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Stability Concepts

The aircraft's response to momentary disturbance is associated with its
inherent degree of stability built in by the designer, in each of the three axes,
and occurring without any reaction from the pilot.

There is another condition affecting flight, which is the aircraft's state of trim
or equilibrium (where the net sum of all forces equals zero).
Some aircraft can be trimmed by the pilot to fly 'hands off' for straight and
level flight, for climb or for descent.

Free flight models generally have to rely on the state of trim built in by the
designer and adjusted by the rigger, while the remote controlled models have
some form of trim devices which are adjustable during the flight.

An aircraft's stability is expressed in relation to each axis:
lateral stability (stability in roll), directional stability (stability in yaw)
and longitudinal stability (stability in pitch).
Lateral and directional stability are inter-dependent.



Stability may be defined as follows:
- Positive stability - tends to return to original condition after a disturbance.
- Negative stability - tends to increase the disturbance.
- Neutral stability - remains at the new condition.
- Static stability - refers to the aircraft's initial response to a disturbance.
- Dynamic stability - refers to the aircraft's ability to damp out oscillations,
which depends on how fast or how slow it responds to a disturbance.

So, a static stable aircraft may be dynamically unstable.
Dynamic instability may be prevented by an even distribution of weight inside the
fuselage, avoiding too much weight concentration at the extremities or at the CG.
Also, control surfaces' max throws may affect the flight stability, since a too much
control throw may cause dynamic instability, e.g. Pilot Induced Oscillations (PIO).

Static stability is proportional to the stabiliser area and the tail moment.
You get double static stability if you double the tail area or double the tail moment.
Dynamic stability is also proportional to the stabiliser area but increases with the
square of the tail moment, which means that you get four times the dynamic stability
if you double the tail arm length.

However, making the tail arm longer or encreasing the stabiliser area will move
the mass of the aircraft towards the rear, which may also mean the need to make
the nose longer in order to minimize the weight required to balance the aircraft...

A totally stable aircraft will return, more or less immediately, to its trimmed state
without pilot intervention.
However, such an aircraft is rare and not much desirable. We usually want an
aircraft just to be reasonably stable so it is easy to fly.
If it is too stable, it tends to be sluggish in manoeuvring, exhibiting too slow
response on the controls.

Too much instability is also an undesirable characteristic, except where an
extremely manoeuvrable aircraft is needed and the instability can be continually
corrected by on-board 'fly-by-wire' computers rather than the pilot, such as a
supersonic air superiority fighter.

Lateral stability is achieved through dihedral, sweepback, keel effect and
proper distribution of weight.
The dihedral angle is the angle that each wing makes with the horizontal (see
Wing Geometry).
If a disturbance causes one wing to drop, the lower wing will receive more lift
and the aircraft will roll back into the horizontal level.

A sweptback wing is one in which the leading edge slopes backward.
When a disturbance causes an aircraft with sweepback to slip or drop a wing,
the low wing presents its leading edge at an angle more perpendicular to the
relative airflow. As a result, the low wing acquires more lift and rises, restoring
the aircraft to its original flight attitude.

The keel effect occurs with high wing aircraft. These are laterally stable simply
because the wings are attached in a high position on the fuselage, making the
fuselage behave like a keel.
When the aircraft is disturbed and one wing dips, the fuselage weight acts like
a pendulum returning the aircraft to the horizontal level.

The tail fin determines the directional stability.
If a gust of wind strikes the aircraft from the right it will be in a slip and the fin
will get an angle of attack causing the aircraft to yaw until the slip is eliminated.




Longitudinal stability depends on the location of the centre of gravity, the
stabiliser area and how far the stabiliser is placed from the main wing.
Most aircraft would be completely unstable without the horizontal stabiliser.

Non-symmetrical cambered airfoils have a higher lift coefficient, but they also
have a negative pitching moment (Cm) tending to pitch nose-down, and thus
being statically unstable, which requires the counter moment produced by the
horizontal stabiliser to get adequate longitudinal stability.
The stabiliser provides the same function in longitudinal stability as the fin does
in directional stability.

Symmetrical (zero camber) airfoils have normally a zero pitching moment,
resulting in neutral stability, which means the aircraft goes wherever you point it.
Reflexed airfoils (with trailing edge bent up) have a positive pitching moment
making them naturally stable, they are often used with flying wings (without the
horizontal stabiliser).

It is of crucial importance that the aircraft's Centre of Gravity (CG) is located
at the right point, so that a stable and controllable flight can be achieved.
In order to achieve a good longitudinal stability, the CG should be ahead of the
Neutral Point (NP), which is the Aerodynamic Centre of the whole aircraft.
NP is the position through which all the net lift increments act for a change in
angle of attack.
The major contributors are the main wing, stabiliser surfaces and fuselage.

The bigger the stabiliser area in relationship to the wing area and the longer
the tail moment arm relative to the wing chord, the farther aft the NP will be and
the farther aft the CG may be, provided it's kept ahead of the NP for stability.



The angle of the fuselage to the direction of flight affects its drag, but has little
effect on the pitch trim unless both the projected area of the fuselage and its
angle to the direction of flight are quite large.

A tail-heavy aircraft will be more unstable and susceptible to stall at low speed
e. g. during the landing approach.
A nose-heavy aircraft will be more difficult to takeoff from the ground and to
gain altitude and will tend to drop its nose when the throttle is reduced. It also
requires higher speed in order to land safely.

The angle between the wing chord line and the stabiliser chord line is called
the Longitudinal Dihedral (LD) or decalage.
For a given centre of gravity, there is a LD angle that results in a certain
trimmed flight speed and pitch attitude.
If the LD angle is increased the plane will take on a more nose up pitch attitude,
whereas with a decreased LD angle the plane will take on a more nose down
pitch attitude.
There is also the Angle of Incidence, which is the angle of a flying surface
related to a common reference line drawn by the designer along the fuselage.
The designer might want this reference line to be level when the plane is flying
at level flight or when the fuselage is in it's lowest drag position.
The purpose of the reference line is to make it easier to set up the relationships
among the thrust, the wing and the stabiliser incidence angles.
Thus, the Longitudinal Dihedral and the Angle of Incidence are interdependent.

Longitudinal stability is also improved if the stabiliser is situated so that it lies
outside the influence of the main wing downwash.
Stabilisers are therefore often staggered and mounted at a different height in
order to improve their stabilising effectiveness.

It has been found both experimentally and theoretically that, if the aerodynamic
force is applied at a location 1/4 from the leading edge of a rectangular wing
at subsonic speed, the magnitude of the aerodynamic moment remains nearly
constant even when the angle of attack changes.
This location is called the wing's Aerodynamic Centre AC.
(At supersonic speed, the aerodynamic centre is near 1/2 of the chord).



In order to obtain a good Longitudinal Stability the Centre of Gravity CG
should be close to the main wings' Aerodynamic Centre AC.
For wings with other than rectangular form (such as triangular, trapezoidal,
compound, etc.) we have to find the Mean Aerodynamic Chord - MAC,
which is the average for the whole wing.
The MAC calculation requires rather complicated mathematics, so a simpler
method called 'Geometric Mean Chord' GMC or 'Standard Mean Chord' SMC
may be used as shown on the drawings below.
MAC is only slightly bigger than GMC except for sharply tapered wings.
Taper ratio = tip chord/root chord.



To calculate MAC of a tapered wing, the following simplified equation
may be used:
MAC = root chord * 2/3 * ((1+T+T2)/(1+T))
Where T is the wing's taper ratio.

The MAC distance from the center line may be calculated as follows:
distance = half span * (1+2*T)/(3*(1+T))


For a delta wing the CG should be located 10% ahead of the geometrically
calculated AC point as shown above.

The MAC of an elliptical wing is 85% of the root chord and is located at 42.4% of
the half wingspan from the root chord.
Elliptical wing's area = pi * wingspan * root chord/4

The AC location for biplanes with positive stagger (top wing ahead of the bottom
wing), is found according to the drawing below.

For conventional designs (with main wing and horizontal stab) the CG location
range is usually between 28% and 33% from the leading edge of the main
wing's MAC, which means between about 5% and 15% ahead of the aircraft's
Neutral Point NP.
This is called the Static Margin, which is expressed as a percentage of MAC.
When the static margin is zero (CG coincident with NP) the aircraft is considered
"neutrally stable".
However, for conventional designs the static margin should be between 5% and
15% of the MAC ahead of the NP.

The CG location as described above is pretty close to the wing's Aerodynamic
Center AC because the lift due to the horizontal stab has only a slightly effect on
the conventional R/C models.

However, those figures may vary with other designs, as the NP location depends
on the size of the main wing vs. the stab size and the distance between the main
wing's AC and the stab's AC.
The simplest way of locating the aircraft's NP is by using the areas of the two
horizontal lifting surfaces (main wing and stab) and locate the NP proportionately
along the distance between the main wing's AC point and the stab's AC point.
For example, the NP distance to the main wing's AC point would be:
D = L · (stab area) / (main wing area + stab area) as shown on the picture below:


There are other factors, however, that make the simple formula above inaccurate.
In case the two wings have different aspect ratios (different dCL/d-alpha) the NP
will be closer to the one that has higher aspect ratio.
Also, since the stab operates in disturbed air, the NP will be more forward than
the simple formula predicts.

The figure below shows a somewhat more complex formula to locate the NP but
would give a more accurate result using the so called Tail Volume Ratio, Vbar.
This formula gives the NP position as a percentage (%) of the wing's MAC aft of
the wing's AC point.



For those who are not so keen on formulas and calculations there is the
Aircraft Center of Gravity Calculator, which automatically calculates the CG
location as well as other usuful parameters based on the formula above.

For Canards check the link below:
Canard Center of Gravity Calculator

 
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