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

$\left[\begin{array}{c}{}^{}\\ p^{}\\ r^{}\\ {}^{}\end{array}\right]=\left[\begin{array}{cccc}-0\mathrm{.}44& 0& -1\mathrm{.}0& 0\mathrm{.}3\\ -8\mathrm{.}02& -20\mathrm{.}4& 2\mathrm{.}19& 0\\ 1\mathrm{.}5& -0\mathrm{.}35& -2\mathrm{.}76& 0\\ 0& 1& 0& 0\end{array}\right]\left[\begin{array}{c}\\ p\\ r\\ \end{array}\right]$

$($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. If a certain eigenvalue is

$=+i$

the damping ratio $()$ and natural frequency $({}_n)$ are given by

$=\frac{-}{\sqrt[\phantom{2}]{{}^2+{}^2}}$

${}_n=\sqrt[\phantom{2}]{{}^2+{}^2}$

$($c$)$ Suitably modify the

lateral.py file on Catcourses used for the in$-$class demonstration last lecture to compute the dynamics of this system for the following initial conditions. Note that the dphi$\_$dt equation we did in class has a small typo. dphi$\_$dt should be set equal to p and not r

$=5$

$p=10ra\frac{d}{s}$

$r=20ra\frac{d}{s}$

$=0$

$($d$)$ Plot the time variation of all four state variables. Make sure to adjust the end time suitably so that you do not show a long time $($after all the perturbations have died off$)$