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!