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.

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.

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.

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.

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.

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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.

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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.

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)

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

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.

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

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:

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?

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.

First off, the logic issue.

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.


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.

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

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

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.

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.

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.

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.

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.

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.

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.

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:

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)

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?

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.

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?"

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.