In structural dynamics, the damped natural frequency \(\omega_d\) of a SDOF system with critical damping ratio \(\xi\) and undamped natural frequency \(\omega_n\) is:

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Architecture Engineering QUESTION #2341

Question 1

\(\omega_d = \omega_n (1 + \xi)\)
\(\omega_d = \omega_n \sqrt{1 - \xi^2}\)✔️
\(\omega_d = \omega_n \sqrt{1 + \xi^2}\)
\(\omega_d = \omega_n (1 - \xi)\)

Correct Answer Explanation

The damped natural frequency is \(\omega_d = \omega_n \sqrt{1 - \xi^2}\). For typical structural damping (\(\xi = 0.02\) to \(0.05\)), \(\omega_d \approx \omega_n\). At critical damping (\(\xi = 1.0\)), \(\omega_d = 0\) and the system returns to equilibrium without oscillating.

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