VIBRATION & ISOLATION

Single-DoF vibration is governed by m̈x + cẋ + k·x = F(t). Natural frequency ωn = sqrt(k/m). For harmonic excitation at frequency ω, the transmissibility TR = (force transmitted to base) / (force applied) depends on the frequency ratio r = ω/ωn and damping ratio ζ. Vibration isolation requires r > sqrt(2); below that ratio the isolator amplifies rather than reduces force. Isolator natural frequency should be at most 1/3 of the lowest forcing frequency.

• ωn = sqrt(k/m)
• f_n = (1/2π)·sqrt(g/δ_static)
• ζ = c / (2·sqrt(k·m))
• TR = sqrt(1 + (2ζr)²) / sqrt((1-r²)² + (2ζr)²)
• r > sqrt(2) for isolation

References & StandardsISO 2017, Rao "Mechanical Vibrations" Ch. 3

Mount vibrating equipment (pumps, compressors, generators) on isolators when the structure or surrounding equipment can't tolerate force or motion. Pick spring rate so f_n is well below the lowest excitation frequency. Add damping to limit amplitude during run-up through resonance. For seismic, design to a target natural frequency in the dominant earthquake band.