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  \ ~---    &,                     
┏━━━━┳━━━━━━━━━━━┓
┃        ┃                      ┃
┣━━━━╋━━━━━━━━━━━┫
┃        ┃                      ┃
┣━━━━╋━━━━━━━━━━━┫
┃jr .    ┃r\l,        Horizonta ┃
┗━━━━┻━━━━━━━━━━━┛
     zo
b) Airplane disturbed ui angle of attack
                                                                                             \-~ A +
c) Airplane disturbed in bank angle
Fig.4.17    Airplanein steady and disUnrbed flight condiOons.
EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 381
axis or the zero-lift line, whereas 0o is the angle between the local horizontal and
the velocit)t vector or the flight path.
    Now let us evaluate various derrvatives like Cxu, Czu, etc., appearing in Eqs.
(4.417-4.419).
     The forces acting on the airplane are lift L, drag D, thrust T, and weight W as
shown. For steady equilibrium flight,
                              Fxo = T ~ Do - W sin0o = o                    (4.423)
 and, for disturbed flight,
                         Fx - Fxo + AF.r                                            (4.424)
                                       - T - D - W sin(0o + AO)                                  (4.425)
                                       = T - (Do + A/:5) - W sin(0o + AO)                   (4.426)
so that
                                                AFx = -AD - W cos OoAO                               (4.427)
For simplicity, we ignore thrust variations during the disturbance. Suppose the air-
craftis disturbed in forward speed. We have U  - Uo + AU, so that au ='aAU.
Then,
aAF,   aFi    aAD   aD
aAU - aAU = -aAU = -au              (4.428)
_ a(;:pU2scD)
   au
= -pUSCD - gpU2Saacju
(4.429)
(4.430)
With U ~ Uo, we have
                            a Fx                                           D
　
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