Every rotating machine — pumps, compressors, motors, HVAC fans — generates periodic forces that try to shake the structure it's mounted to. If the machine's forcing frequency matches a natural frequency of the support, you get resonance: amplitudes grow, bolts loosen, bearings die early, and the building shakes. Vibration isolation is the discipline of decoupling the source from the structure so this never happens.

The core trick: mount the machine on a soft spring (rubber pads, coil springs, air mounts) so the assembly has a natural frequency far below the operating frequency. The math comes from a simple single-degree-of-freedom model. The natural frequency of a mass on a spring is:

• f_n = (1/2π) × √(g/δ_st)
• where δ_st is the static deflection of the mount under the machine's weight

A handy rule of thumb: f_n (Hz) ≈ 15.76 / √δ_st(mm). So a mount that compresses 6 mm under load has a natural frequency around 6.4 Hz.

For isolation to work, the ratio of forcing frequency to natural frequency (r = f/f_n) must be greater than √2 ≈ 1.41. Below that, the mount actually amplifies vibration. Practical designs target r ≥ 3, which gives ~90% isolation (only 10% of the force is transmitted). Higher r is better but means softer mounts and more sag.

Real-world example: A 1750 RPM rooftop air handler vibrates at 1750/60 = 29.2 Hz. To get r = 3, you need f_n ≤ 9.7 Hz, which means static deflection of at least (15.76/9.7)² ≈ 2.6 mm. Builders typically spec spring isolators with 25 mm deflection for rooftop equipment — giving f_n around 3.2 Hz and isolation efficiency above 98%. That's why you can stand next to a roaring rooftop unit and feel nothing through the deck below.

Things that bite engineers:

• Startup/shutdown — the machine passes through resonance every time it spools up. Brief but real; flexible piping connectors are mandatory.
• Damping vs. isolation tradeoff — more damping helps at resonance but hurts isolation at high frequencies.
• Rigid attachments — one hard-piped conduit or copper line can short-circuit the entire isolation system. Use flex connectors everywhere.
• Floor flexibility — if the supporting floor itself is springy, the simple model breaks down; you need a two-mass analysis.