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.
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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.