1. Static Stability

Three cases for static stability: following a pitch disturbance, aircraft can be either unstable, neutral, or stable. https://handwiki.org/wiki/index.php?curid=2080342

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.

2. Longitudinal Stability

The longitudinal stability of an aircraft, also called pitch stability,^[1] refers to the aircraft's stability in its plane of symmetry,^[1] about the lateral axis (the axis along the wingspan).^[2] One important aspect of the handling qualities of the aircraft, it is one of the main factors determining the ease with which the pilot is able to maintain trim.^[1]

If an aircraft is longitudinally stable, a small increase in angle of attack will create a negative (nose-down) pitching moment on the aircraft so that the angle of attack decreases. Similarly, a small decrease in angle of attack will create a positive (nose-up) pitching moment so that the angle of attack increases.^[2]

Unlike motion about the other two axes and in the other degrees of freedom of the aircraft (sideslip translation, rotation in roll, rotation in yaw), which are usually heavily coupled, motion in the longitudinal degrees of freedom is planar and can be treated as two-dimensional.^[1]

3. 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.^[2]

Most successful aircraft have positive longitudinal stability, providing the aircraft's center of gravity lies within the approved range. Some aerobatic 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.

4. Center of Gravity

The longitudinal static stability of an aircraft is significantly influenced by the distance (moment arm or lever arm) between the centre of gravity (c.g.) and the aerodynamic centre of the airplane. The c.g. is established by the design of the airplane and influenced by its loading, as by payload, passengers, etc. The aerodynamic centre (a.c.) of the airplane can be located approximately by taking the algebraic sum of the plan-view areas fore and aft of the c.g. multiplied by their blended moment arms and divided by their areas, in a manner analogous to the method of locating the c.g. itself. In conventional aircraft, this point is aft of, but close to, the one-quarter-chord point of the wing. In unconventional aircraft, e.g. the Quickie, it is between the two wings because the aft wing is so large. The pitching moment at the a.c. is typically negative and constant.

The a.c. of an airplane typically does not change with loading or other changes; but the c.g. does, as noted above. If the c.g. moves forward, the airplane becomes more stable (greater moment arm between the a.c. and the c.g.), and if too far forward will cause the airplane to be difficult for the pilot to bring nose-up as for landing. If the c.g. is too far aft, the moment arm between it and the a.c. diminishes, reducing the inherent stability of the airplane and in the extreme going negative and rendering the airplane longitudinally unstable; see the diagram below.

Accordingly, the operating handbook for every airplane specifies the range over which the c.g. is permitted to move. Inside this range, the airplane is considered to be inherently stable, which is to say that it will self-correct longitudinal (pitch) disturbances without pilot input.^[3]

5. Analysis

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: https://handwiki.org/wiki/index.php?curid=2083369

[math]\displaystyle{ W=L_w+L_t }[/math]

where W is the weight, [math]\displaystyle{ L_w }[/math] is the wing lift and [math]\displaystyle{ L_t }[/math] is the tail force.

For a thin airfoil at low angle of attack, the wing lift is proportional to the angle of attack:

[math]\displaystyle{ L_w=qS_w\frac{\partial C_L}{\partial \alpha} (\alpha+\alpha_0) }[/math]

where [math]\displaystyle{ S_w }[/math] is the wing area [math]\displaystyle{ C_L }[/math] is the (wing) lift coefficient, [math]\displaystyle{ \alpha }[/math] is the angle of attack. The term [math]\displaystyle{ \alpha_0 }[/math] is included to account for camber, which results in lift at zero angle of attack. Finally [math]\displaystyle{ q }[/math] is the dynamic pressure:

[math]\displaystyle{ q=\frac{1}{2}\rho v^2 }[/math]

where [math]\displaystyle{ \rho }[/math] is the air density and [math]\displaystyle{ v }[/math] is the speed.^[4]

5.1. Trim

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.^[5] 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 mass 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:

[math]\displaystyle{ L_t=q S_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) }[/math]

where [math]\displaystyle{ S_t\! }[/math] is the tail area, [math]\displaystyle{ C_l\! }[/math] is the tail force coefficient, [math]\displaystyle{ \eta\! }[/math] is the elevator deflection, and [math]\displaystyle{ \epsilon\! }[/math] 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:

[math]\displaystyle{ M=L_w x_g-(l_t-x_g)L_t\! }[/math]

where [math]\displaystyle{ x_g\! }[/math] is the location of the center of gravity behind the aerodynamic center of the main wing, [math]\displaystyle{ l_t\! }[/math] 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.

5.2. 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 [math]\displaystyle{ \alpha }[/math]:

[math]\displaystyle{ \frac{\partial M}{\partial \alpha}=x_g\frac{\partial L_w}{\partial \alpha}-(l_t-x_g)\frac{\partial L_t}{\partial \alpha} }[/math]

Note: [math]\displaystyle{ \frac{\partial M}{\partial \alpha} }[/math] 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:

[math]\displaystyle{ M=h(L_w+L_t)\! }[/math]

Applying the increment in angle of attack:

[math]\displaystyle{ \frac{\partial M}{\partial \alpha}=h\left(\frac{\partial L_w}{\partial \alpha}+\frac{\partial L_t}{\partial \alpha}\right) }[/math]

Equating the two expressions for moment increment:

[math]\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}} }[/math]

The total lift [math]\displaystyle{ L }[/math] is the sum of [math]\displaystyle{ L_w }[/math] and [math]\displaystyle{ L_t }[/math] so the sum in the denominator can be simplified and written as the derivative of the total lift due to angle of attack, yielding:

[math]\displaystyle{ h=\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}{c S_w} }[/math]

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

[math]\displaystyle{ V_t=\frac{l_t S_t}{c S_w} }[/math]

is known as the tail volume ratio. Its rather complicated coefficient^{[clarification needed]}, 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:

[math]\displaystyle{ h=x_g-0.5 cV_t\! }[/math]

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 [math]\displaystyle{ -h }[/math], so that positive stability is associated with positive static margin.)

6. 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.^[2] (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.^[2]

For a tailless aircraft [math]\displaystyle{ V_t=0 }[/math], 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.

7. Longitudinal Dynamic Stability

An aircraft's static stability is an important 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.

The longitudinal dynamic stability of an aircraft determines whether it will be able to return to its original position. https://handwiki.org/wiki/index.php?curid=2080342