$(10$ points$)$ This problem is best done using the aid of computer programming such as Python or MATLAB. The longitudinal aircraft equations of motion, linear state space form, of a particular aircraft at a particular flight condition, are given by

$\left[\begin{array}{c}{u}^{}\\ w^{}\\ q^{}\\ {}^{}\end{array}\right]=\left[\begin{array}{cccc}-0\mathrm{.}15& 0\mathrm{.}08& 0\mathrm{.}05& -0\mathrm{.}12\\ -0\mathrm{.}073& -0\mathrm{.}13& 1& 0\\ 0\mathrm{.}82& 7\mathrm{.}37& -0\mathrm{.}15& 0\\ 0& 0& 1& 0\end{array}\right]\left[\begin{array}{c}u\\ w\\ q\\ \end{array}\right]$

$($a$)$ Determine the eigenvalues of this system $($which are same as the eigenvalues of the system matrix$)$

$($b$)$ Determine the damping ratio and natural frequency of each of the modes.

$($c$)$ Suitably modify the

phugoid.py file on Catcourses and used for the in$-$class demonstration last lecture to compute the dynamics of this system for the following initial conditions

$u=0$

$w=15\frac{m}{s}$

$q=8ra\frac{d}{s}$

$=0$

Use freestream velocity of $125\frac{m}{s}.$

$($d$)$ Plot the time variation of all four state variables