Roll Transfer Function
The simplest application is probably the roll channel, uncoupled from the other degrees of freedom. From Eq. (7.56) we select the first component equation and scan Fig. 7.1 for the existing roll derivatives
Because we neglect cross-coupling, there remains just
InP — Lpp + LspSp
and the transfer function is
P(s) _ L5p/Iu
Sp(s) s – Lp/1\
The mass properties are commonly lumped into the derivative, and the nomenclature LL is used for L to avoid confusion with the designation for lift. Furthermore, because we emphasize here the aircraft, we replace Sp by the aileron symbol 8a:
P(s) _ LLSa
8a(s) s — LLp
I shall use this transfer function for the roll autopilot design in Sec. 10.2.2.2. LLga and LLp are the dimensioned roll control and damping derivatives, evaluated at the reference flight. They are related to the nondimensional derivatives C/Ja and Cip by LLSa = {qSb/lu)Cha and LLP = {qSb/In)(b/2V)Cip, where q is the dynamic pressure, S the reference area, b the wing span, and V the vehicle speed.