Question $1$: Equations of motion of lateral dynamies $(\%50)$ Suppose a missile flies in the horizontal plane with velocity $V^b=\left[\begin{array}{c}{u}_0\\ v\end{array}\right]$ as shown in the figure. Here $u_0$ is constant, but $v$ is variable, $$ is heading $($yaw$)$ angle, $r$ is yaw rate, $N$ is aerodynamic moment in $z_b$ direction, $A_y$ is acrodynamic force in $y_b$ direction, $r$ is rudder angle, $T$ is thrust and ${}_T$ is thrust misalignment angle and $L$ is distance between c$.$g$.$ and missile's motor. Models of yaw moment and side force are:

$N=N_0+N_{v}v+N_{r}r+N_{r}r$

$A_y=A_0+A_{v}v+A_{r}r$

Here $N_0,N_y,N_r,N_r,A_0,A_v,A_r,{}_T,L$ and $u_0$ are supposed to be constants. We assume $m$ is mass and $l_{zz}$ is moment of inertia about $z_b$ axis.

A$)$ Drive equations of motion. $($use force in $y_b$ direction in and moment in $z_b$ direction$).$

B$)$ Write equations of motion in state space form. Suppose that $x=\left[\begin{array}{c}v\\ r\end{array}\right],u=\left[\begin{array}{c}r\\ T\end{array}\right].$

C$)$ Linearize equations of motion and write them in state$-$space form. Suppose that $x=\left[\begin{array}{c}v\\ r\end{array}\right],u=\left[\begin{array}{c}r\\ T\end{array}\right].$

D$)$ Optain the natural frequency and damping ratio of the system.