The machine that shakes the building

A fan on a roof runs at 900 rpm — 15 Hz — and its rotor is not perfectly balanced, so it applies a rotating force of a few hundred newtons to whatever it stands on. Bolted directly to the slab, that force goes into the building and is felt three floors down.

The remedy is to put the fan on springs, and the remedy has a trap in it.

The crossing at root two #

The transmissibility is

$$T = \sqrt{\frac{1 + (2\zeta\beta)^2}{(1-\beta^2)^2 + (2\zeta\beta)^2}}$$

and setting $\beta^2 = 2$ makes $(1-\beta^2)^2 = 1$, so the numerator and denominator become identical and $T = 1$ exactly, for every $\zeta$. It is not an approximation and it is not a coincidence of the algebra — it is the frequency at which the spring’s force and the inertia force are equal and opposite in a way that leaves the transmitted force unchanged.

The practical content is a design rule with a hard edge:

Below √2 the mount amplifies. At a ratio of 1.2 with 5% damping the transmissibility is 2.21 — the floor receives more than twice what it would have received without any springs. At the resonance itself it is 10.

Above √2 the mount isolates, and increasingly: 0.34 at a ratio of 2, 0.13 at 3, 0.047 at 5.

So a mount is either right or badly wrong, and the boundary is not where intuition puts it. A mount whose natural frequency is anywhere near the machine’s running speed is worse than no mount at all, and the natural frequency of a machine on springs is not something a supplier’s catalogue always makes obvious.

The static deflection is the specification #

There is one number a mount designer actually works with, and it comes from a rearrangement of the frequency formula. The natural frequency of a mass on a spring that deflects $\delta$ under its own weight is

$$f_n = \frac{1}{2\pi}\sqrt{\frac{g}{\delta}} \approx \frac{15.8}{\sqrt{\delta}}$$

with $\delta$ in millimetres — the same $18/\sqrt{\delta}$ family as a floor’s frequency, with a different constant because the mass is concentrated rather than distributed.

That makes the specification concrete. To get the 15 Hz fan to a ratio of 3, its mounts must have a natural frequency of 5 Hz, which needs a static deflection of 10 mm. To get to a ratio of 5, the mounts need 25 mm.

A vibration isolator is specified by how far it sags, and everything else follows. That is why isolators are catalogued by deflection, why a heavier machine on the same springs isolates better, and why the commonest installation error — a mount so stiff that it barely deflects — produces a machine that is bolted to the floor through an expensive spring.

Damping, which helps in one place and hurts in the other #

The reason is in the numerator of the transmissibility. A dashpot connects the machine to the floor directly, and a dashpot’s force is proportional to velocity, so it grows with frequency: at high ratios the dashpot is the main path by which force reaches the floor, and the spring has stopped being the problem.

That produces a genuine engineering trade with no clever answer:

A machine that starts and stops often has to pass through its mount’s resonance twice per cycle of operation. Passing through quickly helps — the build-up needs cycles, not seconds — and damping helps.

A machine that runs continuously wants as little damping as possible.

The usual compromise is a mount with 5–10% damping, and for the run-up problem the better answer is often a snubber: a device that does nothing at all during normal running and limits the motion only when the amplitude exceeds a threshold, so that the damping is present at resonance and absent at running speed.

The reason passing through quickly helps is the general one about rise times: a load applied over a time comparable with the natural period produces almost no dynamic effect at all, and a machine that accelerates through its mount’s resonance in a fraction of the build-up time never develops the response the steady-state curve threatens.

The same curve, read backwards #

That reading explains the identity the transmissibility curve hides: the fraction of a force that gets through a mount equals the fraction of a motion that gets through it.

Isolating a machine from a floor and isolating an instrument from a floor are the same calculation read in opposite directions. In the first, the source is the machine and the receiver is the building; in the second, the source is the building and the receiver is an electron microscope, an optical table or a recording studio. The equation does not distinguish them, so an isolation table for a microscope is specified in exactly the same way — by static deflection, with the same √2 warning, and with the same requirement that the mount’s frequency be well below whatever is shaking the floor.

Which is why the two problems have the same failure. A microscope table with a 5 Hz mount, in a building whose floor has a 5 Hz mode, amplifies exactly what it was bought to remove.

Two-stage isolation, and why it is used #

When a single mount cannot reach the ratio needed — because the machine’s speed is low, or the isolation demanded is severe — the answer is to isolate twice: machine on springs, on an inertia block, on a second set of springs.

The reason it works is worth stating, because it looks like a series arrangement and behaves like one only in part. Above both resonances the transmissibility of two stages falls as the fourth power of the frequency ratio rather than the second — so where a single stage at a ratio of 3 gives 0.13, two stages give something nearer 0.02. The price is a second resonance to pass through and a system with two closely spaced modes, which is the arrangement a mass on a spring on a spring always produces.

The inertia block, and why the floor is not rigid #

Everything above assumes the floor beneath the mounts does not move. It does, and the assumption fails in the case that matters most.

The mount’s isolation is really the ratio of the machine’s stiffness to the floor’s. If the floor is very stiff compared with the mounts, the calculation above holds. If the floor is a long-span composite deck with its own 5 Hz mode and a modest modal mass, the “rigid support” the springs are reacting against is itself a spring, and the two-degree-of-freedom system that results does not behave like either.

The standard remedy is a inertia block: a concrete raft, several times the machine’s mass, sitting on the isolators with the machine bolted to it. It works in three ways at once, and only the first is obvious.

It lowers the natural frequency for a given spring, because the mass is larger — improving the ratio.

It reduces the machine’s own motion, because the unbalanced force now accelerates a much larger mass. That matters for the machine’s alignment and bearings rather than for the building.

It presents a large mass to the floor, so that the floor’s own modal mass is no longer small compared with what is standing on it, which is the condition under which the rigid-support assumption becomes reasonable again.

The same floor plate that a walker excites at particular frequencies is the support a machine mount reacts against, so a mount tuned without reference to the floor’s own modes is a calculation about a support that does not exist. That is the argument for the block stated the other way round: it does not make the mount better, it makes the support real.

Why a slow machine is the hard one #

The specification by static deflection has a consequence the essay has not yet drawn out, and it decides which machines can be isolated at all.

The frequency depends on the sag and only on the sag: $f_n \approx 15.8/\sqrt{\delta}$, with no mass in it anywhere. So the achievable mount frequency is bounded by the sag anybody will tolerate, and the sag has practical ceilings — a machine that sits 50 mm lower when it is filled, whose pipework has to take that up, which rocks on its mounts and needs snubbers for wind and seismic cases, is a machine nobody wants. Twenty-five to fifty millimetres is about the limit in ordinary practice.

Twenty-five millimetres of sag is 3.2 Hz. To reach a frequency ratio of 3 from there, the machine has to run at 9.5 Hz — 570 rpm. Anything slower cannot be isolated to that ratio by any spring anybody is willing to install.

That is why low-speed machinery is the difficult case and high-speed machinery is nearly free. A 3,000 rpm pump is at 50 Hz and needs only 6 mm of sag to be at a ratio of 6; a 300 rpm compressor is at 5 Hz and cannot be given a ratio of even 2 without 100 mm of movement. The problems in a plant room cluster at the slow end, and no amount of care with the mount selection changes it.

There is a correction to make to the inertia block while the point is fresh, because the frequency formula makes one of its three benefits conditional.

An inertia block lowers the natural frequency for a given spring, which is what the earlier section said and is true. But springs are chosen to carry the load, so a machine on a five-times-heavier block is usually put on five times the spring stiffness — and the sag, and therefore the frequency, come out the same. The block does not lower the mount frequency unless the sag is allowed to increase, and the sag is exactly what was bounded above.

Which leaves the block’s other two benefits, and they are the real ones. It reduces the machine’s own motion, because the unbalanced force is now accelerating a far larger mass — that is what protects the machine’s bearings and its alignment. And it presents a large mass to the floor, so that the floor’s own modal mass is no longer small by comparison and the rigid-support assumption the whole transmissibility curve rests on becomes defensible again.

There is one route that does lower the frequency without more sag, and it is worth naming because it is what a difficult installation reaches for. An air spring carries its load on a volume of gas rather than on a deflected element, so its stiffness is set by the volume and the pressure rather than by how far it has sagged — which breaks the link between the two and puts frequencies of 1 to 2 Hz within reach at a working height that does not change. That is the reason air mounts appear under low-speed machinery, under vibration-sensitive instruments, and under vehicles that have to ride softly and sit level.

So an inertia block is not a way of getting a lower mount frequency. It is a way of making the machine behave and of making the calculation legitimate — and the frequency, which is what the isolation depends on, still comes from the sag and from nothing else.

The commonest way it is got wrong #

Three failures account for most unsatisfactory installations, and none of them is a calculation error.

The mount is chosen for the load rather than for the deflection. A supplier’s mount rated for the machine’s weight may deflect 3 mm under it, giving a natural frequency of 9 Hz, which for a 15 Hz machine is a ratio of 1.67 — barely past √2, transmitting 57% of the force. The right question is never “will it carry the machine” but “how far will it sag”.

The mounts are not equally loaded. A machine whose centre of gravity is not at the centre of its mount pattern sits with some springs compressed further than others, so the mount frequencies differ, the machine rocks, and the rocking modes are lower than the bouncing one. Levelling a machine by adjusting its mounts is a dynamic operation dressed as a fitting one.

Something rigid was left connected. Conduit, drain, duct, or a handrail welded to both the machine frame and the building. The mount’s transmissibility of 0.05 and the pipe’s transmissibility of 1.0 are in parallel, and parallel paths add.

All three share a shape worth noticing: the calculation was right and the installation was not, and no analysis of the design would have found any of them. This is a field where commissioning matters more than modelling, which is the same conclusion the tuned mass damper reached from a different direction.

What the picture cannot show #

Six degrees of freedom, not one. A machine on mounts can bounce, rock in two planes, and yaw, and each of those has its own frequency. The bouncing mode is the one the curve above describes; the rocking modes are usually lower and are the ones that produce the complaint. A mount layout is chosen to keep the rocking frequencies out of trouble, which is a geometry problem rather than a stiffness one.

The excitation is not one frequency. A rotating machine produces its running speed, twice it from misalignment, blade-passing frequencies, gear-mesh frequencies, and bearing defect frequencies — a whole comb. A mount that isolates the fundamental isolates the higher ones better, which is the one part of this that is easy.

Pipes, ducts and conduit bypass everything. A perfectly isolated machine connected to the building by a rigid pipe is not isolated. Flexible connections at every service are as important as the mounts and are more often forgotten, and the mechanism is the series-spring argument — the stiffest parallel path wins, and a steel pipe is very stiff.

Static deflection is not the whole of a mount. A rubber mount’s stiffness depends on frequency and temperature; a steel spring’s does not, but a steel spring transmits high frequencies through its own internal resonances, which is why spring mounts are usually fitted with a rubber pad in series.

Why this is a serviceability problem with a strength tail #

Almost everything in this essay is about comfort and function rather than about collapse, and it is worth saying where the exception is.

A badly isolated fan makes a building unpleasant. A badly isolated reciprocating machine — a compressor, a press, an engine — applies forces large enough to matter structurally, and a resonance can put a genuine multiple of them into a floor that was designed for a static equivalent. The tell is that the unbalanced force of a rotating machine goes as the square of the speed, so a machine that is acceptable at half speed can be ten times worse at full speed while nothing else about it changed.

There is a second tail, and it is fatigue. A mount running near its resonance transmits an amplified force at a rate of 15 per second, which is half a million cycles a day. Nothing about the magnitude has to be dramatic for that to matter to a welded detail, and machine-induced fatigue cracking in supporting steelwork is a recognised failure that leaves no trace on any static calculation.

Where the ladder goes #

The immediate rung down is the machine foundation as a whole: a heavy reciprocating machine on the ground, where the “floor” is soil, the support stiffness is a soil property and the damping is radiation into the half-space rather than anything material. It is the one place in structural engineering where the damping is both large and calculable.

The rung sideways is what to do when isolation is impossible because the machine must be rigidly connected — in which case the remedy moves to the receiver, and becomes an added mass on the structure rather than a spring under the source.

And the rung up is the observation that this whole essay is one figure read carefully. The transmissibility curve contains the isolation rule, the damping trade, the specification by deflection, and the identity between force isolation and motion isolation. It is the most economical picture in the field, and the crossing at √2 is the reason it repays being drawn rather than remembered.

It is also the field’s clearest instance of a general habit worth carrying out of it. The rule “put it on springs” is not wrong; it is a rule with a domain, and the domain has a boundary at a specific and calculable place. Almost every rule of thumb in this collection is like that — the equal-displacement rule, the check-the-furthest-bolt rule, “raise the frequency above 4 Hz” — and the useful question about any of them is never whether it is true but where it stops.