Question $3[50$ points$]$

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$\_(1)=2$kg

m$\_(2)=5$kg

k$\_(1)=10($N$)/($m$)$

k$\_(2)=20($N$)/($m$)$

k$\_(3)=5($N$)/($m$)$

c$\_(1)=2$N$*($s$)/($m$)$

c$\_(2)=3$N$*($s$)/($m$)$

c$\_(3)=1\mathrm{.}0$N$*($s$)/($m$)$

f$\_(1)($t$)=0$

f$\_(2)($t$)=5$N for t$>=0$

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

mechanical system using modal analysis. $(15)$

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