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LATERAL FREQUENCY RESPONSE
The computation of the lateral frequency response is carried out with (5.14,3). The column vector on the r. h.s. is conveniently expressed as
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(10.5,1) |
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where |
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Denoting the 4×4 matrix on the l. h.s. of (5.14,3) by F, we get the 4×2 transfer-function matrix r a. G = P-1Q (10.5,2) The elements of G are specifically |
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In addition, from the supplementary relations given in (5.14,3) the transfer functions for ip and yE are
NUMERICAL EXAMPLE
The frequency-response functions for the jet transport in horizontal flight at 30,000 ft altitude and CLe = .25 were calculated from the above equations. All the aerodynamic transfer functions were replaced by the corresponding derivatives, i. e. Gyfi = 0 , GlSa = O^etc. Thus we have neglected terms such as sGtg. The numerical values are the same ones used in the previous examples. The results for some of the state variables are shown in Figs. 10.11 and 10.12. Figure 10.11 shows the responses in /3, <f>, and r to rudder input. The principal feature is the peak at the frequency of the Dutch roll, which because of the relatively light damping of this mode, is substantial. For example, a 1° rudder amplitude produces about 4|° /3 amplitude and 6|° roll amplitude. At zero frequency /3, p, and r are finite, but <f> and ip are infinite. That is, the computed steady state associated with rudder input is a constant rotational motion wss = ipss – f – krss. Since the equations were linearized with respect to ф and are therefore not valid for large ф, this steady state is spurious. The slopes of the high-frequency asymptotes can he predicted from the structure of the general transfer-function matrix. For the given rudder input it yields slopes of —1 for /3, r, and —2 for ф. These slopes are reached approximately by a> = .1 for r and ф, but not for /3. This is because the coefficient of the cubic term in the numerator of GpS contains the small aerodynamic derivative Gyt.
Figure 10.12 shows similar results for aileron angle input. The absence of the control term Cy makes the high-frequency asymptote of GpSJ a line of slope —2 instead of —1.
All the amplitude curves on both figures show a rapid reduction of response once the frequency exceeds that of the lateral oscillation mode.
The sharp dip in |6>5jJ at со = .0025 is characteristic of a zero in the transfer function lying close to the imaginary axis at this frequency.
