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Ba = l711( - Cz8e Cmq CI + Cm8e ITIl + Cm8t Czq Cl)
- Cxu Cz8e /).I + Cx8e Czu l).l
Cct = Cxu (Cz8e Cmq CI - Cm8etTll - Cmbt Czq CI)
- Cxh.Czu Cmq Cl + mi Cx& Cm,, + C.r8r Cmu Czq Cl
+ Cxq Cl Czu Cm8, - Cxq Cl Cmu Cz8t
Da = CxO(Czu Cm8e - Cmu Czb,)
The transfer function for pitch angle is given by
AO(s)
A7(s~ =
mis - Clu -(Cxcr + Cxa,C7S) C18e
-Czu (FirllS - CzaCIS ~ Cza,) Cz8t
-Cmu -(Cmacis + Cma,) Cmb,
Along(S)
Expanding the determinant in the numerator, we obtain
Na = AOS2 + Bos + Co
where
Ao = lT7.1 (tni Cm8, - Cza Cl Cm8e + Cz8e Cmdr CI)
559
(6.118)
(6.119)
(6.120)
(6.121)
(6.122)
(6.123)
(6.124)
(6.125)
Bo = mi (Czat C,na - Czo Cm8e) - Cxu (ITI I Cm8e - Cza Cl Cm8e + Cz8,Cmci,Cl)
+ Cl Cxd ( - Cz Cmbe + Cmu Cz8,) + Cx8e Czu Cma CI
+ CX8e Cm mi - Cne Cmu Czaci (6.126)
Ca = -Cxu (Cz8e Cma, - Cza Cmb,) + Cxli;t(Cmcr Czu - Cmu Cz.cr)
+ Cxa ( - Czu Cm8, + Cmu Cz8,)
We have q -. O, A = AO or q(s) = sO(s), A - sAO(s) so that
q(s) s AO(s)
A8e(s) = A8 (s)
(6.127)
(6.128)
I
560 PERFORMANCE, STABILITY, DYNAMfCS, AND CONTROL
Transfer function for short-period approximation. Taking the Laplace
transform of Eqs. (6.41) and (6.42), we have
(mis - Cza CIS - Cza)Aa(s) - [s(m} + Czqq) + CzO]AO(s) = Cz8,A5e(S)
(6.129)
-(Cmcr + Cm tcis)Aa(s) + (Iy1S2 - Cmqcis)AO(s) = Cm8,A8e(s) (6.130)
Dividing throughout by A8e(S) and using Cramer's rule, we obtain the transfer
function for angle of attack.
CzBe -[S(ITII + CzqCi)+ Cz011
Act(s) . Cm8, s(lyls-CmqCI) "I
A8e = s(ml~zlr) zr Srri+~zqci)+Cz0]1
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