Understanding the Transfer Function

The given transfer function is:

\(G\left( s \right) = \frac{{36}}{{{s^2} + 4.2s + 36}}\)

This is the transfer function of a typical second-order linear time-invariant (LTI) system. These systems are commonly represented by a standard form.

Standard Second-Order System Form

The standard form of a second-order system transfer function is generally given by:

\(G(s) = \frac{{K\omega_n^2}}{{{s^2} + 2\zeta\omega_n s + \omega_n^2}}\)

Where:

• \(\omega_n\) is the undamped natural frequency.
• \(\zeta\) is the damping ratio.
• \(K\) is the DC gain (assuming the numerator is \(K\omega_n^2\)).

The term \(\omega_n\) represents the frequency at which the system would oscillate if there were no damping (\(\zeta = 0\)).

Determining the Value of \(\omega_n\)

To find the value of \(\omega_n\) for the given transfer function, we need to compare its denominator with the denominator of the standard second-order form.

Given denominator: \(s^2 + 4.2s + 36\)

Standard denominator: \(s^2 + 2\zeta\omega_n s + \omega_n^2\)

By comparing the constant terms in the denominators, we can equate them:

\(\omega_n^2 = 36\)

To find \(\omega_n\), we take the square root of 36:

\(\omega_n = \sqrt{36}\)

Since natural frequency is a positive value, we consider the positive root:

\(\omega_n = 6\)

We can also compare the coefficient of the 's' term to find the damping ratio, though it's not required by the question:

\(2\zeta\omega_n = 4.2\)

Substituting the value of \(\omega_n = 6\):

\(2\zeta(6) = 4.2\)

\(12\zeta = 4.2\)

\(\zeta = \frac{4.2}{12} = 0.35\)

The damping ratio is 0.35, indicating the system is underdamped.

The question specifically asks for the value of \(\omega_n\).

Based on our comparison and calculation, the value of \(\omega_n\) is 6.

Final Answer Confirmation

Comparing our calculated value with the provided options:

The value \(\omega_n = 6\) matches option 4.