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Fig. 6.9 Free (longitudinal) response of statically unstable 'version of the general
aviation airplane.
AIRPLANE RESPONSE AND CLOSED-LOOP CONTROL 557
Transfer functions of a comp/ete system. Assunung that an ele'vator is the
only longitudinal control and taking the Laplace transform of Eqs. (6.1-6.3), we
obtain
(ITIIS - Cxu)Le:(S) - (Cxa + Cxdcls)Acr(s) - (Cxq CIS + CxO)AO(s) = Cne A8e(S)
(6.106)
-Czu~i(s) + (mis ~ Czd CIS - Cza)Aa(s) - s(mi + Czqq)AO(s) = Cz8e A8e(S)
(6.107)
-Cm iu(s) - (Cmd CIS + Cma,)Ad(s) + s(lyi S - Cmq Cl )AO(s) = CmBe A8e(S)
(6.108)
Here, s is the Laplace variable, and a bar over the symbol denotes its Laplace
transform. DMding throughout by A8e(s) and using Cramer's rule, we obLain
li(s)
A8 =
C18e -(Cxa + CTdcls) -(Cxq CIS + Cx0)
Cz8t (mis - CzrqS - Cza,) -S(lFII + CzqCl)
-Cz"
-Cmu
(mis - Cz&cis - Czu) -s(mi + CzqCl)
-(CmuCIS + Cmr) s(l),is - CmqC1)
(6.109)
Let Nu denote the determinant in the numerator and Along(S) denote the determi-
nant in the denominator of Eq. (6.109). Then, expanding the determinant in the
numerator, we obtain
Nr = Ar,s3 + Br,s2 + C"s + Dri . (6.110)
where
Ai = I>.i (C.Bt Ull - CXbe Cztic] + Cxd Cl Cz8,)
of the damping effect provided by the third oscillatory mode but subsequently
starts building up due to the real positive root A4. Along with this, pitch rate
also builds up as shown. For such airplanes, it is desirable to have an automatic
stability augmentation system that can take care of this problem so that the air-
plane has the conventional short-period and phugoid modes in the closed-loop
sense. We will discuss the design of such a pitch augmentation system later in this
chapter.
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558 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
Bu = - Cx8,(Czcr I).l + UI1CmqCI - Czd Cmq C2j + rril Cma CI + Czq Cmcr C;)
_ Cxa C21 Cmq + C26e I).l CXO. + C d.Cl Cmbe(,711 + Czq CI)
+ Cxq C21C28e Cmci + Cxq CI CmSe (UIl - Cz&ci)
q7 = Cx8e(CZcr Cm:q CI - mi - Czq CI) - Cmq Cz&C;rcr Cl + Cxa C,U8e(ml + Czq CI)
+ Cxq CI C26e Cma, - Cxq Cl Cm3e Czcr + Cx8Cz& Cmdt CI
+ c\ocr,iieGrZI - CzdCI)
Du = CxO(Czbe Cma - Cm8,Cza)
Furthermore, expanding the determinant in the denominator, we obtain the longi-
tudinal characteristic equation
Aiong(S) = Abs4 + B8S3 + C8S2 + D8s + E8 (6.111)
where the coefficients Ab, B8, Cal, D8, and E8 are identical to those given in Eqs.
(6.21-6.25) and are reproduced here in the following:
A8 = FtZII).l(ITIl ~ CzaCI) (6.112)
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