A certain system has two coupled subsystems. One subsystem is a rotational system with the equation of motion

$80\frac{(d)}{dt}+10=T(t)$

where $T(t)$ is the torque applied by an electric motor, as shown in the figure. The second subsystem is a field$-$controlled motor. The

model of the motor's field current $if$ in amperes is

$0\mathrm{.}001\frac{d{i}_f}{dt}+5i_f=v(t)$

where $v(t)$ is the voltage applied to the motor. The motor torque constant is $K_T=50N*\frac{m}{A}.$ Obtain the damping ratio $,$ time constants,

and undamped natural frequency ${}_n$ of the combined system.

The damping ratio is determined to be

The time constant of the rotational system is determined to be

$8$ sec $.$

The time constant of the motor's field current is determined to be

The undamped natural frequency of the combined system is determined to be

rad$/$s$.$