Problem $1(35$ points $)$

A spring$-$mass$-$damper system is described by the following parameters:

Mass, $(m)=1kg$

Damping coefficient, $(c)=4N\frac{s}{m}$

Spring constant, $(k)=5\frac{N}{m}$

External force, $F_{\text{e}\text{x}\text{t}}(t)=cos(2t)$

a$)$ Derive the differential equation that describes the motion of the mass$-$spring$-$damper system. Show every step of the process. Then, obtain the transfer function between $F(t)$ and $x(t).(5$ points$)$

b$)$ Obtain the response of the system, $x(t).(15$ points$)$

Hint :

When applying the Laplace transform to $cos(2t)$ the resulting expression can be decomposed using partial fractions that involve complex numbers.

You don't need to include the solution steps for calculating the coefficients.

c$)$ Create a Simulink model of the spring$-$mass$-$damper $($do not use the obtained TF in part a$).$ Simulate the system response for the given external force $cos(2t)$ for a simulation time of $20$ seconds. Plot the displacement $x(t)$ over time.

d$)$ Re$-$run the simulation for the damping coefficient values of $c=1N\frac{s}{m}$ and $c=16N\frac{s}{m}.$ Compare the results with the original simulation.

$($Plot Simulink combined $3$ graphs. $c$ and d totally $5$ points. Hand$-$drawn graphs will not be accepted.$)$

e$)$ Create a comparison chart that shows how the system's behavior changes at steady state $($that is$,$ after the transient phase of the oscillations are completed and system oscillates at steady state$)$ Focus on the following parameters: $(10$ points$)$

$\backslash$table$[[$Parameter$,c=1N\frac{s}{m},c=4N\frac{s}{m},c=16N\frac{s}{m},$Comparison$/$Observation$],[$Oscillation$,,,,$How does the peak displacement change?$],[$Amplitude$,,,,$Compare the frequency of oscillations.$],[$Frequency$,,,,$How does the phase behavior change?$],[$Phase Difference,,,,$]]$