# 12.4.5: General differential equations system

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

For a point mass model, the general differential equations system governing the motion of an aircraft is stated as follows:

$-mg \sin \gamma + T \cos \epsilon \cos v - D = m (\dot{V}),\label{eq12.4.4.1}$

$mg \cos \gamma \sin \mu + T \cos \epsilon \sin v - Q = - mV (\dot{\gamma} \sin \mu + \dot{\chi} \cos \gamma \cos \mu),$

$mg \cos \gamma \cos \mu - T \sin \epsilon - L = - mV (\dot{\gamma} \cos \mu - \dot{\chi} \cos \gamma \sin \mu),$

$\dot{x}_e = V \cos \gamma \cos \chi + W_x,$

$\dot{y}_e = V \cos \gamma \sin \chi + W_y,$

$\dot{z}_e = - V \sin \gamma,$

$\dot{m} + \phi = 0.\label{eq12.4.4.7}$

If we assume the following hypothesis:

##### Hypothesis 12.9 Symmetric flight

We assume the aircraft has a plane of symmetry, and that the aircraft flies in symmetric flight, i.e., all forces act on the center of gravity and the thrust and the aerodynamic forces lay on the plane of symmetry. This leads to non sideslip, i.e., $$\beta = ν = 0$$, and non lateral aerodynamic force, i.e., $$Q = 0$$, assumptions.

##### Hypothesis 12.10 Small thrust angle of attack

We assume the thrust angle of attack is small $$\epsilon \ll 1$$, i.e., $$\cos \epsilon \approx 1$$ and $$\sin \epsilon \approx 0$$. For commercial aircrafts, typical performances do not exceed $$\epsilon = \pm 2.5 [deg] (\cos 2.5 = 0.999)$$; in taking off rarely can go up to $$\epsilon = 5 - 10 [deg]$$, but still $$\cos 10 = 0.98$$

ODE system ($$\ref{eq12.4.4.1}$$-$$\ref{eq12.4.4.7}$$) is as follows:

$-mg \sin \gamma + T - D = m (\dot{V}),$

$mg \cos \gamma \sin \mu = -m V (\dot{\gamma} \sin \mu - \dot{\chi} \cos \gamma \cos \mu),\label{eq12.4.4.9}$

$mg \cos \gamma \cos \mu - L = -m V (\dot{\gamma} \cos \mu + \dot{\chi} \cos \gamma \sin \mu),\label{eq12.4.4.10}$

$\dot{x}_e = V \cos \gamma \cos \chi + W_x,$

$\dot{y}_e = V \cos \gamma \sin \chi + W_y,$

$\dot{z}_e = -V \sin \gamma,$

$\dot{m} + \phi = 0.$

Operating Equation ($$\ref{eq12.4.4.9}$$) $$\cdot \cos \mu$$ - Equation ($$\ref{eq12.4.4.10}$$) $$\cdot \sin \mu$$ it yields:

$L \sin \mu = m V\dot{\chi} \cos \gamma.$

Operating Equation ($$\ref{eq12.4.4.9}$$) $$\cdot \sin \mu$$ + Equation ($$\ref{eq12.4.4.10}$$) $$\cdot \cos \mu$$ it yields:

$L \cos \mu - mg \cos \gamma = m V \dot{\gamma}.$

12.4.5: General differential equations system 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.