$1.(30$ points$)$

For the above problem, assume: $\backslash ($ m$=1\backslash$mathrm$\{$~kg$\},\backslash$mathrm$\{$c$\}=1\backslash$mathrm$\{$Ns$\}/\backslash$mathrm$\{$m$\},\backslash$mathrm$\{$k$\}=40\backslash$mathrm$\{$~N$\}/\backslash$mathrm$\{$m$\}\backslash ).$ Answer the following:

Figure $1$: Spring mass damper system.

$($a$)$ Obtain the analytical equation for the force time history.

$($b$)$ Approximate the force with Fourier Series. Plot a comparison of this approximate force with the analytical force, and show that the approximation improves with the number of terms in the Fourier Series.

$($c$)$ Plot the magnitudes of the Fourier Coefficients, $\backslash (\backslash$left$|$a$\_\{$p$\}\backslash$right$|\backslash )$ and $\backslash (\backslash$left$|$b$\_\{$p$\}\backslash$right$|\backslash ),$ versus $\backslash ($ p $\backslash ).$

$($d$)$ Calculate and plot the steady state $($periodic$)$ response of the system for different values of $\backslash (\backslash$mathrm$\{$p$\}\_\{\backslash$text $\{$max $\}\}\backslash )$ that is$,$ for retaining different number terms in the Fourier Series.

$($e$)$ What is the ideal number of terms to retain in the Fourier Series? Answer the question with logical reasoning and relevant plots.