$3\mathrm{.}29$ Derive a mathematical model for the translatory mechanical system of

Figure $3\mathrm{.}67($a$).$ Use MATLAB and Simulink to plot the time variation of

coordinate $x$ for $m=0\mathrm{.}5kg,k_1=100\frac{N}{m},k_2=60\frac{N}{m},k_3=90\frac{N}{m},$

$c_1=10N\frac{s}{m},$ and $c_2=6N\frac{s}{m},$ when the system is subjected to:

$($i$)$ The initial conditions $x(0)=0\mathrm{.}03m$ and $v(0)=-0\mathrm{.}5\frac{m}{s}.$

$($ii$)$ The force $f$ defined as in Figure $3\mathrm{.}67($b$)$ and acting on the mass $m$ along

the positive $x$ direction $($the force is not shown in Figure $3\mathrm{.}67($a$)).$

Consider zero initial conditions.

$($a$)$

$($b$)$