Problem $3(20\%)$

The magnitude Bode plot for the closed$-$loop transfer function $T(s)$ of a vibration system is given in Fig. $3.$

Figure $3$: Magnitude Bode plot of $T(s)$

a$)$ Given that the transfer function is a standard second order system,

i$.(4\%)$ Find the damping ratio of the system.

ii$.(4\%)$ Find the resonant and natural frequency of the system.

iii. $(3\%)$ Find the closed$-$loop transfer function $T(s).$

iv$.(5\%)$ Estimate the percentage overshoot and settling time of the system for a unit step input.

b$)(4\%)$ When the vibration system is subjected to a sinusoidal input, peak response at resonant frequency could lead to strong vibration. The strong vibration may damage mechanical components of the system. Explain how the peak response could be reduced.