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.