# 12.3.2: Forces acting on an aircraft


##### 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.

12.3.2: Forces acting on an aircraft is shared under a CC BY-SA 3.0 license and was authored, remixed, and/or curated by Manuel Soler Arnedo via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.