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.
Mechanical Engineering
Theory of machines
Difficulty: Easy
Choose an option
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Ar = 1
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Br = √2 (≈ 1.414)
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Cr = 2
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Dr < 1 for all damping
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Er > 2 for all damping
Answer
Correct Answer: r = √2 (≈ 1.414)
Explanation
Concept/ApproachTransmissibility 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).
ResultThus, independent of damping (ζ), T = 1 at r = √2.
Final Answerr = √2 (≈ 1.414)