Approximate Lateral Transfer Functions
Approximate transfer functions that can be written out explicitly, and that reveal the main aerodynamic influences in a particular frequency range, can be very useful in designing control systems. In Sec. 6.8 we presented two approximate secondorder systems that simulate the complete fourthorder system insofar as the characteristic modes are concerned. These same approximations can be used to get approximate transfer functions for control response.
SPIRAL/ROLL APPROXIMATION
When aerodynamic control terms are added to (6.8,9) and the Laplace transform is taken, the result is
In (7.10,1) the <31 and Я derivatives are as defined in Sec. 6.8, and the 5£s derivatives are
(7.10,2)
where 8 is either <5„ or 8r. From (7.10,1) we get the desired transfer functions. The denominators are all the same, obtained from (6.8,11) as
Cs2 + Ds + E
and the numerators are (again using 8 for either 8a or Sr):
NvS = a3s3 + a2s2 + axs + a0 Мф8 = bys + b0 NrS = d2s2 + dts + d0 Nps =
7.10 Approximate Lateral Transfer Functions 251
The coefficients in these relations are:
a3 = a2 = —QlgliEp + Яг) — u0Ns
а, = – ад) – u0(%sXp – + Xsg
fl() = g&rX8 – £sNr) (7.10,5)
б, = <tJ820; b0 = u0(XsNv – $VXS) + ~ ад)
d2 = <*)SXV – dx = WuJfP – %РЮ d0 = g(iesXv – $VNS)
DUTCH ROLL APPROXIMATION
Following the analysis of Sec. 6.8 and adding control terms to the aerodynamics the reduced system equations are
v = qjvv – u0r + Д%
r = Xvu + Nrr + ANC
From (7.10,6) we derive the canonical equation
With the system matrices given by (7.10,7) the approximate transfer functions are found in the form of (7.2,8) (see Exercise 7.7) with
f(s) = s2 – (°DV + Nr)s + ((HuNr + u0Nv) 
(7.10,9) 
Nu8„ = ~UO^Sa 

NrSa = NSts – V5o 

AU = – (^Л + u0NSr) 

NrSr = Nss – в)иЯ8г – ®>SNV) 
(7.10,10) 
The accuracy of the preceding approximations is illustrated for the example jet transport on Figs. 7.26 and 7.27. Two general observations can be made: (1) The Dutch Roll approximation is exact in the limit of high frequency, and (2) the spiral/roll approximation is exact as ю » 0. In this respect the situation is entirely analogous to that of the longitudinal case, with the spiral/roll corresponding to the phugoid, and the Dutch Roll to the shortperiod mode. There are ranges of frequency in the middle where neither approximation is good. We repeat that lateral approximations must be used with caution, and that only the exact equations can be relied on to give accurate results.
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