Vibration isolator resonance

Why can putting a machine on soft rubber mounts make it shake the floor harder than bolting it down did?

A pump bolted straight to a concrete plinth hums through the building, so the obvious remedy is to soften the connection and lift it onto rubber mounts. The hum at running speed does fall away — but now every start-up shakes the structure hard enough to bring complaints from two floors up, and one speed on the way to full load is worse than anything the rigid installation produced.

Nothing about the machine changed; the out-of-balance force in the rotor is what it always was. So how can reducing the stiffness of the path to the floor increase what travels along it?

Ask what a mount actually creates. A mass on a spring is not a softer connection; it is a new oscillator with a natural frequency of its own. The rigid bolted case is that same oscillator with an enormously stiff spring, its natural frequency far above anything the machine emits. Fitting rubber does not remove that frequency. It drags it down, into the range the machine works in.

We can put a number on where it lands, because the mount's static squash under the machine's own weight already encodes its stiffness. The natural frequency is the square root of stiffness over mass, and the weight doing the squashing is that same mass times g, so the mass cancels: it is the square root of g over the static deflection. A mount sinking 25 mm gives the square root of 9.81 over 0.025 — 19.8 radians per second, or 3.15 Hz. Bolted to concrete the deflection is microns and the figure is hundreds of hertz.

Now the question becomes a comparison, not a stiffness. Call r the ratio of driving frequency to that natural frequency. Well below r = 1 the disturbing force reaches the floor essentially unchanged. Near r = 1 the pushes arrive in step with the mount's own rhythm and the motion builds, limited only by damping — the floor now receives more than the machine emits. Well above it the mass cannot keep up, moves little, and the spring transmits little.

Where is the crossing? Not, as intuition suggests, at r = 1. Working the transmissibility expression through, the transmitted force equals the applied force exactly at r = the square root of 2, and that point is the same for every damping value. Isolation is therefore not a property a mount possesses; it is a property of the operating point relative to the mount.

Our 3.15 Hz mount under a machine at 1500 rpm — 25 Hz — gives r near 7.9, and the transmitted fraction falls to about two per cent. A stiffer mount tuned to 20 Hz gives r = 1.25, inside the amplification region, and transmits roughly 1.75 times the applied force. That mount is worse than no mount — and it is exactly what a well-meaning fitter chooses when told to use something firm enough that the machine will not wobble.

And there is no way to reach r = 7.9 without passing through r = 1. Every start and every coast-down traverses the resonance. That is the run-up complaint.

The load-bearing claim is that transmissibility depends on the ratio of driving to natural frequency, crosses unity at the square root of two for any damping, and exceeds unity throughout the band below it. The test is a speed sweep: instrument the plinth, run the machine slowly up to speed, and record force reaching the floor against rpm. The account predicts a pronounced peak at a speed set only by the mount and the mass, a crossing back through the rigid-mounting level at about 1.41 times that speed, and steady improvement above. It also predicts an awkward asymmetry — a more heavily damped mount must lower the run-up peak and raise the transmitted force at running speed, because a damper is a second force path in parallel with the spring and that path does not soften with frequency. The refuting observation would be added damping improving both at once, or a crossover that moved when damping changed.

The misconception worth naming is that the mount absorbs vibration. It does not, except for the small part damping dissipates as heat. It is a filter: above the crossover the machine keeps its motion and the floor is simply not told about it, which is why an isolated machine often visibly shakes more than a bolted one while being heard less.

Which constraint really binds? Not isolation: any required attenuation can be bought with a softer mount. What binds is everything softness costs — static deflection large enough to strain pipework, a machine that rocks during load changes, alignment that will not hold, seismic cases a compliant support handles badly. The design lands on the softest mount those allow, and the run-up excursion is the accepted failure: not eliminated but made survivable by keeping the peak brief, since amplitude takes many cycles to build and a briskly accelerating machine never reaches the steady swing the curve shows.

Two honest limits. The single-mass, single-spring picture treats the machine as rigid and moving in one direction, whereas a real one has six rigid-body modes on its mounts, several of which can land in the working range separately. And it assumes the floor is immovable; on a suspended slab whose natural frequency is near the mount's, the two interact and the result can be far worse than the calculation promises.

• Why a heavy inertia block under the machine lowers the resonance and shrinks the run-up swing at once.
• How a tuned mass damper differs from an isolator: one drains an oscillation, the other declines to transmit it.

Every term the collection defines is gathered in the glossary.