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.