Missile longitudinal dynamics can be described with the short-period approximation of the longitudinal equations of motion. These are typically written in state space form. Sign conventions in these models dictate that a positive pin deflection results in a negative pitching moment.
Short Period Dynamics:
Short Period Dynamics:
$$\begin{matrix}
\dot{x}=Ax+Bu \\
y = Cx+Du
\end{matrix}$$
$$x=
\begin{bmatrix}
\alpha \\ q
\end{bmatrix}$$
$$u=\delta_p$$
$$y=
\begin{bmatrix}
A_{zm} \\ q_m
\end{bmatrix}$$
Where:
The fundamental equations of motion govern the aircraft dynamics. The longitudinal missile force diagram can be described with 4 variables. The translational acceleration component, the AOA $\alpha$, the flight path angle $\gamma$
The relationship between these angles can be described by the following equation
$$\begin{matrix}
\alpha=\theta-\gamma \
\dot{\alpha}=\dot{\theta}-\dot{\gamma}
\end{matrix}$$
The angular acceleration is the applied aerodynamic moment divided by the moment of inertia.
$$\ddot{\theta}=\frac{M(\alpha,\delta)}{J}$$
The small-angle approximation of the flight-path rate is
$$\dot{\gamma}=\frac{A_z cos(\alpha)}{V}\approx\frac{A_z}{V}$$
The normal acceleration is
$$A_z=\frac{F_z(\alpha,\delta)}{m}$$
[[Missile Longitudinal Autopilots]]
[[Linearized State-Space Equations for Missile Longitudinal Dynamics]]
- mracekMissileLongitudinalAutopilots2005
- jacksonOverviewMissileFlight2010
Backlinks:
[[Longitudinal Flight Dynamics]]
[[Missile Longitudinal Autopilot Topology]]
[[Moment of Inertia]]
[[Moment of Inertial of a Missile]]
State-Space Model