Question

The dynamic response amplitude \( |H(\omega)| \) of a single degree of freedom system subjected to support motion is given by the following expression: \[ |H(\omega)| = \sqrt{\frac{1 + 4 \zeta^2 \left( \frac{\omega}{\omega_n} \right)^2 }{\left[ 1 - \left( \frac{\omega}{\omega_n} \right)^2 \right]^2 + 4 \zeta^2 \left( \frac{\omega}{\omega_n} \right)^2} } \] Where the damping ratio is \( \zeta \), the excitation frequency is \( \omega \), and the natural frequency of the system is \( \omega_n \). The amplitude \( |H(\omega)| \) increases with an increase in damping ratio (\( \zeta \)) if the excitation frequency (\( \omega \)) is ________ the natural frequency (\( \omega_n \)) of the system.

• A. equal to
• B. 0.75 times
• C. \( \frac{\sqrt{3}}{2} \) times
• D. greater than \( \sqrt{2} \) times

Hint:

When analyzing dynamic response amplitude under support motion, check how damping affects the system at various excitation frequencies. For frequencies significantly above the natural frequency, the response amplitude can increase with damping.

Answer 1

The Correct Option is D

Step 1: Understand the behavior of \( |H(\omega)| \) as a function of damping ratio \( \zeta \). 
We are given: \[ |H(\omega)| = \sqrt{\frac{1 + 4 \zeta^2 \left( \frac{\omega}{\omega_n} \right)^2 }{\left[ 1 - \left( \frac{\omega}{\omega_n} \right)^2 \right]^2 + 4 \zeta^2 \left( \frac{\omega}{\omega_n} \right)^2} } \] Let \( r = \frac{\omega}{\omega_n} \). We want to analyze how \( |H(\omega)| \) behaves with respect to \( \zeta \) for different values of \( r \). 
Step 2: Observe the trend. 
When \( r = 1 \), i.e., excitation frequency equals natural frequency, the expression simplifies but the behavior becomes complex due to the resonance peak.
When \( r>\sqrt{2} \), the numerator increases faster with \( \zeta \) compared to the denominator. Hence, \( |H(\omega)| \) increases with increasing \( \zeta \).
Step 3: Analyze each option.
(A) Incorrect: At \( r = 1 \), the denominator becomes minimal, and amplitude tends to reduce with increasing \( \zeta \).
(B) Incorrect: At \( r = 0.75 \), increasing \( \zeta \) leads to a lower amplitude.
(C) Incorrect: At \( r = \frac{\sqrt{3}}{2} \), amplitude still decreases with damping.
(D) Correct: For \( r>\sqrt{2} \), the increase in \( \zeta \) results in a larger amplitude.