Question

The equation for resonant peak of second order system whose transfer function \( \frac{\omega_n^2}{s^2+2\zeta\omega_n s + \omega_n^2} \) is given by (The question asks for the equation *for* the resonant peak \(M_r\), not the resonant frequency \(\omega_r\).)

• A. \( \frac{\omega_n}{2\zeta} \)
• B. \( \frac{1}{2\zeta\sqrt{1-2\zeta^2}} \) (This looks like \(M_r = 1/(2\zeta\sqrt{1-\zeta^2})\) perhaps, or related to resonant frequency. Option b in image: \( \frac{1}{2\zeta\sqrt{(1-2\zeta^2)}} \) )
• C. \( \frac{1}{2\omega_n\sqrt{(1-2\omega_n^2)}} \)

Hint:

For a standard second-order system \(H(s) = \frac{\omega_n^2}{s^2+2\zeta\omega_n s + \omega_n^2}\):
Resonant frequency \(\omega_r = \omega_n \sqrt{1-2\zeta^2}\) (exists for \(0<\zeta<1/\sqrt{2}\)).
Resonant peak magnitude \(M_r = |H(j\omega_r)| = \frac{1}{2\zeta\sqrt{1-\zeta^2}}\) (for \(0<\zeta<1/\sqrt{2}\)).
Be aware of common formulas and potential variations or errors in question statements/options.

Answer 1

The Correct Option is B

The option in the image that is check-marked is (b): \( \frac{1}{2\zeta \sqrt{1 - 2\zeta^2}} \)

Standard Formula

The standard formula for resonant peak \( M_r \) is:

\[ M_r = \frac{1}{2\zeta \sqrt{1 - \zeta^2}} \]

This is valid for \( 0 < \zeta < \frac{1}{\sqrt{2}} \approx 0.707 \).

Resonant Frequency

The resonant frequency \( \omega_r \) is given by:

\[ \omega_r = \omega_n \sqrt{1 - 2\zeta^2} \]

This also only exists for \( 0 < \zeta < \frac{1}{\sqrt{2}} \).

Evaluation of Option (b)

Option (b) is given as:

\[ \frac{1}{2\zeta \sqrt{1 - 2\zeta^2}} \]

This is not the standard formula for resonant peak \( M_r \), but the square root term here matches the one from the resonant frequency formula \( \omega_r \). Hence, it appears the option confuses the formulas.

Boundary Case Check

• As \( \zeta \to 0 \), \( M_r \to \infty \) — this matches both the standard and incorrect formulas.
• At \( \zeta = \frac{1}{\sqrt{2}} \), the term \( 1 - 2\zeta^2 = 0 \), making the denominator zero, so \( M_r \to \infty \) — which is incorrect since \( M_r = 1 \) at that point.

Conclusion

Given the standard definition of resonant peak magnitude \( M_r \), the correct formula is:

\[ \frac{1}{2\zeta \sqrt{1 - \zeta^2}} \]

Option (b) is incorrect for this definition. There is likely an error in the options or a misinterpretation of what the question asks. If it asks for \( \frac{\omega_r}{\omega_n} \), then the square root term in (b) would be appropriate, but not for \( M_r \).

Final Note: Option (b) is marked, but it does not match the standard formula for resonant peak. The question or options may be flawed.