12.3.2: Forces acting on an aircraft
- Page ID
- 78412
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.