Question

A second-order system is described by the equation

\(\frac{d^2 x}{dt^2}+5 \frac{dx}{dt}+7x=7y \)

The frequency and damping ratio respectively are:

1. 1 rad / sec and 5
2. 5 rad / sec and 7
3. 1 rad / sec and √7
4. √7 rad / sec and 0.94

Answer 1

Correct Answer - Option 4 : √7 rad / sec and 0.94

Concept:

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

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

The characteristic equation is given as:

\({s^2} + 2ζ {ω _n} + ω _n^2 = 0\)       ----(1)

Where,

ζ is the damping ratio

ωn is the undamped natural frequency

Calculation:

\(\frac{d^2 x}{dt^2}+5 \frac{dx}{dt}+7x=7y \)

Taking the Laplace transform of both sides:

s²X(s) + 5sX(s) + 7X(s) = 7Y(s)

\(TF = \frac{{Y\left( s \right)}}{{X\left( s \right)}} = \frac{{7}}{{{s^2} + 5s +7}}\)

Characteristics equation of given system is:

s² + 5s + 7 = 0

From equation (1):

ω_n² = 7

ω_n = √7

2ζω_n = 5

ζ = 0.94

Hence, option 4 is correct.