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Vibration isolation: Frequency ratio for unity transmissibility (general damping). In a base-excited isolation system, transmissibility (force or motion) equals unity at a specific frequency ratio r = ω/ω<sub>n</sub>, independent of damping. Determine that frequency ratio.

Difficulty: Easy

r = 1
r = √2 (≈ 1.414)
r = 2
r < 1 for all damping
r > 2 for all damping

Correct Answer: r = √2 (≈ 1.414)

Explanation:

Concept/Approach
Transmissibility T for a damped single-degree system is T = √(1 + (2ζr)²) / √((1 − r²)² + (2ζr)²).Setting T = 1 and simplifying gives (1 + (2ζr)²) = (1 − r²)² + (2ζr)² ⇒ 1 = (1 − r²)² ⇒ r² = 2 (taking r > 0).

Result
Thus, independent of damping (ζ), T = 1 at r = √2.

Final Answer
r = √2 (≈ 1.414)

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