Longitudinal static stability
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Longitudinal static stability is the stability of an aircraft in the longitudinal, or pitching, plane under steady-flight conditions. This characteristic is important in determining whether a human pilot will be able to control the aircraft in the longitudinal plane without requiring excessive attention or excessive strength.
Static stability 
As any vehicle moves it will be subjected to minor changes in the forces that act on it, and in its speed.
- If such a change causes further changes that tend to restore the vehicle to its original speed and orientation, without human or machine input, the vehicle is said to be statically stable. The aircraft has positive stability.
- If such a change causes further changes that tend to drive the vehicle away from its original speed and orientation, the vehicle is said to be statically unstable. The aircraft has negative stability.
- If such a change causes no tendency for the vehicle to be restored to its original speed and orientation, and no tendency for the vehicle to be driven away from its original speed and orientation, the vehicle is said to be neutrally stable. The aircraft has zero stability.
For a vehicle to possess positive static stability it is not necessary for its speed and orientation to return to exactly the speed and orientation that existed before the minor change that caused the upset. It is sufficient that the speed and orientation do not continue to diverge but undergo at least a small change back towards the original speed and orientation.
Longitudinal stability 
The longitudinal stability of an aircraft refers to the aircraft's stability in the pitching plane - the plane which describes the position of the aircraft's nose in relation to its tail and the horizon. (Other stability modes are directional stability and lateral stability.)
If an aircraft is longitudinally stable, a small increase in angle of attack will cause the pitching moment on the aircraft to change so that the angle of attack decreases. Similarly, a small decrease in angle of attack will cause the pitching moment to change so that the angle of attack increases.
The pilot's task 
The pilot of an aircraft with positive longitudinal stability, whether it is a human pilot or an autopilot, has an easy task to fly the aircraft and maintain the desired pitch attitude which, in turn, makes it easy to control the speed, angle of attack and fuselage angle relative to the horizon. The pilot of an aircraft with negative longitudinal stability has a more difficult task to fly the aircraft. It will be necessary for the pilot devote more effort, make more frequent inputs to the elevator control, and make larger inputs, in an attempt to maintain the desired pitch attitude.
Most successful aircraft have positive longitudinal stability, providing the aircraft's center of gravity lies within the approved range. Some acrobatic and combat aircraft have low-positive or neutral stability to provide high maneuverability. Some advanced aircraft have a form of low-negative stability called relaxed stability to provide extra-high maneuverability.
Center of gravity 
The longitudinal static stability of an aircraft is significantly influenced by the position of the center of gravity of the aircraft. Such potential cg adjustments are typically imagined as increments forward or aft from the "aerodynamic center" of the wing which is a position, typically near 1/4 chord aft of the leading edge, where changes in angle of attack do not change pitching moment. The pitching moment is not typically zero there, just constant.
As the center of gravity moves forward the longitudinal static stability of the aircraft increases. The reason is twofold:
- the moment arm between the horizontal stabilizer increases and
- the contribution of the wing's lift to pitching moment is also stabilizing in consequence of ITS changing moment arm.
Similarly, if the center of gravity is moved aft, the longitudinal static stability of the aircraft decreases. Moving the cg aft far enough will destabilize the airplane, at a point when the wing's upsetting moment exceeds the restoring moment of the tail.
Stability boundaries for a particular aircraft include limitations on the most forward and aft locations permitted for the center of gravity. No attempt should be made to fly an aircraft if its center of gravity is outside the approved range, or will move outside the approved range during the flight.
Near the cruise condition most of the lift force is generated by the wings, with ideally only a small amount generated by the fuselage and tail. We may analyse the longitudinal static stability by considering the aircraft in equilibrium under wing lift, tail force, and weight. The moment equilibrium condition is called trim, and we are generally interested in the longitudinal stability of the aircraft about this trim condition.
Equating forces in the vertical direction:
where W is the weight, is the wing lift and is the tail force.
For a symmetrical airfoil at low angle of attack, the wing lift is proportional to the angle of attack:
where is the wing area is the (wing) lift coefficient, is the angle of attack. The term is included to account for camber, which results in lift at zero angle of attack. Finally is the dynamic pressure:
where is the air density and is the speed.
The tailplane is usually a symmetrical airfoil, so its force is proportional to angle of attack, but in general, there will also be an elevator deflection to maintain moment equilibrium (trim). In addition, the tail is located in the flow field of the main wing, and consequently experiences a downwash, reducing the angle of attack at the tailplane.
For a statically stable aircraft of conventional (tail in rear) configuration, the tailplane force typically acts downward. In canard aircraft, both fore and aft planes are lifting surfaces. The fundamental requirement for static stability is that the aft surface have greater leverage in restoring a disturbance than the forward surface have 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, w.r.t changes in angle of attack will be negative: pitch up to a larger angle of attack and the resultant pitching moment will 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. Violations of this basic principle are exploited in some high performance combat aircraft to enhance agility; artificial stability is supplied by electronic means.
The tail force is, therefore:
where is the tail area, is the tail force coefficient, is the elevator deflection, and is the downwash angle.
Note that for a rear-tail configuration, the aerodynamic loading of the horizontal stabilizer (in ) is less than that of the main wing, so the main wing should stall before the tail, ensuring that the stall is followed immediately by a reduction in angle of attack on the main wing, promoting recovery from the stall. (In contrast, in a canard configuration, the loading of the horizontal stabilizer is greater than that of the main wing, so that the horizontal stabilizer stalls before the main wing, again promoting recovery from the stall.)
There are a few classical cases where this favourable response was not achieved, notably some early T-tail jet aircraft. In the event of a very high angle of attack, the horizontal stabilizer became immersed in the downwash from the fuselage, causing excessive download on the stabilizer, increasing the angle of attack still further. The only way an aircraft could recover from this situation was by jettisoning tail ballast or deploying a special tail parachute. The phenomenon became known as 'deep stall'.
Taking moments about the center of gravity, the net nose-up moment is:
where is the location of the center of gravity behind the aerodynamic center of the main wing, 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.
Static stability 
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 :
Note: 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:
Applying the increment in angle of attack:
Equating the two expressions for moment increment:
The total lift is the sum of and so the sum in the denominator can be simplified and written as the derivative of the total lift due to angle of attack, yielding:
Where c is the mean aerodynamic chord of the main wing. The term:
is known as the tail volume ratio. Its rather complicated coefficient, the ratio of the two lift derivatives, has values in the range of 0.50 to 0.65 for typical configurations, according to Piercy. Hence the expression for h may be written more compactly, though somewhat approximately, as:
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 , so that positive stability is associated with positive static margin.)
Neutral point 
A mathematical analysis of the longitudinal static stability of a complete aircraft (including horizontal stabilizer) yields the position of center of gravity at which stability is neutral. This position is called the neutral point. (The larger the area of the horizontal stabilizer, and the greater the moment arm of the horizontal stabilizer about the aerodynamic center, the further aft is the neutral point.)
The static center of gravity margin (c.g. margin) or static margin is the distance between the center of gravity (or mass) and the neutral point. It is usually quoted as a percentage of the Mean Aerodynamic Chord. The center of gravity must lie ahead of the neutral point for positive stability (positive static margin). If the center of gravity is behind the neutral point, the aircraft is longitudinally unstable (the static margin is negative), and active inputs to the control surfaces are required to maintain stable flight. Some combat aircraft that are controlled by fly-by-wire systems are designed to be longitudinally unstable so they will be highly maneuverable. Ultimately, the position of the center of gravity relative to the neutral point determines the stability, control forces, and controllability of the vehicle.
For a tailless aircraft , the neutral point coincides with the aerodynamic center, and so for longitudinal static stability the center of gravity must lie ahead of the aerodynamic center.
Longitudinal dynamic stability 
An aircraft’s static stability is an important, but not sufficient, measure of its handling characteristics, and whether it can be flown with ease and comfort by a human pilot. In particular, the longitudinal dynamic stability of a statically stable aircraft will determine whether or not it is finally able to return to its original position.
See also 
- Clancy, L.J. (1975) Aerodynamics, Chapter 16, Pitman Publishing Limited, London. ISBN 0-273-01120-0
- Clancy, L.J. (1975), Aerodynamics, Pitman Publishing Limited, London. ISBN 0-273-01120-0
- Hurt, H.H. Jr, (1960), Aerodynamics for Naval Aviators Chapter 4, A National Flightshop Reprint, Florida.
- Irving, F.G. (1966), An Introduction to the Longitudinal Static Stability of Low-Speed Aircraft, Pergamon Press, Oxford, UK.
- McCormick, B.W., (1979), Aerodynamics, Aeronautics, and Flight Mechanics, Chapter 8, John Wiley and Sons, Inc., New York NY.
- Perkins, C.D. and Hage, R.E., (1949), Airplane Performance Stability and Control, Chapter 5, John Wiley and Sons, Inc., New York NY.
- Piercy, N.A.V. (1944), Elementary Aerodynamics, The English Universities Press Ltd., London.
- Stengel R F: Flight Dynamics. Princeton University Press 2004, ISBN 0-691-11407-2.