$5\mathrm{.}1$: A single degree of freedom spring$-$mass system with no damping shown in Fig. $1($a$)$ is subjected to a periodic force $F(t)$ shown in Fig. $1($b$).$

a$)$ Find the Fourier series expansion of $M(t).$ Plot $M(t)$ using the first $2,4,$ and $6$ terms of the expansions.

b$)$ Find the steady state response ${}_p(t)$ due to $M(t).$

$($a$)$

$($b$)$

Fig. $1$