elevator deflection in the regime in which the aircraft spends most of its time is unlikely to be efficient. Thus in choosing that angle it makes sense to take account of the lift coefficient required of the tail. Where the horizontal tail surface is variable incidence, the relationship between incidence angle and lift coefficient is direct.

In your very first post on this thread, you say that "tail incidence is constrained by the desire for a minimum drag configuration at high speed" and that as a result we end up with "near-zero tail incidence". In that, there is the very reasonable premise that a wing producing lift and a tail producing almost no lift gives a longitudinally stable system.

But an inquiring mind might ask why that is the case. They might further ask whether such a situation continued to offer stability if the tail lift were to change to a small positive or negative value, or how far aft one could move the CG whilst preserving stability? They might even ask if one could have an aircraft with significant longitudinal dihedral that stable. It's those questions that this criterion is designed to address.

More importantly, I'm not sure what the new criteria is supposed to represent. Is it a criteria for stability?

Yes, it is. If the pitching moment (nose-down positive) is:

M = a*CL + b*CLt

and we're contrained to be in trim (M=0) then

dM/da = b*CLt * (dCLt/da /CLt - dCL/da /CL)

and if we require that to be positive then it leads to the criterion I quoted for CLt > 0.

dCL/da /CL < dCLt/da /CLt

You have correctly noticed, of course, that we have an issue at CLt = 0, but then it's clear from the full expression above that at that point we simply need positive lift slope from the tail surface, and if CLt < 0 while CL remains > 0, dM/da is also always positive. I'll leave you to consider the inequality for inverted flight. ;)

Do you use this criterion in choosing the tail setting angle? Clearly not. But I do feel that it has some relevance in answering the question "Why would you design longitudinal dihedral into an aircraft?"

[Mad (Flt) Scientist]

OK, sorry, but that's still got nothing to do with longitudinal dihedral

You're using the trim balance to deduce the c.g. position relative to the aerodynamic centre and neutral point (for the special case when the zero lift pitching moment is zero). But there's nothing, again, in any of that that relates to the setting angle of wing and tailplane.

I happily agree that if the cg is aft of the wing AC, as the tail lift for trim increases then the aircraft is also less stable. But it's less stable because the cg is further aft, not because the tail lift has increased. The tail lift is changing because of the cg movement, but it's the cg movement that causes the change in stability, not the tail lift change.

The original case put for LD is that it's the lift itself that is the lift ratio (or angle ratio) that causes stability. What you are now talking about is tail lift varying as a symptom of the variation in cg that also causes the stability to change. That's not the same thing at all.

The CLt=0 case is very important. The proportionality argument goes haywire at that point, yet in reality doubling the tail area of an aircraft with zero tail trim lift will make it very much more stable - something the proportionality argument cannot address.