If the damping ratio ζ is equal to 0 then what will be the maximum overshoot?

Question

Question

This question was previously asked in

DFCCIL Executive S&T 2016 Official Paper

Answer 1

Concept:

The transfer function of the standard second-order system is:

\(TF = \frac{{C\left( s \right)}}{{R\left( s \right)}} = \frac{{ω _n^2}}{{{s^2} + 2ζ {ω _n}s + ω _n^2}}\)

Characteristic equation: \({s^2} + 2ζ {ω _n} + ω _n^2 = 0\)

ζ is the damping ratio

ωn is the undamped natural frequency

\({M_p} = {e^{\frac{{ - ζ \pi }}{{\sqrt {1 - {ζ ^2}} }}}}\) ----(1)

Calculation:

Given:

ζ = 0

From Equation (1);

M_p = 1 i.e.

Mp = 100%

Note:

Mp is the maximum peak overshoot of the closed-loop transfer function

\(M_p \ \alpha \ \frac{1}{ζ}\)