EQUATIONS OF MOTION

The equations of motion are obtained by substituting the generalized forces and the kinetic energy in Lagranges equations, i. e

Je К + IeJ®J = 11 e + fellp + ki2J — «W*02 – PexiPr — І) («)

I’jk + IjSj = k2lP + k2*J (b)

(11.3.12)

The inertia terms on the r. h.s. of (a) are the only nonlinear ones, and in view of the assumptions already made, linearization of these is in order. a0 is the a component of the acceleration of the vehicle mass center and is given by (5.3,18). Without the Earth rotation terms, and for small distur­bances, we get

a0 — w — qu

From (4.3,4), in the linear case, w = Vv. x and и = V, so that the linear expression for the acceleration is

<4 = У A – я К

and (11.3,12) become

Je 4 + Ie. jbj = 11 e + Kip + lciiJ — meeeVe(± — q} + Pexq (a)

U + ^A = V + V (Ь)

(11.3.13)

These equations, when combined with the vehicle equations of motion, convert 8e from a nonautonomous to an autonomous variable, add Qj to the autonomous set, and introduce P and J as nonautonomous variables. The aerodynamic force He is a function of the state variables, i. e. [cf. (6.5,2)]

He = He0 + Hv AF + Ha Да + Я*« + HQq + H, Аде + Щ Se (c)

(11.3.13)

and provides aerodynamic coupling (feedback) between the vehicle motion and the control force. Similarly the terms containing a and q in (11.3,13a) provide inertial coupling between vehicle and control dynamics.