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6.3.3 Lataral-Directional Transfer Functions
Equations (6.168~6.170) for lateral-directional motion based on the assumption
of small disturbance can be expressed as
d
(m, dd _ bi CyBdt - Cyp) Ap - (bi Cyp It + Cy*) A4
+ (mi -- bi Cyr)d~tt-r = Cy8" A8a + Cy8r A8r (6.229)
578 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
cj,
m
1,
o:
t7,
(D
o
u
_ Dutch Rdl Approramation
-_ Complek Fiitth-Order System
Fig. 6.19 Free response (lateral-directional) of the general aviaLion airplane.
(-C,p - bi CtB~) Ap + (-C,pb~t + Ixi dd ) A*
-(C,,b,~+IXZ,~)Ayr=C1&A8a+CiSrA8r (6.230)
(-C p - b,C,,~t) Ap + (_C,,pbidd _ Ixz, dd ) A#
+(-C,,,b,gt+IZ,~,)A/t=Cn31A8o+CnrA8r (6.231)
In the following, we will derive the transfer function for aileron detlection assuming
that the rudder is held fixed, i.e., A8r - 0. Using these transfer functions, the
airplane response to a given aileron input can be obtained. To obtain the airplane
response to rudder inputs, we need the transfer functions for rudder deflection,
which can be obtained by replacing A8a by A8r, Cy3r by Cy8r, Cl8a by CW, and
~
a)
-a
m
a
C
3
d
>-
AIRPLANE RESPONSE AND CLOSED-LOOP CONTROL 579
_ SpiraIApproximation
- Complete Fifth-Ordcr System
Fig. 6.20 Free response (Iateral-directional) of the general aviation airplane.
Cn8a by Cn&- The airplane response to combined aileron and rudder deflections
can be obtained by a linear addition of the two responses because the system is
assumed to be iinear.
Taking the Laplace transform of Eqs, (6.229-6.231), we obtain
(mis - bi C:.ps - C:.p)AB(s) - (bi CypS + C),4)A~(s)
+ (mi - bi C,-r)SA~(s) = C).8a A8a(s) (6.232)
(-Cip - bi C,Bs)Ap(s) + (-ClpblS + Ix s2)A~(s)
- (Clrbl S + Ixzis2)Ap(s) = C18a A8a(s) (6.2:33)
(-Cnp - bi C,Bs)Ap(s) + (-Cnpbi S - /xzis2)A4(s)
+(-CnrbiS + Izls2)AV/(s) = Cn&, A8a (S) (6.2:34)
580 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
1-5
0.5
Dutch Roll Approximation
-- Complete Fitth-Order System
┏━┳━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┓
┃ ┃ ,..II.II ┃
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