Given:

$M=20kg,k=400\frac{N}{m}$

$F_0=90N,C=125\frac{N*s}{m}$

$=6\frac{\text{r}\text{a}\text{d}}{s},$

$($a$)$ Form the differential equation of the spring mass damper system which is subject to the

forcing function $F=F_{0}cos{}_{d}t,$ where ${}_d=6ra\frac{d}{s}(20$ points$)$

$($b$)$ Find an analytical solution of the governing differential equation and plot in MATLAB.

$(30$ points$)$

$($c$)$ Find a numerical solution of the differential equation and compare the analytical solution

with the numerical solution. $(20$ points$)$

$($d$)$ Find the transfer function of the differential equation by taking Laplace Transform $(20$

points$)$

$($e$)$ Find the poles of the transfer function $(10$ points$)$