AN EXAMPLE

We shall calculate the longitudinal response of the jet transport used in previous examples, cruising at 30,000 ft through turbulence of 5000 ft scale. In this situation the point approximation is valid, and (13.3,7 and 11) give the needed input function. For the aerodynamic derivatives we use the same numerical values as in Sec. 9.1. The system equations are used in Laplace transfrom form, i. e. (5.14,2) with cj, T and ye both zero.

We noted in Sec. 11.2 that the phugoid oscillation could be suppressed by the pilot by a simple feedback of pitch-attitude to the elevator deflection. We provide for this in the following equations by including the control equation Д<5в = —Кв. On combining the equations we get:

A(s)y(«) = B(s)

where (йд, wg) are Laplace transforms of the nondimensional gust velocities (ug, wg), and

(Cjy – 0®F – 2/»)		0	-CWe	0 “
-(CLy+ 2 CWJ	-ft, + <h>, + 2ys)	2	0
~Cmy	— ^ma s^mot)	^mQ ^VS^	0
0	0	1	—s	0
. 0	0	0	к	1 .

у = [AV Да q Ав Ade]T

(13.5,3)

(13.5,4)

The required frequency-response functions are found by substituting s = ilc( — ifljFgt*) and solving the resulting complex algebraic equation for the ratios Уі(ік)Ійд and «/г(г&)/й>9.

A response variable of interest not directly included in the above is the load factor. It is defined by An = ALjW. The lift increment AL is taken

as the sum of two parts, that due to aircraft motion (Да, AF) and that due to atmospheric motion (ug, wg). The result obtained is

(«)

(13.5,5)

(b)

Some of the more interesting transfer functions and output spectra are plotted in Figs. 13.10 and 13.11. In Fig. 13.10 we show the squares of the moduli of the transfer functions for speed, angle of attack, pitch attitude, and load factor for vertical gust input. Both stick-fixed and controlled motion are shown. All the motion responses fall off rapidly at high wave number (or high frequency), hut the load factor response tends to the constant value associated with flight on a rectilinear path at constant speed (i. e. no motion response). At wave numbers above 10~2 the load factors are pro­gressively more approximate because of the neglect of unsteady aerodyna­mics. Much more accurate values could he obtained by the simple expedient of multiplying these by a reduction factor for finite wings in sinusoidal gusts—obtained from the generalized Sears function as given by Filotas (ref. 7.16). [The appropriate factor is actually Filotas’ S(kv A)2.]

The effect of the simple elevator-control law (the simple gain is not, of course, the optimum control law for reducing gust response) is seen, as expected, to eliminate the phugoid peaks and substantially to reduce the pitch response for all frequencies lower than that of the short-period mode. With respect to a response, the airplane is seen to act like a low-pass filter, with cut-off frequency at the short-period mode.

On combining these transfer functions with the input spectrum, we get the output spectra, e. g. for speed response

etc. These are shown on Fig. 13.11. (Note that these are two-sided spectra— twice the area gives the mean square.) It is seen that the point approximation is quite adequate in this example for giving the responses in the motion variables (AF, Да, в) but is less satisfactory for the load factor, for which a substantial fraction of the mean-square value is contributed by frequencies above the limit of validity of this approximation. The use of a corrected transfer function as noted above would improve the accuracy of this result appreciably.

If the gust input vector were extended to include the gust-gradient term

(b)

Fig. 13.10 Transfer functions for response to wg input. Jet transport cruising at 30,000 ft and 500 mph with pitch feedback.

(b)

Fie. 13.11 Power spectra of response to wg input. Jet transport cruising at 30,000 ft and 500 mph with pitch feedback.

dwjdx = qg, then the right-hand side of (13.5,1) would read

where qg is the Laplace transform of qa = qgt*, and B'(.s) would he like B(s) but with the additional column [0 — CL — CM 0 0]T. In this case the general response theorem (3.4,49) would hare to be used to calculate outputs, since the cross spectra of both ug and wg with qg are not zero (ref. 13.10). This would entail the calculation not only of the moduli of the transfer functions, but of their real and imaginary parts. [An alternative but equivalent method for this case was given in Dynamics of Flight—Stability and Control (Sec. 10.6), that does not use the input cross spectra.]