Stand near a running washing machine on a spin cycle and you can feel the floor buzz. Stand near a precision lab next door to a heavy press, and that same buzz can blur a microscope image or spoil a measurement. The vibrating machine is the source; the building is the victim. Between them sits a deceptively simple design problem: how do you let a machine shake without letting the shaking spread?
The answer is almost never to make the mounting stiffer. It is usually to make it softer, in a controlled way. This article explains why, introduces transmissibility as the number that quantifies the leak, works a clean example, and lists the traps that catch engineers who tighten bolts when they should be loosening springs.
Why this calculation matters
Rotating and reciprocating machinery — motors, pumps, compressors, fans, engines — produces oscillating forces at its operating frequency. Left unmanaged, those forces travel straight into the supporting structure, where they become noise, fatigue loading, and disturbance to anything sensitive nearby. Isolation mounts are the standard fix, but a mount sized by guesswork can easily make things worse.
That is the part beginners miss. A poorly chosen isolator does not simply fail to help; it can amplify the force passed to the floor several times over. The difference between isolation and amplification comes down to one ratio, and getting that ratio on the correct side of a threshold is the whole job. A few minutes with the transmissibility relationship saves you from installing mounts that resonate.
The core formula
Model the machine as a mass on a spring — a single-degree-of-freedom system. It has a natural frequency f_n set by its mass m and the mount stiffness k:
f_n = (1 / (2*pi)) * sqrt( k / m )
The machine drives the system at a forcing frequency f. What matters is the ratio of the two:
r = f / f_n
The key output is transmissibility, TR — the fraction of the disturbing force that actually reaches the floor. For an undamped isolator above resonance:
TR = 1 / | r^2 - 1 |
Read this carefully, because it has three distinct regions. When r is below 1, the mount is stiffer than the forcing and TR is close to 1 — the force passes straight through, no isolation. When r is near 1, the denominator collapses toward zero and TR becomes very large — this is resonance, and it amplifies the force, sometimes dramatically. Only when r climbs above sqrt(2), about 1.41, does TR drop below 1 and genuine isolation begins. Beyond that, the larger r gets, the more force is blocked.
This is why the cure is a softer mount, not a stiffer one. Softening the mount lowers f_n, which raises r, which pushes you deeper into the isolation region. Stiffening does the opposite and can drag you back toward resonance.
Damping helps you survive the resonance peak — important during run-up and shut-down when the machine sweeps through f_n — but in the isolation region heavy damping slightly worsens transmissibility. Practical isolators balance the two.
A worked example
Consider a machine running at a forcing frequency f = 30 Hz, mounted on isolators chosen to give a natural frequency f_n = 10 Hz.
Step 1 — compute the frequency ratio.
r = f / f_n = 30 / 10 = 3
Step 2 — check that we are in the isolation region. Isolation requires r greater than sqrt(2) ≈ 1.41. Here r = 3, comfortably above the threshold, so the mount will isolate rather than amplify.
Step 3 — compute transmissibility for the undamped case.
TR = 1 / | r^2 - 1 | = 1 / | 9 - 1 | = 1 / 8 = 0.125
So only about 12.5% of the vibrating force reaches the floor; roughly 87.5% is isolated. With a frequency ratio of 3 — a mount three times softer in frequency terms than the forcing demands — the machine can shake freely while the building barely notices.
Notice how much leverage the ratio carries. If the mount had been stiffer, with f_n = 30 Hz, then r = 1 and the system would sit exactly on resonance — the worst possible choice. The same machine, the same floor, only a different spring, and the outcome flips from "isolated" to "amplified."
Common mistakes
Making the mount stiffer to "hold the machine steady." This is the most common and most damaging instinct. A stiffer mount raises f_n, lowers r, and moves you toward resonance. For isolation you generally want a soft mount and a low natural frequency.
Designing for r below sqrt(2). Anything in the region r less than about 1.41 transmits as much force as a rigid bolt or more. If your frequency ratio lands there, the mounts are doing nothing useful — and near r = 1 they are actively harmful.
Ignoring start-up and coast-down. A machine that runs steadily above resonance still has to pass through f_n every time it starts and stops. Without enough damping, that brief pass can produce a large transient. Damping is what you size for those moments, even if it costs a little isolation at running speed.
Assuming more damping is always better. In the isolation region, above sqrt(2), extra damping slightly raises transmissibility. Damping tames the resonance peak; it does not improve steady-state isolation. The right amount is a compromise, not a maximum.
Forgetting static deflection and load sharing. A very soft mount sags under the machine's weight, and the sag must stay within the mount's travel. With multiple mounts, an uneven load means each one sees a different deflection and a different f_n. Both effects need a check, not just the transmissibility number.
Try the interactive NovaSolver calculator
Trading off mass, frequency, and stiffness in your head is hard; seeing the resonance peak and the isolation region on a chart makes it intuitive. The Vibration Isolation Mount Design Calculator on NovaSolver back-calculates the mount stiffness you need from the equipment mass, the excitation frequency, and a target natural frequency. Enter those along with a damping ratio and the number of mounts, and it returns transmissibility, insertion loss in decibels, frequency ratio, total and per-mount stiffness, static deflection, and load per mount — with a live curve that marks the danger zone near resonance.
Related calculators
- Advanced vibration isolation — for cases that need damped transmissibility, multi-stage mounts, or base-excitation models.
- SDOF response calculator — to see the full time-domain response of the spring-mass-damper system behind an isolator.
- Resonance frequency calculator — to pin down the natural frequency you must keep your forcing away from.
You can browse the full set in the vibration tools hub.
Closing note
Vibration isolation rewards a counterintuitive instinct: when a machine shakes the floor, soften the connection rather than tighten it. The whole design collapses to one ratio — forcing frequency over natural frequency — and one threshold, sqrt(2), above which isolation begins. Keep the ratio comfortably above that, add just enough damping to survive run-up, check the static deflection, and a noisy machine can run hard while the building stays quiet.
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