The static stability refers to the initial response of an aircraft after it is subjected to a disturbance.
Aircraft stability
Définition
Stability
Aircraft stability is the ability of our aircraft to maintain and/or return to its original flight path after any disturbance that it could have met during flight. This is subdivided into two parts, static stability and dynamic stability.
Static stability
For instance, if the aircraft tends to come back to the original attitude, we can say it has a positive static stability.
This is the preferred option.
If it tends to deviate from the original attitude, it has a negative static stability.
And if it tends to maintain its new attitude it has a neutral static stability.
Dynamic stability
Now moving on to dynamic stability, which is the response of the aircraft over a certain time period, and which can also be positive, negative or neutral.
When the airplane is deviated from its path by a disturbance but comes back to it, it has a positive dynamic stability.
This is the preferred option.
On the contrary, if the aircraft is deviated from its original position and amplifies the difference with it over time, it gives a negative dynamic stability.
Finally, if the aircraft maintains the same oscillation after a certain time period, it is is qualified has having neutral dynamic stability.
How to improve dynamic stability
In order to improve the stability of the aircraft, it should be designed to have positive static stability, so the centre of pressure should be behind the centre of gravity. The wing should be swept back and di or anhedral. Attention should also be given to the tail section design.
If necessary, the pilot or auto-pilot can input commands to dampen the oscillations by activating the elevators, rudder and ailerons. To achieve this, two hardware exist on board, the AFCS (or automatic flight control system) and the autopilot, as mentioned before.
During aircraft flight, dynamic is the pilot’s and engineer’s concern against any variation coming from a pilot command or weather perturbation issues.
This variation has several impacts on the aircraft’s behaviour during the flight, such as the flight airspeed and altitude variation, the angle of attack and vertical velocity variation and the lateral velocity variation.
Longitudinal stability
Description
The longitudinal static stability refers to the response of the aircraft after a disturbance in pitch, which will be achieved with the wing profile and the horizontal stabilizers.
The longitudinal motion includes the axial velocity u, the vertical velocity w, the pitch rate q, and the pitch attitude theta.
To understand this stability, we must take a closer look at the wing air foil.
The weight force is applied at the centre of gravity, while the lift force is applied at the centre of pressure.
If we move the centre of pressure ahead of the centre of gravity, and there is a disturbance in pitch, there will be a tendency to deviate from the original attitude, giving us negative longitudinal static stability.
If we move the centre of pressure behind the centre of gravity, and apply a disturbance in pitch, a couple will bring the aircraft nose done, creating a tendency to return to return to the original attitude. This is positive longitudinal static stability.
But to prevent the aircraft nose to go down to much, we use the horizontal stabilizer. It will create a moment bringing the nose up. Overall, this will give us a positive longitudinal stability and keep level flight.
For longitudinal dynamic stability, we are still studying the response to a pitch disturbance. In this case, the oscillations are more or less quickly damped.
Aircraft and control matrices
Short period mode
A stable short-period pitching oscillation
Définition
Short-period mode
This mode is a fast mode characterized by a high frequency of oscillation and a relatively high damping factor. This mode is characterized by high variation of the angle of attack: it is a rapid pitching around the center of gravity.
This mode is characterized by:
- an angle of attack and pitch rate that oscillate rapidly
- a pitch angle that oscillates slowly
- an airspeed and altitude that remain almost constant because the time scale is short
A retenir :
The short-period mode is induced by a disturbance in pitch, be it a quick control input (elevator deflection) or a gust, causing a rapid change in the angle of attack and triggering aerodynamic forces on the horizontal tail and wing. The aircraft responds to the new angle of attack with a pitching moment, which causes the aircraft to pitch up or down. As it pitches, the tail and wing generate damping forces proportional to the pitch rate, creating quick oscillations in angle of attack and pitch rate.
Short-period mode approximation
A short-period free response implies that:
The solution of this subsystem is obtained by finding its characteristic equation:
det(xI - A = 0) <=> x² + bx + c = 0
Which admits two roots.
This mode is characterized by:
Phugoid mode
The nose pitches up and climb: that movement increases altitude and decreases speed. As a reaction, the nose is starting to pitch down, which decreases altitude and increases speed.
The development of a stable phugoid
Définition
Phugoid mode
This mode is a long period oscillation with low damping mode. In this mode, there is a large variation of pitch angle but the variation in the angle of attack is nearly zero. It is a very low period with oscillations that are often undistinguishable by the pilot. There are low oscillations. This mode is excited by thrust engine.
It is characterized by:
- a very fast time scale (fractions of a seconds - 2 seconds)
- usually highly stable
- pure rolling, quickly returns to level if trimmed properly
- controlled by ailerons
A retenir :
As said before, the phugoid is an oscillation between speed and altitude. When the aircraft pitches up slightly, it starts to climb so the airspeed decreases (kinetic to potential energy), and as airspeed drops, there is less lift so the aircraft descends again, gaining speed (potential to kinetic energy). Because the aircraft has mass and momentum, it overcorrects slightly, leading to the next half of the oscillation.
Phugoid mode approximation
Curves of longitudinal motion
Transfer functions
The response of our parameters to elevator variation can be written as:
We have the general form of the transfer functions:
Same goes for a throttle variation.
Stability augmentation system
Stability augmentation systems (SAS) were the first feedback control system designs intended to improve dynamic stability characteristics of an aircraft. It is also referred as dampers, stabilizers, and stability augmenters. These systems generally feedback an aircraft motion parameter, such as pitch rate, to provide a control deflection that opposed the motion and increased damping characteristics.
In order to correct a mode, it is necessary to choose a new natural frequency and a new damping factor for that mode.
A retenir :
Level 1: Flying qualities are clearly adequate for the mission flight phase and the desired performance is achievable with no more than minimal pilot compensation.
Level 2: Flying qualities are adequate to accomplish the mission flight phase, but some increase in pilot workload or degradation in mission effectiveness, or both, exists.
Level 3: Flying qualities such that the air vehicle can be controlled in the context of the mission flight phase, even though pilot workload is excessive, or mission effectiveness is inadequate, or both. The pilot can transition form category A flight phase tasks to category B or C flight phases, and category B and C flight phase tasks can be completed.
First, we must make sure the system is state controllable, by making sure the matrix V (as follows) has the same rank as our system (4).
On one hand, we calculate a new characteristic equation for the system:
On the other hand, we calculate the augmented matrix with state feedback as follows:
After the calculation of the components of K, we can make an identification of a, b, c and d, to finally find the augmented matrix.
Lateral stability
Description
The lateral static stability corresponds the response to a disturbance in roll and subsequently in yaw.
Lateral motion includes side velocity v, roll rate p, yaw rate r, and bank angle phi.
If the aircraft is subjected to a disturbance in roll, then it tends to roll with the wind. For this disturbance, stability is provided by the vertical stabilizer and the wing structure as the aircraft rolls due to the disturbance.
Positive lateral static stability is achieved because when lift increases on a wing, lift decreases on the other wing.
The directional static stability corresponds to a disturbance in yaw.
The nose will yaw in the same direction as the wind, and the tail will yaw in the opposite direction because the drag is increased at the tail.
As said before, the lateral stability is influenced by the interconnected roll and yaw disturbances. Indeed, if there is a yaw modification, there is more lift on a wing, creating a roll disturbance.
The biggest risk is a spiral instability, which is characterized by the increase of the bank angle.
The Dutch roll happens when the aircraft tries to come back to its original attitude, it is a combination of yaw and roll.
The lateral dynamic stability is characterized by 3 modes being the spiral, the rolling and the Dutch Roll modes.
Aircraft and control matrices
Roll damping mode
Définition
Roll mode
A short-term, fast response mode where the aircraft returns to equilibrium in roll (bank angle) after a disturbance in roll.
It is characterized by:
- very fast time constant (seconds)
- dominated by roll rate dynamics
- typically stable in most aircraft
- high damping (returns to level flight quickly unless pilot commands otherwise)
A retenir :
An aircraft with a positive roll rate will lower the starboard wing, and raise the port wing. This creates a spanwise-varying upwash and downwash on the two wings as a consequence of the tangential motion. This will always create an increase in incidence in the ‘downgoing’ wing, and a decrease in incidence in the ‘upgoing wing’. The local lift varies proportionally to the incremental incidence, and this creates a restorative moment. Thus, for any roll disturbance, the aircraft will tend to arrest the consequential rolling motion via what is often terms ‘roll damping’. So the roll mode is always stable.
The roll mode is mainly governed by aileron effectiveness and roll damping.
Roll mode approximation
Spiral mode
Définition
Spiral mode
Spiral mode is a long-term, slow dynamic mode where the aircraft slowly diverges in roll and yaw unless corrected — essentially a gradual spiral dive.
It is characterized by:
- slow mode (can take tens of seconds to develop)
- may be stable or unstable (diverging spiral)
- often needs pilot intervention or autopilot correction to maintain level flight
- dangerous if unnoticed (can lead to spiral dives)
A retenir :
The spiral mode results from an imbalance between dihedral effect (the aircraft’s tendency to return to level flight) and yaw stability. If the aircraft rolls slightly and there's not enough restoring moment, it keeps rolling and turning—leading to a spiral.
Spiral mode approximation
Dutch roll mode
Définition
Dutch roll mode
The Dutch roll mode is a coupled oscillation between yaw and roll, like a side-to-side waddle.
It is characterized by:
- oscillatory mode (side-to-side and rolling)
- usually moderately damped, but can be lightly damped or unstable
- more noticeable in swept-wing or high-speed aircraft
- can be uncomfortable for passengers, especially in turbulence
- often suppressed by a yaw damper
A retenir :
For that, a disturbance causes a yaw motion. But during yaw, the aircraft also rolls as one wing moves forward faster (more lift) and the other backward (less lift). Now, the aircraft is banked and it turns, which introduces a sideslip in the opposite direction, feeding into the yaw and roll motions again.
Dutch roll mode approximation
Curves of lateral motion
Applied to each of the modes with their eigenvalues and vectors:
Transfer functions
Applied to a rudder and an aileron variation, we have:
Mathematical reminders
Valeurs et vecteurs propres d'une matrice
Fonction de transfert
