A DMF can be modeled, starting from the engine crankshaft and ending at the clutch$-$to$-$drivetrain shaft:

A shaft 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 $(J_c)$ 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 $(d_4).$

Assuming a crankshaft torque input ${}_{cr}(t)$ and the information about a DMF provided above,

$($a$)$ Draw the system schematic corresponding to the illustration and description

above.

$($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 ${}_{cl}(t).$