Consider the torsional mechanical system in Figure CP4.7(a). The torque due to the twisting of the shaft is -kθ; the damping torque due to the braking device is -bθ; the disturbance torque is td(t); the input torque is r(t); and the moment of inertia of the mechanical system is J . The transfer function of the torsional mechanical system is

A closed-loop control system for the system is shown in Figure CP4.7(b). Suppose the desired angle θd = 0°, k = 5, b = 0.9, and J = 1.
(a) Determine the open-loop response θ(t) of the system for a unit step disturbance (set r(t) = 0).
(b) With the controller gain K0 = 50, determine the closed-loop response, θ(t) to a unit step disturbance.
(c) Plot the open-loop versus the closed-loop response to the disturbance input. Discuss your results and make an argument for using closed-loop feedback control to improve the disturbance rejection properties of the system.
Figure CP4.7
(a) A torsional mechanical system.
(b) The torsional mechanical system feedback control system.

1/J G(s) Elastic shaft As) Mechanical t Disturhance torque system Controller I/J Braking rir) urquedevice fa) (hi