Vibration Isolation from the Foundation Up: Base Excitation Transmissibility Explained

The problem: vibration coming through the floor

Most vibration problems people picture involve a rotating machine shaking its own base. Base excitation is the mirror image of that: the floor is already moving, usually because of a nearby compressor, a forging press two bays over, or highway traffic under a building slab, and the question is how much of that motion reaches the sensitive equipment sitting on top of it. A microscope stage, a coordinate measuring machine, or a wafer stepper does not care whether the shaking originates locally or arrives through the foundation. It only cares about the amplitude it ends up seeing.

The usual fix is a spring-damper isolator between the equipment and the floor: elastomer pads, coil springs, or an air spring table. Done correctly, the isolator can knock incoming vibration down by more than 95%. Done incorrectly, it can amplify the problem, because the same mount that isolates at high frequency acts as a mechanical amplifier near its own resonance. Getting this right is a matter of picking the isolator's natural frequency relative to the disturbance frequency, and that relationship is captured entirely in one dimensionless curve: the transmissibility function.

Setting up the model

Treat the isolated equipment as a single mass sitting on a spring and damper, with the base of that spring-damper pair being driven by the floor's motion rather than a force applied directly to the mass. Four quantities define the whole problem:

• System natural frequency fₙ — set by the isolator's stiffness and the supported mass ( fₙ = (1/2π)·√(k/m) )
• Damping ratio ζ — how much energy the isolator dissipates per cycle, typically 0.02–0.10 for steel springs and 0.05–0.15 for elastomer pads
• Excitation frequency f — the dominant frequency of the floor motion you are trying to block
• Base vibration amplitude — how much the floor itself is moving, in millimeters

Everything downstream, including how much the equipment actually moves, falls out of the ratio between the excitation frequency and the isolator's natural frequency.

The transmissibility formula

Define the frequency ratio first, since the rest of the analysis is written in terms of it:

r = f / fn

Transmissibility, the fraction of base motion that reaches the mounted mass, is:

T = sqrt(1 + (2*zeta*r)^2) / sqrt((1 - r^2)^2 + (2*zeta*r)^2)

This single equation explains almost every practical observation about isolators. At r much less than 1 (excitation far below the isolator's natural frequency), T approaches 1 — the mount is rigid at that frequency and does nothing useful. At r = 1, the denominator's first term vanishes and T spikes toward a peak controlled almost entirely by damping; this is the resonance amplification everyone is warned about. Past r = √2, T finally drops below 1, and it keeps falling as r increases — this is the isolation region, and it is why isolator designers push the natural frequency as low as practical relative to the disturbance.

The amplitude the equipment actually experiences is just the base amplitude scaled by T, and isolation efficiency is a more intuitive way to express the same number:

Isolation efficiency (%) = (1 - T) * 100

A worked example: isolating an instrument rack

Suppose a floor near a stamping line vibrates at 25 Hz with an amplitude of 0.20 mm — measured with an accelerometer during a normal production shift. An instrument rack needs to sit near that floor, and the goal is to keep its residual motion low enough not to disturb a laser alignment stage on top of it.

A soft elastomer isolator is selected with a natural frequency fₙ of 5 Hz and a damping ratio ζ of 0.05, typical for a medium-durometer pad carrying a moderate load.

First, the frequency ratio:

r = f / fn = 25 / 5 = 5.0

That puts the system well into the isolation region (r > 1.414), which is the whole point of choosing a soft mount. Plugging into the transmissibility formula:

numerator   = sqrt(1 + (2*0.05*5)^2) = sqrt(1 + 0.25) = 1.118
denominator = sqrt((1 - 25)^2 + 0.25) = sqrt(576 + 0.25) = 24.005
T = 1.118 / 24.005 ≈ 0.0466

So only about 4.7% of the floor's motion reaches the rack. Isolation efficiency works out to roughly 95.3%, and the mounted-mass amplitude is:

0.20 mm * 0.0466 ≈ 0.0093 mm

That is under 10 microns residual motion from a 200-micron input — a clean result for a laser stage. As a sanity check on how bad things would be without isolation, or if the isolator's natural frequency drifted up toward the excitation frequency, it is worth looking at the resonance amplification reference value: at r = 1 and ζ = 0.05, transmissibility spikes to roughly 10×. That is the fate of any isolator whose natural frequency is accidentally tuned close to the disturbance — a cheap rubber pad that seems fine at rest can turn into an amplifier once the floor starts shaking at the wrong frequency.

Practical guidance and common pitfalls

The most common mistake is picking an isolator stiffness based only on static deflection under load, without checking where fₙ lands relative to the actual disturbance spectrum. A mount that looks soft enough on paper can still land its natural frequency inside the dominant excitation band once the real payload mass is accounted for, especially when equipment is added or removed later in its life.

Damping is a double-edged parameter. More damping tames the resonance peak, which matters if the excitation frequency sweeps through fₙ during startup or shutdown, but too much damping degrades high-frequency isolation, because the (2ζr) term in the numerator grows with both ζ and r. For equipment that only ever sees a narrow, steady excitation band well above resonance, a lightly damped mount usually isolates better. For equipment that has to pass through resonance during a startup ramp, extra damping is worth the tradeoff.

A second pitfall is ignoring multi-frequency floor spectra. Real floors rarely vibrate at one clean tone; a 25 Hz line frequency component might sit alongside a lower-frequency structural sway. An isolator tuned for one frequency can still transmit a troublesome amount of energy at another. It is worth running the transmissibility check at several candidate excitation frequencies, not just the dominant one, before committing to a stiffness.

A third pitfall shows up with very low natural frequencies. Pushing fₙ down to improve isolation at a fixed excitation frequency means a softer spring, and a softer spring means more static deflection under the same supported weight. Below roughly 1–2 Hz, static deflection can become large enough that the equipment sits noticeably lower than intended, or that lateral stability becomes a problem — air springs and pneumatic isolators exist largely to get very low natural frequencies without the sag that a coil spring soft enough to match would produce. It is easy to chase a target transmissibility number on paper and end up specifying an isolator that is mechanically impractical to build.

Finally, remember that the frequency ratio cuts both ways. If a facility upgrades its stamping line and the disturbance frequency shifts, r changes, and an isolator that used to sit safely at r = 5 could end up back near r = 1.

Try it yourself

Running these numbers by hand is fine for a single check, but comparing several candidate isolators, or sweeping across a range of excitation frequencies to find the worst case, is much faster with a tool built for it. The Base Excitation Transmissibility Simulator lets you adjust natural frequency, damping ratio, excitation frequency, and base amplitude directly and watch the frequency ratio, transmissibility, mounted-mass amplitude, and isolation verdict update in real time — useful for narrowing down a mount stiffness before it goes on an order form. If you are also chasing sensor-side effects on the same structure, the accelerometer frequency response tool is a natural next stop.