The longitudinal static stability of conventional aircraft

The movement of the centre of lift on a cambered wing has a destabilising effect. Figure 11.4 shows a cambered-section wing that is balanced or trimmed

Constant

moment

Aerodynamic centre

Fig. 11.2 Centre of pressure and aerodynamic centre

(a) The centre of lift or centre of pressure moves forward as the wing angle of attack increases.

(b) For an aerofoil, the situation shown in (a) above can be represented by a constant moment or couple (shown here in its true directional sense) and a lift force acting through a fixed point known as the aerodynamic centre.

N. B. The sign convention is that nose-up pitching is positive. A cambered aerofoil as shown above therefore gives a mathematically negative moment about the aerodynamic centre

at one angle of attack, by arranging the line of action of the lift to pass through the centre of gravity. If the angle of attack increases due to some upset, then the lift force will move forward, ahead of the centre of gravity, as shown. This will tend to make the front of the wing pitch upwards. The more it does so, the greater will be the upsetting moment. Such a wing is, therefore, not inherently stable on its own.

The condition for longitudinal static stability is that a positive (nose-up) change of angle of attack should produce a negative (nose-down) change in pitching moment.

where M is the pitching moment, and a is the pitch angle.

The conventional method for making an aircraft longitudinally stable, is to introduce a secondary surface, which is called a tailplane in the British con­vention, or more aptly, a horizontal stabiliser in American terminology.

To give some idea of how the tailplane works, we will consider two simple cases, one very stable, and the other highly unstable. Fig. 11.5(a) shows an aircraft trimmed for steady level flight. For simplicity we consider a case where the thrust and drag forces both pass through the centre of gravity, and thus produce no moment. We also ignore the forces and moments on the fuselage. The tail is initially producing a downward force, and hence, a nose-up pitching moment about the centre of gravity, whereas the wing lift produces a nose – down moment. The couple Mo is drawn nose up in Fig. 11.5, which is the normal mathematical convention.

If the aircraft is tipped nose-up by some disturbance as in Fig. 11.5(b), then the tail downforce and its moment will decrease, and the wing lift and its moment will increase. The moments are, therefore, no longer in balance, and

Fig. 11.5 Longitudinal static stability – a highly stable case In this simplified example, the thrust and drag forces pass through the centre of gravity, and effects due to the fuselage are ignored. (a) Aircraft trimmed in pitch LW x a – Mo = Lt x b (b) If the angle of attack increases due to some upset, the tail-down force will decrease, and wing lift will increase. This will result in a nose-down pitching moment, tending to restore the aircraft to its original attitude. N. B. Mo is shown here in the mathematically positive (rather than true) sense

there is a net nose-down moment, which will try to restore the aircraft to its original attitude. This aircraft is thus longitudinally statically stable.

Note, that in the simplified description above, we have ignored the inertia of the aircraft, and we have neglected the flexibility of the aircraft and its controls. Most importantly, we have ignored the effect of the fuselage which normally has a significant destabilising effect. It should further be noted that our simple example does not include any effects due to wing sweep. We should also have taken account of the fact that as the wing angle of attack and lift increases, so will the downwash at the tail. The increased downwash at the tail means that the tail downforce does not fall off as sharply with changing angle of attack as would otherwise have been expected. The restoring force is thus weakened.

Fig. 11.6 Longitudinally unstable arrangement – negative longitudinal dihedral necessitated by having centre of gravity much too far aft Although the aircraft was initially trimmed, any increase in angle of attack will produce an unstable nose-up pitching moment. (a) Aircraft trimmed with wing at 2°. Angle of attack and tail at 4°. (b) Aircraft attitude increased by 2°. Wing now at 4° and tail at 6°. The wing lift will double, but the tail lift will only increase by 50%. The aircraft becomes untrimmed, with a nose-up pitching moment, and it will diverge from its initial attitude. N. B. Just having the centre of gravity behind the wing aerodynamic centre does not make the aircraft unstable; it depends on how far aft the CG is

Wing downwash on the tail generally has a destabilising effect. The influence can be reduced by mounting the tail high relative to the wing.

Figure 11.6 shows a case of an aircraft which is trimmed, yet in a longitudin­ally unstable condition. The wing is initially at 2° angle of attack, and the tail is at 4° angle of attack. If we look at what happens when the angle of attack increases by 2°, due to a disturbance, then we see that the wing angle of attack will double. Since lift is directly proportional to angle of attack, it follows that the wing lift will double. In contrast, increasing the tail angle of attack by 2° from 4° to 6° represents only a 50% increase, with a corresponding 50% increase in tail lift (which is further reduced by the effects of downwash). The resulting forces will, therefore, produce a nose-up pitching moment, and the aircraft will continue to diverge from its original attitude.