Skip to main content
Engineering LibreTexts

12.3.2: Forces acting on an aircraft

  • Page ID
    78412
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    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.

    • Was this article helpful?