Problem $4$:

The system in the figure is underdamped and subjected to the pulsating load sketched below. The load is periodic with period $T.$ The Fourier series for the signal $F(t)$ is already calculated for you and is given by

$F(t)=\frac{F_0}{2}+{\displaystyle \underset{n=1}{\overset{}{}}},3,5,$dots$\frac{2F_0}{n}sin(nt)$

with $=2\frac{}{T}.$

a$)15$ points: Determine symbolically the steady$-$state response $x(t)$ to the load, using all harmonics.

b$)15$ points: Given $m=10kg,k=4000\frac{N}{m},c=40N\frac{s}{m},F_0=10N,T=1s,$ determine the numerical value of the magnitude and phase of harmonic $1$ and $3$ of the response. Comment on your results.