Discussion
Match the quantity with its expression.
1. Natural circular frequency 2. Critical damping coefficient 3. Damped frequency
P. \( 2\sqrt{km} \) Q. \( \sqrt{k/m} \) R. \( \omega_n\sqrt{1-\xi^2} \)
Answer: C
Correct answer: 1-Q, 2-P, 3-R
The three expressions build on one another. Natural circular frequency is \( \sqrt{k/m} \), depending only on stiffness and mass.
Critical damping is \( 2\sqrt{km} \) - note the product rather than the ratio, and the factor of two. Damped frequency is \( \omega_n\sqrt{1-\xi^2} \), the undamped value reduced by an amount that is negligible at practical damping. Recognising which combination of \( k \) and \( m \) each uses settles the first two at once.
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