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      = (Lo + AL)sin Aar _ (Do + AD)cos Aa + T -  W sin Oo
         (4.444)
Similarly,
so that
                    AF\- - Lo Aa - AD
aAFx 8Fx=Lo_aAD
 aA~ = a~ =     - d~ = ~0 -- 8aD
Clu, - CL - CDct
Fzo = -Lo + W cos Oo
(4.445)
(4.446)
(4.447)
(4.448)
Fz - Fzo + AFz
      = -(Lo + AL)cos Aaf - (Do + AD)sin Aa- + W cos 00      (4.449)
      - -Lo - AL - DoAcy + W cos 00                                          (4.450)
    AFz ~ -AL - DoAa
aAFz  aFz   aAL
aA -= aa =-a~L _Do
= - ?dL _ Do
(4.451)
(4.452)
(4.453)
 EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 383
or
It can be shown that
CZCL = -CLa - CD
Cxcr - CLa Acy - CDrr ~ -CDa
Cxq = CLq Act - CDq ~ -CDq
Cza = -CLd, - CDd Aa     -CLa,
                             Czq = -CLq - CDq AcY ~ -CLq
Usually, CDcir and CDq are small and ignored.
(4.454)
(4.455)
(4.456)
(4.457)
(4~458)
     Lateral-directiona/ equations of motion for sma// disturbance.    Substitu-
 ting for ACy, ACi, and ACn from Eqs. (4.412), (4.414), and (4.416) in Eqs.
 (4.394), (4.396), and (4.398), we obtain
  (m, dd  _ b,CyB dd  _ Cyp) Ap - (b,Cyp dd  + Cy*) A4 + (m, dd  _ b,Cyr dd) A k
          = Cy8r A8r + Cyasa A8a                                                                        (4.459)
                      (-C,p - bi CiB dd )AB + (-bi Cipdd + /x, dd   )A4
          +(_b,C,rdd _/z,dd )Ayr=Cl8rA8r+Cl8aA8a  (4.460)
                  (-: C,,p - b,Cnp dd )Ap + (-bi Cnp dd  _ /xz,dd )A4
          +(_biCnrdd+Iz,dd )A =Cn8rA8r+Cn&1A8a  (4.461)
where
d4
dt =P             -    (4.462)
d~
dt =r                (4.463)
bi
Ixl =
(4.464)
(4.465)
   b
= 2U
  Ix
 2~p U2Sb
384            PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
     Iz
Izl = ;-p U2Sb
    Ixz
lxzl = 2pU2Sb
(4.466)
(4.467)
Now let us evaluate various derivatives like Cyp, C),4, etc., appearing in Eqs.
(4.459-4.461).
      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
　
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