Responses to Elevator and Throttle

RESPONSE TO ELEVATOR

When the only input is the elevator angle ASe the system reduces to

where bfj are the elements of B. Solving for the ratios GuSe(s) = Au/A8e and so on yields the four transfer functions.1 Each is of the form (7.2,8). For this case the char-

‘In the subscripts for the transfer function symbols, the symbol Д is omitted in the interest of sim­plicity.

acteristic polynomial is the left side of (6.2,6) (with s replacing A), and the numerator polynomials are

NuSe = -0.000188s3 – 0.2491s2 + 24.68s +11.16

NwSe = -17.85s3 – 904.0s2 – 6.208s – 3.445

NqSe = – 1.158s3 – 0.3545s2 – 0.003873s (?’7’2)

NeSe = -1.158s2 – 0.3545s – 0.003873

There are two other response quantities of interest, the flight path angle у and the load factor nz. Since в0 = 0, Ав = в, and Ay = Ав — Да, it readily follows that

GySe = Gm – GaSe (7.7,3)

We define nz to be (see also Sec. 3.1)

nz = -Z/W (7.7,4)

It is equal to unity in horizontal steady flight, and its incremental value during the re­sponse to elevator input is

Anz = —AZJW = —(ZuAu + Zww + Zqq + Z^w + ZSeASe)/W

After taking the Laplace transform of the preceding equation and dividing by A8e, we get the transfer function for load factor to be

Gnzse = ~(ZUGU S’ + ZwGwSe + ZqGqSe + sZivG„K + Z Se)/W (7.7,5)

The total gain and phase of the frequency responses calculated by (7.5,6) from five of the above six transfer functions (for Ли, w, q, Ay, and Anz) are shown in Figs. 7.14 to 7.18.

The exact solutions show that the responses in the “trajectory” variables и and у are dominated entirely by the large peak at the low-frequency Phugoid mode. Be­cause of the light damping in this mode, the resonant gains are very large. The peak |G„J of nearly З X 104 means that a speed amplitude of 100 fps would result from an elevator angle amplitude of about 100/(3 X 104) rad, or about 0.2°. Similarly, at reso­nance an amplitude of 10° in у would be produced by an elevator amplitude of about 5°. For both of these variables the response diminishes rapidly with increasing fre­quency, becoming negligibly small above the short-period frequency. The phase an­gle for u, Fig. 7.14b, is zero at low frequency, decreases rapidly to near -180° at the phugoid frequency (very much like the lightly damped second-order systems of Fig. 7.13) and subsequently at the short-period frequency undergoes a second drop char­acteristic of a heavily damped second-order system. The “chain” concept of high-or­der systems (Sec. 7.2) is well exemplified by this graph.

By contrast, the attitude variables w and q show important effects at both low and high frequencies. The complicated behavior of w near the phugoid frequency indi­cates the sort of thing that can happen with high-order systems. It is associated with a pole/zero pair of the transfer function being close to one another. Again, above the short-period frequency, the amplitudes of both w and q fall off rapidly.

The amplitude of the load factor Anz has a very large resonant peak at the phugoid frequency, almost 100/rad. It would not take a very large elevator amplitude at this frequency to cause structural failure of the wing!

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