Question $1$: Equations of motion of lateral dynamics $(\%50)$

Suppose a missile flies in the horizontal plane with velocity $Vb=$

$\left[u_{0}v\right]$ 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 aerodynamic 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 $N_0,N_v,N_r,N_r,A_0,A_v,A_r,{}_T,L$ and $u_0$ are supposed to

be constants. We assume $m$ is mass and $I_{zz}$ 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 $x=\left[vr\right],u=\left[rT\right].$

C$)$ Linearize equations of motion and write them in state$-$space form. Suppose that $x=\left[vr\right],u=\left[rT\right].$

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