The rock that sends the waves back

The damping that is radiated found the one place in this subject where damping is computed rather than assumed. A machine block on the ground pushes on the soil, the soil carries the push away as waves, and the waves never come back — so the block loses energy exactly as if it were attached to a dashpot, and Lysmer’s analogues say how large. For a 150 tonne block 5 m across on ordinary firm soil the vertical mode comes out at 47 per cent of critical, the horizontal at 29 and the rocking at 13: more damping, in the first two, than any device a structural engineer could install.

Every one of those numbers assumed the waves had somewhere to go. The analogues are fitted to a half-space — soil extending downward and outward without end — and a half-space has no boundary to reflect anything from. Real ground has rock under it, at five metres or fifty, and rock is stiff enough that a wave arriving at it from soft soil above is sent almost entirely back.

This essay puts the rock in. The result is not that the damping is a little lower. It is that below a definite frequency the damping is not there at all, and a block whose mode falls below that frequency is a lightly damped resonator whatever the half-space said.

A layer has a frequency below which nothing travels #

A layer of soil of depth $H$ over rock is a bounded region. A wave launched downward from the footing reaches the rock, reflects, comes back up to the surface, reflects again, and is trapped between the two boundaries. What decides whether energy can leave the footing is whether a wave can travel sideways along the layer, out to distances where it no longer matters.

For waves trapped between a free surface and a rigid base, sideways travel is only possible above the layer’s lowest natural frequency. Below it the layer can be set moving, but only as a standing pattern that stores energy and returns it every cycle; nothing is carried away. The lowest frequency of a soil column fixed at its base and free at its top is a quarter-wavelength resonance:

$$f_s = \frac{V_s}{4H}$$

for shear waves, which is what a footing sliding horizontally generates. For a footing moving vertically or rocking, the waves are a mixture of compression and shear that Lysmer’s analogue treats as travelling at $V_{La} = 3.4\,V_s/\pi(1-\nu)$, and the cut-off is $V_{La}/4H$. For this soil $V_s$ is 178 m/s and $V_{La}$ is 296.

Below the cut-off, radiation damping is zero. Not small — zero, in the idealised layer, because the mechanism that produced it does not exist. Above it the dashpot rises toward its half-space value over a band of frequency and then ripples around it; the figures here take the rise as a straight ramp over a fifth of the cut-off and ignore the ripple, which is the usual engineering idealisation of the exact layered solution.

Each mode meets its cut-off at a definite depth #

The layer does two things to the block, and they are very unequal. It stiffens the footing, because the rock stops the deep soil deforming: for a footing of radius $r$ on a layer of depth $H$ the vertical stiffness grows by a factor $1 + 1.28\,r/H$, the horizontal by $1 + 0.5\,r/H$ and the rocking by only $1 + r/6H$. On a 4.5 m layer under this 2.5 m footing that is 71 per cent more vertical stiffness, and it lifts the vertical frequency from 12.5 Hz to 16.3.

And it imposes a cut-off, which goes as $1/H$ and starts from nothing. On a 25 m layer the vertical cut-off is under 3 Hz, far below every mode. At 10 m it is 7.4 Hz. At 6 m it is 12.3. At 4.5 m it is 16.4 Hz — and the vertical mode, which the layer has pushed up to 16.3, is now just below it.

So the modes and the cut-offs cross, and each mode crosses at its own depth. The vertical mode loses its radiation on layers shallower than 4.5 m, the rocking mode on layers shallower than 4.7 m. The horizontal mode’s cut-off uses the shear wave, which is slower, and it does not reach the horizontal frequency of 11.2 to 12.8 Hz on any layer at least one and a half footing radii deep, which is as shallow as the stiffness factors can be trusted.

The whole of it, once, for the vertical mode #

The critical depth can be worked out by hand, and doing it once shows which of the two effects is doing the work.

On a half-space the vertical stiffness is $4Gr/(1-\nu) = 4 \times 60 \times 2.5/0.65 = 923$ MN/m, and with 150 tonnes on it the natural frequency is $\sqrt{923 \times 10^6/150{,}000}/2\pi = 12.5$ Hz. The cut-off for vertical motion on a layer of depth $H$ is $V_{La}/4H = 296/4H$. Ignore the stiffening for a moment and set the two equal: $H = 296/(4 \times 12.5) = 5.9$ m.

That is a quarter of the wavelength of a vertical wave at the mode’s own frequency, and it is the rule of thumb worth keeping: the rock matters when it is shallower than a quarter wavelength at the foundation’s own frequency. A wave at 12.5 Hz in this soil is 23.7 m long; a quarter of it is 5.9 m.

Now put the stiffening back. On a layer 4.5 m deep the vertical stiffness is multiplied by $1 + 1.28 \times 2.5/4.5 = 1.71$, which lifts the frequency by $\sqrt{1.71}$ to 16.3 Hz, and the cut-off at that depth is $296/18 = 16.4$ Hz. The two have met. The stiffening moved the critical depth from 5.9 m to 4.5 m — in the block’s favour, because a stiffer footing has a higher frequency and needs a thinner layer before the cut-off catches it — but it did not move it far. The quarter-wavelength estimate is on the safe side by about a metre and a half, and it needs nothing but the half-space frequency and the soil’s wave speed.

The damping does not fade; it goes #

Read against depth, the damping of the vertical mode is nearly flat from 25 m down to 6 m, falling slowly from 49 per cent as the stiffening layer raises $K$ against a fixed dashpot. Then, over a metre or so of depth, it falls off a cliff: from above 40 per cent to the soil’s own 5.

That shape is the whole practical content. A half-space calculation is not a slightly optimistic version of the layered one; over most depths it is right, and over a narrow band it is wrong by a factor of nine. A site investigation that stopped at 6 m and found uniform soil has not established that the half-space is a fair model. It has established that the rock, if it is there, is deeper than 6 m — and whether it is at 7 m or 4.5 m decides whether the block is damped at 45 per cent or 5.

The rocking mode shows the same cliff from a lower start. On a half-space it had only 13 per cent of radiation damping, which the earlier essay identified as the reason rocking is the mode a machine foundation is checked for. On a thin layer it has none, and 5 per cent is what is left.

What the response looks like when the damping goes #

The vertical response at the top of this page is the extreme case. On a half-space the vertical mode is so heavily damped that its “resonance” is a gentle hump at 1.1 times the static deflection, at 8.8 Hz — well below the natural frequency of 12.5, because at 47 per cent of critical the peak of a damped oscillator moves down and flattens out. On a 10 m layer the curve is barely different. On a 6 m layer a real peak appears, 1.74. On a 4.5 m layer the peak is 5.84, at 16.3 Hz: a sharp resonance of the kind every other essay in this field is about, from a mode that on a half-space did not have one.

The rocking mode is less dramatic only because it started with less. On a half-space its peak is 2.86; on 10 and 6 m layers, 2.78 and 3.03, the stiffening and the loss nearly cancelling; on 4.5 m, 9.15. The frequency has hardly moved, from 15.0 Hz to 15.7. Everything that changed is the height of the peak.

Plotted against depth, the peaks tell the same story as the damping from the other side. The vertical peak grows by a factor of 5.3 and the rocking by 3.2 as the layer thins through its critical depth. For a machine that runs at a fixed speed well away from the resonance this may not matter; for one that runs up through its resonance at every start, or one whose operating speed is close to it, it is the difference between a foundation nobody notices and one that shakes the plant.

The ground amplifies most where it can radiate least #

The horizontal cut-off $V_s/4H$ is a familiar number in a different part of this subject. It is the fundamental frequency of the soil column that the ground has a period of its own described: the frequency at which a layer of soil over rock resonates when an earthquake shakes its base, and at which the ground surface moves several times more than the rock beneath it.

That is not a coincidence, and it is the connection worth carrying. Both are statements about the same standing wave. A layer excited from below at its fundamental frequency builds up a standing wave that amplifies the motion at the surface, because energy arriving from the rock cannot escape upward and is not carried sideways. A layer excited from above, by a footing, below that same frequency cannot carry energy away sideways for the same reason. The property that makes a soft site dangerous in an earthquake — its inability to shed energy near its own period — is the property that takes a machine foundation’s damping away.

So the sites where a machine block’s radiation damping is least reliable are precisely the soft, shallow, rock-bottomed sites that seismic codes single out for amplified design spectra. A designer who has classified the site for earthquake purposes has, without noticing, been told the depth below which the foundation’s damping cannot be trusted.

The heavy block needs deeper ground #

The traditional rule for a machine block is to make it three to five times as heavy as the machine, so that its natural frequencies sit well below the running speed. The earlier essay found that the rule costs damping on a half-space, because the damping ratio falls as the square root of the mass. On a layer it costs something worse.

A heavier block has lower natural frequencies, and a lower frequency falls under the cut-off of a deeper layer. The 150 tonne block loses its vertical radiation below 4.5 m of soil. At five times the mass on the same footing — 750 tonnes, the kind of block a large compressor sits on — it loses it below 11.7 m, and its rocking radiation below 10.8. Eleven metres of soft soil over rock is not an unusual site. It is a river terrace, an estuary margin, a reclaimed harbour.

The escape the essay on radiated damping found — spread the block rather than deepen it, because damping is bought with footprint — still works, and for a second reason now. A wider footing at the same mass has higher frequencies, not lower, and keeps them above a deeper cut-off. Footprint buys damping twice over on a layered site: once through the half-space mass ratio, and again by holding the modes above the layer’s cut-off.

What a designer does when the rock is shallow #

Where the quarter-wavelength check says a mode will fall under its cut-off, there are four things to do, and they are the four levers every other essay in this field has used, priced differently here.

Raise the mode above the cut-off. A wider, lighter block has higher frequencies, and on a thin layer the stiffening helps rather than hurts. This is the remedy that keeps the radiation, and it has a limit: the frequency must still sit clear of the machine’s running speed, which is what the frequency below both checks showed can move the other way.

Design for the damping that is left. Take the soil’s own few per cent and check the resonance as the only thing that stops it would check any lightly damped structure. It is the honest answer when the block cannot be changed and the machine runs well away from resonance.

Take the machine’s force off the block. Springs under the machine turn the block into a seismic mass and the question into a transmissibility, which does not depend on radiation at all — the block only has to be heavy.

Add damping that does not depend on the ground. A mass tuned to the offending mode supplies what the layer took away, and it can be sized from the material damping alone, because that is the number that will still be true in ten years.

Embedment, the remedy the essay before this one examined, is weaker here than it looked. The side soil is part of the same layer, and the waves it radiates meet the same rock; its waves are subject to the same cut-off, and below it the sides store energy as the base does and give back only their stiffness.

The block cut from the layer #

The free body is the block, cut from the soil at its base, as in the earlier essays; nothing new crosses the cut. The soil beyond the cut is a spring whose constant is a property of the ground, and a dashpot whose constant is a property of where the ground ends. What changes is the relation between the motion of that surface and the traction on it. On a half-space it is a spring and a dashpot whose constants are Lysmer’s. On a layer the spring is stiffened by Kausel’s and Gazetas’s factors for a stratum over a rigid base, and the dashpot is switched off below the layer’s cut-off and ramped back to its half-space value over a fifth of it.

Every response is the steady amplitude under a harmonic force of fixed size, divided by the static deflection the same force would give on the half-space, so the layer’s stiffening shows as a lower response at low frequency and the lost damping as a higher peak. The soil’s own material damping is taken as 5 per cent throughout and treated as hysteretic, a constant fraction of the stiffness, which is how soil behaves at the small strains a machine produces.

The exact layered solution, a rock that gives, a layer that varies #

The exact layered solution. Above the cut-off, rigorous solutions for a disc on a stratum do not settle smoothly onto the half-space line; they overshoot and ripple, with peaks at the layer’s higher resonances. The ramp drawn here gets the step right and the detail wrong, and a machine running just above a cut-off can see more damping or less than it shows.

A rock that is not rigid. Real bedrock is stiffer than the soil above it, not infinitely stiff, so some energy does pass into it and some radiation survives below the cut-off. The cut-off is sharp in proportion to the impedance contrast between soil and rock, and on a site where the “rock” is merely dense gravel the cliff is a slope.

A layer that is not uniform. Soil stiffens with depth, so a real profile has no single $V_s$ and no single cut-off, and the frequency below which radiation fails is smeared over a band.

Shallow layers. The stiffness factors are valid for a layer at least one and a half footing radii deep and the figures stop there. A block on a thin crust over rock is a block on a stiff spring with almost no damping at all, and needs a different model rather than an extrapolation of this one.

The depth to rock is known #

That the depth to rock is known. It is the one number in the calculation that no amount of care with the block, the machine or the soil’s modulus can supply, and it is the number a site investigation is least often asked for when the question is a machine foundation rather than a pile length. Every figure above is a function of $H$, and the practical instruction they add up to is short: find out how deep the rock is, compute the layer’s cut-off for each mode, and if any mode’s own frequency is below it, design that mode for the soil’s material damping and nothing more.

Still open: the forge hammer, where there is no steady state to be damped #

Every response here is a steady state under a harmonic force — a rotating machine running at a fixed speed, with the damping deciding how high its resonance stands. A forge hammer is not like that. It delivers a blow — a weight that is dropped, many times a minute — the block rings down, and the next blow arrives before or after the ringing has died depending on the hammer’s rate. The transmissibility curve that governs a rotating machine has nothing to say about a blow, and what matters instead is how far the block moves on the first impact and how much of that motion is still there when the second arrives. Whether a spring mount that isolates a rotating machine does anything for a hammer — and what the loss of radiation damping on a layered site does to a train of blows rather than to a resonance — is the question after this one.