The damping that is radiated

The machine that shakes the building is about a machine on a floor, isolated from it by springs, and it ends by pointing downward: the same calculation with the floor replaced by the ground, where the support stiffness is a soil property and the damping is something else entirely.

This is that calculation, and the something else is the reason it is worth a page.

Forty-seven per cent of critical is not a number that appears anywhere else in this collection. Damping is the only thing that stops it makes the case that every response here is quoted with a damping assumption attached and that the assumption is a measurement with a factor of two in it. That is true of every structure and false of this one, and the reason is that none of this is dissipation.

Which free body produced the number #

The free body is the block, cut from the soil at the underside of the footing, with the machine’s unbalanced force on top of it.

Crossing the cut is a distribution of normal and shear tractions from the soil. Nothing else — no dashpot, no material with a loss factor, no friction surface. The soil under a foundation is very nearly elastic at the strains a machine produces, and an elastic material has no damping in it whatever.

What it has is extent. When the block moves down, it launches a compression wave into the half-space; when it moves back, it launches another. Those waves travel away and do not come back, because a half-space is infinite and has no boundary to reflect them from. Energy leaves the free body at a rate proportional to the velocity of the surface it left through, and a force proportional to velocity is a dashpot as far as the block is concerned.

So radiation damping is a geometrical coupling rather than a material property, and it can be computed. Lysmer’s analogues put a spring and a dashpot on each mode of a rigid circular footing, fitted to the exact elastodynamic solution, and every number on this page comes from them:

$$K_z = \frac{4Gr}{1-\nu}, \qquad \zeta_z = \frac{0.425}{\sqrt{B_z}}, \qquad B_z = \frac{(1-\nu)m}{4\rho r^3}$$

Three quantities appear: the footing’s size, the soil’s density and its shear modulus. There is no number in it that anybody has to measure by shaking something and watching it stop.

Why the three modes differ so much #

The hierarchy in the figure — vertical, then horizontal, then rocking — is not a curiosity. It is the whole design problem, and its cause is visible without any arithmetic.

Vertical motion pushes the entire footprint the same way at once. The whole half-space under the block is compressed and released together, so a large volume of soil is involved and the wave field carries a great deal of energy away.

Horizontal motion is the same argument with shear waves, which are slower and carry less energy for the same velocity, so the damping is somewhat lower.

Rocking is different in kind. One half of the footing goes down while the other goes up, so at any distance the two wave fields are out of phase and cancel. Very little reaches the far field, and very little energy is lost. A rocking footing is a dipole and a heaving one is a monopole, which is exactly the distinction that makes a loudspeaker need a baffle.

Which produces the practical rule: a machine foundation is checked for rocking. The vertical mode has so much damping that resonance is not a meaningful concept for it, and the rocking mode behaves like the ordinary lightly damped oscillator everything else in this field assumes.

The trade that mass makes #

Every curve in the first figure falls as the block gets heavier, and that is the one result here that surprises people who have designed foundations by rule.

The traditional rule is a mass ratio: make the block three to five times the mass of the machine. Its purpose is to lower the natural frequency well below the machine’s operating speed, and it does that — frequency goes as $\sqrt{K/m}$ and $K$ does not change when the block gets heavier.

What it also does is lower the damping, and by the same square root: the mass ratio $B$ is proportional to $m$ and $\zeta$ goes as $1/\sqrt{B}$. Five times the mass is 47 per cent down to 21, and the rocking mode from 13 per cent to 2.

So a heavy block is well isolated and lightly damped, and a light one is poorly isolated and heavily damped. Which of those is wanted depends on whether the machine runs steadily above the resonance — in which case the isolation is what matters and the damping is a nuisance at start-up — or whether it runs through the resonance every time it starts, in which case the damping is what carries it through.

The escape from the trade is in that figure and it is the useful design move: damping is bought with footprint and lost with mass, because $B$ goes as $m/r^3$. A block spread wider at the same weight is better on both counts, and it is why machine bases are the shape they are — broad, shallow, and much larger in plan than the machine standing on them. It is the same reasoning that decides a footing on a compressible soil, arrived at from a dynamic argument rather than a settlement one.

The soil, which is the uncertain half #

Two soil numbers enter and their uncertainties are very different.

Density is known to a few per cent, always.

The shear modulus is not. It falls with strain amplitude by a factor of two or three between the very small strains of a wave and the strains under a footing at working load, and the value to use is a small-strain modulus from a seismic test rather than anything a triaxial gives. On a job with no such test it is estimated from an SPT count, and the estimate has a factor of two in it.

But the damping goes as $\sqrt{G}$ through the wave speed and inversely as $\sqrt{m}$, so a factor of two in $G$ is a factor of 1.4 in $\zeta$ — and the frequency, which goes as $\sqrt{K/m} \propto \sqrt{G}$, moves by the same 1.4. The uncertain quantity enters both answers as a square root, which is why a machine foundation calculation with a badly known soil is still worth doing and a damping assumption with a factor of two in it is not.

Putting the two halves together #

The design of a machine block is two questions and this page has answered the second of them. Setting them side by side is what makes the answer usable.

Where is the natural frequency? From the soil spring and the block’s mass: 12.5 Hz vertically on the block drawn, 15.0 in rocking. A machine running at 1,500 rpm forces at 25 Hz, so it is above both — the arrangement every machine foundation aims for, because above resonance the transmissibility falls with frequency, and the crossing at √2 that the rung below is built around is the boundary between helping and hurting.

And what happens on the way there? The machine passes through 12.5 and 15 Hz every time it runs up, and the amplification at those instants is what the damping decides. At 47 per cent there is no peak to pass through at all. At 13 per cent there is one of 4.1, and it lasts as long as the run-up takes.

That pairing is the whole of the design conversation, and it is worth stating in the general form because it recurs: isolation is a steady-state property and damping is a transient one. Mass alone improves the first and spoils the second; footprint improves both. Which is why the useful instruction is not “make the block heavier” but “make it heavier by making it wider”, and why a block deepened rather than spread is the arrangement that goes wrong.

What a hammer does instead #

Everything above is harmonic — a machine with an unbalanced rotor, forcing at a frequency. The other half of the subject is a machine that hits things, and the same three numbers are used in a completely different way.

A forge hammer or a press delivers an impulse. There is no frequency ratio and no transmissibility; what matters is the peak displacement of the block after the blow and how fast it comes back to rest before the next one. Both are read off the same spring, mass and dashpot, and the damping is now doing the whole job rather than deciding a peak.

At 47 per cent of critical the block returns to rest in a fraction of a cycle, and at 13 per cent in a few — so for an impulsive machine the vertical mode’s enormous damping is the design’s principal asset rather than an incidental comfort. It is also why hammer foundations are made broad and shallow while press foundations are made heavy: the first is designed to stop moving and the second to move as little as possible.

The load that is over before it has moved is the same distinction made about a structure and a blast: whether an action is a force or an impulse is decided by comparing its duration with the structure’s own period, and everything downstream of that comparison is different.

What this says about the assumption everywhere else #

There is a wider point in having one calculable case in a field of assumed ones, and it is worth taking.

A structure’s 2 to 5 per cent is a compound of things nobody can separate: friction in connections, cracking in concrete, movement of non-structural partitions, and some radiation into its own foundations. The last of those is the term this page computes, and on a building it is small — a building’s own foundations are stiff relative to it and its modes are not the modes of a rigid block.

But it is not zero, and it is the one term in the compound that behaves like arithmetic rather than like a fitted number. That is why a soil-structure analysis reports a damping higher than the fixed-base value, why the increase is largest for squat stiff buildings, and why the same analysis reports that the effect goes the wrong way for tall ones — the rocking mode of a tall building on soil is a dipole with almost no radiation in it, exactly as the block’s is.

The one place damping can be computed is also the place that explains why the rest of it cannot.

Where the model stops #

The footing is on the surface. A buried block gets extra damping from the shear on its sides, which raises every one of the three numbers — by 50 per cent or more for a fully embedded block — and it is one of the few conservatisms in the method worth keeping rather than exploiting.

The half-space is uniform and infinite. A stiff layer at shallow depth reflects the waves back, which turns the radiation into a resonance of the layer, and below a cut-off frequency there is no radiation at all: waves cannot propagate away in a layer thinner than a quarter wavelength. That is the single most important limitation of the whole method, and it means a site with rock a few metres down behaves nothing like this.

Nothing is nonlinear. Soil under a heavily loaded footing at large amplitude has hysteretic material damping as well, which adds to the radiation term — and it also softens, which lowers the frequency, and the two effects push the answer in opposite directions.

And the block is rigid. A large mat with a flexible machine frame on it has modes of its own, and the analogues describe a rigid body on a half-space rather than a structure on one.

What the pictures cannot show #

They cannot show the machine, which is the source of everything.

The whole calculation is a response to an unbalanced force, and that force is a manufacturer’s number: a residual imbalance in kilogram-millimetres, multiplied by the square of the running speed. It grows with wear, it is worst at start-up while the machine passes through its own critical speeds, and it is the least reliable input in the analysis by a wide margin. The damping is the well-known quantity here and the load is not, which is the reverse of the usual arrangement in this collection.

They also cannot show what the waves do when they arrive somewhere. Radiation damping is energy leaving the foundation, and it leaves in the direction of whatever is nearby: a neighbouring building, a sensitive instrument, a railway. Ground-borne vibration is the same waves seen from the other end, and a foundation designed for maximum radiation damping is a foundation designed to be a good transmitter.

The assumption the figure rests on #

That the dashpot is frequency-independent.

It is not, quite. The exact elastodynamic solution gives an impedance whose real and imaginary parts both vary with the dimensionless frequency $a_0 = \omega r/V_s$, and Lysmer’s analogue is a constant spring and dashpot fitted over $a_0$ from 0 to about 1.5 — which covers a machine at 10 to 25 Hz on an ordinary soil and does not cover a footing under an impact.

Past that range the real impedance falls and the analogue overstates the stiffness; the rocking mode’s dashpot in particular is a poor fit at low frequency, which is why the rocking damping is sometimes quoted as frequency-dependent while the vertical is not.

The general form of that caution is one this collection meets repeatedly: a lumped element standing in for a continuum is fitted over a range, and using it outside the range is the commonest way to get a plausible wrong answer. It is the same relationship a Winkler spring has to the elastic half-space it replaces, in the frequency domain rather than the spatial one.

The history, which is a wave problem borrowed by engineers #

The elastodynamics came first and the engineering long afterwards, which is the reverse of the usual order in this collection.

Lamb solved the problem of a point force oscillating on the surface of an elastic half-space in 1904, as a question in seismology: what fraction of the energy goes into surface waves and what into body waves. The answer — most of it into the Rayleigh wave, travelling out along the surface — is why an earthquake is felt at a distance and it had nothing to do with foundations.

Reissner applied it to a circular footing in 1936 and got the shape of the answer wrong in an instructive way: he assumed a uniform pressure under the footing rather than a rigid displacement, which gives a different impedance and a damping that varies more strongly with frequency than the real one. The rigid-footing solutions came in the 1950s and 1960s, and Lysmer’s analogue — the constant spring and dashpot fitted to them — in 1966.

What that sequence produced is a design method whose central quantity was never measured on a foundation at all. The damping in a machine block was computed from a wave theory forty years before anybody tried to design one with it, and the field checks that followed found the analogue good to ten or twenty per cent, which is better than any structural damping value has ever been known to.

The ladder from here #

Later rungs on this anchor: embedment, and the side shear that raises every impedance. The layered site, where a stiff stratum reflects the waves and radiation stops below a cut-off. Frequency-dependent impedances taken seriously, with the cone models that reproduce them from a one-dimensional argument. Coupled sway and rocking, which for a block with its centre of gravity above the base is a two-degree-of-freedom system whose modes are neither. Transient rather than harmonic loading, which is what a forge hammer is and where the peak displacement rather than the amplification governs. And the same analogues used the other way round, as the springs and dashpots under a building in a seismic soil-structure calculation — where they are called something else and the same three formulas appear.