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      Suppose the airplane is disturbed in bank angle A4 as shown in Fig. 4.17c. We
have
Fy  = Fyo + AFy =  W cos OoA4 + Yaero + AYacro               (4.468)
where Yaem denotes the aerodynamic side force. We have Fyo = Yauo = 0 so that
AFy = W cos 0o A4 + AYauo
aAFy     a Fy
 a A~y ~  ajy = W cos0o
Similarly, it can be shown that
Cy4 = CL COS Oo
Cyp = (Cyp)aero
Cyp = (CyB)aero
Cyp = (Cyp)aero
Cyr = (Cyr)aero
(4.469)
(4.470)
(4.471)
(4.472)
(4.473)
(4.474)
(4.475)
and so on. Here, it is assumed that the side force Yauo has no dependence on the
bank angle ~.
    It is important to bear in mind that Eqs. (4.417-4.419) and (4.459-4.461) are
applicable for small disturbance motion when the airplane is disturbed from a
steady, unaccelerated flight with zero sideslip. If this is not the case, then the
assumed initial conditions in Eqs. (4.370-4,373) will be different, and we have to
derive different sets of equations of motion. This concept will be illustrated with
the help of an example later in this section.
                                           Example 4.7
     A high-performance aircraft executes a velocity vector roll at a rate of 90 deg/s
while flying at 100 m/s and an angle of attack of30 deg. Determine the acceleration
of the aircraft with respect to an Ezulth-fixed inertial system.
    So/ution.   We assume that the x axis of the Earth-fixed system points in the
direction of the velocity vector, y axis to the right and z axis vertically downward,
EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 385
 to form a right-hand system. Then
                     p = s2 cos cr     q -.0     r - S2 smu
                    U = Uocos af     V -.O     W - Uosina
                       U-0'    V -.0    W -.0
The body axes components of the acceleration with respect to the Earth-fixed
system are
                                ax =U +qW -r V
                              ay -V +rU - pW
                          az=W+pV -qU
    Substituting, we get
                                            ax - O
                               ay = S2 sin aUo cos a - 92 cos aUo sina - 0
                                   az -O
     Thus, the aircraft does not experience any acceleration with respect to an Earth-
fixed axes sytem.
Example 4.8
   hSo/ution.   At t - 0, the two coordinate systems coincide. Let (2 -. koS21 :
kol = kil. Furthermore,
                    ~o= [~]
We have
t31,1 = [.:,]
                                                        cos g2it      sin {2it
　
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