04. Full Longitudinal Dynamics
Nd787 C5 L3 A03 L Full Longitudinal Dynamics V2
Incorporating Forces and Moments
When we include forces and moments, the equations of motion look like this:
\begin{aligned}
\dot{x}_I &= u \cos \theta + w \sin \theta \\
\dot{z}_I &= -u \sin \theta + w \cos \theta \\
\dot{\theta} &= q \\
\dot{u} &= F_X/m - qw \\
\dot{w} &= F_Z/m + qu \\
\dot{q} &= M/I_{yy} \\
\end{aligned}
Where we define F_X and F_Z as follows:
\begin{aligned}
\begin{aligned}
F_X =& \ L \sin \alpha - D \cos \alpha + T -mg \sin \theta \\
F_Z =& -L \cos \alpha - D \sin \alpha + mg \cos \theta \\
\end{aligned}
\end{aligned}
You might notice that the equations above depend on the angle of attack \alpha. Does that mean we need to include it in our state vector?
No! It turns out that \alpha can be calculated from quantities that we're already tracking.
From the diagram above you can see:
\alpha = \theta - \gamma
The pitch angle \theta is already known, and the flight path angle \gamma can be calculated as:
\gamma = \arctan\left( \frac{\dot{z}_I}{\dot{x}_I} \right)
This means we can express the angle of attack in terms of known state variables:
\alpha = \theta - \arctan\left( \frac{-u \sin \theta + w \cos \theta}{u \cos \theta + w \sin \theta}\right)