A spring$-$mass system is shown in Fig. $2,$ where the each mass is $5$ kg$,$ and each spring

has a stiffness of $7\mathrm{.}5$ N$/$m$.$ Assume negligible damping.

x$2$ x$1$

m$1$ m$2$

f$1($t$)$ f$2($t$)$

Figure $2$: Schematic of spring$-$mass system for Q$2$

$($a$)$ Calculate the natural frequencies of the system. $[5]$

$($b$)$ Force f$1($t$)=$ sin$($t$)($in Newtons$)$ is applied on mass m$1,$ while force f$2($t$)=$

$2$ cos$(2$t$)($in Newtons$)$ is applied on mass m$2.$ Calculate the particular solution. $[7]$

$($c$)$ If f$($t$)$ is periodic but non$-$harmonic and can be written as a Fourier series

f$($t$)=$

$\backslash$infty X

n$=1$

$1$

$2$n$2+1$

sin $($n$$t$).$

For what value$($s$)$ of the applied frequency $$ will there be a resonance?