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where
Na.spo = AO.sp:os + Ba.spo
Aa.spo = C" be(ml - CzLi,Cl) + Cmdrq Cz8e
         BO.spo = Cma,Cz8e - Cza Cm8e
(6.143)
(6.144)
(6.145)
We will be making use of these transfer functions in the design of flight control
systems to be disc:ssed later in this chapter.
     Transfer function for a phugoid approximation.    rfaking the Laplace trans-
form of Eqs. (6.58) and (6.59) and rearranging; we get
                         (mis - Cx")ri(s) - (Cx0 + Cxqcis)AO(s) = Cx& A8e(S)         (6.146)
                   -Cz"u(s) - LCz0 + s(m] + Czqci)]AO(s) = C28e A8e(S)       (6.147)
Dividing throughout by A8e(S) and using Cramer's rule, the transfer function for
forward velocity can be obtained as
                          C.rB,          -(Cr0 + Cxq CIS)
LI(S)   Cz8r -[CzO+S(ITII+CzqCI)]   (6.148)
A8e~s~=ms-Cx"  -(C., Cxq") )]l
                     -Cz"       -[Cz0 + s(mi + Czq:
Let Nu.ipo and Aipo denote the determinants in the numerator and denominator of
Eq. (6.148). Expanding these determinants, we get
and
            Nr,.ipo = Au.iws + Br,.ipo
Au.ipo = -[CxBt(ml + Czq Cl) - CZBe C):q CI]
           Bi.lpo = Cz8e Cx0 - Cx&Cz0
                                 Alpo = A8.lpoS2 + B8JpoS + C8.lpo
                                      A8.lpo = -ITI,1("11 + CzqCi)
                              B8.lpo = -lFZ I Cz0 + Cxu(tTI l  + Czq CI ) - Czu Cxq Cl
                                       C/S.lpo = C,  Cz0 - Czu Cx0
The transfer function for pitch angle is given by
O(s)
A8e~S~ =
Im'_Cz Cx"
Aspo
(6.149)
(6.150)
(6.151)
(6.152)
(6.153)
(6.154)
(6.155)
 (6.15 6)
562           PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
Expanding the determinant in the numerator of Eq. (6.156), we obtain
 No.bo = Ao.iws + BO.ipo
       Aa.lpo = mi Czb.
B8.h)o = Cx&Cw - C^,"Cz&
(6.157)
(6.158)
(6.159)
  For the general aviation airplane, substitution of various parameters in Eqs.
(6.109), (6.123), and (6.117) gives the following longitudinal transfer functions
for the complete (fourth-order) system:
  ,i(s)      .                    -0.~22S2 + 0.(~~ _] s + 1.5997
        _-
  A8e =  O 3-. l-39 S4+~9~02 S3+4 9935 S2+0~6 -2 s+0 2296
          (6.160)
AO(s)                             -4.4367 S2 - 8.8084 s - 0.4507
   A~  =  O 3~39 S4+~9002 s~+4 9935 s -+0~6~2 s+0 2296
            (6.161)
Aa(s)                 -0.0602 S3  - 4.4954 S2 - 0.2037 s - 0.3103
  A8~ = O 3-. /39S4+~ 90. 2S3+4 9935s2+0~642s+0 2296
           (6.162)
Aa (s)                   -0.0273 S2 - 2.0366 s
　
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