Problem $2$ Consider a translational mechanical system is shown below. Note that $M_3$ is NOT massless.

$($a$)$ The equation of motion for $x_1(t)$ is $f-f_{v1}(x_1^{}-x_2^{})-f_{v2}(x_1^{}-x_2^{})-k(x_1-x_3)=m_1{x}_1^{}.$ Find the equation of motion $($EOM$)$ in the time$-$domain for $x_2$ and $x_3,$ respectively. $(30$ pts $)$

$($b$)$ Apply Laplace transform to the EOMs and present the equations in $s-$domain in matrix form. Note that while the notation here is quite concise, it is very clear that $x_1=x_1(s)=L(x_1(t)),$ etc.

$[?]\left[\begin{array}{c}{x}_3\\ x_2\\ x_1\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ F\end{array}\right]$

The following Python code is constructed based on Eq$.(4)$ and let $Q$ be the matrix that has a question mark in Eq$.4.$ Inspect Eq$.(4)$ carefully and select the code for troopa. $(30$ pts $)$

$\backslash$table$[[$Import sympy as sym$,1],[$Erom sympy$.$matrices import Matrix,$2],[,3],[$fv$1,$ fv$2,$ fv$3,$ fv$4=$ sym$,$symbols$(\text{'}$f$\_$v$1,$ f$\_$v$2,$ f$\_$v$3,$ f$\_$v$4\text{'}),4],[,5],[$Goo $=$ Matrix$($goomba$),6],[$Koo $=$ Matrix $($koopa$),7],[$Troo $=$ Matrix$($troopa$),8],[$Q $-$ Matrix.vstack $($Goo$,$ Koo, Troo$),9]]$

$($i$)\left[\begin{array}{c}-k\\ -(fv1+fvv2)**s\\ k+m1**(s****2)+(fvv1+fv2)**s\end{array}\right]$

$($ii$)[[k+m1**(s****2)+(fv1+fv2)**s$