The forging that cushions the blow

The pad that makes the blow worse made a forge hammer’s foundation two masses — the anvil on its pad, the block on the ground — and struck the anvil with a fixed impulse. A stiff pad sent most of the blow into the anvil’s own ringing; a pad softened towards the block’s frequency moved the block more; a pad tuned and damped like a tuned mass damper tamed a train of blows but did nothing for the blow itself. Every blow there was the same impulse, delivered by a tup whose momentum was fixed before it landed.

That essay ended on what a forging does to that assumption. The work between the dies is a plastic body: it absorbs part of each blow in its own deformation — more when it is hot and soft, less as it cools, almost none when the dies meet with nothing between them, the dead blow hammer operators try to avoid. The share of the tup’s energy that reaches the anvil rises through a forging sequence, and the last blows on a cooling part are the hardest the foundation receives. Whether a foundation should be designed for the dead blow, and how far the ordinary blows fall below it, was the question.

A third body between the tup and the anvil #

The hammer is the earlier essays’: a 30 t anvil on a pad, the pad stiff enough that the anvil alone would ring at four times the frequency of the whole 150 t foundation, the foundation on soil at 12.5 Hz with nearly half of critical damping from the waves it radiates. The tup weighs 2 t and arrives at 7 m/s, carrying 14.0 kN·s of momentum and 49 kJ of energy.

Between them lies the forging. It is modelled as what it mechanically is: a body that resists the tup elastically for the first fraction of a millimetre and then flows, carrying a force set by its flow stress times the area of contact, for as long as the tup keeps pressing. A hot steel billet under a broad die might flow at a few hundred kilonewtons; the same billet near the end of a sequence, cooled by the dies and nearly finished, at several meganewtons. When the dies meet with nothing between them the contact is steel on steel: very stiff, flowing not at all, and returning the tup with about three fifths of its arrival speed. The tup, the anvil on its pad and the block on its soil are integrated together through the blow and the half-second of ringing after it.

The three blows are three different loads. Through a hot forging the force rises to the flow force at once and stays there while the tup squashes the part, for nearly fifty milliseconds — a flat-topped pulse at 0.3 MN, much longer than the anvil’s own period on its pad. Through a cooler forging the plateau is five times higher and five times shorter. With the dies meeting there is no plateau at all: a spike of 18 MN lasting two milliseconds. They are the same tup at the same speed, and the anvil sees three different blows.

The impulse is the tup’s momentum until there is nothing to forge #

A forging that flows stops the tup and keeps it. The tup leaves no faster than it arrived — a rebound of a few hundredths — and the impulse the anvil receives is the tup’s momentum and no more: 14.1 kN·s through the hottest forging, 14.9 through one flowing at 1.5 MN. A part too cold or too hard to flow behaves like the dies themselves, returning the tup at 0.61 of its arrival speed, and the anvil receives the tup’s momentum plus its rebound, $m_t v_0(1 + e) = 22.5$ kN·s: 1.6 times as much.

The weight that was dropped found the same factor for a falling weight that bounces, and the same weight, dropped again found that a plastic landing changes what the structure keeps: a rebound is momentum given back, and every unit given back to the tup is a unit taken again from the anvil. What the forging adds is a reason for the rebound to change from blow to blow. Hot, the part is a lump of putty that keeps the tup; cold, it is a spring that throws it back.

The pad, which the forging protects #

The impulse rises by 1.6 times. The force through the anvil’s pad rises by far more, because the forging changes the blow’s duration as well as its size.

The anvil on its pad is a mass on a spring with a period of about twenty milliseconds. The blow that has no frequency found that a mount softens a blow only when the blow is short against the mount’s period; the converse is that a blow long against the period does not ring the mount at all, but loads it as slowly as a weight. A hot forging’s blow, 48 to 141 milliseconds long, is long against the pad’s twenty, and the pad simply carries the flow force across to the block: 0.15 MN through a part flowing at 0.1 MN, 0.46 through one at 0.3. The dead blow, two milliseconds long, is an impulse to the pad, and the anvil rings on it: 5.86 MN, nearly forty times the softest blow.

The transition is the shock spectrum’s, the curve twice the deflection for the same load drew: a pulse of a given peak force produces up to twice the static response in a spring whose period is longer than the pulse, and only the static response in one whose period is much shorter. Here the pulse’s peak is the flow force and its length is the squash, so a softer forging lowers the peak and lengthens the pulse at once, and both move the pad down the spectrum. A forging flowing at 1.5 MN, whose ten-millisecond blow is half the pad’s period, sits on the spectrum’s shoulder and the pad carries 2.32 MN, half again its flow force.

The forging is a pad nobody designed, and in front of the steel pad that was designed, it is the better one. The dashed line is what a fixed-impulse calculation predicts for each blow’s impulse delivered all at once: 3.70 MN for the hottest forging that actually puts 0.15 through the pad, 3.93 for the cooler one that puts 2.32. The fixed-impulse calculation is right only for the dead blow, which is the one blow it does not need a factor for. For every blow that forges something, it overstates the pad force by up to twenty-five times.

The block, which feels the impulse #

The block, ringing at 12.5 Hz with a period of eighty milliseconds, is between the two regimes. A cooler forging’s ten-millisecond blow is short against its period and it responds to the impulse; the hottest forging’s blow, over a hundred milliseconds long, is long against even the block’s period, and the block barely moves: 0.13 mm. With the dies meeting it moves 1.42 mm, eleven times as far, and the ground under it is pushed with 2.86 MN. Across the sequence an operator actually works — hot at the start, cooler at the end, as drawn below — the block’s movement varies two and a half times, from 0.39 mm to 0.95.

The block is what the neighbours feel, through the ground the rock that sends the waves back carried, damped by the waves it sends out as the damping that is radiated found, and it is what settles under repeated blows. Its design case, like the pad’s, is the dead blow.

The energy a hot forging keeps #

The energy tells the story most plainly. A hammer exists to put energy into a forging, and a hot forging takes it: 99 per cent of the tup’s 49 kJ goes into squashing the part, and the foundation receives a fraction of one per cent. As the part cools it takes less, the tup keeps more for its rebound, and the foundation’s share grows. With the dies meeting, none of it does any work at all: 37 per cent leaves with the tup and 63 per cent, 31 kJ, goes into the anvil and the block, to be passed on through the pad and radiated into the ground.

The heaviest blow the foundation receives is the one that forges nothing. That is not a paradox once the forging is seen as a body with a job. Every joule the forging absorbs is a joule the foundation does not; a forging that is doing its job is the hammer’s best isolator, and the blow that skips it is the worst.

The last blows on a cooling part #

A forging is worked in a sequence of blows while it cools, and its flow stress rises as it does — tenfold over the sequence drawn. The pad force rises with it, blow by blow: 0.46 MN on the first, 1.29 on the fifth, 3.56 on the tenth. The eleventh blow, if the operator strikes once more after the part is finished or misjudges the die gap, puts 5.86 MN through the pad, sixty per cent more than the hardest working blow.

So the foundation’s load is not a single blow repeated. It is a rising sequence that ends near the dead blow and occasionally reaches it, and a design for the average blow is a design for the middle of the sequence. Hammer foundations have long been designed with a factor on the tup’s momentum for the rebound, $1 + e$ with $e$ taken as a half or more; the figures say why that is right for the pad and the block — it is the dead blow — and why it is very conservative for the forging the hammer spends most of its life doing.

Two pads in series #

The forging and the designed pad are two springs in series between the tup and the block, and they divide the work in a way the figures make plain. For a working blow the forging is the softer by far: its flow force caps what passes, and the steel pad beneath barely matters. A pad half as stiff, at 25 Hz, carries 0.49 MN under the hot blow; one twice as stiff, at 100 Hz, carries 0.45. For the dead blow the forging is gone and the pad is everything: the soft pad carries 2.99 MN and the stiff one 11.3, with the block’s movement 1.70 and 1.19 mm.

So the pad is designed for the dead blow alone, and its whole job is a job the forging does for every other blow. That changes what a pad is for. The pad that makes the blow worse weighed a pad’s tuning against the block’s movement under a fixed blow; with the forging in the picture, the pad’s tuning matters only on the blows that strike nothing, and the trade it offers — a softer pad for a smaller force, paid for in the block’s movement — is a trade about the hammer’s worst moment, not its working life.

Controlling the worst blow #

If the foundation is designed by the dead blow, the cheapest way to lighten the foundation is to make the dead blow lighter, and modern hammers can. A hammer whose blow energy can be set blow by blow — hydraulic and counterblow hammers, and drop hammers with controlled lift — can strike the last, finishing blows of a sequence softly. A dead blow at 5 m/s instead of 7 puts 4.18 MN through the pad instead of 5.86 and moves the block 1.02 mm instead of 1.42; at 3.5 m/s, 2.93 MN and 0.71 mm. Both scale with the tup’s speed, because the impulse does. The finishing blows on a cold part need little energy anyway, since a cold part barely flows; striking them gently costs the forging nothing and the foundation a third or a half of its worst load.

Die stops — hard blocks that meet before the dies do and take the dead blow on a larger area — change the contact but not the impulse, since the tup’s momentum and rebound are the same whatever it lands on. What they buy is the dies’ life, not the foundation’s.

The dead blow, by hand #

The dead blow’s impulse is the tup’s momentum plus its rebound: $2{,}000 \times 7 \times (1 + 0.61) = 22{,}500$ N·s. Delivered to the 30 t anvil in two milliseconds it starts the anvil at $22{,}500/30{,}000 = 0.75$ m/s; the pad, of stiffness $k_p = 30{,}000 \times (2\pi \times 50)^2 = 2.96 \times 10^9$ N/m, stops it in a quarter-period over a compression of $0.75/(2\pi \times 50) = 2.4$ mm, carrying $2.96 \times 10^9 \times 0.0024 = 7.1$ MN — the figure’s 5.86 once the block’s own movement under the anvil and the pad’s damping are counted.

The hot blow’s pad force is simpler still: a blow long against the pad’s period is carried across as a static force, and the static force is the forging’s flow force. Through a forging flowing at 0.3 MN the pad carries a little more than 0.3 MN, the figure’s 0.46 once the anvil’s small ringing is added.

One flow force and rigid dies #

The forging is rigid-plastic with one flow force. A real forging hardens as it is squashed and its contact area grows, so its resisting force rises through the blow; the blow is less flat-topped and a little shorter than drawn. And the flow stress depends on the rate of squashing — hot steel is markedly rate-sensitive — so a faster tup meets a stiffer forging.

The dead blow’s restitution is assumed. Die on die, the tup’s rebound depends on the dies’ size, hardness and the film of scale between them; three fifths is a plausible value and nothing in the figures needs it to be exact, because the dead blow’s impulse moves only as $1 + e$.

The tup and the anvil are rigid. A tall tup and a deep anvil carry stress waves that take a fraction of a millisecond to cross them, and a dead blow of two milliseconds is long enough to treat them as rigid bodies only roughly; the peak contact force of a dead blow is shaped by those waves, though its impulse is not.

The foundation is linear. The pad is a spring and a dashpot, and the soil a spring and the dashpot its radiation gives it. A timber or rubber pad hardens under a large compression and a soil yields under a large one; both make the dead blow somewhat worse than drawn, and a pad that bottoms out under a dead blow makes it much worse.

The operator and the dies #

They cannot show the operator, who decides how many blows a part receives and whether the last one lands on the part or on the dies, and who is the only control the foundation has over its worst case. They cannot show the dies themselves, which are designed to survive the dead blow because they suffer it first. And they cannot show the workshop around the hammer, where the dead blows are the ones that rattle the windows and the hot blows the ones nobody notices.

Still open: the hammer that strikes before the block has settled #

Every blow here starts with the foundation at rest. A production hammer strikes forty or sixty times a minute, and the blow that has no frequency found that a train of fixed blows builds up when their interval is a whole number of the block’s periods. With the forging in the picture the blows are not fixed: a sequence of soft blows on a hot part, rising to hard ones as it cools, with the occasional dead blow, each starting from wherever the last left the block. Whether the build-up of a real forging sequence, with blows that grow as the part cools, is governed by the last few hard blows alone or by the whole rising sequence — and whether the soft early blows, which barely move the block, leave it still enough that the dead blow at the end meets a foundation at rest — is the question the single blow leaves for a hammer at work.