Problem $3(20$ points$)$

The following EOMs describe the behavior of an electric motor attached to a torsional spring and damper:

$L{z}^{}+Rz+v_b=v_{in}(t)$

$I{}^{}=-k-b{}^{}+$

Importantly, the equations are coupled by the fact that $=k_{t}z$ and $v_b=k_v{}^{}.$

Here, $L,R,k_v,$I,$k,b,$ and $k_t$ are constants. $z$ is the current in the motor, and $$ is the angular displacement of the motor $($clearly then, ${}^{}$ is the motor's angular velocity$).$ Finally, the input to the system is $v_{in}^{}(t).$

Your tasks:

A Substituting in the coupling terms and $($:$v_b=k_v({}^{})$