Question

For a dynamic system with single degree of freedom, choose the CORRECT statement(s).

• A. Damped natural frequency is greater than the undamped natural frequency.
• B. Dynamic response is always greater than static response at all excitation frequencies.
• C. Damping ratio can be estimated using logarithmic decrement method.
• D. A non-zero phase lag is always observed between the excitation and response for a damped system.

Hint:

Recall how damping changes the natural frequency and how the magnification factor behaves at high frequency ratios.

Answer 1

The Correct Option is C, D

Step 1: Check the frequency statement.
For a damped single degree of freedom system, $\omega_d = \omega_n \sqrt{1 - \zeta^2}$, and since $\zeta$ lies between 0 and 1 for an underdamped system, $\omega_d$ is always less than $\omega_n$.
So the claim that damped frequency is greater than undamped frequency is false.

Step 2: Check the response magnitude statement.
The magnification factor $M = \dfrac{1}{\sqrt{(1-r^2)^2 + (2\zeta r)^2}}$, where $r = \omega/\omega_n$, falls below 1 for large $r$.
This means the dynamic response becomes smaller than the static response at high excitation frequencies, so this statement is also false.

Step 3: Check the damping ratio and phase statements.
The logarithmic decrement $\delta = \ln(x_1/x_2)$ between two successive free vibration peaks connects directly to $\zeta$ through $\zeta = \delta/\sqrt{4\pi^2+\delta^2}$, so estimating damping ratio this way is valid.
The phase angle $\phi = \tan^{-1}\left(\dfrac{2\zeta r}{1-r^2}\right)$ is nonzero whenever $\zeta > 0$ and the system is excited, confirming a lag always exists for a damped system.

Final Answer:
Only the damping ratio and phase lag statements hold true.
\[ \boxed{\text{C, D}} \]