The Millennium Bridge problem

This whole ordeal started with me being too stubborn to use pre-recorded sounds from the internet for an app project. How hard could it be to create a generative bell *ding* sound?

Each note is synthesised from Jean-Claude Risset's additive model of a struck bell; a set of inharmonic partials that each decay at their own rate, so the force of the strike sets how bright the note is and how long it rings. To get that force, I simulated the bell and its clapper (the little thing inside the bell, swinging at its walls) as two pendulums hanging inside a phone based on realtime data from the device's gyroscope and accelerometer.

While testing, I found that the bell would, almost at random, build up an increasingly loud ringing up to a point where I'd have to mute it. I had thought that my calculations were wrong, but it seemed to be a real physical effect based on frequency and resonance. I had to understand it, and that led me all the way to London.

On 10 June 2000, London opened a new footbridge across the Thames: a shallow suspension bridge of steel cables and an aluminium deck, linking St Paul's Cathedral to Tate Modern. Between 80,000 and 100,000 people crossed it that day, up to 2,000 at a time. As the opening crowd set off, the deck began to sway from side to side. People stopped walking and held the handrails. Two days later the bridge closed, and it stayed closed until February 2002.

Video later showed the south span moving about 50 mm at 0.8 Hz and the centre span about 75 mm at 1 Hz, with sideways accelerations that Arup, the bridge's engineers, estimated at 200 to 250 milli-g.

Nothing broke. The bridge was strong enough; the trouble was a sideways force that, in Arup's words, "had not been anticipated during design". David Newland, a Cambridge engineer who advised the bridge's trust on the fix, wrote that some allowance had been made for sideways forces, but nobody had expected "that pedestrians would so easily fall into step or that the lateral force per person would be as great".

This post is about how that happens, which takes resonance, feedback and damping to explain. I'll also take some detours: a bridge that fell under marching soldiers, a Seoul skyscraper shaken by an aerobics class, a bridge in Washington State that tore itself apart in a moderate wind, and a pair of clocks from the 1660s that kept falling into step. Then back to the bell.

A small push at the right moment

Anything that springs back when you displace it has a natural frequency: the rate at which it oscillates if you disturb it and let go. A playground swing has one, and so does a wine glass, a skyscraper and every span of a bridge. Push such a thing at its natural frequency and each push arrives just as the last one's motion is heading the same way, so the energy accumulates.

Damping limits the growth. Friction, air resistance and flexing joints all turn some of the motion into heat. Engineers describe damping with the damping ratio ζ, the fraction of the damping that would stop the structure from oscillating at all. For a lightly damped structure, a steady rhythm at the natural frequency produces roughly 1/(2ζ) times the motion the same force would cause if applied slowly:

Arup measured the Millennium Bridge's damping at 0.6 to 0.8% of critical, slightly more than the 0.5% the designers had assumed. At that level a sustained rhythm at the right frequency is amplified about seventy times, so even a modest push becomes a large motion if it is regular enough.

People can supply that regularity indoors, too. On 5 July 2011 the upper floors of the 39-storey Techno Mart tower in Seoul shook for about ten minutes, and the building was evacuated for two days. Investigators traced the shaking to a Tae Bo class of about twenty people on the 12th floor, working out to Snap!'s "The Power". When they repeated the workout with instruments in place, vibration on the 38th floor came out ten times greater than usual. A later paper on the investigation put the class's jumping rhythm at about 2.7 Hz, matching one of the building's vertical modes. Twenty people weigh almost nothing next to a skyscraper, but they jumped together, at one of the building's own frequencies.

How a walking body pushes

Walking pushes the ground in two ways. Each footstep presses down, so the vertical force repeats at your step rate. Your body also rocks from one foot to the other, and that sideways sway repeats once per stride, at half your step rate. Bridge engineers have described this since at least the 1980s. A 2023 study pooling smartphone and wearable accelerometer data from 20 datasets found the predominant step rate between 1.7 and 2.2 Hz, and between 1.4 and 2.3 Hz in some of them, which puts the sideways sway between about 0.7 and 1.15 Hz.

The vertical push has been understood for a long time. On 12 April 1831 a company of 74 men of the 60th Rifle Corps marched four abreast across the Broughton Suspension Bridge near Manchester, about sixty of them on the deck at once. One or two began whistling a marching tune, the men fell into step, and they felt the bridge vibrate in time with their footsteps and played along with it. A bolt in one of the stay chains snapped; its iron was either bad or had been made brittle in forging. The bridge came down, and about forty soldiers ended up in the river or against the chains. About twenty were injured and nobody died. Afterwards the British Army ordered soldiers to break step when crossing a bridge.

The sideways push is weaker, and for a long time it was treated as noise. But the sideways natural frequencies of long, slender footbridges tend to fall in that same 0.7 to 1.15 Hz band. A 2021 paper in Nature Communications lists fifteen bridges where crowds caused large sideways motion, from a German footbridge in 1972 to Norway's Lardal footbridge in 2001 and a bridge in Porto in 2020:

Almost all of them sit inside the band where people sway; the one far outside it, Cragside at 2.8 Hz, was a test in which people moved in step on purpose. After the Millennium Bridge, Arup concluded that the same thing "could occur on any bridge with a lateral frequency below about 1.3Hz loaded with a sufficient number of pedestrians".

Why a crowd does not cancel out

Resonance alone does not explain the Millennium Bridge. A crowd walks at many slightly different rates with random timing, so on a still deck most of their sideways forces cancel. The combined force of a hundred people walking at random grows only with the square root of their number, about ten people's worth, which the bridge's own damping can absorb.

That stops being true once the deck moves, because the people on it change how they walk. Arup's explanation was that "it is more comfortable for pedestrians to walk in synchronisation with the natural swaying of the bridge, even if the degree of swaying is initially very small", because that makes the motion predictable and helps them keep their balance. Their footfalls then arrive at the bridge's frequency, and in the phase that adds energy.

Arup measured the effect. In December 2000 they put people on the bridge in small groups, up to 275 in all, had them walk circuits around two poles, and recorded the deck with accelerometers and video. They found that each pedestrian added a sideways force proportional to the deck's velocity, about 300 Ns/m over 0.5 to 1.0 Hz. A force that pushes in the direction of motion, in proportion to its speed, is exactly what damping is, with the sign reversed. Every walker on the deck subtracts a little from the bridge's damping. Arup's formula for the number of walkers at which the total reaches zero is

N = 8π·ζ·f·M / k,

with ζ the damping ratio, f the frequency, M the mode's effective mass and k the 300 Ns/m per person. Below that number the bridge settles down after any disturbance. Above it the total damping is negative, and the swaying grows by itself until people can barely walk. In a test on the north span, with people added a group at a time, the deck's sway grew only slightly until 166 were walking, then jumped so violently that the test was stopped.

You can watch this happen in a published model of the bridge and its walkers. Eckhardt, Ott, Strogatz and colleagues treat the bridge's sideways mode as a single damped oscillator and each walker as an oscillator with its own natural rhythm, pushing the deck sideways and nudged earlier or later by the deck's acceleration. Abdulrehem and Ott later published values they describe as representative of the Millennium Bridge, for the first sideways mode of its north span: a modal mass of 113 tonnes at 1.03 Hz with 0.75% damping, a sideways force of 25 N per walker, and natural rhythms spread about 7% around the bridge's frequency. How strongly a walker's rhythm responds to the deck has not been measured directly, so Eckhardt and colleagues chose it to reproduce Arup's 300 Ns/m, which makes the model's threshold match Arup's formula by design. I ran it, adding ten walkers every minute:

For nine minutes nothing happens. The deck moves a few millimetres at most, and the walkers' rhythms stay scattered, with a synchrony score close to what random timing gives. Then, at around a hundred walkers, the deck starts to move, the walkers fall into step with it, and both grow together. By two hundred walkers the simulated deck sways 65 mm with sideways accelerations near 275 milli-g, the same order as the centre span's motion on opening day, though the model's sway would keep growing with more walkers, where real people stop and hold on. The model comes in well below the 166 of Arup's test because it is tuned to Arup's formula, which Arup itself called conservative: for the north span the formula predicts about 70. The shape still matches the test, though, with a long stretch where adding people changes nothing and then a sudden onset. With damping at the level the retrofit aimed for, the same two hundred walkers produce about 1 milli-g.

The swaying is also slow to stop. Run the crowd back down, removing ten walkers a minute, and it persists well below the number that started it:

Across six simulated crowds the swaying started at 80 to 110 walkers and stopped only at 30 to 60. Most of that gap is delay. The crowd above has a threshold of about 63 walkers, and at every fixed size I tried, it ended up in the same state whether its deck started still or already swaying. Near the threshold, though, the walkers' push and the damping almost cancel, so the sway takes minutes to grow or die away, longer than the minute between groups. Adding or removing people more slowly narrows the gap. One of the six crowds did have a narrow range, 85 to 90 walkers, where it could either sway or stay still depending on how it got there. Belykh and colleagues found a similar loop in a different model in 2017, with the wobble starting at 165 walkers and stopping only below 135.

The wind did it at Tacoma

Negative damping does not need people. On 7 November 1940 the Tacoma Narrows Bridge in Washington State, known as Galloping Gertie for the way its deck rose and fell in moderate winds, twisted itself apart in a wind of about 40 mph. In its last phase the deck twisted about its centre line at 0.2 Hz, one side rising as the other fell, with growing amplitude until the main span broke.

Many physics textbooks used to call this resonance, as if the wind had gusted at the bridge's natural frequency. In 1991 K. Yusuf Billah and Robert Scanlan wrote a paper in the American Journal of Physics pointing out that this is wrong. The wind did not need a rhythm. It supplied energy in a way that depended on the motion itself: each twist of the deck changed the airflow so that the aerodynamic forces pushed the twist further. Once the wind passed roughly 20 mph that effect outweighed the structure's damping, and by about 35 mph the twisting grew without limit. Engineers call this flutter.

The Millennium Bridge swayed for the same reason, with people instead of air. On their own, the wind and the walkers would do little. Once the structure moves, its motion decides how they push, and past a certain wind speed or crowd size they put energy in faster than the damping takes it out. Below that point nothing looks wrong. Resonance alone cannot explain either bridge, because neither the wind nor the crowd pushed in time with the structure until its own motion made them.

Clocks, metronomes and a crowd

In 1665 Christiaan Huygens, the inventor of the pendulum clock, noticed that two of his clocks mounted next to each other on the same support kept swinging in exactly opposite directions, and fell back into that pattern after being disturbed. He reported it by letter to the Royal Society, whose minutes called it "an odd kind of sympathy". The coupling was the support: each pendulum moved it very slightly, and the other felt that motion.

The modern version of the experiment costs a few pounds. James Pantaleone put metronomes on a board resting on two empty cans, so the board can roll from side to side. Start the metronomes out of step and they soon tick together, because each one nudges the board and the board nudges all of them. Pantaleone's 2002 paper describes the setup as a mechanical version of the Kuramoto model, the standard model of how a population of oscillators with slightly different natural rhythms falls into step through a shared coupling. In 2005 Steven Strogatz and colleagues applied the same idea to the Millennium Bridge, with walkers as the metronomes and the deck as the board. As Strogatz put it, "if a few of them get into sync by accident, the bridge would become unstable".

But the synchronisation story may have it backwards. In 2009 John Macdonald showed that synchrony is not needed. A person walking on a moving floor keeps their balance by adjusting where they put each foot, and that adjustment alone can feed energy into the deck, whether or not their steps line up with anyone else's. In 2021 Igor Belykh, Macdonald, Allan McRobie and colleagues concluded that the "increased coherence of pedestrians' foot placement is a consequence of, not a cause of" the instability. They also found that, depending on the model, it starts most easily when the bridge's frequency is about 1.1 or 1.3 times the walkers' sideways rhythm, not at an exact match. On this account the crowd first acts as negative damping, the deck starts to move, and only then do people fall into step, which makes it worse.

For an engineer it hardly matters which comes first. Either way the bridge feels a force that grows with its own motion, and the remedy is damping.

Fixing it

Arup had two options: stiffen the bridge until its sideways frequencies rose above the walking band, or add damping. The stiffening studies set a target of 1.5 Hz for every mode with a significant sideways component. The centre span's first sideways mode was at 0.49 Hz, and tripling a frequency takes nine times the stiffness in theory and, once the added mass is counted, well over ten times in practice. Almost all of the span's stiffness came from the tension in its cables, so even fully bracing the deck would have raised that frequency by only a few percent. The extra structure would have been very costly and would have changed the look of the bridge.

So they added damping. The sideways fix was 37 viscous dampers, most of them under the deck or at the piers, aiming for 15 to 20% of critical damping in the sideways modes. More than fifty tuned mass dampers went in as well, mostly as a precaution against vertical motion. The work cost about £5m. In January 2002, 700 Arup staff walked across to test it, and on 30 January 2,000 people crossed three times at different paces. The sway that had reached 75 mm was down to a few millimetres, and the bridge reopened on 22 February 2002.

A tuned mass damper is a smaller oscillator attached to a structure and tuned to the same frequency. When the structure starts to move, the damper swings against it and turns the motion into heat. The best known hangs inside Taipei 101: a 660-tonne steel ball, 5.5 m across, suspended between the 92nd and 87th floors where visitors can watch it. During Typhoon Soudelor on 8 August 2015 it swung a metre, the largest movement ever recorded for it.

Brooklyn Bridge Park's Squibb Park Bridge, a timber footbridge, opened in March 2013. Its bounce grew more pronounced over time, it began moving from side to side, and it closed on 11 August 2014. Repairs costing $3.4 million, including dampers, kept it shut until April 2017.

The same band, in your pocket

Back to the bell. It swung toward whichever way was down, read from the accelerometer, and in its first build its natural frequency was 0.89 Hz with a damping ratio of 7%. An accelerometer cannot tell gravity from acceleration, so when you walk, the "down" it reports tilts back and forth with your sideways sway, which is the correct physics for anything really hanging inside the phone. In simulation, a sway of 0.1 g tilted the bell's resting angle back and forth by up to about 6°, and resonance turned that into a swing of about 25°. The clapper, at 1.66 Hz with 2.6% damping, sat on the footstep band and rattled against the bell even when the bell barely moved. Raising the bell's damping ratio to 50% fixed it, the same remedy Arup chose.

So that is what was going on, on the bridge and in my phone. The bridge had less than one percent of damping, and once the deck moved, the crowd fed it energy faster than it could lose it. My bell had no crowd and no feedback, just a lightly damped pendulum pushed by the rhythm of my own walking. Both were fixed with more damping.

Just felt like sharing this.

How I simulated

The crowd simulations use the model of Eckhardt et al. (2007) as written in Abdulrehem and Ott (2009). The bridge's sideways mode obeys M·ÿ + 2ζΩM·ẏ + MΩ²·y = F·Σ cos θᵢ, and each walker's phase follows θ̇ᵢ = ωᵢ − β·ÿ·cos θᵢ. I used their values for the north span: M = 113 tonnes, Ω = 2π × 1.03 Hz, ζ = 0.75%, F = 25 N and β = 1.75 s/m, with walker frequencies drawn from a Lorentzian distribution centred on the bridge's frequency with a half-width of 7.2% of it. I cut the distribution off at five half-widths, because its long tails would otherwise give a few walkers absurd rhythms. Walkers joined with random phases in groups of ten every minute, and I integrated with fourth-order Runge–Kutta at 4 ms steps. With these values the model's predicted threshold for an infinite crowd is 63 walkers, or 72 without the cut-off. β has not been measured directly: Eckhardt et al. chose it so that the model's push per walker matches Arup's 300 Ns/m, which is why the uncut threshold equals Arup's formula. To tell memory from delay, I also held crowds at fixed sizes for an hour, starting once from a still deck and once from a swaying one. The retrofit case changes only ζ, to 20%. The bell numbers come from rerunning my original simulation code against a synthetic walking signal.

Sources

• P. Dallard et al., "The London Millennium Footbridge", The Structural Engineer 79(22), 2001. Opening day, damping, crowd tests, 300 Ns/m, the threshold formula, stiffening versus damping. PDF
• D. E. Newland, "Vibration of the London Millennium Bridge: Cause and Cure", International Journal of Acoustics and Vibration 8(1), 2003. Dates, sway amplitudes, sideways sway at half the walking pace. PDF
• D. E. Newland, "Vibration of the London Millennium Footbridge: Part 2 – Cure", Ninth International Congress on Sound and Vibration (ICSV9), 2002. The north-span crowd test (166 walkers) and the 2,000-person test. Archived copy
• "Arup: we have fixed the Millennium Bridge", Building, 1 February 2002. Link
• Cornell Chronicle, 2 November 2005, on S. H. Strogatz et al., "Crowd synchrony on the Millennium Bridge", Nature 438, 2005. Link
• B. Eckhardt, E. Ott, S. H. Strogatz, D. M. Abrams, A. McRobie, "Modeling walker synchronization on the Millennium Bridge", Physical Review E 75, 2007. PDF
• M. M. Abdulrehem, E. Ott, "Low dimensional description of pedestrian-induced oscillation of the Millennium Bridge", Chaos 19, 2009. arXiv
• J. H. G. Macdonald, "Lateral excitation of bridges by balancing pedestrians", Proc. R. Soc. A 465, 2009. Link
• I. Belykh et al., "Foot force models of crowd dynamics on a wobbly bridge", Science Advances, 2017. Link
• I. Belykh et al., "Emergence of the London Millennium Bridge instability without synchronisation", Nature Communications, 2021. Link
• M. Straczkiewicz, E. J. Huang, J.-P. Onnela, "A 'one-size-fits-most' walking recognition method for smartphones, smartwatches, and wearable accelerometers", npj Digital Medicine, 2023. Link
• "Broughton Suspension Bridge", Wikipedia link; reports from the Manchester Guardian and Manchester Chronicle, reprinted in The Philosophical Magazine 9, 1831, pp. 384–389 link.
• "Aerobic exercise blamed for tremor at Techno-Mart", The Korea Times, 19 July 2011 link; "Techno Mart", Wikipedia link; "Global vertical resonance phenomenon between steel building and human rhythmic excitations", Journal of Constructional Steel Research, 2013 link.
• K. Y. Billah, R. H. Scanlan, "Resonance, Tacoma Narrows bridge failure, and undergraduate physics textbooks", American Journal of Physics 59(2), 1991; "Tacoma Narrows Bridge (1940)", Wikipedia link.
• M. Bennett, M. F. Schatz, H. Rockwood, K. Wiesenfeld, "Huygens's clocks", Proc. R. Soc. A 458, 2002; "Christiaan Huygens", Wikipedia link.
• J. Pantaleone, "Synchronization of metronomes", American Journal of Physics 70(10), 2002.
• "Taipei 101", Wikipedia link; Taiwan News on the damper link.
• "Squibb Park Bridge", Wikipedia. Link