ELIMINATION OF STEADY-STATE RUDDER ANGLE

The solution presented above contains a feature which could possibly he undesirable—i. e. there is a steady-state rudder angle associated with constant yaw rate r. This means that the autopilot would generate a rudder deflection during steady turns, with 8r > 0 for right turns and vice versa. This is opposite to the rudder deflection wanted in the turn (see Sec. 10.4), and hence we have the autopilot opposing the human pilot. If this situation occurred with any frequency, the pilot rating of the aircraft would be ad­versely affected. On the other hand, CWe = 4.0 represents a very low speed, presumably associated only with landing and take-off, and not ordinarily with turning flight. Thus it would depend on factors somewhat outside the scope of this example whether this steady-state behavior of the autopilot presented a problem or not.

In cruising flight this problem would be more serious, and it would be desired to eliminate it. We illustrate here how it could be done.

The steady-state response of the rudder system can be eliminated by incorporating what amounts to a high-pass filter with zero static gainf in the rudder loop, as shown in Fig. 11.8. The feedback element if23r-s/(l + ts) f A “washout” circuit.

Fig. 11.8 Stability augmentation system for STOL airplane.

has zero static gain (see Sec. 3.2), so that 6r is zero when r = const. The frequency response of this element is

(11.4,8)

so that for сот —> oo, G(im) —»■ K23. Thus by proper choice of r, the filter can be made to behave like a simple gain of K23 above a chosen frequency cov To analyze the system with the filter incorporated, we could find the overall transfer function of the closed-loop system and calculate the roots of the characteristic equation, or alternatively we can modify (11.4,7) to correspond to Fig. 11.8. The latter procedure is by far the simpler in the present instance. The only respect in which (11.4,7) does not apply is in the last of the equations, which now must correspond to

(11.4,9)

[12 + (1 + 12t).s + r. s2] ДST = K2Srsf

The corresponding differential equation is

12 Д<5Г + (1 + 12т) 6r + т8г = K23rf

After conversion to nondimensional form, this becomes

On defining a new variable £, we can replace this second-order equation by a pair of first-order ones, i. e.

The last of (11.4,7) has now to be replaced by the pair (11.4,12). In doing so we eliminate Dr from (11.4,12) by using the third equation of (11.4,7). The result is shown as (11.4,13).

Computations made with (11.4,13) show that the effect of the autopilot in correcting the spiral instability is very much reduced by the filter unless r is very large (Pig. 11.9), in which case the effectiveness of the washout circuit is impaired. As has been pointed out previously, however, a slow divergence of the spiral mode is not unacceptable, so a compromise solution is possible without excessive values of r. Por example, with К1г = 15, К23 = 20 and г = 10 sec the modal characteristics are

Spiral: fdouble = 18.1 sec Oscillation: T — 11.4 sec, NX/i = .56 cycles

0.020

Pig. 11.9 Effect of washout circuit on lateral roots. Ku = 15, K23 — 20.