The pad that makes the blow worse

The blow that has no frequency struck a single foundation block with a forge hammer and found that a blow is not a vibration problem in the ordinary sense: there is no steady state, the force passed to the ground and the movement of the block multiply to a constant, and a mount softens a blow only if the blow is short against the block’s own period. The block there was one mass. A real hammer foundation is two.

The tup does not strike the block. It strikes the anvil, and the anvil — a steel mass often a fifth or more of the whole foundation — sits on a pad of hardwood, rubber or cork on top of the concrete block. The pad makes the foundation a system of two masses and two springs: the anvil on its pad, the block on its ground. Such a system has two ways of ringing, the blow lands on the smaller mass, and the pad’s stiffness decides how long a push the block receives and how the blow’s energy divides between the two ways.

The pad is usually described as a cushion, which invites a hope. A small mass on a spring on a large mass is also a tuned mass damper, and a tuned mass damper, correctly tuned and damped, takes energy out of the large mass’s motion. Could the anvil and its pad be arranged to do that for the block — so that the pad’s own ringing absorbs the blow rather than passing it on? The answer turns on where the blow enters, and it is no for the blow and a qualified yes for what follows it.

Two masses, one blow #

The foundation is the 150-tonne block of the earlier essay with a 30-tonne anvil as part of it, struck with 18 kN·s — the momentum of a two-tonne tup arriving at about nine metres per second. On a rigid seat the anvil and block move as one mass, and the earlier result applies: on soil that gives it a frequency of 12.5 Hz and damping of nearly half of critical, the block moves 0.86 mm.

Put a pad under the anvil and describe it by one number, its tuning: the frequency at which the anvil alone would ring on the pad, over the foundation’s frequency on its ground. A hardwood pad under a heavy anvil is typically several times stiffer than the ground under the whole foundation, a tuning somewhere around four to ten; a rubber pad can be much softer.

The picture has one message and it is the reverse of the cushion’s. Every pad an anvil can realistically sit on makes the block move more than a rigid seat would. At a tuning of four, the block on soil moves 30 per cent more. As the pad softens toward the foundation’s own frequency the penalty grows to a peak — 1.58 times on soil, near a tuning of two; 1.77 times on springs, where the ground supplies almost no damping, near a tuning of 1.2 — and only well below the foundation’s frequency does the curve fall under one: below 0.6 on soil, where the anvil would move 11 mm on every blow, and below 0.8 on springs.

Where the energy goes #

The energy accounts for the stiff end of the curve. The blow gives the anvil momentum $I$ and kinetic energy $I^2/2m_a$. On a rigid seat, anvil and block share that momentum at once, and kinetic energy carried by the same momentum in five times the mass is a fifth of the original: four-fifths of the blow never reaches the foundation’s motion at all. On a stiff pad the same thing happens with a delay — the anvil rattles on its pad at a high frequency, holding four-fifths of the energy, and the pad’s own damping takes it out within a few milliseconds. The foundation’s slow mode receives the anvil’s share of the mass, 0.20, and that is exactly the rigid-seat case.

As the pad softens, the two modes stop being “the anvil” and “the foundation”. Their shapes mix, the lower mode acquires a larger share of the blow’s energy — 0.31 at a tuning of two, 0.76 at one — and that is the mode in which the block moves. The energy that the stiff pad sent harmlessly into the anvil’s rattle, a softer pad sends into the block.

What the block actually does #

The two pads get to their worse answers in different ways, and both are visible in the block’s movement.

On the pad at four times the foundation’s frequency the block rides the anvil. The anvil rings on its pad at about 50 Hz, holding most of the blow’s energy, and each time it presses down it pushes the block; the block’s movement is the foundation’s slow swing with a 50 Hz ripple on top of it. The first crest of the ripple arrives 13 ms after the blow, close to where the swing would have peaked, and the two add. On a rigid seat the ripple does not exist, because the anvil’s rattle is so fast and so quickly damped that the block feels only its average.

On the tuned pad the block receives the blow as a single slow push, lasting about half a period of the anvil on its pad, and it peaks later — at 35 ms — and a little higher, 1.18 mm. Then it goes on ringing, long after the rigid seat has settled. That is the second effect of tuning, and on a block that sits on soil it is the more surprising one. The ground damps the block heavily, nearly half of critical. But near tuning, one of the two modes is the anvil and the block swinging against each other, and in that mode the ground hardly moves: its damping comes almost entirely from the pad. A block that the ground would have stilled in a single swing now rings for two hundred milliseconds with the anvil.

Why a tuned pad cannot absorb the blow #

The hope was that a tuned pad would behave as a tuned mass damper does, and the reason it cannot is a matter of where the load goes in.

A tuned mass damper is a small mass hung on a spring from a large one, and it is driven by the large mass’s motion. When the large mass is forced near its resonance, the small one swings a quarter-cycle behind it, and the force its spring exerts on the large mass is then in the direction that opposes the large mass’s velocity. The damper takes energy out because it is late, and it can be late only if the load arrives at the large mass first.

Here the blow arrives at the small mass. The anvil is not driven by the block’s motion; the block is driven by the anvil’s. Whatever the anvil does on its pad, the block receives it — there is no other path from the blow to the ground — and tuning the pad to the foundation’s frequency tunes the delivery of the blow to the block’s resonance. It is the blow that has no frequency given one: a push lasting about half the foundation’s period, which is the pulse length a foundation answers most.

What the pad is actually for #

If the pad does not protect the block’s motion, it protects something else, and the figure says what: the force under the anvil. On a seat twenty times stiffer than the ground, the anvil and block meet with 23 MN — spread over an anvil base of a few square metres, a bearing stress of several newtons per square millimetre on the concrete, repeated on every blow, at the one place in the foundation where a crack will be driven by thousands of impacts a day. A pad at four times reduces the force to 4.9 MN; at twice, to 2.3.

The price is the anvil’s movement on its seat, which grows as the force falls: from a third of a millimetre at twenty times to one and a half at four and nearly seven when tuned. An anvil that moves several millimetres on every blow is an anvil the smith is working on as it moves, loosening its keys and wedges and working the pad to pieces. The pad is a trade between the force under the anvil and the movement of the anvil, and the block’s own motion is a third party that loses in both directions: 1.05 times its rigid-seat movement at a tuning of ten, 1.30 at four, 1.58 at two.

That gives the pad’s design a shape that has nothing to do with isolation. It should be as soft as the anvil’s movement allows, because that is what limits the bearing stress on the concrete beneath it. And it should stay well stiffer than the foundation’s frequency — several times — because the whole of the region between a tuning of about one and about three is where the block pays most for it.

The seat’s own product #

The trade has an exact form, and it is the earlier essay’s result in a new place. That essay found that for a block struck once, the force passed to the ground times the block’s movement is a constant, $I^2/m$, whatever the mount. The seat under the anvil obeys the same law, one level up.

While the pad is much stiffer than the ground, the ground is too slow to matter during the anvil’s first compression of its pad, and the anvil and block behave as two free masses meeting through a spring. The motion that compresses the pad is their relative motion, which has the reduced mass $\mu = m_a m_b / M$ and starts at the anvil’s velocity $I/m_a$. A mass on a spring given a velocity reaches a peak force of mass times velocity times frequency, and a peak movement of velocity over frequency, so

$$F_{\text{seat}} \approx I\,\omega_p \sqrt{\frac{m_b}{M}}, \qquad \delta_{\text{anvil}} \approx \frac{I}{m_a\,\omega_p}\sqrt{\frac{m_b}{M}}, \qquad F_{\text{seat}}\,\delta_{\text{anvil}} \approx \frac{I^2}{m_a}\,\frac{m_b}{M}.$$

The pad’s stiffness has cancelled out of the product. For this hammer it is $18{,}000^2 / 30{,}000 \times 0.8 = 8.6$ kN·m — twice the kinetic energy of the relative motion, which the pad must store at its peak whatever it is made of. At a tuning of four the formulas give 5.1 MN and 1.7 mm against the full calculation’s 4.9 and 1.5; at twenty, 25 MN and 0.34 mm against 23.4 and 0.31, the difference in each case being the pad’s damping, which takes a little off the first compression.

So a pad chooses a point on a hyperbola, and nothing about its material moves the hyperbola. The only ways to lower both the force under the anvil and the anvil’s movement at once are a smaller blow or a heavier anvil — the second being why anvils are made as heavy as they are. A weight that is dropped is an energy to be stored somewhere, and the pad decides only whether it is stored as force or as movement.

A train of blows on a block on springs #

The block on soil is damped so heavily by the ground that one blow is over before the next arrives, and the story ends there. A block on springs, the other arrangement the earlier essay examined, is different. Its mount has a frequency of 4 Hz and a damping of five per cent, a blow’s ringing takes a second or more to die, and a hammer striking every quarter of a second lands each blow on the ringing of the last — the forced-vibration problem of the machine that shakes the building, arriving in pulses rather than as a sine.

On the ordinary pad a train builds up wherever the interval between blows is close to a whole number of the foundation’s periods: at four blows a second, each blow lands in step with the swing the last one started, and the steady peak climbs to 16.4 mm — nearly four times one blow. A tuned undamped pad makes that worse everywhere, 24.1 mm at worst.

Tune the pad a little below the foundation — to 0.85 — and give it thirty per cent of critical damping, and the build-up almost disappears: the worst steady peak is 6.9 mm, 58 per cent below the ordinary pad’s, while one blow on its own moves the block 4.61 mm, practically the same as the ordinary pad’s 4.47. This is the tuned mass after all, doing what it does. It cannot touch the first blow, which it delivers; but once the block is swinging and the anvil is swinging against it, the pad’s damping sits in the relative motion of the two masses and drains the swing before the next blow arrives. A tuned pad is a tuned mass for the ringing, not for the blow.

The design rules for a tuned mass point to roughly this pad. For a mass ratio of a quarter — the anvil against the rest — the classical choices tune the small mass to $1/(1+\mu) = 0.8$ of the large one’s frequency and damp it at about $\sqrt{3\mu/8(1+\mu)^3} \approx 0.22$. Those rules are derived for a harmonic force on the large mass and they are only a guide here, and the calculation says so: a softer pad still, at 0.6 and damped at 0.3, brings the worst steady peak down to 4.7 mm, because a pad well below the foundation’s frequency begins to isolate as well as absorb.

The anvil pays #

The tuned, damped pad on the spring-mounted block moves its anvil 16.9 mm on every blow. The ordinary pad moves it 1.6. The softer pad that does better still moves it by more than two centimetres. None of those is an anvil a forging can be made on.

So the answer to the question the pad was asked is short. A pad cannot absorb a blow, because the blow reaches the block through it. Tuned and damped, it can take the build-up out of a train of blows on a lightly damped block — and it does so only at a softness that turns the anvil into the thing that moves. On a block on soil, where the ground already provides half of critical damping, there is no build-up for a pad to remove, and tuning only costs.

The two masses, followed in time #

The foundation is two masses: the anvil, and the rest of the block, together 150 tonnes. The anvil sits on a spring and dashpot for the pad; the block sits on another for the ground or the springs, the ground’s stiffness chosen so that the two masses together ring at the foundation’s frequency. The blow is an impulse on the anvil. The two equations of motion are stepped forward in time by the average-acceleration rule, which neither adds nor removes energy numerically, at two hundredths of the shorter of the two periods, over twenty-four periods of the whole foundation. The block’s movement, the ground force and the pad’s force are read at every step. The undamped modes and their shares of the blow’s energy come from the two-by-two eigenvalue problem directly. Because everything is linear, the steady peak under a train of blows is the single response added to itself shifted by whole intervals, which is exact rather than simulated. With a rigid seat the calculation reproduces the single-mass block of the earlier essay, and with the mass divided differently between anvil and block it does not change — the checks that the two-mass model contains the one-mass one.

A pad that is linear, a blow that is instant, a ground that is a spring #

The pad is linear. Timber across the grain and rubber both stiffen as they are compressed, and a real pad is softer for small blows than for large ones. A pad also creeps under the anvil’s weight and ages under its blows, which moves its tuning over the years — toward stiffer, for timber that is crushed, which moves it out of the worst region rather than into it.

The blow is an impulse. The tup’s contact with the work and the anvil lasts a millisecond or two, very short against every period here. A die that holds the workpiece longer turns the blow into a push, and the argument of the earlier essay about pulse length then applies to the anvil’s own mode as well.

The ground is a spring and a dashpot. On soil, both depend on frequency, and the damping the ground provides is radiated waves whose amount depends on the block’s size and on what lies beneath. The anti-phase mode that rang on for two hundred milliseconds is lightly damped here because the block barely moves in it; a model that also let the block rock would give that mode some of the ground’s damping back.

Still open: the forging, which is a third mass #

Every blow here is the same impulse, delivered by a tup whose momentum is fixed before it lands. A forging is not a fixed load. The work between the dies is a plastic body that absorbs part of each blow in its own deformation — more when it is hot and soft, less as it cools and hardens, and almost none when the dies meet with nothing between them, the dead blow that hammer operators try to avoid. The share of the tup’s energy that reaches the anvil therefore rises through a forging sequence, and the last blows on a cooling part are the hardest the foundation receives. Whether the foundation should be designed for the dead blow, which is a misuse, or for the last working blow, which is not, and how much of the difference the pad’s nonlinearity takes up, is a question about the forging rather than the foundation.