Figure $2$ illustrates a viscously damped two$-$degree$-$of$-$freedom system, used to model and analyze vibrations in mechanical structural systems. The system has the following parameters, with zero initial conditions:

$m_1=2kg$

$m_2=5kg$

$k_1=10\frac{N}{m}$

$k_2=20\frac{N}{m}$

$k_3=5\frac{N}{m}$

$c_1=2N*\frac{s}{m}$

$c_2=3N*\frac{s}{m}$

$c_3=1\mathrm{.}0N*\frac{s}{m}$

$f_1(t)=0$

$f_2(t)=5N$ for $t0$

$3\mathrm{.}1$ Determine the forced response of this viscously damped two$-$degree$-$of$-$freedom mechanical system using modal analysis.

$3\mathrm{.}2$ Plot the displacement responses, $x_1(t)$ and $x_2(t),$ of both masses over the first $10$ seconds following the application of the force $f_2(t).$ Clearly label the displacement responses for each mass and indicate any steady$-$state behavior observed in the graph.

Figure $2$: Viscously damped two$-$degree$-$of$-$freedom system.