The crowd that is also the structure

A floor that is strong and unusable is one that a single person walking across it can set moving at the rate it likes to move. That essay reached a conclusion that sounds like a paradox and is not: a floor’s worst case is one walker on an empty bay, because every extra person adds far more damping than force, and occupied damping cannot be designed against since the floor must be acceptable on the day nobody is there.

A grandstand reverses both halves of that. Its load case is a crowd, not a person, and the crowd is not walking: it is jumping in time to music, which puts in forces several times larger. And a grandstand is never loaded empty. The people who jump arrive with the people who sit, so the spectators’ mass and damping are present on every occasion the load is, and they are part of the structure being checked rather than a bonus that might not turn up.

This essay takes the jump, the crowd and the seated spectators in turn, on a stand of 30 tonnes modal mass with an empty frequency of 6 Hz and 2 per cent damping — a stand that satisfies the familiar rule of keeping an empty grandstand’s vertical frequency high.

A jump is a pulse, and gravity fixes its size #

Walking’s forces are measurements, and the walking essay said so: the four harmonics of a footfall come from people walking over force plates, and nothing derives them. Jumping has a cleaner description, because the body is in the air for part of every beat.

Call the fraction of each beat the feet spend on the ground the contact ratio. While they are down, the force rises and falls roughly as a half sine; while they are up, it is zero. And the jumper finishes each beat at the height they started it, so over one beat the stand must push up on them exactly as hard, on average, as gravity pulls down. The average force is one body weight, whatever the jump.

That single condition fixes the pulse’s height. A half sine that occupies half of each beat and averages one body weight must peak at π/2 over one half, which is 3.14 times body weight. Occupying two thirds, it peaks at 2.36. A more springy jump with a shorter contact has a taller pulse, because the same momentum has to be delivered in less time — the same trade a dropped load makes between force and duration.

Harmonics that follow from the pulse alone #

A force that repeats every beat is a sum of sine waves at the beat rate and its multiples, and the pulse’s shape decides how much of each there is. For a half sine with contact ratio α, the amplitude of the n-th harmonic, as a fraction of body weight, is

$$r_n = \frac{2\,\lvert\cos(\pi n\alpha)\rvert}{\lvert 1 - 4n^2\alpha^2\rvert},$$

and every number in the figure is that expression, checked against a numerical integration of the pulse.

Two things in it matter for a stand. The first is size. At a contact ratio of one half, the first harmonic is 1.57 body weights — nearly four times walking’s 0.4 — and the second is 0.67, nearly seven times walking’s 0.1. The harmonic a 6 Hz stand has to worry about from a 3 Hz jump is the second, and it arrives at a strength walking only reaches in its first.

The second is the gap. At a contact ratio of exactly one half, $\cos(3\pi/2)$ is zero, and the pulse has no third harmonic at all. A crowd jumping at 2 Hz with that style of jump puts nothing into a 6 Hz stand through its third multiple, where a crowd with a slightly longer contact does. The harmonics are not a smooth sequence that shrinks, as walking’s are; they are set by where the pulse’s own shape happens to cancel, and a change of jumping style moves the cancellations.

A crowd is not forty jumpers in step #

Forty people jumping to the same beat do not land at the same instant. Each lands a little early or late, by a few hundredths of a second, and the forces of people landing at different times partly cancel.

How much they cancel depends on the harmonic. A timing error of 50 ms is a fifth of a period at 4 Hz but nearly half a period at 9 Hz, so the same scatter that leaves the first harmonic of a 2.25 Hz beat almost in step leaves the fourth almost random. With the scatter spread normally about the beat, the share of a harmonic that stays in step is $e^{-(2\pi n f \sigma)^2/2}$, and what falls out of step adds like noise, growing as the square root of the number of people rather than as the number.

At 50 ms of scatter and 2.25 Hz, each jumper effectively delivers 79 per cent of the first harmonic, 40 per cent of the second and only 16 per cent of the fourth. That last figure is the floor: a harmonic completely out of step still delivers one over the square root of 40 of each person’s full value, and no amount of scatter takes it lower. A crowd is a low-pass filter on its own jumping, and the harmonics that reach the stiffest stands are the first it loses.

The scatter itself is the uncertain number, and the figure shows how much rides on it. A tight crowd scattering by 20 ms keeps 96, 86 and 55 per cent of the first, second and fourth harmonics; a loose one scattering by 100 ms keeps 40 per cent of the first and has already reached the floor on the second and fourth. Between those two crowds the second harmonic, the one that reaches a 6 Hz stand from a 3 Hz beat, changes by more than a factor of five.

So jumping sits between the two kinds of load this field already knows. A machine’s force is known exactly, frequency and amplitude, and walking’s is known statistically, from measurements of many walkers. A jump’s pulse is derived exactly, and a crowd’s timing is statistical, so the size of each harmonic is certain for one person and uncertain for forty.

That is the grandstand’s version of the crowd effect on a footbridge, and it runs the other way from it. On a bridge that sways sideways, walkers fall into step with the deck and the coordination grows with the motion. On a grandstand the coordination comes from the music, not from the structure, and it is best at the lowest harmonic, which is the one a well-designed stand keeps away from.

On a grandstand the crowd is always there #

The walking essay’s reasoning about occupied damping rested on one fact: an office floor has to work empty. A grandstand has no such day. The load exists only when the stand is full, and the spectators who are not jumping — seated, or standing still — are on it every time the jumpers are.

A seated or standing body is not a dead weight. It is a mass carried on legs and soft tissue, with a natural frequency near 5 Hz and damping of about a third of critical, which is enormous beside the stand’s 2 per cent. Attached to the stand, every spectator is a tuned mass damper that nobody tuned. On an office floor that is a favour that cannot be counted, one more item on the list of behaviour that is real, measurable and not to be relied upon. On a grandstand it is part of the structure, present in every load case, and ignoring it does not make the check conservative — it makes it a check of a different structure.

The jumpers themselves are treated differently. A person in the air for half of every beat is not attached to anything for that half, so the jumpers are force only, and the spectators are structure only.

The spectators split the stand into two modes #

With half its modal mass in seated spectators, the stand is no longer one oscillator. It is the stand and the spectators’ bodies, joined through the bodies’ legs, and it has two modes. A frequency is a question of which mass is moving, and the spectators are mass that moves on its own spring rather than mass that rides the stand.

Neither is the stand’s own 6 Hz or the bodies’ own 5 Hz. The lower is at 3.94 Hz and the upper at 7.61 Hz, one below both and one above both. That straddling is not a coincidence of these numbers. When one oscillator is hung on another by a spring, its motion either adds to the other’s, which softens the pair, or opposes it, which stiffens it. It is the same interlacing a machine block’s sway and rocking produce, arriving here through a spring rather than through the height of the mass.

The two modes take the spectators’ damping with them. The lower has about 14 per cent of critical and the upper about 29, both several times the stand’s own 2 per cent, and the tallest peak of the occupied stand is 0.06 m/s² per kilonewton against the empty stand’s 0.83. Measured this way, the spectators have made the stand more than ten times quieter.

But the lower mode is now at 3.94 Hz. In it, most of the moving mass is the spectators’ bodies, which move far more than the stand does. Its frequency sits in exactly the range that the second harmonic of a crowd jumping at 2 Hz reaches, a range the empty stand at 6 Hz had been designed to stay clear of.

As the seats fill, the modes move apart #

The split depends on how full the seats are. With a light crowd, a twentieth of the stand’s mass, the lower mode is already at 4.78 Hz and the upper at 6.28 Hz. By the time the seated crowd weighs as much as the stand’s modal mass — which a lightweight stand can approach — they are at 3.49 and 8.59 Hz.

The response to a crowd jumping at 2.25 Hz follows the lower mode through the second harmonic’s 4.5 Hz. Empty, the stand reaches 0.26 m/s². As the seats fill, the lower mode comes down to meet the harmonic and the response rises, to 0.35 m/s² when the seated crowd is 30 per cent of the stand’s mass. Past that the mode goes on falling, away from the harmonic, and at half the stand’s mass the response is back down to 0.31 m/s².

So the occupancy that governs is neither empty nor full. It is whatever fraction puts the lower mode on the harmonic of the rate the crowd is jumping at, and that is a combination of the stand, the crowd and the music that no single check selects.

The worst case falls and the common case rises #

Put the rate back in, and the opening figure reads plainly. Empty, the stand’s worst rate is 3.00 Hz, where the second harmonic lands on 6 Hz, and it reaches 2.54 m/s², about a quarter of gravity. Occupied, that peak is gone — the stand has no mode at 6 Hz any more — and nothing anywhere from 1.5 to 3.5 Hz exceeds 0.49 m/s². Checked on its worst case, the occupied stand is five times better than the empty one.

Checked across the rates crowds actually jump at, it is not. From 1.64 to 2.32 Hz — 98 to 139 beats a minute, the tempo of most popular music — the occupied stand responds more than the empty one, and at 2.00 Hz by a factor of 1.6. In that band the second harmonic falls between 3.3 and 4.6 Hz, onto the lower occupied mode, where the empty stand had nothing to excite.

That is the argument in one sentence. The empty-stand check both overstates the worst case and misses the rates that govern, because the structure it checks is not the one the crowd stands on.

A different jump moves the bad rates #

A crowd that bounces rather than jumps, with its feet down two thirds of each beat, changes almost every number. The second harmonic falls from 0.67 to 0.16 of body weight and the empty stand’s worst response falls from 2.54 to 0.62 m/s². The third harmonic appears, at 0.13, where the half-contact jump had none.

The bands where occupation makes things worse move with it, to 2.32 to 2.58 Hz and 3.18 to 3.50 Hz, because they are now set by different harmonics meeting the same two modes. A designer who checked one jumping style and found the occupied stand safe at 2 Hz has not checked the crowd that bounces at 3.4.

What a designer checks, given all that #

Three things follow that a single empty-stand check cannot provide.

The occupied modes have to be computed. The spectators’ mass fraction decides where the lower mode goes, and it goes below both the stand and the body. A stand designed to keep its empty frequency above a threshold has done nothing about a mode that the threshold never mentions.

The rate has to be swept, not chosen. The worst rate for the occupied stand is not the empty stand’s worst rate, and the bands where occupation hurts are narrow and move with the jumping style. Every rate a crowd might sustain, and every plausible contact ratio, has to be tried against the occupied modes.

And the empty stand still has to be checked. A stand can be jumped on by a partial crowd, or during a rehearsal with the seats empty, and in that case the empty stand’s sharp 6 Hz resonance is the governing one. Both structures are real, on different days, and each has its own worst case.

The empty check carries an assumption the occupied one mostly escapes. The empty stand’s 2.54 m/s² at 3 Hz is inversely proportional to its own 2 per cent damping, a number that is assumed rather than designed, like every structure’s, and a stand with 1 per cent would respond twice as hard. The occupied stand’s modes take most of their damping from the spectators, so its response depends far less on that guess and far more on how full the seats are.

What the model leaves out #

One mode of the stand, and one body. A real stand has more than one period, and a crowd spread over it excites each mode according to where the jumpers and the spectators are. The spectators are modelled as one lumped body with one frequency, when a seated and a standing person differ, and a crowd is a distribution of both.

Steady jumping. The response drawn is the steady state. A crowd starts and stops with the music, and a few beats may not be enough for a lightly damped mode to build up to the steady value, which flatters the empty stand more than the occupied one.

Jumpers who are only force. A jumper on the ground for half the beat is attached for that half, and for that time adds mass to the stand. The simplification is conservative for the empty stand and uncertain for the occupied one.

Perception as an acceleration. A quarter of gravity on a stand full of people is alarming in a way the response factor for an office was not written to describe. The criteria for a crowd are about safety as much as comfort, and a stand that moves visibly can change how its crowd jumps.

Still open: whether the crowd’s timing depends on the stand’s motion #

Every number here treats the jumpers’ timing as set by the music and independent of the stand. On a footbridge that assumption fails catastrophically, because walkers adjust their stride to a deck that sways under them. Whether a crowd jumping on a stand that bounces noticeably tightens its timing to the stand’s motion — which would restore the higher harmonics that scatter had stripped — or loosens it, because a moving floor is harder to jump on in time, is a question about people rather than structure. It decides whether the timing filter that protects a stiff stand is a property of the crowd, or something the stand can take away.