The blow that has no frequency

Every machine foundation in the essays before this one has carried a machine that runs. The machine that shakes the building was an unbalanced rotor turning at a steady speed; its force was a sine wave, the block’s response settled to a sine wave of the same frequency, and the whole design turned on one curve — transmissibility against the ratio of running speed to natural frequency, passing through one at $\sqrt 2$ whatever the damping. The essays after it computed where that natural frequency was and how much damping the ground supplied, on a half-space and then on a layer over rock, and each of them assumed a steady state.

A forge hammer never reaches one. It lifts a tup of a tonne or two, drops or drives it onto a workpiece on the anvil, and the anvil and its block are struck: a large momentum delivered in a few milliseconds. The block leaps, rings, and — depending on how quickly the ringing dies and how soon the next blow comes — is struck again. There is no frequency of excitation for a transmissibility curve to be drawn against, and the damping the ground supplies is no longer the quantity that limits a resonance. A blow has no frequency, and the rules that curve teaches do not transfer.

What a blow is, to a block #

A blow short against the block’s natural period is described completely by its momentum. Whatever the shape of the force in those few milliseconds, the block has not had time to move before it is over; all that has happened is that it has been given a velocity, $v_0 = I/m$, where $I$ is the impulse the blow delivered and $m$ the block’s mass. Everything after that is free vibration from a standing start at speed.

That is the same regime a blast load lives in, and the same argument: when the load has come and gone before the structure has travelled any distance, its peak force is irrelevant and its impulse is everything. What is different here is the question. A blast essay asks how far the structure moves. A hammer foundation asks that and something else — how much force the block passes on to the ground, because the ground is where the neighbouring buildings are.

The block used throughout is the 150 tonne block of the earlier essays, 5 m across, on soil where its vertical mode is at 12.5 Hz. The blow is 18 kN·s — a two-tonne tup arriving at six metres a second and rebounding at half that speed — which sets the block moving at 0.12 m/s. Undamped, it then oscillates with amplitude $v_0/\omega = I/(m\omega) = 1.53$ mm, and the spring under it passes a peak force of $k$ times that, which is $I\omega = 1.41$ MN.

The force and the movement multiply to a constant #

Those two results contain the first rule, and it is a rule the transmissibility curve has no counterpart for. The peak force is $I\omega$ and the peak movement is $I/(m\omega)$. Multiply them:

$$F_{\max}\,x_{\max} = \frac{I^2}{m}$$

The mount’s stiffness has cancelled. The product of the force the block passes on and the distance it moves is fixed by the blow and the mass alone, and all a mount can do is choose where on that curve to sit.

A soft mount slides the block down its curve: less force to the ground, more movement of the block. Springs at 4 Hz under the 150 tonne block pass 0.42 MN against the soil’s 1.32, and the block moves 4.42 mm against 1.42. That is a real reduction in what reaches the neighbours, bought with three times the movement — and movement at a hammer is not free, because the anvil moving under the die spoils the forging, and a block that moves several millimetres needs flexible connections to everything around it.

The only way off the curve is mass. A heavier block lowers the product $I^2/m$, and at the same stiffness both the force and the movement fall. This is the one place among machine foundations where the textbook rule of a heavy block is right for the reason it is given: for a rotating machine the block’s mass mattered only through the frequency it produced, but for a hammer the mass is in the product directly, and nothing else can reduce both halves at once.

Damping has an optimum, and it is not the most #

The second rule is the one that contradicts the first essay’s intuition directly.

For a rotating machine at resonance, damping is the only thing that limits the response, and more of it is always better. For a blow it is not. The force the mount passes to the ground is the spring’s and the dashpot’s together, $kx + c\dot x$, and the dashpot’s share is proportional to the block’s velocity. A struck block has its largest velocity at the very instant it is struck — that is what being struck means — so a heavily damped mount passes a large force at once, before the spring has compressed at all: $c\,v_0 = 2\zeta\omega I$.

At low damping the spring’s force dominates and a little damping helps, by eating into the first swing. At high damping the dashpot’s instant force dominates and more damping hurts. In between there is a minimum, at about a quarter of critical, where the force is 0.81 of the undamped value; by half of critical it is back to 1.0, and beyond that the damped mount passes more force than no damping at all.

The consequence for the foundations the earlier essays computed is striking. On a half-space the block’s vertical mode had 47 per cent of critical, the damping that made its resonance vanish; on a thin layer over rock it had 5 per cent, and its resonance stood five times higher. Under a blow those two foundations pass almost the same force — 0.947 and 0.931 of the undamped value, 1.34 MN and 1.32. The radiation damping that transformed the rotating machine’s response does nothing for the first blow.

What it does do is visible in the figure below. On the half-space the block is still within a cycle and a half; on the thin layer it rings for the whole interval before the next blow. Movement falls steadily with damping, and so does what is left when the next blow arrives. That is where damping earns its keep on a hammer, and it is a question about trains of blows rather than about one.

A mount softens a blow only if the blow is short #

The impulse idealisation assumes the blow is over before the block moves. Real blows last a few milliseconds to a few tens, and a mount soft enough to have a long period stays in the impulsive regime; a stiff one may not. The figure asks what a blow of finite length does, by giving it a half-sine shape of fixed impulse and varying how long it lasts against the mount’s period.

The shape of the answer is the hammer’s version of the transmissibility curve, and it is drawn on a different axis. Below about a third of a period the mount softens the blow, passing on a peak force smaller than the blow’s own, because it has not had time to push back before the blow is over. Above a third of a period it amplifies it: the mount’s response and the blow’s force overlap, and at around eight-tenths of a period the force to the ground reaches 1.65 times the blow’s peak at 5 per cent damping. Only a blow very long against the period comes back down toward one, where the mount simply transmits a slow push.

So the criterion for isolating a hammer is not the ratio of a running speed to a natural frequency. It is the ratio of the blow’s duration to the mount’s period, and it asks for the period to be at least three times the blow. For a block on soil at 12.5 Hz, whose period is 80 ms, a forging blow of 10 ms is well inside it; a blow cushioned by a soft workpiece or an elastic pad under the anvil, lasting 30 ms, is not.

When one blow arrives before the last has died #

A hammer works at a rate — forty blows a minute for a large drop hammer, more than a hundred for a small power hammer — and every blow arrives on a block still ringing from the last. Whether that matters depends on how much of the ringing survives one interval, which is the damping’s decay per cycle raised to the number of cycles between blows.

Where the interval is a whole number of periods the new blow lands in step with the ringing and adds to it; the train’s steady movement then builds toward a limit set by how much decays between blows. At 5 per cent damping with blows exactly one period apart, the block settles at 3.7 times the movement one blow produces. At 2 per cent, 8.2. At 47 per cent — the half-space’s radiation — 1.04: each blow finds the block essentially still. Between the whole numbers the train can even be quieter than a single blow, because the new blow lands against the ringing rather than with it.

For a block on soil at 12.5 Hz, none of this matters in practice. Its period is 80 ms, and a hammer at a hundred blows a minute strikes every 600 ms — seven or eight periods, over which even 5 per cent damping has let the ringing fall by a factor of ten. The build-up is a hazard of soft mounts, and that is where the springs of the first figure come back.

The springs that soften every blow #

Springs at 4 Hz under the block cut every blow’s force to the ground by a factor of three. They also bring the block’s natural frequency down to 240 cycles a minute, and a hammer running at 120 blows a minute strikes on every second cycle, at 80 on every third. At 5 per cent damping a train at 120 a minute builds the block’s movement to 2.14 times a single blow’s — 9.5 mm instead of 4.4 — and at 2 per cent, which is what steel springs alone supply, to 4.5 times.

So the spring mount carries a hazard the rotating machine’s never did. For a rotor, the isolated block sits far below the running speed and the danger is only in passing through resonance at start-up. For a hammer, the “running speed” is the blow rate, which is low, and a mount soft enough to isolate each blow can put the block’s own frequency at a whole multiple of it. That is why spring-mounted hammer foundations are built with dampers in parallel with the springs: not to reduce the force of each blow, which damping cannot do by much and eventually undoes, but to kill the ringing before the next blow arrives. The damper that achieves that without passing too much force at impact sits at about a fifth to a quarter of critical — exactly the band the single-blow figure found to be best on its own account.

The whole of it, once, for the springs #

The spring-mounted block can be worked through by hand, and the hand calculation shows where each number comes from.

Springs at 4 Hz under 150 tonnes have a stiffness $k = m\omega^2 = 150{,}000 \times (2\pi \times 4)^2 = 94.7$ MN/m. The blow sets the block moving at 0.12 m/s. Undamped, the block would move $v_0/\omega = 0.12/25.1 = 4.77$ mm and the springs would pass $k$ times that, 0.452 MN. With 5 per cent damping the first swing is trimmed to 0.927 of that movement and 0.931 of that force — 4.42 mm and 0.42 MN — which are the figures read off the curve above.

The train at 120 blows a minute puts 0.5 s between blows, which is exactly two periods of a 4 Hz block. In that time the ringing decays by $e^{-\zeta\omega T_b} = e^{-0.05 \times 25.1 \times 0.5} = 0.53$. Each blow lands in step with what is left of all the earlier ones, so the steady movement is one blow’s times $1 + 0.53 + 0.53^2 + \dots = 1/(1 - 0.53) = 2.14$, which is the number the rate figure gives. At 2 per cent damping the decay per interval is 0.78 and the sum is 4.5.

That geometric series is the whole of the build-up, and it gives the design rule in one line: the ringing left when the next blow arrives, $e^{-2\pi\zeta n}$ for $n$ cycles between blows, must be small. For the block on soil, with $n$ about seven and 5 per cent damping, it is 0.11 and the series is 1.12; for the springs, with $n = 2$, it is 0.53. Damping and the number of cycles between blows trade against each other, and a soft mount is a mount that has made $n$ small.

One block, two design questions #

It is worth setting the two kinds of machine side by side, because the same block answers them oppositely.

For a rotating machine, the mount should be soft — natural frequency well below the running speed, past $\sqrt 2$, and clear of every coupled mode — and damping should be modest, enough to carry the machine through resonance at start-up and no more, because above $\sqrt 2$ damping raises the transmitted force. The block’s mass matters through the frequency it sets.

For a hammer, the mount should be soft enough that its period is long against the blow, and not so soft that its frequency meets the blow rate; damping should sit near a quarter of critical, where one blow’s transmitted force is least and a train’s ringing is killed; and the block’s mass matters in its own right, because it is the only thing that lowers the product of force and movement. A foundation designed for one machine and inherited by the other — a forging shop that replaces a press with a hammer, or the reverse — is a foundation designed for the wrong question.

The weight that was dropped made a related point about a single impact on a beam: the stiff structure is the one that suffers, because the energy has to be absorbed over a short distance. The hammer foundation is that result applied to what is passed on rather than what is carried. A stiff block is struck no harder than a soft one, but it passes the blow to the ground at $I\omega$, and $\omega$ is its stiffness.

The block, and the two forces that cross its base #

The free body is the block, cut from its mount at its base. Across the cut pass two forces: the spring’s, proportional to the block’s movement, and the dashpot’s, proportional to its velocity. Their sum is the force the ground receives, and it is the quantity every figure here reports as “force to the ground”. The blow acts on the top of the block and is either an impulse — a velocity given at the first instant — or a half-sine of the same impulse lasting a stated time.

The block is a single mass on a single vertical spring. For the impulse the response is the closed-form free vibration of a damped oscillator from rest at velocity $I/m$; for the half-sine it is the same oscillator stepped through the pulse numerically; and for a train of blows it is the sum of the single-blow responses, shifted by the interval and summed until the oldest contribute nothing.

The anvil, the blow’s own scatter and the ground’s frequencies #

The anvil. A real forge hammer foundation is two masses — the anvil on an elastic pad, and the block under it — and the pad lengthens the pulse the block receives, moving it along the axis of the pulse figure toward the region where a mount amplifies rather than softens. Designing the pad is designing the pulse, and the single-mass block here cannot show it.

The blow’s own uncertainty. The impulse depends on the tup’s speed and on how much it rebounds, which depends on how hot and how soft the workpiece is. A cold blow on a nearly finished forging rebounds more and delivers a larger impulse than a first blow on a hot billet; the whole curve scales with it.

The ground’s frequency dependence. The half-space damping of 47 per cent and the layer’s cut-off belong to steady vibration at the block’s frequency. A blow excites everything at once, and the ground’s response to the high frequencies in a short blow is not its response at 12.5 Hz.

Settlement. Millions of blows are a cyclic load on the soil, and a block that settles unevenly tilts its anvil; that is a question about the soil under repeated load rather than about vibration.

A mount that is linear #

That the mount is linear. Every conclusion here — the constant product, the damping optimum, the build-up — belongs to a spring whose force is proportional to its compression and a dashpot whose force is proportional to its speed. Soil at the strains a large hammer produces is neither: it stiffens under compression, softens under repetition, and dissipates energy in hysteresis rather than viscosity. For a block on springs and dampers the idealisation is close. For a block on the ground it is the right shape and an uncertain size, and a measured decay after one blow is the calibration it needs.

Still open: the anvil and the pad as a two-mass system #

The anvil sits on a pad of timber, rubber or cork, and the pad makes the foundation a system of two masses and two springs: anvil on pad, block on ground. Such a system has two natural frequencies and two ways of ringing, the blow lands on the smaller mass, and the pad’s stiffness decides both how long the pulse lasts when it reaches the block and how the energy of the blow divides between the two modes. A tuned mass is also a small mass on a spring on a large one, and whether the anvil and its pad can be arranged to act as one for the block — so that the pad’s own ringing absorbs the blow instead of passing it on — is a question about choosing the pad’s stiffness for the block rather than for the anvil.