Problem $3(14$ points$)$: Mechanical Rotary System

The system in the image below is a heavy wheel connected to a shaft so that the wheel and shaft can smoothly spin as a unit when the shaft is inserted into the hole in the stationary mounting block. The exploded view on the left shows the configuration of the wheel$-$shaft part and the view on the right shows the part inserted into the stationary block. The hole is sized to allow the part to spin but there is friction between the shaft and the block. The combined spinning inertia is $\backslash ($ J $\backslash ),$ and the friction can be modeled by linear rotary damping $\backslash ($ B $\backslash ).$ The shaft is stiff and has no compliance.

You want to know how the friction slows down the wheel when the wheel has an initial angular velocity $\backslash (\backslash$omega$(0)\backslash ),$ that is$,$ the initial condition response.

$($a$,12$ points$)$ Write the system differential equation for the wheel velocity $\backslash (\backslash$omega$($t$)\backslash ),$ when there is no external torque cource on the wheel.

$($b$,2$ points$)$ Assume $\backslash ($ J$=1,$ B$=1,\backslash$omega$(0)=4\backslash ).$ Sketch the initial condition response for $\backslash (\backslash$omega$($t$)\backslash )$ on the provided axis. Mark the value of the time constant on the horizontal axis and the value of the starting velocity on the vertical axis.