Longitudinal stability

For a tailless aircraft, the neutral point coincides with the aerodynamic center, and so for such aircraft to have longitudinal static stability, the center of gravity must lie ahead of the aerodynamic center.[13]

For missiles with symmetric airfoils, the neutral point and the center of pressure are coincident and the term neutral point is not used.

An unguided rocket must have a large positive static margin so the rocket shows minimum tendency to diverge from the direction of flight given to it at launch. In contrast, guided missiles usually have a negative static margin for increased maneuverability.

The force from the tailplane is proportional to its angle of attack, including the effects of any elevator deflection and any adjustment the pilot has made to trim-out any stick force. In addition, the tail is located in the flow field of the main wing, and consequently experiences downwash, reducing its angle of attack.

In a statically stable aircraft of conventional (tail in rear) configuration, the tailplane force may act upward or downward depending on the design and the flight conditions.[14] In a typical canard aircraft both fore and aft planes are lifting surfaces. The fundamental requirement for static stability is that the aft surface must have greater authority (leverage) in restoring a disturbance than the forward surface has in exacerbating it. This leverage is a product of moment arm from the center of gravity and surface area. Correctly balanced in this way, the partial derivative of pitching moment with respect to changes in angle of attack will be negative: a momentary pitch up to a larger angle of attack makes the resultant pitching moment tend to pitch the aircraft back down. (Here, pitch is used casually for the angle between the nose and the direction of the airflow; angle of attack.) This is the "stability derivative" d(M)/d(alpha), described below.

The tail force is, therefore:

${\displaystyle L_{t}=qS_{t}\left({\frac {\partial C_{l}}{\partial \alpha }}\left(\alpha -{\frac {\partial \epsilon }{\partial \alpha }}\alpha \right)+{\frac {\partial C_{l}}{\partial \eta }}\eta \right)}$

where ${\displaystyle S_{t}\!}$ is the tail area, ${\displaystyle C_{l}\!}$ is the tail force coefficient, ${\displaystyle \eta \!}$ is the elevator deflection, and ${\displaystyle \epsilon \!}$ is the downwash angle.

A canard aircraft may have its foreplane rigged at a high angle of incidence, which can be seen in a canard catapult glider from a toy store; the design puts the c.g. well forward, requiring nose-up lift.

Violations of the basic principle are exploited in some high performance "relaxed static stability" combat aircraft to enhance agility; artificial stability is supplied by active electronic means.

There are a few classical cases where this favorable response was not achieved, notably in T-tail configurations. A T-tail airplane has a higher horizontal tail that passes through the wake of the wing later (at a higher angle of attack) than a lower tail would, and at this point the wing has already stalled and has a much larger separated wake. Inside the separated wake, the tail sees little to no freestream and loses effectiveness. Elevator control power is also heavily reduced or even lost, and the pilot is unable to easily escape the stall. This phenomenon is known as 'deep stall'.

Taking moments about the center of gravity, the net nose-up moment is:

${\displaystyle M=L_{w}x_{g}-(l_{t}-x_{g})L_{t}\!}$

where ${\displaystyle x_{g}\!}$ is the location of the center of gravity behind the aerodynamic center of the main wing, ${\displaystyle l_{t}\!}$ is the tail moment arm. For trim, this moment must be zero. For a given maximum elevator deflection, there is a corresponding limit on center of gravity position at which the aircraft can be kept in equilibrium. When limited by control deflection this is known as a 'trim limit'. In principle trim limits could determine the permissible forwards and rearwards shift of the centre of gravity, but usually it is only the forward cg limit which is determined by the available control, the aft limit is usually dictated by stability.

In a missile context 'trim limit' more usually refers to the maximum angle of attack, and hence lateral acceleration which can be generated.

The nature of stability may be examined by considering the increment in pitching moment with change in angle of attack at the trim condition. If this is nose up, the aircraft is longitudinally unstable; if nose down it is stable. Differentiating the moment equation with respect to ${\displaystyle \alpha }$:

${\displaystyle {\frac {\partial M}{\partial \alpha }}=x_{g}{\frac {\partial L_{w}}{\partial \alpha }}-(l_{t}-x_{g}){\frac {\partial L_{t}}{\partial \alpha }}}$

Note: ${\displaystyle {\frac {\partial M}{\partial \alpha }}}$ is a stability derivative.

It is convenient to treat total lift as acting at a distance h ahead of the centre of gravity, so that the moment equation may be written:

${\displaystyle M=h(L_{w}+L_{t})\!}$

Applying the increment in angle of attack:

${\displaystyle {\frac {\partial M}{\partial \alpha }}=h\left({\frac {\partial L_{w}}{\partial \alpha }}+{\frac {\partial L_{t}}{\partial \alpha }}\right)}$

Equating the two expressions for moment increment:

${\displaystyle h=x_{g}-l_{t}{\frac {\frac {\partial L_{t}}{\partial \alpha }}{{\frac {\partial L_{w}}{\partial \alpha }}+{\frac {\partial L_{t}}{\partial \alpha }}}}}$

The total lift ${\displaystyle L}$ is the sum of ${\displaystyle L_{w}}$ and ${\displaystyle L_{t}}$ so the sum in the denominator can be simplified and written as the derivative of the total lift due to angle of attack, yielding:

${\displaystyle {\frac {h}{c}}={\frac {x_{g}}{c}}-\left(1-{\frac {\partial \epsilon }{\partial \alpha }}\right){\frac {\frac {\partial C_{l}}{\partial \alpha }}{\frac {\partial C_{L}}{\partial \alpha }}}{\frac {l_{t}S_{t}}{cS_{w}}}}$

Where c is the mean aerodynamic chord of the main wing. The term:

${\displaystyle V_{t}={\frac {l_{t}S_{t}}{cS_{w}}}}$

is known as the tail volume ratio. Its coefficient, the ratio of the two lift derivatives, has values in the range of 0.50 to 0.65 for typical configurations.[15] Hence the expression for h may be written more compactly, though somewhat approximately, as:

${\displaystyle h=x_{g}-0.5cV_{t}\!}$

${\displaystyle h}$ is known as the static margin. For stability it must be negative. (However, for consistency of language, the static margin is sometimes taken as ${\displaystyle -h}$, so that positive stability is associated with positive static margin.)[7]^: 8