For this FBD below:

$($b$)$ Derive the equations of motion for this rotational mechanical system.

$($c$)$Derive the state$-$space representation for the system, where the output of interest is the clutch$-$to$-$drivetrain shaft angular velocity

$\backslash$omega cl$($t$)$

Details: Ashaft supported by bearings, with friction represented by a rotational damper of coefficient $($d$1),$ connected to the primary rotating mass $($J$1).$

$$ The primary rotating mass $($J$1)$ is connected the secondary rotating mass $($J$2)$ by a torsional spring of coefficient $($k$1)$ in parallel with a torsional damper of coefficient $($d$2).$

$$ The secondary rotating mass $($J$2)$ is connected to the clutch assembly which can be represented by a torsional spring of coefficient $($k$2)$ in parallel with a torsional damper of coefficient $($d$3)$ connected to the clutch rotating mass $($Jc$)$ which is then

connected to the clutch$-$to$-$drivetrain shaft.

$$ The clutch$-$to$-$drivetrain shaft is supported by bearings with friction represented by

a rotational damper of coefficient $($d$4)$