Concise and other forms of EoMs (Working Engineer Version)#

In the past two sections, a good Undergraduate level of description of the Concise and Non-Dimensional forms of the Equations of Motion was shown. Let’s now present things slightly differently - using more nomenclature, just to fuck things up for you properly.

We’re also going to change the meaning of a couple of symbols until I’m smart enough to figure out a way to make this next section consistent with the last one (some large find/replace operation, I suspect).

Let’s think about where we ended up, in the preceding, the Concise Form of the Longitudinal Equations of Motion was given as (75):

\[\begin{split}\begin{align}\begin{bmatrix} \dot{u}\\\dot{w}\\\dot{q}\\\dot{\theta}\end{bmatrix} = \begin{bmatrix} X_u & X_w & 0 & -g\cdot\cos\theta_0\\ Z_u & Z_w & U_0 & -g\cdot\sin\theta_0\\ M_u^* & M_w^* & M_q^* & M_\theta^*\\ 0 & 0 & 1 & 0 \end{bmatrix}\begin{bmatrix} {u}\\{w}\\{q}\\{\theta} \end{bmatrix} + \begin{bmatrix} 0\\Z_{\delta_e}\\M_{\delta_e}^*\\0 \end{bmatrix}\left[\delta_e\right]\end{align}\end{split}\]
\[\dot{\vec{x}} = A\vec{x} + B\vec{u}\]

Which is but one means of presenting the equations of motion in state space formulation. Concise in this representation means (I think):

  • Each term in the A and B matrix is a single term

  • Each output is in real engineering units