In vibration isolation for a lightly damped single-degree-of-freedom system, what is the force transmissibility (transmitted force divided by exciting force) when the frequency ratio r = ω/ωₙ equals 2?
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A0.5
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B1
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C1.5
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D2
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E0.33
Answer
Correct Answer: 0.33
Explanation
Introduction / Context
Force transmissibility T in vibration isolation quantifies how much dynamic force passes through a mount from a vibrating machine to its foundation. Designers aim for T < 1 (isolation) by operating above resonance (r > √2) with appropriate damping.
Given Data / Assumptions
- Single-degree-of-freedom mass–spring–damper.
- Light damping (typical isolation mounts).
- Frequency ratio r = ω/ωₙ = 2.
Concept / Approach
The standard formula for force transmissibility is T = √(1 + (2ζr)²) / √((1 − r²)² + (2ζr)²), where ζ is damping ratio. For light damping (say ζ ≈ 0.05–0.1), r = 2 places the system well into the isolation region, making T significantly below unity and near one third.
Step-by-Step Solution
1) Substitute r = 2 into the formula: numerator √(1 + (4ζ)²); denominator √((1 − 4)² + (4ζ)²) = √(9 + (4ζ)²).2) For ζ = 0.05: numerator ≈ √(1 + 0.04) ≈ 1.02; denominator ≈ √(9 + 0.04) ≈ 3.00; hence T ≈ 1.02 / 3.00 ≈ 0.34.3) For ζ = 0.1: numerator ≈ √(1 + 0.16) ≈ 1.08; denominator ≈ √(9 + 0.16) ≈ 3.03; hence T ≈ 0.36.4) Light-damping values cluster near one third, commonly rounded to 0.33.Verification / Alternative check
As ζ → 0, T → 1/|1 − r²| = 1/3 ≈ 0.333 for r = 2, matching the isolation-limit estimate.
Why Other Options Are Wrong
- 1, 1.5, 2: imply no isolation or amplification; inconsistent with r = 2 in light damping.
- 0.5: higher than the theoretical undamped value 1/3; too conservative.
Common Pitfalls
- Using displacement transmissibility instead of force transmissibility; forms differ near resonance.
- Assuming heavy damping always improves isolation; excessive ζ increases numerator and can degrade high-frequency isolation.
Final Answer
0.33