The release the rope cannot slow

The counterweight that knows the load gave a luffing crane a load cell in its hoist and a motor on its counterweight, and found that knowing the payload is worth less than it looks. The counterweight can follow the payload as the rope takes it up, so the step at pick-up becomes a ramp. But a payload can also be let go, by a parting sling, an opening grab or a magnet losing its plate, and the crane is then thrown backwards by whatever the counterweight was holding and half as much again as the jib and the mast spring back. The best share for the counterweight to balance is a hedge between lifting the payload and losing it. With a hoisting factor $\varphi_2 = 1.2$ and a rebound factor $\beta_3 = 0.5$ it is 0.35 of the payload, for a mast designed for 0.85 of the payload’s moment.

That essay ended on the hope that the release, too, could be made slow. A pick is slow because the crane chooses to take it slowly. If a device could slow the release the same way — a brake paying out against the falling rope force, a damped spring at the jib’s head — the rebound factor would fall towards nothing, and the best share would climb, the essay hoped, back past a half towards the full payload.

Both halves of that hope can be tested. The first is about where a release’s speed comes from. The second is about what a slow release is worth to the counterweight when it has one.

The rope holds only what the hook holds #

The rope’s force at the jib’s head is the hook’s load, and nothing else. When a sling parts, the hook has nothing on it and the rope’s force falls to the weight of the hook block within the time it takes the stretched rope to shorten, a few hundredths of a second. A drum brake cannot hold the force up, because a drum can only pay out rope, and an empty hook does not pull any out. A spring in the rope at the jib’s head carries the rope’s force and loses it at the same moment: the only mass it has to hold back is the hook block’s, and a hook block a few per cent of the payload’s weight on a spring stiff enough to carry the payload swings back in a small fraction of a second.

The only way to keep a force in the rope after the payload has gone is to keep a mass on it, and a mass large enough to slow the release is a second payload. How fast a release happens belongs to the payload. A grab of sand opens over a second or two and its load leaves a little at a time, so the rope’s force falls as a ramp. A magnet’s plate stays on until the moment it falls, and then it is gone. A parted sling is gone at once. Crane standards tabulate exactly that difference — 0.5 for a grab or a slow-release device, 1.0 for a magnet or a fast release — and the payload, not the crane, decides which row applies. It is the sudden removal of the weight that was dropped turned round: a load that leaves at once overshoots exactly as one that arrives at once does.

What the crane does own is how it answers a release. A mast and a jib that were bent forward by the payload spring back as a structure with its own period and its own damping, and a release counts as slow or sudden only against those.

The swing that follows a release #

The crane here is the one the counterweight that follows the jib began with: a 50 m, 300 kN jib lifting 60 kN at 48.3 m, the payload’s moment 2,898 kN·m. Its mast and jib together rock backwards as one mode, with a period of 2 s and a damping ratio of 0.02, which is what a steel structure of that size gives with nothing added. Its counterweight holds the hedge of the earlier essay, 0.35 of the payload, so while it carries the payload the mast holds 0.65 of the payload’s moment forwards, and once the payload has gone it settles at 0.35 backwards.

Let go at once, the crane swings straight through its new rest and reaches 1.29 times the payload’s moment backwards half a period later, which is the static 0.35 and a rebound of 0.94 on top — a moment the foundation must also keep inside its footprint, backwards, where weight is the only thing resisting it. It then rocks for many periods, losing about an eighth of each swing, because a damping of 0.02 takes a long time to do anything. Let go over 1.2 s, the swing is smaller and later, 0.82: the rope’s force falls while the crane is already moving back, so part of the release arrives when the crane is moving the same way and does not drive it. With a damping ratio of 0.20 the release at once reaches 0.88 and the rocking is gone after a period.

The two smaller swings are reached by different means. One is the payload going more slowly; the other is the crane answering more sluggishly. The question is which of those a designer can count on.

Slow is a fraction of a period #

The rebound of a release taking a time $t_r$ depends on $t_r$ only through $\tau = t_r/T$, its share of the crane’s period. A release at once leaves a rebound of 0.94 with this damping, and 1 with none. The rebound falls as the release lengthens, reaching almost nothing at $\tau = 1$: a rope force falling steadily over exactly one period starts the crane swinging and then stops driving it at the moment its swing has come back to where it began. At one and a half periods it rises again to 0.19, and at two it is gone again.

Those zeros are real and useless. A grab’s release time varies with what it is carrying and how it opens, and a design cannot place itself on a zero it may miss by a fifth of a period. What a design can rely on is the worst rebound for any release at least as slow as the one it expects, which is the envelope through the lobes. Undamped, the envelope is $1/\pi\tau$. It falls below the standard’s 0.5 for a grab at six tenths of a period, and below 0.25 at eight tenths.

The standard’s 0.5 for a grab is a release taking about six tenths of the crane’s period. Read the other way, that is the condition the standard’s row implies: a grab-like release earns its halved rebound only on a crane that answers slowly enough for the grab to count as slow.

A stiff crane makes every release slow #

The same release of 1.2 s is a different load on cranes of different stiffness. On a crane that rocks with a period of 1 s it is more than a whole period long and leaves a rebound of 0.19. On the crane here, with a period of 2 s, it leaves 0.47, about the standard’s value. On a crane with a period of 4 s, a tall tower with a long jib, it is three tenths of a period and leaves 0.81, nearly a magnet’s. The release is the same in every case; what changes is how long the crane takes to answer it.

That turns the device the earlier essay asked for into a property the crane already has. A crane cannot slow its payload’s release, but it can be stiff, and a stiffer mast makes every release slower in the only measure that matters. Stiffening a mast to halve its period costs steel at its base, which is where the counterweight was meant to save it, so the trade is a real one; but it is a trade between quantities a designer controls, rather than a reliance on a device that must work in the instant the crane has already failed.

Twice the deflection for the same load found the same rule for a load arriving: a load brought on over a quarter of the structure’s period is still very nearly sudden, and what decides is the rise time divided by the period, never either one alone. It found the zero at one whole period too, the free vibration the start of the ramp sets going cancelled by the one its end sets going. A release is the same arithmetic in reverse. The crane does not care whether the load is coming or going, only how fast against its own clock — and a design, which cannot aim at the zero, has to read the envelope instead.

What damping can buy, and what it costs #

The other thing a crane owns is its damping. For a release at once, the rebound is the first overshoot of a damped oscillator, $e^{-\pi\zeta/\sqrt{1-\zeta^2}}$. With the 0.02 a welded steel crane has of its own, that is 0.94: damping that low barely touches the first swing, and the only thing that stops it is a slow decay over many periods after the damage is done. Bringing a release at once down to the standard’s value for a grab takes a damping ratio of 0.22, and down to 0.25 takes 0.40.

Those are very large numbers for a structure. A damper added to a frame acts through a brace, and the damper is only as good as its brace: with a brace as stiff as the frame it braces, the most any viscous damper can add is 20.7% of critical, and with a brace a quarter as stiff, 5.9%. A crane’s mast is its own brace, and the brace a damper would need in order to add 22% to a mast’s backward rocking would have to be stiffer than the mast. That stiffness, spent on the mast itself, would have shortened the period instead.

A damper does have one advantage over a device on the rope: it is passive and it is always there. It works in the instant of a release because it worked in the instant before. It is not, though, enough on its own.

The share stops at half the surge #

Suppose the release is slow, by whatever means. What does the counterweight gain?

The earlier essay’s hedge balances lifting against losing. Lifting off, the mast carries the hoisting factor less the share balanced, $\varphi_2 - \beta$. Let go, it carries the share balanced plus the rebound, $\beta + \beta_3$, backwards. The best share makes the two equal:

$$\beta^* = \frac{\varphi_2 - \beta_3}{2}, \qquad m^* = \frac{\varphi_2 + \beta_3}{2}.$$

With a slow release, $\beta_3$ is now computed from the crane’s period and damping and the release time rather than read from a table, and it falls towards nothing as the release lengthens. The best share climbs, but not towards the whole payload. It climbs towards $\varphi_2/2$: 0.60 for a hoisting factor of 1.2.

The reason is that a slow release removes the rebound and leaves the release. A counterweight holding a share of the payload throws the mast backwards by exactly that share once the payload has gone, however gently it went, because the counterweight is still there and the payload is not. A slow release changes the dynamic part of the backward moment and leaves the static part alone. So the best balance between lifting and losing, with no rebound at all, still holds only half of what the lift asks — the hoisting factor split down the middle between the forward moment at lift-off and the backward moment once the load has gone.

Let go at once on this crane, the best share is 0.13 and the mast is designed for 1.07 times the payload’s moment, worse than the standard’s grab row because 0.94 is worse than 0.5. Let go over 1.2 s the share is 0.36 and the moment 0.84, which is the earlier essay’s hedge, because 1.2 s on this crane is about the standard’s grab. Over 6 s the share is 0.56 and the moment 0.64, close to the limit. A slow release moves the hedge back up to where the static crane’s half was, and no further. The earlier essay’s hope — past a half towards the full payload — was the step from the static share to the full balance, and that step is the release itself, not its rebound.

What the knowledge is worth at the tip #

For the same mast — designed for the linked crane’s 2,898 kN·m with 60 kN at the tip and the standard’s grab — the hedged crane may lift 56.1 kN at the tip if its payloads can go at once, 71.7 if they go over 1.2 s and 94.3 if they go over six seconds. Damped to 0.20, the same three are 69.5, 81.5 and 97.1. Nothing the release does allows more than 100 kN, the payload at which the hedge reaches half the surge.

The linked crane, with its counterweight at half, gains from a slow release too. Let go at once it is allowed only 41.7 kN, because a release at once and a counterweight at half throw the mast back by 1.44 times the payload’s moment. Once the release takes more than about 1.7 s, the release stops governing it and the lift-off surge does, and it is allowed 85.7 kN whatever the release does after that.

So the load cell is worth most where the release is worst. Let go at once, the hedged crane lifts 35% more than the linked one; let go over six seconds, 10% more. A slow release and a load cell are partly the same purchase, because each one buys back the backward moment the other leaves. A crane whose payloads are always grabs of bulk material, emptied slowly, gains little from a driven counterweight. A crane lifting steel with a magnet gains the most from it — and needs it most.

The motor cannot catch the swing #

One more device is close at hand: the counterweight’s own motor. A crane that knows the payload has gone, because its load cell has just read nothing, could drive its counterweight inwards to take off the backward moment.

It cannot do it in time. The drive that follows the jib through a luff moves about 150 kN·m of moment a second. The backward moment it would have to take off is the counterweight’s whole share of the payload, 0.35 of 2,898 kN·m, or about 1,000 kN·m, and the crane reaches its worst backward swing half a period after a release at once — one second here. In that second the drive takes off at most 150 kN·m, a seventh of what it needs, and it follows its load cell half a second behind, so for half of that second it has not yet begun. A drive a hundred times faster would be a different machine, and moving a heavy counterweight that fast would put its own inertia into the mast — the mass that helps by being late, run the wrong way round. Against a release, a counterweight is a fixed weight.

The release, by hand #

Let go at once with a damping of 0.02, the rebound is $e^{-0.02\pi/\sqrt{1-0.02^2}} = e^{-0.0629} = 0.94$. The best share is $(1.2 - 0.94)/2 = 0.13$ and the design moment $(1.2 + 0.94)/2 = 1.07$. The same mast allows $2{,}898/(1.07 \times 48.3) = 56.1$ kN.

Let go over 1.2 s on a crane of period 2 s, $\tau = 0.6$ and the undamped rebound is $\sin(0.6\pi)/(0.6\pi) = 0.951/1.885 = 0.50$, which is the standard’s grab. The best share is $(1.2 - 0.5)/2 = 0.35$, the earlier essay’s.

With no rebound at all, the share is $1.2/2 = 0.60$ and the design moment 0.60, so the mast allows $2{,}898/(0.60 \times 48.3) = 100$ kN. That is the most any release allows, and balancing the whole payload instead would give a design moment of $\max(1.2 - 1,\ 1 + 0) = 1.0$ and allow 60 kN — no more than the linked crane with the standard’s grab.

A crane reduced to one swing #

One mode answers the release. The mast and the jib are treated as one oscillator with one period, and the factor of two belongs to one mode: a structure with several does not overshoot by one factor everywhere. A real crane has a jib that bounces, a mast that sways and a counterweight that rocks on its own supports, and a release excites all of them. The mast’s base moment is mostly in the lowest, which is the one taken here.

The period is the same before and after. Losing the payload makes the crane lighter and its period shorter, a little, since most of the moving mass is the jib, the counterweight and the mast. A shorter period after the release makes the release count as slightly slower, so the rebounds here are slightly on the safe side.

Damping is left out of the lift-off. The hoisting factor of 1.2 is kept whatever the damping, although a heavily damped crane would surge less at lift-off too. That helps the damped crane in a way the tip payloads do not show.

The release is a straight ramp. A grab’s load leaves faster at first and slower as the jaws open wide, and the shape changes the lobes in the rebound figure though not its envelope much. A magnet’s plate is the straight ramp of zero length.

The whole payload is let go. A partial release lets go of a share of the payload, and the best share rises towards the static half as the released share falls, as the earlier essay found.

Still open: the counterweight that is held back by a brake #

Nothing on the rope can slow a release, and nothing on the counterweight can catch one. What could act at the right moment is a brake between the counterweight and its track. A counterweight on rollers, held by a brake that slips at a set force, stays where the drive put it as long as the mast is carrying a payload. When the payload goes, the backward moment is the counterweight’s moment, and a counterweight free to roll inwards against a friction brake would give some of it up as it moved — dissipating energy in the slip, which is damping placed exactly where the release puts its moment. Whether a slipping counterweight can take enough out of the first backward swing to halve it, and what it does to the mast during a lift when the same brake must not slip, is a question about whether a counterweight is a moment and not a weight even in the instant it is needed least.