Step-Function Response

The response of the airplane to a sudden movement of the elevator is shown by the step response. This requires a solution in the time domain as distinct from the pre­ceding solution, which was in the frequency domain. Time domain solutions are com­monly obtained simply by integrating (7.1,4) by a Runge-Kutta, Euler, or other inte­gration scheme, the choice being dependent on the order of the system, accuracy required, computer available, and so on. The software used[18] for the example to follow does not integrate the equations, but instead uses an alternative method. It inverts the transfer function using the Heavyside expansion theorem (A.2,10). For the same jet airplane and flight condition as in the preceding example, the control vector for ele­vator input is с = [Д5е 0]T and A and В are as before. (Note that only the first col­umn of В is needed.) Time traces of speed, angle of attack, and flight path angle are shown in Figs. 7.19 and 7.20 for two time ranges when the elevator displacement is one degree positive, that is, down.

It is seen from Fig. 7.19, which shows the response during the first 10 sec, that only the angle of attack responds quickly to the elevator motion, and that its variation is dominated by the rapid, well-damped short-period mode. By contrast, the trajec­tory variables, speed, and flight path angle, respond much more slowly. Figure 7.20, which displays a 10 min time span, shows that the dynamic response persists for a very long time, and that after the first few seconds it is primarily the phugoid mode that is evident.

The steady state that is approached so slowly has a slightly higher speed and a slightly smaller angle of attack than the original flight condition—both changes that would be expected from a down movement of the elevator. The flight path angle is seen to be almost unchanged—it increased by about one-tenth of a degree. The reason for an increase instead of the decrease that would be expected in normal cruising flight is that at this flight condition the airplane is flying below its minimum-drag speed.

If the reason for moving the elevator is to establish a new steady-state flight con­dition, then this control action can hardly be viewed as successful. The long lightly damped oscillation has seriously interfered with it. Clearly, longitudinal control, whether by a human or an automatic pilot, demands a more sophisticated control ac­tivity than simply moving it to its new position. We return to this topic in Chap. 8.

Phugoid Approximation

We can get an approximation to the transfer functions by using the phugoid ap­proximation of Sec. 6.3. The differential equation is (6.3,6) with control terms added, that is,

л8е m Z* m MSe 0

frequency responses calculated with the above approximate transfer func­tions are shown on Figs. 7.15 to 7.18. It is seen that the phugoid approximation is ex­act at very low frequencies and the short-period approximation is exact in the high – frequency limit. For frequencies between those of the phugoid and short-period modes, one approximation or the other can give reasonable results.