The magnitude of frequency response of an underdamped second-order system is 5 at 0 rad/sec and peaks at \(\frac{10}{\sqrt{3}}\) at 5√2 rad/sec. The transfer function of the system is

Concept:

DC gain = 5

\({M_r} = \frac{10}{\sqrt 3} = \frac{1}{{2\xi \sqrt {1 - {\xi ^2}} }}\)

100 [4ε² (1 – ε²)] = 3

400 ε²– 400 ε⁴ = 3

400x² – 400 x + 3 = 0  

x² – x + 0.0075 = 0

ξ² = 0.99244 , ξ² = 0.00755

ξ = 0.99 , ξ = 0.08689

If resonant peak > 1 then

 \(\zeta \le \frac{1}{{\sqrt 2 }} = 0.707\)

∴ ξ = 0.08689

\(5\sqrt 2 = {\omega _n}\sqrt {1 - 2{\xi ^2}} \)

\(5\sqrt 2 = {\omega _n}\sqrt {1 - 2 \times 0.00755} \)

ωn = 7 rad/s

\({s^2} + 2\xi {\omega _n}s + \omega _n^2\)

= s² + 1.21 s + 49

Now \({\left. {\frac{K}{{{s^2} + 1.21 s + 49}}} \right|_{s = 0}} = 5\) 

K = 49 × 5 = 245

So the required transfer function = \(\frac{{245}}{{\left( {{s^2} + 1.21s + 49} \right)}}\)