Isolating a machine from the floor

vibration isolationisolator selectiontransmissibilityspring mount sizinghow much vibration gets through

Choosing isolators for a machine: the unbalance force, the mount's natural frequency, transmissibility, and the isolation you can claim.

Rotating Unbalance Force

F = m_u \, e \, \omega^{2}

The rotating force a small unbalanced mass throws off, growing with the SQUARE of speed. The reason a rotor that runs smoothly at half speed can tear its mounts apart at full speed.

Natural Frequency from Static Deflection

f_n = \frac{1}{2\pi} \sqrt{\frac{g}{\delta_{st}}}

The natural frequency read off the amount the mount sinks under the weight it carries — no spring rate needed, because the weight already measured it. The fastest sanity check in isolation work.

Undamped Natural Frequency

f_n = \frac{1}{2\pi} \sqrt{\frac{k}{m}}

The frequency a mass on a spring returns to when nothing is driving it: stiffness over mass, square-rooted, divided by 2π. Every other number in vibration work is measured against this one.

Damping Ratio from the Damping Coefficient

\zeta = \frac{c}{2\sqrt{k m}}

The dashpot's rating measured against the one value that would stop the oscillation dead: ζ = c/c_c, where c_c = 2√(km). The number that decides whether a system rings, creeps back, or does something in between.

Force Transmissibility

TR = \sqrt{\frac{1 + (2\zeta r)^{2}}{(1 - r^{2})^{2} + (2\zeta r)^{2}}}

The fraction of a machine's shaking force that reaches the floor through its mounts, as a function of the frequency ratio and the damping. Below a ratio of √2 it is greater than 1 — the isolator amplifies.

Vibration Isolation Efficiency

I = 1 - TR

Transmissibility restated the way a specification writes it: the percentage of the shaking force the mounts keep off the floor. A negative answer is the honest report that the mounts are making it worse.

How they fit together

The job is to stop a running machine from shaking the structure it sits on, and it is worth knowing at the outset that this set contains the most reliably counter-productive piece of engineering in the building trades. Rotating unbalance force is what you are fighting, and its ω² is why the problem grows so fast with speed: the same residual unbalance that is a nuisance at 900 rpm is four times the force at 1800. It is also the argument for balancing before isolating, since halving the eccentricity halves the force for the price of a balancing run, while isolators only redirect what is already there.

Two formulas give the mount's natural frequency and it is worth having both. Natural frequency from static deflection is the practical one, because isolator catalogues are organised by deflection under load — you are picking a mount that squashes 25 mm, not one with a stated k. Undamped natural frequency from k and m is the same answer from the design side. They agree, and having both means you can check a supplier's claim against the mass you are actually putting on it. Damping ratio then characterises the mount material: a steel spring is around 0.005, natural rubber 0.05, and a damped elastomer or a viscous mount higher still.

Transmissibility is where the set earns its place, and the rule is worth memorising before any of the arithmetic. An isolator only isolates when the frequency ratio r exceeds √2, roughly 1.41. Below that it makes things worse, and at r = 1 it is a resonant amplifier that can multiply the force by ten or more. A soft mount under a slow machine is not a partial success — it is an amplifier, and floors have been cracked this way. The design target is r of 3 or more, meaning a natural frequency at a third of running speed or below, which for a 1800 rpm machine means fn under 10 Hz and a fairly deflected mount. Damping is the second counter-intuitive part: more damping helps at resonance and hurts above √2, so a heavily damped mount is the right choice for a machine that must run up through its resonance on every start, and the wrong one for a machine that lives at constant speed well above it. Isolation efficiency is simply 1 − TR and exists to put the result in a sentence a client will read: 95% isolation sounds like a result, whereas a transmissibility of 0.05 sounds like a measurement.