Flight Dynamics Formulation

9.8.1 Variable and vector definitions

In this section we will adopt the following simplified notation for the aircraft motion parameters and aero­dynamic forces and moments. This notation is fairly standard in the discipline of flight dynamics, stability, and control.

R = { x } U = { U } = j p j Fb = j X j Mb = j M j

The components of the first three vectors above, together with the three Euler angles, are grouped in the following state vector with 12 components:

x(t) = { xe ye ze ф 6 ф и v w p q r }T (9.24)

As treated in Section 6.3, the aerodynamic forces X, Y, Z and moments L, M, N are functions of this state

vector, and also of the control vector 8, which for a typical aircraft consists of aerodynamic control surface deflections and engine forces.

8(t) = { 8a 5e 8r Sf 5T }T (9.25)

The first four components correspond to aileron, elevator, rudder, and flap deflections, and the last compo­nent is an engine-thrust variable. Unconventional aircraft may have other types of control variables.

9.8.2 General equations of motion

The 12 state vector components (9.24) are governed by 12 ODEs in time. These are the six kinematic equations for the aircraft position rate (9.11) and the Euler angle rates (9.16), and the six dynamic equations for the linear momentum rate (9.19) and the angular momentum rate (9.19). Using the new simplified notation defined above, these 12 equations are written out fully as follows.

Xe = (cos 6 cos ф) и + (sin ф sin 6 cos ф — cos ф sin ф) v + (cos ф sin 6 cos ф + sin ф sin ф) w

ye = (cos 6 sin ф) и + (sin ф sin 6 sin ф + cos ф cos ф) v + (cos ф sin 6 sin ф — sin ф cos ф) w (9.26)

Ze = (— sin 6) и + (sin ф cos 6) v + (cos ф cos 6) w

ф=

p

+ q sin ф tan

6 + r

cos ф tan 6

6=

q

cos ф — r sin

ф

(9.27)

ф=

q

sin ф/ cos 6 +

r cos ф/ cos 6

X

mg sin 6 =

m (U

+ qw —

rv)

Y +

mg

sin ф cos 6 =

m (V

+ ru —

pw)

(9.28)

Z +

mg

cos ф cos 6 =

m (w

+ pv —

-qu)

L = Ixx p + Ixy q + Ixz r + (Izz — Iyy) qr + Iyz (q2 — r2) + Ixz pq — IXy pr + hz q — hy r

M = Ixy p + Iyy q + Iyz r + (Ixx — Izz) rp + Ixz (r2 — p2) + Ixy qr — Iyz qp + hx r — hz p (9.29)

N = Ixz p + Iyz q + Izz r + (Izz — Ixx) pq + Ixy (p2 — q2) + Iyz rp — Ixz rq + hy p — hx q

When the three linear momentum equations (9.28) are multiplied by 1/m, and the three angular momentum

equations (9.29) are multiplied by the inverse of the moment of inertia tensor, Ї, they become explicit expressions for the linear and angular velocity rates, U = … , p = … , etc. AH the 12 equations (9.26)- (9.29) then collectively have the classical state-space evolution form with 12 equation components:

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