Longitudinal static stability
Encyclopedia
Longitudinal static stability is the stability of an aircraft in the longitudinal, or pitching, plane during static (established) conditions. This characteristic is important in determining whether an aircraft will be able to fly as intended.
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
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.
, 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.
increases and the longitudinal static stability of the aircraft also increases. As the center of gravity moves aft, the longitudinal static stability of the aircraft decreases.
The limitations specified for an aircraft type and model include limitations on the most forward position, and the most aft position, 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.
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 force
s 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.
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 coefficient of lift of the fore surface be greater than that of the aft surface; but even this general statement obviously does not apply to tailless aircraft. 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.
:
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.)
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.
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 stabilityDirectional stability
Directional stability is stability of a moving body or vehicle about an axis which is perpendicular to its direction of motion. Stability of a vehicle concerns itself with the tendency of a vehicle to return to its original direction in relation to the oncoming medium when disturbed away from...
and lateral stability.)
If an aircraft is longitudinally stable, a small increase in angle of attack
Angle of attack
Angle of attack is a term used in fluid dynamics to describe the angle between a reference line on a lifting body and the vector representing the relative motion between the lifting body and the fluid through which it is moving...
will cause the pitching moment
Pitching moment
In aerodynamics, the pitching moment on an airfoil is the moment produced by the aerodynamic force on the airfoil if that aerodynamic force is considered to be applied, not at the center of pressure, but at the aerodynamic center of the airfoil...
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 autopilotAutopilot
An autopilot is a mechanical, electrical, or hydraulic system used to guide a vehicle without assistance from a human being. An autopilot can refer specifically to aircraft, self-steering gear for boats, or auto guidance of space craft and missiles...
, 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
Fuselage
The fuselage is an aircraft's main body section that holds crew and passengers or cargo. In single-engine aircraft it will usually contain an engine, although in some amphibious aircraft the single engine is mounted on a pylon attached to the fuselage which in turn is used as a floating hull...
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
Center of gravity
In physics, a center of gravity of a material body is a point that may be used for a summary description of gravitational interactions. In a uniform gravitational field, the center of mass serves as the 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
Relaxed stability
In aviation, relaxed stability is the tendency of an aircraft to change its attitude and angle of bank of its own accord. An aircraft with relaxed stability will oscillate in simple harmonic motion around a particular attitude at an increasing amplitude....
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. As the center of gravity moves forward the moment arm between the horizontal stabilizerStabilizer (aircraft)
In aviation, a stabilizer provides stability when the aircraft is flying straight, and the airfoil of the horizontal stabilizer balances the forces acting on the aircraft....
increases and the longitudinal static stability of the aircraft also increases. As the center of gravity moves aft, the longitudinal static stability of the aircraft decreases.
The limitations specified for an aircraft type and model include limitations on the most forward position, and the most aft position, 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.
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 equilibriumMechanical equilibrium
A standard definition of static equilibrium is:This is a strict definition, and often the term "static equilibrium" is used in a more relaxed manner interchangeably with "mechanical equilibrium", as defined next....
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 force
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...
s 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
Angle of attack
Angle of attack is a term used in fluid dynamics to describe the angle between a reference line on a lifting body and the vector representing the relative motion between the lifting body and the fluid through which it is moving...
, the wing lift is proportional to the angle of attack:
where
is the wing area is the (wing) lift coefficientLift coefficient
The lift coefficient is a dimensionless coefficient that relates the lift generated by a lifting body, the dynamic pressure of the fluid flow around the body, and a reference area associated with the body...
,
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.Trim
The tailplaneTailplane
A tailplane, also known as horizontal stabilizer , is a small lifting surface located on the tail behind the main lifting surfaces of a fixed-wing aircraft as well as other non-fixed wing aircraft such as helicopters and gyroplanes...
is usually a symmetrical airfoil, so its force is proportional to angle of attack, but in general, there will also be an elevator
Elevator (aircraft)
Elevators are flight control surfaces, usually at the rear of an aircraft, which control the aircraft's orientation by changing the pitch of the aircraft, and so also the angle of attack of the wing. In simplified terms, they make the aircraft nose-up or nose-down...
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
Downwash
In aeronautics downwash is the air forced down by the aerodynamic action of a wing or helicopter rotor blade in motion, as part of the process of producing lift....
, reducing the angle of attack at the tailplane.
For a statically stable aircraft of conventional (tail in rear) configuration, the tailplane
Tailplane
A tailplane, also known as horizontal stabilizer , is a small lifting surface located on the tail behind the main lifting surfaces of a fixed-wing aircraft as well as other non-fixed wing aircraft such as helicopters and gyroplanes...
force typically acts downward. In canard aircraft, both fore and aft planes are lifting surfaces. The fundamental requirement for static stability is that the coefficient of lift of the fore surface be greater than that of the aft surface; but even this general statement obviously does not apply to tailless aircraft. 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 stallStall (flight)
In fluid dynamics, a stall is a reduction in the lift coefficient generated by a foil as angle of attack increases. This occurs when the critical angle of attack of the foil is exceeded...
before the tail, ensuring that the stall is followed immediately by a reduction in angle of attack
Angle of attack
Angle of attack is a term used in fluid dynamics to describe the angle between a reference line on a lifting body and the vector representing the relative motion between the lifting body and the fluid through which it is moving...
on the main wing, promoting recovery from the stall. (In contrast, in a canard
Canard (aeronautics)
In aeronautics, canard is an airframe configuration of fixed-wing aircraft in which the forward surface is smaller than the rearward, the former being known as the "canard", while the latter is the main wing...
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
Center of gravity
In physics, a center of gravity of a material body is a point that may be used for a summary description of gravitational interactions. In a uniform gravitational field, the center of mass serves as the center of gravity...
, the net nose-up moment is:
where
is the location of the center of gravity behind the aerodynamic centerAerodynamic center
The torques or moments acting on an airfoil moving through a fluid can be accounted for by the net lift applied at some point on the foil, and a separate net pitching moment about that point whose magnitude varies with the choice of where the lift is chosen to be applied...
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 derivativeStability derivatives
Stability Derivatives, and also Control Derivatives, are measures of how particular forces and moments on an aircraft change as other parameters related to stability change . For a defined "trim" flight condition, changes and oscillations occur in these parameters...
.
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
Static margin
Static margin is a concept used to characterize the static stability and controllability of aircraft and missiles.*In aircraft analysis, static margin is defined as the distance between the center of gravity and the neutral point of the aircraft....
. 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
Static margin
Static margin is a concept used to characterize the static stability and controllability of aircraft and missiles.*In aircraft analysis, static margin is defined as the distance between the center of gravity and the neutral point of the aircraft....
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
Chord (aircraft)
In aeronautics, chord refers to the imaginary straight line joining the trailing edge and the center of curvature of the leading edge of the cross-section of an airfoil...
. 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
Fly-by-wire
Fly-by-wire is a system that replaces the conventional manual flight controls of an aircraft with an electronic interface. The movements of flight controls are converted to electronic signals transmitted by wires , and flight control computers determine how to move the actuators at each control...
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
Tailless aircraft
A tailless aircraft traditionally has all its horizontal control surfaces on its main wing surface. It has no horizontal stabilizer - either tailplane or canard foreplane . A 'tailless' type usually still has a vertical stabilising fin and control surface...
, 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.