$5\mathrm{.}2$: The spring$-$mass$-$damper system shown in Fig $2($a$)$ is excited by a periodic displacement $y(t),$ given in Fig. $2($b$),$ applied to the free end of spring $k_2.$

a$)$ Find the Fourier series expansion of $y(t).$

b$)$ Assuming $k_l=k_2=k,$ find the steady$-$state response $x_p(t)$ due to $y(t).$

Fig. $2$