# 12.4.1: Dynamic relations

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Therefore, the dynamic equations governing the translational motion of the aircraft are uncoupled:

$-mg \sin \gamma + F_{T_x} + F_{A_x} = m (\dot{u} - rv + qw),$

$mg \cos \gamma \sin \mu + F_{T_y} + F_{A_y} = m (\dot{v} + ru - pw),$

$mg \cos \gamma \cos \mu + F_{T_z} + F_{A_z} = m (\dot{w} - qu + pv),$

The aerodynamic forces, expressed in wind axes, are as follows:

$(\vec{F}_A)_w = \begin{bmatrix} -D \\ -Q \\ -L \end{bmatrix},$

where $$D$$ is the aerodynamic drag, $$Q$$ is the aerodynamic lateral force, and $$L$$ is the aerodynamic lift.

The propulsive forces, expressed in wind axes, are as follows:

$(\vec{F}_T)_w = \begin{bmatrix} T \cos \epsilon \cos v \\ T \cos \epsilon \sin v \\ -T \sin \epsilon \end{bmatrix},$

where $$T$$ is the thrust, $$\epsilon$$ is the thrust angle of attack, and $$ν$$ is the thrust sideslip.

##### Theorem 12.6 Fixed engines

We assume the aircraft is a conventional jet airplane with fixed engines. Almost all existing aircrafts worldwide have their engines rigidly attached to their structure.

12.4.1: Dynamic relations 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.