Vibration Isolation

The single-degree-of-freedom model

Almost every isolation problem starts from the same idealization: a rigid mass m sitting on a spring of stiffness k and a viscous damper of coefficient c. The machine (the mass) is shaken either by an internal force — an unbalanced rotor, a reciprocating piston — or by motion of the floor underneath. The equation of motion for the mass displacement x under a harmonic disturbance is the classic forced, damped oscillator:

m·ẍ + c·ẋ + k·x = F(t)

where:
  m = isolated mass            (kg)
  c = damping coefficient      (N·s/m)
  k = mount stiffness          (N/m)
  F(t) = disturbing force      (N), e.g. F₀·sin(ωt)

From this fall two numbers that decide everything. The undamped natural frequency f_n = (1/2π)·√(k/m) is the frequency at which the mass would bounce freely on the spring. The damping ratio ζ = c / (2√(km)) measures how quickly free oscillations die out. The disturbing frequency is f, and the single most important parameter in isolation is the dimensionless frequency ratio r = f / f_n.

Transmissibility: the master curve

The quantity an isolator is judged by is transmissibility — the ratio of transmitted to applied force (or transmitted to applied motion; the two are mathematically identical). Solving the oscillator for steady-state harmonic input gives:

          √[ 1 + (2ζr)² ]
T(r) = ──────────────────────────────
        √[ (1 − r²)² + (2ζr)² ]

undamped limit (ζ → 0):   T = 1 / |1 − r²|

Read the curve from left to right. At very low r (stiff mount, slow disturbance) T ≈ 1 — the force passes straight through as if the spring weren't there. As r climbs toward 1 the denominator collapses and T spikes: this is resonance, where an undamped mount would transmit an infinite force and even a lightly damped one can transmit 5× to 20× the input. At r = √2 (≈ 1.41) every curve, for any damping, passes through exactly T = 1 — the universal crossover. Only beyond √2 does T finally drop below 1 and the mount begins to isolate, eventually rolling off toward zero at high r.

This is the counterintuitive heart of the subject: a soft mount is not automatically a good mount. Put a machine on springs whose natural frequency happens to match a running harmonic, and you have built a resonator, not an isolator. Isolation is a high-frequency phenomenon that only switches on once you have placed the natural frequency safely below the disturbance.

The √2 rule and choosing f_n

Because isolation requires r > √2, the design recipe is simply "make the natural frequency low enough." Engineers rarely stop at √2, though — they push r to 3–5 to get well down the roll-off. The link between natural frequency and the physical mount is wonderfully direct. Under its own weight the mass deflects the spring by a static amount δ = mg/k, and substituting into f_n = (1/2π)√(k/m) eliminates both m and k:

f_n = (1/2π) · √(g/δ)

      ≈ 0.4985 / √δ     (δ in metres  → f_n in Hz)

practical form:  f_n ≈ 15.76 / √δ   (δ in millimetres → f_n in Hz)

So the only thing that sets the natural frequency is how far the load sags the mount. A mount that deflects 1 mm gives f_n ≈ 15.8 Hz; 10 mm gives ≈ 5 Hz; 25 mm gives ≈ 3.2 Hz. This is why low-frequency isolation always means soft, deeply-sagging springs — and why a stiff rubber pad that compresses only a fraction of a millimetre can never isolate low frequencies, no matter how "vibration-absorbing" the marketing claims.

Worked example: isolating a 1500 rpm fan

A 200 kg centrifugal fan runs at 1500 rpm. Its unbalance produces a once-per-revolution disturbing force. We want at least 90% isolation. What mount stiffness and static deflection do we need?

Disturbing frequency:
  f = 1500 rpm / 60 = 25 Hz

Target: 90% isolation → T = 0.10 (undamped, conservative)
  T = 1/(r² − 1)  for r > √2
  0.10 = 1/(r² − 1)  →  r² − 1 = 10  →  r² = 11  →  r ≈ 3.32

Required natural frequency:
  f_n = f / r = 25 / 3.32 = 7.5 Hz

Required mount stiffness (total, all mounts):
  k = (2π f_n)² · m = (2π·7.5)² · 200
    = (47.1)² · 200 = 2219 · 200
    ≈ 4.44 × 10⁵ N/m

Static deflection check:
  δ = mg/k = 200·9.81 / 4.44×10⁵
    = 1962 / 4.44×10⁵
    ≈ 4.4 mm

Cross-check with rule of thumb:
  f_n ≈ 15.76/√δ_mm = 15.76/√4.4 ≈ 7.5 Hz  ✓

The static-deflection shortcut confirms the result: with δ in millimetres, f_n ≈ 15.76/√δ_mm, so 15.76/√4.4 ≈ 7.5 Hz — exactly what the stiffness route gave. This is no coincidence; the shortcut is just the exact form f_n = (1/2π)√(g/δ) rewritten for millimetres, since (1/2π)√(g)·√1000 ≈ 15.76. Beware the common "5/√δ" version floating around: that constant only works for δ in centimetres, and silently under-predicts f_n by a factor of √10 if you feed it millimetres. Always anchor on the exact equation and keep the units straight.

So four spring mounts of ~111 kN/m each, each sagging about 4.4 mm under the fan, place f_n at 7.5 Hz, r at 3.3, and deliver 90% isolation at running speed — provided the fan can spin up through its 7.5 Hz resonance quickly and with enough damping to keep the transient bounded.

The damping trade-off

Damping is where intuition fails most people. Adding damping unambiguously helps near resonance — it caps the peak height, which matters every time the machine coasts down through f_n on shutdown. But above √2, damping hurts isolation, because the (2ζr)² term in the transmissibility numerator grows with frequency. The undamped mount rolls off at −40 dB/decade; a heavily damped mount degrades toward −20 dB/decade.

	Undamped (ζ ≈ 0)	Light damping (ζ ≈ 0.1)	Heavy damping (ζ ≈ 0.5)
Peak T at resonance	∞ (limited only by nonlinearity)	≈ 5.0	≈ 1.15
T at r = 3 (running speed)	0.125	0.14	0.39
High-frequency roll-off	−40 dB/decade	≈ −40 dB/decade	−20 dB/decade
Resonance behaviour on spin-up	Dangerous — large transient	Manageable transient	Well-controlled
Best for	Constant-speed, never crosses fn	General industrial machinery	Variable-speed sweeping through fn
Typical hardware	Air springs, soft coil	Coil + rubber, neoprene	Friction / wire-rope, viscous dampers

The standard compromise lands at ζ = 0.05–0.15: enough to keep the resonance peak in the 3×–10× range during start/stop, while preserving steep roll-off at operating speed. Where a machine sweeps through its natural frequency on every cycle (variable-frequency drives, vehicle suspensions), engineers accept higher ζ — or switch to a semi-active or active mount whose damping changes with operating point, getting the best of both regimes.

Types of isolator

• Steel coil springs. Large static deflection (10–100 mm) gives very low f_n (1–4 Hz), excellent for heavy rotating machinery. Almost no internal damping, so they need a parallel viscous or friction damper and are prone to surge resonances in the coils themselves.
• Elastomeric (rubber) mounts. Combine stiffness and material damping (ζ ≈ 0.05) in one part. Compact and cheap; ideal for f_n of 8–20 Hz. Stiffness drifts with temperature and creeps over years; rubber hardens and the isolation degrades.
• Air springs / pneumatic isolators. An air column gives extremely low f_n (down to ~1 Hz) and the natural frequency stays nearly constant as load changes — the foundation of optical-table and semiconductor-tool isolation. Need a compressor and leveling valves.
• Wire-rope (cable) isolators. Loops of stranded cable that dissipate energy through inter-strand friction. Tolerant of shock, temperature, and chemicals; common on military and aerospace equipment.
• Pneumatic + active (servo) isolators. Sensors and actuators cancel disturbance in real time, beating the passive √2 limit and isolating below f_n. Used where nanometre stability is required, at high cost and complexity.

Failure modes and trade-offs

• Running near resonance. The cardinal sin: a natural frequency that coincides with a running speed or a strong harmonic. The mount amplifies instead of isolating, mounts fatigue, and the structure shakes apart. Always check r against every excitation harmonic, not just 1× speed.
• Too-stiff mounts. Choosing a mount because it "feels solid" defeats the purpose — if r < √2 there is no isolation at all. Soft is the whole point.
• Rocking and coupled modes. The 1-DOF model assumes pure vertical bounce. Real machines also rock and sway, with their own (often lower) natural frequencies. Mounts placed below the center of gravity invite rocking resonances that the vertical analysis never reveals.
• Static overload and creep. Elastomer mounts creep under sustained load, lowering δ over years, raising f_n, and quietly degrading isolation until a machine that was fine on day one is shaking the floor a decade later.
• Insufficient damping through spin-up. A constant-speed design with near-zero damping is fine at speed but builds a huge transient every time it passes through f_n on start and coast-down, hammering the mounts and anything nearby.
• Short-circuiting the mount. A rigid pipe, conduit, or grout bridge that bypasses the isolator transmits force directly and renders the whole installation useless — a classic field defect on isolated pumps and chillers.

Force isolation versus motion isolation

Two distinct goals share one curve. In force (source) isolation the machine is the troublemaker — a generator, compressor, or punch press — and the mount protects the surrounding structure from the force it pumps out. In motion (base) isolation the floor is the troublemaker — traffic, footfall, nearby machinery — and the mount protects a delicate payload such as a microscope, interferometer, or wafer stage. Because the transmissibility expression is identical for transmitted force and transmitted motion, the same √2 rule, the same damping trade-off, and the same hardware serve both. An optical table floating on air springs and a diesel generator on coil mounts are solving mirror images of the same equation.