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DR Mode
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Aileron Control Input |
Rudder Control Input |
|
|
Side velocity |
—5.107e-015 sA3 + 8.798 sA2 – 67.23 s — 13.56 |
13.48 sA3 + 424.4 sA2 + 5 21. 5 s — 7.05 2 |
|
sA4 + 1.589 sA3 + 1.78 sA2 + 1.915 s + 0.01238 |
sA4 + 1.589 sA3 + 1.78 sA2 + 1.915 s + 0.01238 |
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|
Roll rate |
— 1.62 sA3 — 0.5858 sA2 — 2.201 s — 3.123 e-0 1 7 |
0.392 sA3 — 0.2813 sA2 — 1.865 s — 1 .0 0 1 e-0 1 6 |
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sA4 + 1.589 sA3 + 1.78 sA2 + 1.915 s + 0.01238 |
sA4 + 1.589 sA3 + 1.78 sA2 + 1.915 s + 0.01238 |
|
|
Yaw rate |
—0.0188 sA3 + 0.03101 sA2 + 0.003289 s — 0 . 147 6 |
—0.864 sA3 — 1.127 sA2 — 0.05891 s — 0 .1 255 |
|
sA4 + 1.589 sA3 + 1.78 sA2 + 1.915 s + 0.01238 |
sA4 + 1.589 sA3 + 1.78 sA2 + 1.915 s + 0.01238 |
|
|
Roll angle |
4.441e-016 sA3 — 1.62 sA2 — 0.5858 _ — 2.20 1 |
8.882e-016 sA3 + 0.392 sA2 — 0.2 8 10 s — 1.8 65 |
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Sa4 + 1.589 sA3 + 1.78 sA2 + 1.915 s + 0.01238 |
sA4 + 1.589 sA3 + 1.78 sA2 + 1.915 s + 0.01238 |
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TABLE 5.2 Lateral Modes Transfer Functions for the Chosen Aircraft |
 |
 |
 |
and assuming no control inputs, the simplified form of a state-space model for the DR oscillatory mode can be expressed as [2]
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TABLE 5.3 Lateral-Directional Characteristics of the DC-8 Aircraft
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Solving for the eigenvalues of the characteristic equation yields the following expressions for the natural frequency and damping ratio for this oscillatory mode:
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YpNr — NpYr + Щщ Frequency Vdr = u0 |
(5.27) |
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(Yp + NrU0 1 Damping ratio Zdr = | U0 2vnDR |
(5.28) |
Example 5.6
The state-space model of the DR mode of a supersonic fighter aircraft is given as
With the derivatives incorporated into this model we obtain
-0.139 2.218/75- 1.125 -0.571
Solution
Following the procedure outlined in previous examples, four TFs are obtained (Table 5.4). The frequency responses are shown in Figure 5.7. The unit step responses for sideslip and yaw rate are shown in Figure 5.8; the lightness of the damping is evident from these responses.
