12.3.2: Forces acting on an aircraft
- Page ID
- 78412
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The external actions acting on an aircraft can be decomposed, without loss of generality, into propulsive, aerodynamic and gravitational, notated respectively with subindexes (\((\cdot)_T, (\cdot)_A, (\cdot)_G\)):
\[\vec{F} = \vec{F}_T + \vec{F}_A + \vec{F}_G, \nonumber \]
\[\vec{G} = \vec{G}_T + \vec{G}_A, \nonumber \]
The gravitational force can be easily expressed in local horizon axes as:
\[(\vec{F}_G)_h = \begin{bmatrix} 0 \\ 0 \\ mg \end{bmatrix}, \nonumber \]
where \(g\) is the acceleration due to 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:
\[(\vec{F}_G)_w = L_{wh} (\vec{F}_G)_h = \begin{bmatrix} -mg \sin \gamma \\ mg \cos \gamma \sin \mu \\ mg \cos \gamma \cos \mu \end{bmatrix}. \nonumber \]
Introducing the propulsive, aerodynamic and gravitational actions in System (12.3.1.10-12.3.1.15):
\[-mg \sin \gamma + F_{T_x} + F_{A_x} = m (\dot{u} - rv + qw), \nonumber \]
\[mg \cos \gamma \sin \mu + F_{T_y} + F_{A_y} = m (\dot{v} + ru - pw), \nonumber \]
\[mg \cos \gamma \cos \mu + F_{T_z} + F_{A_z} = m (\dot{w} - qu + pv), \nonumber \]
\[L_T + L_A = I_x \dot{p} - J_{xz} \dot{r} + (I_z - I_y) qr - J_{xz} pq, \nonumber \]
\[M_T + M_A = I_y \dot{q} - (I_z - I_x) pr - J_{xz} (p^2 - r^2), \nonumber \]
\[N_T + N_A = I_z \dot{r} - J_{xz} \dot{p} + (I_x - I_y) pq - J_{xz} qr. \nonumber \]
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.