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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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动力机械和机身手册3(43)
