$[30$ pts$]$ A DMF is a rotating mechanical device that is used to provide continuous

rotational energy in systems where the energy source is not continuous, the same way as

a conventional flywheel acts, but damping any violent variations in torque or rotational

speed that could cause unwanted vibrations.

Figure $1$: DMF working principle

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$)[10$ pts$]$ Draw the system schematic corresponding to the illustration and description

above.

$($b$)10pts$ Derive the equations of motion for this rotational mechanical system.

$($c$)10pts$ Derive the state$-$space representation for the system, where the output of

interest is the clutch$-$to$-$drivetrain shaft angular velocity ${}_{cl}(t).$