Problem $3.$ A mass$-$spring$-$damper system of mass

$m=1kg,$ damping coefficient $b=12N\frac{s}{m},$ and

stiffness $k=100\frac{N}{m}$ is moved by constant force

$F_0=10N$ for the duration of $t_1=1s,$ and by zero

force afterwards. The spring is unstretched at the

position $x=0.$

The EoM is given by:

$m{x}^{}+b{x}^{}+kx=F(t)$

whereas the ICs are $x(0)=0$ and $x^{}(0)=0.$

a$)$ Determine the following: static deformation,

undamped natural angular frequency, damping ratio,

damped natural angular frequency, damped natural

frequency, and time period.

b$)$ Calculate the step response by solving the EoM for

tin$[0,t_1].$ Calculate $x_1=x(t_1)$ and $v_1=x^{}(t_1).$

c$)$ Calculate the free response by solving the EoM for

$t{t}_1,$ using $x(t_1)=x_1$ and $x^{}(t_1)=v_1$ as IC$.$

d$)$ Simulate the motion in Matlab by solving the EoM

numerically. Plot the analytical and the numerical

solution $x(t)$ over tin$[0,2]s$ in the same graph.

Read the overshoot and the rise time off of the figure.

Submit the figure and your codes.