Longitudinal dynamic stability – pitching oscillations

Let us look first in greater detail at the motion we considered above in which we disturb the aircraft in pitch and then release it (Fig. 12.2). If the aircraft is

statically stable then the resulting pitching moment will be nose down, tending to return the aircraft to its original attitude. This restoring moment is very nearly directly proportional to the disturbance for a conventional aircraft operating at moderate Mach number. The way in which it is produced by the tailplane was described in Chapter 11. The resulting motion for a typical air­craft is shown in Fig. 12.2 and consists of a heavily damped oscillation in pitch, accompanied by very little change in height or speed. This motion has come to be called the ‘Short Period Pitching Oscillation’, or SPPO.

If the motion was solely caused by the restoring moment due to increased tail angle of attack, then the oscillation would continue with the same ampli­tude. It would then be said to be neutrally stable dynamically or ‘undamped’. During the motion, however, another effect is caused by the tailplane which is not apparent when we simply consider the ‘static’ forces due to the change in attitude.

Consider the instant in the motion when the aircraft is pitching, nose up, through its original attitude (Fig. 12.3). This pitching motion increases the angle of attack on the tail and hence produces a moment which opposes the nose-up pitching. Note that this effect depends on the rate of change of the attitude of the aircraft, or its angular speed. This speed is greatest at the time when the aircraft passes through the equilibrium position. It opposes the over­shoot (Fig. 12.3), thus tending to damp out the oscillatory motion. Because the oscillations eventually disappear the motion is dynamically as well as statically stable.

A further damping effect is provided because the angle of attack of the air­craft is increasing with time. The increase in the strength of the wing trailing vortex system, caused by the angle of attack increase, takes some time before it

Fig. 12.3 Damping of pitching oscillation Angular velocity causes a lift force on tail which opposes the rotation in pitch and damps oscillation

makes itself felt at the tail. The tail lift is, therefore, a little greater than it would be if the angle of attack were held steady, and this again contributes to the damping effect.

The combined effect of these damping terms is usually very pronounced, and the motion is heavily damped, usually not lasting more than one or two cycles in a typical conventional aircraft configuration.

In the above paragraphs a very simplified view has been taken of the SPPO, since the emphasis has been on the major factors influencing the motion. In reality, as the angle of attack changes during the pitching motion of the aircraft, the lift will change, also in an oscillatory fashion. Thus the pitching motion will be combined with a vertical motion. A more detailed analysis of the motion shows that this has a slight influence on frequency but significantly increases damping.

Another, even more subtle, factor which we have ignored is the effect that the pitching motion has on the wing itself. This is explained by Fig. 12.4. As the wing rotates, the relative motion through the air produces a downwash over the front of the section and an upwash over the rear. This has the effect of changing the moment produced by the wing section, and this again will add slightly to the damping of the motion. If the wing is swept, this effect will be more pronounced because the distance between the root and tip sections will mean that an upwash will be produced at the tip and a downwash at the root.

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