Stability and Control — Page 286, Lesson 347

Let’s start with the big idea. When we talk about longitudinal dynamic stability, we are no longer just asking whether the aeroplane returns to its original attitude. We are asking how it returns, and over what time. That is what the phrase "time history response" means. It is the record of how the displacement amplitude — how far the aeroplane has been pushed away from its trimmed state — varies with time after the disturbance. So let me define the two outcomes clearly, because they are the foundation. Dynamic stability exists when the amplitude of the motion decreases with time. The aeroplane oscillates, but each swing is smaller than the last. Dynamic instability exists when the amplitude increases with time — each swing grows larger, which is obviously unacceptable. Now, an aeroplane must demonstrate positive dynamic stability for the major longitudinal motions. That means it must actually damp the motion, not just fail to grow it. But there is a quantitative requirement here, and this is where the professional standard comes in. The aeroplane must reduce the amplitude of motion at a certain rate. That rate is specified by the time necessary for the amplitude to reduce to one-half of its original value. We call this the time to damp to half-amplitude. So if the aeroplane is disturbed and its oscillation amplitude halves in, say, a certain number of seconds, that time is the measure of its dynamic stability. Shorter time to damp to half-amplitude means stronger damping. Now, why do we limit ourselves to longitudinal motion? Because the aeroplane in free flight has six degrees of freedom. Let me unpack that. There are three rotations — roll, pitch, and yaw — and three translations — horizontal, vertical, and lateral. That is six independent ways the aeroplane can move. For longitudinal dynamic stability, we can limit the degrees of freedom to pitch rotation, plus vertical and horizontal translation. So we are ignoring roll, yaw, and lateral translation for this analysis. And there is a good reason we can do that. The aeroplane is usually symmetrical from left to right. Because of that symmetry, there is no need to consider coupling between longitudinal and lateral or directional motions. In plain terms, a pitch disturbance does not excite a roll or yaw response in a symmetric aeroplane, so we can study the longitudinal motion on its own. That leaves us with the principal variables in longitudinal motion. There are three. First, the pitch attitude of the aeroplane — that is the angle of the fuselage relative to the horizon. Second, the angle of attack, which will differ from the pitch attitude by the inclination of the flight path. So if the flight path is climbing or descending, the angle of attack and the pitch attitude are not the same thing — the difference between them is the flight path angle. Third, true airspeed, which we abbreviate TAS. Now, the longitudinal dynamic stability of an aeroplane generally consists of two basic modes of oscillation. The first is the long period oscillation, which we call the phugoid. The second is the short period motion. The longitudinal motion of the aeroplane may consist of a combination of these two modes, but their characteristics are sufficiently distinct that each oscillatory tendency may be studied separately. That is the key point — we can separate them and analyse each one on its own. So let me summarise what we have. Dynamic stability is about amplitude decreasing with time, measured by the time to damp to half-amplitude. We restrict the analysis to pitch rotation plus vertical and horizontal translation, justified by left-right symmetry. The variables are pitch attitude, angle of attack, and true airspeed. And the motion splits into two distinct modes — the phugoid and the short period motion. That sets us up perfectly to look at each mode in detail next.