Question

In vibration isolation system, if \(\omega_r\) is less than \(\sqrt{2}\), then for all values of the damping factor, the transmissibility is

• A. less than unity, but greater than zero
• B. equal to unity
• C. greater than unity
• D. zero

Hint:

For effective vibration isolation, \(\omega_r>\sqrt{2}\) is preferred.

Answer 1

The Correct Option is C

In a vibration isolation system, the concept of transmissibility is crucial. Transmissibility, \( T \), refers to how well a system isolates vibrations and is defined as the ratio of the output amplitude to the input amplitude. 

The expression for transmissibility in terms of the frequency ratio \( \omega_r = \frac{\omega}{\omega_n} \) (where \( \omega \) is the forcing frequency and \( \omega_n \) is the natural frequency of the system) and damping factor \( \zeta \) is given by:

\[ T = \sqrt{\frac{1 + (2\zeta\omega_r)^2}{(1 - \omega_r^2)^2 + (2\zeta\omega_r)^2}} \]

For \( \omega_r < \sqrt{2} \), we can analyze the condition for transmissibility:

When damping is low, the peak occurs at frequencies lower than the natural frequency, making \( \omega_r \) important. For \( \omega_r < \sqrt{2} \), especially as it approaches 1, we generally find \( T > 1 \).

For higher damping levels, while the amplitude is decreased, transmissibility \( T \) remains greater than 1 due to the dominance of the denominator, specifically the \((1-\omega_r^2)^2\) term when close to resonance.

Thus, considering all damping factors and given \( \omega_r < \sqrt{2} \), the transmissibility \( T \) remains greater than unity.

The correct conclusion: greater than unity.