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?

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

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