A simplified version of this system can be modeled as a two$-$degree$-$of$-$freedom, where the excitation is the waves$$ motion while the bottom of the device is fixed to the seabed. Two magnets moving inside a coil, creating current due to their oscillation based on Faraday$$s law. In this design, a buoy floating on the surface pulls the plate connected to the spring k$1$ and damper c$1.$ Here, the damping value is the effective damping coefficient which include mechanical damping $($cm$)$ and electrical damping $($ce$)$ caused by the mechanical$-$to$-$electrical conversion.

$1)$Consider the values listed in table $1$ for a small$-$scale machine and find the un$-$damped natural frequencies and mode shapes for this system. $($c$1=0,$ c$2=0,$ c$3=0,$ y$=0)($using analytical hand calculations $-$ characteristic equation$)(20$ points$)$

$2)$Calculate and plot system response for$$un$-$damped, free vibration using Matlab, $($write a code in Matlab to calculate the response and plot the system response$).($c$1=0,$ c$2=0,$ c$3=0,$ y$=0,$ x$01=0\mathrm{.}1,$ x$02=0\mathrm{.}1,$ v$01=1,$ v$02=0)(20$ points$)$

$3)$Construct the equations of motion for damped system and create the model of the system in Simulink. Consider the dampers and base excitation in your Simulink model. Plot the movements of each magnet $($each mass$)$ for an excitation caused by a wave following y$($t$)=0\mathrm{.}5$ sin $(0\mathrm{.}8$t$)$ motion. $($Initial conditions are zero for this case.$)$ Which mas has a larger amplitude? $(20$ points$)$

$4)$To get the most electricity from this harvester, the goal is to maximize the displacements. Consider the same parameters listed in table $1$ and the same input from the waves. We can change the value of k$2$ as well as c$2.$ Investigate a combination of the c$2$ values $(5$ N$.$s$/$m and $12$N$.$s$/$m$)$ and k$2$ values $(200$ N$/$m and $800$ N$/$m$)(4$ cases in total$).$ Which case will increase the movements of both magnets to get the maximum power output. Consider Steady$-$State response for each mass and use estimated value from Simulink graphs. Make a table and compare the max$($x$1)+$max$($x$2)$ values. Which combination of c$2$ and k$2$ gives you the max combined value.$(15$ points$)$

$5)$We want to choose an installation site for this harvester with waves moving in the form of y$($t$)=0\mathrm{.}5$ sin $($f$*$t$).$ Try to find the best value for the frequency f for this specific design with parameters listed in table $1($ i$.$e$.,$ in which environment this design will produce maximum electricity?$)(15$ points$)$

Answer all question please with all details, simulink graph and matlab code