12.3.2: Forces acting on an aircraft

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Hypothesis 12.3 Forces acting on an aircraft

The external actions acting on an aircraft can be decomposed, without loss of generality, into propulsive, aerodynamic and gravitational, notated respectively with subindexes (\((\cdot)_T, (\cdot)_A, (\cdot)_G\)):

\[\vec{F} = \vec{F}_T + \vec{F}_A + \vec{F}_G,\]

\[\vec{G} = \vec{G}_T + \vec{G}_A,\]

The gravitational force can be easily expressed in local horizon axes as:

\[(\vec{F}_G)_h = \begin{bmatrix} 0 \\ 0 \\ mg \end{bmatrix},\]

where \(g\) is the acceleration due to gravity.

Theorem 12.4 Constant gravity

The acceleration due to gravity in atmospheric flight of an aircraft can be considered constant (\(g = 9.81[m/s^2]\)), due to a small altitude of flight when compared to the radius of earth. Therefore, the little variations of \(g\) as a function of \(h\) are neglectable.

To project the force due to gravity into wind-axes reference frame:

\[(\vec{F}_G)_w = L_{wh} (\vec{F}_G)_h = \begin{bmatrix} -mg \sin \gamma \\ mg \cos \gamma \sin \mu \\ mg \cos \gamma \cos \mu \end{bmatrix}.\]

Introducing the propulsive, aerodynamic and gravitational actions in System (12.3.1.10-12.3.1.15):

\[-mg \sin \gamma + F_{T_x} + F_{A_x} = m (\dot{u} - rv + qw),\]

\[mg \cos \gamma \sin \mu + F_{T_y} + F_{A_y} = m (\dot{v} + ru - pw),\]

\[mg \cos \gamma \cos \mu + F_{T_z} + F_{A_z} = m (\dot{w} - qu + pv),\]

\[L_T + L_A = I_x \dot{p} - J_{xz} \dot{r} + (I_z - I_y) qr - J_{xz} pq,\]

\[M_T + M_A = I_y \dot{q} - (I_z - I_x) pr - J_{xz} (p^2 - r^2),\]

\[N_T + N_A = I_z \dot{r} - J_{xz} \dot{p} + (I_x - I_y) pq - J_{xz} qr.\]

The three aerodynamic momentum of roll, pitch and yaw \((L_A,M_A,N_A)\) can be controlled by the pilot through the three command surfaces, ailerons, elevator and rudder, whose deflections can be respectively notated by \(\delta_a, \delta_e, \delta_r\). Notice that such deflection have also influence in the three components of aerodynamic force, and therefore the 6 equations are coupled and must be solved simultaneously.